ONLINE ISSN 2189-8871

TECHNICAL REPORTS OF THE METEOROLOGICAL RESEARCH INSTITUTE No.80

Reference Manual for the Meteorological Research InstituteCommunity Ocean Model version 4(MRI.COMv4)
BY
Hiroyuki Tsujino, Hideyuki Nakano, Kei Sakamoto, Shogo Urakawa
Mikitoshi Hirabara, Hiroshi Ishizaki, and Goro Yamanaka,(Oceanography and Geochemistry Research Department)

気象研究所技術報告
第 80 号
気象研究所共用海洋モデル第４版(MRI.COMv4)
解説
辻野博之・中野英之・坂本圭・浦川昇吾
平原幹俊・石崎廣・山中吾郎（気象研究所海洋・地球化学研究部）

気象研究所
METEOROLOGICAL RESEARCH INSTITUTE, JAPANMarch 2017

METEOROLOGICAL RESEARCH INSTITUTE
Established in 1946
Director-General: Dr. Kiyoharu Takano
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Director: Mr. Ko Koizumi
Director: Dr. Tomoaki Ose
Director: Mr. Isao Takano
Director: Dr. Izuru Takayabu
Director: Mr. Osamu Suzuki
Director: Dr. Kenji Maeda
Director: Dr. Hitoshi Yamasato
Director: Dr. Tsurane Kuragano

1-1 Nagamine, Tsukuba, Ibaraki, 305-0052 JapanTECHNICAL REPORTS OF THE METEOROLOGICAL RESEARCH INSTITUTE
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気象研究所技術報告 第80号 平成29年3月
TECHNICAL REPORTS OF THE METEOROLOGICALRESEARCH INSTITUTE No.80, March 2017
doi:10.11483/mritechrepo.80

Reference Manual for the
Meteorological Research InstituteCommunity Ocean Model version 4(MRI.COMv4)
気象研究所共用海洋モデル第４版(MRI.COMv4)
解説

Hiroyuki Tsujino, Hideyuki Nakano, Kei Sakamoto, Shogo Urakawa
Mikitoshi Hirabara, Hiroshi Ishizaki, and Goro Yamanaka,(Oceanography and Geochemistry Research Department)

辻野博之・中野英之・坂本圭・浦川昇吾
平原幹俊・石崎廣・山中吾郎（気象研究所 海洋・地球化学研究部）

Preface
It has long been recognized that the role of the ocean in the earth’s climate system is conclusively important in such
issues as the global warming, the long-term variability in the air-sea coupled system like ENSO, associated meteorologicalextreme phenomena, and so forth. In these situations, modeling of the ocean has become an indispensable method ofstudying the climate variability and predicting its future state as well as studying the mechanisms of the oceanic variability
itself.
The Oceanography and Geochemistry Research Department of the Meteorological Research Institute (MRI) developed
its original, general-purpose numerical ocean model, the Meteorological Research Institute Community Ocean Model(MRI.COM), in the early 2000s for both the research work in MRI and operational work in the Japan Meteorological
Agency (JMA) by combining two ocean models developed for their research work. The ocean modeling activities aremaintained under MRI research programs "Development of a high-resolution (eddy-resolving) ocean general circulation
model and study on formation, maintenance, and variation mechanisms of water masses based on the model" (fiscal years
2003 through 2007), "Development of an ocean environmental model and assimilation system and study on variation
mechanisms of the ocean environment – feasibility study –" (fiscal year 2008), "Development of ocean environmental
forecasting methods" (fiscal years 2009 through 2013), and "Research on advanced ocean models" (starting from fiscal
year 2014). The MRI.COM was revised as its fourth version. The present publication is the new MRI.COM manual
corresponding to the MRI.COM (version 4). The manual has been thoroughly revised from the previous version publishedin 2010.
The ocean modeling study in MRI began in the late 1970s for investigating the variability of the Kuroshio south ofJapan. First, an ocean model with the primitive-equation system developed by former Prof. K. Takano in UCLA, USA, wasintroduced. Another ocean model was then introduced slightly later in 1981. It was similar to the former but developedby an ocean research group in the University of Tokyo. Since that time, the two ocean models with diﬀerent codes havebeen improved in parallel in MRI for various purposes. The former model from UCLA has been vigorously optimized to
exhibit a high computational eﬃciency in vector machines, and has been used in experiments with long-term integrations.
The latter model from the University of Tokyo incorporated many options from the early stage, such as a surface mixed
layer model, an isopycnal diﬀusion scheme, and a simple sea ice process, for various research and operational purposes.
In the early 1990s, the first coupled ocean-atmosphere model experiment was conducted through cooperation between
the Oceanographic Research Department (at the time) and the Climate Research Department, MRI, to simulate El Nino
phenomenon. Since then, construction of a climate model synthesizing atmosphere, ocean, sea ice, and land surface has
been strongly desired both for research and operational work associated with climate warming projection and seasonal
forecasts, including the ENSO cycle prediction. To this end, development of a new, general-purpose ocean model,
MRI.COM, which could provide the oceanic part of the synthetic climate model, has been initiated based on the two ocean
models used so far to achieve eﬃciency in model improvement and management and to integrate their merits. In designing
the new model, the main frame of the former model and the various physical options of the latter model were transferredto the new model, and many newly developed physical processes and schemes were added.
Since 2010 in which the first English version of the MRI.COM manual was published, the model has been continuously
updated through further improvements in physical processes and addition of new processes. One of the most pronounced
improvements in MRI.COM (version 4) is the introduction of a new vertical coordinate system (z ∗ coordinate system)
which better represents the shallow area in coastal modeling. The introduction of a new two-way nesting scheme which
accurately conserves water and heat has the benefit in climate modeling. This makes it possible for an ocean model toseamlessly handle climate change and coastal disaster prevention. Recent achievements have been fruited in this manual.
MRI.COM has been developed along with its own usage as a part of the climate model and the ocean data assimilation
system in MRI as well as stand-alone experiments. MRI.COM has already been in operation at JMA for 9 years,
contributing to providing reliable information such as seasonal prediction and ocean prediction. Based on our experiences,–i–we believe MRI.COM is one of the best ocean models in the world. I express special thanks to the present and past
participants in the model development for their great deal of eﬀorts and help. We hope MRI.COM and the present manual
will contribute to research work in the fields of climatology, oceanography, and environmental sciences in domestic andforeign institutions as well as to the research and operational work in MRI and JMA.

Tsurane Kuragano
Director
Oceanography and Geochemistry Research Department– ii –

前文
地球温暖化や ENSO 等の大気海洋結合系の長期変動やそれに伴う異常気象等、地球の気候システムにおける、海洋の役割の重要性は古くから認識されてきた。このような状況の下、海洋数値モデリングは、海洋自体の変動メカニズムの解明のみならず、気候変動の研究やその将来予測を行うにあたっての不可欠な手段となっている。
気象研究所海洋・地球化学研究部では、2000 年代前半にそれまで研究業務に開発・使用されてきた二種類の海洋モデルを統合して、高い汎用を持つ気象研究所共用海洋モデル (MRI.COM) を独自に開発し、気象研究所の研究業務や気象庁における現業運用に供してきた。その後も海洋モデル開発研究は、気象研究所経常研究、「高解像度（渦解像）
海洋大循環モデルの開発とそれによる水塊の形成、維持、及び変動機構の解明」（平成１５年度〜平成１９年度）、「海
洋環境モデル・同化システムの開発と海洋環境変動機構の解明に関する研究―フィージビリティ・スタディ― 」（平成２０年度）、「海洋環境の予測技術の開発」（平成２１年度〜平成２５年度）、「海洋モデルの高度化に関する研究」（平成２６年度〜平成３０年度）において継続されている。今回これまでの海洋モデルよりも更に汎用性を高くした
MRI.COM（第 4 版）を開発したことから、本技術報告は新しいバージョンの海洋モデルの英語版解説書として、これまでの解説書（2010 年出版）の内容を大幅に改訂して出版するものである。
海洋・地球化学研究部におけるモデリング研究は、海洋変動のメカニズムを解明することを目的に 1970 年代の終
わり頃に開始された。その初期段階で、米国 UCLA で高野健三教授により開発されたプリミティブ方程式系モデルが導入された。一方、その直後に東京大学の海洋グループによって開発された別のプリミティブ方程式系モデルも導入された。それ以降、海洋モデルとしてコードの全く異なる２系列のモデルが併存し、それぞれのモデルに独自の改良が加えられ、目的に応じて利用に供されてきた。UCLA 系列モデルの特徴は、当時のベクトル計算機に適合させた計
算効率性の高さであり、水平的高解像度実験や全球深層循環実験等に使用された。一方、東大系列モデルの特徴は、海面混合層や等密度面拡散、海氷過程といった多彩な物理過程をオプションとして含んでいることであり、ENSO や中層水形成等、表層・中層の時間変動性をターゲットとする種々の研究や気象業務にも幅広く用いられた。

1990 年代初期、エルニーニョ現象再現のための初めての大気・海洋結合モデル実験が、海洋研究部（当時）と気候研究部との共同研究として行われて以来、ENSO サイクルはもちろんのこと、地球温暖化予測、季節予報等に関連した研究および気象業務での利用にとって、大気・海洋・海氷・陸域等を総合した気候モデル構築の必要性が急速に高まってきた。このため、海洋研究部では、モデル開発・管理の効率化とそれぞれのモデルの長所の統合を目的として、
従来の 2 系列の海洋モデルをもとに広範な種々の目的に供し得る新たな汎用的海洋モデルを開発することとした。2系列モデルの統合に当たっては、海洋モデルとしての大枠は UCLA 系列のものを用い、東大系列の多彩な物理過程モデルを融合させるとともに、最新の物理過程やスキームを取り入れることとした。
前回の英語版解説書が出版された 2010 年以後も物理プロセスのさらなる改良と新たなプロセスの付加が続けられた。MRI.COM（第 4 版）について特筆すべきは、沿岸モデリングにおける浅海域の再現性向上のために、新しい鉛直座標系（ z ∗ 座標系）を導入したことである。また、気候モデリングに資するため保存性を向上させたネスティングス
キームを導入する等の改良が行なわれ、気候変動から沿岸防災までをシームレスに扱える海洋モデルとなっている。
本解説書には最近の成果が結実している。

MRI.COM はすでに海洋モデル単独実験のみならず、気候モデル実験の海洋パートおよび海洋データ同化システムのモデルパートとしても数多くの研究上の実績を積み上げてきたもので、気象庁ですでに９年間現業運用され、季節予報や海況予報などの確度の高い情報発信に貢献してきた。その経験から、本モデルは世界に幾つかある他の海洋モ
デルに十分伍して行ける性能を持っていると確信している。長年にわたる海洋モデル開発関係者の多大な努力と協力に深く感謝の意を表する。今後、本モデルと解説書が気象庁と気象研究所における気象業務や研究活動のみならず、
日本国内、諸外国における気候、海洋、環境科学の研究の推進に大きく貢献することを祈念している。

海洋・地球化学研究部長
倉賀野 連– iii –

Abstract
About this manual and MRI.COM
This technical report is a manual of the Meteorological Research Institute Community Ocean Model (MRI.COM).
MRI.COM is an ocean general circulation model developed and maintained at the Meteorological Research Institute(MRI) of the Japan Meteorological Agency (JMA). It has been used for studying large scale oceanic phenomena and as
the oceanic part of the coupled climate models developed at MRI. Recently, it is expanding its coverage to coastal small
scale phenomena.
The current version of MRI.COM is version 4. Version 1 (developed around 2000) was intended to present a prototype.
Eﬀorts were devoted to combining the two ocean models used until then in MRI. For this reason, users at that time tended tobe restricted to MRI research scientists who were committed to the development. Thus, users were deeply knowledgeable
about the model.
Version 2 (early 2000s) was intended for use in the operational forecasting system in JMA. Since the number of users
without direct experience in developing models was expected to increase, the developers decided to write a detailed manual
for that version. The Japanese version was published in 2005 (Ishikawa et al., 2005) and eventually became the prototypeof this manual.
Version 3 was intended for use as an oceanic component of the Earth System Model of MRI (MRI-ESM1; Yukimoto
et al., 2011). One of the reasons for creating a new version was that the definition of vertical grid arrangement was
modified during the development. MRI participated in the Coupled Model Intercomparison Project Phase 5 (CMIP5)
using MRI-ESM1, and its results on the future projection were expected to be used by a wide range of communities, so a
detailed description of its oceanic component was prepared in English (Tsujino et al., 2010).
Version 4 is intended for use as the oceanic part of the next operational forecasting systems of JMA as well as a new
Earth System Model of MRI (MRI-ESM2) prepared for the Coupled Model Intercomparison Project Phase 6 (CMIP6).
This version includes the following improvements.
• Tide producing potential and inverse barometer eﬀect of surface atmospheric pressure are introduced into thedepth-integrated (barotropic) equations, considering an application to coastal processes. The linear response of thesystem to the tide producing potential is calculated separately from the other processes, so that self-attraction andloading aﬀect only tidal processes.
• A vertically rescaled height coordinate (z ∗ coordinate; Adcroft and Campin, 2004) has been introduced.
• Nesting function has been expanded. Oﬀ-line (parent and child models are executed serially) one-way, on-line(parent and child models are executed in parallel) one-way and two-way nesting become possible. In on-line
two-way nesting, sea water volume and scalar quantities can be conserved for a system of parent and child models.
• Program files are classified into several packages to ease the maintenance. Each package is made as independent as
possible from other ones and is controlled by a driver program, which is named as package_ctl.F90.
• The user interface is entirely revised. Specifically, monitoring and sampling functions have been expanded, several
input and output files use a format diﬀerent than the previous versions so that their contents can be checkedeasily, and the block names and the contents of namelists are renamed so that they are more self-explanatory. See
READMEs for details.
Note that the purpose of this manual is to present a detailed description of a particular model system. The mathematicalexpressions of processes, the parameterization methods, and the numerical algorithms presented here follow those adopted
in the latest code. They are largely state-of-the-art, but this does not necessarily mean that they are the complete reflectionsof physical, mathematical, and numerical integrity. Every method is subject to possible sophistication. We welcome
critical comments and suggestions from any reader or user, which are necessary for further improvement.
For a more general or detailed description of OGCMs, please refer to textbooks by Griﬃes (2004) and Kantha andClayson (2000). The former thoroughly describes the fundamentals of OGCMs, and the latter concisely summarizes themodeling of various oceanic processes such as tide and sea ice.
–v–

Organization
Chapter 1 introduces OGCMs and MRI.COM. It also presents the classification of OGCMs and the status of MRI.COM
with respect to the state-of-the-art OGCMs.
Part I describes the model configuration. Governing equations are derived in Chapter 2, and the spatial grid arrangement
is explained in Chapter 3.
Part II describes the solution procedures of diagnostic equations. Chapter 4 presents definition of the equation of stateof sea water. Chapter 5 presents definition of the continuity equations for unit grid cells.
Part III describes the solution procedures of momentum equation. The method of solving the barotropic and thebaroclinic part of the momentum equation are presented in Chapters 6 and 7, respectively.
Part IV describes the method of solving the advection-diﬀusion equation for tracers. Chapter 8 presents advection
schemes. Subgrid-scale parameterizations for horizontal and vertical mixing are explained in Chapters 9 and 10, respectively. Biogeochemical models are presented in Chapter 11 and inert tracers are explained in Chapter 12. Chapter 13
explains usage of the tracer package.
Part V describes boundary processes. Surface fluxes are presented in Chapter 14. Surface mixed-layer models arepresented in Chapter 15. Bottom boundary layer parameterization is explained in Chapter 16. Sea ice part is explained in
Chapter 17. How to construct and run a pair of nested-grid models is presented in Chapter 18.
Part VI contains miscellaneous topics. Basics of the finite diﬀerence method are presented in Chapter 19, generalorthogonal curvilinear coordinates and related calculus are introduced in Chapter 20, and finally user’s guide to construct
and run a model is presented in Chapter 21.
Each chapter is almost independent from other chapters. Thus the readers might be able to understand the contents ofeach chapter without referring to other chapters. However, reading Part I will give the readers the background to help
understand the remainder of this manual.
The following are some comments about the notations used throughout this manual. The characters and expressions
in Courier fonts are adopted from program codes. The subscripts and indices used in discrete equations are intended toexpress staggered grid arrangements. They do not necessarily correspond to the array indices in program codes.

– vi –

概要
MRI.COM と本解説書について
本技術報告は、気象研究所共用海洋モデル (MRI.COM) の解説書である。MRI.COM は気象庁気象研究所で開発、維持されてきた海洋大循環モデルである。本モデルは、大きなスケールの海洋現象に関する研究や気象研究所で開発された気候モデルの海洋部分として使用されてきた。最近では、その適用可能範囲を沿岸の小スケール現象に広げつつある。

MRI.COM の最新バージョンは４である。バージョン１（2000 年頃）は基本型を作成することを目的としたものであった。それまで気象研究所で使用されてきていた２系統の海洋モデルを統合することに精力が注がれた。それ故、当時の利用者は開発に直接携わる気象研究所の研究者に限定されていた。言い換えれば、利用者が本モデルに関する深い知識を有していた。
バージョン２（2000 年代前半）は、気象庁の現業システムへの供用を目的としたものであった。モデル開発に直接関与しない利用者数の増加が見込まれたため、当時の開発者らはこのバージョンに対する詳細な説明書の執筆を決め、日本語による解説書が 2005 年に出版された (Ishikawa et al., 2005)。この日本語版は、その英語版である本解説書の原型となっている。
バージョン３は、気象研究所地球システムモデル (MRI-ESM1; Yukimoto et al., 2011) の海洋部分への供用を主な目的としたものである。バージョン更新の理由のひとつは、開発段階で、鉛直格子点の定義位置を変更したことであった。気象研究所は MRI-ESM1 を用いて結合モデル相互比較プロジェクト第５フェーズ（CMIP5）に参加した。同モ
デルによる将来予測結果等は、幅広い分野の人々に使用されることが見込まれたため、著者らは、その海洋部分についての詳細な説明を英語で執筆することとした (Tsujino et al., 2010)。
バージョン４（本バージョン）は気象庁における次期現業海洋予測システム、並びに気象研究所が参画する結合モデル相互比較プロジェクト第６フェーズ（CMIP6）用の新地球システムモデル（MRI-ESM2）の海洋部分としての供用を意図したものである。本バージョンでは以下の改良がおこなわれている。
• 起潮力ポテンシャルおよび海面気圧による押し下げ・吸い上げ効果が順圧モードの方程式に導入された。これ
らは沿岸海洋予測への適用を考慮したものである。起潮力ポテンシャルに対する系の線形応答は他のプロセスから独立して解かれる。そうすることによって、self-attraction 及び loading 効果は潮汐過程にのみ影響を及ぼすようになっている。
• 海面水位変動を海面から海底までスケーリングする新しい鉛直座標 (z ∗ coordinate; Adcroft and Campin, 2004)を導入した。
• ネスティング機能が拡充された。オフライン（親と子モデルが逐次実行される）単方向、オンライン（親と子モデルが並列同時実行される）単方向、双方向ネスティングが可能となった。オンライン双方向ネスティング
では、海水体積とトレーサー積算値が親モデルと子モデルのシステムで保存させることが可能となっている。
• プログラムを全面的に書き換え、各プロセスをパッケージ化し、保守を容易にした。各パッケージは可能な限
り他のパッケージに依存しないようにし、package_ctl.F90 という名前のファイルに含まれる、ドライバープ
ログラムのみにより制御される。
• ユーザーインターフェイスが全面的に改訂された。具体的には、モデルの挙動監視機能とサンプリング機能が拡張された。いくつかの入出力ファイルでは以前のバージョンとは異なるフォーマットが使用され、中身が容易に確認できるようになった。NAMELIST ブロック名および変数名を改め、より自明となるようにした。詳しくは各種 README ファイルを参照されたい。
読者には、本解説書の目的が、特定のモデルシステムに関する詳細な説明を与えることである点に注意していただ
きたい。本解説書に示す、海洋に生じる現象の数学的表現、パラメタリゼーションの方法、数値アルゴリズムは、最新のプログラムコードに則ったものである。これらは概ね最先端の知見を反映したものであるが、それがすなわち物
理的、数学的、計算機科学的手法の完全性を表現していることを意味するわけではない。あらゆる手法も改良の対象となり得る。従って、著者らは読者ならびに利用者からの忌憚なきコメントや助言を歓迎する。これらはモデルのさらなる改良には不可欠なものである。

– vii –

海洋大循環モデルに関するより包括的で詳細な解説に興味のある読者は Griﬃes (2004) や Kantha and Clayson(2000) による教科書を参照することを勧める。前者には海洋大循環モデルの原理的な面についての詳細な記述が、後者には潮汐や海氷といった、様々な海洋現象のモデリングの関する知見が簡潔にまとめられている。

本解説書の構成
第 1 章では、海洋大循環モデルと MRI.COM を紹介する。海洋大循環モデルの分類と、最先端の海洋モデルに対する
MRI.COM の位置づけについて述べる。
第 I 部では、モデルの基本設定について述べる。第 2 章では支配方程式の定式化を行う。空間格子配置、及び単位格子の面積や体積計算方法の解説を第 3 章で行う。
第 II 部では、診断式について説明を行う。第 4 章は海水の状態方程式の定義を示す。第 5 章は単位格子に対する連続方程式の差分式の定義を示す。
第 III 部ではモデルの運動方程式の解説を行う。運動方程式の順圧成分と傾圧成分の解法を第 6 章と第 7 章でそれぞれ述べる。
第 IV 部ではトレーサーに対する移流拡散方程式の解法を述べる。トレーサー移流スキームについて第 8 章で解説する。サブグリッドスケールの水平方向と鉛直方向の混合に対するパラメタリゼーションについて、第 9 章および第10 章でそれぞれ解説する。幾つかの生物地球化学モデルを第 11 章で解説する。パッシブトレーサーの幾つかを第 12章で解説する。第 13 章はトレーサーパッケージの使用法を説明する。
第 V 部では境界プロセスの解説を行う。海面フラックスの取り扱いを第 14 章で、幾つかの海面混合層モデルを第15 章で、海底境界層モデルを第 16 章で、海氷モデルを第 17 章で解説する。入れ子モデルの作成と使用法について第18 章で解説する。
第 VI 部では上記に分類できない項目について取り上げる。差分解法の基本について第 19 章で、一般直交曲線座標とそれに関連した計算法について第 20 章で、最後にモデルの作成と実行方法の解説を第 21 章で行う。
各章はほぼ他の章から独立しており、読者は他の章を参照しなくとも各章の内容を理解できるはずである。但し、第 I 部を読んでおくと、それが背景的知識となって、本解説書の残りの部分の理解が容易になるので、参考にしていただきたい。
最後に本解説書で用いる表記法についての注意点を述べる。本文中タイプライター (Courier) 活字体が用いられている部分はプログラムコードからの抜粋である（綴り間違えなどではない）。差分式に現れる添え字や指数はスタガー
ド（千鳥状）格子配置を表現するように意図している。しかし必ずしもプログラムコードの配列番号とは対応していないので注意願いたい。

– viii –Contents
Chapter 1

1.1
1.2
1.3
1.4

Part I

OGCMs and MRI.COM

1

What do OGCMs cover? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Classification of OGCMs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
About MRI.COM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Future of OGCMs and MRI.COM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1
2
3
3

Configuration

Chapter 2

2.1
2.2
2.3
Chapter 3

3.1
3.2
3.3
3.4
3.5
3.6

Part II

5

Governing Equations

7

Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7
14
19

Spatial grid arrangement

21

Horizontal grid arrangement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Vertical grid arrangement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Indices and symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Calculation of horizontal grid cell area and width . . . . . . . . . . . . . . . . . . . . . . . . . . .
Calculation of vertical cell thickness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21
22
23
23
26
28

Diagnostic Equations

33

Chapter 4

4.1
4.2
Chapter 5

5.1
5.2
5.3
5.4

Part III
Chapter 6

6.1
6.2
6.3
6.4
6.5
6.6
Chapter 7

7.1
7.2
7.3

Equation of State

35

Basics of the equation of state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
An equation of state used by MRI.COM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35
36

Continuity equation

41

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Finite diﬀerence expression for the T-cell continuity equation . . . . . . . . . . . . . . . . . . . . .
Finite diﬀerence expression for the U-cell continuity equation . . . . . . . . . . . . . . . . . . . .
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41
41
44
45

Equations of Motion

47

Equations of motion (barotropic component)

49

Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Time integration on barotropic time levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Prognostication of state variables at the baroclinic time level . . . . . . . . . . . . . . . . . . . . .
Horizontal diﬀusivity of sea surface height . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Inclusion of tidal eﬀect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49
50
53
57
57
61

Equations of motion (baroclinic component)

63

Advection terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Pressure gradient term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

63
73
75

– ix –

7.4
7.5

Part IV

Coriolis term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Evolution of tracers

Chapter 8

85

Finite volume or flux form method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
Weighted upcurrent scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
QUICK scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
QUICKEST for vertical advection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
UTOPIA for horizontal advection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
Second Order Moment (SOM) scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
MPDATA scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

Chapter 9

SGS parameterization of lateral mixing of tracers

9.1
9.2
9.3
9.4

107

Introduction and Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
Horizontal diﬀusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
Isopycnal diﬀusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
Gent and McWilliams parameterization for eddy-induced transport . . . . . . . . . . . . . . . . . . 112

Chapter 10

10.1
10.2
Chapter 11

11.1
11.2
11.3
11.4
11.5
11.6
11.7
Chapter 12

12.1
12.2
12.3
Chapter 13

13.1
13.2
13.3

Part V83

Tracer advection schemes

8.1
8.2
8.3
8.4
8.5
8.6
8.7

80
80

SGS parameterization of vertical mixing of tracers

Biogeochemical model

125

Inorganic carbon cycle and biological model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
Carbon cycle component . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
Obata and Kitamura model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
NPZD model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
Usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
Program structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
Inert tracers

137

Ideal age tracer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
CFCs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
SF6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
Package Structure and Usage

139

Package structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
Handling the initial state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
Configuration of tracers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

Boundary Processes

Chapter 14

14.1
14.2
14.3
14.4
14.5
14.6
14.7
14.8
14.9
14.10

119

Vertical diﬀusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
Convective adjustment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

147

Sea surface fluxes

149

Momentum flux (surface stress) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
Heat flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
Shortwave radiation flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
Longwave radiation flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
Latent and sensible heat fluxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
Freshwater flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
Treatment in the temperature equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
Treatment in the salinity equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
Bulk transfer coeﬃcient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
Specification of input data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167–x–14.11
14.12
Chapter 15

15.1
15.2
15.3
15.4
Chapter 16

16.1
16.2
16.3
16.4
16.5
16.6
Chapter 17

17.1
17.2
17.3
17.4
17.5
17.6
17.7
17.8
17.9
17.10
17.11
Chapter 18

18.1
18.2
18.3
18.4
18.5
18.6
18.7

Part VI

Surface data objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
Technical details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
Turbulence Closure Models

175

Statistical closure model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
Mellor and Yamada model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
Noh and Kim (1999) model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
Generic length scale model by Umlauf and Burchard (2003) . . . . . . . . . . . . . . . . . . . . . 184
Bottom Boundary Layer (BBL)

189

General description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
Grid arrangement and exchanges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
Pressure gradient terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
Eddy eﬀects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
Usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
Usage notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
Sea ice

195

Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
Thermodynamic processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
Remapping in thickness space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
Advection and Diﬀusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
Ridging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
Adjustment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
Technical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
Usage notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
Nesting

225

Feature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
Grid and Topography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
Intermodel transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
Conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
Sponge region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
Usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236
Program structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242

Miscellaneous

Chapter 19

19.1
19.2
19.3
19.4
19.5
19.6
Chapter 20

20.1
20.2
20.3
20.4
20.5
Chapter 21

245

Basics of the finite diﬀerence method247

Diﬀusion equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
Finite diﬀerence expressions for time derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
Finite diﬀerence expression for space derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
Finite diﬀerencing of advection-diﬀusion equation . . . . . . . . . . . . . . . . . . . . . . . . . . 250
Implicit method for vertical diﬀusion equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
Summary of schemes used by MRI.COM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
Generalized orthogonal curvilinear coordinate grids

253

Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
Generation of orthogonal coordinate system using conformal mapping . . . . . . . . . . . . . . . . 253
Rotation of vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
Mapping a quantity from geographic coordinates to transformed coordinates . . . . . . . . . . . . . 256
Vector operation and diﬀerentiation in generalized orthogonal coordinates . . . . . . . . . . . . . . 258
User’s Guide

261– xi –

21.1
21.2
21.3
21.4
21.5
21.6
21.7

Model setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
Preparation of input data files for execution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262
Restart file . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264
Execution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268
Monitoring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270
MRI.COM eXecution Environment (MXE) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270
Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271

Bibliography

277

Index

283

– xii –

Chapter 1

OGCMs and MRI.COM
This chapter outlines the ocean general circulation models (OGCM) and the status of MRI.COM.

1.1 What do OGCMs cover?
OGCMs have been supposed to simulate relatively large-scale phenomena such as global-scale thermohaline circulations,basin-scale wind-driven circulations, and mesoscale eddies (Figure 1.1). Small-scale processes that are either unresolved
or neglected might be incorporated in some forms of subgrid-scale (SGS) parameterizations. The current basin or global
scale OGCMs may cover phenomena from thermohaline circulations to mesoscale eddies, but it is still almost impossible toconduct a simulation long enough to achieve a quasi-steady state of a thermohaline circulation with a horizontal resolution(∼ several km) that is suﬃciently high to resolve mesoscale eddies, even with the present computation resources. For
these reasons, the standard practice in the ocean model community is to use a low horizontal resolution (a few hundred
kilometers) model to study global thermohaline circulations and to use a limited-domain model to study an eddying
ocean. Some research projects seek to conduct a several-hundred-year integration of a high resolution model that resolves
mesoscale eddies using enormous resources, but such a resource is not available to everyone.
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\U

\U \U

Figure1.1 Various oceanic phenomena in terms of their space and time scales and coverage of the ocean generalcirculation model. the figure of oceanic phenomena is adopted from von Storch and Zwiers (2001), but somephenomena are added.

–1–1.2 Classification of OGCMs
In recent years, the OGCMs are used to directly simulate some of the previously unresolved small scale phenomena.
These include submesoscale currents (McWilliams, 2016) and the generation of internal gravity waves due to the interactions of tides and mesoscale currents with topographic features. Because large scale features such as mesoscale eddies,boundary currents, and oceanic fronts are prerequisite for the spontaneous emergence of these fine-scale features, anapproach of embedding (nesting) subdomains with refined grids in a basin scale model is often taken. Use of a common
OGCM code is preferable in these approaches.

1.2 Classification of OGCMs
Most OGCMs used by ocean research scientists and by operational centers for forecasting climate and oceanic statesnumerically solve almost the same set of equations for the Boussinesq and hydrostatic ocean. The fundamental equations
consist of the momentum equation for continuous fluid, the advection-diﬀusion equation for temperature and salinity, theequation of state of sea water, and the mass conservation equation, collectively called primitive equations (Chapter 2).
If necessary, equations for additional processes such as surface mixed-layer physics, sea ice, and bottom boundary layer
physics are added.
Most OGCMs applied to ocean-climate studies adopt the finite diﬀerence or finite volume method on structured meshto discretize the equations. Recently, the finite element method on horizontally unstructured mesh is also applied to bothcoastal ocean simulation and global ocean simulation (e.g., Chen et al., 2011; Danilov, 2013). MRI.COM takes the former
approach. The spectral approach widely used by atmospheric models would have diﬃculty treating lands that completely
block ocean circulation in the zonal direction, and thus this approach is not usually adopted in general-purpose ocean
models.
Ocean models adopting horizontally structured mesh are classified by how they discretize the vertical direction. The
choice of the vertical coordinate leads to fundamental diﬀerences among the models. There are three classes: zcoordinate models or z-models adopting depth as the vertical coordinate, σ-coordinate or terrain following models
adopting fractional depth between the sea surface and the sea floor as the vertical coordinate, and ρ-coordinate or
isopycnal models adopting isentropic surfaces (iso-potential density surfaces) as the vertical coordinate. Each class has
its advantages and disadvantages and recent eﬀorts are directed toward adopting generalized vertical coordinates, i.e.,
remedying each model’s disadvantages by using advantages of other classes. Readers are referred to the book by Griﬃes(2004) for more general discussion about OGCMs.

1.2.1

Z -coordinate models ( z -models)The first z-coordinate general circulation model was developed by Dr. Kirk Bryan and his colleagues at the GeophysicalFluid Dynamics Laboratory (GFDL) in the 1960’s (e.g., Bryan, 1969). This model is sometimes referred to as the BryanCox-Semtner model or GFDL model. The z-models utilize the character of the ocean that the local pressure is expressed
as a function of depth by zero-order approximation, which makes implementing the equation of state straightforward.
Implementation of bottom topography and drawing of results are also straightforward. The models of this class aremost widely used in the community because of their versatility. Such models were first used as components of coupledatmosphere-ocean models.
The descendant of the GFDL model is called the Modular Ocean Model (MOM; Griﬃes, 2012) and is one of the mostwidely used z-models. Massachusetts Institute of Technology general circulation model (MITgcm; Marshall et al., 1997)
and Nucleus for European Modelling of the Ocean (NEMO; Madec and the NEMO team, 2008) are also widely used.
Most climate centers participating in climate model intercomparison projects use the z-coordinate models; MRI.COM
also adopts z-coordinate. In Japan, the Center for Climate System Research Ocean Component Model (COCO) at the
University of Tokyo (Hasumi, 2006) and the Research Institute of Applied Mechanics Ocean Model (RIAMOM) are also
in this class.
The major disadvantages of this class of models are as follows:• The vertical resolution in shallow seas and near the sea floor tends to be low and the processes that arise near thecoast and the sea floor tend to be poorly reproduced.
• Numerical inaccuracy in the tracer transport algorithm immediately leads to spurious diapycnal mixing of thetransported properties, while the diapycnal mixing is supposed to be very small in the ideal ocean interior.
The first disadvantage is expected to be remedied by the σ-models and the second by the ρ-models. However, z-model’sdisadvantages are not completely overcome; these substitutes have their own diﬃculties.

–2–

Chapter 1

OGCMs and MRI.COM

1.2.2 Sigma-coordinate models (σ -models)The first sigma-coordinate model was developed by Dr. George Mellor and his colleagues at Princeton University. Since
the number of vertical grid points is invariable throughout the model domain, σ-models are widely used for coastal ocean
simulations.
Two major sigma-models are widely used in the community: The Princeton Ocean Model (POM; Mellor, 2004) and
the Regional Ocean Modeling System (ROMS; Shchepetkin and McWilliams, 2003; 2005).
The major disadvantages of this class of models are as follows:• An accurate representation of the horizontal pressure gradient is diﬃcult near steeply sloping bottom topography.
• The lateral mixing along the same vertical layer near the continental slope region might lead to mixing of theshoreward light water and the seaward dense water.
These problems might prohibit using σ-models in long-term integrations of the global ocean.

1.2.3 Isopycnal-coordinate models ( ρ-models)The first isopycnal-coordinate model was developed by Dr. Rainer Bleck at the University of Miami. The development of
this class of models is based on the fact that sea water moves along isopycnal surfaces in the interior. Thus, the characterof a water mass is well maintained in the ocean interior. Since many theoretical studies of physical oceanography use
an isopycnal-coordinate framework, the ρ-models have the great advantage of providing good correspondence between
theory and numerical models.
A major ρ-model widely used in the community is the Miami Isopycnic Coordinate Ocean Model (MICOM; Bleck andBoudra, 1986) developed at the University of Miami.
The major disadvantages of this class of models are as follows:• Implementation of surface mixed layer models into a ρ-model is in itself inappropriate.
• Since the density levels are prescribed, this class of models might not be appropriate for studying a drastic climatechange that could lead to great variations in density of major water masses.
The Hybrid Coordinate Ocean Model (HYCOM; Bleck et al., 2002) has been developed in an eﬀort to remedy some ofthese disadvantages.

1.3 About MRI.COM
MRI.COM is a z-coordinate model and uses the finite volume method on structured mesh to discretize the governing
equations. The horizontal grid arrangement is Arakawa’s B-grid (Arakawa, 1972). Coast lines are defined by the
periphery of the grid cell centered by the velocity point, i.e., the lines connecting the tracer points. This arrangement issuitable for the discrete expressions for the side boundary conditions for velocity, and transport through a narrow passage
can be achieved with a single grid cell.
The program structure of MRI.COM is presented along with a typical computational cost of each process in Figure 1.2.
Though the program code should ideally use MKS units, MRI.COM basically uses cgs units for historical reasons. The
sea ice model, the mixed-layer model, some surface bulk formulae, and biogeochemical models are coded in MKS units
and converted into cgs units before their outputs are used by the main part. Details are described in each chapter.

1.4 Future of OGCMs and MRI.COM
As OGCMs acquire scientific and numerical integrity, they expand the area of their usage and begin to fulfill social as well
as scientific needs. The developer of an OGCM thus has the responsibility to ensure its scientific and numerical integrity
and to acknowledge its limitations. Feedback from users of various fields and continuous eﬀorts to overcome limitations
will certainly improve the OGCMs.
The advance of OGCMs has kept pace with that of super-computers. The mainstream of super computing is shiftingfrom vector computation to massively parallel computation with distributed memory. To rewrite the current vector-friendly
program codes to adapt to this shift is immediately needed for some OGCMs. With increasing computing power, everhigher resolution simulations will be achieved. The result will be more strongly aﬀected by how subgrid-scale processes
are parameterized and thus subgrid-scale schemes should be selected carefully.
To continue to stand as a multi-purpose model, an OGCM should be easily coupled with other component models and–3–1.4 Future of OGCMs and MRI.COM[program ﬁle name]exchange withatmosphere

initialization[time.F90, gridm.F90, gridg.F90, topo.F90]

chap 21

typical cost
4%
11%

surface ﬂuxes

bulk formulasea icechap 14chap 17[surﬄux.F90][bulk.F90][ice̲main̲cat.F90, etc.]

1%

continuity equation(determine vertical transport)

chap 5[continuity.F90]

6%equation of motion(baroclinic component)

chap 7[clinic.F90]

11%equation of motion(barotropic component)

chap 6[surface.F90][surface̲integ.F90][surface̲bcli.F90]

29%

advection of tracers

chap 8chap 13[tracer.F90][som̲adv.F90]

22%

diﬀusion of tracers(isopycnal and diapycnal)

chap 9chap 10[trcimp.F90][ipcoef.F90][ipycmix.F90]

2%calculation of density
stability check and adjustment

chap 4chap 10[density.F90][stratiﬁcation.F90][cnvajs.F90]

2%

preparation for the next time step chap 19[rewrit.F90][surface.F90]

12%

mixed layer model
determine vertical mixing coeﬀ.

chap 15[my25.F90][nkoblm.F90][gls.F90]

1%

output to ﬁleswith speciﬁed intervals

chap 21[history.F90]

termination

Figure1.2 Structure and typical computational cost of components of MRI.COM. Note that the sum of computationalcosts does not become 100% owing to round-oﬀ at each component.

data assimilation schemes. Other component models include wave, river, iceberg, and ice-shelf models. In the interaction
with these models, physical quantities should be exchanged by following conservative lows. Having an interface touniversal couplers and an adjoint code should be mandatory for such a multi-purpose OGCM.
These are the main subjects for developing MRI.COM in the coming years.

–4–Part I

Configuration

Chapter 2

Governing Equations
In this chapter, the governing equations for the general ocean circulation are formulated. These equations are usually calledprimitive equations. A discrete form of these equations is also presented to explain the fundamental solution methods.
The detailed numerical methods are presented in later chapters.

2.1 Formulation
2.1.1 Coordinate System
The fundamental purpose of developing this ocean circulation model (MRI.COM) is to use it for realistic simulationsof oceanic circulations in various circumstances. To achieve this, we must carefully choose a coordinate system before
formulating the governing equations.
In the lateral direction, the governing equations need to be formulated on a sphere. Originally, spherical coordinates
were adopted, and the equations were discretized on a geographical (latitude-longitude-depth) grid. A problem arises
for a global model because the North Pole is a singular point in the geographic coordinate system. Since the zonal grid
widths within five latitudinal degrees from the Pole become less than a tenth of those in middle to low latitudes, a shorttime step is required owing to the limitation of the Courant-Friedrichs-Lewy (CFL) condition.∗ This becomes a burden
on performing long-term integration.
One simple means to remove this North Pole singularity is to shift both poles to land. In this case, one may use thespherical coordinate program codes without major modification by only adjusting the Coriolis parameter. Unfortunately,
there are not many pairs of points on land that are symmetric about the Earth’s center.† Even if the most ideal pair with
poles on Greenland and Antarctica (near the Ross Sea) was chosen, it is only 5 degrees from the coastline to the newly
shifted pole. One might also be concerned that the Equator is not represented as a line on the shifted grid arrangement.‡
To resolve these issues, the model equations are represented on generalized orthogonal coordinates instead of sphericalcoordinates. Users can select the coordinate system according to their purposes. For example, the resolution of a target
region can be intentionally enhanced by placing a pole of the distorted grid near the target region. Of course, a regionalmodel without the North Pole may be arranged on geographic coordinates since spherical coordinates are one form ofgeneralized orthogonal coordinates. Now our model equations are formulated on generalized orthogonal coordinates.
Chapter 20 summarizes the concepts and calculus related to generalized orthogonal coordinates.
In the vertical direction, the depth coordinate was adopted from the first stage of the development. No attempt has
been made to apply other options such as terrain following or density coordinates. In the earliest stage, the sea surface
was assumed to have a rigid-lid on it. Then, the sea surface was allowed to move freely. When the free surface was first
introduced, the movement of sea surface was absorbed in the first layer of the model. The problem with this treatment is that
it is not possible to take the first layer thickness thinner than about 4 meters whereas a finer vertical resolution is requirednear the sea surface. This is because the contrast of mean sea level in the global ocean may reach 3 meters. To resolve thisproblem, the upper several layers were allowed to undulate following the sea surface evolution as in the sigma-coordinate
model (Hasumi, 2006). A problem with this approach is that there is a transition in the vertical coordinate, which would
make analytical treatment awkward in some situations. For MRI.COM version 4, we have adopted a vertically rescaledheight coordinate system, where a sea level undulation is reflected throughout the water column (Adcroft and Campin,
2004). This vertical coordinate is named z ∗ coordinate.
In this section, we first formulate the governing equations on Cartesian coordinates for brevity. Then a coordinatetransformation in the lateral direction is applied to the governing equations. Approximations and boundary conditions
The time step, ∆t, needs to satisfy v∆t/∆x ≤ 1, where v is velocity and ∆x is the grid width.
Greenland and Antarctica, China and Argentina, Kalimantan and Columbia.
‡ If the grid size is fine enough, the Kelvin wave in the shifted-pole model will propagate along the Equator as the theory suggests.

∗

†

–7–

2.1 Formulation
are discussed using equations on generalized orthogonal coordinates. Equations are further transformed to introduce
generalized vertical coordinates. Readers are referred to Griﬃes and Adcroft (2008) for the thorough discussion on theformulation of primitive equations for ocean circulation models.

2.1.2 Primitive equations in Cartesian coordinatesa. General governing equations

Evolution of state variables of ocean circulation (v, ρ, θ, S, p), where v = (u, v, w) is the velocity vector, ρ is density, θ ispotential temperature, S is salinity, and p is pressure, is obtained by solving the following simultaneous equations written
in Cartesian coordinates.
Momentum equation in a vector form isρ∂v+ ρ(v · ∇)v + 2Ω × ( ρv) = −ρ∇Φ − ρ∇ΦT − ∇p + ∇ · τ,
∂t

(2.1)

where Ω is the rotation vector of the Earth, Φ is the geopotential, ΦT is the tide producing potential, τ is the frictional stresstensor. In the rest of this chapter, the tide producing potential is neglected for brevity. Implementation of tide producing
potential is thoroughly discussed in Chapter 6.
Mass conservation equation is∂ρ+ ∇ · ( ρv) = 0.
(2.2)∂tEquations for potential temperature and salinity are∂θ+ ρv · ∇θ = −∇ · ( ρFθ ) + ρQ θ
∂t

(2.3)

∂S
+ ρv · ∇S = −∇ · ( ρFS ) + ρQ S,
∂t

(2.4)

ρ
and
ρ

where Fθ,S are tracer fluxes due to subgrid-scale transport and mixing parameterizations and Q θ,S are sources of tracers
due to nudging, convective adjustment (Section 10.2), shortwave absorption (Section 14.3), etc. Here, tracer concentration
is expressed as concentration per unit mass of sea water.
Equation of state of sea water determines in situ density of sea water. Density is given as a function of potential
temperature, salinity, and pressure:ρ = ρ(θ, S, p).
(2.5)
The equation of state is usually given as a polynomial fit to the available measurements. A detailed description of this will
be presented in Chapter 4.
The above set is the most general set of equations governing the evolution of oceanic state.
b. Boussinesq approximation

Because the density of sea water varies only by 5% throughout the water column and the horizontal density variations areless than 1%, most ocean general circulation models use the Boussinesq approximation. In the Boussinesq approximation,
the density (ρ) in the non-linear product of density times velocity (ρv) that appears in the momentum equation (2.1) is
replaced by a reference density (ρ0 ). The momentum equation becomes∂v
ρ1
1+ (v · ∇)v + 2Ω × v = − ∇Φ − ∇p + ∇ · τ.
∂t
ρ0
ρ0
ρ0

(2.6)

Note that the in-situ density (ρ) is retained for the geopotential term.
Further, the sea water is treated as incompressible. Mass conservation equation (2.2) becomes the volume conservation
equation:∇ · v = 0.
(2.7)
The tracer concentrations are now concentration per unit volume instead of unit mass:∂θ+ v · ∇θ = −∇ · Fθ + Q θ ,
∂t
–8–

(2.8)

Chapter 2

Governing Equations

and∂S+ v · ∇S = −∇ · FS + Q S .
∂t

(2.9)

The above expression is the most general form under the Boussinesq approximation. This form is used to formulate an
oceanic non-hydrostatic model and a quasi-hydrostatic model (Marshall et al., 1997).

2.1.3 Primitive equations in generalized orthogonal coordinates
We consider the momentum equation first. On a generalized orthogonal coordinate system (µ, ψ, r) whose unit vectors
are eµ , eψ , and er , the momentum equation for velocity v = ueµ + veψ + wer , where u = hµ µ̇, v = hψ ψ̇, w = hr ṙ, isrepresented by
(
)
(
)
∂hµ
∂hψ
∂hµ
1 ∂p
v
w
∂hr
∂u
µ
+ v · ∇u + f ψ w − f v = −
−
u−
v −
u−
w + Ffric,(2.10)∂tρ0 hµ ∂ µ hµ hψ ∂ψ∂µhr hµ ∂r
∂µ
(
)
(
)
∂hψ
∂hψ
∂hµ
∂v
1 ∂p
w
∂hr
u
ψ
+ v · ∇v + f u − f µ w = −
−
v−
w −
v−
u + Ffric,(2.11)∂tρ0 hψ ∂ψ hψ hr ∂r∂ψh µ hψ ∂ µ
∂ψ
(
)
(
)
∂hµ
∂hψ
∂w
1 ∂p
ρ ∂Φ
u
∂hr
v
∂hr
r
+ v · ∇w + f µ v − f ψ u = −
−
−
w−
u −
w−
v + Ffric, (2.12)∂t
ρ0 hr ∂r
ρ0 ∂r
hr h µ ∂ µ∂rhψ hr ∂ψ∂r
where hµ , hψ , and hr are scale factors, which measure the width in the original coordinate of the unit length in the newcoordinate. Metric terms appear on the r.h.s.. Ffric is frictional force obtained as the divergence of frictional stress tensor.
The radial distance from the Earth’s center is represented by r and the gravitational acceleration is in the negative directionof r.
The Coriolis force is represented by2Ω × v = (2Ωψ w − 2Ωr v)eµ + (2Ωr u − 2Ωµ w)eψ + (2Ωµ v − 2Ωψ u)er ,(2.13)

where Ω = Ωµ eµ +Ωψ eψ +Ωr er is the rotation vector of the Earth. We designate f µ = 2Ωµ , f ψ = 2Ωψ , and f = f r = 2Ωr .
We apply the following two approximations which are relevant to the momentum equation.
a. Shallow ocean approximation

Shallow ocean approximation employs the fact that the vertical thickness of the ocean is far smaller than the radius of the
Earth. Since the vertical scale of motion of a water particle is far smaller than the Earth’s radius (a), the radial distance r
as a scalar quantity is replaced by the Earth’s radius a. The new vertical coordinate (z) is the distance (positive upward)
from the geoid (sea surface height in the state of rest) and ∂/∂r is replaced by ∂/∂z. We set a constant vertical scale
factor hr (= hz = 1). Horizontal scale factors are independent of vertical coordinate (∂hµ /∂r = ∂hψ /∂r = 0). As aresult, to conserve angular momentum under this approximation, we drop the metric terms that involve w for the horizontal
components and all the metric terms for the vertical components.
We also assume that the gravitational acceleration is constant (g). This assumption results in a specific expression ofgeopotential as Φ = gz.
b. Hydrostatic approximation

For horizontal motions with a scale larger than a few kilometers, hydrostatic balance is maintained in the vertical direction.
The vertical momentum equation becomes:
∂p
0=−
− ρg.
(2.14)∂z
By hydrostatic approximation, we must drop all the remaining Coriolis terms that do not involve f to conserve angularmomentum (Phillips, 1966).
We also separately treat horizontal and vertical strain for calculating frictional stresses. The vertical stress is usually
parameterized as the vertical diﬀusion of momentum:
Ffric =1
∂ ( ∂u )1∇·τ =νV+ ∇ · τ horizontal strain,
ρ0
∂z
∂z
ρ0

where νV is the vertical viscosity (essentially vertical diﬀusion of momentum).

–9–(2.15)

2.1 Formulationc. Equations solved by a standard version of MRI.COM

With the above approximations, the momentum equation is now written as:
{
}
(
)
∂hψ
∂(hψ uu) ∂(hµ vu)
∂hµ
∂u
1
∂(wu)
v
+
+
+
+
u−
v − fv∂t hµ hψ∂µ∂ψ∂z
hµ hψ ∂ψ∂µ1 ∂p1
∂ ( ∂u )=−+(∇ · τhorizontal strain )u +νV,
ρ0 h µ ∂ µ ρ0
∂z
∂z
}
)
{
(
∂(hψ uv) ∂(hµ vv)
∂hψ
∂hµ
∂v
1
∂(wv)
u
+
+
+
+
v−
u + fu∂t hµ hψ∂µ∂ψ∂zh µ hψ ∂ µ
∂ψ
1 ∂p
1
∂ ( ∂v )=−+(∇ · τhorizontal strain )v +νV.
ρ0 hψ ∂ψ ρ0∂z∂z(2.16)(2.17)

Continuity equation is written as:1h µ hψ

{∂(hψ u) ∂(hµ v)
+
∂µ
∂ψ

}

∂w
= 0.
∂z

+

The equations for potential temperature and salinity are written as:{}∂(hψ uθ) ∂(hµ vθ)
∂θ
1
∂(wθ)
=−
+
−
− ∇ · Fθ + Q θ ,∂th µ hψ
∂µ∂ψ∂z{}∂(hψ uS) ∂(hµ vS)
∂S
1
∂(wS)
=−
+
−
− ∇ · FS + Q S .
∂th µ hψ
∂µ∂ψ∂z(2.18)(2.19)(2.20)

2.1.4 Boundary conditions
a. Momentum equation■ Upper surface ( z = η ):At the sea surface, momentum flux enters the ocean in the form of wind stress (or stress fromsea ice where sea ice exists):(τ µ, τψ )∂(u, v)=.
(2.21)
νV∂z z=η
ρ0Note that z is defined positive upward (the surface wind stress is positive into the ocean).
In the model algorithm, this is treated as a body force to the first level velocity,µψ

(Fsurf, Fsurf ) =(τ µ, τψ ),
ρ0 ∆z 1(2.22)

2

where ∆z 1 is the thickness of the first layer, and τ µ and τψ are zonal and meridional stress, respectively.
2
Surface fresh water flux is assumed to have zero velocity.
■ Bottom ( z = −H ):µψ

Bottom friction (τb in zonal and τb in meridional direction) exists at the sea floor (z = −H (µ, ψ)).
µ

νV

ψ

(τ , τ )∂(u, v)=− b b .
∂z z=−H (µ,ψ)ρ0(2.23)In the model algorithm, this is treated as a body force to the bottom level (k = kbtm) velocity,µµψ

(Fbottom, Fbottom ) =ψ(τb , τb )
ρ0 ∆zkbtm− 12where ∆zkbtm− 1 is the thickness of the bottom layer.
2■ Side wall:

No slip condition is applied (u = v = 0).

– 10 –,(2.24)

Chapter 2

Governing Equations

b. Temperature and Salinity■ Upper surface ( z = η ):At the sea surface, heat and fresh water are exchanged with atmosphere and sea ice. Salt is also
exchanged below sea ice. All these exchanges are expressed as surface fluxes and become surface boundary conditions.
The surface boundary conditions for temperature and salinity are expressed as follows:∂θθ= Fsurf,∂z z=η∂SS
κV= Fsurf,∂z z=η
κV(2.25)(2.26)α ) is defined positive downward (positive into the ocean).
where surface flux (Fsurf
Sθ) and sediment trap (Fbottom) may aﬀect■ Bottom ( z = −H ): At the sea floor (z = −H (µ, ψ)), geothermal heating (Fbottom

temperature and salinity:

κV

∂θ
θ
= Fbottom,∂z z=−H (µ,ψ)

κV

∂S
S
= Fbottom,∂z z=−H (µ,ψ)(2.27)

αwhere bottom flux (Fbottom) is defined positive upward (positive into the ocean).

■ Side wall:

For any tracer, the adiabatic condition is applied at the side wall:
∂θ
= 0,
∂n

∂S
= 0,
∂n

(2.28)

where n denotes the direction perpendicular to the wall. River discharge is expressed as a part of the surface fresh waterflux.
c. Continuity equation■ Upper surface ( z = η ):At the sea surface, vertical velocity is generated because a water parcel moves following thefreely moving sea surface. Surface fresh water flux is explicitly incorporated into the boundary condition for the continuity
equation.
dη
∂η
1 ∂η
1 ∂η
w=
− (P − E + R + I) =+u+v
− (P − E + R + I),(2.29)dt∂thµ ∂ µhψ ∂ψwhere P is precipitation, E is evaporation, R is river discharge, and I is fresh water exchange with sea ice component.
■ Bottom ( z = −H ):At the sea floor, vertical velocity is generated because the water parcel moves following the bottom

topography:( 1 ∂H
1 ∂H )
w=− u+v.
hµ ∂ µhψ ∂ψ(2.30)■ Evolution equation for sea surface height (η ):Using these boundary conditions, we obtain evolution equation for sea
surface height η by vertically integrating the continuity equation (2.18),{}∂(hψ (H + η)⟨u⟩) ∂(hµ (H + η)⟨v⟩)
1∂η++= P − E + R + I,
(2.31)∂th µ hψ
∂µ∂ψ∫η
1where ⟨(u, v)⟩ = H+η
(u, v)dz.
−H
d. Mixing at the surface boundary layer

Near the sea surface, strong vertical mixing may occur for stably stratified situations because of active turbulence. These
processes occur on a small scale (< several meters) and cannot be resolved in a large scale hydrostatic model with typicalhorizontal and vertical resolutions. These processes are parameterized as enhanced vertical viscosity and diﬀusivity. The
user chooses whether to set a high vertical viscosity and diﬀusivity a priori or to use a surface boundary layer model.
MRI.COM supports several surface boundary layer models. In the surface boundary layer models, vertical viscosity anddiﬀusivity are calculated every time step. See Chapter 15 for details.
– 11 –

2.1 Formulation

2.1.5 Generalized vertical coordinates
From MRI.COM version 4, the standard choice for the vertical coordinate is z ∗ , which was first introduced by Adcroft andCampin (2004). Before showing a specific expression of governing equations in z ∗ coordinate, we consider generalized
vertical coordinates (s). Note that generalized vertical coordinates employed by ocean models are not orthogonal.
Horizontal velocities are not perpendicular to the vertical coordinate but perpendicular to the local gravitational field. The
generalized vertical coordinate surface s = s(µ, ψ, z, t) is expressed as a smooth function of the original coordinate andtime. We introduce a new scale factor z s , which measures the thickness in the original depth coordinate of the unit length
in the new coordinate:∂z.
(2.32)zs ≡∂s µψtWe further introduce vertical velocity ṡ in generalized vertical coordinates.
Using a transformation rule presented by Adcroft and Campin (2004), we write the governing equations in generalized
vertical coordinates as follows:{}()∂(z s hψ uu) ∂(z s hµ vu)∂hµ∂hψ∂(z s u)1∂(z s ṡu)
v
+
+
+
+ zs
u−
v − zs f v∂th µ hψ∂µ∂ψ∂shµ hψ ∂ψ∂µ1 ∂pρ ∂(gz)11 ∂ ( νV ∂u )= −z s− zs+ z s (∇ · τhorizontal strain )u + z s,(2.33)
ρ0 h µ ∂ µρ0 h µ ∂ µρ0z s ∂s z s ∂s{}()∂(z s hψ uv) ∂(z s hµ vv)
∂hψ
∂hµ
1
∂(z s ṡv)∂(z s v)
u
+
+
+
+ zs
v−
u + zs f u∂th µ hψ∂µ∂ψ∂sh µ hψ ∂ µ∂ψ
1 ∂pρ ∂(gz)11 ∂ ( νV ∂v )= −z s(2.34)− zs+ z s (∇ · τhorizontal strain )v + z s,ρ0 hψ ∂ψ
ρ0 hψ ∂ψρ0z s ∂s z s ∂s
ρ
1
∂z s
+
∂t
h µ hψ{∂(gz) ∂p
+
= 0,
∂s
∂s

∂(z s hψ u) ∂(z s hµ v)+∂µ∂ψ(2.35)}+∂(z s ṡ)
= 0,
∂s

{
}
∂(z s hψ uθ) ∂(z s hµ vθ)∂(z s θ)1∂(z s ṡθ)+++= −z s ∇ · Fθ + z s Q θ ,∂th µ hψ∂µ∂ψ∂s{}∂(z s hψ uS) ∂(z s hµ vS)∂(z s S)1∂(z s ṡS)+++
= −z s ∇ · FS + z s Q S,∂th µ hψ
∂µ∂ψ∂s(2.36)(2.37)(2.38)andρ = ρ(θ, S, p).

(2.39)

Evolution equation for sea surface height is obtained by vertically integrating the continuity equation (2.36) andconsidering the boundary condition. It has the same form as in the depth coordinate system:][∂(hψ U) ∂(hµ V )1∂(η + H)++
= P − E + R + I,
(2.40)∂th µ hψ
∂µ∂ψ
where(U, V ) =∫ s(z=η)[z s (u, v)]ds.

(2.41)s(z=−H )2.1.6z ∗ coordinate

a. Definition and boundary condition

Definition of the new vertical coordinate z ∗ is as follows:z ∗ = σ(µ, ψ, z, t)H (µ, ψ) =z − η(µ, ψ, t)
H (µ, ψ),H (µ, ψ) + η(µ, ψ, t)– 12 –(2.42)

Chapter 2

Governing Equations

where z = −H (µ, ψ) is sea floor and η is the free surface displacement. σ is a symbol of the conventional terrain following
vertical coordinate. In z ∗ coordinate, σ is further scaled by sea floor depth H (µ, ψ), which makes z ∗ coordinate moresimilar to the depth coordinate rather than the terrain following coordinate.
∂z
The scaling factor z s = ∂z
∗ is
∂z
H +η
=
.
(2.43)∗∂z
H
The vertical velocity in this coordinate system is expressed as w ∗ . This has the following relation with the verticalvelocity w of the depth coordinate z,
w ∗ ≡ Dt z ∗ =

(
)
H (
z∗ )
z∗ η
w− 1+
Dt η + 2 vh · ∇H ,
H +η
H
H

(2.44)

where Dt represents the material time derivative operator in any coordinate system,
Dt ≡

∂
∂
u ∂
v ∂
w ∂
+v·∇=
+
+
+
.
∂t
∂t hµ ∂ µ hψ ∂ψ z s ∂s(2.45)

Horizontal boundary conditions are unchanged by this coordinate transformation. Vertical boundary conditions needsome discussion.
Sea floor (z = −H) in z ∗ coordinate is also −H.
z = −H (µ, ψ) ⇒ z ∗ = −H (µ, ψ).

(2.46)

Thus the kinematic boundary condition at sloping bottom is
wz=−H = −vh · ∇H ⇒ wz∗∗ =−H = −vh · ∇H,
where vh = (u, v, 0) is the horizontal component of velocity at z = −H (µ, ψ).
Sea surface (z = η) in z ∗ coordinate isz = η(x, y, t) ⇒ z ∗ = 0.

(2.47)(2.48)

Sea surface is fixed in time in z ∗ frame. In other words, model domain and grid cells are logically fixed in time. We do
not have to concern about the moving sea surface and vanishing of the first layer thickness § .
The kinematic boundary condition at the sea surface is
wz=η = Dt η − (P − E + R + I) ⇒ wz∗∗ =0 = −H(P − E + R + I).
H +η(2.49)For example, precipitation (P > 0) penetrates the ocean as a scaled downward vertical velocity.
The governing equations in z ∗ vertical coordinate are in the same form as equations (2.33) through (2.40), with z s
replaced by zz ∗ = H+ηH .
b. Pressure gradient term

Horizontal momentum equations in generalized vertical coordinates (2.33), (2.34) involve both pressure gradient andgeopotential gradient term. Pressure gradient error appears when these terms do not cancel each other. That said, pressure
gradient error is not a big issue for z ∗ coordinate because horizontal gradient of a constant z ∗ -surface is usually very small.
However, source of errors must be kept as small as possible. For this purpose, we first separate density into constant andits deviationρ = ρ0 + ρ′ (x, y, z ∗, t).
(2.50)
Pressure is also separated in the same manner (p = p0 + p′) and the hydrostatic relation (2.35) is separated into two
equations(H +η)

ρ0 and p0 (z ∗ = 0) = 0,
H
(H +η)∂z ∗ p′ = −g
ρ′ and p′ (z ∗ = 0) = pa,H∂z ∗ p0 = −g

(2.51)(2.52)§ However, sea surface is not allowed to touch the see floor (H + η ≤ 0), which is a local problem. This is a restrictive condition inherent to thiscoordinate system.

– 13 –

2.2

Numerical Methods

where pa is atmospheric pressure¶ . Specific expression for p0 is obtained by integrating (2.51)
p0 (z ∗ ) = p0 (x, y, z, t) = −g ρ0

H +η ∗
z = g ρ0 (η − z) = g ρ0 η − ρ0 Φ.
H(2.53)

Rewriting pressure gradient term by using this, we have
ρ1
ρ0 + ρ ′1∇z ∗ (p0 + p′ ) + ∇z ∗ Φ = g∇η − ∇z ∗ Φ + ∇z ∗ p′ +∇z ∗ Φ
ρ0
ρ0
ρ0
ρ0
1
ρ′
= g∇η + ∇z ∗ p′ + ∇z ∗ Φ.
ρ0
ρ0(2.54)(2.55)

It is noticed that time-independent terms are removed. Because geopotential is Φ = gz, the momentum equation isexpressed asg ρ′
1∇z ∗ z = F(2.56)∂t vh + vh · ∇z ∗ vh + w ∗ ∂z ∗ vh + f ẑ × vh + g∇η + ∇z ∗ p′ +ρ0ρ0Perturbation pressure is obtained by integrating (2.52) asp′ (z ∗ ) = pa + g∫ 0z∗

ρ′ zz ∗ dz ∗ = pa + g∫ ηz(z ∗ )

ρ′ dz.

(2.57)

Then (2.56) becomes∂t vh + vh · ∇z ∗ vh + w ∗ ∂z ∗ vh + f ẑ × vh( ∫ η
) g ρ′
11∇z ∗ z = F.
ρ′ dz ++ ∇z ∗ (pa + g ρ0 η) + ∇z ∗ gρρρ0z(z ∗ )
|0
{z
} |0
{z
}
fast

(2.58)

slow

We separate the pressure gradient term into barotropic (fast) and baroclinic (slow) component in preparation for the
split-explicit solution method for equations of motion.

2.2 Numerical Methods
2.2.1 Discretization and finite volume method
To solve the primitive equations formulated in the previous section, the equations are projected on a discrete lattice andthen advanced for a discrete time interval.
Because primary choice of the vertical coordinate of MRI.COM is z ∗ , a logically fixed (but actually moving) Eulerian
lattice is arranged. A detailed description of the grid arrangement is given in Chapter 3. The equations are then volume
integrated over a discrete model grid cell. This approach is called a finite volume approach or sometimes a flux form
expression in this manual.
A vertically integrated expression for the primitive equations is useful for describing the solution procedure. These arecalled semi-discrete equations (Griﬃes, 2004). The body force and metric terms will be simply multiplied by the grid
width. The material transport and subgrid-scale flux terms need some attention.
In this section the vertical coordinate of z ∗ is written as a general symbol s. The material transport of any quantity αthat commonly appears in the prognostic equations,}{∂(hψ z s uα) ∂(hµ z s vα)∂(z s α)1∂(z s ṡα)+++(2.59)∂th µ hψ
∂µ∂ψ∂s¶ Ice-loading eﬀect (e.g., Campin et al., 2008) has not been included in MRI.COM– 14 –

Chapter 2

Governing Equations

is vertically integrated over a (k − 21 )-th grid cell bounded by s k−1 and s k to give{}∫ sk−1∫ sk−1∫ sk−1∂(hψ z s uα) ∂(hµ z s vα)
1∂(z s ṡα)∂(z s α)
ds +
+
ds +
ds
∂t
h
h
∂
µ
∂ψ
∂s
µ
ψ
sk
sk
sk
{
∫
∫
∫
)
)
)}
1
∂ ( sk−1∂ ( sk−1∂ ( sk−1=(z s α)ds +hψ z s uα ds +hµ z s vα ds∂t sk
h µ hψ ∂ µ s k∂ψ sk( ∂s k−1 z s u(s k−1 ) ∂s k−1 z s v(s k−1 ) ∂s k−1
)
− zs
+
+
− z s ṡ(s k−1 ) α(s k−1 )
∂t
hµ
∂µ
hψ
∂ψ
)
( ∂s k z s u(s k ) ∂s k z s v(s k ) ∂s k++
− z s ṡ(s k ) α(s k ).
+ zs∂thµ ∂ µhψ ∂ψThe first line on the r.h.s. is expressed in a semi-discrete form as
}
{
)
)
)
∂ (
1
∂ (
∂ (
∆zα 1 +hψ ∆zuα 1 +hµ ∆zvα 1 ,
k− 2
k− 2
k− 2
∂t
h µ hψ ∂ µ∂ψ

(2.60)(2.61)

where any quantity is assumed to have a uniform distribution within a grid cell.
Using ṡ ≡ Dt s and (2.45), the last two lines are reduced to the diﬀerence between vertical advective fluxes:
the last two lines of (2.60) = z s ṡ(s k−1 )α(s k−1 ) − z s ṡ(s k )α(s k ).

(2.62)For the sea surface (k = 1; s0 = 0) and the bottom (k = kbtm; skbtm = −H), kinematic conditions (2.49) and (2.47) areused to givethe last two lines of (2.60) = −(P − E + R + I)α(0) − z s ṡ(s1 )α(s1 )(2.63)
at the surface andthe last two lines of (2.60) = z s ṡ(skbtm−1 )α(skbtm−1 ) − 0
at the bottom (bottom term vanishes to give no advective fluxes through the sea floor).
Similarly, the vertical integral of the divergence of the subgrid-scale fluxes gives{}∫ sk−1∫ sk−1∂(hψ z s Fµ ) ∂(hµ z s Fψ )
1
∂Fs
+
ds +
ds
h
h
∂
µ
∂ψ
∂s
µ
ψ
sk
sk
{
}
)
)
∂ (
1
∂ (
+
=
hψ ∆zFµhµ ∆zFψk− 12
k− 12
h µ hψ ∂ µ∂ψ( Fµ (s) ∂s Fψ (s) ∂s)( Fµ (s) ∂s Fψ (s) ∂s)−+− Fs (s)++− Fs (s) .
k−1
khµ ∂ µhψ ∂ψhµ ∂ µhψ ∂ψ

(2.64)(2.65)

In summary, the material transport and subgrid-scale flux parts are integrated for a vertical grid cell to give thesemi-discrete expression on the r.h.s.,{}∫ sk−1∫ sk−1∫ sk−1∂(hψ z s uα) ∂(hµ z s vα)∂(z s α)1∂(z s ṡα)
ds +
+
ds +
ds
∂t
h
h
∂
µ
∂ψ
∂s
µ ψ
sk
sk
sk
}
{
∫ sk−1∫ sk−1∂(hψ z s Fµ ) ∂(hµ z s Fψ )
∂Fs
1
+
ds +
ds
+
h
h
∂
µ
∂ψ
∂s
µ ψ
sk
sk
{
}
)
)
)
∂ (
1
∂ (
∂ (
=
∆zα 1 +hψ ∆zuα 1 +hµ ∆zvα 1 + z s ṡ(s k−1 )α(s k−1 ) − z s ṡ(s k )α(s k )
k−
k−
k− 2
∂t
h µ hψ ∂ µ
∂ψ
2
2
}
{
)
)
1
∂ (
∂ (
+
+
+ Fs (s k−1 ) − Fs (s k ).
(2.66)hψ ∆zFµ
h
∆zF
µ
ψ
k− 12
k− 12
h µ hψ ∂ µ∂ψThe quantityαFsurf= (P − E + R + I)α(0) +( Fµ (s0 ) ∂s0
hµ

∂µ

+

)
Fψ (s0 ) ∂s0− Fs (s0 ) = (P − E + R + I)α(0) − Fs (0)hψ ∂ψ(2.67)

taken from (2.63) and (2.65) may be regarded as a surface forcing term and corresponds to the surface flux (positive
downward) given in the previous section. The first term on the r.h.s. of (2.67) is the tracer transport by the fresh water
flux, and the second term is the microstructure flux calculated by subgrid-scale parameterizations such as bulk formula.
Similarly, geothermal heating may be incorporated as microstructure flux from the sea floor (Fh (−H) · ∇h H + Fs (−H)).
– 15 –

2.2

Numerical Methods

2.2.2 Momentum equation
a. Mode splitting and explicit solution method for the barotropic mode

Here we consider to solve the momentum equation with hydrostatic and Boussinesq approximation. Equations are (2.33),(2.34), and (2.35). To integrate these equations in time, we should know the instantaneous vector field, pressure, andgeopotential. For the vector field, we use one at the previous time level. We obtain the pressure field by integrating thehydrostatic equation vertically, in which the sea surface height (z = η) is needed. The sea surface height is also needed forgeopotential. To obtain the surface height, we should solve vertically integrated continuity equation (2.40).
The rise and fall of the sea level causes external gravity waves whose phase speed is two orders of magnitude greaterthan that of other waves. This will impose tight limits on the time intervals due to the CFL condition. We want to separate
or remove external gravity waves, because they are usually not important when a target phenomenon has a longer timescale.
Historically, external gravity waves were removed from the model by prohibiting the vertical movement of the sea
surface (rigid-lid approximation). In this case, the vertically integrated equations result in a vorticity equation in the formof the Poisson equation, solved by relaxation methods. The surface pressure is then diagnostically obtained as the pressure
pushing up the lid.
After the sea surface was allowed to move vertically, the problem of fast external modes was resolved by separating thebarotropic mode from the baroclinic mode. The barotropic mode is solved explicitly with a short time step. The baroclinic
mode can take a longer time step by reflecting a state of the barotropic mode in which high-frequency components arefiltered out. Since this free surface option is more suitable for parallel computation than the rigid-lid approximation, only
the free surface option is supported by MRI.COM. The essence of the split-explicit free surface formulation is explained
below. See Chapter 6 for details.
b. Barotropic mode

If we put
U=

∫ 0
−H

z s uds =N∑

uk− 1 ∆zk− 1 ,
2

2

V=

k=1

∫ 0
−H

z s vds =N∑

vk− 1 ∆zk− 1 ,22(2.68)

k=1

and separate fast and slow modes of the pressure gradient as in (2.58), then the vertically summed semi-discrete momentum
equations are∂U(η + H) ∂(pa + g ρ0 η)− fV = −+ X,∂t
ρ0 h µ∂µ∂V(η + H) ∂(pa + g ρ0 η)+ fU = −+ Y,∂tρ 0 hψ
∂ψ

(2.69)(2.70)

where pa is the atmospheric pressure at sea surface. Density ρ has been separated into mean ρ0 and perturbation ρ′, and
X = − ∇H ·N(∑(∆z(u, v)u)k− 1

2

k=1

−

(
)
N [
∑
∂hψ ]
∂hµ
v
−
u−
v
∆zk− 1k− 21
2hµ hψ ∂ψ
∂µ
k=1

∫ 0
N
N [
N
∑
∑
]
g ∑[ ′ ]1 1µµµz s g ρµ ds ′ ∆zk− 1 −ρ zµ ∆zk− 1 +(∆zFhorz )k− 1 + Fsurf ∆z 1 + Fbottom ∆z N − 1
2
2
2
2
2
ρ
h
ρ
h
0
µ
0
µ
s 1
k=1
k=1
k=1
k−

(≡

)

N
∑

2

(2.71)

Fµ ),

k=1

(
)
N [
) ∑
∂hµ
∂hψ ]
u
(∆z(u, v)v)k− 1 +u−v∆zk− 1k− 21
2
2hµ hψ ∂ψ
∂µ
k=1
k=1
∫
N
N
N [
0
∑
∑
]
g ∑[ ′ ]1 1ψψψz s g ρψ ds ′ ∆zk− 1 −ρ zψ ∆zk− 1 +(∆zFhorz )k− 1 + Fsurf ∆z 1 + Fbottom ∆z N − 1
−
2
2
2
2
2
ρ
h
ρ
h
0
ψ
0
ψ
s
1
k=1
k=1
k=1

Y = − ∇H ·

N
(∑

k−

(≡

N
∑

2

Fψ ).

(2.72)

k=1– 16 –

Chapter 2

Governing Equations

The vertically integrated continuity equation is given by∂η
1 ( ∂(hψ U) ∂(hµ V ) )++= (P − E + R + I).
∂th µ hψ
∂µ∂ψ

(2.73)

We solve these equations for U, V , and η with a short time step constrained by the phase speed of the external gravity
waves. On returning (U, V ) to the baroclinic mode, the eﬀect of high-frequency phenomena with time scale shorter thanthe baroclinic time step is filtered out by a weighted averaging, which is symbolically expressed as follows:1⟨U⟩n+1 = U n + ∆t cl f ⟨⟨V ⟩⟩n+ 2 − ∆t cl⟨⟨ (η + H) ∂(pa + ρ0 gη) ⟩⟩n+ 21+ ∆t cl X n,
ρ0 h µ∂µ⟨⟨ (η + H) ∂(pa + ρ0 gη) ⟩⟩n+ 21
1+ ∆t clY n,⟨V ⟩n+1 = V n − ∆t cl f ⟨⟨U⟩⟩n+ 2 − ∆t cl
ρ 0 hψ
∂ψ

(2.74)(2.75)

∑M ∗
∑M ∗
∑
1
1
m′ − 12 , with m ∈ {1, M ∗ } representing the barotropic time
where ⟨·⟩n+1 ≡ m=1
bm (·) m− 2 and ⟨⟨·⟩⟩n+ 2 ≡ m=1
bm m
m′ =1 (·)
level, n representing the baroclinic time level, ∆t cl being the baroclinic time step, and bm being weighting factors explainedin Chapter 6.
c. Baroclinic mode

To solve the baroclinic mode, we can omit to obtain absolute pressure by using the method described below.
Velocity is decomposed into a barotropic component and a baroclinic component as follows:u =û + ū,(2.76)v =v̂ + v̄,(2.77)

where ū and v̄ are barotropic components and û and v̂ are baroclinic components.
We consider updating a new velocity (u ′, v ′) using a momentum equation where the fast pressure gradient term is
dropped:
u

′

k− 12

v

′

∆z n+11 − un 1 ∆z n 1
k− 2

k− 2

k− 2

∆t cl
n+1 − v n ∆z n
∆z
1
1
1
1

k− 2

k− 2

k− 2

k− 2

∆t cl

1

= f [vk− 1 ∆zk− 1 ]n+ 2 + Fµn,2(2.78)

2

1= − f [uk− 1 ∆zk− 1 ]n+ 2 + Fψn,2

(2.79)2where)[ v ( ∂hµ∂hψ ]∆z(u, v)u 1 −u−v∆zk− 1k− 2
k− 12
2
hµ hψ ∂ψ
∂µ
∫
N
0
[ 1 1
]
g ∑[ ′ ]µµµ−z s g ρµ ds ′ ∆zk− 1 −ρ zµ ∆zk− 1 + (∆zFhorz )k− 1 − Fvertk−1 + Fvertk ,222
ρ0 h µ s 1
ρ0 hµ k=1k−2)
[ u ( ∂hµ
(
)
∂hψ ]
n
+
∆zk− 1Fψ = −∇ H · ∆z(u, v)v
u−
v
k− 12
k− 12
2
hµ hψ ∂ψ
∂µ
∫
N
0
[ 1 1
]
g ∑[ ′ ]ψψψ−ρ zψ ∆zk− 1 + (∆zFhorz )k− 1 − Fvertk−1 + Fvertk .
z s g ρψ ds ′ ∆zk− 1 −222
ρ 0 hψ s 1
ρ0 hψ k=1
Fµn = −∇ H ·

(

)

k−

(2.80)(2.81)

2

Summing over the whole water column gives
∑N

∑N
′
n+1
n
n
k=1 (uk− 1 ∆z k− 1 ) − k=1 (uk− 1 ∆z k− 1 )
2

∑N

′

k=1 (vk−

2

2

2

∆t cl
n+1 ) − ∑ N (v n ∆z n )
1 ∆z
1
1
1
k=1
2

k− 2

k− 2

k− 2

∆t cl

= f

N
∑

(2.82)

N∑1[uk− 1 ∆zk− 1 ]n+ 2 + Y n .

(2.83)

2

2

k=1

= −f

2

k=1

– 17 –

1[vk− 1 ∆zk− 1 ]n+ 2 + X n,

2

2.2

Numerical Methods

X n and Y n are removed by using the equations from the barotropic mode (2.74), (2.75) to give∑N

k=1 (⟨u⟩n+1 ∆z n+1 ) − ∑ N (u′n+1
k=1 k− 1 ∆z k− 1 )
k− 12
2
2

∆t cl
∑N

k=1 (⟨v⟩

n+1 ∆z n+1 ) − ∑ N (v ′n+1
k=1 k− 1 ∆z k− 1 )
k− 12
2
2

∆t cl

1

= f ⟨⟨V ⟩⟩n+ 2 − f= − f ⟨⟨U⟩⟩

n+ 21

N
∑
⟨⟨ (η + H) ∂(pa + ρ0 gη) ⟩⟩n+ 12
1,[vk− 1 ∆zk− 1 ]n+ 2 −22
ρ0 h µ∂µk=1(2.84)N∑⟨⟨ (η + H) ∂(pa + ρ0 gη) ⟩⟩n+ 121+f[uk− 1 ∆zk− 1 ]n+ 2 −.
22
ρ 0 hψ
∂ψ
k=1(2.85)

where ⟨u⟩n+1 = ⟨U⟩n+1 /(η n+1 + H) and ⟨v⟩n+1 = ⟨V ⟩n+1 /(η n+1 + H).
This is combined with (2.78) and (2.79) to give()
z′
u 1 − u′ + ⟨u⟩n+1 ∆z n+11 − un 1 ∆z n 1[] n+ 21
z
k− 2
k− 2
k− 2
k− 2
1
n+ 21
n+1
n+1
= f vk− 1 ∆zk− 1− f [v]n+ 2 ∆zk−
∆zk−
1
1 + f ⟨⟨v⟩⟩
2
2
∆t cl
2
2
∆z n+11 ⟨⟨k−(η + H) ∂(pa + ρ0 gη) ⟩⟩n+ 12− n+1 2(2.86)+ Fµn
ρ0 h µ
∂µ
η
+H
(
)
z
′
v 1 − v ′ + ⟨v⟩n+1 ∆z n+11 − v n 1 ∆z n 1[] n+ 21
z
k− 2
k− 2
k− 2
k− 2
1
n+1
n+1
n+ 12
= − f uk− 1 ∆zk− 1+ f [u]n+ 2 ∆zk−
∆zk−
1 − f ⟨⟨u⟩⟩
1
2
2
∆t cl
2
2
∆z n+11 ⟨⟨k−(η + H) ∂(pa + ρ0 gη) ⟩⟩n+ 12(2.87)− n+1 2+ Fψn,
ρ 0 hψ
∂ψη+Hzwhere (...) denotes the thickness weighted vertical average.
1

z

1

1

z

1

Since f [v]n+ 2 ∆z n+11 ∼ f ⟨⟨v⟩⟩n+ 2 ∆z n+11 , f [u]n+ 2 ∆z n+11 ∼ f ⟨⟨u⟩⟩n+ 2 ∆z n+11 , and the fourth terms on the r. h. s. are
k− 2

k− 2
k− 2
z
z
the surface pressure gradient, we may regard u ′ − u ′ + ⟨u⟩n+1 and v ′ − v ′ + ⟨v⟩n+1 as the real updated velocity for timek− 2level n + 1, the baroclinic component is expressed as û = u ′ − u ′ and v̂ = v ′ − v ′.
To summarize, we first solve for (u ′, v ′ ) using (2.78) and (2.79), and then compute the baroclinic component by û = u ′ −u ′and v̂ = v ′ − v ′. The absolute velocity is obtained by u = û + ū and v = v̂ + v̄.

2.2.3 Continuity equation
The vertical component of velocity is obtained by vertically integrating the continuity equation (2.36) from top to bottom.
By using a flux form (setting α = 1 in the r.h.s. of (2.66)), the surface boundary condition (2.49) may be naturally included.
The vertical integration for the k-th vertical level is performed as follows:()()∂ hψ ∆zk− 1 uk− 1∂ hµ ∆zk− 1 vk− 1 
[
]

1 
2
2
2
2 

,
(z s ṡ)k = (z s ṡ)k−1 + ∆s k− 1 (∂t z s )k− 1 ++(2.88)
22h µ hψ ∂µ∂ψ
where ∆zk− 1 is the width of the (k − 21 )-th layer and ∆s k− 1 is the logical width of s of the (k − 21 )-th layer. Note also that22∂t z s is independent of depth.

2.2.4 Temperature and salinity equation
We solve for potential temperature instead of in situ temperature, because the potential temperature is conserved throughvertical movement.

– 18 –

Chapter 2

Governing Equations

a. A semi-discrete expression

The equation for potential temperature and salinity is an advection-diﬀusion equation (2.37) and (2.38). Its semi-discrete
expression is as follows:()∂(θ k− 1 ∆zk− 1 ) = − ∇ H · ∆zhψ uθ, ∆zhµ vθ
− (z s ṡθ)k−1 + (z s ṡθ)k
k− 12
2
2
∂t
(
)
θ
θ
θ
− ∇ H · ∆zhψ Fµθ , ∆zhµ Fψθ
1 − Fs k−1 + Fs k + Q ∆z k− 1 ,(2.89)()∂(Sk− 1 ∆zk− 1 ) = − ∇ H · ∆zhψ uS, ∆zhµ vS 1 − (z s ṡS)k−1 + (z s ṡS)k
k−
2
2
∂t
) 2
(
S
S
S
S
S
− ∇ H · ∆zhψ Fµ , ∆zhµ Fψ
1 − Fs k−1 + Fs k + Q ∆z k− 1 .

(2.90)

k− 2

2

k− 2

2

Several options for discretizing each term on the r.h.s. are detailed in Chapters 8 through 10.
b. Treating the unstably stratified layer

Since the hydrostatic approximation is used, an unstable stratification should be removed somehow. Generally, we assume
that vertical convection occurs instantaneously to remove unstable stratification. We call this convective adjustment, whichis explained in Section 10.2.
One might also choose to mix tracers by setting the local vertical diﬀusion coeﬃcient to a large value such as
10 000 cm2 s−1 where stratification is unstable. In this case, the tracer equation should be solved using the partial implicit
method, which is described in Section 19.5.

2.3 Appendix
2.3.1 Acceleration method
It usually takes several thousand years before the global thermohaline circulation reaches a steady state under (quasi-)steady
forcing. The limiting factor for the time step is the phase speed of external gravity waves (∼ 200 m/s) for the barotropic
mode and that of internal gravity waves (∼ a few m/s) for the baroclinic mode. A several-thousand-year integration will
not be a workable exercise as long as we are restricted by this criterion in determining the time step. Bryan (1984) proposeda method to accelerate the ocean’s convergence to equilibrium. In this method, the phase speed of waves is reduced bymodifying the governing equations, and a thermally balanced state is quickly reached by reducing the specific heat.
Specifically, they are achieved by multiplying a constant to the tendency terms (α for momentum and γ for tracers) toincrease inertia and to reduce specific heat. When a steady state is reached in these equations, we expect that the samebalance as the undistorted equations will be obtained, because the only diﬀerence between the distorted and undistorted
equations are tendency terms, which are expected to be zero in the steady state.
The modified momentum equation is given by{}()∂(hψ uu) ∂(hµ vu)
∂hµ
∂hψ
∂u
1
∂(wu)
v
α +
+
+
+
u−
v − fv∂t hµ hψ∂µ∂ψ∂z
hµ hψ ∂ψ
∂µ
(
)
1 ∂p
1
∂
∂u
=−
+
(∇ · τhorizontal strain )u +νV,
(2.91)
ρ0 h µ ∂ µ ρ0
∂z
∂z
}
{
(
)
∂(hψ uv) ∂(hµ vv)
∂hψ
∂hµ
∂(wv)
1
u
∂v
+
+
+
v−
u + fuα +∂t hµ hψ∂µ∂ψ∂zh µ hψ ∂ µ
∂ψ
(
)
1 ∂p
1
∂
∂v
(2.92)=−+(∇ · τhorizontal strain )v +νV.
ρ0 hψ ∂ψ ρ0∂z∂z
The modified temperature and salinity equations are given by{}∂(hψ uθ) ∂(hµ vθ)
∂θ
1
∂(wθ)
γ
=−
+
−
− ∇ · Fθ + Q θ ,∂th µ hψ
∂µ∂ψ∂z{}∂(hψ uS) ∂(hµ vS)
∂S
1
∂(wS)
γ
=−
+
−
− ∇ · FS + Q S .
∂th µ hψ
∂µ∂ψ∂zHere, equations are written in depth (z) coordinate for brevity.
– 19 –(2.93)(2.94)

2.3

Appendix

These modifications are equivalent to changing time to t ′ = t/α and the Brunt-Vaisala frequency to N ′2 = N 2 α/γ. In′
this case, the equivalent depth for the n-th mode of the vertical mode decomposition becomes Hn = Hn /α.
The dispersion relation for the free inertia-gravity waves becomes:
ω2 =f 2 ( gHn ) 2 2+(k + l ).
αα2(2.95)

Since the angular frequency ω is inversely proportional to α 1/2 , the phase speed becomes low for large α. The model
can be run with a long time step.
The dispersion relation for Rossby waves becomes:[
f 2 ] −1.
ω = − βk α(k 2 + l 2 ) +gHn(2.96)

Again, a large α yields a low phase speed.
In standard practice, a value from several tens to a few hundred is used as α, a value of one is used near the sea surface,and a value about a tenth is used near the bottom as γ.
It should be noted that when α is too large, the model field is prone to baroclinic instability. Since this should not occur
in nature, an integration of the model should be performed carefully by checking outputs during the integration.

2.3.2 Physical constants
On Table 2.1, we list basic physical constants used for MRI.COM. These are defined in param.F90. Physical constants orformulae used to calculate surface fluxes and sea ice processes are explained in Chapters 14 and 17, respectively.
Table2.1 Physical constants used in the model

radius of the Earthacceleration due to gravity
angular velocity of the Earth’s rotation
the absolute temperature of 0 ◦C
the average density of sea water
the specific heat of sea watervalue
6375.0 × 105 cm
981.0 cm · s−2
π/43 082.0 radian · s−1
273.15 K
1.036 g · cm−3
3.99 × 107 erg · g−1 · K−1(1.0 erg · g−1 · K−1= 1.0 × 10−4 J · kg−1 · K−1 )– 20 –

variable name in MRI.COM
RADIUS
GRAV
OMEGA
TAB
RO
CP

Chapter 3

Spatial grid arrangement
The model ocean domain is defined as a three-dimensional aggregate of rectangular grid cells limited by surfaces of
constant values on model’s logical coordinate system. Just above the bottom, vertical thickness of the cell can be locallychanged (so called partial cell). The horizontal and vertical lengths of the cells are regarded as the horizontal and verticalgrid sizes, respectively. In MRI.COM, the grid size can be varied spatially in each direction (variable grid size). Note thatthe introduction of z ∗ coordinate results in the temporally variable actual vertical grid sizes, though the logical grid size
is fixed in time.

3.1 Horizontal grid arrangement
MRI.COM adopts somewhat unique horizontal grid arrangement, which is illustrated in Figure 3.1a. Horizontal components of velocity and bottom depth are defined at the center of the grid cell (×), and tracers such as temperature and salinity,
density, and sea-surface height (SSH) are defined at the four corners of the cell (◦). Hereafter, for simplicity, the velocity
point is referred to as the U-point; the grid cell centered on the U-point, the U-cell (Figure 3.1c); the tracer point as the
T-point; and the grid cell centered on the T-point, the T-cell (Figure 3.1c). The T-cells are staggered from the U-cells by ahalf grid size and consists of partial cells along the coast lines (Figure 3.2). The coast lines are defined by the peripheryof the U-cells, i.e., the lines connecting the T-points. This type of horizontal grid arrangement is called Arakawa’s B-gridarrangement (Arakawa, 1972). Although Arakawa’s B-grid arrangement is also used in MOM (NOAA-GFDL, USA) and
COCO (AORI of U. Tokyo and JAMSTEC, Japan), the primary cell is the T-cell in those models and the coast lines aredefined by the lines connecting the U-points (Figure 3.1b).
In the case of the variable grid size, the T-points are defined just at the centers of the T-cells as seen in Figure 3.1c,but the U-points are not at the centers of the U-cells. The U-points are arranged so that the U-cell boundary stands at the
mid-point between two neighboring U-points.
See Section 3.6.2 for the placement and numbering of the grid at the edge of the model domain.

o : θ, S, ηx : u, v, Ho : θ, S, ηx : u, v, Ho : θ, S, ηx : u, v, H
U-cell boundary
T-cell boundary

Figure3.1 Horizontal grid arrangement. (a) MRI.COM (◦ : θ, S, η, × : u, v, H), (b) MOM and COCO (◦ : θ, S, η, H,
× : u, v), (c) Variable grid size in MRI.COM– 21 –

3.2 Vertical grid arrangement

(b)

(a)
Ti-2, j+1

Ti-2, jTi-1, j+1

Ti-1, jTi-1, j-1

Ti, j+1

Ti, j

Ti, j-1

Ti+1, j+1
Vi-3/2, j+1/2

Vi-1/2, j+1/2

Vi+1/2, j+1/2

Vi-3/2, j-1/2

Vi--2, j-1/2

Vi+1/2, j-1/2

Ti+1, jTi+1, j-1

Figure3.2 Horizontal grid lattices in relation to topography. (a) Tracer lattice. (b) Velocity lattice. Land distribution(shade) is common for (a) and (b).

3.2 Vertical grid arrangement
A variable grid size is usually used for the vertical grid arrangement, i.e., fine near the surface and coarse at depth. As
illustrated in Figure 3.3a, tracers (◦) and velocity (×) are defined at just the mid-depth level of the cell, and the verticalmass fluxes W (△, 2) are defined at the boundary of the cell. There are two kinds of W , one for the T-cell (W T ; △) andanother for the U-cell (W U ; 2). Their horizontal locations are the T-points and the U-points depicted in Figure 3.1a.
In order to express the gentle bottom slope as smoothly as possible, the thickness of the deepest U-cell at each horizontal
location is variable (so called partial cell), with a limitation that it must exceed a fraction of around 10 percent of thenominal thickness of the layer to avoid violating the vertical CFL condition (Figure 3.3b). Otherwise, as presented in
Figure 3.3c, the gentle bottom slope is expressed by wide, flat bottoms and cliﬀs here and there with height of vertical gridsize ∆z. In these regions, the vertical velocity is concentrated at the cliﬀs, resulting in relatively strong fictitious horizontal
currents there.
(a) k
0
1/2
1

sea sufacedz(1)= Δz1/2

k-1
k-1/2
k

dz(k)

= Δzk-1/2

k+1/2

dz(k+1) = Δzk+1/2

k+1
k+3/2

dz(k+2) = Δzk+3/2

KM-1/2KM

dz(KM)○䚷θ, S×䚷u, vT䕧䚷W
U□䚷W= ΔzKM-1/2

(b)

(c)

Figure3.3 Vertical grid arrangement. (a) Placement and numbering of vertical grid. (b) Smooth bottom slope withpartial bottom cells. (c) Stair-like bottom slope.

– 22 –

Chapter 3

Spatial grid arrangement

3.3 Indices and symbols
The conventions for indexing and the definitions of symbols used in finite diﬀerence expressions of the equations throughout
this document are given here.
The actual distance corresponding to an increment of ∆µ in the zonal direction of generalized orthogonal coordinatesis expressed as follows:∆x ≡ hµ ∆µ,(3.1)
where hµ is the scaling factor. The actual meridional distance is defined similarly:∆y ≡ hψ ∆ψ.

(3.2)

The vertical distance is expressed by ∆z. For a discretized grid cell, the horizontal area is expressed by either ∆S or ∆A(= ∆x∆y) and the volume is expressed by ∆V (= ∆x∆y∆z).
The subscript indices expressing the horizontal grid position in the finite diﬀerence expression of the equations areusually integers for the T-points, i.e., (i, j) and are increased by a half for the U-points (i + 12 , j + 21 ) (Figure 3.2). In somecases vice versa, with a notice.
In the vertical subscript index for the finite diﬀerence expression, the upper level of a grid cell, where the vertical massflux is defined, is numbered as k (k = 0 being the sea surface), the levels of the T-point and U-point are numbered as k − 12
(Figure 3.3a). In some cases, the T-point and U-point levels may be numbered as k, with a notice.

3.4 Calculation of horizontal grid cell area and width
When equations are solved in MRI.COM, the temporal variations of physical quantities are calculated as a budget of theirfluxes through the boundaries of the U-cells or T-cells (finite volume method). For this method, it is necessary to know
the area and volume of the grid cells. These are numerically calculated for generalized orthogonal coordinate grids andanalytically for geographic coordinate grids.

3.4.1 Generalized orthogonal coordinates
The longitude and latitude (λ, ϕ) of grid points on the sphere are defined by user as a function of the model coordinates(µ, ψ),
λ = λ(µ, ψ), ϕ = ϕ(µ, ψ).
For example, the distance from a T-point (µ(i), ψ( j)) to a point a half grid size to the east (µ(i + 21 ), ψ( j)) (variable namein the model: dx_bl; Figure 3.4a) is approximated numerically as follows taking µ1 = µ(i), µ2 = µ(i + 12 ), and ψ1 = ψ( j):
M
∑
[ (
) (
) (
) (
)]
L λ µ1 + (m − 1)δ µ, ψ1 , ϕ µ1 + (m − 1)δ µ, ψ1 , λ µ1 + mδ µ, ψ1 , ϕ µ1 + mδ µ, ψ1 .

(3.3)

m=1

Here, L[λ 1, ϕ1, λ 2, ϕ2 ] is the distance between the two points (λ 1, ϕ1 ) and (λ 2, ϕ2 ) on the sphere along a great circle andδ µ = (µ2 − µ1 )/M (divided by M ∼ 20 between µ1 and µ2 ).
Similarly, a quarter grid area (a_bl; Figure 3.4b) surrounded by four points (µ(i), ψ( j)), (µ(i + 12 ), ψ( j)), (µ(i +11112 ), ψ( j + 2 )), and (µ(i), ψ( j + 2 )) is, taking ψ2 = ψ( j + 2 ) and δψ = (ψ2 − ψ1 )/N (divided by N ∼ 20 between ψ1 andψ2 ), calculated as:
N ∑
M
∑
[ (
(
1) )
L λ µ1 + (m − 1)δ µ, ψ1 + n − δψ ,2
n=1 m=1((1) )λ µ1 + mδ µ, ψ1 + n − δψ ,
2
×
[ (
)
(
1)
L λ µ1 + m − δ µ, ψ1 + (n − 1)δψ ,
2
(
)
(
1)
λ µ1 + m − δ µ, ψ1 + nδψ ,2– 23 –((1) )ϕ µ1 + (m − 1)δ µ, ψ1 + n − δψ ,2((1 ) )]
ϕ µ1 + mδ µ, ψ1 + n − δψ
2
(
)
(
1)
ϕ µ1 + m − δ µ, ψ1 + (n − 1)δψ ,
2
(
)]
(
1)
ϕ µ1 + m − δ µ, ψ1 + nδψ .
2

(3.4)

3.4

Calculation of horizontal grid cell area and width

As depicted in Figure 3.4, (a_bl)i, j is the area of the lower left quarter of the central U-point. Those for the lower right(a_br)i, j , upper left (a_tl)i, j , and upper right (a_tr)i, j quarters are obtained similarly.
The unit area centered on U-point ((areau)i, j ) is then expressed as:(areau)i, j = (a_bl)i, j + (a_br)i, j + (a_tl)i, j + (a_tr)i, j ,(3.5)

and the area centered on T-point ((areat)i, j ) as
(areat)i, j = (a_bl)i, j + (a_br)i−1, j + (a_tl)i, j−1 + (a_tr)i−1, j−1 .

(3.6)

Following the conventions for indexing introduced in Section 3.3, the above equations are expressed in later chapters as
follows:(areau)i+ 1 , j+ 1 = (a_bl)i+ 1 , j+ 1 + (a_br)i+ 1 , j+ 1 + (a_tl)i+ 1 , j+ 1 + (a_tr)i+ 1 , j+ 1 ,(3.7)(areat)i, j = (a_bl)i+ 1 , j+ 1 + (a_br)i− 1 , j+ 1 + (a_tl)i+ 1 , j− 1 + (a_tr)i− 1 , j− 1 .

(3.8)

2

2

2
2

2

2

2

2

2
2

2
2

2
2

2
2

2
2

3.4.2 Geographic coordinates
For grids in the geographic coordinate system, we use more precise and computationally lighter analytical solutions. Let
us examine the situation of T-cell divided in quarters (Figure 3.4). The area of the northeastern quarter (anhft, thesame as that of the northwestern quarter) is obtained by the latitudinal integration of the thick line in Figure 3.4b, where∆ϕ = ϕ( j + 12 ) − ϕ( j − 12 ).
Using the latitude of T-point ϕ( j), the zonal width of the grid unit for T-points ∆λ = λ(i + 12 ) − λ(i − 21 ), and the Earth’s
radius a, the length of the thick line along the latitude circle (∆s) is expressed as:∆s = a∆λcos ϕ.
2(3.9)

Integrating this in the latitudinal direction, we obtain the following.
{ ()}∫ ϕ+ ∆ϕ
2
a2 ∆λ
a2 ∆λ
∆ϕ
(anhft)i, j =∆sadϕ =
cos ϕdϕ =sin ϕ +− sin ϕ
2
2
2
ϕ
ϕ
(
)
∆ϕ
∆ϕ
=a2 ∆λ cos ϕ +
sin
4
4
(
)
∆ϕ
∆ϕ
∆ϕ
=a2 ∆λ cos ϕ cos− sin ϕ sin
sin
4
4
4
(
)
∆ϕ
∆ϕ
∆ϕ
=a2 ∆λ cos ϕ cossin
1 − tan ϕ tan
4
4
4
(
)
2
a
∆ϕ
∆ϕ
= ∆λ cos ϕ sin
1 − tan ϕ tan.
2
2
4∫ ϕ+ ∆ϕ2(3.10)

Similarly, the area of the southeastern quarter of the T-cell (variable name in the model: ashft, the same as that of thesouthwestern quarter) is expressed as:
(
)
a2
∆ϕ
∆ϕ
(ashft)i, j = ∆λ cos ϕ sin
1 + tan ϕ tan.
(3.11)
2
2
4
At the north and the south poles, where ϕ = ±90◦ , we obtain the following by changing (3.10) and (3.11) to the following
forms.
()∆ϕ∆ϕ
a2
cos ϕ − sin ϕ tan(3.12)(anhft)i, j = ∆λ sin
2
2
4
(
)
∆ϕ
∆ϕ
a2
cos ϕ + sin ϕ tan.
(ashft)i, j = ∆λ sin(3.13)
2
2
4– 24 –

Chapter 3

Spatial grid arrangement(a)dy_tri, j

dy_tli, j

Ui, j

dx_tli, j

dx_tri-1, j

dy_bli, j

dy_bri-1, j

dx_tri, j

dy_bri, j

Ti, j
dx_bri-1, j

dx_bri, j

dx_bli, j

dy_tri, j-1

dy_tli, j-1

dy_tri-1, j-1

dx_tli, j-1

dx_tri-1, j-1

U-cell boundary
T-cell boundary(b)anhfti, j

a_tri, ja_tli, jUi, janhfti-1, j

a_bli, jTi, ja_bri, j∆sashfti, j

ashfti-1, j

Figure3.4 Variables that define a grid unit. (a) Distance. (b) Area. Grid indices (i, j) follow the convention described in Section 3.3.

At the north pole:(anhft)i, j =0
(ashft)i, j =(3.14)a22∆λ sin

∆ϕ
∆ϕ
tan
.
2
4

(3.15)At the south pole:a2∆ϕ∆ϕ∆λ sintan
22
4(ashft)i, j =0.
(anhft)i, j =In our model(a_bl)i, j = (anhft)i, j ,(a_br)i, j = (anhft)i+1, j ,(a_tl)i, j = (ashft)i, j+1,

(a_tr)i, j = (ashft)i+1, j+1,

and the areas of the grid cells centered on the U-points and T-points are calculated by (3.5) and (3.6).

– 25 –(3.16)(3.17)

3.5

Calculation of vertical cell thickness

3.5 Calculation of vertical cell thickness
a. Distribute sea level variation to the four sub-divided parts of a T-cellIf the sea level variation η i, j is known at a T-point, the thickness of four divided cells that comprise a T-cell is determined
as follows:(dzu_bl)i+ 1 , j+ 1 ,k− 1 = (η i, j + Hi+ 1 , j+ 1 )
2

2

2

2

2

(dzu_cnst)i+ 1 , j+ 1 ,k− 122(dzu_tl)i+ 1 , j− 1 ,k− 1 = (η i, j + Hi+ 1 , j− 1 )
2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

(3.18)(dzu_cnst)i+ 1 , j− 1 ,k− 12

2

2(3.19)

Hi+ 1 , j− 12

(dzu_cnst)i− 1 , j+ 1 ,k− 12

2

2(3.20)

Hi− 1 , j+ 1
2

(dzu_tr)i− 1 , j− 1 ,k− 1 = (η i, j + Hi− 1 , j− 1 )

2

2

2

(dzu_br)i− 1 , j+ 1 ,k− 1 = (η i, j + Hi− 1 , j+ 1 )2

Hi+ 1 , j+ 1

2

(dzu_cnst)i− 1 , j− 1 ,k− 1
2

2

2

Hi− 1 , j− 1
2,

(3.21)

2

where H is the depth of sea floor and dzu_cnst is the logical definition of vertical cell thickness in z ∗ frame, which does
not vary in time. Volume of the four cells becomes(volu_bl)i+ 1 , j+ 1 ,k− 1 = (a_bl)i+ 1 , j+ 1 (dzu_bl)i+ 1 , j+ 1 ,k− 1(3.22)(volu_tl)i+ 1 , j− 1 ,k− 1 = (a_tl)i+ 1 , j− 1 (dzu_tl)i+ 1 , j− 1 ,k− 1(3.23)(volu_br)i− 1 , j+ 1 ,k− 1 = (a_br)i− 1 , j+ 1 (dzu_br)i− 1 , j+ 1 ,k− 1(3.24)(volu_tr)i− 1 , j− 1 ,k− 1 = (a_tr)i− 1 . j− 1 (dzu_tr)i− 1 , j− 1 ,k− 1 .

(3.25)

2
2
2
2

2
2
2
2

2

2

2

2

2

2

2

2

2
2
2

2

2
2
2

2

2
2
2

2

2
2
2

2

Using these, the new volume of T-cells and U-cells is obtained. There is no leak in volume by using this method. Figure
3.5 illustrates the procedure.
zxηFigure3.5 Illustration of a vertical slice through a set of grid cells in the x-z plane for z ∗ coordinate. The center pointin each cell (•) is a velocity point. The cross (×) is a tracer point.

– 26 –

Chapter 3

Spatial grid arrangementb. U-cell

Thickness of a U-cell (dzu) is obtained by dividing U-cell’s volume by U-cell’s horizontal area. U-cell’s volume is a sumof the four sub-divided cells whose volume varies following sea level variation on T-cells where they belong to. Using(3.22) through (3.25), we have,(volu)i+ 1 , j+ 1 ,k− 1 = (volu_bl)i+ 1 , j+ 1 ,k− 1 + (volu_br)i+ 1 , j+ 1 ,k− 1
2

2

2

2

2

2

2

2

2

+ (volu_tl)i+ 1 , j+ 1 ,k− 1 + (volu_tr)i+ 1 , j+ 1 ,k− 1 .
2

2

2

2

2

2

(3.26)

Then the thickness is computed by/(dzu)i+ 1 . j+ 1 ,k− 1 ≡ (volu)i+ 1 , j+ 1 ,k− 1 (areau)i+ 1 , j+ 1 ,
2

2

2

2

2

2

2

(3.27)

2that is, the thickness of a U-cell is the average of the thickness of the four sub-divided T-cells.
c. T-cell

Volume of a T-cell is also calculated as a sum of the four sub-divided cells. Using (3.22) through (3.25), we have(volt)i, j,k− 1 = (volu_bl)i+ 1 , j+ 1 ,k− 1 + (volu_tl)i+ 1 , j− 1 ,k− 1
2

2

2

2

2

2

2

+ (volu_br)i− 1 , j+ 1 ,k− 1 + (volu_tr)i− 1 , j− 1 ,k− 1 .
2

2

2

2

2

2

(3.28)

Because horizontal cross section of a T-cell is not uniform vertically owing to the presence of partial cells, thickness of a
T-cell cannot be determined identically. Here, the thickness dzt_cnst is determined as the diﬀerence between top andbottom face of the T-cell,((dzt_cnst)i, j,k− 1 = max (dzu_cnst)i+ 1 , j+ 1 ,k− 1 , (dzu_cnst)i+ 1 , j− 1 ,k− 1 ,
2
2
2
2
2
) 2 2
(dzu_cnst)i− 1 , j+ 1 ,k− 1 , (dzu_cnst)i− 1 , j− 1 ,k− 1 .
(3.29)
2

2

2

2

2

2

d. Depth anomalies

We defined vertical cell thickness of four divided cells that comprise a T-cell by (3.18) through (3.21). In the same manner,
we may define the anomalies of the actual depth at the center and bottom of the four divided cells, with −η at the sea
surface (s = 0). Note that depth is defined positive downward.
For the center,(dpu_bl)i+ 1 , j+ 1 ,k− 1 = −2

2

Hi+ 1 , j+ 1 − (dpu_cnst)i+ 1 , j+ 1 ,k− 1
2

2

2

2

(dpu_tl)i+ 1 , j− 1 ,k− 1 = −
2

2

2

2

2

2

2

η i, j(3.31)

η i, j(3.32)

η i, j(3.33)

η i, j(3.34)

η i, j(3.35)

η i, j(3.36)

η i, j ,(3.37)

2

2

Hi− 1 , j+ 1 − (dpu_cnst)i− 1 , j+ 1 ,k− 1
2

2

2

2

2

Hi− 1 , j+ 1
2

2

2

Hi+ 1 , j− 12

(dpu_tr)i− 1 , j− 1 ,k− 1 = −(3.30)222

η i, j2

Hi+ 1 , j− 1 − (dpu_cnst)i+ 1 , j− 1 ,k− 12(dpu_br)i− 1 , j+ 1 ,k− 1 = −2

Hi+ 1 , j+ 1

2

2

Hi− 1 , j− 1 − (dpu_cnst)i− 1 , j− 1 ,k− 1
2

2

2

2

2

Hi− 1 , j− 1

2

2

2

and for the bottom,(depu_bl)i+ 1 , j+ 1 ,k = −2

Hi+ 1 , j+ 1 − (depu_cnst)i+ 1 , j+ 1 ,k
2

2

2

2

(depu_tl)i+ 1 , j− 1 ,k = −
2

2

2

2

2

2

2

2

Hi− 1 , j+ 1
2

2

2

Hi− 1 , j+ 1 − (depu_cnst)i− 1 , j+ 1 ,k

2

(depu_tr)i− 1 , j− 1 ,k = −2

Hi+ 1 , j− 1
2

2

2

Hi+ 1 , j− 1 − (depu_cnst)i+ 1 , j− 1 ,k

2

(depu_br)i− 1 , j+ 1 ,k = −2

Hi+ 1 , j+ 1

2

2

Hi− 1 , j− 1 − (depu_cnst)i− 1 , j− 1 ,k2

2

2

Hi− 1 , j− 1

2

2

– 27 –

2

2

3.6

Usage

where dpu_cnst is the logical depth at the center of a U-cell (dashed line in Figure 3.3a) and depu_cnst is the logicaldepth at the bottom of a U-cell (solid line in Figure 3.3a).
The depth anomaly of a T-point is obtained as an area average. For example, at the center,{(dpt)i, j,k− 1 = (a_bl)i+ 1 , j+ 1 (dpu_bl)i+ 1 , j+ 1 ,k− 1 + (a_tl)i+ 1 , j− 1 (dpu_tl)i+ 1 , j− 1 ,k− 1
2
2
2
2
2
2
2
2
2
2
}2 /
+ (a_br)i− 1 , j+ 1 (dpu_br)i− 1 , j+ 1 ,k− 1 + (a_tr)i− 1 . j− 1 (dpu_tr)i− 1 , j− 1 ,k− 1 (areat)i, j,k+ 1(3.38)
2

2

2

2

2

2

2

2

2

2

2

and for the bottom,{(dept)i+ 1 , j+ 1 ,k = (a_bl)i+ 1 , j+ 1 (depu_bl)i+ 1 , j+ 1 ,k + (a_tl)i+ 1 , j− 1 (depu_tl)i+ 1 , j− 1 ,k
2
2
2
2
2
2
2
2
2
2
}/
+ (a_br)i− 1 , j+ 1 (depu_br)i− 1 , j+ 1 ,k + (a_tr)i− 1 . j− 1 (depu_tr)i− 1 , j− 1 ,k (areat)i, j,k+ 1 .
2

2

2

2

2

2

2

2

2

(3.39)In the above calculation, we use only full cells, that is, we do not include partial cells except for the bottom cell (k = ktbtm).
This is reflected in the use of (areat)i, j,k+ 1 instead of (areat)i, j,k− 1 .
2
2
Using this, the variable thickness of a T-cell is calculated as follows:(dzt)i, j,k− 1 = (dept)i, j,k − (dept)i, j,k−1 + (dzt_cnst)i, j,k− 1 .
2(3.40)

2

3.6 Usage
3.6.1 Choice of horizontal coordinate system
For horizontal coordinate system, either spherical or generalized orthogonal curvilinear coordinates must be chosen.
a. Spherical coordinates

For a model that does not include the North Pole, spherical coordinates with geographical longitude-latitude axes will bea standard choice.
If spherical coordinates are chosen, model option SPHERICAL must be added to the list of options (the line start withOPTIONS =) specified in configure.in.
The geographical position of north and south pole should not be necessarily at the Earth’s North and South Pole. If themodel’s north pole is displaced, this is specified by namelist nml_poles (Table 3.1).
Table3.1 Namelist nml_poles for SPHERICAL option. The geographical location of the model’s south pole isdetermined automatically by the specification of the north pole.

variable namenorth_pole_lon
north_pole_lat

units
degreedegreedescription
geographical longitude of north polegeographical latitude of north pole

usage
default is 0°
default is 90°N

b. Generalized orthogonal curvilinear coordinates

Singularity at the North Pole may be avoided by coordinate transformation within the framework of generalized orthogonal
curvilinear coordinates.
Two options are available:• Tripolar grid (TRIPOLAR) combining geographical lat-lon south of around 60◦ N and transformed coordinates to thenorth.
• Joukowski conversion (JOT) applied to the whole sphere of the Earth.
When either TRIPOLAR or JOT option is chosen, that option must be added to the list of options (the line startwith OPTIONS =) specified in configure.in. In addition, namelist nml_poles must be specified, but in this case thegeographical locations of the two new singular points instead of the Earth’s North and South Pole should be given (Table
3.2). See Chapter 20 for details.

– 28 –

Chapter 3

Spatial grid arrangement

Table3.2 Namelist nml_poles for TRIPOLAR or JOT option. See also Figure 3.6b for the locations of the two singilar
points to be specified.

variable namenorth_pole_lon

units
degreenorth_pole_lat

degreesouth_pole_lon

degreesouth_pole_lat

degreedescription
geographical longitude of one of the two new
singular points
geographical latitude of one of the two new
singular points
geographical longitude of the other of the two
new singular points
geographical latitude of the other of the two
new singular points

usage
specification is required
specification is required
specification is required
specification is required

3.6.2 Definition of model domain
To determine the model domain, the region of interest must be first filled with U-cells, where sea floor depths are given.
This is the lightly shaded region of Figure 3.6, which is called the core region. For a closed basin model, this core region
must be surrounded by land cells (Figure 3.6a). For a model domain with cyclic boundary condition (Figure 3.6b), twohalo U-cells must be added to both western and eastern sides. When a zonally cyclic condition is used, CYCLIC option
must be specified in configure.in. For a tri-pole model with TRIPOLAR or JOT option, three U-cells must be added as
halos to the northern end of the core region. The total number of grid points (imut and jmut) must include halo cells.
The total numbers of grid points in the three directions (imut, jmut, and km) must be given to configure.in. Minimum
information that must be given to configure.in for compiling a global tri-polar grid model will look like as follows.
An example specification given to configure.in for a global tri-polar grid model
OPTIONS="TRIPOLAR CYCLIC"
IMUT=364
JMUT=368
KM=60





Further horizontal grid information is given to the model by namelist at run time. It is necessary to specify the westernand southern end of model’s core region and the X and Y axis grid spacing. They are given by namelist nml_horz_grid(Table 3.3). When the horizontal grid spacing is given by a file, the file is read by the model at run time as follows.
Format of horizontal grid spacing data (file_dxdy_tbox_deg).
real(8) :: dxtdeg(imut), dytdeg(jmut)
integer(4),parameter :: lun = 10
open(lun, file=file_dxdy_tbox_deg, form=unformatted )read( lun ) i, j
if ( ( i == imut ).and.( j == jmut ) ) thenread( lun ) dxtdeg
read( lun ) dytdeg
end ifclose(lun)Here, dxtdeg and dytdeg are longitudinal and latitudinal width of T-cells, respectively (Figure 3.6c). Note that theycan vary only in the direction of their own axis.
The vertical water column is filled by U-cells with specified thickness. The start point of the vertical grid is always set
to be zero (sea surface). The vertical grid index increases downward and the vertical grid width (∆z k− 1 = dz(k)) is given
2either by a file or a namelist. Which one to select is determined by namelist nml_vert_grid (Table 3.4) at run time.
When the vertical grid spacing is given by a file, the file is read by the model at run time as follows.

– 29 –

3.6

Usage

Table3.3 Namelist nml_horz_gridvariable namelon_west_end_of_core
lat_south_end_of_core
dx_const_deg
dy_const_deg
file_dxdy_tbox_deg



units
model longitude
in degreemodel latitude in
degreedegreedegreecharacterdescription
the longitude of the western endof model’s main region
the latitude of the southern endof model’s main region
uniform X-axis grid spacing
uniform Y-axis grid spacing
X and Y axis grid spacing are
given by this fileusage
specification is required
specification is requireddefault is zerodefault is zeroIf either X or Y axis grid spacing
is not uniform, prepare this file.

Format of vertical grid spacing data (file_dz_cm).

real(8) :: dz(km)integer(4),parameter :: lun = 10
open(lun, file=file_dz_cm, form=unformatted)
read(lun) k
if ( k == km ) thenread(lun) dz
endif
close(lun)
Table3.4 Namelist nml_vert_gridvariable namefile_dz_cm

unitscharacterdz_const_cm

cm

description
vertical grid spacing is given by this fileuniform vertical grid spacingusage
If the vertical grid spacing is not uniform,
prepare this file. dz_const_cm must not
be given.
default is zero

3.6.3 Grid cell area and line elements
When spherical coordinates are chosen (SPHERICAL), grid cell area and line elements are calculated analytically by themodel (Section 3.4.2). When generalized orthogonal curvilinear coordinates are chosen (TRIPOLAR or JOT), they should
be read from file (Section 3.4.1). The file name must be given by namelist nml_grid_scale (Table 3.5).
Table3.5 Namelist nml_grid_scalevariable namefile_scaleunitscharacterdescription
grid cell area and line elements are givenby this fileusage
specification is required for TRIPOLAR or
JOT.

The file that contains the grid cell and line elements (see also Figure 3.4 for positions) is read by the model at run time
as follows.

– 30 –

Chapter 3

Spatial grid arrangement

(a)

i=2

i=imutj=jmutj=2

lat_south_end_of_core

lon_west_end_of_core(b)(north_pole_lon, north_pole_lat)(south_pole_lon, south_pole_lat)

i=imut/2+1

i=3

i=imut-2

j=jmutj=jmut-2

Earth’ s North Polej=2

lat_south_end_of_core

lon_west_end_of_core(c)Ti+1, j+1

Ti, j+1

Ui-1, jUi, jdytdeg(j)dxtdeg(i)Ti, jUi-1, j-1

Ti+1, jUi, j-1

Figure3.6 Illustration to explain how to determine the model domain and how to give information to the model.
Schematic of a model with (a) closed domain, (b) zonally cyclic (CYCLIC) and northern end folding (TRIPOLAR or
JOT) conditions. The light shades are model’s region of interest (core region). The dark shades are land that must be
attached around the core region. The white cells in (b) are halos. (c) Definition of the most basic arrays (dxtdeg anddytdeg) that define grid spacing. Grid indices (i, j) follow array indices in program codes. The indices of the T-points
in the vicinity of boundaries are shown in (a) and (b).

– 31 –

3.6
Format of grid cell area and line elements (file_scale; if not SPHERICAL)

Usage


real(8) :: a_bl(imut,jmut), a_br(imut,jmut), a_tl(imut,jmut), a_tr(imut,jmut)
real(8) :: dx_bl(imut,jmut), dx_br(imut,jmut), dx_tl(imut,jmut), dx_tr(imut,jmut)
real(8) :: dy_bl(imut,jmut), dy_br(imut,jmut), dy_tl(imut,jmut), dy_tr(imut,jmut)
integer(4),parameter :: lun = 10
open(unit=lun, file=file_scale, form=unformatted)
read(unit=lun) a_bl ! U-box area of bottom-left 1/4 gridread(unit=lun) a_br ! U-box area of bottom-right 1/4 gridread(unit=lun) a_tl ! U-box area of top-left 1/4 gridread(unit=lun) a_tr ! U-box area of top-right 1/4 gridread(unit=lun) dx_bl ! U-box length of bottom-left 1/4 gridread(unit=lun) dx_br ! U-box length of bottom-right 1/4 gridread(unit=lun) dx_tl ! U-box length of top-left 1/4 gridread(unit=lun) dx_tr ! U-box length of top-right 1/4 gridread(unit=lun) dy_bl ! U-box length of bottom-left 1/4 gridread(unit=lun) dy_br ! U-box length of bottom-right 1/4 gridread(unit=lun) dy_tl ! U-box length of top-left 1/4 gridread(unit=lun) dy_tr ! U-box length of top-right 1/4 gridclose(lun)– 32 –

Part II

Diagnostic Equations

Chapter 4

Equation of StateThe in situ density is needed to calculate the pressure gradient term in the momentum equation. As indicated in (2.5), theequation of state represents in situ density as a function of pressure, temperature, and salinity. Here we present the specificform of the equation of state. We still adhere to the 1980 International Equation of State of Seawater (EOS-80; UNESCO,1981). Thermodynamic Equation Of Seawater - 2010 (TEOS-10; IOC, SCOR, and IAPSO, 2010) has not been adopted.

4.1 Basics of the equation of state
The standard equation of state provided by UNESCO (1981) represents density (kg m−3 ) as a function of in situ temperature∗t (◦C), salinity S (practical salinity scale (pss) ∼ parts per thousand (ppt)), and pressure P (bar). Note that in situ temperatureis used, not potential temperature. Density (ρw ) of pure water (S = 0) under sea level pressure is given as a function oftemperature (t):ρw (t) = 999.842594 + 6.793952 × 10−2 t − 9.095290 × 10−3 t 2
−4 3

−6 4

(4.1)
−9 5

+ 1.001685 × 10 t − 1.120083 × 10 t + 6.536332 × 10 t .
Density at the sea surface (ρsurf = ρ(t, S, 0)) is expressed using sea surface temperature and salinity:ρsurf = ρw

(4.2)

+ (0.824493 − 4.0899 × 10−3 t + 7.6438 × 10−5 t 2 − 8.2467 × 10−7 t 3 + 5.3875 × 10−9 t 4 )S+ (−5.72466 × 10−3 + 1.0227 × 10−4 t − 1.6546 × 10−6 t 2 ) S 2
3+ 4.8314 × 10−4 S 2 .
The in situ density (ρ = ρ(t, S, P)) is converted from ρsurf using the following equation,ρ = ρsurf /(1 − P/K ),(4.3)

where K (S, t, P) is the secant bulk modulus. Its value at pure water, Kw , is given by
Kw = 19652.21 + 1.484206 × 102 t − 2.327105t 2 + 1.360477 × 10−2 t 3 − 5.155288 × 10−5 t 4 .

(4.4)

The value at the sea surface (K0 ) is given by
K0 = Kw + (54.6746 − 0.603459t + 1.09987 × 10−2 t 2 − 6.1670 × 10−5 t 3 )S+ (7.944 × 10

−2

−2

−4 2

(4.5)

3
2

+ 1.6483 × 10 t − 5.3009 × 10 t ) S ,∗ EOS-80 provided by UNESCO (1981) was based on temperature on the International Practical Temperature Scale of 1968 (IPTS-68; t ) and68

the Practical Salinity Scale 1978 (PSS-78). After that the International Committee for Weights and Measures adopted a new temperaturescale (the International Temperature Scale of 1990 (ITS-90); t90 ). It has been recommended that t90 should be converted to t68 by a relation
t68 = 1.00024 ∗ t90 when it is used for EOS-80 (Saunders, 1990). However, this conversion has not been applied for temperature in any operationof MRI.COM.

– 35 –

4.2

An equation of state used by MRI.COM

Table4.1 Coeﬃcients for the lapse rate of sea water (Eq. 4.8).

a0
a1
a2
a3

+3.5803 × 10−5
+8.5258 × 10−6

−6.8360 × 10−8
+6.6228 × 10−10b0b1+1.8932 × 10−6
−4.2393 × 10−8

c0
c1
c2
c3

+1.8741 × 10−8
−6.7795 × 10−10+8.7330 × 10−12−5.4481 × 10−14

d0
d1
e0
e1
e2

−1.1351 × 10−10+2.7759 × 10−12−4.6206 × 10−13+1.8676 × 10−14−2.1687 × 10−16

and the value at pressure P is given by
K = K0

(4.6)
−3

−4 2

−7 3

+ P (3.239908 + 1.43713 × 10 t + 1.16092 × 10 t − 5.77905 × 10 t )+ P (2.2838 × 10−3 − 1.0981 × 10−5 t − 1.6078 × 10−6 t 2 ) S+ P (1.91075 × 10−4 ) S 2

3+ P2 (8.50935 × 10−5 − 6.12293 × 10−6 t + 5.2787 × 10−8 t 2 )+ P2 (−9.9348 × 10−7 + 2.0816 × 10−8 t + 9.1697 × 10−10 t 2 ) S.
When potential temperature (θ) is available, it should be converted to in situ temperature. The conversion equation is
obtained as follows using the adiabatic lapse rate Γ(t, S, P):∫ P
t(θ 0, S, P) = θ 0 +Γ(t, S, P ′ )dP ′ .
(4.7)
P0

A polynomial for the adiabatic lapse rate Γ(t, S, P) is given by UNESCO (1983):
Γ(t, S, P) = a0 + a1 t + a2 t 2 + a3 t 3+ (b0 + b1 t)(S − 35)(4.8)+ {c0 + c1 t + c2 t 2 + c3 t 3 + (d 0 + d 1 t)(S − 35)} P+ (e0 + e1 t + e2 t 2 ) P2 .
Coeﬃcients are given on Table 4.1.

4.2 An equation of state used by MRI.COM
4.2.1 Formulation
Since potential temperature (θ) is a prognostic variable in MRI.COM, it is desirable that the potential temperature can bedirectly used for the equation of state in the model. If this is possible, it will not be necessary to convert between potential
temperature and in situ temperature. Accordingly, the equation of state of MRI.COM is a polynomial of almost the same
form as UNESCO, but it is now the function of θ, S, and P and has a modified set of parameters. The parameters are
determined by the least square fit for a realistic range of potential temperature and salinity following the method of Ishizaki(1994).
In MRI.COM version 4, considering the treatment of brackish waters in coastal modeling as well as the practical demandof numerical stability, we slightly modified the polynomial form and revised the set of coeﬃcients relative to the previous
version. The process of determining coeﬃcients will be described below. We allow salinity to take slightly negativevalues that could be caused by numerical errors. So we use a slightly wider range (−2 ≤ θ ≤ 40 ◦C, −2 ≤ S ≤ 42 pss, and
0 ≤ P ≤ 1000 bar) than the previous version.
First, density is calculated at the sea surface (potential density or σθ ) using equations (4.1) and (4.2). We introduce a3
3
minor modification from (S) 2 to (|S|) 2 . It has been confirmed that this modification leads to a smooth density profile at
S = 0.
The pressure dependent part, or specific volume K (θ, S, P) is given by
K (θ, S, P) = e1 (P) + e2 (P)θ + e3 (P)θ 2 + e4 (P)θ 3 + e5 (P)θ 4+ S( f 1 (P) + f 2 (P)θ + f 3 (P)θ 2 + f 4 (P)θ 3 )3+ (|S|) 2 ( f 5 (P) + f 6 (P)θ + f 7 (P)θ 2 ),– 36 –(4.9)

Chapter 4

Equation of State

where
e1 (P) = ec1 + (gc1 + hc1 P)P,f 1 (P) = fc1 + (gc5 + hc4 P)P,e2 (P) = ec2 + (gc2 + hc2 P)P,f 2 (P) = fc2 + (gc6 + hc5 P)P,

e3 (P) = ec3 + (gc3 + hc3 P)P,f 3 (P) = fc3 + (gc7 + hc6 P)P,

e4 (P) = ec4 + gc4 P,
e5 (P) = ec5,

(4.10)

f 4 (P) = fc4,
f 5 (P) = fc5 + gc8 P,
f 6 (P) = fc6,
f 7 (P) = fc7 .

The set of coeﬃcients in the above equation is computed using a least square fit and listed on Table 4.2. In the previous
version, uniform 1.0 ◦C, 1.0 pss, 10 decibar bins were used for the least square fit in the range of −2 ≤ θ ≤ 40 ◦C,
0 ≤ S ≤ 42 pss, and 0 ≤ P ≤ 1000 bar. But it is found that maximum error tends to occur in the range of low potential
temperature and low salinity, and increasing the number of bins in this range is favorable for reducing the error (notshown). Therefore, potential temperature and salinity bins are changed from 1.0 ◦C and 1.0 pss to 0.1 ◦C and 0.1 pss for−2 ≤ θ ≤ 10 ◦C and −2 ≤ S ≤ 10 pss, respectively. Using 151×151×101 combinations of the above range of potential
temperature, salinity, and pressure, in situ temperature is first computed using (4.7). Density is then calculated by the
UNESCO equations using in situ temperature and salinity. The above coeﬃcients are determined using these data ofdensity, potential temperature, salinity, and pressure by the least square method. They are given on Table 4.2.
Table4.2 Coeﬃents for the equation of state of sea water. See equation (4.10).

ec1
ec2
ec3
ec4
ec5

19659.35
144.5863−1.722523
1.019 238 × 10−2
−4.768 276 × 10−5

fc1
fc2
fc3
fc4
fc5
fc6
fc7

52.85624−3.128 126 × 10−1
6.456 036 × 10−3
−5.370 396 × 10−5
3.884 013 × 10−1
9.116 446 × 10−3
−4.628 163 × 10−4

gc1
gc2
gc3
gc4
gc5
gc6
gc7
gc8

3.185918
2.189 412 × 10−2
−2.823 685 × 10−4
1.715 739 × 10−6
6.703 377 × 10−3
−1.839 953 × 10−4
1.912 264 × 10−7
1.477 291 × 10−4

hc1
hc2
hc3
hc4
hc5
hc6

2.111 102 × 10−4
−1.196 438 × 10−5
1.364 330 × 10−7
−2.048 755 × 10−6
6.375 979 × 10−85.240 967 × 10−10With the new set of coeﬃcients, the maximum density error (thick dashed line) relative to the UNESCO equation isless than 1.6 × 10−3 kg m−3 when pressure is less than 5000 dbar (Figure 4.1). This maximum error is less than that(3 × 10−3 kg m−3 ) of McDougall et al. (2003)’s equation of state. The error is also smaller than that using the previous
version (version 3 and earlier) in this pressure range (thin dashed line). Though the relatively larger maximum errors arefound when the pressure is greater than 8000 dbar, they occur only when S ≃ 40 pss or S ≃ 0 pss (not shown) and would
not cause a serious problem. The standard deviation (thick solid line) is less than 1 × 10−3 kg m−3 when pressure is lessthan 8000 dbar.

4.2.2 Implementation
In MRI.COM, in situ density is computed by (4.3) using (4.2) for ρsurf and (4.9) for K. For Pressure (P), a horizontally
uniform value is used on each vertical level where densities are evaluated, instead of using the actual pressure at eachlocation. This is based on the assumption that horizontal variation of pressure on a constant depth (z ∗ ) surface does not
aﬀect the required accuracy of density.
In the program code of MRI.COM, the in situ density is evaluated at the depth where the tracer is defined and the depthof the boundary of tracer cells. In the standard case, pressure at the depth where tracer is defined (Pk− 1 ) and that at the2depth of the cell boundary (Pk ) are calculated as follows:
106 Pk− 1 = ρ0 gzk− 1 ,(4.11)

10 Pk = ρ0 gzk ,(4.12)

2

2

6

where ρ0 is the reference density, g is the acceleration due to gravity (Table 2.1), and zk− 1 and zk are the actual depths in2

the state of rest. Note that a factor 106 converts pressure in cgs units (dyn cm−2 ) to bar.
If CALPP option is chosen, the time variation of these horizontally uniform pressures is considered. In this case, pressuresare evaluated as follows:– 37 –

4.2

An equation of state used by MRI.COM

Figure4.1 Deviation from the UNESCO equation for the old (version 3 and earlier) and new (version 4) set ofcoeﬃcients. Unit is 10−3 kg m−3 . Thin lines represent the equation of state using the old coeﬃcients and thick lines
represent those of the new ones. The solid (long dashed) lines represent the standard deviation (maximum error) fromthe UNESCO equation in the range of −2° ≤ θ ≤ 10 ◦C and 10 ≤ S ≤ 40 pss.

1
g∆z 1 ρ 1 xy,
2
2
2
k
∑
ρl− 3 xy + ρl− 1 xy
2
2
6
6
10 Pk− 1 = 10 P 1 + g
∆zl−1
2
2
2
l=2
106 P 1 =(4.13)

2

106 Pk = gk∑∆zk− 1 ρk− 1 xy .
22(k ≥ 2),(4.14)(4.15)

k=1

where ρk− 1 xy represents horizontally averaged density at (k − 21 )th level. In addition, the density averaged for the entire
2
model domain ( ρ̄) is used as the reference density in the momentum equations instead of ρ0 .
These quantities are evaluated at a time step interval specified by namelist nml_calpp (Table 4.3) and should be stored
in a restart file to be used in the next run (Table 4.4). In the restart file, horizontally averaged density at tracer levels(ρk− 1 xy ), the density averaged for the entire model domain ( ρ̄), pressures used for the equation of state (Pk− 1 and Pk ) are2
2
stored. The restart file has the following format:– 38 –

Chapter 4

Equation of State

Format of the restart file for the equation of state (nmlrs_density).



read(8):: dmn(km), ddmna, pm(km+1), pd(km)integer(4):: k, reck
real(8):: density_array(3*km+2)
character(14) :: date = ’20010101000000’ !- for 0:00z1JAN2001
integer(4),parameter :: nu = 10 ! device number
open(nu, file=’result/rs_density.’//date, form=’unformatted’, && access=’direct’,recl=(3*km+2)*8)
read(nu, rec=1) density_arrayclose(nu)reck = 0
do k = 1, km
reck = reck + 1
dmn(k) = density_array(reck) ! horizontally averaged density
end doreck = reck + 1
ddmna = density_array(reck)! density averaged over the entire model domain
do k = 1, km + 1
reck = reck + 1
pm(k) = density_array(reck) ! pressure at the tracer cell boundaries
end dodo k = 1, km
reck = reck + 1
pd(k) = density_array(reck) ! pressure at the tracer levels
end do

Table4.3 namelist nml_calpp

variable namenstep_calpp_interval

units
1

description
time step interval with which horizontally averaged density and pressure are evaluated forthe equation of state

usage
required if CALPP

Table4.4 namelist nml_density_run

variable namel_rst_density_inunits
logicaldescription.true. :Read restart files specified by
nmlrs_density for the initial condition.
.false. : Start condition is calculated by the 3-D
density field of the initial state.

– 39 –

usage
default = l_rst_in of
nml_run_ini_state,
see Table 21.5.

Chapter 5

Continuity equation
5.1 Introduction
The mass (volume) fluxes, which are fundamental for estimating the advection of momentum and tracers, are calculated on
the basis of the finite diﬀerence expression of the continuity equation. Owing to the use of staggered grid arrangement, thefinite diﬀerence expression of the continuity equation (2.88) diﬀers for the T-cell and U-cell (Figure 5.1). In MRI.COM,
the mass continuity for the T-cell is primary and that for the U-cell is derived from the former by an averaging operation.
By this, we can avoid spurious vertical mass fluxes for the U-cell continuity, which appear when the U-cell continuity is
calculated independently of the T-cell continuity, with the largest error magnitude increasing as the grid size decreases(Webb, 1995).

5.2 Finite diﬀerence expression for the T-cell continuity equation
Finite diﬀerence expressions of the free surface equation (2.40),[]∂(hψ U) ∂(hµ V )∂(η + H)1++
= P − E + R + I,∂th µ hψ
∂µ
∂ψ

(5.1)

the continuity equation (2.36),
1
∂z s
+
∂t
h µ hψ

{∂(z s hψ u) ∂(z s hµ v)
+
∂µ
∂ψ

}
+

∂(z s ṡ)
= 0,
∂s

(5.2)

and the tracer equations (2.37, 2.38){}∂(z s hψ uθ) ∂(z s hµ vθ)∂(z s θ)1∂(z s ṡθ)+++= −z s ∇ · Fθ + z s Q θ ,∂th µ hψ∂µ∂ψ∂s{}∂(z s hψ uS) ∂(z s hµ vS)∂(z s S)1∂(z s ṡS)+++
= −z s ∇ · FS + z s Q S,∂th µ hψ
∂µ
∂ψ
∂s

(5.3)
(5.4)

must be mutually consistent in order to keep sign-definiteness of tracers. Among them, the continuity equation isfundamental. The continuity equation is used to diagnostically obtain vertical velocity ṡ. Actually, in MRI.COM, verticalvelocity is not computed but vertical transport (wlwl) is extensively used. Vertical transport is mathematically expressed
as W T ≡ z s ṡ∆AT (s), where ∆AT (s) is the horizontal area of a T-cell∗ . To obtain W T , we horizontally integrate the semi
discrete expression for the continuity equation (2.88) using the surface boundary condition for w ∗ .
Using the surface boundary condition for the vertical velocity in z ∗ coordinate (2.49), we have
WzT∗ =0 = z s ṡ s=0 ∆ATs=0 =H +η ∗
wz ∗ =0 ∆ATz∗ =0 = −(P − E + R + I)∆ATz∗ =0 .
H

(5.5)

The surface flux of a state variable caused by the surface fresh water transport (r.h.s. of the above) is treated as the surface
boundary condition for the advection term of that state variable.
We now integrate the semi-discrete expression for the continuity equation (2.88) over the oceanic part of the verticallyk-th T-cell. The integral over a T-cell of the tendency term of the scaling factor (2nd term on r.h.s.) is expressed as a

∗ Horizontal cross section of a T-cell is treated as constant throughout the cell, that is, vertically bounding walls are assumed to move with the cell

– 41 –

5.2) and the average thickness for the k-th vertical cell in z ∗ coordinate

multiple of the average cross sectional area (∆ATi, j,k− 12(∆si, j,k− 1 ):Finite diﬀerence expression for the T-cell continuity equation

2[]
Wi,Tj,k = Wi,Tj,k−1 + ∆ATi, j,k− 1 ∆si, j,k− 1 (∂t z s )i, j
2

2

+
+

(dzu · u)i+ 1 , j+ 1 ,k− 1 + (dzu · u)i+ 1 , j− 1 ,k− 1
2

2

2

2

2

2

2
(dzu · v)i+ 1 , j+ 1 ,k− 1 + (dzu · v)i− 1 , j+ 1 ,k− 1
2

2

2

2

2

2

2

∆yi+ 1 , j −2∆x i, j+ 1 −(dzu · u)i− 1 , j+ 1 ,k− 1 + (dzu · u)i− 1 , j− 1 ,k− 1
2

2

2

2

2

2

2
(dzu · v)i+ 1 , j− 1 ,k− 1 + (dzu · v)i− 1 , j− 1 ,k− 1
2

2

2

2

2

2

2

2

∆yi− 1 , j
2∆x i, j− 1 , (5.6)
2where it is assumed that the actual thickness of a U-cell (dzu) also works as a land-sea mask. The second term on the righthand side is regarded as an actual temporal variation of the volume of a T-cell, that is,∆ATi, j,k− 1 ∆si, j,k− 1 (∂t z s )i, j = ∂t (volt)i, j,k− 1 ,
2

2

(5.7)

2

where equation (3.28) is used. Conversely, by using W T calculated from (5.6), calculation of the volume flux divergence
for a T-cell results in the volume change rate of the T-cell.
To check consistency with the free surface equation, we insert (5.7) into (5.6) and then sum them up from the surface
to the bottom to have
Wi,Tj,ktbtm = 0 = − (P − E + R + I)∆ATi, j, 1 +T∂(Vcolumn)i, j

Ui+ 1 , j+ 1 + Ui+ 1 , j− 1

+

2
2
2
2
∆yi+ 1 , j
2∂t2
Vi+ 1 , j+ 1 + Vi− 1 , j+ 1
Ui− 1 , j+ 1 + Ui− 1 , j− 1
Vi+ 1 , j− 1 + Vi− 1 , j− 1
2
2
2
2
2
2
2
2
2
2
2
2
−
∆yi− 1 , j +∆x i, j+ 1 −∆x i, j− 1 .
2
2
2
2
2
2
2

(5.8)

T
Because the tendency of the volume of a T-point water column Vcolumn
is determined by the undulation of sea level, we
have,
T∂(Vcolumn)i, j∂η i, j= (areat)i, j, 1
= (ws)i, j + (transport_wflux)i, j ,
(5.9)
2
∂t
∂t

where
(ws)i, j = −( Ui+ 12 , j+ 12 + Ui+ 12 , j− 12+2
Vi+ 1 , j+ 1 + Vi− 1 , j+ 1
2

2

2

Ui− 1 , j+ 1 + Ui− 1 , j− 1

∆yi+ 1 , j −

2

2

2

2

∆x i, j+ 1 −

2

2

2

2
Vi+ 1 , j− 1 + Vi− 1 , j− 1
2

2

2

2

2

2

∆yi− 1 , j
2∆x i, j− 1
2)

(5.10)

and(transport_wflux)i, j = (P − E + R + I)i, j ∆ATi, j, 1 .

(5.11)

2

It is confirmed that (5.8) has an integrated form of the free surface equation (2.40).
As a preparation for explaining the U-cell continuity equation, the finite diﬀerence expression of the continuity equation
for the vertically k-th T-cell (5.6) is rewritten in a concise form as follows, by defining the mass fluxes passing througheach side of the grid cell (Figure 5.1):
T
T
MCTi, j,k− 1 ≡ Ui−− Ui++ Vi,Tj− 1 − Vi,Tj+ 1 + Wi,Tj,k − Wi,Tj,k−1
1
1
,j
,j
2

2

2

2

2

= ∂t (volt)i, j,k− 1 ,(5.12)

2

where

T
∗
Ui+
= ui+
1
1 ∆yi+ 1 , j ∆z,
,j
,j
2

∗
=
ui+
1
,j
2

2

2

1
(u 1 1 + ui+ 1 , j− 1 ),2
2
2 i+ 2 , j+ 2

Vi,Tj+ 1 = vi,∗ j+ 1 ∆x i, j+ 1 ∆z,
2vi,∗ j+ 1 =
2

2

2

1
(v 1 1 + vi− 1 , j+ 1 ).
2
2
2 i+ 2 , j+ 2

(5.13)(5.14)

The finite diﬀerence analog of the continuity for the partial T-cell along the coastline (Figure 5.1b) is defined as follows:
Vi,Tj+ 1 =2

1 ∗v 1 ∆x 1 ∆z,
2 i, j+ 2 i, j+ 2

Vi,Tj− 1 =– 42 –21 ∗v 1 ∆x 1 ∆z,
2 i, j− 2 i, j− 2

(5.15)

Chapter 5

Continuity equation
(a)

(b)

i, j+1i, j+1

i+1, j+1
U-cellv*i, j+1/2

v*i, j+1/2

i+1/2, j+1/2

vm i, jvmi, ju*i-1/2, j

u*i+1/2, j
i, j umi, ji-1, jT-celli+1, ji, j

u*i+1/2, j

v*i, j-1/2

v*i, j-1/2

i, j-1(c)

i+1/2, j+3/2(d)i+1, j+1

i, j+1v*i, j+1/2

v*i, j+1/2

i-1/2, j+1/2

U-celli+3/2, j+1/2

vmi, ju*i+1/2, j

u*i-1/2, ji, j

u*i-1/2, jumi, jvmi, ju*i+1/2, j

i, j umi, ji-1, jv*i, j-1/2

T-celli+1, jv*i,j-1/2
i+1/2, j-1/2

i, j-1Figure5.1 Horizontal arrangement of variables for the continuity equation. (a) Relationship between T-cell and U-cell(standard form). (b),(c) Relationship between T-cell and U-cell near the coast. (d) Diagonal square grid cell and mass
fluxes.

vi,∗ j+ 1 = vi+ 1 , j+ 1 ,2

2vi,∗ j− 1 = vi+ 1 , j− 1 ,

2

2

2

∗
= 0.
ui−
1
,j

(5.16)1 ∗v 1 ∆x 1 ∆z,
2 i, j− 2 i, j− 2

(5.17)

2

2

For the corner part of land as shown in Figure 5.1(c), it is given as follows:
T
Ui−
=
1
,j
2

1 ∗
u 1 ∆y 1 ∆z,
2 i− 2 , j i− 2 , j

Vi,Tj− 1 =∗ui−= ui− 1 , j+ 1 ,1,jvi,∗ j− 1 = vi+ 1 , j− 1 .

2

2

2

2

2

2

(5.18)

2

The boundary condition for Wi,Tj is as follows:()
Wi,Tj,0 = (∆AT ∂t η)i, j = Ui− 1 , j ∆yi− 1 , j −Ui+ 1 , j ∆yi+ 1 , j +Vi, j− 1 ∆x i, j− 1 −Vi, j+ 1 ∆x i, j+ 1 − (P − E + R + I)i, j ∆ATi, j, 1 (5.19)
2

at the surface (where U =

∑N

2

2

k=1 uk− 12 ∆z k− 12 and V =

2

2

2

2

2

2

∑N

k=1 vk− 12 ∆z k− 12 ), andWi,Tj,ktbtm = 0
at the bottom (k = ktbtm).

– 43 –(5.20)

5.3

Finite diﬀerence expression for the U-cell continuity equation

5.3 Finite diﬀerence expression for the U-cell continuity equation
The finite diﬀerence expression of the continuity equation for a U-cell (i + 21 , j + 12 ) is defined using those for T-cells as
follows (Figure 5.1(a, b, and c)):MCU≡
i+ 12 , j+ 12 ,k− 12

MCTi, j,k− 1

2Ni, j,k− 1

+

2

=

MCTi+1, j,k− 1+

2

Ni+1, j,k− 1

MCTi, j+1,k− 1

2∂t (∆V T )i, j,k− 1
2Ni, j,k− 1+

Ni, j+1,k− 1

MCTi+1, j+1,k− 1

2∂t (∆V T )i+1, j,k− 1
2

Ni+1, j,k− 1

2

+

2

+

2

Ni+1, j+1,k− 1

2∂t (∆V T )i, j+1,k− 1
2Ni, j+1,k− 1

2+∂t (∆V T )i+1, j+1,k− 1
2

Ni+1, j+1,k− 1

2,(5.21)

2

where Ni, j,k− 1 is the number of sea grid cells around the T-point (i, j) in the (k − 12 )th layer. Usually, N = 4 for T-cells
2
away from land (Figure 5.1a), but N < 4 for the partial T-cells along coast lines (Figure 5.1b,c). This equation means thatthe mass convergence in a U-cell consists of the sum of the contributions from four surrounding T-cells.
The standard form of the mass continuity, which applies for U-cells (i + 12 , j + 12 ) away from coast lines, is as follows:
1
MCU≡ (MCTi, j,k− 1 + MCTi+1, j,k− 1 + MCTi, j+1,k− 1 + MCTi+1, j+1,k− 1 )i+ 12 , j+ 12 ,k− 12
4
2
2
2
2
1
= [∂t (∆V T )i, j,k− 1 + ∂t (∆V T )i+1, j,k− 1 + ∂t (∆V T )i, j+1,k− 1 + ∂t (∆V T )i+1, j+1,k− 1 ].
2
2
2
2
4
This can be rewritten as
1 U
1 U
1 U
1 U
U
U
U
(U + Ui,Uj+1 ) − (Ui+1,
j + Ui+1, j+1 ) + (Vi, j + Vi+1, j ) − (Vi, j+1 + Vi+1, j+1 )2 i, j
2
2
2
U
U
+ Wi+
≃ ∂t (∆V U )i+ 1 , j+ 1 ,k− 1 .
−Wi+11, j+ 1 ,k−1
, j+ 1 ,k
2

2

2

2

2

2

2

(5.22)(5.23)

Note that the l.h.s. does not give the exact volume tendency that would be obtained by following the definition of thevolume of a U-cell (3.26). This results in a local violation of momentum conservation, though the momentum is globallyconserved. We keep using the definition of the volume of a U-cell given by (3.26) because we do not want to use thesimple average of the volumes of T-cells similar to (5.21) for defining the control volume of a U-cell. This comes fromthe need for a simple algebraic expression. Terms Ui,Uj , Vi,Uj , and W U 1 1 are defined as follows:i+ 2 , j+ 2 ,kUi,Uj = umi, j ∆yi, j ∆z,umi, j =1 ∗∗(u 1 + ui−
1 ),2 i+ 2 , j
2, j

U
Wi+
=
1
, j+ 1 ,k
2

2

Vi,Uj = vmi, j ∆x i, j ∆z,(5.24)

1 ∗(v 1 + vi,∗ j− 1 ),2 i, j+ 2
2(5.25)

)
1( T
T
T
T
W
+ Wi+1,j,k + Wi, j+1,k + Wi+1, j+1,k .
4 i, j,k(5.26)

vmi, j =Note that W U is obtained by an averaging operation on W T and also that the following equations are derived from (5.13),(5.24), and (5.25) for the standard form of the U-cell continuity:
1 T
TUi,Uj = (Ui+
+ Ui−
1
1 ),
2
2, j
2, j
1
Vi,Uj = (Vi,Tj+ 1 + Vi,Tj− 1 ).
2
2
2

(5.27)

The advecting velocity for momentum is based on (U U , V U , W U ).
All the above relationships hold for the cases with variable grid sizes (Figure 3.1c).
The l.h.s. of the standard form of the continuity equation (5.23) expresses the convergence of mass fluxes along thehorizontal coordinate axes and it is completed as far as the continuity equation is concerned. However, when the masscontinuity is used to calculate the momentum advection, the l.h.s. of (5.23) is rewritten as follows to express the convergenceof the diagonal mass fluxes to the coordinate axes, and is used together with its original form (5.23):
1 U
1 U
1 U
1 U
U
U
U
(U + Vi,Uj ) − (Ui+1,
j+1 + Vi+1, j+1 ) + (Ui, j+1 − Vi, j+1 ) − (Ui+1, j − Vi+1, j )2 i, j
2
2
2
U
U
−Wi+
+ Wi+
≃ ∂t (∆V U )i+ 1 , j+ 1 ,k− 1
1
1, j+ 1 ,k−1
, j+ 1 ,k2

2

2– 44 –

2

2

2

2(5.28)

Chapter 5

Continuity equation

Let us explain the meaning taking the first term on the l.h.s. of (5.28),
Ui,Uj + Vi,Uj =

}
1{ T
T
(Ui− 1 , j + Vi,Tj− 1 ) + (Ui++ Vi,Tj+ 1 ) ,
1
,
j
2
2
2
2
2

(5.29)as an example, where (5.27) is used. If the flow is horizontally nondivergent, the horizontal mass fluxes U T 1 and V T

i− 2 , ji, j− 121
1
1
1
in the first term on the r.h.s. are expressed by streamfunction at two pairs of U-points, (i − 2 , j + 2 ) and (i − 2 , j − 2 ), and(i − 21 , j − 12 ) and (i + 12 , j − 12 ), respectively. Then, their sum corresponds to the net mass flux crossing the diagonal section
connecting the two U-points (i − 12 , j + 12 ) and (i + 12 , j − 21 ) (Figure 5.1d). The second term on the r.h.s. expresses the same

quantity, though the route is diﬀerent. Thus, multiplying by a factor of two, the l.h.s. of (5.28) means the horizontal massconvergence in the diagonal square defined by four U-points (i − 12 , j + 21 ), (i + 12 , j − 12 ), (i + 32 , j + 12 ), and (i + 12 , j + 32 ).
Multiplying by a factor of 21 , the l.h.s. of (5.28) itself means the horizontal mass convergence in the U-cell (i + 21 , j + 12 ),whose area is a half of that of the diagonal square.
Taking the first four terms on the l.h.s. of (5.23) as Ai+ 1 , j+ 1 and those of (5.28) as Bi+ 1 , j+ 1 , the standard form of the2222
continuity equation for the U-cell used for the calculation of the momentum advection is generally expressed as:UU
α Ai+ 1 , j+ 1 + β Bi+ 1 , j+ 1 − Wi++ Wi+≃ ∂t (∆V U )i+ 1 , j+ 1 ,k− 1 ,11, j+ 1 ,k−1
, j+ 1 ,k(5.30)

α + β = 1.

(5.31)

2

2

2

2

2

2

2

2

2

2

2

where
As shown later in Chapter 7, (α, β) = (2/3, 1/3) for the generalized Arakawa scheme and (α, β) = (1/2, 1/2) for the
standardized form derived from the continuity equation generalized for arbitrary bottom topography. What Webb (1995)
proposed corresponds to (α, β) = (1, 0).

5.4 Summary
To summarize the solution procedure, MRI.COM uses (5.8) to obtain W T under the boundary conditions (5.19) and (5.20).
Then, using W T , W U is obtained by (5.26).

– 45 –

Part III

Equations of Motion

Chapter 6

Equations of motion (barotropic component)
Historically, iterative methods have been used to solve the barotropic part of the momentum equations by applying therigid-lid approximation. The number of iterations to get convergence of a solution is of order N, where N is the larger
number of grid points of the two horizontal directions. Thus the number of iterations increases as the number of grid
points increases. This means that the iterative process could occupy a large part of the total CPU time. This is a severeburden and should be remedied.
An alternative is to replace the rigid lid with a free surface. The number of short barotropic time steps within a long
baroclinic time step is roughly decided by the ratio between external and internal gravity wave speeds, around 70 to 100,
which becomes smaller than N when the number of grid points of the model is large.∗ In addition, this method is moresuitable for parallel computation than the iterative methods. Thus, for fine-resolution models, the free-surface formulation
has numerous advantages over the iterative methods. The free-surface formulation was adopted in the Bryan-Cox-Semtner
numerical ocean general circulation models by Killworth et al. (1991).
However, the free-surface formulation has a problem when it is used with a mixed-layer model. To appropriately resolve
the surface mixed layer, the uppermost layer should be less than a few meters thick. This leads to a serious problem, sincea free-surface model does not work when the sea surface is below the bottom of the uppermost layer and the thickness of
this layer vanishes. This occurs in world ocean models, because the diﬀerence between the maximum and minimum of
the sea surface height becomes several meters.†
To remedy this problem, σ-coordinate has been adopted near the sea surface until MRI.COM version 3. This method iscalled σ-z formulation and is introduced by Hasumi (2006). The thicknesses of the several upper layers (z ≤ −HB ) varyas the sea surface height does whereas the vertical position of the layers below (z = −HB ) is kept fixed (Figure 6.1a).
There were two major shortcomings in the implementation of the σ-z formulation in MRI.COM. First, because we
set HB as a constant, the sea-floor cannot be shallower than HB . We typically take HB as 20 - 30 meters, thus therepresentation of circulation around coastal areas with shallow sea-floor may not be accurate. Second, because a hybridof vertical coordinates is used, some sort of dynamical analyses may be awkward.
To resolve these problems, a vertically rescaled height coordinate (z ∗ ) introduced by Adcroft and Campin (2004), where
undulation of the sea surface is distributed to all vertical grid cells in a water column, is adopted in MRI.COM version 4.
In this chapter, we explain the free-surface formulation of MRI.COM with z ∗ vertical coordinate.

6.1 Governing equations
As described in Chapter 2, the prognostic variables in the free-surface model are the surface elevation (η) and the verticallyintegrated velocity (U and V ). The prognostic equations are obtained by integrating momentum and continuity equations
vertically.
The momentum equations (2.69) and (2.70) are re-written:∂U(η + H) ∂(pa + ρ0 gη)− fV = −+ X,∂t
ρ0 h µ∂µ∂V(η + H) ∂(pa + ρ0 gη)+ fU = −+ Y,∂tρ 0 hψ
∂ψ

(6.1)
(6.2)

A North Pacific model with 1/4° × 1/6° resolution, which was the first meso-scale eddy permitting model developed by the ocean modelinggroup of MRI about 20 years ago, had 742 grid points in the zonal direction. This figure prompted us to make a transition from the rigid-lid tofree surface formulation.
† For example, the maximum and minimum sea-surface heights in a 1° × 1° world ocean model are about 1 m (subtropical gyres) and −2 m (theRoss Sea), respectively.

∗– 49 –

6.2

Time integration on barotropic time levels(b) z*(a) σ-zz=-HB

Figure6.1 Schematic of (a) the near surface σ-coordinate layers (σ-z coordinate) and (b) z ∗ vertical coordinate.

where
X≡

N
∑

Fµ = −∇ H ·

k=1
N [
∑

−

k=1

Y≡

N
∑

1 1
ρ0 h µ

∫ 0

sk− 1

−

k=1

]
z s g ρµ ds ′ ∆zk− 1 −22Fψ = −∇ H ·

k=1
N [
∑

(
)
N [
) ∑
∂hµ
∂hψ ]
v
(∆z(u, v)u)k− 1 −u−v∆zk− 1k− 21
2
2hµ hψ ∂ψ
∂µ
k=1
k=1

N
(∑

1 1
ρ 0 hψ

(6.3)
(

N
(∑

∫ 0

N
N
∑
g ∑[ ′ ]µµµρ zµ ∆zk− 1 +(∆zFhorz )k− 1 + Fsurf ∆z 1 + Fbottom ∆z N − 12222
ρ0 hµ k=1
k=1

N [
) ∑
∂hψ
∂hµ
u
(∆z(u, v)v)k− 1 +u−v
2hµ hψ ∂ψ
∂µ
k=1
k=1

sk− 1
2

]
z s g ρψ ds ′ ∆zk− 1 −
2

)]
k− 12

∆zk− 1
2

NN∑g ∑[ ′ ]ψψψρ zψ ∆zk− 1 +(∆zFhorz )k− 1 + Fsurf ∆z 1 + Fbottom ∆z N − 1 .
2222
ρ0 hψ k=1
k=1

(6.4)

Note that the surface atmospheric pressure pa is omitted in the remainder of this chapter. In MRI.COM, the surface
atmospheric pressure is not included unless explicitly specified by SLP option.
The continuity equation (2.73) is re-written:{}∂(hψ U) ∂(hµ V )1∂η++= (P − E + R + I),(6.5)∂th µ hψ
∂µ∂ψ
where P is precipitation (positive downward), E is evaporation (positive upward), R is the river discharge rate (positive
into the ocean), and I is the mass exchange with the sea ice model (positive into the ocean).
Figure 6.2 illustrates the grid arrangement of the free-surface formulation. The variable η is defined at T-points, and thevariables U and V are defined at U-points. Forcing terms (X and Y ) in Eqs. (6.1) and (6.2) are calculated in the subroutine
for the baroclinic component and defined at U-points.

6.2 Time integration on barotropic time levels
Figure 6.3 presents schematics of the time integration of the barotropic mode in the free-surface formulation. When thetime integration of the baroclinic mode is performed from step n (t = t n ) to step n + 1 (t = t n+1 , ∆t = t n+1 − t n ), thecorresponding time integration of the barotropic mode is carried out from step n to a step sometime beyond step n + 2
with the barotropic time interval ∆t tr using the vertically integrated values (X, Y ) at t = t n calculated in the program thatsolves the baroclinic mode. A weighted average of the barotropic mode over the integration period is used to represent thevertically integrated velocity at t = t n+1 .
The "Euler forward-backward" scheme is a stable and economical numerical scheme for linear gravity wave equations
without advection terms (Mesinger and Arakawa, 1976), and this is adopted for the governing equations in the free-surface
formulation of MRI.COM. This scheme is more stable than the leap-frog scheme. The time step can be doubled for– 50 –

Chapter 6

Equations of motion (barotropic component)
Δxη i-1, j+1

η i, j+1

η i+1, j+1

U i-1/2, j+1/2

U i+1/2, j+1/2

η i+1, jη i, jη i-1, jΔyU i+1/2, j-1/2

U i-1/2, j-1/2

η i-1, j-1

η i+1, j-1

η i, j-1

Figure6.2 Grid arrangement of the free-surface formulation

the linear gravity wave equations. In this numerical scheme, either the continuity equation or the momentum equation
is calculated first, and then the estimated values are used for calculating the remaining equations. In the procedure of
MRI.COM, the surface elevation is first calculated using the continuity equation; the calculated surface elevation is thenused to calculate the pressure gradient terms of the momentum equations.
Killworth et al. (1991) recommended using the Euler backward (Matsuno) scheme for the free surface model except for
the tidal problem. The Euler-backward scheme damps higher modes and is more stable. However, the computer burden
increases considerably because this scheme calculates the equations twice for one time step. In MRI.COM, stable solutions
are eﬃciently obtained by using the Euler forward-backward scheme because the time filter is applied for the barotropicmode.
The finite-diﬀerence expression of the continuity equation (Eq. 6.5) is′(η i, j − η i, j )
∆t tr

+

[
]
ψ
µ
1
(δµ hψ U )i, j + (δψ hµ V )i, j = (P − E + R + I)i, j ,(hµ hψ )i, j(6.6)

where the subscripts are labeled on the basis of T-points. The variable η i, j is located at T-points, and the variables Ui+ 1 , j+ 1
2

2

and Vi+ 1 , j+ 1 are located at U-points. (They are located at (i + 12 , j + 12 ) on the basis of T-points; see Figure 3.3). The
22
finite-diﬀerencing and averaging operators are defined as follows:δµ Ai ≡µAi ≡Ai+ 1 − Ai− 1
2

2

∆µi
Ai+ 1 + Ai− 1222

δµ Ai+ 1 ≡,2

Ai+1 − Ai,∆µi+ 1
2

,

Ai+ 1
2

µ

Ai+1 + Ai
≡
.
2

(6.7)

The same applies to ψ. In the program codes, the above equation is multiplied by the area of a T-cell (∆ST i, j = (∆x∆y)i, j =(hµ ∆µhψ ∆ψ)i, j ):′(6.8)
(η i, j −η i, j ) · ∆ST i, j
) (
)}
(
{
ψ
ψ
µ
µ
= ∆t tr · (P − E + R + I)i, j · ∆ST i, j − ∆yi+ 1 , j U i+ 1 , j − ∆yi− 1 , j U i− 1 , j − ∆x i, j+ 1 V i, j+ 1 − ∆x i, j− 1 V i, j− 1 .
2

2

2

2

2

2

2

2

Averaging operators are defined the same way as the previous ones. This equation is used to obtain the new surface′
elevation, η i, j .
′After obtaining η i, j , the momentum equations, Eqs. (6.1) and (6.2), are solved. A longer time step can be used when

– 51 –

6.2

Time integration on barotropic time levels

the semi-implicit scheme is applied for the Coriolis terms in the momentum equations. Their finite-diﬀerence forms are(U′

i+ 12 , j+ 12− Ui+ 1 , j+ 1 )
2

2

∆t tr
(V

′

i+ 12 , j+ 12− Vi+ 1 , j+ 1 )
2

f (V
−

′

i+ 12 , j+ 12

+

∆t tr

+ Vi+ 1 , j+ 1 )
2

2

2
f (U

2

′

i+ 12 , j+ 12′

i+ 12 , j+ 12

both sides by ∆t tr , we have

=−

2

2

(hµ )i+ 1 , j+ 1
2

+ Ui+ 1 , j+ 1 )2

2

2

Next, we solve these equations for Uµ,ψ

g(Hi+ 1 , j+ 1 + η ′ i+ 12 , j+ 21 )′i+ 12 , j+ 12

2

2

2

(6.9)

2

2

µ,ψ

=−

g(Hi+ 1 , j+ 1 + η ′ i+ 12 , j+ 21 )22(hψ )i+ 1 , j+ 1
2

and V

ψ

δµ η ′ i+ 1 , j+ 1 + Xi+ 1 , j+ 1µδψ η ′ i+ 1 , j+ 1 + Yi+ 1 , j+ 1 .
2

2

2(6.10)

2

2. Let the r.h.s. of the above equations be GX and GY . Multiplying

f ∆t tr ′(Vi+ 1 , j+ 1 + Vi+ 1 , j+ 1 ) = ∆t tr GXi+ 1 , j+ 1 ,
2
2
2
2
2
2
2
′
f ∆t tr ′(Vi+ 1 , j+ 1 − Vi+ 1 , j+ 1 ) +(Ui+ 1 , j+ 1 + Ui+ 1 , j+ 1 ) = ∆t tr GYi+ 1 , j+ 1 ,
2
2
2
2
2
2
2
2
2
2
2
′

(Ui+ 1 , j+ 1 − Ui+ 1 , j+ 1 ) −2

2

2(6.11)

2(6.12)

leading tof ∆t tr ′f ∆t tr
Vi+ 1 , j+ 1 = Ui+ 1 , j+ 1 +V 1 1 + ∆t tr GXi+ 1 , j+ 1 ,
2
2
2
2
2
2 i+ 2 , j+ 2
2
2
2
2
′
f ∆t tr ′f ∆t tr
Vi+ 1 , j+ 1 +Ui+ 1 , j+ 1 = Vi+ 1 , j+ 1 −Ui+ 1 , j+ 1 + ∆t tr GYi+ 1 , j+ 1 .
2
2
2
2
2
2
2
2
2
2
2
2
′

Ui+ 1 , j+ 1 −Letting the r.h.s. of the above equations be RX and RY , we may derive expressions for U′

i+ 12 , j+ 21(6.13)(6.14)and V′

i+ 21 , j+ 12

as follows:}/{{( f ∆t tr ) 2 }′f ∆t tr
RYi+ 1 , j+ 1,
Ui+ 1 , j+ 1 = RXi+ 1 , j+ 1 +
1+
2
2
2
2
2
2
2
2
{
}/{
( f ∆t tr ) 2 }′f ∆t tr1+
Vi+ 1 , j+ 1 = RYi+ 1 , j+ 1 −RXi+ 1 , j+ 1
.
2
2
2
2
2
2
2
2

(6.15)(6.16)

baroclinic time level
n+1

n

n+2

Δtcl
depth-integrated
properties
n

n

n+1

X,Y

n+1

U ,V
weighted average (n+1)

weighted average (n+2)

0.05
0.04

am

0.03

bm

0.02
0.01
0
-0.01

2M

M
Δttr

M*

barotropic time level

Figure6.3 Schematic figure of the time integration of the barotropic mode and its time-filtering procedure. am andbm are weights for t = n + 2 and t = n + 1, respectively.

– 52 –

Chapter 6

Equations of motion (barotropic component)

6.3 Prognostication of state variables at the baroclinic time level
In the split-explicit method, the state at the next baroclinic time level is obtained by using the solution of the barotropic
mode explained in the previous section. To do this, an appropriate time-filtering is necessary. The velocity at the baroclinic
time level uses the time-filtered barotropic state as the depth averaged velocity.

6.3.1 Weighted averaging to obtain a barotropic state at the baroclinic time level
The main purpose of solving the barotropic submodel on proceeding from the baroclinic time level of t = t n to t = t n+1
is to obtain the barotropic state variables η, U, and V at the baroclinic time level of t = t n+1 . To achieve this, startingfrom the baroclinic time level of t = t n , the governing equations of the barotropic submodel are integrated slightly beyondt = t n+2 to compute the barotropic state variables as a weighted average of those at barotropic time levels.
We follow the method adopted by the Regional Oceanic Modeling System (ROMS; Shchepetkin and McWilliams, 2005)
to determine a weighting function for averaging. The method of ROMS is actually designed to compute the state variables
at t = t n+2 as well as a flow field at t = t n+1 which is consistent with surface height at t = t n+2 under the discretization of(6.5) by the leap-frog scheme,
η n+2 − η n= −∇ · U n+1 + (P − E + R + I).
(6.17)
2∆t clIn the framework of the time integration of MRI.COM, only flow field at t = t n+1 , (U n+1, V n+1 ), is used among theresults of the barotropic submodel. The surface height at baroclinic time levels are computed by (6.17). This is to make thesurface height equation be consistent with the continuity equations in a vertical column (Leclair and Madec, 2009). (Bysetting the runtime parameter l_global_local_cnsv_ssh = .false., an averaged state from the barotropic submodel
may be used as η n+1 instead of (6.17), although this should be used with care.)In the old versions of MRI.COM, simple averaging of (2 × ∆t cl /∆t tr + 1) barotropic time steps between the two baroclinic
time intervals [t n, t n+2 ] is used to obtain state variables at t n+1 . From version 4, we use a weighted average as explained by
Shchepetkin and McWilliams (2005). Derivation of the weighting function for (U n+1, V n+1 ), bm , is briefly summarized
below.
First, a weighting shape function to calculate variables at t = t n+2 is denoted with {am } (Note that the target is att = t n+2 , not t = t n+1 ). This must satisfy discrete normalization and centroid conditions,M∗∑am ≡ 1,m=1

M∗∑mam ≡ 2M,

(6.18)

m=1

where M is the ratio between barotropic-baroclinic time step (M = ∆t cl /∆t tr ), and M ∗ is the last index at which am > 0,
where 2M ≤ M ∗ . The weighted averaging is designed so that aliasing between barotropic and baroclinic modes issuppressed as well. With a set of {am }, state variables at the baroclinic time level t n+2 is computed asη n+2 =

M∗∑
m=1

am η m, U n+2 =M∗∑

amU m, V n+2 =

m=1

M∗
∑

am V m,(6.19)

m=1

where η m , U m , and V m are instantaneous state variables at the barotropic time level t = t m .
To be consistent with the continuity equation at baroclinic time levels, the vertically integrated continuity equation mustsatisfyη n+2 − η n= −∇ · Un+1 .
(6.20)
2∆t clThe flow field at the time level t = t n+1 can be determined accordingly. Assuming that the vertically integrated continuity
equation is advanced in time as1
η m+1 − η m(6.21)= −∇ · Um+ 2 ,∆t tr
we may obtain the expression for η m as
η m = η 0 − ∆t tr

m−1
∑
m′ =0

– 53 –′∇ · Um + 2 .
1(6.22)

6.3

Prognostication of state variables at the baroclinic time level

Inserting this into (6.19) and after some manipulations, which is detailed in Shchepetkin and McWilliams (2005), we have
η

n+2

M∗
∑

= η − 2∆t cl ∇ ·0′bm′ Um − 2 ,1(6.23)

m′ =1

where η 0 = η n and

M∗

1 ∑
am .
bm′ =
2M m=m′(6.24)

By comparing (6.20) and (6.23), vertically integrated flow field at the baroclinic time level t = t n+1 is obtained as
U

n+1

=

M∗
∑

bm U

m− 12

, V

n+1

=

M∗
∑

1

bm V m− 2 .

(6.25)m=1

m=1

We require that a set of {bm } satisfies

M∗
∑

M∗
∑

bm ≡ 1,m=1

mbm ≡ M,

(6.26)

m=1

as well as (6.18).
Specific shape of the weighting function (am ) for τ = m/2M, (1 ≤ m ≤ M ∗ ) is given as follows:
A(τ) = A0{( τ )p [τ01−( τ )q]
τ0

−r

τ}
,
τ0

(6.27)

where p = 2, q = 2, r = 0.2346283, and A0 and τ0 are chosen to satisfy normalization conditions (6.18) and (6.26)
iteratively. The initial guess for A0 and τ0 is given as follows:
A0 = 1,τ0 =(p + 2)(p + q + 2).
(p + 1)(p + q + 1)

(6.28)

Approximate shape of {am } and {bm } are shown by red and blue lines of Figure 6.3.
In the following, the discretized momentum equation for the 3-D velocity field under the split-explicit method will bederived. This is to clarify the diﬀerence from the one for the 3-D velocity under a normal, fully explicit discretizing
method. Let us consider reconstructing the flow field of (6.25) from the momentum equation for the barotropic submodel.
In the following, we adopted a simple time stepping scheme (forward scheme) and conceptual notations for representing
time levels. Because specific expressions will depend upon the choice of time stepping algorithm, the following equations
do not necessarily correspond to the time stepping methods used in the model. But this discrepancy is not essential.
At barotropic time levels, momentum equations are advanced in time as follows:1

U m+1 = U m + ∆t tr f V m+ 2 − ∆t tr[ (η + H) ∂(pa + ρ0 gη) ] m+ 21+ ∆t tr X,
ρ0 h µ∂µ[ (η + H) ∂(pa + ρ0 gη) ] m+ 211
V m+1 = V m − ∆t tr f U m+ 2 − ∆t tr+ ∆t trY,
ρ 0 hψ
∂ψ(6.29)(6.30)

1

where [·]m+ 2 is an approximate value of (·) at the time level of m + 12 , which is dependent on the choice of the time stepping
scheme.
′Successive summation of (6.29) and (6.30) over m = [0, m − 1] yields
U m = U 0 + ∆t tr

m−1
∑

′

1

′

1

f V m + 2 − ∆t tr

m′ =0

V m = V 0 − ∆t tr

m−1
∑
m′ =0

f U m + 2 − ∆t tr

m−1∑[(η + H) ∂(pa + ρ0 gη) ] m′ + 12
+ m∆t tr X,
ρ0 h µ
∂µ
m′ =0
m−1
∑[

(η + H) ∂(pa + ρ0 gη) ] m′ + 12
+ m∆t trY,
ρ 0 hψ
∂ψm′ =0– 54 –(6.31)(6.32)

Chapter 6

Equations of motion (barotropic component)

Applying the time averaging procedure of (6.25) yields⟨U⟩n+1 ≡M∗∑

bm U m

(6.33)

m=1


M∗ 
m
M∗
M∗ 
m [
∑
∑
∑
∑
∑
] m′ − 21 



∂(p
+
ρ
gη)
(η
+
H)
a
0
m′ − 21 + bm= U + ∆t tr
fV
mbm ∆t tr X, (6.34)
bm − ∆t tr
ρ0 h µ
∂µ
m=1 
m′ =1
m=1 
m′ =1

 m=1
M∗
∑
n+1
⟨V ⟩
≡
bm V m(6.35)
0

m=1


M∗ 
m
M∗ 
M∗
m [


∑
∑
∑
∑
∑



(η + H) ∂(pa + ρ0 gη) ] m′ − 12 m′ − 21 = V − ∆t tr
fU+
mbm ∆t trY, (6.36)bmbm − ∆t tr
ρ 0 hψ
∂ψ
 m=1

′
m=1 
m′ =1
m=1 

m =1

0

which are rearranged to have a form
1

⟨U⟩n+1 = U 0 + ∆t cl f ⟨⟨V ⟩⟩n+ 2 − ∆t cl⟨⟨ (η + H) ∂(pa + ρ0 gη) ⟩⟩n+ 12+ ∆t cl X n,
ρ0 h µ∂µ⟨⟨ (η + H) ∂(pa + ρ0 gη) ⟩⟩n+ 12
1⟨V ⟩n+1 = V 0 − ∆t cl f ⟨⟨U⟩⟩n+ 2 − ∆t cl+ ∆t clY n,
ρ 0 hψ
∂ψ

(6.37)(6.38)∑M ∗∑1m′ − 21 .
bm m
where ⟨⟨·⟩⟩n+ 2 ≡ m=1
m′ =1 (·)
On the other hand, the baroclinic momentum equation where surface pressure gradient term is dropped is
u

′

k− 12

v

′

∆z n+11 − un 1 ∆z n 1
k− 2

k− 2

k− 2

1

= f [vk− 1 ∆zk− 1 ]n+ 2 + Fµn,∆t cln+1 − v n ∆z n
∆z
1
1
1
1

k− 2

k− 2

k− 2

2

k− 2

(6.39)

2

1= − f [uk− 1 ∆zk− 1 ]n+ 2 + Fψn .

∆t cl

2

(6.40)

2This is vertically summed up to give
∑N

∑N
′
n+1
n
n
k=1 (uk− 1 ∆z k− 1 ) − k=1 (uk− 1 ∆z k− 1 )
2

∑N

′

k=1 (vk−

2

2

= f

2

∆t cl
n+1 ) − ∑ N (v n ∆z n )
1 ∆z
1
1
1
k=1
2

k− 2

k− 2

N
∑

1

[vk− 1 ∆zk− 1 ]n+ 2 + X n,

(6.41)

N∑1[uk− 1 ∆zk− 1 ]n+ 2 + Y n .

(6.42)

2

2

k=1

k− 2

= −f

∆t cl

2

2

k=1

X n and Y n are removed from (6.37), (6.38), (6.41), and (6.42) to give∑N

k=1 (⟨u⟩n+1 ∆z n+1 ) − ∑ N (u′n+1
k=1 k− 1 ∆z k− 1 )
k− 12
2
2

∆t cl
∑N

k=1 (⟨v⟩

n+1 ∆z n+1 ) − ∑ N (v ′n+1
k=1 k− 1 ∆z k− 1 )
k− 12
2
2

∆t cl

1

= f ⟨⟨V ⟩⟩n+ 2 − f

1N∑⟨⟨ (η + H) ∂(pa + ρ0 gη) ⟩⟩n+ 12
1[vk− 1 ∆zk− 1 ]n+ 2 −22
ρ0 h µ∂µk=1(6.43)= − f ⟨⟨U⟩⟩n+ 2 + f

N∑⟨⟨ (η + H) ∂(pa + ρ0 gη) ⟩⟩n+ 12
1[uk− 1 ∆zk− 1 ]n+ 2 −.
22
ρ 0 hψ
∂ψ
k=1(6.44)– 55 –

6.3
This is combined with (6.39) and (6.40) to give()
z′
u 1 − u′ + ⟨u⟩n+1 ∆z n+11 − un 1 ∆z n 1
k− 2

k− 2

k− 2

k− 2

∆t cl

] n+ 21
[
z
1
n+ 21
n+1
n+1
− f [v]n+ 2 ∆zk−
∆zk−
= f vk− 1 ∆zk− 1
1 + f ⟨⟨v⟩⟩
1
2

−
(
v

′

k− 21

)
z
− v ′ + ⟨v⟩n+1 ∆z n+11 − v n 1 ∆z n 1
k− 2

∆t cl

Prognostication of state variables at the baroclinic time level

k− 2

k− 2

2

∆z n+11k− 2

2

2⟨⟨ (η + H) ∂(pa + ρ0 gη) ⟩⟩n+ 12
ρ0 h µη n+1 + H

∂µ

+ Fµn

(6.45)[] n+ 21
z
1
n+1
n+ 12
n+1
= − f uk− 1 ∆zk− 1+ f [u]n+ 2 ∆zk−
∆zk−
1 − f ⟨⟨u⟩⟩1
2−∆z n+11k− 2
η n+1 + H

22⟨⟨ (η + H) ∂(pa + ρ0 gη) ⟩⟩n+ 12
ρ 0 hψ

∂ψ2+ Fψn,(6.46)zwhere (...) denotes the thickness weighted vertical average. This is the momentum balance for the total velocity fieldunder the mode-splitting scheme. In comparison with (6.39) and (6.40), the correction terms appear in the tendency andCoriolis terms owing to the use of the split-explicit method. In the tendency term on the l.h.s., the vertically averagedzzvelocity (u′ , v ′ ) is replaced by the prediction of the barotropic model (⟨u⟩n+1 , ⟨v⟩n+1 ). Likewise on the r.h.s., the Coriolis
11term takes a form in which vertical average for [uk+ 1 ]n+ 2 is replaced by ⟨⟨u⟩⟩n+ 2 .
2

6.3.2 Update of 3-D state variables at the baroclinic time level
Vertically integrated transport at time t = t n+1 is obtained by a weighted averaging of those at barotropic time levels as
explained in the previous section. Surface elevation at time t = t n+1 is computed by using vertically integrated transport
at time t n , which has been calculated in the previous time step and stored, and updated with the leap frog scheme:
η n+1 − η n−1+ ∇ · U n = P − E + R + I,
2∆t

(6.47)

as explained in Section 6.3.1.
With this surface elevation, the height of the vertical column and the volume of the grid cells in the column may be
determined (see Chapter 3). Once the volume is determined, 3D state variables at time t = t n+1 may be computed. The
velocity field may be computed as follows:• A provisional baroclinic velocity field (u ′ n+1, v ′ n+1 ) is computed using the tendency terms computed by thebaroclinic module.
• The provisional velocity (u ′ n+1, v ′ n+1 ) is integrated over the whole column and the vertical mean of the provisional
velocity is computed.
• The vertical mean velocity is subtracted from the provisional velocity first and the actual mean velocity based on(U n+1, V n+1 ) is added instead in order to yield the final velocity field (un+1, v n+1 ).
These operations correspond to the expression for the velocity at the (n + 1)th time step in the tendency term of (6.45),z′(6.46), for example, uk−1/2 − u′ + ⟨u⟩n+1 .
Because (6.47) is consistent with the continuity equation for the T-cell, computation of tendency of tracer due toadvection is both conservative and constancy preserving. This method guarantees that tracer is conserved both globallyand locally.
Note that body forcing for the uppermost layer such as wind-forcing and restoration of temperature and salinity to theprescribed values may act diﬀerently for grid cells with diﬀerent width. The body force for the uppermost layer becomes( ∂u ∂v )1 (τµ, τψ ),,1 = ... +k=∂t ∂t
ρ0 ∆z 1
2

(6.48)

2

where (τµ, τψ ) are the wind stress at the surface (momentum flux), and ∆z 1 is the variable thickness of the uppermost
2
layer. Thus, the uppermost layer is more accelerated when this layer is thinner than the standard value.

– 56 –

Chapter 6

Equations of motion (barotropic component)

When the restoring condition is applied at the surface, the corresponding temperature and salinity fluxes are
Fzθ = −
FzS = −1(θ − θ ∗ )∆z 1 ,
2
γθ

Fzθ
∂θ
= ... +,
∂t k= 1∆z 1

(6.49)

1(S − S ∗ )∆z 1 ,
2
γs

FzS
∂S
= ... +.
∂t k= 1∆z 1

(6.50)

22

2

2

Thus, the temperature and salinity are more strongly restored to the prescribed values when this layer is thinner than thestandard value.

6.4 Horizontal diﬀusivity of sea surface height
The null mode, or checkerboard pattern, would appear in the sea surface height field commonly in the Arakawa B-gridmodels. For the purpose of suppressing this mode, a weak horizontal diﬀusion of sea surface height may be included in
the vertically integrated continuity equation,{}∂(hψ U) ∂(hµ V )1∂η++= D(η) + (P − E + R + I)
(6.51)∂th µ hψ
∂µ∂ψ
where D is the diﬀusion operator (Chapter 12.9 in Griﬃes (2004)). The diﬀusion operator mixes a sea surface height ineach direction of the model coordinates with the harmonic scheme. The specific form of the harmonic-type diﬀusivity isrepresented as follows:
{
(
)
(
)}
∂ hψ κ H ∂η1∂ hµ κ H ∂η
D (η) =+
(6.52)h µ hψ ∂ µhµ ∂ µ∂ψhψ ∂ψ ∂hψ Fµη ∂hµ Fψη 
1 


+
(6.53)=
h µ hψ
∂µ∂ψ where κ H is the horizontal diﬀusion coeﬃcients and the diﬀusive fluxes are represented byFη = −( κ H ∂η κ H ∂η ),.
hµ ∂ µ hψ ∂ψ

(6.54)

Therefore, continuity equation may be rewritten as follows:ηη∂[hψ (U + Fµ )] ∂[hµ (V + Fψ )] 


∂η
1 
 = (P − E + R + I).

+
+


∂t
h µ hψ
∂µ∂ψη

(6.55)

η

If the horizontal diﬀusivity is non-zero for sea surface height, U ∗ = U + Fµ and V ∗ = V + Fψ are treated as the verticallyintegrated transport velocity, which are divided by the column height to replace the vertical mean velocity based on (U, V )which have been contained in the 3-D velocity field used to advect momentum and tracers. It should be noted that additionalηηrestart files are necessary when horizontal diﬀusion is applied to SSH. This is because (Fµ , Fψ ) are needed to obtain sea
surface height for the new time step. MRI.COM provides namelist nmlrs_ssh_dflx_x and nmlrs_ssh_dflx_y for thispurpose.

6.5 Inclusion of tidal eﬀect
This section explains the tide option (TIDE) in MRI.COM.

6.5.1 Tide producing term in the momentum equation
Tidal forcing can be introduced into the barotropic part of the momentum equations (6.1) and (6.2) as follows (e.g., Schiller,
2004):]∂U(η + H) ∂(pa + ρ0 gη) g(η + H) ∂ [− fV = −−(1 − α)η − βη eq + X,∂t
ρ0 h µ
∂µ
hµ
∂µ
]
∂V
(η + H) ∂(pa + ρ0 gη) g(η + H) ∂ [+ fU = −−(1 − α)η − βη eq + Y,∂t
ρ 0 hψ∂ψhψ∂ψ
– 57 –(6.56)(6.57)

6.5 Inclusion of tidal eﬀect
Table6.1 Constants of major tidal modes

n
1
2
3
4
5
6
7
8

Tidal Mode
K1 , declination luni-solar
Q1 , principal lunar
P1 , principal solar
Q1 , elliptical lunar
M2 , principal lunar
S2 , principal solar
N2 , elliptical lunar
K2 , declination luni-solar

K, m
0.141565
0.100514
0.046843
0.019256
0.242334
0.112841
0.046398
0.030704σ, 10−4 /sec
0.729210.67598
0.725230.64959
1.405191.454441.37880
1.45842χ, degh0 + 90h0 − 2s0 + 90
−h0 − 90h0 − 3s0 + p0 − 90
2h0 − 2s00
2h0 − 3s0 + p0
2h0Here h0 , s0 , and p0 are the mean longitudes of the sun and moon and the lunar perigee atGreenwich midnight: h0 = 279.69668 + 36000.768930485T + 3.03 × 104T 2 , s0 = 270.434358 +
481267.88314137T − 0.001133T 2 + 1.9 × 10−6T 3 , p0 = 334.329653 + 4069.0340329575T −0.010325T 2 − 1.2 × 10−5T 3 , where T = (27392.500528 + 1.0000000356D)/36525, D = d +365(y − 1975) + Int[(y − 1973)/4], d is the day numbe of the year (d = 1 for January 1), y ≥ 1975
is the year number, and Int[x] is the integral part of x.

where α defines self-attraction and loading of the ocean tides, β = 1 + k − h defines eﬀect of tide-generating potential andcorrection due to earth tides with k and h being Love numbers, and η eq is equilibrium tide, or astronomical tide-generating
potential. In MRI.COM, we usually set α = 0.88 and β = 0.7.
The eight primary constituents that consist of the four largest equilibrium tides of the diurnal and semi-diurnal species(K1 , O1 . P1 , Q1 , M2 , S2 , N2 , K2 ) may be included in the tide producing potential η eq of MRI.COM. Following Schwiderski(1980), we write the n-th (n = [1, 4]) diurnal equilibrium tide asη eq,n = Kn cos2 ϕ cos(σn t + χ n + 2λ)= Kn cos2 ϕ[cos(σn t + χ n ) cos 2λ − sin(σn t + χ n ) sin 2λ],(6.58)and the n-th semi-diurnal component (n = [5, 8]) asη eq,n = Kn sin 2ϕ cos(σn t + χ n + λ)
= Kn sin 2ϕ[cos(σn t + χ n ) cos λ − sin(σn t + χ n ) sin λ].

(6.59)The meaning of the mathematical symbols is as follows:
t
λ
ϕ
K
σ
χ

universal standard time in seconds
east longitudelatitudeamplitude of partial equilibrium tide in meters
frequency of partial equilibrium tide in sec−1
astronomical argument of partial equilibrium tide relative to Greenwich midnight

And constants are listed on Table 6.1.

6.5.2 Separation of linear responses to tidal forcing
By including tidal forcing, the balance in the basic fields solved without tides will be modified through non-linear termsof the equations. In addition to the non-linear terms, self-attraction and loading terms in (6.56) and (6.57) will modify the
relationships between the sea surface height gradient and the barotropic currents. That is, the solution without astronomical
tide-generating potential (η eq = 0) will be diﬀerent than that solved with standard equations without self-attraction andloading terms.
To resolve the possible problem that the terms specialized for tides aﬀect the basic fields unintentionally, our tide scheme
calculates the tidal fields separately from the basic fields within the barotropic submodel as explained by Sakamoto et al.
(2013). The essence of the solution method is explained below. First, we rewrite the basic equation of the barotropic
submodel in the following form that highlights the terms that may be specialized for tides:
τ btm∂U+ f k × U = −g(η + H)∇(η − βη eq − η SAL ) + Fhorz +
+ X′,
∂t
ρ0

(6.60)

where Fhorz is the vertically integrated horizontal viscosity parameterization terms and τ btm is the bottom friction terms that
may be specialized for tidal currents. They are separated from X defined by (6.3) and (6.4) and X′ ≡ X − Fhorz − τ btm /ρ0 .
– 58 –

Chapter 6

Equations of motion (barotropic component)

Note also that atmospheric pressure term is dropped for brevity. Continuity equation is∂η+ ∇ · U = Fw,∂t(6.61)

where Fw ≡ P − E + R + I is surface fresh water forcing that will be enter the system through the basic fields.
Next, the variables are decomposed into the linear tidal component and the basic component,
U = Ub + Ult(6.62)

η = η b + η lt

(6.63)

F= Fhorz+ Fhorzltbτ btm = τbbtm + τltbtm,(6.64)

horz(6.65)

where "lt" refers to linear tidal component and "b" refers to basic fields. Now we decompose (6.60) and (6.61) to obtaingoverning equations for each of the two components.
For linear tidal component:τ∂Ult+ f k × Ult = −g(η + H)∇(η lt − βη eq − η SAL ) + Fhorz+ lt ,
lt∂t
ρ0(6.66)∂η lt+ ∇ · Ult = 0,∂t

(6.67)

btm

where we assume that self-attraction and loading acts only to linear tidal component in such a way as η SAL = (1 − α)η lt .
For the basic component:
τ∂Ub+ f k × Ub = −g(η + H)∇η b + Fhorz+ b + X′,b∂t
ρ0(6.68)∂η b+ ∇ · Ub = Fw .
∂t

(6.69)

btm

In the above decomposition, the linear tidal component represents only the linear response to tidal forcing, and thesecondary eﬀects of tide, such as tidal advection and internal tides, are represented by the basic component. The linear
terms in the barotropic equations, such as the Coriolis force and the Laplacian horizontal viscosity with a constant viscosity
coeﬃcient may be split into the basic and linear tidal components naturally. The bottom friction term (τ btm ) is non-linear
and should be treated carefully. The bottom friction term (see Section 7.3.6 for details) is expressed asτ btm = −ρ0 CD |u|Tθ u,
where CD is a drag coeﬃcient and Tθ is a matrix representing horizontal veering with an angle of θ,()
cos θ − sin θTθ =.
sin θ cos θ(6.70)(6.71)

Sum of the two components are used for the scalar part (|u|) and only the vector part is decomposed:τbbtm = −ρ0 CD |ub + ult |Tθ ub,(6.72)

τltbtm = −ρ0 CD |ub + ult |Tθ ult .

(6.73)

Similarly, when the horizontal viscosity parameterization is non-linear as in the Smagorinsky scheme, the coeﬃcientpart (e.g., νH in the equations below) is calculated by using the full velocity and the decomposed fields are applied to theviscosity operator. When the horizontal tension DT and shear DS of the velocity field,∂ ( u )∂ ( v )− hµ,h µ ∂ µ hψhψ ∂ψ hµ∂ ( u )∂ ( v )DS = hµ+ hψ,hψ ∂ψ hµh µ ∂ µ hψDT = hψ– 59 –(6.74)(6.75)

6.5 Inclusion of tidal eﬀect
is used to calculate horizontal viscosity as))
1 ∂ ( 21 ∂ ( 2hψ νH DT + 2
hµ ν H DS ,2hψ h µ ∂ µhµ hψ ∂ψ())
1 ∂
1 ∂ ( 2Fvhorz = 2hψ2 νH DS − 2hµ νH DT ,hψ h µ ∂ µhµ hψ ∂ψ
Fuhorz =(6.76)(6.77)

where νH is the horizontal viscosity coeﬃcient, ub and ult is used for DT and DS to calculate Fhorz
and Fhorz
by using
ltb(6.76) and (6.77).
By formulation, Fhorz , τ btm , and X′ are computed at the baroclinic time level of n and kept fixed during the integrationof barotropic submodel. However, when the barotropic field varies significantly during the integration, which is expected
when tidal forcing is included, Fhorz and τ btm should be allowed to vary with the evolution of barotropic velocity field. In
this case, Fhorz
and τltbtm in (6.66) may be replaced by the viscosity parameterization suitable to the tidal velocity fields
ltbtm(represented by Fhorz
tide and τtide , respectively). Then, (6.66) and (6.68) becomes,
τtide
∂Ult
+ f k × Ult = −g(η + H)∇(αη lt − βη eq ) + Fhorz,
tide +∂tρ0(6.78)

τ∂Ub+ f k × Ub = −g(η + H)∇η b + X − Fhorz− lt .
lt∂t
ρ0(6.79)

btm

btm

A sequence of operations explained in Section 6.3.1 is applied to both basic and tidal components. But now we use
surface height at time level t = t n as the height of column applied to the pressure gradient term during the barotropic
subcycle. The weighted average of the barotropic fields is written as follows:⟨⟨⟩⟩n+ 211⟨Ult ⟩n+1 = U0lt − ∆t cl f k × ⟨⟨Ult ⟩⟩n+ 2 − ∆t cl g(η n + H) ∇(αη lt − βη eq )1

n+ 2
+ ∆t cl ⟨⟨Fhorz+
tide ⟩⟩

1
btm n+ 12∆t cl ⟨⟨τtide⟩⟩ ,ρ0(6.80)(⟨⟨⟩⟩n+ 12τltbtm ) n1+ ∆t cl X − Fhorz−.
⟨Ub ⟩n+1 = U0b − ∆t cl f k × ⟨⟨Ub ⟩⟩n+ 2 − ∆t cl g(η n + H) ∇η blt
ρ0(6.81)

These are summed and then combined with the baroclinic part (6.39) and (6.40) to give′(⟨u⟩n+1 + u

k− 12

z

− u′ )∆z n+11 − un 1 ∆z n 1
k− 2

∆t cl

k− 2

k− 2

1

1

1

z

n+1
n+ 2
n+1
= − f k × ⟨⟨u⟩⟩n+ 2 ∆zk−
+ f k × [u]n+ 2 ∆zk−
1 − f k × [uk− 1 ∆z k− 1 ]12

n+1− g∆zk−1
2

ηn + H⟨⟨η n+1 + H2∇(η b + αη lt − βη eq )1n+1
+ ∆zk−
1
2

2

n+ 2 − (Fhorz ) n⟨⟨Fhorztide ⟩⟩lt

η n+1 + H

2⟩⟩n+ 211

n+1+ ∆zk−
1
2

btm ⟩⟩n+ 2 − (τ btm ) n⟨⟨τtidelt

ρ0 (η n+1 + H)n+1 n
+ ∆zk−
1F .
2(6.82)

There is no strong restriction on the form of the viscosity parameterization applied to the tidal velocity fields. However,by using a common form for both baroclinic and barotropic fields, depth integrated viscosity terms in (6.82) would cancel
when barotropic and baroclinic mode are synchronous. In MRI.COM, the bottom friction for the linear tidal component
is calculated as follows:
Ub + Ult
Ult
btm
τtide
= −ρ0 CDTθ.
(6.83)
η+H
η+H
Horizontal viscosity parameterization is the Laplacian operator with the viscosity coeﬃcient fixed in time.
The solution method to predict the two components of the barotropic mode is schematically illustrated by Figure 6.4.
On starting the barotropic submodel, the total barotropic mode is split into the basic and the linear tidal components (Ub ,η b , Ult , and η lt ) and the time evolution is calculated separately. A simple sum of the two modes is returned to the mainpart of OGCM. For the next step, the total and the linear tidal components are retained. The basic component is calculated
by subtracting the tidal part from the total.

– 60 –

Chapter 6

Equations of motion (barotropic component)
Eq. of Motion, Eq. of Continuityu

Baroclinic Eq.

uʼ

step N

step N+1

Barotropic Eq.

u
η
U
η

Ub
ηb

Basic Eq.

Ult
ηlt

Linear tidal Eq.

Ubʼ
ηbʼ

uʼ
ηʼ
Uʼ
ηʼ

Ultʼ
ηltʼ

Figure6.4 Schematic figure of the separating basic and linear tidal component of the barotropic mode

6.5.3 Nesting experiment with tide
The tide scheme can work together with the nesting schemes in MRI.COM. This subsection describes briefly how the tidescheme works with on-line nesting and oﬀ-line one-way nesting methods. See Chapter 18 for the general explanation of
the nesting schemes.
For on-line nesting, the tide models communicate each other between the parent low-resolution model and the childhigh-resolution model in order to predict tides at the next step in the same manner as the barotropic model. The lateralboundaries of the child model receive Ult and η lt from the parent model every steps of the barotropic mode in the bothcases of one-way and two-way nesting. On the other hand, in the case of two-way nesting with the "replace" configuration,
the internal region of the parent model receives them from the child model.
For oﬀ-line one-way nesting, the communication process of the tide model is also the same as that of the barotropic
model. The parent model saves Ult and η lt on bands corresponding to the lateral boundaries of the child model. The
child model reads the files and interpolates them spatially and temporally to the lateral boundaries every barotropic steps.
It should be noted that other lateral boundary data output by the parent model, such as three-dimensional velocity, sea
surface height and x and y transports, also includes the linear tidal component. That is, the parent model saves u, Ub + Ult
and η b + η lt .
As a special function for oﬀ-line one-way nesting, the child model can be executed under tidal forcing even when theparent model does not include tides. That is, users can run the parent model without TIDE, while the child model with
it. In this case, in advance, users have to create files for the child model boundaries of Ult and η lt in the same format as
the parent model output. The MXE (see Section 21.6) package oﬀers a tool for it based on the Matsumoto et al. (2000)’s
dataset in the directory prep/nest/offline/. In addition, boundary files output by the parent model do not include thetidal component, which is diﬀerent from the case that the parent model includes tides. Users have to specify namelist tohandle this diﬀerence (l_parent_include_tide = .false. in nml_submodel_tide).

6.6 Usage
Behavior of the barotropic model at run time is specified by the three namelist blocks shown on Tables 6.2
through 6.4. For the initial condition of the model, restart files must be prepared for five variables: sea surface
height, X- and Y-ward barotropic transports, and X- and Y-ward transports due to SSH diﬀusion (unnecessary if
l_global_local_cnsv_ssh = .false.).
In addition, following model options are available.
TIDE: Tide producing forcing is activatedSpecify parameters of the tide model by namelist nml_tide_model, the initial condition by nml_tide_run, andforced tidal constituents by nml_tide_forcing. If a specific state is used for the initial condition, three restart
files must be prepared. See Section 6.5 for details.
SLP: Sea surface is elevated/depressed according to surface atmospheric pressure
Give atmospheric sea-level pressure, pa in (6.1) and (6.2), to the model by namelist nml_force_data (See Table
14.4).
FSVISC: Explicit viscosity is added to the barotropic momentum equation
Specify the horizontal viscosity coeﬃcient by namelist nml_barotropic_visc_horz. See Section 6.2 for thetime discretization method.
– 61 –

6.6

Usage

See docs/README.Namelist for namelist details.
Table6.2 Namelist nml_barotropic_model for the specifying basic features of the barotropic model (see Section 6.6)

variable
dt_barotropic_sec
l_global_local_cnsv_ssh

units
sec
logicaldescription
time step interval
how to predict sea surface height(Section 6.3.1)

usage
requireddefault = .true.

Table6.3 Namelist nml_barotropic_run for starting the barotropic model (see Section 6.6)

variablel_rst_barotropic_inunits
logicall_rst_barotropic_dflx_in

logicaldescription
read initial restart for sea surface
and transport or notread initial restart for diﬀusive
flux of SSH or not (Section 6.4)

usage
default = l_rst_in
default = l_rst_barotropic_in,
valid only whenl_global_local_cnsv_ssh = .true.

Table6.4 Namelist nml_barotropic_diff for specifying SSH diﬀusion (see Section 6.6)

variablessh_diff_cm2psunits
cm2 s−1

description
diﬀusivity for SSH (Section 6.4)

– 62 –

usage
default = zeroChapter 7

Equations of motion (baroclinic component)
This chapter explains the advection terms (Section 7.1), the pressure gradient terms (Section 7.2), and the viscosity terms(Section 7.3) of the hydrostatic momentum equation. The basic equations (2.33) and (2.34) in Chapter 2 are re-written:}()∂(z s hψ uu) ∂(z s hµ vu)∂hµ∂hψ∂(z s ṡu)
v
+
+
+ zs
u−
v − zs f v∂µ∂ψ∂shµ hψ ∂ψ
∂µ
∫ η
′ ∂z
)
1 ∂(pa + g ρ0 η)
1 ∂ (
g
ρ
= − zs− zsg
ρ′ dz − z sρ0 h µ∂µρ0 h µ ∂ µρ0 h µ ∂ µz(z ∗ )1
1 ∂ ( νV ∂u )+ z s (∇ · τhorizontal strain )u + z s,ρ0z s ∂s z s ∂s{}()∂(z s hψ uv) ∂(z s hµ vv)∂hψ∂hµ∂(z s v)
1∂(z s ṡv)
u
+
+
+
+ zs
v−
u + zs f u∂th µ hψ∂µ∂ψ∂sh µ hψ ∂ µ∂ψ∫η
)1 ∂(pa + g ρ0 η)
1 ∂ (g ρ′ ∂z= − zs− zsg
ρ′ dz − z sρ 0 hψ∂ψρ0 hψ ∂ψ
ρ0 hψ ∂ψz(z ∗ )1
1 ∂ ( νV ∂v )+ z s (∇ · τhorizontal strain )v, +z s.
ρ0z s ∂s z s ∂s

1∂(z s u)+∂th µ hψ

{

(7.1)

(7.2)

One of the unique characteristics of MRI.COM’s momentum advection terms is that there are oblique exchanges ofmomentum between U-cells that share only a corner. This scheme enables the flow field around and over the bottom
topography to be naturally expressed. Furthermore, quasi-enstrophies, (∂u/∂ y) 2 and (∂v/∂ x) 2 , for the U-cells awayfrom land are conserved in calculating the momentum advection for horizontally non-divergent flows. The description of
momentum advection in Section 7.1 is based on Ishizaki and Motoi (1999).
The discrete expressions for the viscosity terms in the momentum equations are based on generalized orthogonal
coordinates. A harmonic operator is used as the default assuming a no-slip condition on the land-sea boundaries. A
biharmonic operator (VISBIHARM option) and a parameterization of viscosity as a function of deformation rate (SMAGOR
option) may also be used.

7.1 Advection terms
Chapter 5 demonstrated that the mass fluxes used for calculating momentum advection are identical to those for the masscontinuity of the U-cell, and that they are obtained by an averaging operation (5.21) of those for the T-cell mass continuity(5.12) to (5.18). This is the preliminaries for constructing the general mass flux form over an arbitrary bottom and coastal
topography. Its vertical part can express diagonally upward mass fluxes over bottom relief and its horizontal part can
express horizontally diagonal mass fluxes along coast lines (Ishizaki and Motoi, 1999).
Here we explain how to obtain the mass fluxes used in the momentum advection and how to get the finite diﬀerence
expression of the advection terms.
The horizontal subscript indices of variables are integers for the T-point (i, j), and therefore, (i + 12 , j + 12 ) for the U-point.
In the vertical direction, integer k is used for the level of the vertical mass fluxes and the level for the T- and U-points a
half vertical grid size lower is expressed by k + 12 (Figure 3.3(a)).

– 63 –

7.1

Advection terms

7.1.1 Vertical mass fluxes and its momentum advection
According to the definition (5.12) and (5.21) in Chapter 5, the vertical mass flux at the upper surface, level k, of the U-cellU(i + 12 , j + 12 , k + 12 ), W i+ 12 , j+ 12 ,k , is defined by surrounding W T as
U

W i+ 12 , j+ 12 ,k =Wi,Tj,kNi, j,k+ 1

+

2

T
Wi+1,
j,k

Ni+1, j,k+ 1+

2

Wi,Tj+1,k+Ni, j+1,k+ 1
2

T
Wi+1,
j+1,k

Ni+1, j+1,k+ 1

,

(7.3)

2

where Ni, j,k+ 1 is the number of sea grid cells around the T-point Ti, j in layer k + 21 .
2

On the other hand, the vertical mass flux at the bottom surface, level k, of the U-cell (i + 12 , j + 12 , k − 12 ), W U 1

i+ 2 , j+ 12 ,kdefined as
U
Wi+
=
1
, j+ 1 ,k2

2

Wi,Tj,kNi, j,k− 1
2

+

T
Wi+1,
j,k

Ni+1, j,k− 1
2+

Wi,Tj+1,k+Ni, j+1,k− 1

2

T
Wi+1,
j+1,k

Ni+1, j+1,k− 1

.

is

(7.4)

2

U

Though W T are continuous at the boundary of vertically adjacent T-cells, W i+ 21 , j+ 12 ,k and W U 1

i+ 2 , j+ 21 ,kseem to bediscontinuous at the boundary when N are vertically diﬀerent, for example, Ni, j,k+ 1 < Ni, j,k− 1 , over the bottom relief.
22
However, this apparent discrepancy can be consistently interpreted by introducing diagonally upward or downward mass
fluxes as shown below.
a. One-dimensional variation of bottom relief

We first consider a case in which the bottom depth varies in one direction like a staircase and a barotropic current flows
from the left to the right over the topography. Figure 7.1(a) indicates the mass continuity for T-cells to conserve barotropy
in shallow regions (the U-points are just intermediate between T-points) . Figure 7.1(b) depicts the mass continuity for
U-cells, derived from those for adjacent T-cells. Except for fluxes just on the bottom slope, each flux is obtained as a meanvalue of neighboring fluxes for T-cells. Just on the bottom slope, we must introduce a flux that flows along the slope toensure mass continuity. The lowermost U-cells at the slope have nonzero vertical flux at the bottom. The barotropy of theflow and the distribution of vertical velocity are thereby kept reasonable for U-cell fluxes.
(a)

(b)

U, V

U, V

U, V

U, V

3

3

4.5

9

3

3

U, V

U, V

U, V

3

3

3.75

6.75

3

3

4.5

1.5

0.75

4.5
3

3

U, V
9

3
2.25

3.75
1.5

3

3

3

1.5

Figure7.1 (a) Two-dimensional mass fluxes for T-cells on a stair-like topography. (b) Two-dimensional mass fluxes
for U-cells on the same topography.

b. Two-dimensional variation of bottom depth

The diagonally upward or downward mass fluxes introduced in the previous simple case are generalized for flows overbottom topography that varies two-dimensionally. For simplicity, we consider a two-layer case without losing generality.
First, we consider three examples of bottom relief, and then generalize the results.
■Example 1 Consider a case in which all cells are sea cells in the upper and lower layers except for cell d in the lower
layer (cell dl ) (Figure 7.2). We use suﬃxes l and u to designate the lower and the upper layer. The central T-point andT-cell are represented by A. The vertical mass flux W T should be continuous at the interface between cells Al and Au ,though the area of cell Al (3/4 measured in grid area units) diﬀers from that of Au (1 unit). Let us consider how this T-cellmass flux W T should be distributed to the mass flux W U of neighboring cells represented by a, b, c, and d. In the lower– 64 –

Chapter 7

Equations of motion (baroclinic component)

layer, WT is shared by three cells, al , bl , and cl , so the contribution of W T to each W U is W T /3, but in the upper layer,
it is W T /4 because part of WT should also be shared by W U at cell du . Here W U at the bottom of cell du is no longer
zero. Therefore, W T /4 of the W T /3 shared by each of the three lower sea grid cells al , bl , and cl is purely vertical, andthe remaining W T /12 (= W T /3 − W T /4) flows to cell du through the interface. Gathering these diagonal fluxes fromthe lower three cells, the total amount entering cell du is certainly W T /4 (=W T /12 × 3). The advected momentum value
should be the mean of those at the starting and ending cells of the flux, if the centered diﬀerence scheme is used, which isnecessary to conserve the total kinetic energy.

UPPER

LOWER

du

cu
䃁

cl

dl
䃁

䃁

T

W
Al

Au
䃁

䃁

au

bu

䃁
T

al

bl

T

1/4W

T

䃁

䃁

䃁

T

1/3W

T

1/3W 䋭 1/4W = 1/12W

Figure7.2 First example of land-sea patterns, in which all four upper cells are sea cells, with three sea cells and one
land cell in the lower layer.

■Example 2

Next, consider an example in which only bl is a sea cell in the lower layer, and all four cells are sea cells in
the upper layer (Figure 7.3). In the lower layer, W T is shared only by bl but in the upper layer, it is shared by all four cells.
Therefore, W T /4 of W T at cell bl is carried vertically upward and the remaining 3W T /4 is distributed to the other threecells in the upper layer (au , cu , and du ), each receiving W T /4.
■Example 3 A third example holds that the upper layer also has land area. In this example, cells cl , dl , and du are land
cells and the others are sea cells (Figure 7.4). In the lower layer, W T is shared by two cells (al and bl ) while it is shared bythree cells (au , bu , and cu ) in the upper layer. Therefore, from each of al and bl , W T /3 of W T /2 goes vertically upward
and the remaining W T /6 (= W T /2 − W T /3) goes diagonally upward to cell cu with a total amount of W T /3 (= W T /6 × 2).
c. Generalization

The relationship between the land-sea distribution and the vertically and diagonally upward fluxes stated above is generalized for an arbitrary land-sea distribution. Assume cell dl is a land but cell du is a sea cell and consider the diagonally
upward fluxes coming to cell du . We take Nl as the number of sea cells around point A in the lower layer and Nu as thenumber in the upper layer (1 ≤ Nl < Nu ≤ 4). Each cell in the lower layer carries W T /Nl , and W T /Nu of it goes verticallyupward. The remaining
W T /Nl − W T /Nu = W T (Nu − Nl )/(Nl Nu )(7.5)
should be distributed as diagonally upward fluxes to sea cells in the upper layer at which the lower layer is land. The
number of such upper sea cells is Nu − Nl including cell du . Thus, each diagonally upward flux coming to cell du isW T (Nu − Nl )/(Nl Nu ) × 1/(Nu − Nl ) = W T /(Nl Nu ).

(7.6)

The number of such fluxes coming to the cell du is Nl , so their total is
W T /(Nl Nu ) × Nl = W T /Nu .
– 65 –

(7.7)

7.1

UPPER

LOWER

du

cu
䃁

Advection terms

cl

䃁

dl
T

Au

W

Au
䃁

䃁

䃁

al

au

bu

䃁

bl

T

1/4W

T

䃁
T

T

W

T

1/3( W 䋭 1/4W ) = 1/4W

Figure7.3 Second example of land-sea patterns, in which all four upper cells are sea cells, with one sea cell and threeland cells in the lower layer.

UPPER

LOWER

cu

dl

cl

du
䃁

T

Au
䃁

䃁

bu

T

Al

䃁

䃁

al

bl

au

䃁

W

T

T

1/3W

䃁 1/2W
T

T

1/2W 䋭 1/3W = 1/6W

Figure7.4 Third example of land-sea patterns, in which one of the upper cells is a land cell, with two land and two
sea cells in the lower layer.

Based on these discussions we understand the diﬀerence between (7.3) and (7.4).
We regard the name of each cell such as al also as a land-sea index. If we assume that al = 1(0) when cell al is a sea(land) cell, then the diagonally upward mass flux and momentum flux coming from cell al to cell du are
al W T /(Nl Nu )and al W T (ual + udu )/(2Nl Nu ),(7.8)

where ual and udu are the velocity at cells al and du , respectively. Purely vertical mass flux and momentum flux from cellal to cell au are expressed as
al W T /Nu and al W T (ual + uau )/(2Nu ),(7.9)
respectively, where uau is the velocity at cell au . Similar formulations apply to cells bl and cl .
– 66 –

Chapter 7

Equations of motion (baroclinic component)

Mass and momentum fluxes for W T at other T-points around cell du should be calculated similarly to complete verticallyand diagonally upward momentum advections around cell du . When Nu = Nl , diagonally upward fluxes need not be
considered and only vertical fluxes (7.9) apply.
To summarize, the discrete expression for the vertical flux of zonal momentum that is transported into a U-cell at(i + 12 , j + 12 , k − 12 ) through its bottom, FL i+ 1 , j+ 1 ,k (u), is given as follows:
2

2

T
T
 T

Wi+1,
Wi,Tj+1,k
Wi+1,)  Wi, j,k1(j,kj+1,k ui+ 1 , j+ 1 ,k− 1 + ui+ 1 , j+ 1 ,k+ 1 × 
+
+
+
2
2
2
2
2
2
2
Ni+1, j,k− 1
Ni, j+1,k− 1
Ni+1, j+1,k− 1  Ni, j,k− 1
2
2
2
2 
(
)
ei+ 1 , j+ 1 ,k− 1 1 − ei+ 1 , j+ 1 ,k+ 1
2
2
2
2
2
2
+
×
2
[{
(
)
(
)
ei− 1 , j− 1 ,k+ 1 ui+ 1 , j+ 1 ,k− 1 + ui− 1 , j− 1 ,k+ 1 + ei+ 1 , j− 1 ,k+ 1 ui+ 1 , j+ 1 ,k− 1 + ui+ 1 , j− 1 ,k+ 1

FL i+ 1 , j+ 1 ,k (u) =
2

2

2

2

2

2

2

2

2

2

2

2

(
+ ei− 1 , j+ 1 ,k+ 1 ui+ 1 , j+ 1 ,k− 1 + ui− 1 , j+ 1 ,k+ 1

)}

2

2

2

2

2

2

2

(7.10)

2

Wi,Tj,k×

Ni, j,k+ 1 Ni, j,k− 1
2
2
(
)
(
)
+ ei+ 1 , j− 1 ,k+ 1 ui+ 1 , j+ 1 ,k− 1 + ui+ 1 , j− 1 ,k+ 1 + ei+ 3 , j− 1 ,k+ 1 ui+ 1 , j+ 1 ,k− 1 + ui+ 3 , j− 1 ,k+ 1
2

2

2

2

2

2

2

2

2

{

2

2

2

2

2

2

2

2

2

(

+ ei+ 3 , j+ 1 ,k+ 1 ui+ 1 , j+ 1 ,k− 1 + ui+ 3 , j+ 1 ,k+ 1
2

2

2

2

{

2

2

2

2

2

(

+ ei− 1 , j+ 1 ,k+ 1 ui+ 1 , j+ 1 ,k− 1 + ui− 1 , j+ 1 ,k+ 1
2

2

2

2

2

2

2

2

)}

2

)

2

(
+ ei+ 1 , j+ 3 ,k+ 1 ui+ 1 , j+ 1 ,k− 1 + ui+ 1 , j+ 3 ,k+ 1

)}

2

2

2

2

2

2

2

2

T
Wi+1,
j,k

×

Ni+1, j,k+ 1 Ni+1, j,k− 122()+ ei− 1 , j+ 3 ,k+ 1 ui+ 1 , j+ 1 ,k− 1 + ui− 1 , j+ 3 ,k+ 1
2

2

2

2

2

2

2

2

2

Wi,Tj+1,k×

Ni, j+1,k+ 1 Ni, j+1,k− 1
2
2
(
)
(
)
+ ei+ 3 , j+ 1 ,k+ 1 ui+ 1 , j+ 1 ,k− 1 + ui+ 3 , j+ 1 ,k+ 1 + ei+ 1 , j+ 3 ,k+ 1 ui+ 1 , j+ 1 ,k− 1 + ui+ 1 , j+ 3 ,k+ 1
2

2

2

2

2

2

2

2

2

{

2

2

2

2

2

2

2

2

2

(

+ ei+ 3 , j+ 3 ,k+ 1 ui+ 1 , j+ 1 ,k− 1 + ui+ 3 , j+ 3 ,k+ 1
2

2

2

2

2

2

2

2

)}

2

2

×

2

2

2

2

2

]

T
Wi+1,
j+1,k

Ni+1, j+1,k+ 1 Ni+1, j+1,k− 1
2

2

2

2

,

2

where ei+ 1 , j+ 1 ,k− 1 is the land-sea index for a U-cell (unity for sea and zero for land). Note that diagonally upward/downward2
2
2
momentum fluxes may only occur for a U-cell with the sea floor underneath. The momentum fluxes entering and leaving
the cells at the lower level (k = k + 21 ) are individually added or removed from the cell when calculating them.

7.1.2 Horizontal mass flux and its momentum advectiona. Horizontal mass fluxes

We next consider the generalization of the U-cell horizontal mass fluxes for arbitrary coast lines, in order to derive thehorizontal momentum advection. To do this, we start with the generalization of the T-cell mass continuity (5.12)-(5.18).
Assuming that ei+ 1 , j+ 1 is a land-sea index (unity for sea and zero for land) for U-cell (i + 12 , j + 12 ), the general formulae
2
2
for U T 1 and V T 1 are given asi+ 2 , ji, j+ 2

1∗TUi+
= (ei+ 1 , j− 1 + ei+ 1 , j+ 1 )ui+
1 ∆yi+ 1 , j ∆z
1
,
j
2
2
2
2
2
2
2
2, j
and
1
Vi,Tj+ 1 = (ei− 1 , j+ 1 + ei+ 1 , j+ 1 )vi,∗ j+ 1 ∆x i, j+ 1 ∆z,
2
2
2
2
2
2
2
2
where, u∗ 1 and v ∗i+ 2 , ji, j+ 12

(7.11)

are the zonal and meridional velocity at the eastern and northern side boundary of a T-cell (i, j)

as computed by (5.14), (5.16), and (5.18). Here we neglect the vertical subscript (k − 21 ).
Substituting these formulae into the T-cell mass continuity (5.12), the X (zonal) component of the mass continuity for– 67 –

7.1
U-cell (i + 12 , j + 12 ) (5.21), XMCU 1

i+ 2 , j+ 12

XMCU= ei+ 1 , j+ 1
i+ 1 , j+ 1
2

2

2

{ 1

, multiplied by its own land-sea signature ei+ 1 , j+ 1 , is expressed as2∆yi+ 1 , j+ 1
2

2

2

2

Advection terms

2

∆z

∗
∗
[(e 1 1 + ei− 1 , j+ 1 )ui−
− (ei+ 1 , j− 1 + ei+ 1 , j+ 1 )ui+
1
1 ]
2
2
2
2
2
2
Ni, j i− 2 , j− 2
2, j
2, j
1
∗
∗
+
[(e 1 1 + ei+ 1 , j+ 1 )ui+
− (ei+ 3 , j− 1 + ei+ 3 , j+ 1 )ui+
1
3 ]
2
2
2
2
2
2
Ni+1, j i+ 2 , j− 2
2, j
2, j
1
∗
∗
+
[(e 1 1 + ei− 1 , j+ 3 )ui−
− (ei+ 1 , j+ 1 + ei+ 1 , j+ 3 )ui+
]
1
1
2
2
2
2
2
2
Ni, j+1 i− 2 , j+ 2
2 , j+1
2 , j+1}
1∗∗[(ei+ 1 , j+ 1 + ei+ 1 , j+ 3 )ui+
− (ei+ 3 , j+ 1 + ei+ 3 , j+ 3 )ui+
]
+
1
3
,
j+1
,
j+1
2
2
2
2
2
2
2
2
Ni+1, j+122
∆yi+ 1 , j+ 1
{
[
1
2
2
∗
= ei+ 1 , j+ 1
∆z ×
(e 1 1 + ei− 1 , j+ 1 )ui−
1
2
2
2
2
2
Ni, j i− 2 , j− 2
2, j
( 1
1 )
∗
+ −
+
(ei+ 1 , j− 1 + ei+ 1 , j+ 1 )ui+
1
2
2
2
2
Ni, j Ni+1, j
2, j
]
1
∗
(e 3 1 + ei+ 3 , j+ 1 )ui+−322
Ni+1, j i+ 2 , j− 2
2, j
[ 1
∗
+
(e 1 1 + ei− 1 , j+ 3 )ui−
1
2
2
Ni, j+1 i− 2 , j+ 2
2 , j+1
(
1
1 )
∗
+ −
(e 1 1 + ei+ 1 , j+ 3 )ui++
1
2
2
Ni, j+1 Ni+1, j+1 i+ 2 , j+ 2
2 , j+1]}
1∗−(e 3 1 + ei+ 3 , j+ 3 )ui+.
322
Ni+1, j+1 i+ 2 , j+ 2
2 , j+1

×(7.12)

Here, recalling
Ni, j = ei− 1 , j− 1 + ei+ 1 , j− 1 + ei− 1 , j+ 1 + ei+ 1 , j+ 1 ,
2

2

2

2

2

2

2

2

(7.13)

we have,
(

−

1
1 )
+
(ei+ 1 , j− 1 + ei+ 1 , j+ 1 )2
2
2
2
Ni, j Ni+1, j1
1=(e 1 1 + ei− 1 , j+ 1 ) −(e 3 1 + ei+ 3 , j+ 1 )2
2
2
2
Ni, j i− 2 , j− 2
Ni+1, j i+ 2 , j− 2
(7.14)

and(−1
Ni, j+1+

1
Ni+1, j+1)

(ei+ 1 , j+ 1 + ei+ 1 , j+ 3 )
2

2

=

2

2

1
1
(e 1 1 + ei− 1 , j+ 3 ) −(e 3 1 + ei+ 3 , j+ 3 ).
2
2
2
2
Ni, j+1 i− 2 , j+ 2
Ni+1, j+1 i+ 2 , j+ 2
(7.15)– 68 –

Chapter 7

Equations of motion (baroclinic component)

Thus, based on (5.24) and (5.25),∆yi+ 1 , j+ 1

XMCU
= ei+ 1 , j+ 1
i+ 1 , j+ 1
2

2

2

−
+

2

2

2

2

1

∆z

{[ 1
Ni, j

∗
∗
(ei− 1 , j− 1 + ei− 1 , j+ 1 )(ui−
+ ui+
1
1 )
,j
,j
2

2

2

2

∗
∗
(ei+ 3 , j− 1 + ei+ 3 , j+ 1 )(ui+
+ ui+
1
3 )
,j
,j

Ni+1, j
[ 1

2

2

2

2

2

2

2

]

2

∗
∗
(ei− 1 , j+ 1 + ei− 1 , j+ 3 )(ui−
+ ui+
)
1
1
2
2
2
2
2 , j+1
2 , j+1

Ni, j+1]}
1∗∗(ei+ 3 , j+ 1 + ei+ 3 , j+ 3 )(ui+
−
+
u
)
1
3
i+ 2 , j+1
2
2
22
Ni+1, j+1
2 , j+1[( 1)
1
Ui,Uj +
UU
= ei+ 1 , j+ 1 ei− 1 , j+ 1
2
2
2
2 N
Ni, j+1 i, j+1
i, j
( 1
)
1
1 U
U
U
− ei+ 3 , j+ 1
Ui+1,
Ui+1,
U
j +
j+1 +ei− 12 , j− 122
2 N
Ni+1, j+1
Ni, j i, j
i+1, j]1
1
1
UU− ei+ 3 , j+ 3
Ui+1,
Ui,Uj+1 − ei+ 3 , j− 1
Ui+1,
j+1 + ei− 12 , j+ 32
j .
2
2 N
2
2
Ni, j+1
Ni+1, ji+1, j+1(7.16)

Adding the Y (meridional) component, YMC (the expression is omitted here), to the above formula, we obtain thehorizontal part of the U-cell mass continuity, HMC, as follows:= XMCU+ YMCU
HMCU
i+ 1 , j+ 1
i+ 1 , j+ 1
i+ 1 , j+ 1
2

2

2

2

2

2

= ei+ 1 , j+ 1
2
2
[
( 1
)
( 1
)
1
1
U
U
× ei− 1 , j+ 1
Ui,Uj +
Ui,Uj+1 − ei+ 3 , j+ 1
Ui+1,
+
Ui+1,
j
j+1
2
2 N
2
2 N
Ni, j+1
Ni+1, j+1i, ji+1, j
)
)
( 1
(
1
1
1
U
U
Vi,Uj +
Vi+1,
− ei+ 1 , j+ 3
Vi,Uj+1 +Vi+1,+ ei+ 1 , j− 1
j
j+1
2
2 N
2
2 N
Ni+1, jNi+1, j+1
i, ji, j+11
1UU+ ei− 1 , j− 1
(Ui,Uj + Vi,Uj ) − ei+ 3 , j+ 3
(Ui+1,
j+1 + Vi+1, j+1 )
2
2 N
2
2 N
i, j
i+1, j+1]
1
1
UU+ ei− 1 , j+ 3
(Ui,Uj+1 − Vi,Uj+1 ) − ei+ 3 , j− 1
(Ui+1,
j − Vi+1, j ) .
2
2 N
2
2 N
i, j+1
i+1, j(7.17)

Assuming mass fluxes ME, MN, MNE, MSE as follows:
MEi, j+ 1 = ei+ 1 , j+ 1 ei− 1 , j+ 1

( 1)
Ui,Uj+1 ,Ni, j+1)
1U
Vi,Uj +
Vi+1,,
MNi+ 1 , j = ei+ 1 , j+ 1 ei+ 1 , j− 1
j
2
2
2
2 N
Ni+1, j2
i, j
1
MNEi, j = ei+ 1 , j+ 1 ei− 1 , j− 1
(Ui,Uj + Vi,Uj ),
2
2
2
2 N
i, j
1
(Ui,Uj − Vi,Uj ),MSEi, j = ei− 1 , j+ 1 ei+ 1 , j− 1
2
2
2
2 N
i, j
2

2

2

2

2

Ni, j
( 1

1

Ui,Uj +

(7.18)then,
HMCU=MEi, j+ 1 − MEi+1, j+ 1 + MNi+ 1 , j − MNi+ 1 , j+1
i+ 1 , j+ 1
2

2

2

2

2

2

+ MNEi, j − MNEi+1, j+1 + MSEi, j+1 − MSEi+1, j .
Here, ME and MN are axis-parallel mass fluxes, and MNE and MSE are horizontally diagonal ones (Figure 7.5).

– 69 –(7.19)

7.1

MSE i, j+1

MN i+1/2, j+1

MNE i+1, j+1

ME i, j+1/2

X

ME i+1, j+1/2

MN i+1/2, j

MSE i+1, j

Advection termsi, j
MNE i, j

Figure7.5 Distribution of generalized mass fluxes for U-cell (i + 1/2, j + 1/2)

If we derive the formula for the standard case from (7.17) (all of N are 4),
1 [1 U
1 U
U(U + Ui,Uj+1 ) − (Ui+1,
j + Ui+1, j+1 )2 2 i, j
2
]
1 U
1
U
+ (Vi,Uj + Vi,Uj+1 ) − (Vi+1,
j + Vi+1, j+1 )2
2
1 U
1 [1 UUU+(U + Vi, j ) − (Ui+1, j+1 + Vi+1,
j+1 )2 2 i, j
2
]
1
1 U
U
+ (Ui,Uj+1 − Vi,Uj+1 ) − (Ui+1,
j − Vi+1, j ) .
2
2

HMCU
=
i+ 1 , j+ 1
2

2

(7.20)

This expression means that the horizontal mass flux convergence is a mean of those of the axis-parallel mass fluxes(5.23) and of the diagonal ones (5.28). However, their weighting factors α and β are both 1/2 in the present case, while(α, β) = (2/3, 1/3) for the generalized Arakawa scheme, which conserves quasi-enstrophy such as (δv/δx) 2 and (δu/δy) 2
in a horizontally non-divergent flow.
b. Horizontal momentum advection

For the standard case away from land, we have the freedom to choose weights (α : β) for averaging the convergences of
the axis-parallel and the horizontally diagonal mass fluxes, as long as α + β = 1, as seen in (7.20). In MRI.COM α = 2/3
and β = 1/3 are chosen for the standard case so that the momentum advection terms lead to the generalized Arakawa
scheme. In this case the zonal momentum advection term is expressed by convergence of the horizontal momentum fluxes– 70 –

Chapter 7

Equations of motion (baroclinic component)

Table7.1 Definition of a land-sea index, the index identifying each case (column A), the coeﬃcient of Ui,Uj in theaxis-parallel mass flux (column B), and the coeﬃcient of Ui,Uj in the horizontally diagonal mass flux (column C), foreight combinations of indices a, b, and c in Figure 7.2b (see the main text). Cell d is assumed to be a sea cell, and themomentum advection by means of Ui,Uj into and from cell d is generalized. Note that Ui,Uj is identically zero for cases
5 and 8 (b = c = 0).

CASE
n
1
2
3
4
5
6
7
8

Land-sea index
a b c
1 1 1
1 1 0
1 0 1
0 1 1
1 0 0
0 1 0
0 0 1
0 0 0

A
abc
ab(1 − c)ab(1 − b)c(1 − a)bc−(1 − a)b(1 − c)
(1 − a)(1 − b)c−

B
Coeﬃcient of Ui,Uj(axis-parallel)
+
1/3
0
1/3
1/3
0
0
1/2
0

C
Coeﬃcient of Ui,Uj(horizontally-diagonal)
×
1/6
1/3
0
1/3
0
1/2
0
0

as follows:
2 [11(ui− 1 , j+ 1 + ui+ 1 , j+ 1 ) (Ui,Uj + Ui,Uj+1 )
2
2
2
2
3 2
2
1
1 U
U
− (ui+ 1 , j+ 1 + ui+ 3 , j+ 1 ) (Ui+1,
j + Ui+1, j+1 )
2
2
2
2
2
2
1
1
U
+ (ui+ 1 , j− 1 + ui+ 1 , j+ 1 ) (Vi,Uj + Vi+1,
j)
2
2
2
2
2
2
]
1
1
U
− (ui+ 1 , j+ 1 + ui+ 1 , j+ 3 ) (Vi,Uj+1 + Vi+1,
)
j+1
2
2
2
2 2
2
1 [1
1
+
(u 1 1 + ui− 1 , j− 1 ) (Ui,Uj + Vi,Uj )22 2
3 2 i+ 2 , j+ 2
1
1 U
U− (ui+ 1 , j+ 1 + ui+ 3 , j+ 3 ) (Ui+1,
j+1 + Vi+1, j+1 )
2
2
2
2 2
2
1
1
+ (ui+ 1 , j+ 1 + ui− 1 , j+ 3 ) (Ui,Uj+1 − Vi,Uj+1 )
2
2
2
2 2
2
]
1
1 U
U
− (ui+ 1 , j+ 1 + ui+ 3 , j− 1 ) (Ui+1,− Vi+1,
)
j
j
2
2
2
2 2
2

CADi+ 1 , j+ 1 (u) =22(7.21)

This scheme under Arakawa’s B-grid arrangement conserves the quasi-enstrophies ((∂u/∂ y) 2 and (∂v/∂ x) 2 ) in a horizontally non-divergent flow.
To merge the generalized Arakawa scheme for the standard case into the general form of the horizontal mass flux
expressed in Figure 7.5 and related momentum flux, let us examine the axis-parallel and horizontally diagonal mass flux
associated with Ui,Uj , taking topography into account. Look at Figure 7.2b, where letters a, b, c, and d designate theland-sea index and names of U-cells. Cell d is assumed to be a sea cell (d = 1). We analyze two kinds of mass fluxes
associated with Ui,Uj under diﬀerent combinations of a, b, and c (eight cases), as indicated in the first column in Table 7.1.
Column (A) corresponds to an index, which is unity for its own combination and zero for all other combinations. Column(B) lists the coeﬃcient of Ui,Uj in the axis-parallel mass flux of the U-cell mass continuity (7.17). Column (C) indicatesthe coeﬃcient of Ui,Uj in the horizontally diagonal mass flux of (7.17).
The generalized coeﬃcient of Ui,Uj in the axis-parallel mass flux (c1 ) is obtained by summing the product of A and B
over the eight cases. Similarly, the generalized coeﬃcient in the horizontally diagonal mass flux (c2 ) is obtained by the– 71 –

7.1

Advection terms

summing the product of A and C. That is,
c1 =

8
∑

An Bn =

1c(ab − a − b+3)
6

An Cn =

1b(3−a − c).
6

n=1

and
c2 =

8
∑
n=1

(7.22)

Then, the axis-parallel and the horizontally diagonal flux of zonal momentum (u) related with Ui,Uj , multiplied by theland-sea index d, are11d(uc + ud )c1Ui,Uj = (uc + ud ) cd(ab − a − b+3)Ui,Uj ,
2
2
6
and
1
1
d
(ub + ud )c2Ui,Uj = (ub + ud ) bd(3−a − c)Ui,Uj ,2
2
6(7.23)

respectively.
The resultant momentum fluxes are as follows:
1
FEi, j+ 1 (u) = (ui− 1 , j+ 1 + ui+ 1 , j+ 1 )MEi, j+ 1 ,
2
2
2
2
2
2
2
1
FNi+ 1 , j (u) = (ui+ 1 , j− 1 + ui+ 1 , j+ 1 )MNi+ 1 , j ,
2
2
2
2
2
2
2
1
FNEi, j (u) = (ui+ 1 , j+ 1 + ui− 1 , j− 1 )MNEi, j ,
2
2
2
2
2
1
FSEi, j (u) = (ui− 1 , j+ 1 + ui+ 1 , j− 1 )MSEi, j ,
2
2
2
2
2

(7.24)

where1
MEi, j+ 1 = (CXNi, j Ui,Uj + CXSi, j+1 Ui,Uj+1 ),6
2
1U
MNi+ 1 , j = (CYEi, j Vi,Uj + CYWi+1, j Vi+1,
j ),
6
2
1
MNEi, j = CNEi, j (Ui,Uj + Vi,Uj ),6
1
MSEi, j = CSEi, j (Ui,Uj − Vi,Uj ),6(7.25)

andCXNi, j =ei+ 1 , j+ 1 ei− 1 , j+ 1 (ei+ 1 , j− 1 ei− 1 , j− 1 − ei+ 1 , j− 1 − ei− 1 , j− 1 + 3),
2

2

2

2

2

2

2

2

2

2

2

2

CXSi, j =ei+ 1 , j− 1 ei− 1 , j− 1 (ei+ 1 , j+ 1 ei− 1 , j+ 1 − ei+ 1 , j+ 1 − ei− 1 , j+ 1 + 3),
2

2

2

2

2

2

2

2

2

2

2

2

CYEi, j =ei+ 1 , j+ 1 ei+ 1 , j− 1 (ei− 1 , j+ 1 ei− 1 , j− 1 − ei− 1 , j+ 1 − ei− 1 , j− 1 + 3),
2

2

2

2

2

2

2

2

2

2

2

2

CYWi, j =ei− 1 , j+ 1 ei− 1 , j− 1 (ei+ 1 , j+ 1 ei+ 1 , j− 1 − ei+ 1 , j+ 1 − ei+ 1 , j− 1 + 3),
2

2

2

2

2

2

2

2

2

2

2

2

CNEi, j =ei+ 1 , j+ 1 ei− 1 , j− 1 (3 − ei− 1 , j+ 1 − ei+ 1 , j− 1 ),
2

2

2

2

2

2

2

2

CSEi, j =ei− 1 , j+ 1 ei+ 1 , j− 1 (3 − ei+ 1 , j+ 1 − ei− 1 , j− 1 ).
2

2

2

2

2

2

2

(7.26)

2

Finally, convergence of the horizontal momentum fluxes is written as
CADi+ 1 , j+ 1 (u) =FEi, j+ 1 (u) − FEi+1, j+ 1 (u) + FNi+ 1 , j (u) − FNi+ 1 , j+1 (u)
2

2

2

2

2

2

+ FNEi, j (u) − FNEi+1, j+1 (u) + FSEi, j+1 (u) − FSEi+1, j (u).
(7.27){}∂(z s hψ uu)∂(z h vu)
This is the discrete expression for the advection term of the zonal momentum hµ1hψ+ s∂ψµunder the∂µfinite volume method (equations being integrated over a U-cell).
– 72 –

Chapter 7

Equations of motion (baroclinic component)

7.2 Pressure gradient term
In the split-explicit solution method, the pressure gradient term of the horizontal momentum equation is separated into
barotropic (fast) and baroclinic (slow) terms (Eq. 2.58). Its form ispressure perturbation

geopotentialz}|{ z }| {[ ∫ η] g ρ′11′∇s (pa + g ρ0 η) + ∇s g
ρ dz +∇s z .
ρ0
ρ0
ρ0
z(s)
|
{z
} |
{z
}
fast

(7.28)

slow

Here, we use symbol s to indicate z ∗ for brevity, and so ∇s means ∇z ∗ . These three terms correspond to the first, second,
and third terms in the r.h.s. of (7.1) and (7.2). From the perspective of the momentum conservation, the finite diﬀerenceof the pressure gradient terms should be expressed so that the pressure at the interface of adjacent cells is common, giving
only boundary pressures after horizontal integration. This is not diﬃcult for the barotropic mode. The baroclinic mode
should be considered carefully. Among the baroclinic terms in (7.28), the former is called the pressure perturbation term
and the latter is called the geopotential term.
Because sea-floor depth is defined at U-cells in MRI.COM, it is not necessary to consider the horizontal gradient of
the sea floor in calculating the pressure gradient for a bottom U-cell unlike σ-coordinate models and possibly the depthcoordinate models that employ staggered grid arrangements diﬀerent from MRI.COM. This makes the finite diﬀerence
expression simple, and the pressure gradient error may be expected to be small. However, this simplicity does not hold forthe bottom boundary layer (BBL). The treatment of the pressure gradient term for BBL is explained in Chapter 16.

zx

Figure7.6 Illustration of a vertical slice through a set of grid cells in the x-z plane for z ∗ coordinate. The center pointin each cell (•) is a velocity point, and the pressure gradient term is calculated at this point. The cross (×) is a tracer
point, and pressure is calculated here. The triangle (△), whose z ∗ value is that of the adjacent U-point of the bottom
U-cell, is the hydrostatic pressure point which is used to evaluate pressure gradient for the bottom U-cell.

a. Discretization of the pressure perturbation term

Figure 7.6 shows the grid points on the x-z plane relevant to calculating pressure gradient terms. Pressures are definedon T-points. Because the actual vertical integration distances are diﬀerent between adjacent T-points for z ∗ coordinate,
it is appropriate to diﬀerentiate pressures after computing them at T-points rather than vertically integrating the pressure
gradient as in z coordinate. However, the round-oﬀ error due to the diﬀerentiation of vertically integrated quantity would
increase with depth. To avoid this, we add contribution from the integration over one vertical grid to the pressure gradientterm at the upper vertical level, as explained below.
– 73 –

7.2

Pressure gradient term

Introducing pressure perturbation p′, the pressure perturbation term in Eq. (7.28) is expressed as
[ ∫ η
]
1
1
∇s g
ρ′ dz = ∇s p′,
ρ0
ρ
0
z(s)
∫ η
′
where p =g
ρ′ dz.

(7.29)(7.30)z(s)A simple discrete expression would be given as follows (1/ρ0 is omitted):p′i+1, j+1, k− 12′2

i+1, j, k− 122(∇s p )i+ 1 , j+ 1 ,k− 1 = x̂
2

+p′

p′

−

pi,′ j,k− 1 = g
2

k−1
∑

p′

+p′

i+1, j+1, k− 1
2i, j, k− 1
2

2

+p′

−
2

( ρ′ )i, j,l− 1 (dzt)i, j,l− 1 + g( ρ′ )i, j,k− 12(dzt)i, j,k− 1
2

2

2

l=1

i+1, j, k− 12

+p′i, j, k− 122

∆yi+ 1 , j+ 1

2

2

p′

i, j+1, k− 1
2

2

+ ŷ

∆x i+ 1 , j+ 1

2

2

where

i, j+1, k− 1
2,

(7.31)

2.

(7.32)

However, because density perturbation (ρ′) would generally take similar values in adjacent grids, the pressure perturbation(p′) would also take similar values in adjacent grids. The round-oﬀ error caused by diﬀerentiation would increase withthe depth. To avoid this round-oﬀ error, we use another discrete expression given as follows:{}(∇s p′ )i+ 1 , j+ 1 ,k− 1 = (∇s p′ )i+ 1 , j+ 1 ,k− 3 + (∇s p′ )i+ 1 , j+ 1 ,k− 1 − (∇s p′ )i+ 1 , j+ 1 ,k− 3
2

2

2

2

2

2

2

= (∇s p′ )i+ 1 , j+ 1 ,k− 3 + x̂
2

2

2

where
∆pi,′ j,k−1 ≡ pi,′ j,k− 1 − pi,′ j,k− 3 = g( ρ′ )i, j,k− 3
2

2

2

2

,

(7.33)

2(dzt)i, j,k− 1
2

2

2

2

2

i+ 21 , j+ 21′′∆pi+1, j, k−1 +∆pi, j, k−1−∆yi+ 1 , j+ 1

2

2

∆x

2

′
′
∆pi+1,+∆pi,
j+1, k−1
j+1, k−1

+ ŷ

2

′
′
∆pi+1,∆p′+∆pi+1,+∆p′
j+1, k−1
j, k−1− i, j+1, k−12 i, j, k−1
2+ g( ρ′ )i, j,k− 1(dzt)i, j,k− 1
2

2

2

.

(7.34)

b. Discretization of the geopotential term

The geopotential term is also discretized so that a round-oﬀ error is minimized. Considering that the time-dependent partof the actual depth z = z(s) is due to the perturbation (z ′ (s)) from the state of rest (z0 (s)) caused by the sea level variation,
taking diﬀerentiation for z(s) at depth is not quite accurate owing to the numerical round-oﬀ. Then we separate the actual
depth as z(s) = z0 (s) + z ′ (s) and only use z ′ (s) for horizontal gradient of geopotential, that is,
g ρ′
g ρ′
∇s z =∇s z ′ (s).
ρ0ρ0(7.35)

Here, we used the fact that the depth of the constant s-surface is flat at the state of rest, i.e. ∇s z0 (s) = 0. (Again, we cannot
use this simplicity for the bottom boundary layer. See Chapter 16 for details.) Then, the term is diﬀerentiated as[ g ρ′
ρ0∇s z ′=]gρ0

i+ 12 , j+ 12′′′′
ρ′i+1, j +ρ′i, j zi+1,
ρ′i+1, j+1 +ρ′i, j+1 zi+1,
j −z i, j
j+1 −z i, j+1+2∆xi+ 1 , j
2∆xi+ 1 , j+1
{[
2

2

2

]
x̂ +

′
′
′
′
ρ′i, j+1 +ρ′i, j zi,
ρ′i+1, j+1 +ρ′i+1, j zi+1,
j+1 −z i, j
j+1 −zi+1, j+2∆yi, j+ 1
2∆yi+1, j+ 1
[
2

2

2

]}
ŷ .
(7.36)

Here, we follow the expression presented in Section 3.3.1 of the reference manual of GFDL-MOM.
c. The sea-floor grid (except for bottom boundary layer)

Owing to the introduction of the partial cell, the grid width of the bottom U-cell may depend on the horizontal position.
The pressure gradient term in (7.28) is the gradient of pressure on constant s(z ∗ )-surface. Thus, the pressure gradient
force on a partial U-cell is evaluated using the pressure perturbation p′ obtained by integrating to the depth where velocity– 74 –

Chapter 7

Equations of motion (baroclinic component)

is defined on z ∗ coordinate (the central depth of the bottom U-cell). These are the points with triangle (△) symbols in
Figure 3.5.
To calculate the pressure gradient, the actual depth of the vertical level where velocity is defined for the bottom U-cell(s = sU) must be obtained at all four corner T-points of a U-cell. Here, these are determined using the ratio of the half
k=kbtmwidth of a U-cell 12 ∆sU to the width of a T-cell ∆sT :T
TzU
kbtm = zkbtm−1 − ∆zkbtm

∆sU
kbtm
2∆sTkbtm.

(7.37)The depth anomaly used for computing geopotential gradient anomaly may be obtained in the same way:
′T
′T
z ′U
kbtm = z kbtm−1 − ∆z kbtm

∆sU
kbtm
2∆sTkbtm.

(7.38)

The finite diﬀerence expression for horizontal gradient is formally the same as (7.33) and (7.36). However, the vertical
integration to obtain p′ is performed to the depth where velocity is defined for the bottom U-cell. Geopotential term isevaluated using a depth anomaly of s(z ∗ )-surface. Thus, on the same vertical level, the algorism is diﬀerent depending onwhether it is bottom cell or not.

7.3 Viscosity
The viscosity in an ocean general circulation model seeks to attenuate numerical noise rather than parameterizing thesubgrid-scale momentum transport. The momentum advection scheme should conserve the total kinetic energy in thegeneral three-dimensional flows and the total enstrophy in the two-dimensional flows. Therefore, spatially and temporally
centered discretization should be used, although this inevitably produces near-grid-size noise accompanying numericaldispersion. In eddy-resolving models, the current velocity and the numerical noise are greater than those of eddy-lessmodels. A biharmonic viscosity scheme has been widely used to reduce numerical noise while maintaining the eddy
structure.
The viscosity term is represented by V and is calculated separately in the lateral and vertical directions, i.e., the fourthand fifth terms, respectively, on the r.h.s. of (7.1) and (7.2). For horizontal viscosity, the harmonic (default) or biharmonic(VISBIHARM option) scheme can be selected. Anisotropy of viscosity with respect to the flow direction can be applied(VISANISO option) when harmonic viscosity is chosen. The viscosity coeﬃcient is a constant by default but can be
determined as a function of local velocity gradients and grid-size (SMAGOR option).
For vertical viscosity, the harmonic scheme is used and the local coeﬃcient is the larger one of the background constantand the value calculated from a turbulence closure scheme. A parameterization of bottom friction (Weatherly et al., 1980)
is adopted at the lowest layer.

7.3.1 Horizontal viscosity
The specific form for harmonic viscosity is shown here. Horizontal tension DT and shear DS are defined as follows:
( )
( )
∂
∂
u
v
− hµ
,
(7.39)DT = hψh µ ∂ µ hψhψ ∂ψ hµ
( )
( )
∂
v
u
∂
D S = hψ+ hµ.
h µ ∂ µ hψhψ ∂ψ hµThe viscosity terms are(7.40)Vu =1 ∂ ( 2 )
1 ∂ ( 2 )hσ
hµ σ S ,+Tψhψ2 hµ ∂ µhµ2 hψ ∂ψ

(7.41)Vv =1 ∂ ( 2 )
1 ∂ ( 2 )hψ σ S − 2
hµ σT ,
2hψ h µ ∂ µhµ hψ ∂ψ(7.42)

where σT = νH DT and σS = νH DS , and νH is the horizontal viscosity coeﬃcient. The above representation of theviscous term was derived by Bryan (1969) and is consistent with Smagorinsky (1963).

– 75 –

7.3

Viscosity

If we take hµ = 1, hψ = 1, the coordinate system is Cartesian. In this case, the viscosity term is reduced to the Laplacian
form if the viscosity coeﬃcient is a constant. In the geographic coordinate system, where (µ, ψ) = (λ, ϕ), hλ = a cos ϕ,and hϕ = a, tension and shear are
1 ∂u 1 ∂v v
DT =−− tan ϕ,(7.43)
a cos ϕ ∂λ a ∂ϕ a
1 ∂v 1 ∂u u+
+ tan ϕ.
a cos ϕ ∂λ a ∂ϕ a(7.44)Vu =∂1 ∂2 tan ϕ1σT +σS − σS,a cos ϕ ∂λa ∂ϕa(7.45)Vv =∂1 ∂2 tan ϕ1σS −σT + σT,a cos ϕ ∂λa ∂ϕa(7.46)DS =The viscosity terms in this case arewhere the third term on the r.h.s. is called the metric term.
When biharmonic viscosity is used (VISBIHARM option), the above operation is repeated twice using a viscosity
coeﬃcient νBH . The terms Vu and Vv given by (7.41) and (7.42) are sign-reversed and substituted as u and v in equations(7.39) and (7.40). A biharmonic scheme dissipates noise only on scales near the grid size. This scale selectivity allows the
explicitly represented eddies to survive without unphysical damping in eddy-resolving models, although we must note thata biharmonic operator produces overshootings and spurious oscillations of variables (Delhez and Deleersnijder, 2007). A
biharmonic viscosity scheme is not suitable for coarse resolution models that cannot resolve mesoscale eddies.
A non-slip condition is used for the side boundaries of topography by default in MRI.COM. A free-slip condition
assuming zero viscosity there is also available.

7.3.2 Horizontal anisotropic viscosity (VISANISO)
Smith and McWilliams (2003) proposed a method of making a harmonic viscosity scheme anisotropic in an arbitrary
direction. Setting σT and σS in equations (7.41) and (7.42) to
(

σT
σS

)
=

[( 1

2 (ν0 + ν1 )

0

0
ν1

)

(
+ (ν0 − ν1 )nµ nψ

−2nµ nψnµ2 − nψ2nµ2 − nψ22nµ nψ

)] (

DT
DS

)
,

(7.47)

where n̂ = (nµ, nψ ) is a unit vector in an arbitrary direction and ν0 (ν1 ) is the viscosity coeﬃcient parallel (perpendicular)
to n̂. When VISANISO option is selected, n̂ is set to the direction of local flow in MRI.COM. Given the harmonic viscosity
only in the direction of flow (ν1 = 0), the numerical noise is erased while the swift currents and eddy structures aremaintained.
The following is a note on usage. The behavior of this scheme is specified at run time by namelist nml_visaniso (Table
7.2). The ratio ν1 /ν0 should be given by cc0 (default value is 0.2). The ratio at the lateral boundary should be given by
cc1 (default value is 0.5). When the variable flgvisequator is set as a positive number in the namelist, the ratio ν1 /ν0
is tapered linearly from cc0 at the latitude flgvisequator (in degrees) to vis_factor_equator at the Equator. The
ratio is not tapered when a negative number is set, and the default value of flgvisequator is −1.
Table7.2 Namelist nml_visaniso anisotropic horizontal viscosity

variable

units
1

cc1
figvisequator

1
degreelatitude1

cc0

vis_factor_equator

description
the factor of anisotropy in viscosity; perpendicular / parallel (ν1 /ν0 ) to the flow
the factor (ν1 /ν0 ) at lateral boundary
the factor is tapered toward the Equator; when |LAT| <flgvisequator [deg]the factor at Equator when flgvisequator > 0– 76 –

usage
default = 0.2
default = 0.5
default = −1 (notapering)
default = 0.0

Chapter 7

Equations of motion (baroclinic component)

7.3.3 Smagorinsky parameterization for horizontal viscosity (SMAGOR option)
To give the necessary but minimum viscosity to reduce numerical noise, the viscosity coeﬃcient is made proportional to
the local deformation rate (SMAGOR option; Smagorinsky, 1963; Griﬃes and Hallberg, 2000). When this parameterization
is used with the biharmonic scheme, the scale selectivity of the viscosity scheme becomes more eﬀective.
Defining deformation rate |D|:√
|D| =DT2 + DS2 ,(7.48)

the viscosity coeﬃcients are set as follows:
(

)2

νH =

C∆min
π

νBH =

∆2min
νH ,
8

|D|,

(7.49)(7.50)

where C (smagor_scale) is a dimensionless scaling parameter set by considering numerical stability and ∆min is thesmaller of the zonal and meridional grid widths.
The parameter C should be selected to satisfy the following conditions.
• Restriction of grid Reynolds number:∆minνH > U,(7.51)
2• Restriction on the width of the lateral boundary layer:νH > β∆3min,(7.52)

∆2min
,
2∆t

(7.53)• CFL condition:
νH <where β = df /dy is the meridional√gradient of the Coriolis parameter. Scaling the deformation rate |D| by U/∆min gives
the condition for stability: C > π/ 2 ≈ 2.2 from (7.51) (Griﬃes and Hallberg, 2000).
Behavior of this scheme at run time is specified by namelist nml_smagor, whose components are listed on Table 7.3.
Table7.3 Namelist nml_smagor for Smagorinsky parameterization of horizontal viscosity coeﬃcientvariablesmagor_scalesmagor_diff_ratio
l_smagor_nonzero_bg

units
1
1
logicaldescription
scaling factor coeﬃcient
ratio for diﬀusivity to scaling factor of viscosity
Use background viscosity as given by
nml_baroclinic_visc_horz (Table 7.4)

7.3.4 Discretization of the horizontal viscosity term
Using the notations
δµ Ai, j ≡Ai+ 1 , j − Ai− 1 , j2

2, δψ Ai, j ≡Ai, j+ 1 − Ai, j− 1
2

2

∆µ
∆ψ
δi Ai, j ≡ Ai+ 1 , j − Ai− 1 , j , δ j Ai, j ≡ Ai, j+ 1 − Ai, j− 1 ,
2

and

µ

Ai, j ≡

2

2

1
( A 1 + Ai+ 1 , j ),2
2 i− 2 , j

ψAi, j ≡

– 77 –,

21( A 1 + Ai, j+ 1 ),2
2 i, j− 2

usage
default = −1.0
default = −1.0
default = .false.

7.3

Viscosity

deformation rates are discretized as follows:
( ) ψ
( ) µ
hψi, jhµi, juv
DT i, j =δµ−δψ,hµi, jhψ i, jhψi, jhµ i, j( ) ψ( ) µhµi, jhψi, jv
u
DS i, j =δµ+δψ.
hµi, jhψ i, jhψi, jhµ i, j(7.54)

Horizontal viscosity forces of (7.41) and (7.42) are discretized as follows:1∆Vi+ 1 , j+ 1
2
2


(
)
(
1
1
ψ
µ)
,

2
2
δi ∆y∆zhψ νH DT
δ j ∆x∆zhµ νH DS
×  2
+ 2
1
1
1
1

i+
,
j+
i+ 2 , j+ 222hµ i+ 1 , j+ 1
 hψi+ 1 , j+ 1

2
2
2
2
1
Fy i+ 1 , j+ 1 =2
2∆Vi+ 1 , j+ 1
2
2


(
)
(
1
1
ψ
µ)

 .
2
2
− 2
×  2δi ∆y∆zhψ νH DS
δ j ∆x∆zhµ νH DT
i+ 21 , j+ 12 i+ 21 , j+ 12
hµ i+ 1 , j+ 1
 hψi+ 1 , j+ 1

2
2
2
2

Fx i+ 1 , j+ 1 =2

2(7.55)The non-slip condition at the side boundaries of topography is discretized as follows. When the grid point (i − 12 , j + 12 )is defined as a (vertically partial) land (Figure 7.7a), the velocity gradients at the wall are calculated as follows:( )ui+ 1 , j+ 1∂u
= 2 − 2,
∂ x i, j+ 1
∆x i j( ) 2 v 1 1
i+ , j+∂v= 2 − 2,
(7.56)∂ x i, j+ 1
∆x i j2where ∆x −i j is the length between the points (i, j + 12 ) and (i + 21 , j + 12 ). The contribution of this wall to the force is:
FxW i+ 1 , j+ 1 = −2

2

FyW i+ 1 , j+ 1 = −2

2

1∆Vi+ 1 , j+ 1 hψ 2i+ 1 , j+ 1
2
2
2
2
1
∆Vi+ 1 , j+ 1 hψ 2i+ 1 , j+ 12

2

2

∆yi, j+ 1 ∆ z̃i, j+ 1 hψ 2i− 1 , j νHi− 1 , j
2

2

2

2

ui+ 1 , j+ 1
2

2

∆x −i+ 1 , j+ 1
2∆yi, j+ 1 ∆ z̃i, j+ 1 hψ 2i− 1 , j νHi− 1 , j
2

2

2

2

2

2

vi+ 1 , j+ 1
2

2

∆x −i+ 1 , j+ 1
2,,

(7.57)

2

where ∆ z̃i, j+ 1 is the wall height.
2

7.3.5 Vertical viscosity
Only the harmonic scheme is considered. The vertical momentum flux is assumed to be proportional to the verticalgradient of velocity. For the upper part of a U-cell at the (k − 12 )th vertical level, the momentum flux (positive upward) is
calculated as follows:)(uk− 3 − uk− 1
∂u
2
2
− νv
= −νv k−1,
∂z k−1∆zk−1where ∆zk−1 = (∆zk− 3 + ∆zk− 1 )/2, ∆zk− 1 is the thickness of the U-cell (dzu), and νv is the vertical viscosity coeﬃcient.
222Similarly, the momentum flux in the lower part of the U-cell is calculated as follows:()
uk− 1 − uk+ 1
∂u
2
2
− νv
= −νv k,∂z k∆zkwhere νv k is set to zero if the (k + 12 )th level is the solid Earth. The bottom friction is calculated independently (see thenext subsection). Also note that the variations of the grid thickness due to the partial bottom cell and the undulation of thesea surface are not considered when evaluating fluxes for simplicity.
– 78 –

Chapter 7

Equations of motion (baroclinic component)aui-1/2, j+1/2

ui+1/2, j+1/2
Δyi+1/2, j

Δxi, j+1/2

ui+1/2, j-1/2
Δzi

ui-1/2~Δ

ui+1/2zi-ui-1/2

Δzi+1

~
Δz
i+1

ui+3/2

uk-1/2

uk-1/2~Δz
k+1/2

Δzk+1/2

Δzk

uk+1/2~Δzk-1/2

Δzk-3/2

uk-3/2

Δzk-1

uk-3/2Δzk-1/2b

Figure7.7 (a) A schematic distribution of grids for horizontal viscosity. Upper: Plan view. Lower: Side view. The
shadings denote solid earth. (b) A schematic distribution of grids for vertical viscosity. Side views. Left: The lower
adjacent layer (k + 21 ) has a sea bed. Right: The U-cell (k − 12 ) has a sea bed.

To calculate viscosity, the divergence of the momentum flux is first calculated. The expression for the vertical viscosity
term is( ∂u ))(
νv k−1 (uk− 3 − uk− 1 ) νv k (uk− 1 − uk+ 1 )νv ∂z k−1 − νv ∂u∂ ( ∂u )
∂z k
2
2
2
2
νv
=
=
−
(7.58)
1∂z∂z k− 2
L
L
L
∆zk−1 ∆z 1
∆zk ∆z 1
∆z 1
k− 2

k− 2

k− 2

where the variation of the grid thickness for the U-cell due to the partial bottom cell is now taken into account and is
f 1 (Figure 7.7(b)). Note that the first term on the r.h.s. of equation (7.58)
L that is, ∆z
L 1 = ∆z 1 − ∆zrepresented by ∆z,
k−
k−
k−
2

2

2

is set to zero in calculating the viscosity term for the vertical level of 12 (k = 1). See section 14.1 for sea surface windstress. For the vertical viscosity coeﬃcient νv , MRI.COM uses the larger of the value predicted by a turbulent closure
scheme and a background one.

– 79 –

7.4

Coriolis term

7.3.6 Bottom friction
When a U-cell in the (k − 12 )th layer contains solid earth (Figure 7.7(b) right), the stress from the lower boundary (τxb, τyb )is calculated following Weatherly et al. (1980). The specific expression is as follows:
(

τxb
τyb

)
=

(
√
cos θ 02
2
−ρ0 Cbtm u 1 + v 1
k− 2
k− 2
sin θ 0− sin θ 0cos θ 0)u 1
* k− 2 + ,v 1, k− 2 -where Cbtm is a dimensionless constant. Viscous stress at the lower boundary has a magnitude proportional to the squareof the flow speed at the U-cell and an angle (θ 0 + π) relative to the flow direction.
In MRI.COM,
Cbtm = 1.225 × 10−3
θ 0 = ±π/18 rad(≡ 10°),
where θ 0 is positive (negative) in the northern (southern) hemisphere. The variables are designated in the model as
Cbtm = abtm, cos θ 0 = bcs, and sin(±θ 0 ) = isgn ∗ bsn, where isgn = 1 in the northern hemisphere and isgn = −1 inthe southern hemisphere.

7.4 Coriolis term
Because MRI.COM employs Arakawa B-grid arrangement, both of the horizontal components of velocity are defined at
the same point. Coriolis term is evaluated by using the middle time level for the leap-frog scheme and the start time level
for the Euler-backward scheme.

7.5 Usage
Runtime behavior of the baroclinic mode in the default settings is specified by the five namelist blocks (some of them are
optional) shown on Tables 7.4 through 7.8. For the initial condition of the 3D velocity field, restart file must be prepared
for X- and Y-ward velocities.
In addition, following model options are available.
BIHARMONIC or VISBIHARM: Biharmonic horizontal viscosity is used instead of harmonic viscosity
Specify visc_horz_cm4ps [cm4 sec−1 ] instead of visc_horz_cm2ps [cm2 sec−1 ] in namelist
nml_baroclinic_visc_horz. See Section 7.3.1 for detail.
VISANISO: Anisotropic viscosity coeﬃcients are used
Specify the coeﬃcients by nml_visaniso (Table 7.2). See Section 7.3.2 for detail.
SMAGOR or SMAGHD: Smagorinsky parameterization is used for horizontal viscosity
Specify nml_smagor (Table 7.3) for factors of the Smagorinsky parameterization. See Section 7.3.3 for detail.
VIS9P: Nine points are used in computing viscosity
Five points are used by default.
See docs/README.Namelist for namelist details.
Table7.4 Namelist nml_baroclinic_visc_horz (required) for the horizontal viscosity (see Section 7.5)

variable
visc_horz_cm2psunits
cm2 sec−1

description
horizontal viscosity (νH in Section
7.3.1)

usage
requiredContinued on next page

– 80 –

Chapter 7

Equations of motion (baroclinic component)

variablefile_visc_horz_2d
slip_factor

Table 7.4 – continued from previous page
units
description
usage
cm2 sec−1 2D distribution of viscosity
optional (This overwrites
visc_horz_cm2ps.)
factor
0.d0 means no-slip condition, while default = 0.d0 (only valid with
1.d0 slip (Section 7.3.1)
VIS9P option)

Table7.5 Namelist nml_visc_vert_bg (optional) for vertical viscosity (see Section 7.5)

variable
visc_vert_bg_cm2psunits
cm2 sec−1

description
background vertical viscosity (νv in
Section 7.3.5)

usagedefault = 1.d0

Table7.6 Namelist nml_bottom_friction (optional) for the bottom friction (see Section 7.5)

variable
btmfrc_scale
btmfrc_angle_deg
file_btm_frc_2d

units
1
degreefile namedescription
Cbtm in Section 7.3.6
θ0
2D distribution of Cbtm

usage
default = 1.225d−3
default = 1.d1
optional (This overwrites
btmfrc_scale )Table7.7 Namelist nml_hvisc_add (optional) for additional horizonta viscosity (see Section 7.5)

variablel_hvisc_add_harmonic

units
logicalfile_hvisc_harmonic

cm2 sec−1

description
use additional harmonic viscosity ornot
2D distribution of viscosity

usage
default = .false.
file name

Table7.8 Namelist nml_baroclinic_run (optional) for starting the baroclinic mode (see Section 7.5)

variablel_rst_baroclinic_inunits
logicaldescription
read initial restart for 3D velocity ornot– 81 –

usage
default = l_rst_inPart IV

Evolution of tracersChapter 8

Tracer advection schemes
This chapter describes tracer advection schemes available in MRI.COM. The tracer advection schemes available in
MRI.COM are: the weighted upcurrent scheme (Section 8.2), the Quadratic Upstream Interpolation for Convective
Kinematics (QUICK; Leonard, 1979) as described in Section 8.3 (QUICKADVEC option), a combination of the QUICK with
Estimated Streaming Terms (QUICKEST; Leonard, 1979) for vertical advection (Section 8.4) and the Uniformly ThirdOrder Polynomial Interpolation Algorithm (UTOPIA; Leonard et al., 1993), which is a two-dimensional generalization of
the QUICKEST scheme, for horizontal advection (Section 8.5) with UTZQADVEC option. For QUICKEST and UTOPIA,
a flux limiter that prevents unrealistic extrema is applied (Leonard et al., 1994).
The above schemes seek to improve the accuracy by refining the finite-diﬀerence expression at the cell faces. There areother approaches. The Second Order Moment (SOM; Prather, 1986) scheme seeks to improve the accuracy by considering
the distribution within the cell and is available in MRI.COM through SOMADVEC option (Section 8.6). The Multidimensional
Positive Definite Advection Transport Algorithm (MPDATA; Smolarkiewicz and Margolin, 1998) scheme seeks to cancel
the numerical diﬀusion of the lower order scheme by using anti-diﬀusive velocities and is available in MRI.COM through
MPDATAADVEC option (Section 8.7).
Diﬀerent advection schemes can be used for individual tracers. They should be chosen from among compiled schemes
at run time (Chapter 13).

8.1 Finite volume or flux form method
The equation governing the time change of tracer T is{}∂(z s hψ uT ) ∂(z s hµ vT )∂(z s T )1∂(z s ṡT ) ∂(z s T )+++=+ A(T ) = −z s ∇ · FT + z s QT ,∂th µ hψ
∂µ
∂ψ
∂s
∂t

(8.1)

where A(T ) represents the advection operator for T. The finite diﬀerence form of the l.h.s. of (8.1) is given by firstconsidering a control cell volume and then calculating fluxes through each cell face, and setting their divergence andconvergence to be the time change rate at the grid cell (Figure 8.1). In finite diﬀerence form, this is expressed as follows:
Ti,n+1∆Vi,n+1= Ti,n−1∆Vi,n−1j,k− 1j,k− 1j,k− 1j,k− 122

2

2+ 2 ∆t {FXAi− 1 , j,k− 1 − FXAi+ 1 , j,k− 1 + FYAi, j− 1 ,k− 1 − FYAi, j+ 1 ,k− 1 + FZAi, j,k − FZAi, j,k−1 }, (8.2)
2

2

2

2

2

2

2

2

where ∆V is the volume of the grid cell, and FXA, FYA, and FZA represent (flux due to advection) × (area of the cellboundary). The same consideration may be applied for the discretization of the diﬀusion term.
As explained in Chapter 3, ∆V varies with time and is given by (3.28),∆Vi, j,k− 1 ≡ (volt)i, j,k− 1 = (volu_bl)i+ 1 , j+ 1 ,k− 1 + (volu_tl)i+ 1 , j− 1 ,k− 1
2

2

2

2

2

2

2

2

+ (volu_br)i− 1 , j+ 1 ,k− 1 + (volu_tr)i− 1 , j− 1 ,k− 1 .
2

2

2

2

2

2

(8.3)

The volume of the left-lower quarter of the T-cells at (i, j) (corresponding to the southwestern part in geographic
coordinates) is represented by (volt_tr)i− 1 , j− 1 . Similarly, (volt_tl)i+ 1 , j− 1 is the right-lower (southeastern),2222(volt_br)i− 1 , j+ 1 is the left-upper (northwestern), and (volt_bl)i+ 1 , j+ 1 is the right-upper (northeastern) quarter of a2222T-cell at (i, j).

– 85 –

8.2 Weighted upcurrent scheme
Fluxes due to advection are given as follows:
T
FXAi+ 1 , j,k− 1 = Ui+
T 1
1,1, j,k− 1 i+ , j,k−

(8.4)

FYAi, j+ 1 ,k− 1 = Vi,Tj+ 1 ,k− 1 Ti, j+ 1 ,k− 1 ,

(8.5)

2

2

2

2

2

2

2

2

2

2

2

2

FZAi, j,k = Wi,Tj,k Ti, j,k ,(8.6)

where horizontal volume transport U T and V T are defined as follows, using (5.13) to (5.18):
T
=
Ui+
1
, j,k− 12

2

Vi,Tj+ 1 ,k− 1 =22∆yi+ 1 , j (2

2)
ui+ 1 , j− 1 ,k− 1 ∆zi+ 1 , j− 1 ,k− 1 + ui+ 1 , j+ 1 ,k− 1 ∆zi+ 1 , j+ 1 ,k− 1 ,

(8.7)

)
vi− 1 , j+ 1 ,k− 1 ∆zi− 1 , j+ 1 ,k− 1 + vi+ 1 , j+ 1 ,k− 1 ∆zi+ 1 , j+ 1 ,k− 1 .

(8.8)

2

∆x i, j+ 1 (
2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

Vertical volume transport W T is then obtained by diagnostically solving (5.12). Moreover, the vertical velocity w, which
is necessary for using QUICKEST, is calculated as follows (w is not needed except for QUICKEST):
Wi,Tj,k = wi, j,k × (areat)i, j,k− 1 ,

(8.9)

2

where
(areat)i, j,k− 1 = (a_tr)i− 1 , j− 1 × (aexl)i− 1 , j− 1 ,k− 1 + (a_tl)i+ 1 , j− 1 × (aexl)i+ 1 , j− 1 ,k− 1
2

2

2

2

2

2

2

2

2

2

2

+ (a_br)i− 1 , j+ 1 × (aexl)i− 1 , j+ 1 ,k− 1 + (a_bl)i+ 1 , j+ 1 × (aexl)i+ 1 , j+ 1 ,k− 1 .
2

2

2

2

2

2

2

2

2

2

(8.10)An array aexl is set to be unity if the corresponding U-cell is a sea cell and set to be zero otherwise.
This formulation does not depend on the choice of the advection scheme. The diﬀerence arises from the way ofdetermining cell boundary values of tracer, Ti+ 1 , j,k− 1 , Ti, j+ 1 ,k− 1 , and Ti, j,k . There are several choices for tracer advection22
22
schemes, which are explained in the following sections.
• Weighted upcurrent scheme (adv_scheme%name = ”upc”). This scheme is always available for use. Specify theweighting ratio listed on Table 8.1.
• Quadratic Upstream Interpolation for Convective Kinematics (QUICK; Leonard, 1979) for both horizontal andvertical direction (QUICKADVEC option; adv_scheme%name = ”quick”).
• The combination of the Uniformly Third-Order Polynomial Interpolation Algorithm (UTOPIA; Leonard et al.,1993) for horizontal direction and QUICK with Estimated Streaming Terms (QUICKEST; Leonard, 1979) for thevertical direction (UTZQADVEC option; adv_scheme%name = ”utzqadv”).
• Second Order Moment (SOM; Prather, 1986) schemes for both horizontal and vertical direction (SOMADVEC option;
adv_scheme%name = ”som”).
• Multidimensional Positive Definite Advection Transport Algorithm (MPDATA; Smolarkiewicz and Margolin, 1998)
for all directions (MPDATAADVEC option; adv_scheme%name = ”mpdata”).

8.2 Weighted upcurrent scheme
In the weighted upcurrent scheme (e.g., Suginohara and Aoki, 1991; Yamanaka et al., 2000), the cell boundary value isdetermined by a weighted average of the upcurrent scheme and the centered finite diﬀerence scheme.
The upcurrent scheme employs the upstream value as the cell boundary value: Ti, j,k− 12 ,upcurrent= Ti+1, j,k− 1 ,i+ 12 , j,k− 12

T



2

if
if

Ui+ 1 , j,k− 1 > 0,2
2
Ui+ 1 , j,k− 1 < 0.
2

(8.11)

2

The centered finite diﬀerence scheme uses the average between the two neighboring points of tracer as the cell boundary
value:
Ti+1, j,k− 1 + Ti, j,k− 122
centerTi+.
(8.12)1 =1
2
2 , j,k− 2– 86 –

Chapter 8

Tracer advection schemes

V Ti, j+1/2
UVi+1/2, j+1/2

UVi-1/2, j+1/2
U Ti-1/2, j

U Ti+1/2, jTi, jUVi-1/2, j-1/2

UVi+1/2, j-1/2
V Ti, j-1/2
W Ti, kUVi-1/2, k+1/2

Ti, k+1/2

UVi+1/2, k+1/2

W Ti, k+1
Figure8.1 Grid arrangement around TS-Box (Upper: Views from the upper, Below: views from the horizontal).
Fluxes represented by an arrow are calculated.
Table8.1 Namelist nml_tracer_adv for weighted upcurrent scheme

variable nameweight_upcurrent_horz

units
1

weight_upcurrent_vert

1

description
upstream-side weighting ratio for thehorizontal advection
upstream-side weighting ratio for thevertical advectionusage
1.0: up-current, 0.5: centered diﬀerence
1.0: up-current, 0.5: centered diﬀerence

Taking the ratio of the upcurrent scheme to be α (0 ≤ α ≤ 1), the tracer flux at the eastern face of a grid cell is expressed
as follows:
UT 1+ UT 1− UT 1
UT 1






i+ 2 , j,k− 12
i+ 2 , j,k− 12
i+ 2 , j,k− 12
i+ 2 , j,k− 21FXAi+ 1 , j,k− 1 =α 
T
+
T
i
i+1


2
2
2
2




Ti+1, j,k− 1 + Ti, j,k− 1
22T+ (1 − α)Ui+
1
1
22 , j,k− 2
UT 1
UT 1


1(

(


i+ 2 , j,k− 12 )i+ 2 , j,k− 21 )1T.
= Ui+ 1 , j,k− 1 
1
+
α
T
+
1
−
α
T
i
i+1
T
T


2

2
2 2
U 1
U 1
1
1
i+
,
j,k−
i+
,
j,k−


2
2
2
2

(8.13)(8.14)

Mainly for the sake of computational eﬃciency, we give a parameter β = 12 (α + 1) instead of α at run time. Diﬀerent
parameters for horizontal and vertical directions may be given, which are listed on Table 8.1.

– 87 –

8.3 QUICK scheme

8.3 QUICK scheme
This section explains the QUICK scheme. In the QUICK scheme, the cell boundary value is interpolated by a quadratic
function, using three points, with one of them added from the upstream side (Figure 8.2).

T
Ti+1

quadratic function thatpasses three points

Ti

Ti+1/2

Ti-1
Ui+1/2

i-1

X

i+1

i
T-box

Figure8.2 Schematic for interpolation: Ti represents the value of a tracer at T-point with index i, and Ti+ 1 is the cell2boundary value. In the QUICK scheme, Ti+ 1 is interpolated by a quadratic function that passes Ti and the neighboring
2
T-point values, Ti−1 and Ti+1 .

Originally, cell boundary values in the QUICK scheme are given as follows:∆x i Ti+1, j,k− 1 + ∆x i+1Ti, j,k− 1∆x i+1 ∆x i
ci+ 1 , j,k− 1 ,2
2
4∆y j+1 ∆y j
2
2
Ti, j+ 1 ,k− 1 =−d i, j+ 1 ,k− 1 ,22
22∆y j+1 + ∆y j4∆zk− 1 Ti, j+1,k+ 1 + ∆zk+ 1 Ti, j,k− 1 ∆zk− 1 ∆zk+ 1
2
2
2
2
2
2
Ti, j,k =−ei, j,k ,∆zk+ 1 + ∆zk− 1
4
Ti+ 1 , j,k− 1 =
2

2

2

2

∆x i+1 + ∆x i
∆y j Ti, j+1,k− 1 + ∆y j+1Ti, j,k− 12−(8.15)(8.16)(8.17)

2

where c, d, and e are defined depending on the direction of the mass flux as follows:ci+ 1 , j,k− 1
2

2

=
=

d i, j+ 1 ,k− 1
2

2

=
=

ei, j,k==∆x i δ x δ x Ti, j,k− 12(≡ cp ),x2∆x i∆x i+1 δ x δ x Ti+1, j,k− 1
2(≡ cm ),x2∆x i+1∆y j δ y δ y Ti, j,k− 12(≡ d p ),y2∆y j∆y j+1 δ y δ y Ti, j+1,k− 12(≡ d m ),y2∆y j+1∆zk+ 1 δz δz Ti, j,k+ 1
2
2(≡ e p ),z
2∆zi, j,k+ 1
2∆zk− 1 δz δz Ti, j,k− 12
2(≡ em ),z2∆zi, j,k− 1
2– 88 –

if

UT 1

> 0,

if

UT 1

< 0,

if

VT

> 0,

if

VT

< 0,

if

Wi,Tj,k > 0,ifWi,Tj,k < 0.

i+ 2 , j,k− 12
i+ 2 , j,k− 12

i, j+ 12 ,k− 21
i, j+ 12 ,k− 21(8.18)

Chapter 8

Tracer advection schemes

The finite diﬀerence operators are defined as follows (definitions in y and z directions are the same):δ x Ai ≡x

Ai ≡Ai+ 1 − Ai− 1
2

2

∆x i
Ai+ 1 + Ai− 1
2

2

2

,

Ai+1 − Ai,∆x i+ 1

δ x Ai+ 1 ≡
2

2

,

x

Ai+ 1 ≡
2

Ai+1 + Ai.
2(8.19)

Letting cp , d p , and e p represent their values for positive velocity at the cell boundary and cm , d m , and em represent theirvalues for negative velocity at cell boundary and taking
ca = cm + cp
cd = cm − cp

(8.20)(8.21)

da = dm + d p(8.22)

dd = dm − d p(8.23)

e a = em + e p(8.24)

e d = em − e p,(8.25) ∆x i, j T∆x i+1, j ∆x i, j
i+1, j,k− 12 + ∆x i+1, j Ti, j,k− 21Tc
FXAi+ 1 , j,k− 1 = Ui+
−
1
1
a
1
1 
i+ 2 , j,k− 222∆x i+1, j + ∆x i, j
8
2 , j,k− 2 ∆x i+1, j ∆x i, j
T
+ |Ui+
cd i+ 1 , j,k− 1 ,
1
1|
2
2
8
2 , j,k− 2 ∆yi, j T
1 + ∆yi, j+1T
∆yi, j+1 ∆yi, j
i, j+1,k− 2i, j,k− 12
d a i, j+ 1 ,k− 1 FYAi, j+ 1 ,k− 1 = Vi,Tj+ 1 ,k− 1 
−
2
2
2
2
∆yi, j+1 + ∆yi, j
8
2
2 



(8.26)

we obtain∆yi, j+1 ∆yi, j
d d i, j+ 1 ,k− 1 ,
2
2
8
2
2

 ∆z
∆zi, j,k+ 1 ∆zi, j,k− 1
i, j,k− 12 Ti, j,k+ 12 + ∆zi, j,k+ 12 Ti, j,k− 2122eai, j,k 
FZAi, j,k = Wi,Tj,k −∆zi, j,k+ 1 + ∆zi, j,k− 1
8


2
2
∆zi, j,k+ 1 ∆zi, j,k− 1
2
2+ |Wi,Tj,k− 1 |ed i, j,k .
82+ |Vi,Tj+ 1 ,k− 1 |(8.27)(8.28)

Equation (8.26) can be rewritten as
T
FXAi+ 1 , j,k− 1 ≃ Ui+
1
1 T̃i+ 1 , j,k− 1 + AQ22
2
2
2 , j,k− 2∂ 3Ti+ 1 , j,k− 1
2

∂ x3

2

,

(8.29)

where T̃i+ 1 , j,k− 1 is the value of T at the cell boundary interpolated by the cubic polynomial, and22T
AQ = |Ui+|
1, j,k− 12∆x i+1 ∆x i+ 1 ∆x i

2

2

8

.

(8.30)

Although the time integration for advection is done by the leap-frog scheme, the second term on the r.h.s. of (8.29) has
a biharmonic diﬀusion form, and thus the forward scheme is used to achieve calculation stability (Holland et al., 1998).
A similar procedure is applied for the north-south and vertical directions.
The weighted up-current scheme is used for vertical direction if wi, j,k−1 > 0 and the T-point at (i, j, k + 12 ) is below
the bottom. The upstream-side weighting ratio is given by the user as the namelist parameter specified for the up-current
scheme (Table 8.1).

8.4 QUICKEST for vertical advection
This section describes the specific expression and the accuracy of the QUICK with Estimated Streaming Terms (QUICKEST; Leonard, 1979) for vertical advection.

– 89 –

8.4 QUICKEST for vertical advection
Consider a one-dimensional equation of advection for incompressible fluid∂T∂+(wT ) = 0,∂t∂z(8.31)

where w is a constant. Although the velocities are not uniform in the real three dimensional ocean, we assume a constantvelocity for simplicity.
Following the notation of vertical grid points and their indices (Section 3.2), tracers are defined at the center (k − 12 )of the vertical cells and vertical velocities are defined at the top (k − 1) and bottom (k) faces of the vertical cells. The
following relation holds for the vertical grid spacing:∆zk =∆zk+ 1 + ∆zk− 1
2

2

2

.

(8.32)In QUICKEST, the distribution of tracer T is defined using the second order interpolations, and the mean value during
a time step at the cell face (boundary of two adjacent tracer cells) is calculated. The coeﬃcients for the second orderinterpolation are calculated first. A Taylor expansion of T about point zk gives
2∆zk− 3∆zk− 32
2
Tk− 3 = c0 + c1 *+ ∆zk− 1 + + c2 *+ ∆zk− 1 + + O(∆z 3 ),
2
2
2
, 2
, 2
2
∆z
∆zk− 1k− 21
2
Tk− 1 = c0 + c1
+ c2
+ O(∆z 3 ),22
4
∆z 2 1∆zk+ 1
k+ 2
2
Tk+ 1 = c0 − c1
+ c2
+ O(∆z 3 ),22
42
∆zk+ 3∆zk+ 32
2
Tk+ 3 = c0 − c1 *+ ∆zk+ 1 + + c2 *+ ∆zk+ 1 + + O(∆z 3 ).
2
2
2
, 2
, 2
-

(8.33)(8.34)(8.35)(8.36)

Coeﬃcients c0 , c1 , and c2 can be solved using three of the four equations (8.33), (8.34), (8.35), and (8.36). The three
upstream-side equations are chosen. When w > 0 (w < 0), equations (8.34), (8.35), and (8.36) ((8.33), (8.34), and (8.35))
are used. The solution is as follows:
c0 =
c1 =

Tk− 1 ∆zk+ 1 + Tk+ 1 ∆zk− 1
2

2

2

2∆zk

Tk− 1 − Tk+ 1
2

2

∆zk

−
(T

2

−

∆zk+ 1 ∆zk− 12∆zk− 1 − ∆zk+ 1
2

2

4
2

2

c2,

c2,

(8.37)(8.38))−Tk+ 1
Tk+ 1 −Tk+ 3
k− 1

1
2
2
2
2

−


∆z k +∆z k+1∆z k∆z k+1()
c2 = 
Tk− 3 −Tk− 1
Tk− 1 −Tk+ 1

1
2
2
2
2

 ∆z +∆z−∆z k−1∆z k k−1 k(w > 0),(8.39)(w < 0).

Next, equation (8.31) is integrated over one time step and one grid cell.
∫ t n+1∫ zk−1
dt

tn

dz
zk

The r.h.s. of (8.40) can be written as
−

∂T
=−
∂t

∫ t n+1
tn∫ t n+1∫ zk−1
dt

tn

dz
zk

∂
(wT ).
∂z

dt (wu Tu − wl Tl ),(8.40)(8.41)

where subscript u (l) denotes z = zk−1 (z = zk ). Assuming that w does not depend on time,∫ t n+1
tn

dtTl =∫ 0−wl ∆t[c0n + c1n ξ + c2n ξ 2 + O(∆z 3 )]

dξ
.
wl

(8.42)

Thus expression (8.41) becomes−∆t(wu Tun − wl Tln ) + O(∆z 3 w∆t),– 90 –(8.43)

Chapter 8

Tracer advection schemes

where
Tln =

1
wl ∆t

∫ 0

(c0n + c1n ξ + c2n ξ 2 )dξ−wl ∆tcn
cn= c0n − 1 wl ∆t + 2 wl2 ∆t 2 .
23(8.44)

Using up to the second order terms of a Taylor expansion, the l.h.s. of (8.40) can be written as follows:∫ t n+1∫ zk−1
dt

tn

zk



∆z 2 1
k− 2
∂T

n+1
n
n+1
n
3 
dz
= ∆zk− 1 Tk− 1 − Tk− 1 +(Tzz k− 1 − Tzz k− 1 ) + O(∆z )  ,
2
∂t
24
2
2
2
 2


(8.45)

where
n

∂Tzz
+ O(∆t 2 )∂t k− 1
2
2
]n
2 [
∂
∂
= −∆t 2(wT )+ O(∆t 2 )
1
∂z ∂z
k−

n+1
n
Tzz k−= ∆t
1 − Tzz
k− 1
2

2

∂
2
n
= −∆t (wTzz )k−1 + O(∆t )∂z2 wu Tzz un − wl Tzz ln 
 + O(w∆t∆z) + O(∆t 2 ).
= −∆t ∆zk− 1
2(8.46)The expression for the r.h.s. of (8.45) becomes∆z 2 1 w T n − w T n
k− 2
u zz ul zz l

n
3 
n+1
∆t
∆zk− 1 Tk− 1 − Tk− 1 −+ O(∆z )  + O(w∆t∆z 4 ) + O(∆z 3 ∆t 2 ).
224∆zk− 12
 2

2
Based on (8.43) and (8.47), the discretized forecasting equation is expressed as follows:
∆t
n+1
n
Tk−
−
1 = T
k− 12
∆zk− 1
2
2



∆z 2
w T n − w T n − k− 12 (w T n − w T n )  + O(α∆z 4 ) + O(∆z 2 ∆t 2 ),l l
u zz ul zz l  u u24


(8.47)(8.48)

where
w∆t
< 1,
∆z
n
Tzz l = 2c2 + O(∆z).
α≡(8.49)(8.50)

The accuracy of equation (8.48) is max(O(∆z 4 ), O(∆z 2 ∆t 2 )).

8.5 UTOPIA for horizontal advection
The Uniformly Third Order Polynomial Interpolation Algorithm (UTOPIA; Leonard et al., 1993) is an advection schemethat can be regarded as a multi-dimensional version of QUICKEST. In UTZQADVEC option, horizontally two-dimensional
advection is calculated using UTOPIA. Vertical advection is calculated separately using QUICKEST.
Since grid intervals could be variable in both zonal and meridional directions in MRI.COM, UTOPIA is formulated
based on a variable grid interval. It is assumed that the tracer cell is subdivided by the borderlines of the velocity cells
into four boxes with (almost) identical area.
Consider an equation of advection:
∂T
1 ∂
1 ∂
+
(hψ uT ) +(hµ vT ) = 0.
∂th µ hψ ∂ µhµ hψ ∂ψ
Integrated over a tracer cell and for one time step,∫ ψL +∆ψL /2∫ µ L +∆µ L /2dψdµ( χ n+1 − χ n ) = −∆t(urnTrn ∆yr − ulnTln ∆yl + vunTun ∆x u − vdnTdn ∆x d ),
ψL −∆ψL /2µ L −∆µ L /2

– 91 –(8.51)(8.52)

8.5

UTOPIA for horizontal advection

where χ ≡ hµ hψ T and Trn etc. on the r.h.s. are the face values described later. On the l.h.s. of (8.52), the second-orderinterpolation of χ is used to integrate the terms. The Taylor expansion of χ about L is given as follows (see Figure 8.3 forthe label of the point):
χ = χ L + a10 (µ − µ L ) + a20 (µ − µ L ) 2 + a01 (ψ − ψ L ) + a02 (ψ − ψ L ) 2 + a11 (µ − µ L )(ψ − ψ L ).

(8.53)

Then values at points E, W, N, and S are

NW

N

W

L

E

SW

S

SE

WW

u

r

SS
l

d

Figure8.3 Labels of tracer grid points (upper case characters) and faces (lower case characters).

χ E = χ L + a10 ∆µr + a20 ∆µ2r ,(8.54)

χ L − a10 ∆µl + a20 ∆µ2l ,χ L + a01 ∆ψu + a02 ∆ψu2 ,χ L − a01 ∆ψ d + a02 ∆ψ d2 ,(8.55)

χW =
χN =
χS =

(8.56)(8.57)

where∆ψ L + ∆ψ N,
2
∆ψ L + ∆ψS∆ψ d ≡,
2∆µ L + ∆µ E
∆µr ≡
,
2
∆µ L + ∆µW
∆µl ≡
.
2
∆ψu ≡

(8.58)(8.59)(8.60)(8.61)

Using these known values, the following parameters are obtained,χE − χLχ L − χW
+ ∆µr
∆µr
∆µl
a10 =
,
∆µr + ∆µl
χE − χLχ L − χW
−
∆µr
∆µl
a20 =
,
∆µr + ∆µl
χN − χLχ L − χS
∆ψ d
+ ∆ψu
∆ψu
∆ψ d
a01 =
,
∆ψu + ∆ψ d
χN − χLχ L − χS
−
∆ψu
∆ψ d
a02 =
.
∆ψu + ∆ψ d∆µl

– 92 –(8.62)(8.63)(8.64)(8.65)

Chapter 8

Tracer advection schemes

Substituting (8.53) into the l.h.s. of (8.52) yields
∆µ2L n+1∆ψ 2L n+1nnn ∆µ L ∆ψ L  χ n+1
(a
(a
−
χ
+
−
a
)
+
−
a
)
L
20
02  .
12 20
12 02
 L


(8.66)

Using equation (8.51), the following approximation is allowed:nn  hψr unT n − hψl unT n
hµu vunTµµu− hµd vdnTµµd

r µµr
l µµl
n+1
n
= −∆t 
− a20
+
a20
 ,
∆µ L
∆ψ L


n n
n n
 hψr unT n − hψl unT n

−
h
v
T
T
h
v
µd
µu
r ψψr
u ψψu
l ψψl
d ψψd 
n+1
n
a02
− a02
= −∆t 
+
 ,
∆µ L
∆ψ L



(8.67)(8.68)

n is the value of the second order derivative at the right face r, whose expression is similar to that of c described
where Tµµr
20later. Therefore, under a suitable approximation,

TLn+1 = TLn −∆t n ˜n(u T ∆yr − ulnT˜ln ∆yl + vunT˜un ∆x u − vdnT˜dn ∆x d ),∆SL r r(8.69)

where∆µ2L n∆ψ 2L nTµµl −T ,24
24 ψψl∆ψ 2L n∆µ2L nT˜dn = Tdn −Tµµd −
T.
24
24 ψψdT˜ln = Tln −(8.70)(8.71)

Next, the expressions for Tln and Tdn are required. The term Tln is the average over the hatched area of Figure 8.4, andthe values of T n are given as the second order interpolation about l of Figure 8.3. Similar operations will be used to obtain
the expression for Tdn .

W

L

vl∆t

ul∆t
Figure8.4 Area used to average tracer values for the face lFirst, Taylor expansions of T n about l and d are written as follows:
T n |l =c00 + c10 (µ − µl ) + c20 (µ − µl ) 2 + c01 (ψ − ψ L ) + c02 (ψ − ψ L ) 2 + c11 (µ − µl )(ψ − ψ L ),(8.72)

T |d =d 00 + d 10 (µ − µl ) + d 20 (µ − µl ) + d 01 (ψ − ψ L ) + d 02 (ψ − ψ L ) + d 11 (µ − µl )(ψ − ψ L ).

(8.73)n2

2– 93 –

8.5

UTOPIA for horizontal advection

The T n values at eight points around l are,∆µ2∆µ L
+ c20 L ,
2
4
2
∆µ
∆µ
W
n
+ c20 W ,TW
= c00 − c10
2
4
(
)
(
)2
∆µ E
∆µ E
n
TE = c00 + c10 ∆µ L ++ c20 ∆µ L +
,
2
2
(
)
(
)2
∆µWW
∆µWW
n
TWW
= c00 − c10 ∆µW ++ c20 ∆µW +
,
2
2
∆µ L
TNn = TLn + c01 ∆ψu + c02 ∆ψu2 + c11
∆ψu,
2
∆µ L
TSn = TLn − c01 ∆ψ d + c02 ∆ψ d2 − c11
∆ψ d,
2
∆µW
n
n
+ c01 ∆ψu + c02 ∆ψu2 − c11
TNW
= TW
∆ψu,
2
∆µW
n
n
TSW
= TW
− c01 ∆ψ d + c02 ∆ψ d2 + c11∆ψ d .
2
TLn = c00 + c10(8.74)(8.75)(8.76)(8.77)(8.78)(8.79)(8.80)(8.81)

To obtain all six coeﬃcients, six of these equations (points) are used. The equations are chosen according to the following
flow direction.
uln > 0,vln > 0 ⇒ L, W, WW, S, NW, SW(8.82)

uln < 0,uln > 0,uln < 0,vln > 0 ⇒ L, W, E, N, S, SW
vln < 0 ⇒ L, W, WW, N, NW, SW
vln < 0 ⇒ L, W, E, N, S, NW(8.83)(8.84)(8.85)

From equations (8.74) and (8.75),
n∆µW TLn + ∆µ L TW∆µ L ∆µW
− c20
,
2∆µl
4
n
T n − TW∆µ L − ∆µW
c10 = L
− c20
.
∆µl
2

c00 =

(8.86)(8.87)

When uln > 0, from (8.74) and (8.77),
nn
TLn − TW
T n − TWW
− W
∆µl
∆µll
c20 =
,
∆µl + ∆µll∆µW + ∆µWW
where ∆µll ≡.
2

Using equations (8.80) and (8.81),

When uln < 0, from (8.75) and (8.76),

Using equations (8.78) and (8.79),nnn
T n − TSW
TNW
− TW
− W
∆ψu
∆ψ d
c02 =
.
∆ψu + ∆ψ dnTEn − TLn TLn − TW
−
∆µr
∆µl
c20 =
.
∆µr + ∆µl

TNn − TLn TLn − TSn
−
∆ψu
∆ψ d
c02 =
.
∆ψu + ∆ψ d
– 94 –(8.88)(8.89)(8.90)(8.91)

Chapter 8

Tracer advection schemes

When vln > 0, from (8.79) and (8.81),
nn∆µW (TLn − TSn ) + ∆µ L (TW− TSW)+ c02 ∆ψ d,
2∆µl ∆ψ dn
T n − TW
− TSn + TLn
c11 = SW.
∆µl ∆ψ dc01 =(8.92)(8.93)

When vln < 0, from (8.78) and (8.80),nn∆µW (TNn − TLn ) + ∆µ L (TNW− TW)
− c02 ∆ψu,2∆µl ∆ψunn
T n − TLn − TNW+ TW
c11 = N.
∆µl ∆ψuc01 =(8.94)(8.95)

Next, using equation (8.73), the T n values at eight points around d are
∆ψ 2
∆ψ L
+ d 02 L ,
2
4
∆ψ 2
∆ψS
TSn = d 00 − d 01
+ d 02 S ,
2
4
)
)2
(
(
∆ψ N
∆ψ N
+ d 02 ∆ψ L +,TNn = d 00 + d 01 ∆ψ L +
2
2
)
(
)2
(
∆ψSS
∆ψSS
n
+ d 02 ∆ψS +,TSS= d 00 − d 01 ∆ψS +
2
2
∆ψ L
∆µr ,
TEn = TLn + d 10 ∆µr + d 20 ∆µ2r + d 11
2
∆ψ L
n
TW
= TLn − d 10 ∆µl + d 20 ∆µ2l − d 11
∆µl,
2
∆ψS
n
= TSn + d 10 ∆µr + d 20 ∆µ2r − d 11
∆µr ,
TSE
2
∆ψS
n
= TSn − d 10 ∆µl + d 20 ∆µ2l + d 11
TSW
∆µl .
2
TLn = d 00 + d 01

(8.96)(8.97)(8.98)(8.99)(8.100)(8.101)(8.102)(8.103)

From equations (8.96) and (8.97),∆ψS TLn + ∆ψ L TSn∆ψ L ∆ψS− d 02,2∆ψ d
4
T n − TSn∆ψ L − ∆ψSd 01 = L.
− d 02∆ψ d
2d 00 =(8.104)(8.105)

When vdn > 0, from (8.96) and (8.99),nTLn − TSn TSn − TSS
−
∆ψ d
∆ψ dd
d 02 =,∆ψ d + ∆ψ dd∆ψS + ∆ψSS
where ∆ψ dd ≡.
2

From (8.102) and (8.103),

When vdn < 0, from (8.97) and (8.98),nn
TSE− TSn TSn − TSW
−
∆µr
∆µl
d 20 =.
∆µr + ∆µl

TNn − TL TLn − TSn
−
∆ψu
∆ψ d
d 02 =.
∆ψu + ∆ψ d
– 95 –(8.106)(8.107)(8.108)

8.5
From (8.100) and (8.101),UTOPIA for horizontal advectionnTEn − TLn TLn − TW
−
∆µr
∆µl
d 20 =.
∆µr + ∆µl

(8.109)When udn > 0, from (8.101) and (8.103),nn∆ψS (TLn − TW
) + ∆ψ L (TSn − TSW)+ d 20 ∆µl,2∆ψ d ∆µlnn
+ TSW
T n − TSn − TW.
d 11 = L∆ψ d ∆µl

d 10 =(8.110)(8.111)

When udn < 0, from (8.100) and (8.102),n− TSn )
∆ψS (TEn − TLn ) + ∆ψ L (TSE
− d 20 ∆µr ,2∆ψ d ∆µrn
− TLn + TSn
T n − TSE
d 11 = E.
∆ψ d ∆µr

d 10 =(8.112)(8.113)The value of Tln is the average of T n over the hatched area of Figure 8.4. Defining
ξln =

uln
,
hµl

vln
,
hψl

η ln =(8.114)

we have∫ ψL +∆ψL /2 ∫ µl
1nT n dµdψ
Tl = nξl ∆t∆ψ L  ψL −∆ψL /2 µl −ξln ∆t+∫ ψL −∆ψL /2

ψL −∆ψL /2−ηln ∆t∫ µl + ξlnn (ψ−ψL +∆ψL /2)η[lµl −ξln ∆t{T (ψ) − T (ψ + ∆ψ L )}dµdψ 

]
n

n

1
1
1
= c00 − η ln ∆t c01 +∆ψ 2 + ∗ (η n ∆t) 2 c02
212 L 3 l1
11− ξln ∆t c10 + (ξln ∆t) 2 c20 + † ξln η ln ∆t 2 c11 .
2
3
3(8.115)This is the result for uln > 0 and vln > 0. The result is the same independent of the sign of uln and vln .
Similarly,
11
Tdn = d 00 − η dn ∆t d 01 + (η dn ∆t) 2 d 02
2
]
[ 3
1
1
1
1 n
− ξ d ∆t d 10 +∆µ2L + ‡ (ξ dn ∆t) 2 d 20 + § ξ dn η dn ∆t 2 d 11,
2
12
3
3
where
ξ dn =udn,
hµd

η dn =∗ 1 in COCO2
† 1 in COCO4
‡ 1 in COCO2
§ 1 in COCO
4

– 96 –

vdn.
hψd(8.116)(8.117)

Chapter 8

Tracer advection schemes

Therefore,∆µ2L 
1
1
T˜ln = c00 − ξln ∆t c10 +  (ξln ∆t) 2 −
c20
2
12 
 3

1
1
1
− η ln ∆t c01 + (η ln ∆t) 2 c02 + ξln η ln ∆t 2 c11,233
∆ψ 2L 1
1
T˜dn = d 00 − η dn ∆t d 01 +  (η dn ∆t) 2 −
d 02
2
12 
 3


(8.118)1
1
1− ξ dn ∆t d 10 + (ξ dn ∆t) 2 d 20 + ξ dn η dn ∆t 2 d 11 .
2
3
3

(8.119)Finally, we describe how to derive the boundary conditions. Since the face values of the tracers are calculated throughthe second order interpolation, the value of a tracer at a point over land is sometimes necessary. For that case, the value
should be appropriately decided by using the tracer values at the neighboring points in the sea. Since ocean models
generally assume that there is no flux of tracers across land-sea boundary, the provisional value over land should be given
so as not to create a normal gradient at the boundary.
When the face value of a tracer at boundary l is calculated, W and L are not land, but either N or S may be land, andeither NW or SW may be land. When N or S is land, the land-sea boundary runs at the center of L in the zonal direction.
It is reasonable to assume that the value of land grid N or S must not cause any meridional tracer gradient at L set bysecond order interpolation using the values at grids N, L, and S. Thus, we set(TNn − TLn )∆ψ d2 = (TSn − TLn )∆ψu2 .

(8.120)

When NW or SW is a land grid, the following should be assumed.
n
n
n
n
(TNW
− TW
)∆ψ d2 = (TSW
− TW
)∆ψu2

(8.121)

n
n
n
(TWW
− TW
)∆µ2l = (TLn − TW)∆µ2ll .

(8.122)n(TEn − TLn )∆µ2l = (TW− TLn )∆µ2r .

(8.123)

When WW is a land grid,
When E is a land grid,
Similar boundary conditions are specified for face d.

8.6 Second Order Moment (SOM) scheme
8.6.1 Outline
The Second Order Moment (SOM) advection scheme by Prather (1986) seeks to improve the accuracy by treating thetracer distribution within a grid cell, unlike the scheme that aims to calculate the tracer flux at the boundary of grid cellswith high accuracy. It is assumed that the distribution of tracer f in a grid cell (0 ≤ x ≤ X, 0 ≤ y ≤ Y, 0 ≤ z ≤ Z;
volume V = XY Z) can be represented using second order functions as follows:f (x, y, z) = a0 + a x x + a xx x 2 + ay y + ayy y 2 + az z + azz z 2 + a xy xy + ayz yz + azx zx.

(8.124)

Prather (1986) expressed the above as a sum of orthogonal functions Ki (x, y, z);f (x, y, z) = m0 K0 + m x K x + m xx K xx + my Ky + myy Kyy + mz Kz + mzz Kzz + m xy K xy + myz Kyz + mzx Kzx,

– 97 –(8.125)

8.6

Second Order Moment (SOM) scheme

where the orthogonal functions are given as follows:
K0 = 1,K x (x) = x − X/2,
K xx (x) = x 2 − X x + X 2 /6,
Ky (y) = y − Y /2,
Kyy (y) = y 2 − Y y + Y 2 /6,
Kz (z) = z − Z/2,

(8.126)Kzz (z) = z − Z z + Z /6,
2

2

K xy (x, y) = (x − X/2)(y − Y /2),
Kyz (y, z) = (y − Y /2)(z − Z/2),
Kzx (z, x) = (z − Z/2)(x − X/2),∫

andKi K j dV = 0
The constants for normalization are decided using∫
K x2 dV = V X 2 /12,∫(i , j).

(8.127)∫2
K xx
dV = V X 4 /180,∫
Ky2 dV = VY 2 /12,2
Kyy
dV = VY 4 /180,∫∫
Kz2 dV = V Z 2 /12,∫2Kzz
dV = V Z 4 /180,

∫
2
K xy
dV = V X 2Y 2 /144,∫
2
Kyz
dV = VY 2 Z 2 /144,

2Kzx
dV = V Z 2 X 2 /144.

The moments are set by the following expressions:∫S0 =f (x, y, z)K0 dV = m0V,∫
Sx = (6/X )f (x, y, z)K x (x)dV = m x V X/2,∫
Sxx = (30/X 2 )f (x, y, z)K xx (x)dV = m xx V X 2 /6,∫
Sy = (6/Y )f (x, y, z)Ky (y)dV = my VY /2,∫
Syy = (30/Y 2 )f (x, y, z)Kyy (y)dV = myy VY 2 /6,∫
Sz = (6/Z )f (x, y, z)Kz (z)dV = mz V Z/2,∫
Szz = (30/Z 2 )f (x, y, z)Kzz (x)dV = mzz V Z 2 /6,∫
Sxy = (36/XY )f (x, y, z)K xy (x, y)dV = m xy V XY /4,∫
Syz = (36/Y Z )f (x, y, z)Kyz (y, z)dV = myz VY Z/4,∫
Szx = (36/Z X )f (x, y, z)Kzx (z, x)dV = mzx V Z X/4.

(8.128)All these moments are transported with the upstream advection scheme. The procedure is carried out in one direction
at a time. The second and third procedures use the results of the last procedure. For simplicity, we describe the change– 98 –

Chapter 8

Tracer advection schemesof each moment caused by an advection procedure in one direction (x) in a two dimensional plane (x y) in the following.
You may replace (y, Y ) with (z, Z ).
When velocity c in the x direction is positive, the right part of the grid cell,X − ct ≤ x ≤ X,

0 ≤ y ≤ Y,

0 ≤ z ≤ Z,

(8.129)is removed from the cell and added to the adjacent cell on the right during time interval t. This part is expressed using
superscript R. The remaining part,0 ≤ x ≤ X − ct,

0 ≤ y ≤ Y,

0 ≤ z ≤ Z,

(8.130)is expressed by superscript L. New orthogonal functions KiR (KiL ) are calculated in the part R (L) with the volume
V R = ctY Z (V L = (X − ct)Y Z). The orthogonal functions are given as follows:
K0L = K0R = 1,
K xL = x − (X − ct)/2,

K xR = x − (2X − ct)/2,

L
K xx= x 2 − (X − ct)x + (X − ct) 2 /6,
R
K xx= x 2 − (2X − ct)x + (X − ct)X + (ct) 2 /6,

KyL = KyR = y − Y /2,(8.131)

L
R
Kyy
= Kyy
= y 2 − Y y + Y 2 /6,
LK xy= [x − (X − ct)/2](y − Y /2),
RK xy
= [x − (2X − ct)/2](y − Y /2).

The basic quantities for calculating the moments areR
m0R = m0 + K̄ xR m x + K̄ xx
m xx,

m xR = m x + 2K̄ xR m xx,
R
m xx= m xx,(8.132)

myR = my + 2K̄ xR m xy,
Rmyy= myy,
Rm xy= m xy,where K̄ is the average of the new orthogonal function:
K̄ xL = −ct/2,
LK̄ xx= ct(2ct − X )/6,

K̄ xR = (X − ct)/2,
RK̄ xx= (X − ct)(X − 2ct)/6.

The moments in the right part to be removed are expressed as follows:
S0R = α[S0 + (1 − α)Sx + (1 − α)(1 − 2α)Sxx ],SxR = α 2 [Sx + 3(1 − α)Sxx ],RSxx= α 3 Sxx,

(8.133)

SyR = α[Sy + (1 − α)Sxy ],R
Syy= αSyy,R
Sxy= α 2 Sxy,– 99 –

8.6

Second Order Moment (SOM) scheme

where α = α R = ct/X = V R /V . The moments in the remaining part are expressed as follows:
S0L = (1 − α)[S0 − αSx − α(1 − 2α)Sxx ],SxL = (1 − α) 2 (Sx − 3αSxx ),LSxx= (1 − α) 3 Sxx,

(8.134)

SyL = (1 − α)(Sy − αSxy ),LSyy= (1 − α)Syy,
L= (1 − α) 2 Sxy .
SxyAs the final step of the procedure, the orthogonal functions and moments transported from the adjacent cell and those in
the remaining part of the original cell are combined to create new united moments in the cell. The calculation is terribly
complex, and only the results are presented:
S0 = S0R + S0L,
Sx = αSxR + (1 − α)SxL + 3[(1 − α)S0R − αS0L ],L
R+ (1 − α) 2 Sxx+ 5{α(1 − α)(SxR − SxL ) + (1 − 2α)[(1 − α)S0R − αS0L ]},Sxx = α 2 Sxx

Sy = SyR + SyL,
R
L
Syy = Syy
+ Syy
,
R
L
Sxy = αSxy
+ (1 − α)Sxy+ 3[(1 − α)SyR − αSyL ],whereα = α R = V R /(V R + V L ).

(8.135)

The globally integrated tracer is conserved through these operations.

8.6.2 Flux limiter
Some limiters are necessary to guarantee that a tracer is positive (negative) definite.
a. Prather (1986)

Prather (1986) proposed to set limits for the moments related to the direction of advection. For instance, when the moments
are advected in the x direction,
S0 ≥ 0,Sx′ = min[+1.5S0, max(−1.5S0, Sx )],(8.136)′
Sxx= min[2S0 − |Sx′ |/3, max(|Sx′ | − S0, Sxx )],′Sxy= min[+S0, max(−S0, Sxy )].

It should be noted that the application of this limiter does not completely guarantee that the tracer will be positive (negative)
definite.
b. Merryfield and Holloway (2003)

Flux limiters introduced by Morales Maqueda and Holloway (2006) (hereinafter MH06) extended that of Prather (1986)(hereinafter PR86).
Let us consider the advection in the x direction. As in PR86, MH06 consider the mean tracer distribution (τ̄(x)) in thex direction by integrating in the y and z direction within the grid cell (Equation (2.15) of MH06).
∫ Y∫ Z1dy dz τ(x, y, z)
YZ 00[( x )2]x
1
S0 − Sx + Sxx + 2(Sx − 3Sxx ) + 6Sxx
=
,
V
X
X

τ̄(x) =

– 100 –

(8.137)

Chapter 8

Tracer advection schemes

where x, y, z represent the position relative to the lower south-western corner of the grid cell and X, Y , and Z are grid
widths in the x, y, and z directions, respectively. V is the volume of the grid cell.
First we consider the minimum value of τ̄(x). When Sxx is negative (region I), the minimum is taken either at x = 0 or
x = X because τ̄(x) is convex upward. Even when Sxx is positive, if the value of x that gives the global minimum of thequadratic function is not within 0 ≤ x ≤ X (region II), the minimum is taken either at x = 0 or x = X. Otherwise (regionIII), the minimum value is given as the global minimum of the quadratic function. These are summarized as follows:
S0 − |Sx | + Sxx,
 S − |S | + S ,0xxx
V min τ̄(x) =  S − S x2 +3S x2 x ,6S x x
 0

if Sxx ≤ 0
if |Sx | ≥ 3Sxx ≥ 0(region I),(region II),if 3Sxx ≥ |Sx |(region III).

(8.138)The line of V min τ̄(x) = 0 is plotted in the (|Sx |, Sxx ) space in Figure 8.5. If a pair of (|Sx |, Sxx ) resides in the region
bounded by the blue line and the left vertical axis, the minimum value of τ̄(x) is positive.
2S0

S0 - (S2x + 3S2xx)/(6Sxx) = 0

Sxx

S0

0.5S0

0

S0 - |Sx| + Sxx = 0
-S0
0

S0

1.5S0 1.732S0|Sx|

Figure8.5 Schematic to explain the flux limiter to avoid undershooting. This is a reproduction of Figure 3 ofMorales Maqueda and Holloway (2006). See text for explanation.

PR86 proposed to modify the moments so that the minimum of τ̄ is positive. This is to improve the sign-definiteness ofthe tracer. The limiters of MH06 are also based on this strategy. They proposed to modify the moments in the following
way,
[
[
]]
Sx′ = min 31/2 S0, max −31/2 S0, Sx ,
[
]
(
) 1/2
[ ′
]
S′2

2
x

min S0 + S0 − 3, max Sx − S0, Sxx ,if Sx′ < 23 S0,



′

[
[
]
]
(8.139)Sxx = ()) 1/2(
′2 1/2S′2 min S0 + S02 − S3x, max S0 − S02 − 3x
, Sxx , if Sx′ ≥ 23 S0,[]′
Sxy = min S0, max[−S0, Sxy ] ,
[
]
′
Sxz
= min S0, max[−S0, Sxz ] .
′ , S ′ are modified moments and it is assumed that S ≥ 0. This is slightly diﬀerent from PR86 because an
Here, Sx′ , Sxx0xy√
approximation 3 ∼ 1.5 (see 8.136) was used in PR86.
This modification is explained using Figure 8.5. If a pair of (|Sx |, Sxx ) resides in the region that gives negative minimumof τ̄(x), |Sx | is shifted along the horizontal axis so that it is less than 31/2 S0 . Then Sxx is shifted along the vertical axis sothat the point is within the region that gives the minimum of τ̄(x) to be positive. The modification to Sxy, Sxz is designed
so that a negative value of the tracer is avoided by these moments.

– 101 –

8.6

Second Order Moment (SOM) scheme

MH06 proposed to extend the above modifications for setting the lower and upper bounds for the tracers to improve themonotonicity. The lower and upper limit for τ(x) are set as τa and τb , respectively (τa ≤ τ̄(x) ≤ τb ). The setting of thelower limit is achieved by replacing S0 with S0∗ = S0 − V τa in the above modification. For the upper limit, we interpretthe condition "the maximum of τ̄(x) must be less than τb " as "the minimum of −τ̄(x) must be more than −τb ." In otherwords, the condition is "the minimum of τb − τ̄(x) must be positive."
Using (8.137), τb − τ̄(x) may be expressed as follows:[( x )2]1xτb − τ̄(x) =V τb − S0 + Sx − Sxx − 2(Sx − 3Sxx ) − 6Sxx
VXX[( x )2]
1 ∗x=
S + Sx − Sxx − 2(Sx − 3Sxx ) − 6Sxx,
(8.140)
V 0
X
X
where S0∗ = V τb − S0 . Its minimum is obtained as follows:S0∗ − |Sx | − Sxx,
S0∗ − |Sx | − Sxx,
V min [τb − τ̄(x)] =  S ∗ + S x2 +3S x2 x ,6S x x
 0

if Sxx ≥ 0
if |Sx | ≥ −3Sxx ≥ 0(region I),(region II),if − 3Sxx ≥ |Sx |(region III).

(8.141)

Following Figure 8.5, this is visualized as Figure 8.6.
S0

S*0 - |Sx| - Sxx = 0

Sxx0-0.5S0-S0S*0 + (S2x + 3S2xx)/(6Sxx) = 0

-2S0

S0

0

1.5S0 1.732S0|Sx|

Figure8.6 Schematic to explain the flux limiter to avoid overshooting. The region in (|Sx |, Sxx ) space that may set
the upper bound on tracer distribution. The region bounded by the blue curve and the left vertical axis gives the tracer
value that does not exceed the upper bound.

The modification for the moments corresponding to (8.139) is expressed as follows:[[]]
Sx′ = min 31/2 S0∗, max −31/2 S0∗, Sx ,
[
[
]]
(
)
′2 1/2
∗ − S ′ , max −S ∗ − S ∗2 − S x

min
S
,
S
,

xx
x

3
0
0
0

′

[
[
]]
Sxx
=
(
)
) 1/2
(
′2 1/2S′2 min −S0∗ + S0∗2 − S3x, max −S0∗ − S0∗2 − 3x
, Sxx ,
[]′
Sxy = min S0∗, max(−S0∗, Sxy ) ,
[
]
′
Sxz
= min S0∗, max(−S0∗, Sxz ) .

if Sx′ < 23 S0∗,
if Sx′ ≥ 23 S0∗,

(8.142)

MH06 proposed three modification methods (Method A - C). Following COCO, MRI.COM adopts "Method B" of
Morales Maqueda and Holloway (2006) as proposed by Merryfield and Holloway (2003), which lays emphasis on– 102 –

Chapter 8

Tracer advection schemesmonotonicity and suppresses numerical diﬀusion as well. In this method, the minimum and the maximum value among
the three grid cells (i − 1, i, i + 1) are set as the lower and the upper limit, respectively. Moments are adjusted using
Eqs. (8.139) and (8.142). This method does not guarantee exact monotonicity, but improves monotonicity of the zeroth
moment.

8.6.3 Calculating SOM advection in MRI.COM
It should be noted that the coordinate system is not Cartesian in ocean models. Since the coordinate system covers aspherical surface, the x direction in a grid cell is not identical to that in the adjacent cell, for instance. Thus, the exact
conservation of moments cannot be realized. In addition, a tracer-cell including solid earth (sea floor or lateral boundary)is not a cuboid, so the orthogonal functions cannot be defined precisely for such a cell. Nevertheless, the procedures
described in the previous subsection can be carried out using the volume of seawater in the non-cuboid grid cell, and thezeroth moment S0 (total amount of the tracer) is conserved.
As indicated in the expressions in the previous subsections, the volume-integrated moments (Si ) and the fractionof volume to be removed (α) are used in the SOM advection scheme. There are 10 moments for each tracer. The
fraction α is calculated using volume transports (USTARL, VSTARL, and WLWL), which are calculated in the subroutines
continuity__diagnose_horizontal and continuity__diagnose_vertical. Following Prather (1986), the procedures in three directions are executed in order, not simultaneously. By default, the procedure in the meridional direction(advec_y) is called first, the zonal direction (advec_x) next, and lastly the vertical direction (advec_z). The order ofoperations in the horizontal direction may be flipped every time step if runtime option lsomstrang is set to be true (Strang
splitting). This would improve the overall accuracy of the serially executed advection operations (Skamarock, 2006). Thechange in the tracer value caused by SOM advection is estimated in the subroutine tracer__adv and added to the variable
trcal directly.
Usage

Model option SOMADVEC must be specified for compilation when the SOM advection scheme will be used for any tracer.
Namelist nml_somadv is required (see Table 8.2) at run time. You must also specify which tracer will use this scheme aswell as what kind of specifications will be used for that tracer. See Section 13.3.2 and Table 13.3 for details.
Table8.2 Variables defined in namelist nml_somadv

namefile_som_in
file_som_out
nstep_som_restart_intervallsomstrang
lsommonitor∗ i={x, xx, y, yy, z, zz, xy, yz, zx},

explanationbase name of the restart file to be input
base name of the restart file to be output
interval in time steps with which restart
files are output
The order of calling x and y direction isreversed every time step
flag to monitor the conservation of the
moments

string or value{file_som_in}_i_n ∗{file_som_out}_i_n ∗default is 0 (at the end of this job).true. / .false.
.true. / .false.

n={01, 02, ..., numtrc}; tracer number

8.7 MPDATA scheme
8.7.1 Outline
This section explains the MPDATA scheme. In the MPDATA scheme (e.g., Smolarkiewicz and Margolin, 1998), advection
for T is first solved by using the original volume transport, U T , V T , W T , to obtain a temporary value (T (1) ) using the
upstream scheme. Using this temporary value, an anti-diﬀusive volume transport (U (1) , V (1) , W (1) ) is computed. Here,the superscript (1) means that it is the first approximation to the error to be subtracted. This set of transports is used tocompute a value of the next time step using the upstream scheme starting from the above temporary value.
Original dimensionless form of the anti-diﬀusive velocity given by Smolarkiewicz and Margolin (1998) is written as– 103 –

8.7 MPDATA scheme
follows (their equation 16):
u (1) δt
= |U |(1 − U ) A(1) − U V B (1) − U WC (1),δxv (1) δt
= |V |(1 − V )B (1) − VWC (1) − VU A(1),V (1) ≡
δyw (1) δt
W (1) ≡
= |W |(1 − W )C (1) − WU A(1) − WV B (1),δz
U (1) ≡

(8.143)(8.144)(8.145)where U , V, and W are the dimensionless velocity based on the original flow field and A(1) , B (1) , and C (1) are defined as
[

] (1)
δx ∂T
A ≡
,
2T ∂ x
[
] (1)
δy ∂T
B (1) ≡,
2T ∂ y
[
] (1)
δz ∂T
(1)
C ≡
.
2T ∂z
(1)

(8.146)(8.147)(8.148)In MRI.COM, we formulate them by using the set of original volume transports U T , V T , W T . Considering the definitionof the volume transport (8.7) to (8.9), the dimensionless form of volume transports can be written as
Ui+ 1 , j,k− 1 =
2

2

(

2

2

∆t
),

(8.149)(),∆x i, j+ 1 ∆yi, j+ 1 ∆zi− 1 , j+ 1 ,k− 1 + ∆zi+ 1 , j+ 1 ,k− 1(8.150)

2

2

2

2

2

2

2

2V T 1 1 ∆ti, j+ 2 ,k− 2

2

Wi, j,k =i+ 2 , j,k− 21∆x i+ 1 j ∆yi+ 1 , j ∆zi+ 1 , j− 1 ,k− 1 + ∆zi+ 1 , j+ 1 ,k− 1
2

Vi, j+ 1 ,k− 1 =2U T 1

2

2

Wi,Tj,k ∆t∆zi, j,k (areat)i, j,k+ 1

2

2

2

2

2

.

(8.151)

2

The A(1) through C (1) terms at (i + 12 , j, k − 21 ), which are needed to calculate U (1)1

i+ 2 , j,k− 12

A(1)1=
i+ 2 , j,k− 12∆x i+ 1 , j δ x T (1)1

i+ 2 , j,k− 21x2

2T (1)1

i+ 2 , j,k− 12

C (1)1i+ 2 , j,k− 12=

2T (1)1
i+ 2 , j,k− 12=xy∆yi+ 1 , j δ y T (1)12

i+ 2 , jxy

2T (1)1
i+ 2 , j,k− 12(8.153)+ϵxz∆zi+ 1 , j δz T (1)12

(8.152)+ϵi+ 2 , j,k− 12

B (1)1, are

i+ 2 , jxz(8.154)+ϵ

where ϵ is a small number that prevents zero division when tracer values reach zero. The finite diﬀerence and averaging
operators are defined as follows (definitions in y and z directions are the same):δ x Ai ≡x

Ai ≡Ai+ 1 − Ai− 1
2

2

∆x i
Ai+ 1 + Ai− 1
2

2

2

Ai+1 − Ai,∆x i+ 1,

δ x Ai+ 1 ≡,Ai+1 + Ai
xAi+ 1 ≡
.
2
2

2

2

The A(1) through C (1) terms should be estimated at (i, j + 12 , k − 12 ) for V (1) 1

i, j+ 2 ,k− 12– 104 –and (i, j, k) for Wi,(1)j,k .

(8.155)

Chapter 8

Tracer advection schemes

Then anti-diﬀusive volume transport is written as
U (1)1i+ 2 , j,k− 12

V (1) 1i, j+ 2 ,k− 12T= Ui+(1 − Ui+ 1 , j,k− 1 ) A(1)11, j,k− 1
2

2

2

2

i+ 2 , j,k− 12= Vi,Tj+ 1 ,k− 1 (1 − Vi, j+ 1 ,k− 1 )B (1) 1
2

2

2

2

i, j+ 2 ,k− 12

T
− Ui+
V 1
1
1
, j,k− 1 i+ , j,k−2

2

2

Wi,(1)j,k = Wi,Tj,k (1 − Wi, j,k )Ci,(1)j,k − Wi,Tj,k Ui+ 1 , j,k

xz

2

2

2

B (1)1i+ 2 , j,k− 21

2− Vi,Tj+ 1 ,k− 1 Ui, j+ 1 ,k− 1
2

xy

xy

A(1) 1i, j+ 2 ,k− 12

2

Ai,(1)j,k − Wi,Tj,k Vi+ 1 , j,k

yz

2

T
− Ui+
Wi+ 1 , j,k− 1
1, j,k− 1
2

2

2

xz

2

C (1)1i+ 2 , j,k− 12(8.156)− Vi,Tj+ 1 ,k− 1 Wi, j+ 1 ,k− 1
2

2

2

2

yz

C (1) 1i, j+ 2 ,k− 12(8.157)Bi,(1)j,k .

(8.158)8.7.2 Gauge transformation and flux limiter
Original MPDATA scheme is positive definite and cannot be used for tracers that take negative value. This can be avoided
by adding some constant for the temporary value, T (1) .
The monotonicity of MPDATA scheme can be obtained by using a non-oscillatory option proposed by Smolarkiewicz
and Grabowski (1990). In this option, an anti-diﬀusive volume transport has the upper limit given as follows:[
U (1)1] mon[
V (1) 1

] moni+ 2 , j,k− 12

i, j+ 2 ,k− 12[min(1, β out 1 , β in
)U (1)1
i+1, j,k− 12i, j,k− 2
i+ 2 , j,k− 21

=
out
in
 min(1, β)U (1)1,βi, j,k− 12
i+ 2 , j,k− 21
i+1, j,k− 12ifmin(1, β out 1 , β in
)V (1)

i, j+1,k− 12 i, j+ 12 ,k− 12i, j,k− 2= min(1, β out, β in 1 )V (1)1
i, j,k− 2 i+ 2 , j,k− 12
i, j+1,k− 12U (1)1> 0,

ifi+ 2 , j,k− 12
U (1)1< 0,
i+ 2 , j,k− 12

ifV (1) 1> 0,

if

i, j+ 2 ,k− 12
V (1) 1 1 < 0,i, j+ 2 ,k− 2

 min(1, β out

ifWi,(1)j,k > 0,

ifWi,(1)j,k < 0,] mon 
Wi,(1)j,k=, β in 1 )Wi,(1)j,ki, j,k− 2i, j,k+ 12
min(1, β out 1 , β in 1 )Wi,(1)j,ki, j,k+ 2
i, j,k− 2(8.159)(8.160)(8.161)where
βi,inj,k− 1 ≡2

T max 1 − T (1)i, j,k− 2i, j,k− 21,(8.162)− T min 1
i, j,k− 2i, j,k− 12,
Aout 1 + ϵ
i, j,k− 2(8.163)

Ain 1 + ϵi, j,k− 2T (1)βi,out≡j,k− 12

T max 1 = max(i, j,k− 2

T mini, j,k− 12

= min

(

Ti, j,k− 1 , Ti+1, j,k− 1 , Ti−1, j,k− 1 , Ti, j+1,k− 1 , Ti, j−1,k− 1 , Ti, j,k+ 1 , Ti, j,k− 3 ,
2
2
2
2
2
2
2
)
T (1) 1 , T (1)
T (1)
T (1)
T (1)
T (1) 1 , T (1) 3 ,
1,
1,
1,
1,

(8.164)Ti, j,k− 1 , Ti+1, j,k− 1 , Ti−1, j,k− 1 , Ti, j+1,k− 1 , Ti, j−1,k− 1 , Ti, j,k+ 1 , Ti, j,k− 3 ,
2
2
2
2
2
2
2
)
T (1) 1 , T (1)
T (1)
T (1)
T (1)
T (1) 1 , T (1) 3 ,
1,
1,
1,
1,

(8.165)i, j,k− 2i, j,k− 2

i+1, j,k− 2

i+1, j,k− 2i−1, j,k− 2i−1, j,k− 2

i, j+1,k− 2

i, j+1,k− 2i, j−1,k− 2i, j−1,k− 2i, j,k+ 2

i, j,k+ 2

i, j,k− 2i, j,k− 2

and Ain and Aout are the absolute values of the total incoming and outgoing advection flux at the T-box.

8.7.3 Usage
Model option MPDATAADVEC must be specified for compilation when MPDATA will be used for any tracer. At run time,
you must specify which tracer will use this scheme as well as what kind of specifications will be used for that tracer. See
Section 13.3.2 and Table 13.3 for details.

– 105 –,,

Chapter 9

SGS parameterization of lateral mixing of tracersThis chapter explains subgrid-scale parameterizations for horizontal mixing of tracers.

9.1 Introduction and Formulation
Historically, a harmonic diﬀusion operator is applied in each direction of the model coordinates to express mixing oftracers. In the real ocean, transport and mixing would occur dominantly along neutral (isopycnal) surfaces. Thus,
horizontal mixing along a constant depth surface is generally inappropriate since neutral surfaces are generally slanting
relative to a constant depth surface. Neutral physics schemes are devised as substitutes for the harmonic scheme in thehorizontal direction, while the harmonic scheme continues to be used for vertical diﬀusion.
By default, the diﬀusion operator mixes a tracer in each direction of the model coordinates with the harmonic scheme.
For horizontal diﬀusion, the biharmonic scheme can be used instead of the harmonic scheme. Using (20.22) and (20.23),
the harmonic-type diﬀusivity is represented for any tracer (T) as follows:
{
(
)
(
)}
1
∂ hψ z s κ H ∂T
∂ hµ z s κ H ∂T
∂ ( κV ∂T )−z s ∇ · FT = D(T ) =++h µ hψ ∂ µhµ ∂ µ∂ψhψ ∂ψ
∂s z s ∂s
T
T
T
 ∂Fz
 ∂(hψ z s Fµ ) ∂(hµ z s Fψ ) 
1 
−

+
,
=−
(9.1)


h µ hψ
∂µ∂ψ∂swhere κ H and κV are the horizontal and vertical diﬀusion coeﬃcients and the diﬀusive fluxes are represented by F. D(T )represents the diﬀusion operator for T. When the biharmonic-type is selected for horizontal diﬀusion, the above Laplacian
operation is repeated twice reversing its sign after the first operation.
When the neutral physics schemes are selected, the advection-diﬀusion equation for any tracer (T) is expressed as follows(Gent and McWilliams, 1990):
[ ∂
]
]
DT
∂ [
+ ∇ H · T (κT ∇ H ρ/ρz ) +T∇ H · (−κT ∇ H ρ/ρz ) = D(T ) + QT ,
Dt
∂z
∂z

(9.2)

where the first term on the r.h.s. is the isopycnal diﬀusion, whose form is given byD(T ) = ∇ · (κ I K∇T ),

(9.3)

where
ρ2 + ρ2z
*. y
1
K= 2
. −ρ x ρyρ x + ρ2y + ρ2z −ρ ρx z,

−ρ x ρyρ2x + ρ2z
−ρy ρz−ρ x ρz+−ρy ρz //ρ2x + ρ2y -1 + ( ρy /ρz ) 2
1*. −( ρ x /ρz )( ρy /ρz )=1 + ( ρ x /ρz ) 2 + ( ρy /ρz ) 2−ρ x /ρz,(9.4)−( ρ x /ρz )( ρy /ρz )1 + ( ρ x /ρz ) 2−ρy /ρz−ρ x /ρz+/ ,−ρy /ρz22( ρ x /ρz ) + ( ρy /ρz ) -(9.5)(Redi, 1982). In the above, the Cartesian notation is used for simplicity. The subscript x represents ∂/(hµ ∂ µ) y represents∂/(hψ ∂ψ), and z represents ∂/(z s ∂s), The isopycnal diﬀusion coeﬃcient is represented by κ I . Diapycnal diﬀusion is notconsidered here.

– 107 –

9.2

Horizontal diﬀusion

The second and third terms on the l.h.s. of (9.2) have the form of advection terms with a transport velocity vector(uT , vT , wT ):∂ (1 ∂ρ ∂ρ)κT/,∂zhµ ∂ µ ∂z∂ (1 ∂ρ ∂ρ)
κT
,
vT ≡
/
∂z
hψ ∂ψ ∂z
1 { ∂ ( hψ ∂ ρ ∂ ρ )∂ ( hµ ∂ ρ ∂ ρ ) }
κT
+
κT
,
wT ≡ −//h µ hψ ∂ µhµ ∂ µ ∂z∂ψhψ ∂ψ ∂z
uT ≡

(9.6)
(9.7)
(9.8)

(Gent and McWilliams, 1990). This velocity can be understood as the advection caused by the thickness diﬀusion of an
isopycnal layer, where κT is often referred to as thickness diﬀusivity.
Note that these could be rewritten asG(T ) = ∇ · (κT A∇T )
(9.9)
with
0
A = *. 0, ρ x /ρz00ρy /ρz−ρ x /ρz−ρy /ρz +/ .
0-

(9.10)

Comparing with (9.5), we notice that the isopycnal diﬀusion and the thickness diﬀusion terms are combined to yield aconcise form (Griﬃes, 1998) and (9.2) can be rewritten as:DT= ∇ · {(κ I K − κT A)∇T } + QT .
Dt

(9.11)

Three types of horizontal diﬀusion, harmonic horizontal diﬀusion (default), biharmonic horizontal diﬀusion (TRCBIHARM
option), and isopycnal diﬀusion (ISOPYCNAL option), are available in MRI.COM. When isopycnal diﬀusion (Redi, 1982)
is selected, the parameterization scheme for eddy induced advection by Gent and McWilliams (1990) (GM scheme) isused with it (see Section 9.4 for details of this parameterization).
The following is a guide to selecting a horizontal diﬀusion scheme. Biharmonic diﬀusion is appropriate for a high resolution model that can resolve eddies because it is more scale-selective than harmonic diﬀusion and does not unnecessarily
suppress disturbances in resolved scales. On the other hand, biharmonic diﬀusion is not recommended in eddy-less models
because this would result in numerical instability. Harmonic horizontal diﬀusion is also not recommended because thisscheme would cause unrealistic cross-isopycnal (diapycnal) mixing as mentioned above. Instead, both isopycnal diﬀusion
and the GM scheme should be used there. Using an anisotropic GM scheme can maintain the meso-scale eddy structuresand swift currents by restricting the direction of diﬀusion, and thus may be usable even for a high resolution model.
The finite diﬀerence expression of (9.11) is given by taking the finite volume approach as follows:
Ti,n+1∆Vi,n+1= Ti,n−1∆Vi,n−1+ 2 ∆t {FXDi− 1 , j,k− 1 − FXDi+ 1 , j,k− 1 + FYDi, j− 1 ,k− 1 − FYDi, j+ 1 ,k− 1j,k− 1j,k− 1j,k− 1j,k− 1
2

2

2

2

2

2

2

2

2

+ FZDi, j,k − FZDi, j,k−1 } + (other terms).

2

2

2

(9.12)

Usually, fluxes due to diﬀusion are explicitly represented using a starting time level of the temporal discretization. However,when the flux is very large relative to the grid size and the time step interval, which would often occur for vertical fluxes,
an implicit scheme is used. These issues are discussed in Chapter 19.

9.2 Horizontal diﬀusion
9.2.1 Laplacian diﬀusion
Harmonic horizontal diﬀusion assumes that the diﬀusion flux is a product of the gradient of tracer and the diﬀusion
coeﬃcient (κ H ). The finite diﬀerence expressions for the fluxes are given as follows:y

FXDi+ 1 , j,k− 1 = − κ H ∆yi+ 1 , j ∆zi+ 1 , j,k− 1 δ x Ti+ 1 , j,k− 1 ,
2

2

2

2

2

2

2

x

FYDi, j+ 1 ,k− 1 = − κ H ∆x i, j+ 1 ∆zi, j+ 1 ,k− 1 δ y Ti, j+ 1 ,k− 1 ,2

2

2– 108 –

2

2

2

2

(9.13)(9.14)

Chapter 9

SGS parameterization of lateral mixing of tracers

where
Ti+1, j,k− 1 − Ti, j,k− 1

δ x Ti+ 1 , j,k− 1 ≡22

2∆x i+ 1 , j

2,(9.15).

(9.16)

2

Ti, j+1,k− 1 − Ti, j,k− 1

δ y Ti, j+ 1 ,k− 1 ≡222∆yi, j+ 1

2

2

9.2.2 Biharmonic diﬀusion
Biharmonic horizontal diﬀusion (TRCBIHARM option) assumes that the diﬀusion flux is proportional to the gradient of the
Laplacian of tracer. The finite diﬀerence expressions for the fluxes are given as follows:√y
FXDi+ 1 , j,k− 1 = κ b ∆yi+ 1 , j ∆zi+ 1 , j,k− 1 δ x [Lap (T )]i+ 1 , j,k− 1 ,(9.17)
2
2
2
2
2
2
2
√
x
(9.18)
FYDi, j+ 1 ,k− 1 = κ b ∆x i, j+ 1 ∆zi, j+ 1 ,k− 1 δ y [Lap (T )]i, j+ 1 ,k− 1 ,
2

2

2

2

2

2

2

where κ b is diﬀusivity coeﬃcient andLap (T )i, j,k− 1 =2∆x i, j ∆yi, j√√yx(δ x κ b ∆zi, j,k− 1 δ x Ti, j,k− 1 + δ y κ b ∆zi, j,k− 1 δ y Ti, j,k− 1 ).
2
2
2
2
∆Vi, j,k− 1(9.19)

2

9.2.3 Specification of coeﬃcient
The diﬀusion coeﬃcient of horizontal diﬀusion is specified using the namelists listed on Tables 9.1 and 9.2. This must be
zero if ISOPYCNAL option is selected, unless the horizontal diﬀusion is applied intentionally.
Table9.1 Namelist nml_tracer_diff_horz

variable name
diff_horz_cm2ps
diff_horz_cm4ps
file_diff_horz_2d

units
cm2 s−1
cm4 s−1characterdescription
Laplacian diﬀusion coeﬃcient (κ H )Biharmonic diﬀusion coeﬃcient (κ b )2D distribution of diﬀusion coeﬃcient is read fromthis fileusage
if not TRCBIHARM
if TRCBIHARM

Table9.2 Namelist nml_grid_size_change_mix_coefs

variable namel_grid_size_change_mix_coefs

units
logicaldescription
the given coeﬃcient is multiplied by thefraction of the local grid size to 100 km.

usage

9.3 Isopycnal diﬀusion
In isopycnal diﬀusion, the diﬀusion flux is expressed by high diﬀusivity along the isopycnal surface κ I , low diﬀusivity
across the isopycnal surface κ D , and their product with the gradient of tracer in each direction. Using diﬀusion tensor K,
each flux component is written as F m (T ) = −K mn ∂nT, and then
1 + Sy2 + ϵ Sx2*.
K=. (ϵ − 1)Sx Sy
1 + S2, (1 − ϵ )SxκI(ϵ − 1)Sx Sy
1 + Sx2 + ϵ Sy2(1 − ϵ )Sy(1 − ϵ )Sx+(1 − ϵ )Sy // ,ϵ + S2 -(9.20)

where
∂ρ ∂ρ
*.
+
∂ y //
S = (Sx, Sy, 0) = ..− ∂ x , −, 0/ ,. ∂ρ ∂ρ /S = | S |,
κD
ϵ=
κI

, ∂z

∂z

(9.21)(9.22)(9.23)– 109 –

9.3

Isopycnal diﬀusion(Redi, 1982).
The finite diﬀerence method is based on Cox (1987) except for the small isopycnal slope approximation. The finite
diﬀerence form of three components of the gradient of tracer in calculating the east-west component of flux FXDi+ 1 , j,k− 1
2
2
is given as follows (see Figure 9.1):(δ x T )i+ 1 , j,k− 1 = δ x Ti+ 1 , j,k− 1 ,2

2

2(9.24)2(δ y T )i+ 1 , j,k− 1 = δ y Ti+ 1 , j,k− 1xy,(9.25)(δz T )i+ 1 , j,k− 1 = δz Ti+ 1 , j,k− 1xz.

(9.26)xy,(9.27)

2

2

2

2

2

2

2

2

Similarly, the north-south component FYDi, j+ 1 ,k− 1 is given by22(δ x T )i, j+ 1 ,k− 1 = δ x Ti, j+ 1 ,k− 1
2

2

2

2

(δ y T )i, j+ 1 ,k− 1 = δ y Ti, j+ 1 ,k− 1 ,
2

2

2

2

yz

(δz T )i, j+ 1 ,k− 1 = δz Ti, j+ 1 ,k− 1 ,
2

2

2

2

(9.28)(9.29)and the vertical component FZDi, j,k is given byxz(δ x T )i, j,k = δ x Ti, j,k ,(9.30)yz(δ y T )i, j,k = δ y Ti, j,k ,(9.31)(δz T )i, j,k = δz Ti, j,k .

(9.32)Ti, j+1

Ti+1, j+1

Ti, j

Ti+1, jTi, j-1

Ti+1, j-1

Figure9.1 The way of calculating the gradient at the circle (i + 12 , j, k − 21 ) in isopycnal diﬀusion: the east-west
gradient is indicated by an arrow through the circle, and the north-south gradient is given by averaging four arrows in
the vertical direction.

The density gradient in the calculation of each component of the diﬀusion tensor can be obtained by replacing T in
the above equations with ρ. However, density is calculated at the reference level k − 21 for the east-west and north-south
components, and at the reference level k for the vertical component.
In addition, the upper bound on the isopycnal slope Smax (slope_clip_iso in namelist nml_isopy_slope_clip,
Table 9.4) is set because a nearly vertical isopycnal slope and the resultant low horizontal diﬀusivity could cause numericalinstability. If |S| > Smax , ∂z ρ in all flux components is replaced so as to satisfy |S| = Smax .
– 110 –

Chapter 9

SGS parameterization of lateral mixing of tracers

The vertical flux due to the third diagonal component of the diﬀusion tensor (9.20) is
FZDi, j,k = (areat)i, j,kThus the eﬀective vertical diﬀusivity κ eﬀ isκ eﬀ =

κ I (ϵ + S 2 )δz Ti, j,k .
1 + S2

κ D + κ I S2.
1 + S2(9.33)(9.34)For a steep isopycnal slope S ∼ 1/100 and a canonical value of isopycnal diﬀusion coeﬃcient κ I ∼ 107 cm2 s−1 and atypical value of diapycnal diﬀusion coeﬃcient κ D ∼ 10−1 cm2 s−1 ,κ eﬀ ∼ 103 cm2 s−1 .

(9.35)

This is a fairly large value which warrants use of an implicit scheme (Section 19.5). In MRI.COM, this term is separated
from other terms and solved with other vertical diﬀusion terms using an implicit method.
Griﬃes et al. (1998) noted a problem in the finite diﬀerence expression of the isopycnal diﬀusion as implemented in
the GFDL-model by Cox (1987). The problem is that the down-gradient orientation of the diﬀusive fluxes along theneutral directions does not necessarily guarantee the zero isoneutral diﬀusive flux of locally referenced density (e.g.,potential temperature when it is the only active tracer). This is caused by the nature of the finite diﬀerence method andthe non-linearity of the equation of state. Griﬃes et al. (1998) proposed a remedy, but this remains to be implemented in
MRI.COM.

9.3.1 Tapering around steep isopycnal slopes
As noted above, we set an upper bound on the isopycnal slope used to evaluate isopycnal tracer diﬀusion terms in MRI.COM
in order to prevent numerical instability around steep isopycnal slopes. Griﬃes (2004) shows that such slope clipping
could lead to an unrealistically large tracer flux. Danabasoglu and McWilliams (1995) propose another method that uses
a hyperbolic tangent to exponentially taper isopycnal diﬀusion in steep slope regions. The following factor is multiplied
to the isopycnal diﬀusion coeﬃcient there:{()}
1
Scenter − |S|f steep =1 + tanh.
(9.36)
2
SwidthTwo parameters Scenter and Swidth , which determine a transitional region, are given by center_transition andwidth_transition in namelist nml_tracer_diff_isopy_taper, respectively (Table 9.5). This tapering method
is valid when ISOTAPER option is chosen and used with the other factor f surface introduced in the subsequent section.

9.3.2 Surface tapering
The vertical displacement of a water parcel due to mesoscale eddy stirring (D) is approximately calculated as follows:
D = R|S|,(9.37)

where R is the internal deformation radius and |S| is the isopycnal slope. If the depth of the water parcel (d) is shallower
than D, the boundary constrains its displacement and eddy diﬀusive fluxes (Griﬃes, 2004). The ISOTAPER option enables
the use of the following tapering scheme proposed by Large et al. (1997):
{
(
)}
1
1
f surface =1 + sin π r −,(9.38)
22
where r = max(0, min(1, d/D)). Eddy diﬀusivity is tapered oﬀ to zero toward the sea surface in the region where
0 ≤ d ≤ D. In MRI.COM, the reference level of the depth d is set at the boundary layer depth (BLD): d = −z − BLD. In
MRI.COM, the surface mixed layer depth (MLD) is treated as the BLD (see also the next section). This means that theeddy diﬀusivity is tapered to zero within the mixed layer. The upper boundary of this tapering region can be changed to a
constant level by setting upper_level_isotaper_m in namelist nml_tracer_diff_isopy_taper (Table 9.5).
By default, the horizontal diagonal terms of the isopycnal diﬀusion tensor (9.20) are not multiplied by the two factors
noted above. This means that the isopycnal diﬀusion is rendered the horizontal diﬀusion within the boundary layer
and in the steep slope region. One may apply these tapering methods to the horizontal diagonal terms by setting
l_apply_hdiag = .true. in namelist nml_tracer_diff_isopy_taper (Table 9.5).
– 111 –

9.4 Gent and McWilliams parameterization for eddy-induced transport

9.3.3 Specification of coeﬃcient
The diﬀusion coeﬃcients of isopycnal diﬀusion and GM parameterization explained in the next section are specified using
the namelist listed on Table 9.3. We can use diﬀerent slope maximal limits for isopycnal diﬀusion and GM parameterizationthat are specified using the namelist listed on Table 9.4. Configurations of surface tapering for the isopycnal diﬀusion
scheme with ISOTAPER option are specified using the namelist listed on Table 9.5.
Table9.3 Namelist nml_tracer_diff_isopy

variable name
diff_isopy_cm2ps
diff_diapy_cm2ps
diff_thick_cm2psunits
cm2 sec−1
cm2 sec−1
cm2 sec−1

description
isopycnal diﬀusion coeﬃcient
diapycnal diﬀusion coeﬃcient
coeﬃcient of GM parameterization

usage
if ISOPYCNAL
if ISOPYCNAL
if ISOPYCNAL

Table9.4 Namelist nml_isopy_slope_clip

variable nameslope_clip_iso

units
1

slope_clip_gm

1

description
maximum slope of isopycnal surface for isopycnal diﬀusion
maximum slope of isopycnal surface for GM
parameterization

usage
if ISOPYCNAL
if ISOPYCNAL

Table9.5 Namelist nml_tracer_diff_isopy_taper

variable namel_apply_hdiag

units
logicalcenter_transition1

width_transition1

ai_minupper_level_isotaper_m
l_explicit_vdif

cm2 sec−1mlogicaldescription
apply tapering factors to horizontal diagonal terms (default = .false.)
center of the transition region of taperingwith hyperbolic tangent (default = 0.005)
width of the transition region of taperingwith hyperbolic tangent (default = 0.001)
lower limit of horizontal isopycnal diﬀusion coeﬃcient when l_apply_hdiag =.true. (default = 0.0)
BLD for the sine taper (default = mixed
layer depth (−1.0))
handle the vertical diﬀusion term explicitly(default = .false.)

usage
if ISOTAPER
if ISOTAPER
if ISOTAPER
if ISOTAPER

if ISOTAPER
if ISOTAPER

9.4 Gent and McWilliams parameterization for eddy-induced transport
9.4.1 General features
The Gent and McWilliams (1990) parameterization represents transports of tracers due to disturbances smaller than thegrid size, assuming that a flux proportional to the gradient of the layer thickness exists along the isopycnal surface. The
isopycnal diﬀusion stated above does not produce any flux when the isopycnal surface coincides with the isotherm andisohaline surface. This parameterization, however, produces fluxes in such a case, and acts to decrease the isopycnal slope.
Flux convergence due to diﬀusion is expressed as follows:
R(T ) = ∂m (J mn ∂nT )(9.39)

Diﬀusion tensor J mn is expressed as the sum of the symmetric component K mn = (J mn + J nm )/2 and the anti-symmetriccomponent Amn = (J mn − J nm )/2. Isopycnal diﬀusion has the form of a symmetric diﬀusion tensor. Convergence of a

– 112 –

Chapter 9

SGS parameterization of lateral mixing of tracersm = −Amn ∂ T is as follows:
skew flux caused by the anti-symmetric component Fskewn

R A (T ) = ∂m ( Amn ∂nT )= ∂m ( Amn )∂nT
= ∂n (∂m AmnT ),(9.40)

where Amn ∂m ∂nT = 0 and ∂m ∂n Amn = 0 are used. If we set a virtual velocity u∗n ≡ −∂m Amn , then the flux due to theanti-symmetric component could be regarded as the advection due to this virtual velocity u∗ . In this case, the flux is
Fadv = u∗T and R A (T ) = −u∗ · ∇T since u∗ is divergence free by definition.
The Gent and McWilliams parameterization for eddy-induced transport velocity is given byu∗ = −/ ∂h∂(κ GM ∇ρ h),∂ρ∂ρwhere h is the depth of the neutral surface (ρ = const). This velocity is expressed in the depth coordinate as()
u∗ = −∂z (κ GM Sx ), −∂z (κ GM Sy ), ∇h · (κ GM S)
where

(
)
S = (Sx, Sy, 0) = −ρ x /ρz , −ρy /ρz , 0 .

(9.41)(9.42)(9.43)(Gent et al., 1995).
Griﬃes (1998) showed that the tendency of a tracer due to this parameterization might be expressed using an antisymmetric diﬀusion tensor A00−κ GM Sx00−κ GM Sy +/ ,A = *.
(9.44)
κ
S
κ
S
0
GM y
, GM xso that∂T= · · · + ∇ · (A∇T ).
∂t(9.45)

The flux due to advection can be expressed using a vector streamfunction,Ψ = κ GM ẑ × S = (−κ GM Sy, κ GM Sx, 0),
which produces u∗ in (9.42):(9.46)

Fadv = u∗T = T (∇ × Ψ).

The skew diﬀusive expression for the flux using tensor A in (9.44) is
Fskew = −A∇T = −(∇T ) × Ψ = Fadv − ∇ × (T Ψ).
Thus, the convergence of the flux expressed in tensorial form matches that of the advective expression. In other words, theGent and McWilliams parameterization is realized by only adding A to the tensor of the isopycnal diﬀusion K (Griﬃes,1998).

9.4.2 Dependency of coeﬃcient on space and time
By default, the diﬀusivity coeﬃcient for the Gent-McWilliams parameterization is constant both in time and space, whose
value, [κ GM ]ref , is given by diff_thick_cm2ps in namelist nml_tracer_diff_isopy (Table 9.3). However, it may be
dependent on local horizontal grid size by specifying a namelist (see the next paragraph). Several parameterization may
be used by choosing GMVAR option. User should specify one of l_visbeck, l_eden, l_danabasoglu to be .true. in
namelist nml_gmvar_select (Table 9.8).
a. Simple scheme

If l_grid_size_change_mix_coefs = .true. in namelist nml_grid_size_change_mix_coefs (Table 9.7), the coefficient may be dependent on the horizontal grid size according to the following formulaκ GM = [κ GM ]ref × min(∆x, ∆y)/(100 km),where ∆x and ∆y are local zonal and meridional grid sizes of a U-cell, respectively.
– 113 –

(9.47)

9.4 Gent and McWilliams parameterization for eddy-induced transport

b. Visbeck et al. (1997)

To use the method proposed by Visbeck et al. (1997), l_visbeck = .true. in namelist nml_gmvar_select.
Visbeck et al. (1997) proposed to give the GM coeﬃcient κ GM asκ GM = αwhere α = 0.015,
M2 =

M2 2
l ,
N

g
∇H ρ ,
ρ0(9.48)

N2 = −

g ∂ρ
,
ρ0 ∂z

and l the horizontal length scale of the baroclinic zone, g acceleration of gravity, ρ0 reference density.
Specifically in our model,) 2 (∫ −H0) 2  12 (∫ −H0

g
∂
ρ
∂
ρ

M2 =
dz +
dz  ,
ρ0 D  −H1 ∂ x
∂
y

−H1

and
N2 =

g [σ0 (H1 ) − σ0 (H0 )],ρ0 D(9.49)(9.50)

where H0 = 100 m, H1 = 2000 m, D = H1 − H0 , and σ0 is the potential density. Lower limit for N is set so thatN 2 ≥ 10−9 s−1 .
Using the following formula for the phase speed of the 1st baroclinic mode gravity wave (Sueyoshi and Yasuda, 2009)
1
c1 =
π

∫ 0

(

−H B

g ∂σ0
−
ρ0 ∂z

) 21
dz,

where HB is the depth of sea floor, deformation radius λ 1 is calculated as follows:()
c14
λ 1 = min, 4 × 10m.
f

(9.51)(9.52)

Using a factor r = 7, GM coeﬃcient is determined as follows:κ GM = αM2(r λ 1 ) 2 .
N(9.53)

Lower and upper limits for the coeﬃcient are set as follows:
300 ≤ κ GM ≤ 1500

m2 s−1 .

(9.54)

c. Eden and Greatbatch (2008)

To use the method proposed by Eden and Greatbatch (2008), l_eden = .true. in namelist nml_gmvar_select.
Eden and Greatbatch (2008) and Eden et al. (2009) proposed that the thickness diﬀusivity is given byκ GM = cL 2 σ(9.55)The eddy length scale L is given as the minimum of the Rossby radius L r and Rhines scale L Rhi . This choice for L wasfound to be consistent with independent estimates of eddy length scales from satellite observations and high-resolution
model results (Eden, 2007) and with recent theoretical considerations (Theiss, 2004). L Rhi is estimated from variables of
the coarse resolution model asσ
L Rhi =
(9.56)
β(Eden and Greatbatch, 2008), while L r is given by[c1
L r = min,|f|
– 114 –

√

]
c1
,
2β

(9.57)

Chapter 9

SGS parameterization of lateral mixing of tracers

where c1 denotes the 1st baroclinic gravity wave speed calculated approximately as eq. (9.51). Considering the thermal
wind relation in mid-latitudes, Eden and Greatbatch (2008) proposed that the inverse eddy time scale σ is given by
σ = f (Ri + γ) − 2 .
1(9.58)∂
uh | −2 denotes the local Richardson number. γ(> 0) is introduced to prevent the singularity as N → 0,
Here, Ri = N 2 | ∂zwhich acts eﬀectively as an upper limit for σ and consequently for κ GM . The default values of γ and c in eq. (9.55), are
200 and 2, respectively.

d. Danabasoglu and Marshall (2007)

To use the method proposed by Danabasoglu and Marshall (2007), l_danabasoglu = .true. in namelist
nml_gmvar_select.
Danabasoglu and Marshall (2007), guided by Ferreira et al. (2005) and Ferreira and Marshall (2006), proposed to
specify the vertical variation of κ GM using[ N2 ]
κ GM =[κ GM ]ref,(9.59)
2
Nref
where N is the local buoyancy frequency and [κ GM ]ref is the constant reference value of κ GM within the surface diabatic
layer. Nref is the reference buoyancy frequency obtained just below the diabatic layer, in other words, the first stable N 2
2 is set to 1 for all shallower depths. Between the depth at which N 2 = N 2
below surface diabatic layer. The ratio N 2 /Nref
refand the ocean bottom, we also ensure that
N2
Nmin ≤ 2 ≤ 1.0,
(9.60)
Nref
where Nmin is the lower limit specified by the user (ratio_bvf_min in namelist nml_gmvar_danabasoglu).

9.4.3 Surface tapering
By default, no specific modification is applied to the eddy-induced transport velocity of the Gent-McWilliams parameterization near the surface and the bottom, except for limiting the isopycnal slope to a specified value (slope_clip_gm innamelist nml_isopy_slope_clip). This may result in too strong transport velocity in the first vertical level of the model(sea surface). The problem may be overcome by tapering the transport in the surface mixed layer, where the transport
is made nearly or completely uniform in the vertical direction. This is realized by choosing either SLIMIT or GMTAPER
option.
a. Simple scheme

By choosing SLIMIT option, the Gent-McWilliams coeﬃcient (κ GM ) is linearly reduced from the value at the base of
mixed layer to zero at the sea surface within the mixed layer,
κ GM (z) = κ GM (z = −MLD) × (−z/MLD) for − MLD ≤ z ≤ 0,

(9.61)

where MLD is the depth of the mixed layer. The MLD is defined as the level at which the local potential density is larger
than the surface density by a specified value, given by the user (default value is 0.03 kg m−3 ).
b. Danabasoglu et al. (2008)

By choosing GMTAPER option, a practical scheme proposed by Danabasoglu et al. (2008) is used. This scheme modifies
the Gent-McWilliams vector stream function for eddy induced transport velocity near the surface, aiming to implement anear-surface parameterization proposed by Ferrari et al. (2008). Concept of the near-surface parameterization is as follows(Danabasoglu et al., 2008):• In the turbulent boundary layer (BL), the eddy-induced velocity is set parallel to the boundary and has no vertical
shear, as expected in the mixed layer.
• There is an eddy diﬀusion of buoyancy along the boundary as well as along isopycnals.
• In the interior the parameterization satisfies the adiabatic constraint as in the original scheme.
• The two forms are matched through a transition layer that separates the quasi-adiabatic interior with isopycnicallyoriented eddy fluxes from the near boundary regions.

– 115 –

9.4 Gent and McWilliams parameterization for eddy-induced transport
Two vertical length scales must be estimated to implement this parameterization: the boundary layer depth (BLD) andthe transition layer thickness (TLT). Their sum is defined as the diabatic layer depth (DLD), over which the upper-ocean
eddy fluxes depart from their interior formulas. In MRI.COM, the surface mixed layer depth (MLD) is treated as the BLD.
The MLD is defined as the level at which the local potential density is larger than the surface density by a specified value,
given by the user (default value is 0.03 kg m−3 ). The TLT is defined by the range of isopycnals that can be lifted into theboundary layer by eddy heaving, which is given by the product of the internal deformation radius (R) and the isopycnal
slope (|S|):
D = R|S|.
(9.62)
Thus we calculate D at each grid point and the DLD is obtained as follows:
DLD = BLD + D.

(9.63)

Now the near-surface expression for the eddy-induced vector streamfunction is given in the following. The streamfunctionis split into its boundary layer, ΨBL , and transition layer, ΨTL , expression as follows:
ΨBL =
and
ΨTL =

( z + BLD ) 2
TLTη−zΨ0 for − BLD ≤ z ≤ ηη + BLDΦ+( η−z )
Ψ0 for − DLD ≤ z < −BLD
η + BLD(9.64)(9.65)The two functions Ψ0 and Φ are chosen such that Ψ and its vertical derivative are continuous across the base of BLD
and the base of TLT. These constraints then yieldη + BLD(2ΨI + TLT∂z ΨI )2(η + BLD) + TLT(9.66)TLT(ΨI + (η + DLD)∂z ΨI ),2(η + BLD) + TLT(9.67)

Ψ0 =
and
Φ=−

where ΨI is the interior eddy-induced streamfunction at the base of the transition layer given by the Gent-McWilliams
parameterization,z × ∇H ρ
ΨI = −κ GM
at z = −DLD.
(9.68)∂z ρ
In the implementation, to evaluate both ΨI and ∂z ΨI at z = −DLD, ΨI are evaluated at the vertical grid points thatstraddle z = −DLD.
Danabasoglu et al. (2008) also showed that the model solutions are not very sensitive to their transition layer
thickness. Whether the transition layer is included or not may be specified by l_transition_layer in namelist
nml_gm_transition.

9.4.4 Anisotropic Gent-McWilliams scheme
An anisotropic GM scheme (Smith and Gent (2004), GMANISOTROP option), which gives greater diﬀusivity only in the
direction of the current vector, is also available. Using unit vector n̂ = (n x, ny ) in an arbitrary direction, the two-dimensional
anisotropic diﬀusion tensor is defined as follows:() ()κ A n2x + κ B n2y
κ B n x ny
L M
K2 =
=
,
(9.69)M N
κ B n x ny
κ B n2x + κ A n2ywhere κ A is the diﬀusivity in the n̂ direction, and κ B is that in the direction normal to n̂. This is applied to the antisymmetric tensor in the Gent-McWilliams scheme, and the following expression is obtained (Smith and Gent, 2004),00A′ = *.
, LSx + M Sy00
M Sx + N Sy

−LSx − M Sy
−M Sx − N Sy +/ .
0-

(9.70)In the choice of GMANISOTROP option, n̂ is set in the direction of the local horizontal velocity. The value of κ A is readfrom namelist nml_tracer_diff_isopy (variable name diff_thick_cm2ps). The ratio of κ B /κ A is read from namelist
nml_gmanisotrop (variable name cscl_isotrop). The default value of cscl_isotrop is set to 1/2.
– 116 –

Chapter 9

SGS parameterization of lateral mixing of tracers

9.4.5 Usage Summary
How to specify the overall behavior of the Gent-McWilliams parameterization is summarized as follows.
a. Model options

Model options related to the GM parameterization are listed on Table 9.6
Table9.6 List of model options related to GM parameterization.

option nameGMVAR
SLIMIT
GMTAPERGMANISOTROPAFC

description
Coeﬃcient of GM parameterization is allowed to varyLinearly reduce the coeﬃcient of GM parameterizationfrom the bottom of the mixed layer to the sea surface
Taper GM vector stream function near the sea surface
An-isotropic horizontal variation of GM parameterization
Calculate additional flux by using horizontal gradients ofdensity and velocity (Hirabara et al., 2010)

usage
specify nml_gmvar
cannot be used with GMTAPER
cannot be used with SLIMIT,
GMANISOTROP, AFC
specify nml_gmanisotrop
cannot be used with TRCBIHARM

b. Spatial dependency

The diﬀusion coeﬃcient of GM parameterization may be grid size dependent by using the namelist listed on Table 9.7.
Table9.7 Namelist nml_grid_size_change_mix_coefs

variable namel_grid_size_change_mix_coefs

units
logicaldescription
the given coeﬃcient is multiplied by thefraction of the local grid size to 100 km.

usage
default = .false.

Overall behavior of GM parameterization with GMVAR option should be specified by using the namelists listed on Tables
9.8 through 9.12.
Table9.8 Namelist nml_gmvar_select for GMVAR

variable namel_visbeckl_edenl_danabasoglu

units
logicallogicallogicaldescription
use Visbeck et al. (1997)
use Eden and Greatbatch (2008)
use Danabasoglu and Marshall (2007)

usage
choose only one of the three options
choose only one of the three options
choose only one of the three options

Table9.9 Namelist nml_gmvar_visbeck for GMVAR

variable namestart_depth
base_depth
cscl_gmvar
upper_limit
lower_limit
lcalc_defrad

units
cm
cm
1
cm2 sec−1
cm2 sec−1
logicaldefrad_const

cm

length_factor

1

description
density gradients are averaged from start_depth
density gradients are averaged to base_depth
parameter for GM diﬀusivity calculation
upper limit of thickness diﬀusivity
lower limit of thickness diﬀusivity
flag whether deformation radius is calculated ornotupper limit of deformation radius whenlcalc_defrad = .true.
constant horizontal length scale whenlcalc_defrad = .false.
[horizontal length scale] = [deformation radius] ×length_factor– 117 –

usage
l_visbeck = .true.
l_visbeck = .true.
l_visbeck = .true.
l_visbeck = .true.
l_visbeck = .true.
l_visbeck = .true.
l_visbeck = .true.

l_visbeck = .true.

9.4 Gent and McWilliams parameterization for eddy-induced transport
Table9.10 Namelist nml_gmvar_eden for GMVAR

variable name
c_EG

units
1

gamma_EG
upper_limit
lower_limit

1
cm2 sec−1
cm2 sec−1

description
In this parameterization, Thickness diﬀusivity isparameterized as c_EG × L 2 × sigma_EG.
sigma_EG = f /(Ri + gamma_EG)
upper limit of thickness diﬀusivity
lower limit of thickness diﬀusivity

usage
l_eden = .true.
l_eden = .true.
l_eden = .true.
l_eden = .true.

Table9.11 Namelist nml_gmvar_danabasoglu for GMVAR

variable nameratio_bvf_min

units
1

ratio_bvf_max

1

description
Lower bound for the squared buoyancy frequency
relative to the reference value (default = 0.1)
Upper bound for the squared buoyancy frequency
relative to the reference value (default = 1.0)

usage
l_danabasoglu = .true.
l_danabasoglu = .true.

Table9.12 Namelist nml_gm_transition

variable namel_transition_layer

units
logicaldescription
include transition layer into diabatic
layer (default = .false.)

usage
eﬀective when l_danabasoglu =.true.

c. Anisotropic scheme

Behavior of GMANISOTROP option should be specified using the namelist listed on Table 9.13.
Table9.13 Namelist nml_gm_anisotrop

variable namecscl_isotrop

units
1

description
factor for anisotropy in GM diﬀusivity– 118 –

usage
if GMANISOTROPChapter 10

SGS parameterization of vertical mixing oftracers
This chapter explains subgrid-scale parameterizations of vertical mixing of tracers.

10.1 Vertical diﬀusion
Vertical diﬀusion term takes the form of Laplacian and the vertical diﬀusion flux is proportional to the vertical gradient oftracer. The finite diﬀerence form is as follows:
FZDi, j,k = −κ z (areat)i, j,k+ 1 δz Ti, j,k ,(10.1)

2

where the use of (areat)i, j,k+ 1 implies that the flux occurs only through the oceanic part of the grid interface and2δz Ti, j,k ≡Ti, j,k− 1 − Ti, j,k+ 1
2

∆zk

2

.

(10.2)

Note that, for simplicity, the change of the grid thickness at the bottom and fluctuations of the surface height are notconsidered in the grid distance ∆zk when calculating the gradient.
In most realistic simulations, a backward (implicit) scheme is used in the time integration (VVDIMP option; Section 19.5)
because high diﬀusivity is expected owing to the choice of parameterizations needed for realistic simulations. Otherwise,a forward scheme is used.

10.1.1 Specification of coeﬃcient
Background vertical diﬀusivity, which is horizontally uniform, a function of depth, and fixed in time, should be always
given. Additionally, a three dimensional distribution can be set by selecting VMBG3D option to incorporate locally enhanced
mixing processes induced by interaction between the bottom topography and tidal currents (e.g., St. Laurent et al., 2002).
With this choice, three dimensional distributions for vertical diﬀusivity and viscosity should be prepared in advance.
Tables 10.1 and 10.2 summarizes how to give background vertical diﬀusivity.
Table10.1 Namelist nml_diff_vert_bg. Specify only one of the two variables

variable name
diff_vert_bg_cm2psunits
cm2 s−1

file_diff_vert_1d_cm2ps

description
vertically uniform value of background vertical diﬀusivity
file having vertical 1D distribution

usage
Usable only if a vertically uniform
value is intended
cannot be specified with the above

Table10.2 Namelist nml_vmbg3d. Specify when VMBG3D is selectedvariable namefile_vmix_3d
imvm
jmvm

units

description
file having 3D distribution
east-west data size
north-south data size

usage

Continued on next page

– 119 –

10.2 Convective adjustment

variable namel_vmintpol
slatvmslonvmdlatvmdlonvmunits
logicalTable 10.2 – continued from previous page
description
usage
interpolate input data to model grid points
latitude of the western end
only if l_vmintpol = .true.
longitude of the southern end
only if l_vmintpol = .true.
uniform grid spacing in the meridional direction
only if l_vmintpol = .true.
uniform grid spacing in the zonal direction
only if l_vmintpol = .true.

In addition to the static background profiles, the following processes give varying vertical diﬀusivity coeﬃcients atevery model time step.
• Surface mixed layer models (TURBULENCE option).
• Vertical component of isopycnal diﬀusion (ISOPYCNAL option).
• Enhanced diﬀusivity (= 1.0 m2 s−1 = 104 cm2 s−1 ) where the stratification is unstable (DIFAJS option).
• Enhanced diﬀusivity around rivermouths to avoid too low salinity if river run-oﬀ is received by the model (RUNOFF
option). This scheme is especially needed when positive definiteness is not guaranteed by a tracer advection
algorithm. See Table 10.3 for how to specify the mixing.
The vertical diﬀusion for "this" time step is taken as the largest of the above estimations.
Table10.3 Namelist nml_vmix_river. Specify when RUNOFF is selectedvariable namel_enhance_vmix_rivmouth

units
logicaldiff_max_vmixriv_cm2ps

cm2 sec−1

depth_max_vmixriv_cm

cm

para_vmixriv_1

1

para_vmixriv_2

1

description
diﬀusivity is enlarged around therivermouth
maximum value of the enlarged diffusivity (= κ rivmax )vertical diﬀusion is enlarged fromsurface to this depth, this is also usedby subroutine salinity_limit
parameter for the enlarged vertical
diﬀusion formula (= a). See below.
parameter for the enlarged vertical diﬀusion formula (= b). Enlarged diﬀusion is calculated asκ riv = min(alog10 Wriver +b, κ rivmax ),where Wriver is river discharge ratein cm sec−1

usage
default = .false.
default = 1 × 104 cm2 sec−1
default = 30 × 102 cm
default = 10
default = 7

10.2 Convective adjustment
Convective adjustment is realized by replacing the density (temperature and salinity) that is statically unstable (the upper
density exceeds the lower density) in a water column with the averaged density between neighboring levels (neutralization),
considering that interior convection occurs in that place. Most of the realistic phenomena represented by the convective
adjustment include the developing mixed layer due to surface cooling during winter. Convective adjustment also includes
the case in which dense bottom water flows out the sill and flows down along the slope. Moreover, the convective adjustment
includes the practical eﬀect that it suppresses disturbances caused by the numerical calculation error and smooths the
distribution.
In general, there are three numerical schemes for convective adjustment. In the simplest one, adjustment is done for a pairof two neighboring levels, and then for a pair of another two neighboring levels. By repeating this procedure, it attempts
to neutralize the density in the unstable part. This procedure is simple at each step, but it has a defect that the finite-time
repetition does not necessarily guarantee reaching the complete averaged value. In the second scheme, adjustment isdone by assigning a high vertical diﬀusivity between the two levels that are statically unstable and by solving the vertical
diﬀusion term using an implicit method. This method cannot remove the unstable condition completely in one procedure.
However, it has good calculation eﬃciency for the case where the model has a high vertical diﬀusivity already due to themixed layer or isopycnal diﬀusion schemes and thus needs an implicit method to solve it. In MRI.COM, this scheme isinvoked by specifying DIFAJS option. The vertical diﬀusivity between the unstable grid points is set to 10 000 cm2 s−1 .
– 120 –

Chapter 10

SGS parameterization of vertical mixing of tracers

In the third scheme, the unstable part is first neutralized. The stability at the top and bottom of the neutralized column isthen reexamined. If the unstable condition remains, the part including the already-neutralized column is re-neutralized.
This procedure continues until the instability at the top and bottom of the neutralized column disappears. This method
can remove the unstable part completely and thus is called "Complete Convection," but it requires a number of iterations,the vertical level size minus one, at maximum. The third method, which is the default scheme in MRI.COM, is explained
below (Ishizaki, 1997).

10.2.1 Algorithm
In order to minimize the judgment process ("IF" statement) and replace it by arithmetic calculation, this scheme defines
two integer indices, α k and βk , at the layer boundaries, and six real variables TU k , TLk , SU k , SLk , VU k , and VLk , (k =1, KM − 1), in addition to the vertical rows of temperature, salinity, and density Tk , Sk , Rk , (k = 12 , KM − 12 ) (KM is thenumber of levels; see Figure 10.1). The level at the vertical boundary of a T-cell corresponds to the integer k. The index
α k indicates an unstable part within a water column: α k = 1 if it is unstable at the level between k − 12 and k + 21 , andα k = 0 if it is neutral or stable. The index βk memorizes the mixed part: βk = 1 at the boundary where it is neutral asa result of mixing, and βk = 0 elsewhere. Variables TU k , SU k , and VU k and TLk , SLk , and VLk are temperature, salinityand volume accumulated by multiplying α above the level k and below the level k, respectively, and are expressed by thefollowing recursive relation.
VU 1 = α1V 1 ,2

VU 2 = α2 (V1+ 1 + α1 V 1 ) = α2 (V1+ 1 + VU 1 ),
2

2

2

· ··,
VU k = α k (Vk− 1 + VU k−1 ),2· ··,VU KM−1 = αKM−1 (VKM−1− 1 + VU KM−2 ),(10.3)2andVLKM−1 = αKM−1 VKM− 1 ,2

VLKM−2 = αKM−2 (VKM−1− 1 + αKM−1VKM− 1 ) = αKM−2 (VKM−1− 1 + VLKM−1 ),
2

2

2

· ··,
VLk = α k (Vk+ 1 + VLk+1 ),2· ··,VL1 = α1 (V1+ 1 + VL2 ),(10.4)

2

where Vk+ 1 denotes a volume of the cell at the level k + 12 . In a similar way, other quantities are expressed as follows:2

TU 1 = α1T 1 V 1 ,2

TU k = α k (Tk− 1 Vk− 1 + TU k−1 ),SU 1 = α1 S 1 V 1 ,SU k = α k (Sk− 1 Vk− 1 + SU k−1 ),

2

2

2

2

2

2

2

TLKM−1 = αKM−1TKM− 1 VKM− 1 ,2

T L k = α k (Tk+ 1 Vk+ 1 + TLk+1 ),SLKM−1 = αKM−1 SKM− 1 VKM− 1 ,SL k = α k (Sk+ 1 Vk+ 1 + SLk+1 ),

2

2

2

2

2

2

where Tk+ 1 and Sk+ 1 are temperature and salinity at the level k + 21 .
2
2
According to this definition, if α k = 1 and elsewhere 0, we get
VU k + VLk = Vk− 1 + Vk+ 1 ,2

2

TU k + TLk = Tk− 1 Vk− 1 + Tk+ 1 Vk+ 1 ,
2

2

2

2

SU k + SLk = Sk− 1 Vk− 1 + Sk+ 1 Vk+ 1 ,2

2– 121 –

2

2

2

(10.5)

10.2 Convective adjustment
Surface
α1

k=1/2

T1/2

k=1+1/2

T1+1/2

k-1/2

Tk-1/2

k+1/2

Tk+1/2

k=1

α2

k=2

αk-1

k-1

αk

k

αk+1

k+1

αKM-2
k=KM-2

TKM-1-1/2
k=KM-1-1/2

αKM-1
k=KM-1

TKM-1/2
k=KM-1/2
k=KM

BottomFigure10.1 Reference vertical grid points in Section 10.2

indicating a volume and accumulated temperature and salinity in an unstable part and
TU k + TLk,
VU k + VLk
SU k + SLk
SM k− 1 ,k+ 1 =,2
2
VU k + VLk

TM k− 1 ,k+ 1 =2

2(10.6)

are volume averaged temperature and salinity, respectively.
If the level k constitutes a series of the unstable part, the same equation holds for the averaged temperature and salinity.
For example, let α k−1 = α k = 1 and α k−2 = α k+1 = 0,
VU k−1 + VLk−1 = VU k + VLk= Vk−1− 1 + Vk− 1 + Vk+ 1 ,2

2

2

TU k−1 + TLk−1 = TU k + TLk= Tk−1− 1 Vk−1− 1 + Tk− 1 Vk− 1 + Tk+ 1 Vk+ 1 ,
2

2

2

2

2

2

SU k−1 + SLk−1 = SU k + SLk= Sk−1− 1 Vk−1− 1 + Sk− 1 Vk− 1 + Sk+ 1 Vk+ 1 ,
2

2

2

2

2

2

(10.7)

andTU k + TLk
TU k−1 + TLk−1=,
VU k−1 + VLk−1 VU k + VLk
SU k−1 + SLk−1
SU k + SLk
SM k−1− 1 ,k+ 1 ==.
2
2
VU k−1 + VLk−1 VU k + VLk

TM k−1− 1 ,k+ 1 =2

2

These are averages of the three layer, k − 1 − 12 , k − 12 , and k + 12 .
– 122 –

(10.8)

Chapter 10

SGS parameterization of vertical mixing of tracers

10.2.2 Numerical procedure
In summary, numerical procedures are summarized as follows.
[1] Density is calculated at the intermediate depth between adjacent levels using a pair of temperature and salinity andis judged to be statically stable or unstable. If an instability occurs, α(α 1 ) is replaced by 1, otherwise by 0. At this stage,β( β 0 ) is set to 0, where the superscript denotes the number of the iteration.
After this preprocessing, the following procedure (represented by n-th) is repeated until the instability is removed.
[2] Based on equations (10.3) to (10.5), VU, TU, SU, VL, TL, and SL are calculated using α n for a water column that
includes an unstable part.
[3] The vertical mean TM and SM are calculated for the unstable part using equation (10.6) and substituted for theoriginal values of T and S. This change modifies the density at the intermediate depth in [1].
[4] The value of α n is stored in β n . β n = 1 is set if α n = 1, or α n = 0 and β n−1 = 1, and otherwise β n = 0. This ispresented by the following:
βkn = α kn + βkn−1 (1 − α kn ).
(10.9)[5] The static stability is judged only for βkn = 0. Let α kn+1 = 1 if statically unstable, and 0 otherwise. If there is no
unstable part, the procedure for that water column is completed.
[6] For a water column which still includes an unstable part, modification for α kn+1 is done using βkn by the following.
After the procedure [2], any instability will be found only at the bottom of the part that is neutral as a result of prior mixing.
In that case, the neutral part must be treated as an unstable part, that is, α kn+1 = 1. On the other hand, no more procedureis needed if the upper and lower end is stable, giving α kn+1 = 0. This is done by a recursive formula going down and up in
the following.
γ1 = α1(n+1),

γk = α k(n+1) + (1 − α k(n+1) ) βk(n) γk−1

(n+1)
αKM−1
= γKM−1,(n+1)
α k(n+1) = γk + (1 − γk ) βk(n) α k+1,(10.10)where γ is a work variable, but may be treated as α itself in a FORTRAN program. Then, the procedure goes back to [2].
Table 10.4 shows an example of the case with six levels. Static instability is removed after the three-time iteration. The
second column of α in the table is the result of the corrected α kn+1 using βkn based on equation (10.10), as described in [6].
Note that βk0 = 0, though there is no description in the table.

– 123 –

10.2 Convective adjustment

Table10.4 Example of the convective adjustment procedure
n

1

VU+VL

TU+TL

β

V 1 + V 1 1 + V2 12
2
2
V 1 + V 1 1 + V2 1

T 1 V 1 + T1 1 V1 1 + T2 1 V2 1
2
2
2
2
2
2
T 1 V 1 + T1 1 V1 1 + T2 1 V2 1

1
1

0
0
V5 1

0
0
V4 1 + V5 1

00
T4 1 V4 1 + T5 1 V5 1

0
0
1

V1 1 + V2 1 + V3 1

V 1 + V1 1 + V2 1 + V3 1

VU

VL

1
2

1
1

1
1

V1
2
V 1 + V1 1

3
4
5

0
0
1

0
0
1

0
0
V4 1

1

0

1

V1

2

2

α

k

3

0
1

1
1

2

2

V1 1 + V2 1
2
2
V2 1
2

2

2

2

2

V 1 + V1 1
2

2

2

2

2

V2 1 + V3 12

V 1 + V1 1 + V2 1
2

2

2

2

V3 1
2

2

2

2

2

2

2

2

2

2

2

2

2

2

3
∑

2

2

2

2

2

Tk+ 1 Vk+ 1

1

Tk+ 1 Vk+ 1

1

Tk+ 1 Vk+ 1

1

0
0

0
1

2

k=0
3
∑

2

2

2

k=0
3
∑

2

V 1 + V1 1 + V2 1 + V3 12

2

2

V 1 + V1 1 + V2 1 + V3 1
2

2

2

2

2

2

k=0

4
5

0
0

0
0

0
0

0
0

1

0

1

V1

V1 1 + V2 1 + V3 1 + V4 1 + V5 1

2

3

3
4
5

0
0
1
0

1
1
1
1

2

2

V 1 + V1 1
2

2

2

2

2

2

2

2

2

2

2

2

2

2

V 1 + V1 1 + V2 1 + V3 1 + V4 1
2

2

2

2

2

Vk+ 1
Vk+ 1
Vk+ 1

2

Vk+ 1

Tk+ 1 Vk+ 1

1

Tk+ 1 Vk+ 1

1

Tk+ 1 Vk+ 1

1

Tk+ 1 Vk+ 1

1

2

k=0
5
∑

2

k=0
5
∑

2

k=0

1

2

k=0
5
∑

2

k=0
5
∑

Tk+ 1 Vk+ 1
2

k=0
5
∑

2

k=0
5
∑

2

– 124 –

5
∑

2

k=0
5
∑

2

V5 1

Vk+ 1
2

k=0
5
∑

2

V4 1 + V5 1
2

5
∑

2

V3 1 + V4 1 + V5 1

V 1 + V1 1 + V2 1 + V3 1
2

2

V2 1 + V3 1 + V4 1 + V5 1

V 1 + V1 1 + V2 1
2

0
0

2

k=0

2

2

2

2

2

Chapter 11

Biogeochemical model
There are several options for Biogeochemical models in MRI.COM. These biogeochemical models have been developed forboth ocean-only and coupled ocean-atmosphere-vegetation carbon cycle studies. They feature an explicit representationof a marine ecosystem, which is assumed to be limited by light, temperature, and nutrients availability. This chapter
describes the details of the biogeochemical models.

11.1 Inorganic carbon cycle and biological model
Biogeochemical models are composed of inorganic carbon-cycle and ecosystem component models. In the inorganiccarbon-cycle component, the partial pressure of CO2 at the sea surface (pCO2 ) is diagnosed from the values of dissolved
inorganic carbon (DIC) and Alkalinity (Alk) at the sea surface, which should be aﬀected by the ecosystem component.
The diﬀerence in pCO2 between the atmosphere and ocean determines uptake or release of CO2 from the ocean to theatmosphere and is essential for simulating the CO2 concentration in the atmosphere. Inorganic carbonate chemistry
and partial pressure physics are well understood and can be reproduced with fair accuracy. The ecosystem component
deals with various biological activities, and gives sources and sinks of the nutrients, DIC, Alk, and dissolved oxygen
through these activities. Our knowledge of these activities is far from complete, and they are diﬃcult to estimate even instate-of-the-art models.
There are many biological models and methods for calculating the ecosystem components. One of the simplest biologicalmodels has only one nutrient component (such as PO4 ) as a prognostic variable and calculates neither phytoplankton nor
zooplankton explicitly. In these cases, the export of biologically generated soft tissue (organic matter) and hard tissue(carbonate) to the deep ocean, collectively known as the biological pump, is parameterized in terms of temperature,salinity, shortwave radiation, and nutrients.
A Nutrient-Phytoplankton-Zooplankton-Detritus (NPZD) model is more complex than the above model, but still a
simple biological model. The NPZD model has four prognostic variables (nutrient, phytoplankton, zooplankton, anddetritus). Though parameterized in a simple form, basic biological activities, such as photosynthesis, excretion, grazing,and mortality are explicitly calculated.
More complex models classify phytoplankton and zooplankton into several groups, and deal with many complex
interactions between them. In general, it is expected that the more complex the biological model becomes the more
realistic pattern the model can simulate. However, because of our incomplete knowledge about the biological activities,
the complex models do not always yield better results, even though they require more computer resources.
To simulate the carbon cycle in the ocean, some biological processes should be calculated in the ecosystem component
to obtain DIC at the sea surface. However, the carbon cycle component is not always necessary when our interests are tosimulate the ecosystem itself. The Ocean Carbon-Cycle Model Intercomparison Project (OCMIP) protocols and studies
of Yamanaka and Tajika (1996) and Obata and Kitamura (2003) focus on the former carbon cycle in the ocean, and theecosystem components in these studies are quite simple. Biogeochemical models adopted in MRI.COM are classified inthis category. The latter studies usually use complex biological models such as NEMURO (Kishi et al., 2001). Of course,
this type of model could be adopted as an ecosystem component of the biogeochemical model in the former studies inhopes of better simulation of carbon cycle.
Originally, the carbon cycle component followed the OCMIP protocols (Orr et al., 1999) whose authority is recognizedin the community. Recently, biogeochemical protocols for CMIP6 require that the former OCMIP code should be replacedby mocsy routines (Orr and Epitalon, 2015) to use the equilibrium constants recommended for best practices (Orr et al.,2016). MRI.COM can choose either option by setting a namelist. Here we describe the procedure using the mocsy routines.
MRI.COM has several options for the ecosystem component. At present, MRI.COM can incorporate the Obata andKitamura model (Obata and Kitamura, 2003) or an NPZD model based on Oschlies (2001). The biogeochemical modelof MRI.COM is largely based on Schmittner et al. (2008) when an NPZD model is adopted as an ecosystem component.
Basic units in MRI.COM are cgs, but in these biogeochemical subroutines, we use MKS units for the sake of future– 125 –

11.2 Governing equations
development. We use mol/m3 for the units of nutrients. When the coeﬃcients of their model are applied, they should be
converted to the corresponding units.

11.2 Governing equations
Here we describe the biogeochemical models of MRI.COM. When an NPZD model is incorporated as the ecosystemcomponent, the governing equations are as follows. When Obata and Kitamura model is used instead of the NPZD model,
the first four biogeochemical compartments (DIC, Alk, PO4 , and O2 ) are used.
∂DIC= −A(DIC) + D (DIC) + Sb (DIC) + Jv (DIC) + Jg (DIC),(11.1)∂t∂Alk= −A(Alk) + D(Alk) + Sb (Alk) + Jv (Alk),(11.2)∂t∂[PO4 ]= −A([PO4 ]) + D([PO4 ]) + Sb ([PO4 ]),(11.3)∂t∂[O2 ]= −A([O2 ]) + D ([O2 ]) + Sb ([O2 ]) + Jg ([O2 ]),(11.4)∂t∂[NO3 ]= −A([NO3 ]) + D([NO3 ]) + Sb ([NO3 ]),(11.5)∂t∂[PhyPl]= −A([PhyPl]) + D([PhyPl]) + Sb ([PhyPl]),(11.6)∂t∂[ZooPl]= −A([ZooPl]) + D([ZooPl]) + Sb ([ZooPl]),(11.7)∂t∂[Detri](11.8)= −A([Detri]) + D ([Detri]) + Sb ([Detri]),∂twhere A() and D() represent advection and diﬀusion operator, respectively, and Sb () is source minus sink due to thebiogeochemical activities. The square brackets mean dissolved concentration in mol/m3 of the substance within them.
The terms represented by Jg () and Jv () are the air-sea gas fluxes (Section 11.3.1) and the dilution and concentration eﬀects
of evaporation and precipitation on DIC and Alk (Section 11.3.2), respectively, which appear only at the sea surface. The
term Jg () is calculated based on the OMIP protocol by using the air-sea gas transfer velocity and concentration in theseawater. The term Jv () appears only when the salinity flux is given virtually instead of the increase or decrease of thevolume at the surface layers due to evaporation and precipitation.

11.3 Carbon cycle component
To estimate Jg and Jv , we follow the protocol of OMIP, which is described in detail in Orr et al. (2016). The program tocalculate them is based on the mocsy subroutines. We have modified this subroutine so that it can be used in the calculationof the MRI.COM code.

11.3.1 Air-sea gas exchange fluxes at the sea surface ( Jg )The air-sea gas transfer must be calculated for DIC and [O2 ]. The terms Jg (DIC) and Jg ([O2 ]) appear only in theuppermost layer. When these fluxes are expressed as Fg (DIC) and Fg ([O2 ]), Jg (DIC) and Jg ([O2 ]) are given as follows:
Jg (DIC) =Fg (DIC),∆z 1(11.9)

2

Fg ([O2 ])Jg ([O2 ]) =,∆z 1

(11.10)2

where
Fg (DIC) = KwCO2 ∗ ([CO2 ]sat − [CO2 ]surf ),(11.11)Fg ([O2 ]) = KwO2 ∗ ([O2 ]sat − [O2 ]surf ),(11.12)and ∆z 1 is the thickness of the first layer of the model. Here a standard gas transfer formulation is adopted. KwCO2 and KwO2
2
are their gas transfer velocity, [CO2 ]sat and [O2 ]sat are their saturation concentrations with respect to the atmosphere, and[CO2 ]surf , [O2 ]surf , are their surface concentration.
– 126 –

Chapter 11

Biogeochemical modela. Gas transfer velocity

The coeﬃcients KwCO2 and KwO2 are the air-sea gas transfer (piston) velocity and are diagnosed as follows:() −1/2ScCO2
2U10(1 − f i ),660
( O2 ) −1/2
Sc
O2
2
Kw = aU10(1 − f i ),660

KwCO2 = a(11.13)(11.14)where• f i is the fraction of the sea surface covered with ice,
• U10 is 10 m scalar wind speed,• a is the coeﬃcient of 0.251 cm hr−1 /(m s−1 ) 2 , which will give Kw in units of cm hr−1 when winds speeds (U10 ) arein m s−1 . This is specified in the OMIP protocol,• ScCO2 and ScO2 are the Schmidt numbers for CO2 and O2 .
b. Saturation concentration with respect to the atmosphere

The surface water gas concentration in equilibrium with the atmosphere (saturation concentrations) for gas A has thefollowing relationship.
[A]sat = K0 f A(11.15)= K0 C f p A(11.16)= K0 C f (Pa − pH2 O)x A(11.17)= ϕAx APa
≃ ◦ ϕ0A x APaPa= ◦ [A]0sat,
Pa(11.18)(with errors less than 0.1%)(11.19)(11.20)where K0 is its solubility, f A is its atmospheric fugacity, C f is its fugacity coeﬃcient, p A is its atmospheric partial pressure,x A is its mole fraction in dry air, Pa is total atmospheric pressure, Pa◦ is the standard atmosphere (=1013.25 hPa), pH2 O
is water vapor pressure at saturation, ϕ A is its solubility function, ϕ0A and [A]0sat are its solubility function and saturation
concentration at the reference pressure.
Orr et al. (2016) explain two approaches to calculate the saturation concentration.
The first approach uses equations (11.19) or (11.20), computing solubility function ϕ0A or saturation concentration [A]0sat
at the reference atmospheric pressure first, and then converting them to those at the atmospheric pressure Pa . For gasesthat are often used as tracers in oceanography such as CFCs, ϕ0A can be expressed as a function of in-situ temperature andsalinity by using empirical fit. For oxygen, an empirical fit for [O2 ]0sat is available. This is a rather conventional approach,and is used in MRI.COM except for carbon.
The other approach uses equations (11.15)-(11.17), computing its partial pressure p A ≡ (Pa −pH2 O)x A, then multiplying
by K ′ ≡ K0 C f , or multiplying by C f and then multiplying by K0 if atmospheric fugacity ( f A ) is needed. For typicalgases, K ′, K0 , and C f are available as a function of temperature and salinity. The mocsy routine adopts this approach, and
MRI.COM also follows it for calculating carbon flux at the surface.
c. Computing CO2 concentrations at the surface

In the mocsy routine, [CO2 ]surf is diagnosed every step from DIC, Alk, temperature, salinity, [PO4 ], and silicate concentration at the surface. After [CO2 ]surf is computed, fugacity ( f CO2 ) and partial pressure (pCO2 ) of the ocean surface arecomputed as
f CO2 = [CO2 ]surf /K0,

(11.21)

pCO2 = f CO2 /C f ,(11.22)where K0 and C f are estimated by using temperature and salinity.
– 127 –

11.3 Carbon cycle component
Diagnosis of [CO2 ]surf is the most complex of the above calculations and has the largest computational cost. To be
precise, diagnosis of [CO2 ] actually means diagnosing [CO2 ] + [H2 CO3 ], which are diﬃcult to distinguish analytically.
These two species are usually combined and the sum is expressed as the concentration of a hypothetical species, [CO∗2 ] or
[H2 CO∗3 ]. Here, the former notation is used. The relationship between this [CO∗2 ] and DIC is as follows:
DIC = [CO2 ] + [H2 CO3 ] + [HCO−3 ] + [CO2−3 ]= [CO∗2 ](11.23)+ [HCO−3 ] + [CO2−
3 ].

(11.24)In the OMIP protocol, the following equations are solved to obtain [CO∗2 ].
The equilibrium constants for dissociation reactions are:
K1 =[H+ ][HCO−3 ][CO∗2 ]

[H+ ][H2 PO−4 ][H3 PO4 ]

K2P =

KSi =

[H+ ][CO2−3 ],[HCO−3 ](11.25)[H+ ][B(OH) −4 ],[B(OH) 3 ]

KB =

K1P =

K2 =

[H+ ][HPO2−4 ][H2 PO−4 ](11.26)

K3P =[H+ ][PO3−4 ][HPO2−
4 ][H+ ][SiO(OH) −3 ],[Si(OH) 4 ],(11.27)(11.28)KW = [H+ ][OH− ],(11.29)[H+ ]F [SO2−4 ],−[H SO4 ](11.30)[H+ ]F [F− ],[HF](11.31)

KS =
and

KF =

where [H+ ] is the hydrogen ion concentration in sea water and [H+ ]F is the free hydrogen ion concentration. There isanother scale for the hydrogen ion concentration, the total hydrogen ion concentration [H+ ]T . The subscript T means"total" and F means "free." These three hydrogen ion concentrations are related as follows:
(
)
ST
FT
+
+
[H ] = [H ]F 1 ++,(11.32)
KS KF
(
)
ST
and
[H+ ]T = [H+ ]F 1 +.
(11.33)KS
There are three pH scales corresponding to these three hydrogen ion concentrations.
The equilibrium constants K x are given as a function of temperature, salinity, and pH. Note that the equilibrium constants
are given in terms of concentrations, and that all constants are referenced to the seawater pH scale, except for KS , whichis referenced to the free pH scale.
The total dissolved inorganic carbon, boron, phosphate, silicate, sulfate, and fluoride are expressed as follows:
DIC = [CO∗2 ] + [HCO−3 ] + [CO2−3 ],(11.34)

BT = [B(OH) 3 ] + [B(OH) −4 ],(11.35)

3−
PT = [H3 PO4 ] + [H2 PO−4 ] + [HPO2−
4 ] + [PO4 ],(11.36)

Si T = [Si(OH) 4 ] + [Si(OH) −3 ],(11.37)– 128 –

Chapter 11

Biogeochemical modelandST = [HSO−4 ] + [SO2−
4 ],(11.38)

FT = [HF] + [F− ].

(11.39)Alkalinity used in this calculation is defined as follows:
−
−
2−
3−
−
Alk = [HCO−3 ] + 2[CO2−3 ] + [B(OH) 4 ] + [OH ] + [HPO4 ] + 2[PO4 ] + [SiO(OH) 3 ]

− [H

+

(11.40)]F − [HSO−4 ] − [HF] − [H3 PO4 ].

These expressions exclude the contribution of NH3 , HS− , and S2− .
If we assume that DIC, Alk, [PT ], and [SiT ] are known, this system contains 18 equations with 18 unknowns, so they canbe solved. Detailed solver method has been described in Orr and Epitalon (2015). The concentration [SiT ] is not predictedin the biogeochemical model adopted in MRI.COM but is rather specified by the annual mean value from WOA2013.

11.3.2 Dilution and concentration eﬀects of evaporation and precipitation on DIC and Alk
The dilution and concentration eﬀects of evaporation and precipitation significantly impact the concentrations of somechemical species in seawater. This is particularly true for DIC and Alk, which have large background concentrations
compared with their spatial variability. MRI.COM uses a free surface, so the impact of evaporation and precipitation isstraightforward to model unless SFLUXW or SFLUXR option is used. In these options, salinity flux is diagnosed and appliedinstead of the freshwater flux. In this case, the dilution and concentration eﬀect of evaporation (E) and precipitation (P)should be taken into account. Here, they are parameterized as virtual DIC and Alk fluxes, similar to the virtual salt flux
used in physical ocean GCMs.
In MRI.COM, the tendency of salinity due to the virtual salt flux is given bysflux(i, j) = −(P − E) ∗ S(i, j, 1)/∆z,(11.41)

where S(i, j, 1) and ∆z are the salinity and thickness of the uppermost layer. Note that the variable sflux(i, j) is not thesalinity flux but the time change rate of salinity due to the flux even though the spelling brings up the image of the flux.
In MRI.COM, DIC and Alk are modified by the virtual salt flux as follows:
Jv (DIC(i, j, 1)) = sflux(i, j)/S(i, j, 1) ∗ DIC(i, j, 1),(11.42)Jv (Alk(i, j, 1)) = sflux(i, j)/S(i, j, 1) ∗ Alk(i, j, 1).

(11.43)

Strictly speaking, air-sea fluxes of fresh water impact other species. However, these modifications are not usually applied
because their spatial variabilities are significantly greater than those of DIC and Alk.
In the OMIP protocol, as well as the previous OCMIP protocol, the global averaged salinity Sg is used instead ofS(i, j, 1) in equations(11.42,11.43). In addition, globally integrated Jv (DIC) and Jv (Alk) are set to 0. In MRI.COM, thesemodifications are not the default considering the use in regional ocean models.

11.4 Obata and Kitamura modelThis section was contributed by A. Obata.
The Obata and Kitamura model used in MRI.COM simply represents the source and sink terms of DIC, Alk, [PO4 ], and[O2 ] due to the biogeochemical activities: new production driven by insolation and phosphate concentration in the surface
ocean, its export to depth, and remineralization in the deep ocean. According to the Michaelis-Menten kinetics (Dugdale,1967), phosphorus in the new production exported to depth (ExprodP) is parameterized as r L[PO4 ]2 /([PO4 ] + k), wherer is a proportional factor (r = 0.9 yr−1 ), L is the insolation normalized by the annual mean insolation on the equator, andk is the half-saturation constant (k = 0.377 mmol/m−3 ). The values of r and k are adjusted to reproduce the optimum
atmospheric CO2 concentration and ocean biogeochemical distribution for the preindustrial state of the model. The
relationship between the changes in the chemical composition of seawater and the composition of particulate organic
matter (POM) is assumed to follow the Redfield ratio P : N : C : O2 = 1 : 16 : 106 : −138 ≡ 1 : Rnp : Rcp : −Rop ,where Rab = A/B and "o" represents O2 (Redfield et al., 1963). The rain ratio of calcite to particulate organic carbon(POC) is 0.09, which is in the range proposed by Yamanaka and Tajika (1996). The surface thickness where the export
production occurs is fixed at 60 m. The vertical distribution of POM and calcite vertical flux below a depth of 100 m is
– 129 –

11.5 NPZD model
proportional to (z/100m) −0.9 and exp(−z/3500m) (z is the depth in meters), respectively, following the work of Yamanaka
and Tajika (1996). The remineralization of POM (RemiP for phosphorus) and the dissolution of calcite (SolnCa) at depth
are parameterized by these fluxes. Oxygen saturation is prescribed at the sea surface. The solubility of oxygen is computedfrom the formula of Weiss (1970). Source and sink terms of Sb () representing the above processes are as follows:
Sb (DIC) = Rcp ∗ RemiP + SolnCa − Rcp ∗ ExprodP(11.44)

Sb (Alk) = 2 ∗ SolnCa + Rnp ∗ ExprodP − Rnp ∗ RemiP(11.45)

Sb ([PO4 ]) = RemiP − ExprodP(11.46)

Sb ([O2 ]) = − Rop ∗ Sb ([PO4 ])(11.47)

11.5 NPZD model
The NPZD model used in MRI.COM is constructed on the assumptions that the biological elemental composition ratio isnearly constant (Redfield ratio) and that the concentration of organisms can be estimated by nitrogen or phosphorus. The
prognostic variables of nitrate (NO3 ), phytoplankton (PhyPl), zooplankton (ZooPl), and detritus (Detri) are normalized interms of nitrogen 1 mol/m3 . For example, [PhyPl] represents the concentration of phytoplankton estimated in terms ofnitrogen in one cubic meter (N mol/m3 ). The increase and decrease of carbon can be diagnosed by multiplying by Rcn .
Source and sink terms Sb () calculated in the NPZD model are as follows. Those for DIC and Alk, Sb (DIC) and Sb (Alk),
used for calculating the carbon cycle, are described later in this section.
Sb ([PhyPl]) = Priprod − MortP1 − MortP2 − GrP2Z(11.48)

Sb ([ZooPl]) = assim ∗ GrP2Z − Excrtn − MortZ(11.49)Sb ([Detri]) = [(1 − assim) ∗ GrP2Z + MortP2 + MortZ] − RemiD − wdetri
Sb ([NO3 ]) = MortP1 + Excrtn + RmeiD − PriProd∂Detri∂z(11.50)(11.51)Sb ([PO4 ]) = Rpn ∗ Sb ([NO3 ])(11.52)

Sb ([O2 ]) = − Ron ∗ Rnp ∗ Sb ([PO4 ])(11.53)

There is no input from the atmosphere such as nitrogen fixation in the above equations, so the sum of these five equations
becomes zero at each grid point except for the term for detritus sinking (−wdetri ∂Detri∂z ). The term for detritus sinking
expresses the biological pump, whose role is to remove nutrients from the upper layers and transport them into the deep
ocean where the plankton cannot use the nutrients. When vertically integrated, the sum of each grid is 0 even though
this sinking term is included. The nutrients are transported horizontally through physical processes such as advection anddiﬀusion.
In general, the nitrate limitation is more severe than the phosphate limitation so it is not always necessary to calculate
phosphate. However, in the simpler model of Obata and Kitamura (2003), phosphate is used as a prognostic variable.
So, to be consistent, phosphate is calculated in the ecosystem component of MRI.COM. Next, we elaborate on the above
equations.

11.5.1 Description of each term
• Priprod = J (I, N, P) ∗ [PhyPl]
Primary production expresses photosynthesis (described in detail in the next subsection).
• MortP1 = ϕ P ∗ [PhyPl]
The conversion of mortality phytoplankton directly into nutrients. This term was introduced by Oschlies (2001) toincrease the primary production of subtropical gyre, where the nutrient limit is severe.
• MortP2 = ϕPP ∗ [PhyPl]2
The conversion from phytoplankton to detritus (normal mortality of phytoplankton).
• GrP2Z = G(P) ∗ [ZooPl]2
The grazing of zooplankton. There are a number of parameterizations of grazing. In this formulation, this is given
as G(P) = g ∗ ϵ ∗ [PhyPl]2 /(g + ϵ ∗ [PhyPl]2 ). Among the grazing, the ratio assim is used for the growth ofzooplankton, and the remainder (1 − assim) is converted to detritus.
• Excrtn = d ∗ [ZooPl]
Excretion of zooplankton. The excretion is dissolute and directly returned to nutrients (NO3 ).
– 130 –

Chapter 11

Biogeochemical model• MortZ = ϕ Z ∗ [ZooPl]
The conversion from the zooplankton to detritus (mortality of zooplankton).
• RemiD = ϕ D ∗ [Detri]Remineralization of detritus. This is converted to nutrients through the activity of bacteria.
2TKOCT[2TQFWEVKQP

㧺

2
[

OQTVCNKV[

P

OQ

KQ

TV

GV

CN

ET

KV

GZ

TGOKPGTCNK\CVKQP&

ITC\KPI<GZRQTVOQTVCNKV[

Figure11.1 Schematic of NPZD model

11.5.2 Primary Production
The growth rate of phytoplankton is limited by the irradiance (I) and nutrients. This limitation is expressed in several
ways. Here we adopt an expression with a minimum function:
J (I, N, P) = min (JI , JN , JP ) ,(11.54)

where JI denotes the purely light-limited growth rate, and JN and JP are nutrient-limited growth rates that are functionsof nitrate or phosphate.
The light-limited growth is calculated as follows:
JI = [

Jmax αI]

2 + (αI) 2 1/2
Jmax.

(11.55)Here, Jmax is the light-saturated growth, which depends on temperature based on Eppley (1972) as
Jmax = a · bcθ ,(11.56)

where a = 0.6 day−1 , b = 1.066, and c = 1 (◦C) −1 . Note that the default values in MRI.COM are based on Schmittner
et al. (2008) and diﬀer from these values (see Table 11.1). Equation (11.55) is called Smith-type growth. The coeﬃcient
α in the equation is "the initial slope of photosynthesis versus irradiance (P-I) curve," that is,∂ JI.
I→0 ∂Iα = lim(11.57)

Thus, it represents how sensitive JI is to the irradiance when the light is weak.
Irradiance (I) depends on the angle of incidence and the refraction and absorption in the seawater.
()∫ z̃
I = Iz=0 PAR exp −k w z̃ − k ePdz ,0– 131 –

(11.58)

11.5 NPZD model
where Iz=0 denotes the downward√shortwave radiation at the sea surface, PAR is the photosynthetically active radiation
ratio (0.43) and z̃ = z/ cos θ = z/ 1 − sin2 θ/1.332 is the eﬀective vertical coordinate (positive downward) with 1.33 as
the refraction index according to Snell’s law relating the zenith angle of incidence in air (θ) to the angle of incidence inwater. The angle of incidence θ is a function of the latitude ϕ and declination δ.
For the nutrient-limited growth rate (JN and JP ), we adopt the Optimal Uptake (OU) equation instead of the classic
Michaelis-Menten (MM) equation. For the classic MM equation, the nitrate-limited growth rate is expressed as
JN = JMM =Jmax N,
KN + N

(11.59)where K N is a half-saturation constant for NO3 uptake. In contrast, the Optimal Uptake (OU) equation for a nitrate isexpressed as follows:
V0 N
JN = JOU =,(11.60)√N + 2 VA00 N + VA00
where A0 and V0 are the potential maximum values of aﬃnity and uptake rate, respectively (see Smith et al. (2009) for
details). Optimal Uptake (OU) kinetics assumes a physiological trade-oﬀ between the eﬃciency of nutrient encounter at
the cell surface and the maximum rate at which a nutrient can be assimilated (Smith et al., 2009). The key idea is thatphytoplankton alters the number of its surface uptake sites (or ion channels), which determines the encounter timescale,versus internal enzymes, which assimilate the nutrients once encountered.
We set parameters V0 and A0 so that the rates of uptake, JMM and JOU , are equal at N = K N . In addition, we fix theratio V0 /A0 = αOU , where αOU is determined from fitting the data. This requires
)2
(
√
αOU
Jmax .
V0 = 0.5 1 +KN(11.61)

V0 N.
N + 2 αOU N + αOU(11.62)

Finally, we obtain
JOU =

√

We use αOU = 0.19, which is determined from the fitting of log K N vs log N in the wide range of N by Smith et al. (2009).

11.5.3 Variation of DIC and Alk due to biological activity
Production of DIC and Alk is controlled by changes in inorganic nutrients and calcium carbonate (CaCO3 ), in molar
numbers according to
Sb (DIC) = Sb ([PO4 ])Rcp − Sb ([CaCO3 ]),(11.63)
Sb (Alk) = −Sb ([NO3 ]) − 2 · Sb ([CaCO3 ]).

(11.64)Thus, only these source and sink terms of DIC and Alk are estimated. Since [PO4 ] and [NO3 ] are prognostic variables,
their source and sink are explicitly calculated by the biological model. In contrast, the downward movement of CaCO3 ismuch faster than the modeled downward velocity of water mass, so [CaCO3 ] is not a prognostic variable, and its source(Pr) and sink (Di) are diagnosed by the following equation,
Sb ([CaCO3 ]) = Pr ([CaCO3 ]) − Di([CaCO3 ]).

(11.65)

Following Schmittner et al. (2008), the source term (Pr ([CaCO3 ])) of calcium carbonate is determined by the productionof detritus as follows:
Pr ([CaCO3 ]) = [(1 − assim) ∗ [GrP2Z] + [MortP2] + [MortZ]] RCaCO3 /POC RC:N ,(11.66)

where assim, GrP2Z, MortP2, and MortZ are as described above. The sink term (Di([CaCO3 ])) of calcium carbonate isparameterized as∫)d (exp (−z/DCaCO3 ) ,(11.67)
Di([CaCO3 ]) =Pr ([CaCO3 ])dz ·dzwhich expresses an instantaneous sinking with an e-holding depth of DCaCO3 = 3500 m. In this equation, z is positive
downward. This depth of 3500 m was estimated by Yamanaka and Tajika (1996) to reproduce the observed nutrient profile.
– 132 –

Chapter 11

Biogeochemical modelThis value is standard and also is used in the simple biological model in the protocol of OCMIP. The vertical integral of
the source minus sink should be zero. Thus, when the sea bottom appears before the sum becomes zero, the remaining
calcium carbonate is assumed to be dissolved in the lowermost layer. By using the ratio RCaCO3 /POC = 0.035 used by
Schmittner et al. (2008), the resultant global mean Rain ratio should be roughly consistent with the recently estimatedrange (0.07 to 0.11) based on various observations.

11.6 Usage
In MRI.COM, carbon, dissolved oxygen, and ecosystem model as well as other passive tracers like CFCs can be calculated
together or separately. Options described in configure.in determines what components should be used in MRI.COM.
The combination of the options also determines the number of passive tracers that is necessary for calculation. See detailfor the program tracer_vars.F90.
• When CARBON is used, carbon component is calculated in the model. For the ocean only model, the CBNHSTRUN
option should be used to set the atmospheric CO2. Its internal source or sink is calculated in an ecosystem model.
• When CARBON is used, dissolved oxygen component is calculated.
• When CBNHSTRUN is used, the ‘AtmosphericxCO2’ in ppm should be applied as additional atmospheric forcing.
This is the standard configuration for ocean only run. See Chapter 14 for the detailed setting.
• When O2 is used, dissolved oxygen component is calculated. Its internal source or sink is calculated in an ecosystemmodel.
For ecosystem model, MRI.COM can use NPZD or OBT options.
• When NPZD is used, an NPZD model is used as the ecosystem component. In addition, when CHLMA94 option
is used, the chlorophyll concentration is considered to calculate the shortwave penetration following Morel andAntoine (1994).
The parameters of NPZD are set in namelist nml_bioNPZD. The default values are based on Schmittner et al. (2008)
and listed on Table 11.1. If the parameters of Oschlies (2001) are used, the high nutrient-low chlorophyll (HNLC)
region in the North Pacific is not appropriately expressed. This may be because the parameters of Oschlies (2001)
are calibrated for the North Atlantic biological model. The commonly used unit of time in biological models is[day]. Thus, in the namelist, the time unit of the biological parameter is specified by using the unit [day]. In themodel, the time unit is converted to seconds, [sec].
• When OBT is used, a simple biological model of Obata and Kitamura (2003) is applied as the ecosystem component.
Restart files for the passive tracers should be specified by repeatedly writing nmlrs_ptrc. The order of passive tracers
is determined in tracer_vars.F90. Following is an example when CARBON, O2, and NPZD options are chosen. In thiscase, numtrc_p = 8 should be specified in configure.in.
An example specification of restart files when CARBON, O2, and NPZD options are chosen.
&nmlrs_ptrc fname=’result/rs_dic’/&nmlrs_ptrc fname=’result/rs_alk’/&nmlrs_ptrc fname=’result/rs_o2’ /&nmlrs_ptrc fname=’result/rs_po4’ /&nmlrs_ptrc fname=’result/rs_no3’ /&nmlrs_ptrc fname=’result/rs_PhyPl’/&nmlrs_ptrc fname=’result/rs_ZooPl’/&nmlrs_ptrc fname=’result/rs_Detri’/– 133 –

11.7 Program structure
Table11.1 Parameters used for the NPZD ecosystem component (NPZD).

variable namealphabio
abio
bbio
cbio
PARbiodkcbiodkwbiork1bioNO3
rk1bioPO4alpha_ougbio
epsbiophiphyphiphyqa_npz
phizood_npz
reminaw_detr
fac_wdetr

c_mrtnRcn
Ron
Rnp
dp_euphdp_mrtndp_eprdc
Rcaco3poc
Dcaco3
shwv_intv

description

units

Initial slope of P-I curve
Maximum growth rate parameter
Maximum growth rate = a * b ** (c * T)
Photosynthetically active radiation
Light attenuation due to phytoplankton
Light attenuation in the water
Half-saturation constant for NO3 uptake
Half-saturation constant for PO4 uptake
Fitting constant for Optical Uptake kinetics
Maximum grazing rate
Prey capture rate
Specific mortality/recycling rate
Quadratic mortality rate
Assimilation eﬃciency
Quadratic mortality of zooplankton
Excretion
Remineralization rate
Sinking velocity
Arbitrary parameter for numerical stability.
When the concentration of detritus in the n+1 st level is higher
than fac_wdetr times that in the n th level, w_detr is set to
0 between n and n+1 level.
Dimensionless scaling factor for Martin et al. (1987)
Phi(z) = Phi(zo) * (z/dp_mrtn)**(-c_mrtn)
Molar elemental ratio (C/N)
Molar elemental ratio (O2 /N)
Molar elemental ratio (N/P)
Maximum depth of euphotic zone
Characteristic depth of martin curve
The depth where the bio-export is diagnosed.
This value should be less than dp_mrtn.
CaCO3 over nonphotosynthetic POC production ratio
CaCO3 remineralization e-folding depthInterval for calculating the irradiance and light-limited growthrate. This must be a divisor of the time step for tracer.

11.7 Program structure
ogcm__ini|+-- ptrc_ctl__ini|||+-- ptrc__ini|||+-- cbn__ini|||+-- cbn__ini_history|+-- tracer__ctl__ini|+-- tracer__ini|+-- bio__ini– 134 –

(W m−2 ) −1 day−1
day−1

m−1 (mol m−3 ) −1
m−1
mol m−3
mol m−3
day−1(mol m−3 ) −2 day−1s−1−3(mol m ) −1 day−1(mol m−3 ) −1 day−1
day−1
day−1
m day−1

default value
0.1d0
0.2d0
1.066d0
1.d0
0.43d00.03d30.04d00.7d-30.0d0
0.19d01.575d0
1.6d6
0.014d00.05d30.925d0
0.34d30.01d00.048d0
2.0d0
3.d0

0.858d0

m
m
m

7.d0
10.d0
16.d0
150.d0400.d0126.d0m
min

0.05d03500.d0
10.d0Chapter 11

Biogeochemical model|+-- bio__ini_history

ogcm__run|+-- part_1|||+-- ptrc_ctl__main
|
|
|
|
|
+-- ptrc__surfflux
|
|
|
|
|
|
|
+-- cbn__set_xco2a
|
|
|
|
|
|
|
+-- cbn__calc_virtual_flux
|
|
|
|
|
|
|
+-- cbn__calc_co2_flux
|
|
|
|
|
|
|
+-- co2flux_mocsy
|
|
|
|
|
+-- ptrc__predict_surf
|
|
|
|
|
+-- cbn__predict_surf|||+-- tracer_ctl__predict|||+-- tracer__internal_source|||+-- bio__predict|+-- hist_ctl|+-- hist_ctl__write|+-- cbn__write_history– 135 –

Chapter 12

Inert tracersThis chapter explains inert tracers implemented in MRI.COM. In general, to include an inert tracer in a model integration,
you need to specify the model options and tracer attributes corresponding to that tracer. General explanation on how to
specify attributes of a tracer is given at Section 13.3.

12.1 Ideal age tracer
Ideal age of a water mass is the time in year since it last contacted with the sea surface. This tracer is introduced byEngland (1995).

12.1.1 Source / Sink term
Source and sink of an age tracer a [year], is expressed as follows. In the oceanic interior,∂a1= −A(a) + D(a) +∂t
the total number of seconds of this year

year/sec,(12.1)

where A() and D() represent advection and diﬀusion operator, respectively. Note that the r. h. s. depends on whether it
is leap year or not. The tracer ages slowly in leap years than in normal years.
At the sea surface, the ideal age is set to zero:a(k = 1) = 0.

(12.2)

12.1.2 Usage
Required Model option IDEALAGE
Required Add one to numtrc_p
Required Restart file for passive tracer (nmlrs_ptrc) (See Section 11.6)
Required Name for nml_tracer_data is ’Ideal Age Tracer’ (See Section 13.3)
Optional Namelist nml_tracer_idealage_start to specify the base date and time [year, month, day, hour, minute,second] for the age of water.

12.2 CFCs
MRI.COM can treat CFCs (CFC11 and CFC12) following the protocols of OMIP (Orr et al., 2016). There is no sourceand sink for CFCs in the interior.

12.2.1 Surface boundary condition
The CFCs have air-sea gas fluxes at the sea surface as the source. Their dissolved concentrations [A] [mol m−3 ] evolve
according to the advection (A()) and diﬀusion (D()) in the oceanic interior,∂[A]= −A([A]) + D([A]) + Jg ([A]).
∂t– 137 –

(12.3)

12.3

SF6

Jg () represents source/sink due to the air-sea gas fluxes F ([A]) at the sea surface,
Jg ([A]) =F ([A]),∆z 1(12.4)

2

withF ( A) = k w ([A]sat − [A]),(12.5)

where k w is its gas transfer velocity, [A]sat is the surface gas concentration in equilibrium with the atmosphere at an
atmospheric pressure at the surface (Pa ). k w is a function of wind velocity, surface temperature, and atmospheric pressure
at the sea surface. [A]sat is given by[A]sat = ϕ0A x A,
(12.6)
where x A is its mole fraction in dry air. The combined solubility term ϕ0A is computed using the empirical fit of temperatureand salinity. See Orr et al. (2016) for further details.

12.2.2 Usage
Required Model option CFC
Required Add two to numtrc_p
Required Restart file for passive tracer (nmlrs_ptrc) (See Section 11.6)
Required Names for nml_tracer_data are ’CFC11’ and ’CFC12’ (See Section 13.3)
Required Namelist nml_force_data to specify partial gas pressure (See Section 14.10)
file_data File that contains mole fraction in dry air of CFC11 (and CFC12) [ppt]name CFC11 (and 12)
txyu ’t’

12.3 SF6
The calculation of SF6 also follows the protocols of OMIP (Orr et al., 2016). The formulation of SF6 is nearly the sameas CFCs. Only the coeﬃcients of the empirical fit diﬀer.

12.3.1 Surface boundary condition
See Section 12.2.1, where CFCs should read SF6 .

12.3.2 Usage
Required Model option SF6
Required Add one to numtrc_p
Required Restart file (nmlrs_ptrc) (See Section 11.6)
Required Names for nml_tracer_data is ’SF6’ (See Section 13.3)
Required Namelist nml_force_data to specify partial gas pressure (See Section 14.10)
file_data File that contains atmospheric SF6 [ppt]name SF6
txyu ’t’– 138 –

Chapter 13

Package Structure and Usage
13.1 Package structure
Packages relevant to tracers are listed as follows.

13.1.1 Tracer equation
tracer_ctl.F90:
tracer.F90:
upc_adv.F90:
quick_adv.F90:
utzq_adv.F90:
som_adv.F90:
mpdata_adv.F90:
tracer_vars.F90:+vvdimp/trcimp.F90:+isopycnal/ipcoef.F90:+isopycnal/ipycmix.F90:

Controller of this package
Main program of this package
Upcurrent advection scheme
QUICK advection scheme (QUICKADVEC)
Combination of UTOPIA and QUICKEST advection scheme (UTZQADVEC)
Second order moment advection scheme (SOMADVEC)MPDATA advection scheme (MPDATAADVEC)
Setting of tracer attributes
Solver of the vertical diﬀusion part using the implicit method (VVDIMP)
Calculation of tensor components of neutral physics parameterization (ISOPYCNAL)
Calculation of tendency due to neutral physics parameterization (ISOPYCNAL)

13.1.2 Vertical mixing coeﬃcientsvmixcoef_ctl.F90:
Controller of the vertical mixing package
vmixcoef.F90:
Main program of the vertical mixing package
vmixcoef_vmbg.F90:
Estimation of background vertical diﬀusion coeﬃcient
vmixcoef_vars.F90:
Declaration of variables+runoﬀ/vmixcoef_rivermouth.F90: Estimation of vertical mixing coeﬃcient around the river mouth

13.1.3 Stratification and convective adjustment
strat_adjust_ctl.F90:
stratification.F90:
cnvajs.F90:
strat_adjust_vars.F90:

Controller of stratification and adjustment package
Main program of calculation of stratification
Main program of convective adjustment
Declaration of variables

13.1.4 Reference state and restoring coeﬃcient
restore_cond_ctl.F90:
restore_cond.F90:
force_data.F90:

Controller of reference state and restoring coeﬃcient
Main program of reference state and restoring coeﬃcientService package that handles external forcing data

13.1.5 Passive tracers
ptrc_ctl.F90:
ptrc.F90:

Controller of passive tracer evolution
Main program of passive tracer evolution that mainly treats surface sources and sinks– 139 –

13.2

Handling the initial state

ptrc_subp.F90:

Sub package of passive tracer evolution that describes internal sources and sinks(called from tracer.F90)+ptrc/cfc.F90, sf6.F90, etc. Sub package of passive tracer evolution that describes specialized processesof particular tracers

13.2 Handling the initial state
How to determine the initial state for temperature and salinity is specified in namelist nml_tracer_run. Parameters arelisted on Table 13.1.
Table13.1 Namelist nml_tracer_run

variable namel_rst_tracer_inunits
logicall_rst_uni_strati

logicalstart_rec_uni_strati
end_rec_uni_strati

integer

description.true. : Read restart files specified by
nmlrs_t and nmlrs_s for the initial condition.
.false.: Start condition depends on thel_rst_uni_strati.false.: Start from 3D-distribution at thestarting time of reference data following
nml_tracer_data.
.true. : Start from uniform stratification
created by the reference data following
nml_tracer_data. Time average is conducted based on start_rec_uni_strati
and end_rec_uni_strati.
uniform stratification is created by the average from start_rec_uni_strati data
record to end_rec_uni_strati record.

usage
Default is the same as l_rst_inof nml_run_ini_state.

if l_rst_tracer_in = .false.

if l_rst_uni_strati = .true.

13.3 Configuration of tracers
The attributes of each tracer such as name, advection scheme, restoring condition, reference data, and restoring coeﬃcients,
are stored in the structural type type_tracer_data defined in tracer_vars.F90. The contents of this structural typeare specified by namelist nml_tracer_data, which should be repeatedly defined as many times as the number of tracersthat should be calculated. Tables 13.2 through 13.8 list the variables.

13.3.1 Name
List of eﬀective names is found in subroutine tracer_vars__set_num_and_name of tracer_data.F90
Table13.2 Namelist nml_tracer_data

variable name
name

units
characterdescription
Name of tracer. Two tracers are necessary:"Potential Temperature" and "Salinity."

usage
Case sensitive. For example, "potential
temperature" is not correct.

13.3.2 Advection scheme
Following can be specified as the name of the advection scheme (adv_scheme%name).
• "upc" : weighted UP-Current advection scheme (always available)• "quick" : QUICK advection scheme (QUICKADVEC)• "utzq" : UTOPIA + ZQUICKEST schemes with ultimate limiter (UTZQADVEC)• "som" : Second-Order Moment advection scheme (SOMADVEC)• "mpdata" : MPDATA advection scheme (MPDATAADVEC)– 140 –

Chapter 13

Package Structure and Usage
Table13.3 Namelist nml_tracer_data related to advection scheme

variable nameadv_scheme%nameunitscharacterdescription
Name of the advection
scheme.

adv_scheme%limiter_som_org

logicaladv_scheme%limiter_som_Merryfield03

logicaladv_scheme%lrstin_som

logicaladv_scheme%lrstout_som

logicaladv_scheme%limiter_mpdata_nonoscillatory

logicaladv_scheme%eps_lim_mpdatathat oftracerthat oftracerUse flux limiter for SOM
by Prather (1986)
Use flux limiter for som by
Merryfield and Holloway(2003)The SOM initial state of
moments is read from file
The SOM final state of moments is written to file
Apply flux limiter for MPDATA
Very small value to avoid
zero division
Minimum value for tracer

adv_scheme%min_value_mpdata

usage
Diﬀerent advection
schemes can be set
for individual tracers.
SOMADVEC
SOMADVEC

SOMADVEC
SOMADVECMPDATAADVECMPDATAADVECMPDATAADVEC

13.3.3 Restoring condition
The following are variables related to the restoring condition for a tracer.
Table13.4 Namelist nml_tracer_data related to restoring condition

variable namerestore_conf%l_surf_restore
restore_conf%l_body_restore

units
logicallogicaldescription
restore condition at the surface is applied or not
restore condition in the interior is applied or not

usage
default = .false.
default = .false.

13.3.4 Reference data
a. Three dimensional reference state for restoring

When the field of a tracer is restored to a reference state, the attributes of the reference state should be given by the variables
listed on Table 13.5. This reference state is also used to produce an initial state for that tracer when its restart file is notavailable (See Table 13.1).
Table13.5 Namelist nml_tracer_data related to reference values for body forcing and initial condition

variable nametrcref_conf%file_data

unitscharactertrcref_conf%file_data_grid
trcref_conf%imfrc
trcref_conf%jmfrc
trcref_conf%kmfrc
trcref_conf%interval

character
integer
integer
integer
integer

description
a file name that contains reference
values for body forcing and initialcondition.
a file name of grid
grid size of data in x direction
grid size of data in y direction
grid size of data in z direction
regular time interval of data

– 141 –

usage

needed if linterp = .true.

positive value : unit is sec−1 : monthly−999 : steady forcing
Continued on next page

13.3

variable nametrcref_conf%num_data_max
trcref_conf%ifstart

trcref_conf%lrepeat
trcref_conf%linterp
trcref_conf%linterp_v
trcref_conf%ilinear
trcref_conf%luniform
trcref_conf%luniform_v
trcref_conf%ldouble
trcref_conf%iverbose
trcref_conf%ldefined

Table 13.5 – continued from previous page
units
description
integerthe number of record contained in thefileifstart(6) [ymdhms] of the first record of the
input file

logicallogicallogicalinteger
integerlogicallogicalinteger
integerclimatological data is repeatedly usedinterpolate horizontally or notinterpolate vertically or notinterpolation methoddata is horizontally uniform or notdata is vertically uniform or not
input data is double or notstandard output of progress
the input data is defined or not

Configuration of tracers

usage

1999,1,1,0,0,0 when the firstrecord is the average value ofJan 1999 and its data interval
is monthly.
default = .false.
default = .false.
default = .false.
1 : linear, 2 : spline
default = .false.
default = .false.
default = .false.
1 : extensive, 0 : minimum
default = .false.

Format of tracer reference / restoring data is shown in the following.

Format of tracer reference / restoring data (trcref(_surf)_conf%file_data)

integer(4), parameter :: imn = 12, nu = 99
integer(4) :: imfrc, jmftc, kmfrc! data size
character(128) :: file_data, fname_gridreal(4) :: ttlev(imfrc,jmfrc,kmfrc,imn)real(8) :: alonf(imfrc), alatf(jmfrc), dpf(kmfrc)
logical :: linterp, linterp_v! main data
open (unit=nu,file=file_data,access=direct,recl=4*imfrc*jmfrc*kmfrc)
do m = 1, imn
write(unit=nu,rec=m) ttlev(:,:,:,m)end doclose(nu)! longitude/latitude of main data
if (linterp) then ! If input data is horizontally interpolated in the model.
open (unit=nu,file=file_grid)
write(nu) alonf, alatf
if (linterp_v) then ! If input data is vertically interpolated in the model.
write(nu) dpf
end ifclose(nu)end if– 142 –



Chapter 13

Package Structure and Usageb. Two dimensional reference state for surface restoring

When the surface field of a tracer is intended to be restored to a reference state, the attributes of the surface reference stateshould be given by the variables listed on Table 13.6.
Table13.6 Namelist nml_tracer_data related to reference values for surface restoring forcing.

variable nametrcref_surf_conf%file_data

unitscharactertrcref_surf_conf%file_data_grid
trcref_surf_conf%imfrc
trcref_surf_conf%jmfrc
trcref_surf_conf%interval

character
integer
integer
integertrcref_surf_conf%num_data_max

integertrcref_surf_conf%ifstart

ifstart(6)trcref_surf_conf%lrepeat

logicaltrcref_surf_conf%linterp
trcref_surf_conf%ilinear
trcref_surf_conf%luniform

logicalintegerlogicaltrcref_surf_conf%ldouble
trcref_surf_conf%iverbose
trcref_surf_conf%ldefinedlogicalintegerlogicaldescription
a file name that contains reference values for surface restoring forcing
a file name of grid
grid size of data in x direction
grid size of data in y direction
regular time interval of data

the number of record containedin the file[ymdhms] of the first record of
the input file

climatological data is repeatedly usedinterpolate horizontally or notinterpolation methoddata is horizontally uniform ornotinput data is double or notstandard output of progress
the input data is defined or not

usage

needed if linterp = .true.

positive value : unit is sec−1 : monthly−999 : steady forcing

1999,1,1,0,0,0 when the firstrecord is the average value ofJan 1999 and its data interval
is monthly.
default = .false.
default = .false.
1 : linear, 2 : spline
default = .false.
default = .false.
1 : extensive, 0 : minimum
default = .false.

13.3.5 Restoring coeﬃcienta. Coeﬃcient for three dimensional restoring

When the field of a tracer is restored to a reference state, the attributes of the file that contains restoring coeﬃcients should
be given by the variables listed on Table 13.7. Units of restoring coeﬃcient is sec−1 .
Table13.7 Namelist nml_tracer_data related to restoring coeﬃcient for body forcing.

variable namerstcoef_conf%file_data

unitscharacterrstcoef_conf%file_data_grid
rstcoef_conf%imfrc
rstcoef_conf%jmfrc
rstcoef_conf%kmfrc
rstcoef_conf%interval

character
integer
integer
integer
integer

description
a file name that contains restoring
coeﬃcient for body forcing.
a file name of grid
grid size of data in x direction
grid size of data in y direction
grid size of data in z direction
regular time interval of data

– 143 –

usage
needed if linterp = .true.

positive value : unit is sec
Continued on next page

13.3

variable name

Table 13.7 – continued from previous page
units
description

rstcoef_conf%num_data_max

integer

rstcoef_conf%ifstart

ifstart(6)rstcoef_conf%lrepeat

logicalrstcoef_conf%linterp
rstcoef_conf%linterp_v
rstcoef_conf%ilinear
rstcoef_conf%luniform
rstcoef_conf%luniform_v
rstcoef_conf%ldouble
rstcoef_conf%iverbose
rstcoef_conf%ldefinedlogicallogicalintegerlogicallogicallogicallogicallogicalthe number of record contained in
the file[ymdhms] of the first record of the
input file

climatological data is repeatedlyusedinterpolate horizontally or notinterpolate vertically or notinterpolation methoddata is horizontally uniform or notdata is vertically uniform or not
input data is double or notstandard output of progress
the input data is defined or not

Configuration of tracers

usage−1 : monthly−999 : steady forcing

1999,1,1,0,0,0 when the firstrecord is the average value ofJan 1999 and its data interval
is monthly.
default = .false.
default = .false.
default = .false.
1 : linear, 2 : spline
default = .true.
default = .false.
default = .true.
1 : extensive 0 : minimum
default = .false.

Note that the default settings for luniform and ldouble are diﬀer from those of the other attributes.
b. Coeﬃcient for surface restoring

When the surface field of a tracer is intended to be restored to a reference state, the attributes of the file that contains surface
restoring coeﬃcients should be given by the variables listed on Table 13.6. Units of the surface restoring coeﬃcient issec−1 .
Table13.8 Namelist nml_tracer_data related to restoring coeﬃcient for surface restoring forcing.

variable namerstcoef_surf_conf%file_data

unitscharacterrstcoef_surf_conf%file_data_grid
rstcoef_surf_conf%imfrc
rstcoef_surf_conf%jmfrc
rstcoef_surf_conf%interval

character
integer
integer
integer

rstcoef_surf_conf%num_data_max
rstcoef_surf_conf%ifstart

ifstart(6)rstcoef_surf_conf%lrepeat

logicalrstcoef_surf_conf%linterp

logicalrstcoef_surf_conf%ilinear
rstcoef_surf_conf%luniform

integerlogicalrstcoef_surf_conf%ldouble
rstcoef_surf_conf%iverbose
rstcoef_surf_conf%ldefinedlogicallogicallogicaldescription
a file name that contains
restoring coeﬃcient for forsurface restoring forcing.
a file name of grid
grid size of data in x direction
grid size of data in y direction
regular time interval of data

the number of record contained in the file[ymdhms] of the first recordof the input file

climatological data is repeatedly usedinterpolate horizontally ornotinterpolation methoddata is horizontally uniform
or notinput data is double or notstandard output of progress
the input data is defined or not– 144 –

usage

needed if linterp = .true.

positive value : unit is sec−1 : monthly−999 : steady forcing

1999,1,1,0,0,0 when the firstrecord is the average value ofJan 1999 and its data interval
is monthly.
default = .false.
default = .false.
1 : linear, 2 : spline
default = .false.
default = .false.
1 : extensive, 0 : minimum
default = .false.

Chapter 13

Package Structure and Usage

13.3.6 Example
Following is an example of namelist nml_tracer_data for Salinity. Some systems may not allow blank lines or comment
lines in a namelist. In this case, you should delete them.
&nml_tracer_data
name="Salinity",! advection scheme
adv_scheme%name="som",
adv_scheme%limiter_som_org=.false.,
adv_scheme%limiter_som_Merryfield03=.true.,
adv_scheme%lrstin_som=.false.,
adv_scheme%lrstout_som=.true.,! restore_condition
restore_conf%l_surf_restore=.true.
restore_conf%l_body_restore=.false.
! trcreftrcref_conf%file_data=’../data/file_sclim.grd’,trcref_conf%file_data_grid=’dummy.d’,trcref_conf%imfrc=184,
trcref_conf%jmfrc=152,
trcref_conf%kmfrc=51,
trcref_conf%interval=-1,
trcref_conf%ifstart=1947,12,1,0,0,0,
trcref_conf%num_data_max=14,
trcref_conf%lrepeat=.false.,
trcref_conf%linterp=.false.,
trcref_conf%ilinear=1,
trcref_conf%iverbose=1,! rstcoef
rstcoef_conf%ldefined=.false.,! trcref_surf
trcref_surf_conf%file_data=’../data/file_ssurf.grd’,trcref_surf_conf%file_data_grid=’dummy.d’,trcref_surf_conf%imfrc=184,
trcref_surf_conf%jmfrc=152,
trcref_surf_conf%interval=-1,
trcref_surf_conf%ifstart=1947,12,1,0,0,0,
trcref_surf_conf%num_data_max=14,
trcref_surf_conf%lrepeat=.false.,
trcref_surf_conf%linterp=.false.,
trcref_surf_conf%ilinear=1,
trcref_surf_conf%iverbose=1! rstcoef_surf
rstcoef_surf_conf%file_data=’../data/rstcoef_surf_s.grd’,
rstcoef_surf_conf%file_data_grid=’dummy.d’,rstcoef_surf_conf%imfrc=1,
rstcoef_surf_conf%jmfrc=1,
rstcoef_surf_conf%interval=-999,
rstcoef_surf_conf%num_data_max=1,
rstcoef_surf_conf%iverbose=1,
rstcoef_surf_conf%luniform=.true.,
rstcoef_surf_conf%ldouble=.true.,/– 145 –

Part V

Boundary Processes

Chapter 14

Sea surface fluxes
The governing equations introduced in Chapter 2 need boundary conditions. At the sea surface, they take the form of theflux boundary condition. Momentum fluxes for the equations of motion, heat and material fluxes for the temperature andother tracer equations, and fresh water fluxes for the continuity equation. These surface fluxes are the main driving forceof oceanic motions. This chapter explains how they are treated in MRI.COM.
Ideally, the realistic momentum, heat and fresh water fluxes based on observation should be used when a realistic
simulation is intended. However, because existing sea surface fluxes derived from observations include large errors andthe ocean models have their own errors, it is not common for an ocean-sea ice model in stand-alone mode to be driven
by fluxes without a feedback process. It is a common practice for an ocean-sea ice model to compute fluxes using a bulk
formula with sea surface meteorological elements derived from atmospheric reanalysis data and the surface temperatureand velocity of its own.
For a simulation using a coupled atmosphere-ocean model, all surface fluxes may be given by the atmospheric component
that computes fluxes using the surface states of the oceanic component. Users are referred to Yukimoto et al. (2011) for
the treatment of fluxes in MRI-ESM.
Section 14.1 describes momentum flux. How heat fluxes are treated is described in Sections 14.2 through 14.5.
Treatment of fresh water fluxes is described in Section 14.6, including the computations of the equivalent fluxes underinvariable first layer volume condition (Section 14.6.2). Sections 14.7 and 14.8 explain how heat and fresh water fluxes
are used in temperature and salinity equations. Several options are available for the choice of the bulk formula to calculate
momentum, latent and sensible heat, and evaporative fluxes. They are detailed in Section 14.9 along with a generalformulation for bulk transfer coeﬃcients. How to give the attributes of external forcing data to the model is explained in
Section 14.10. Finally, some technical details are presented in Sections 14.11 and 14.12.
Here is a cautionary note. Although recent satellite observations enable us to obtain sea surface fluxes with highresolution in space and time, even higher accuracy is necessary for practical uses. For example, a bias of several W m−2
in heat flux may greatly aﬀect the thickness of sea ice, meaning that accuracy on the order of several W m−2 is necessary
to clarify climatic change (WGASF, 2000). Eﬀorts in enhancing observations and evaluating sea surface fluxes based on
various methods have been expanded globally. In the future, a high-resolution ocean model and a new scheme for advection
and diﬀusion may be developed to improve the simulation capability. It is noted, however, that increased observation
frequency does not necessarily guarantee improved accuracy of the fluxes (for example, it is unlikely that the accuracy ofa bulk coeﬃcient would be improved). Hence, it should be kept in mind that sea surface fluxes presently involve large
uncertainties.

14.1 Momentum flux (surface stress)
14.1.1 Formulation
Surface forcing to the momentum equation, or surface momentum flux into the ocean, is in the form of wind stress (orstress from sea ice), and is treated as a body force to the first level velocity in the model algorithm(τx, τy )∂(u1, v1 ) = ... +,∂tρ0 ∆z 1

(14.1)

2

where (u1, v1 ) is horizontal velocity at the first level, ∆z 1 is the thickness of the first layer, ρ0 is the reference density of2
sea water (Table 2.1), and τx and τy are zonal and meridional stresses.
Surface stress is an area weighted average of stresses due to surface wind and sea ice:(τx, τy ) = (1 − A)(τAOx , τAOy ) + A(τIOx , τIOy ),– 149 –

(14.2)

14.1 Momentum flux (surface stress)→ and −
where A is the area fraction of sea ice. −τ−AOτ−→IO are air-ocean and ice-ocean stresses, respectively.
−→−−→ is computed by using the wind vector of the surface air, U
Wind stress (air-sea momentum flux) vector, τAO
a , the first−→level velocity (surface current), Us , and a bulk transfer coeﬃcient CD (see Section 14.9 for computation) as follows:
−→
−
→ −→
−
→
−
→ = ρ C |Uτ−AOa D a − αUs |(Ua − αUs ),(14.3)

where ρ a is the density of air and α is the contribution of the surface current to the calculation of relative surface wind.
Usually, α is unity, but it can be tuned to reduce damping eﬀect of the wind stress on the oceanic current (see Table 14.3).
τ−→
See Chapter 17 for the computation of ice-ocean stress (−IO ).

14.1.2 Numerical implementation
On implementing Eq. (14.3), the time level of the surface velocity must be the starting time level of the integration. For
the time level n of the leap-frog scheme,−→n−→n−1 −→n−→−→ n = ρ C |U
τ−AO|(Ua − αUs n−1 ),a D a − αUs(14.4)

and for the first part of the Matsuno scheme,−→n−→n −→n−→n−→ n = ρ C |Uτ−AOa D a − αUs |(Ua − αUs ).

(14.5)The same treatment should be applied to the computation of ice-ocean stress. The surface velocity at the previous timelevel should be passed to the sea ice part for the leap-frog scheme.
In the bulk formulae implemented in MRI.COM, bulk transfer coeﬃcients are computed on the tracer point. Those on
the velocity point are obtained as area-weighted averages from the surrounding tracer points.

14.1.3 Usagea. Model option

By default, wind stress vector is given as external data and each component should be read from separate files. (Windstress data is always required, and thus files filled with zeros are necessary when wind stress is not applied.) The input data
should be in units of [dyn cm−2 ]. When TAUBULK option is specified, the wind stress is calculated by using a bulk formula.
In this option, components of wind vector at 10 m in units of [cm s−1 ] should be prepared as external data, instead of thedefault wind stress data.
b. Input data

Time series of wind stress or surface wind vector data on a structured lattice should be prepared in the single precision,
direct access format with a constant time interval. The data may be interpolated in time and space in the model. Attributesof the data are specified by nml_force_data. The name of the data and the units assumed by the model are listed on
Table 14.1. The name of the data should be "U-wind" and "V-wind" regardless of wind stress or wind vector. See Section
14.10 for details on how to specify external forcing files.
Table14.1 Wind data to be read from namelist nml_force_data.

X-ward wind stressY-ward wind stress
X-ward wind speed
Y-ward wind speed

nameU-windV-windU-windV-windunits
dyn cm−2
dyn cm−2
cm s−1cm s−1usage
if not TAUBULK
if not TAUBULK
if TAUBULK
if TAUBULK

c. Namelist

At the initial stage of the model integration, wind stress or wind may be gradually set up from zero at the start and unity
after a specified period. The set up period is specified in namelist nml_force_windsetup (Table 14.2).

– 150 –

Chapter 14

Sea surface fluxes
Table14.2 Namelist nml_force_windsetup.

variable nameidays_wind_setup

units
day

description
wind set up period in days

usage
default : no tapering on wind

Contribution of the surface current to the calculation of relative surface wind in (14.3) may be tuned by namelist
nml_bulk_wind (Table 14.3). This is because the use of full surface current may lead to too strong damping on swift
surface current in some model settings.
Table14.3 Namelist nml_bulk_wind.

variable namecurrent_contri_factor

units
1

description
Contribution of the surface current to thecalculation of relative surface wind

usage
default = 1.0

14.2 Heat flux
14.2.1 Formulation
There are four main components in surface heat flux: shortwave radiation (QSH ), longwave radiation (QLO ), latent (QLA ),and sensible (QSN ) heat flux. Specific expression for these fluxes is shown in the following sections. In addition to thoseθ ) contribute
fluxes, the heat exchanges with the sea ice (Qice ) and the heat transport due to surface fresh water fluxes (FWF
to surface heat flux. See Section 14.6 for fresh water fluxes and Chapter 17 for details of exchange with sea ice.
The total sea surface heat flux QNET is given byθ
QNET = QSH + QLO + QLA + QSN + Qice + ρ0 cp FWF,(14.6)

where ρ0 and cp are density and specific heat of sea water. Note that downward heat flux is defined as positive.

14.2.2 Numerical implementation
The components of net heat flux (14.6) are broken into three terms,
θ
QNET = QSH + QOTHER + ρ0 cp FWF.

(14.7)

Shortwave radiation (QSH ) penetrates into the ocean interior. Other terms (QOTHER ) are absorbed in the first layer. Freshθ ) needs separate treatment because it should be consistent with the continuity equation andwater transport term (ρ0 cp FWF
included in the solution procedures of the tracer advection term.

14.2.3 Usagea. Model options

The HFLUX option must be selected to treat surface heat fluxes explicitly in the model.
b. Input data

Time series of shortwave and longwave radiation, SST (unnecessary for LWDOWN option), surface air temperature, surface
specific humidity, surface wind speed, and sea-level pressure on a structured lattice should be prepared. The data may beinterpolated in time and space in the model. Attributes of the data are specified by nml_force_data. The name of thedata and the units assumed by the model are listed on Table 14.4. See Section 14.10 for details on how to specify external
forcing files at run time.

– 151 –

14.3

Shortwave radiation flux

Table14.4 Heat flux data to be read from namelist nml_force_data when HFLUX option is selected.

elements
Downward shortwave radiation
Downward longwave radiation

nameShortWave
LongWave

Sea surface temperature
Surface air temperature
Surface air specific humidity
Scalar wind speed
Sea level pressure

SST
TempAir
SphAirScalarWind
SeaLevelPressureunits
erg s−1 cm−2 = ×10−3 W m−2
erg s−1 cm−2

usage

◦C

Net longwave by default(SST is required).
Downward if LWDOWN
unnecessary if LWDOWN

1
cm s−1hPa

unnecessary if TAUBULK
also available for SLP◦C

14.3 Shortwave radiation flux
14.3.1 Formulation
By default, the downward shortwave radiation (Qdown
SH ) is read as external forcing data. A part of the downward shortwave
radiation (α o Qdown)
isreflected
at
the
sea
surface,and the remainder penetrates into the ocean interior as the net shortwaveSH
downradiation flux (QSH (z = η) = (1 − α o )QSH ), where α o is the albedo at the sea surface and η is the position of sea surface
in the fixed vertical coordinate. More than 50% of the radiation (in the near infra-red band) is absorbed within a depth of
1 m below the sea surface, but the remainder (in the visible and ultraviolet bands) penetrates further into the ocean interior
and aﬀects the subsurface temperature structure.
The penetrating shortwave radiation is expressed as a function of depth (z) in a fixed frame QSH (z) and its verticalconvergence ∂QSH (z)/∂z warms the sea water.
∂θ
1 ∂QSH (z)= ... +,∂t
ρ0 c p∂z(14.8)

where ρ0 is the density of sea water and cp is the specific heat of sea water. There are three options regarding how thevertical profile of shortwave penetration is specified.
a. Standard scheme of penetration

According to Paulson and Simpson (1977), the shortwave radiation flux penetrating into the ocean interior is given byusing the fraction of the near infra-red band (R) and the attenuation distances for the near infra-red band (ζ1 ) and thevisible and ultraviolet band (ζ2 ) as,
QSH (z) = QSH (0)[R exp{(−η + z)/ζ1 } + (1 − R) exp{(−η + z)/ζ2 }],(14.9)

where we set R = 0.58, ζ1 = 35 cm, and ζ2 = 2300 cm, using the optical properties of Water Type I based on theclassification by Jerlov (1976). QSH (0) is the net solar radiation flux at the sea surface.
b. A penetration scheme that takes into consideration of the insolation angle

The SOLARANGLE option enables us to include the eﬀect of the insolation angle on the shortwave radiation flux in theoceanic interior. This scheme is based on Ishizaki and Yamanaka (2010). In this case, if the depth (z) on the r.h.s. of
Equation (14.9) is replaced by the penetrating distance from the sea surface, the shortwave radiation is given by
QSH (z) = QSH (0)[R exp{(−η + z)/(ζ1 sin θ w )} + (1 − R) exp{(−η + z)/(ζ2 sin θ w )}],

(14.10)where θ w is the penetrating angle in the ocean interior. When SOLARANGLE option is specified in coupled models, Equation(14.10) is replaced by
QSH (z) = QSHb (z) + QSHd (z),(14.11)
where QSHb (z) and QSHd (z) are the shortwave radiation fluxes due to direct and diﬀuse solar radiation. Those fluxes are
expressed as follows:
QSHb (z) = QSHb (0)[Rb exp{(−η + z)/(ζ1 sin θ w )} + (1 − Rb ) exp{(−η + z)/(ζ2 sin θ w )}],

(14.12)

QSHd (z) = QSHd (0)[Rd exp{(−η + z)/ζ1 } + (1 − Rd ) exp{(−η + z)/ζ2 }],(14.13)– 152 –

Chapter 14

Sea surface fluxes

where Rb and Rd are the ratios of near infra-red to total radiation in direct and diﬀuse solar radiation, respectively. QSHb (0)and QSHd (0) are the net direct and diﬀuse solar radiation fluxes at the sea surface.
c. A penetration scheme that takes into consideration of chlorophyll concentration

Recent studies indicate that solar radiation absorption and local heating within the upper ocean are strongly influenced by
the chlorophyll concentration. By specifying CHLMA94 and NPZD options, we can use a shortwave penetration model withthe chlorophyll concentration (Morel and Antoine, 1994). In this scheme, the shortwave radiation flux penetrating into theocean interior is given by
QSH (z) = QSH (0)[R exp{(−η + z)/ζ0 } + (1 − R)(V1 exp{(−η + z)/ζ1 } + V2 exp{(−η + z)/ζ2 })].

(14.14)

The first exponential is for the infra-red waveband (> 750 nm), which is not influenced by biological materials andattenuates with a distance of ζ0 = 26.7 cm. The second and third exponentials are for the visible and ultraviolet bands (<750 nm). V1 , V2 , ζ1 , and ζ2 are calculated from an empirical relationship as a function of chlorophyll concentration (chl[mg · m−3 ]) as follows:
V1 = 0.321 + 0.008C + 0.132C 2 + 0.038C 3 − 0.017C 4 − 0.007C 5,(14.15)

V2 = 0.679 − 0.008C − 0.132C 2 − 0.038C 3 + 0.017C 4 + 0.007C 5,(14.16)5

ζ1 = 1.540 − 0.197C + 0.166C − 0.252C − 0.055C + 0.042C ,(14.17)

ζ2 = 7.925 − 6.644C + 3.662C − 1.815C − 0.218C + 0.502C ,(14.18)

2

3

2

3

4
4

5

where C = log10 (chl). It is noted that V1 + V2 = 1.
When SOLARANGLE option is added to the options mentioned above, the shortwave radiation is slightly modified by
QSH (z) = QSH (0)[R exp{(−η + z)/(ζ0 sin θ w )}+ (1 − R)(V1 exp{(−η + z)/(ζ1 sin θ w )} + V2 exp{(−η + z)/(ζ2 sin θ w )})].

(14.19)In coupled models, Equation (14.19) is replaced by
QSH (z) = QSHb (z) + QSHd (z),(14.20)where QSHb (z) and QSHd (z) are the shortwave radiation fluxes due to direct and diﬀuse solar radiation. These fluxes are
expressed as follows:
QSHb (z) = QSHb (0)[Rb exp{(−η + z)/(ζ0 sin θ w )}+ (1 − Rb )(V1 exp{(−η + z)/(ζ1 sin θ w )} + V2 exp{(−η + z)/(ζ2 sin θ w )})],(14.21)

QSHd (z) = QSHd (0)[Rd exp{(−η + z)/ζ0 }+ (1 − Rd )(V1 exp{(−η + z)/ζ1 } + V2 exp{(−η + z)/ζ2 })],

(14.22)where Rb and Rd are the ratios of near infra-red to total radiation for direct and diﬀuse solar radiation, respectively.
d. Albedo schemes

There are three options for the sea surface albedo. The first option (albedo_choice = 1 in namelist nml_albedo_ocean)
is a constant value, which should be specified as alb in namelist nml_albedo_ocean. The second option(albedo_choice = 2) is based on Large and Yeager (2009) and is given by
α o = 0.069 − 0.011 cos(2ϕ),(14.23)where ϕ is latitude. The third option (albedo_choice = 3) is based on Baker and Li (1995) and is given by
α o = 0.06 + 0.0421x 2 + 0.128x 3 − 0.04x 4 +(3.12
0.074x ) 5+
x ,5.68 + U 1.0 + 3.0U

where x = 1 − sin θ a (θ a is a height angle of the Sun), and U is the surface wind speed [m s−1 ].

– 153 –

(14.24)

14.3

Shortwave radiation flux

14.3.2 Numerical implementation
Shortwave flux is computed on T-cells, which means that variations of horizontal cross section due to topography (S) must
be taken into account. The vertical profile of the shortwave penetration can be expressed as a function of the vertical depth
from the sea surface. Discrete form of the absorption of shortwave heat for a grid cell (i, j, k − 12 ) is
(
)
n
n−1
n
(θV )i,n+1
1 = (θV )1 + ... + 2∆t Q SH i, j,k−1 Si, j,k− 1 − Q SH i, j,k Si, j,k+ 1 ,j,k−i, j,k−2

2

2

2(14.25)for the leap-frog scheme.

14.3.3 Usagea. Model option• SOLARANGLE must be selected to take the solar penetration angle into consideration• CHLMA94 must be selected to take the eﬀect of chlorophyll concentration into consideration. The chlorophyll
concentration will be taken from either the biogeochemical model with NPZD option or climatology (name =ChlorophyllClimatology) with CHLCLIM option.
When CHLCLIM option is selected, time series of 2-D chlorophyll concentration should be prepared. The data may beinterpolated in time and space in the model. See Section 14.10 for how to specify external forcing files.
b. Namelist

Namelists related to shortwave radiation are listed on Tables 14.5 through 14.7.
Table14.5 Namelist nml_albedo_ocean.

variable name
alb

units
no (real8)

albedo_choiceno (integer)

description
constant albedo
for albedo_choice = 1
choice of albedo schemeusage

1 : constant given by alb
2 : Large and Yeager (2009)
3 : MRI-ESM1

Table14.6 Namelist nml_solarangle.

variable nameinterval_shortwave_sec

units
sec

l_variable_zenith_ang

logicall_const_zenith_ang_mean

logicall_const_zenith_ang_noon
l_diurnal_var_shortwave

logicallogicall_penetr_ang_considered

logicaldescription
time interval of shortwave data
use zenith angle withdaily variation
use average zenith angle
from sunrise to sunset
use zenith angle at noon
daily variational amplitude of solar flux is calculated in this module
introduction of transmitted angle based on
Snell’s law

usage
requireddefault = .true.
valid if l_variable_zenith_ang = .false.
valid if l_variable_zenith_ang = .false.
default = .false., If this is true,
interval_shortwave_sec = 86400
default = .true.

Table14.7 Namelist nml_frac_nearir.

variable nameqsratio_nearIR

units
real(8)description
ratio of near infra-red band to the total
shortwave flux (R)– 154 –

usage
default = 0.58

Chapter 14

Sea surface fluxes

14.4 Longwave radiation flux
14.4.1 Formulation
Net longwave radiation into the ocean is the residual between the downward longwave radiation Qdown
LO (external data) andthe upward black body radiation of sea water with the sea surface temperature (θ surf ),
down
4
Qnet
LO = Q LO − em σ(θ surf + 273.15) .

(14.26)

Here, em is the emissivity for sea water, which should be specified by user, and σ = 5.67 × 10−5 erg · s−1 · cm−2 · K−4 isthe Stefan-Boltzmann constant.

14.4.2 Numerical implementation
There is no special issue on the numerical implementation. The net longwave flux is absorbed in the first layer of themodel. Before used to update the first level temperature, it is summed into an array (qothr) which gathers all heat flux
components except for shortwave. Note that the sea surface (first model level) temperature at the starting time level is used
to evaluate upward longwave flux in any time integration scheme.

14.4.3 Usagea. Model option

By default, the net longwave radiation is read as external forcing data. By choosing LWDOWN option, the downward
longwave radiation is read. LWDOWN option will become a standard (default) choice in the near future.
b. Namelist

Emissivity (em ) of black body radiation from the sea surface may be specified by user at run time with namelist
nml_emissivity_sea_water (Table 14.8).
Table14.8 Namelist nml_emissivity_sea_water.

variable nameemissivity

units
no (real8)

description
emissivity of upward longwave radiationfrom sea water

usage
default = 1.0

14.5 Latent and sensible heat fluxes
14.5.1 Formulation
The bulk method is used to calculate latent and sensible heat fluxes. In the bulk method, latent heat flux (QLA ) and sensible
heat flux (QSN ) are calculated using bulk transfer coeﬃcients CE and CH (see Section 14.9),⃗ |(qs − qa ),QLA = − ρ a LCE |∆U(14.27)⃗ |(Ts − θ a ).
QSN = − ρ a cpa CH |∆U(14.28)Here, ρ a is the air density, L is the latent heat of evaporation, qa is the specific humidity of the surface air, qs is thesaturated specific humidity of the sea surface temperature, θ a is the surface air potential temperature (K), Ts is the absolute⃗ = U⃗a − αU⃗s is the vector diﬀerence between the surface wind and the first levelsea surface water temperature (K), and ∆Uvelocity (surface current). As explained in relation to the wind stress (Section 14.1), α is the contribution of the surface
current to the calculation of the relative wind vector. The quantity cpa is the specific heat of air.

14.5.2 Numerical implementation
There is no special issue on the numerical implementation. The latent and sensible heat fluxes are absorbed in the firstlayer of the model. Before used to update the first level temperature, they are summed into an array (qothr) which gathers
all heat flux components except for shortwave. Note that the sea surface (first model level) temperature at the starting timelevel is used to evaluate fluxes in any time integration scheme.
– 155 –

14.6 Freshwater flux

14.5.3 Usage
Model option

MRI.COM can provide two calculation methods for bulk transfer coeﬃcients. One of them must be specified when HFLUX
option is chosen.
• BULKNCAR for Large and Yeager (2004; 2009)• BULKKONDO2 for Kondo (1975)
See Sections 14.9.2 and 14.9.3 for details.

14.6 Freshwater flux
14.6.1 Formulation
Freshwater flux through the sea surface (F W ) is caused mainly by precipitation (P), evaporation (E), river discharge (R),and formation-melting of sea ice (I):F W = P − E + R + I,
(14.29)Precipitation and river discharge are given as input data. Fresh water flux due to sea ice formation-melting is given by thesea ice model. Evaporation is computed by a bulk formula in the ocean model and it should be consistent with the latent
heat flux:⃗ |(qs − qa ) = −QLA /L.
E = ρ a CE |∆U(14.30)Note that evaporation (E) is defined as positive upward.
Fresh water flux is treated as the surface boundary condition for the surface height equation:∂η= ... + F W .
∂t(14.31)

Fresh water is added to/subtracted from the surface grid cell. Thickness of grid cells in the whole vertical column should
be modified to adapt to this change in the surface height. Concentration of materials (tracer species) in a grid cell should
be updated by taking into account of these volume changes. Knowledge about the contents of materials in the fresh water
flux is required for the computation, which is explained below.

14.6.2 Material fluxes due to fresh water flux
Precipitation, evaporation, and river discharge (WAO = P − E + R) are assumed to have the first level temperature (θ surf )and zero salinity. The water transported through the ice bottom (WIbot ) is also assumed to have the first level temperature.
The water transported from the ice surface (WIsurf > 0) is assumed to have the freezing point temperature (θ freeze ). The
water exchanged with ice is assumed to have low salinity (SI = 4.0 pss). Note that the freezing point temperature (θ freeze )is given by mSI , where m = −0.0543 ◦C/pss.
a. When fresh water fluxes are explicitly added to the systemIf fresh water flux is simply added to the first layer volume, surface temperature and salinity fluxes due to the fresh water
flux are given as follows:θ
FWF= (WAO + WIbot ) · θ surf + WIsurf · θ freeze,(14.32)S
FWF= (WIbot + WIsurf ) · SI .

(14.33)b. When fresh water fluxes are not explicitly added to the system

If the surface fresh water flux is not added to the first layer to avoid a long-term drift of sea level (SFLUXW option),equivalent temperature and salinity fluxes should be imposed to incorporate the eﬀects of surface fresh water flux. To
obtain a formula, the conservation equations for the first layer heat and salinity are considered. Temperature and salinity
are assumed to evolve from the old values (θ old, Sold ) to the new values (θ new, Snew ) due to fresh water flux.

– 156 –

Chapter 14

Sea surface fluxes

If a volume change is allowed,
Vnew θ new = Vold θ old + (WAO θ old + WIbot θ old + WIsurf θ freeze ) · ∆A · ∆t,(14.34)

Vnew Snew = Vold Sold + (WIbot SI + WIsurf SI ) · ∆A · ∆t,(14.35)

Vnew = Vold + (WAO + WIbot + WIsurf ) · ∆A · ∆t,(14.36)

where ∆A is the horizontal area.
If a volume change is not allowed,
θVold θ new = Vold θ old + FWF· ∆A · ∆t,(14.37)

S· ∆A · ∆t,
Vold Snew = Vold Sold + FWF(14.38)θ and F S are temperature and salinity flux for the constant volume case.
where FWF
WF
Removing θ new , Snew , and Vnew from the above equations, we have,

Vold {Vold θ old + (WAO θ old + WIbot θ old + WIsurf θ freeze ) · ∆A · ∆t}θ= {Vold + (WAO + WIbot + WIsurf ) · ∆A · ∆t}(Vold θ old + FWF· ∆A · ∆t),(14.39)Vold {Vold Sold + (WIbot SI + WIsurf SI ) · ∆A · ∆t}
S= {Vold + (WAO + WIbot + WIsurf ) · ∆A · ∆t}(Vold Sold + FWF· ∆A · ∆t).

(14.40)The fluxes for the constant volume case are given by
Vold
WIsurf · (θ old − θ freeze ),Vold + W · ∆A · ∆t
Vold
S
{WAO · Sold + (WIbot + WIsurf ) · (Sold − SI )},FWF=−
Vold + W · ∆A · ∆t
θ
FWF
=−

(14.41)(14.42)where W = WAO + WIbot + WIsurf .
If |Vold | ≫ |W · ∆A · ∆t|, we have,θFWF= − WIsurf · (θ old − θ freeze ),(14.43)S
FWF= − WAO · Sold − (WIbot + WIsurf ) · (Sold − SI ).

(14.44)14.6.3 Numerical implementation
a. Adding/subtracting new sea water

Fresh water flux is applied to the sea level prediction in the baroclinic time interval:
η n+1 = η n−1 + ... + 2∆t(F W ) n + (F W ) n−1,
2(14.45)where (F W ) n−1 and (F W ) n are fresh water fluxes evaluated at time levels n − 1 and n, respectively.
In the same manner, fresh water transport is reflected in the equations for temperature(θV ) n+1
= (θV ) n−1+ ... + ∆t [{(WAO ) n−1 + (WIbot ) n−1 } · (θ surf ) n−1 + (WIsurf ) n−1 · (θ freeze ) n−1 ]
1
1
2

2

+ ∆t [{(WAO ) n + (WIbot ) n } · (θ surf ) n + (WIsurf ) n · (θ freeze ) n ](14.46)and salinity(SV ) n+1
= (SV ) n−1+ ... + ∆t {(WIbot ) n−1 + (WIsurf ) n−1 } · SI + ∆t {(WIbot ) n + (WIsurf ) n } · SI .
1
1
2

2

– 157 –

(14.47)

14.6 Freshwater flux

b. Suppressing long-term drift of sea water volume

Because the global mean of F W should not necessarily be zero and there is no feedback mechanism in an ocean-sea ice
system to balance the fresh water budget, the globally averaged volume of the ocean may continue to increase or decrease.
To avoid this, the integral of the globally averaged freshwater flux is set to zero by selecting WADJ option,W
FADJ= FW − FW ,(14.48)where F W is the global mean of F W (= P − E + R), where E includes sublimation over sea ice. Note that exchange withsea ice is not included in the computation of F W .
Another option for suppressing the long term drift of sea level is to transform all freshwater fluxes to salinity flux, whichis activated by specifying SFLUXW option. Formulae for the temperature and salinity fluxes with this option are given by(14.43) and (14.44).

14.6.4 Usagea. Model options• WFLUX should be specified to explicitly treat fresh water fluxes in the model. In this case, precipitation data must
be prepared as input data.
• RUNOFF enables river discharge to be treated. River run-oﬀ data should be given as external data.
• WADJ imposes conservation of sea water volume of the ocean - sea ice system.
• SFLUXW computes equivalent temperature and salinity fluxes due to fresh water flux
b. Input data

When WFLUX and RUNOFF options are specified, time series of precipitation and river run-oﬀ data must be prepared,
respectively. Data on a structured lattice should be prepared in the single precision direct access format with a constanttime interval. Attributes of the data are specified by nml_force_data. Name and units assumed by the model aresummarized in Table 14.9.
The data may be interpolated in time and space in the model. See Section 14.10 for how to specify external forcing files.
Table14.9 Water flux data to be read from namelist nml_force_data.

element
precipitation
river discharge

namePrecipitation
RiverDischargeRateunits
g s−1 cm−2
g s−1 cm−2

usage
WFLUX
RUNOFFc. Namelist

The river run-oﬀ data around Antarctica may be treated as the discharge of iceberg if sea ice is present in the grid. Namelist
nml_mkflux specifies this behavior (Table 14.10).
Table14.10 Namelist nml_mkflux. See also Table 14.12.

variable namel_glacierunits
logicall_after_rivermouth_spreadlogicaldescription
if a rivermouth grid has sea ice then riverdischarge is converted into iceberg input
from glacierriver discharge is converted into iceberg after a smoothing operation of river discharge
data

usage
default = .false.
default = .false.

When WADJ option is selected, the global summation of surface fresh water fluxes, which is used for determining theoﬀsetting flux, can be made MPI parallel independent by choosing l_strict_waterflux_adjust = .true. in namelist
nml_waterflux_adjust (Table 14.11).

– 158 –

Chapter 14

Sea surface fluxes
Table14.11 Namelist nml_waterflux_adjust.

variable namel_strict_waterflux_adjust

units
logicaldescription
water flux adjustment is made MPI parallel
independent

usage
default = .false.

14.7 Treatment in the temperature equation
14.7.1 Formulation
The contribution of surface forcing (Fzθ ) and shortwave radiation flux (QSH ) to the first layer temperature is expressed asFzθ
1 ∂QSH∂θ
+,
1 = ... +∂t k= 2∆z 1
ρ0 cp ∂z(14.49)2

where ∆z 1 is the thickness of the first layer of a T-cell. The term (Fzθ ) consists of the net sea surface heat flux except2θ ), and restoring of SST (θfor the shortwave radiation flux (QOTHER ), heat transport due to fresh water flux (FWF
surf ) to a∗
specified value (θ ):QOTHER
1
θ
Fzθ =
+ FWF
−
(θ surf − θ ∗ )∆z 1 .
(14.50)2
ρ0 c pγθ
The heat flux (QOTHER ) has been converted to temperature transport using the reference density (ρ0 ) and specific heat (cp )of sea water. The third term on the r.h.s. is a restoring term to a specified SST (θ ∗ ). The parameter γθ is a restoring time(in units of seconds) and should be read from input file (see Chapter 13). The restoring term is converted to temperaturetransport by multiplying the thickness of the first layer of a T-cell (∆z 1 ).
2
By default, surface temperature forcing consists only of the restoring term. By specifying HFLUX option, sea surface
heat fluxes are treated explicitly. The heat transport due to fresh water flux are included when WFLUX option is specified.

14.7.2 Numerical Implementation
The heat transport due to fresh water flux is incorporated in the advection schemes. Shortwave absorption is treatedseparately from other heat flux terms. Heat flux components except for the shortwave radiation are gathered in an array(qothr) and added to the tendency of the first layer temperature.
Treatment of supercooled water needs some explanation. When the temperature of the first layer is below the freezing
point, the temperature is set to the freezing point and this heat comes from the latent heat release of the new sea ice
formation. This is done in the sea ice model. Because the sea ice part uses the forward scheme to proceed in time, thesea surface temperature at the "current" time level must be given to the sea ice part from the ocean part. The result is thatthe temperature below the freezing point at the "current" time level is modified and adjusted to the freezing point by thesea ice part. This treatment is not compatible with the leap-frog time stepping employed by the ocean part. Specifically,treating this temperature adjustment process in a form of heat flux exchange may result in a numerical instability. Though
the modification of sea surface temperature in the sea ice part is inconvenient for some purposes, we keep this treatment
until a revision of time-stepping scheme is considered.

14.8 Treatment in the salinity equation
14.8.1 Formulation
The fresh water flux (FW ) modifies the volume but not the salt content of the surface layer, changing the salinity of thesurface layer (see also Chapter 6). Since fresh water flux is not related directly to the sea surface salinity, the model sea
surface salinity might be far from the observed value. Hence, an adjustment is sometimes needed to restore the model sea
surface salinity to the observed one (see the last term on the r.h.s. of the next equation). Generally, the model salinity is
restored to the observed climatological sea surface salinity since no reliable data set of historical sea surface salinity isavailable at present.
The explicit surface forcing (FzS ) consists of salt transport due to formation and melting of sea ice and restoring of SSS(Ssurf ) to a specified value (S ∗ ):FzS = (WIbot + WIsurf ) · SI −– 159 –

1
(Ssurf − S ∗ )∆z 1 .
2
γs(14.51)

14.8 Treatment in the salinity equation
The water exchanged with ice (WIbot + WIsurf ) is assumed to have low salinity (SI = 4.0 pss). The parameter γs is a
restoring time (in units of seconds) and should be read from input file (see Chapter 13). The restoring term is converted
to salinity transport by multiplying the thickness of the first layer of a T-cell (∆z 1 ).
2
By default, surface salinity forcing consists only of the restoring term. When WFLUX option is specified, the eﬀect of
the fresh water flux on salinity may be included.
With WFLUX option, by default, the restoring term is converted to the fresh water flux. The expression for F W becomes
FW = P − E + R + I +1 Ssurf − S ∗∆z 1 .
2γs Ssurf(14.52)In this case, the salinity flux should be
FzS = (WIbot + WIsurf ) · SI .

(14.53)

However, the restoring term is not converted to the freshwater flux if SFLUXR option is selected.
If SFLUXW option is specified, the fresh water flux is not added to the ocean and its eﬀect on salinity is now explicitly
expressed as the salinity flux:
FzS = −WAO · Ssurf − (WIbot + WIsurf ) · (Ssurf − SI ) −1(Ssurf − S ∗ )∆z 1 ,2
γs(14.54)

where Ssurf is the first level salinity.
The contribution of surface forcing (FzS ) to the first layer salinity is expressed as
FzS
∂S
=
...
+
,
∂t k= 12∆z 1

(14.55)2where ∆z 1 is the thickness of the first layer of a T-cell.
2

14.8.2 Numerical Implementation
The salinity transport due to fresh water flux is incorporated in the advection schemes. The first layer salinity is calculated
by taking into consideration of the volume change of the grid cell due to the continuity equation.

14.8.3 Usagea. Model option• WFLUX treats fresh water fluxes explicitly in the model. Salinity will be aﬀected by fresh water fluxes.
• SFLUXW converts fresh water flux to surface salinity flux
• SFLUXR suppresses conversion of salinity restoring to water flux
• STABLERUN restores sea surface salinity to climatology when salinity is lower than 5 pss
b. Namelist

Namelists related to salinity flux at the sea surface are listed on Tables 14.12 and 14.13.
Table14.12 Namelist nml_mkflux. See also Table 14.10.

variable namers_localunits

description
how salinity for iceberg and snow is extractedusage
1 : taken from the local grid0 : taken from global sea surface

Table14.13 Namelist nml_sss_restore.

variable namesss_rst_diff_max

units
pss

description
the maximum of the diﬀerence between modeland reference sea surface salinity

usage
default = 100 pss
Continued on next page

– 160 –

Chapter 14

Sea surface fluxes

variable namel_nosss_rst_seaicel_nosss_rst_csticel_sss_rst_cnsv
l_strict_sss_rst_cnsv

Table 14.13 – continued from previous page
description
no surface salinity restoring below sea ice
no surface salinity restoring at the coastal gridwith sea ice
logical set globally integrated surface salinity restoringto zerological global summation of surface salinity restoring
flux is done serially so that the result becomes
MPI parallel independentunits
logicallogicalusage
default = .false.
default = .false.
default = .false.
default = .false.

14.9 Bulk transfer coeﬃcientThis section briefly describes how to calculate sea surface fluxes using a bulk formula. For details, readers are referred toKantha and Clayson (2000) and Large and Yeager (2004). Bulk transfer coeﬃcient for air-ice interface is given in Section
17.2.3.

14.9.1 Formulation of bulk formula
Transfer processes through atmosphere and ocean boundaries are governed by complicated turbulent processes. Traditionally, these turbulent fluxes are parameterized as a bulk transfer law. This method does not explicitly solve the interaction
between the atmosphere and the ocean, and attributes all unknown processes to bulk transfer coeﬃcients.
Momentum (τ), sensible heat (HS ), latent heat (H L ), and water vapor (E) fluxes are written in terms of turbulent
components as follows:τ = − ρ a wu = ρ a u∗2,

(14.56)HS = − ρ a cpa wθ = ρ a cpa u∗ θ ∗,(14.57)

H L = − ρ a L E wq = ρ a L E u∗ q∗,(14.58)E = − ρ a wq = ρ a u∗ q∗ = H L /L E ,(14.59)where a bar over two variables denotes a covariance between the turbulent component of vertical velocity (w) and that ofeach physical property, wind speed (u), air potential temperature (θ), and specific humidity (q). ρ a is air density, u∗ isfriction velocity, θ ∗ ≡ HS /( ρ a cpa u∗ ) is the temperature scale in the boundary layer, q∗ ≡ H L /( ρ a L E u∗ ) = E/( ρ a u∗ ) is
the scale of specific humidity in the boundary layer, cpa is the specific heat of air, and L E is the latent heat of water vapor
evaporation.
Sea surface fluxes are also represented using bulk formulae as follows:τ = ρ a CDh |U⃗a − U⃗s | 2,

(14.60)HS = − ρ a cpa |U⃗a − U⃗s |CHh (Ts − θ a ),(14.61)

H L = − ρ a L E |U⃗a − U⃗s |CEh (qs − qa ),(14.62)E = ρ a |U⃗a − U⃗s |CEh (qs − qa ),(14.63)where U⃗a is the wind vector at a height of zh , U⃗s is the current vector at the sea surface, and Ts is the absolute sea surface
water temperature (K). The subscript "a" means the value at a height of z = zh , and the subscript "s" means the value atthe sea surface. The contribution factor (α) of the surface current to the calculation of the relative wind vector explainedin Section 14.1 is omitted here for brevity.
Parameter CDh is called a drag coeﬃcient. CHh and CEh are transfer coeﬃcients for heat and water vapor and are called
the Stanton coeﬃcient and the Dalton coeﬃcient, respectively. These can be estimated by observed atmospheric elements(wind velocity, air temperature, and humidity) at a height in the boundary layer, not by observed turbulent fluxes, using– 161 –

14.9 Bulk transfer coeﬃcientthe following Equations (14.64), (14.65), and (14.66) and the similarity law of Monin-Obukhov mentioned below,
CDh =

u∗2

,
|U⃗a − U⃗s | 2
HS /ρ a cpa
u∗ θ ∗
CHh = −=,|U⃗a − U⃗s |(Ts − θ a )|U⃗a − U⃗s |(Ts − θ a )
E/ρ a
u∗ q∗
CEh =
=
.
|U⃗a − U⃗s |(qs − qa )|U⃗a − U⃗s |(qs − qa )(14.64)(14.65)(14.66)The similarity law of Monin-Obukhov assumes that physical properties in the atmosphere-ocean boundary layer (a layer
with a thickness of several tens of meters located below the lower mixed layer of the atmosphere) have similar verticalprofiles when they are scaled with the stability parameter and sea surface fluxes. Vertical profiles of wind speed (U), airtemperature (θ), and humidity (q) can be written as follows:
(z)
κz ∂U
= ϕM
,
u∗ ∂z
L
(z)
κz ∂θ
= ϕH
,
θ ∗ ∂z
L
(z)
κz ∂q
= ϕE
,
q∗ ∂z
L

(14.67)(14.68)(14.69)where κ = 0.4 is the von Karman constant, and L is the Monin-Obukhov length scaleL=u∗3u3 θ v.
=− ∗κQ b
κ g wθ v(14.70)In (14.70), Q b = −g wθ v /θ v is the buoyancy flux, θ v is the virtual temperature (θ v = θ (1 + 0.6078 q)), and wθ v is thenet heat flux including the water vapor flux:wθ v = wθ (1 + 0.6078 q) + 0.6078 θ wq.

(14.71)

In (14.67) to (14.69), ζ = z/L is the Monin-Obukhov similarity variable, and ϕ M, H,E is a nondimensional function forwind velocity, air temperature, and specific humidity. The nondimensional function is assumed to be a mathematicallysimple function.
Integrating Equations (14.67), (14.68), and (14.69) vertically, we have the following.
[]u∗z
U (z) − Us =ln − ΨM (ζ ) ,(14.72)
κ
z0
[
]
θ∗
z
θ(z) − Ts =ln
− ΨH (ζ ) ,(14.73)
κ
z0T
[
]
q∗
z
(14.74)q(z) − qs =ln
− ΨE (ζ ) ,
κ
z0E
where
ΨM, H,E (ζ ) =

∫ ζ[

′

1 − ϕ M, H,E (ζ )0] dζ ′
ζ′.

(14.75)

In (14.72) to (14.74), z0 , z0T , and z0E are roughness lengths for each physical property. When the stability of the boundary
layer is already known, bulk transfer coeﬃcients can be estimated using these roughness lengths,
CDh = [

CHh,Eh = [
ln

κ2
zh
ln − ΨM (ζh )z0zh− ΨM (ζh )z0][– 162 –

]2,

κ2
ln

zh
z0T,0E(14.76)].
− ΨH,E (ζh )(14.77)

Chapter 14

Sea surface fluxes

In neutral stability (ζ = 0), the bulk transfer coeﬃcient is a function of the roughness length (z0N ,z0TN , and z0EN foreach) only and is expressed as follows:κ2
CDNh = [(14.78)
]2,
zh
ln
z0N

CHNh,ENh = [

zh
ln
z0N

][

κ2
zh

ln

]

(14.79)z0TN,0EN
CDNh]=[
z0TN,0EN
1 1/2
1 − CDNh ln
κ
z0N
1/2
κCDNh

=[
ln

(14.80)].

zh(14.81)z0TN,0EN

Normally, these neutral bulk transfer coeﬃcients are estimated at a height of 10 m. Non-neutral bulk transfer coeﬃcientsat an arbitrary height (zh ) are connected with the neutral bulk transfer coeﬃcients at a height of 10 m by eliminating theroughness length as follows:
CDN10
CDh = [
(14.82)
(
)]2,
1
zh
−1/2
1 + CDN10 CDN10 ln− ΨM (ζh )
κ
z10
) 1/2
CDh
CDN10
(
)] ,
CHh,Eh = [
1
zh
−1/2
ln
− ΨH,E (ζh )1 + CHN10,EN10 CDN10κz10(

CHN10,EN10(14.83)where z10 means z = 10 m.
The neutral bulk transfer coeﬃcients at a height of 10 m (CDN10 , CHN10 , and CEN10 ) are often estimated, according to
the stability, as a function of velocity at 10 m. Various formulations can be used (see the following subsections). To be
more realistic, the bulk transfer coeﬃcients should be regarded as a function of wave age, but its general formulation has
not been achieved yet.
In principle, if we have a complete set of atmospheric elements in the boundary layer, we can get the bulk coeﬃcientsand fluxes because height can be adjusted using the similarity law. This requires an iteration of the calculation as follows(you can select an appropriate way for your data):• Convert wind speed, air temperature, and specific humidity into those at 10 m in the neutral stability and use a bulk
coeﬃcient at 10 m in the neutral stability.
• Convert a bulk coeﬃcient at 10 m into one with a height and a stability at which atmospheric elements are observed.
• Convert air temperature and specific humidity into those at a height where wind speeds are observed, and convert
the bulk coeﬃcient at 10 m in the neutral stability into one with a height and a stability at which wind speeds areobserved.

14.9.2 Large and Yeager (2004; 2009): BULKNCAR
The bulk formula based on Large and Yeager (2004; 2009) closely follows the formulation explained in the previous
subsection. Each flux is estimated at a height where wind speed is observed by transforming air temperature and humidityfrom the height where they are observed (z θ and zq , respectively) to the height where wind speed is observed. To do thisiteration in the program bulk.F90, specify BULKITER option in configure.in. The observed heights are specified in
namelist nml_srfflx_dbase (see Table 14.18).
The bulk transfer coeﬃcient at 10 m in the neutral stability is given as follows:{
10 CDN10 =
3

U10N
2.70
−10U 6 ,U10N + 0.142 + 13.09 − 3.14807 × 10
10N

2.34
– 163 –

U10N < 33 m s−1
U10N ≥ 33 m s−1(14.84)

14.9 Bulk transfer coeﬃcient√
103 CHN10,EN10 = 18.0 CDN10, stable ζ > 0,√
103 CHN10,EN10 = 32.7 CDN10, unstable ζ ≤ 0,

(14.85)(14.86)

Each physical property is estimated at a height where wind speed is observed in the following operations. First, calculate
the virtual temperature θ v as follows:θ v = θ(z θ )(1 + 0.6078 q(zq ))(14.87)
Next, calculate the bulk transfer coeﬃcient at 10 m in the neutral stability assuming that the first guess for the 10 m wind
⃗ (zu )| = |U⃗a (zu ) − U⃗s | (Equations (14.84) to (14.86)). The first guesses of the
speed in the neutral stability is U10N = |∆U
scales for the friction velocity, air temperature, and specific humidity are estimated assuming that these bulk coeﬃcientsare at the observed height and stability,√√τ⃗ (zu )|,
= CDN10 |∆U
ρa
HS
CHN10
=√
(θ(z θ ) − Ts ),θ∗ =ρ a c p u∗
CDN10
E
CEN10
=√
q∗ =
(q(zq ) − qsat (Ts )),ρ a u∗
CDN10u∗ =(14.88)(14.89)(14.90)where qsat (Ts ) is the saturated specific humidity at sea surface temperature Ts .
Next, perform the iteration using the three Monin-Obukhov similarity variables, ζu = zu /L, ζ θ = z θ /L, and ζ q = zq /L,and an integral of the nondimensional profile function for the vertical gradient of each physical property, ΨM (ζ ) for
momentum and ΨH (ζ ) for scalars.
The Monin-Obukhov similarity variables are calculated as follows:][κgz θ ∗
q∗.
(14.91)ζ= 2+u∗ θ v (q(zq ) + 0.6078−1 )The integral of the non-dimensional profile function is expressed asΨM (ζ ) = ΨH (ζ ) = −5ζ,

(14.92)

if it is stable (ζ ≥ 0), and
(

)
(
)
1+X
1 + X2πΨM (ζ ) = 2 ln
+ ln
− 2 tan−1 (X ) + ,
2
2
2
(
)
2
1+X
ΨH (ζ ) = 2 ln,
2
if it is unstable (ζ < 0). In the above,

X = (1 − 16ζ ) 1/4 .

(14.93)(14.94)(14.95)Using these values, convert the wind speed to that at 10 m in the neutral stability, and convert the temperature andspecific humidity to those at a height where the wind speed is observed,√[(] ) −1
CDN10
zu
⃗
ln
U10N = |∆U (zu )| 1 +− ΨM (ζu )
,
κ
z10
[
]
zθ
θ∗
ln
θ(zu ) = θ(z θ ) −+ ΨH (ζu ) − ΨH (ζ θ ) ,
κ
zu
[
]
zq
q∗
q(zu ) = q(zq ) −ln+ ΨH (ζu ) − ΨH (ζ q ) ,κzu– 164 –

(14.96)(14.97)(14.98)

Chapter 14

Sea surface fluxes

where z10 means z = 10 m. Estimate the bulk coeﬃcient at 10 m in the neutral stability using U10N , and then obtain thebulk coeﬃcient at a height (zu ) where wind speed is observed,
CDN10
(
)]2,
1
zu
−1/2
1 + CDN10 CDN10 ln− ΨM (ζu )
κ
z10
(
) 1/2
CDu
CHN10
CDN10
(
)] ,
CHu = [
1zu
−1/2
1 + CHN10 CDN10 ln− ΨH (ζu )
κ
z10
(
) 1/2
CDu
CEN10
CDN10
(
)] .
CEu = [
zu
1
−1/2
ln
− ΨH (ζu )1 + CEN10 CDN10κz10
CDu = [

(14.99)(14.100)(14.101)

Repeat the procedures to calculate the bulk coeﬃcients using these bulk coeﬃcients with temperature and specific humidity
at z = zu , and recalculate virtual temperature (14.87) and the scales for friction velocity, temperature, and specific humidity(Equations of (14.88), (14.89), and (14.90)).

14.9.3 Appendix of Kondo (1975): BULKKONDO2In Appendix 1-3 of Kondo (1975), approximate formulae of the bulk transfer coeﬃcients are given in support of practicalapplications. The BULKKONDO2 option realizes this in MRI.COM. In this scheme, providing that atmospheric elements areobserved at various heights, latent and sensible heat fluxes are estimated at a height where air temperature and humidity
are observed. Momentum fluxes are also estimated at this height. Wind speed and bulk transfer coeﬃcients should becorrected to ones at the height where the air temperature and humidity are observed. In the program of bulk.F90, users
may specify BULKITER option in configure.in to do this iteration. The observed heights are specified in namelist
nml_srfflx_dbase (see Table 14.18).
⃗ (zu )| in
The following procedure is executed starting from the observed wind speed relative to the surface current |∆Uunits of [m/s] at a height of zu [m]:1. Estimate the neutral bulk transfer coeﬃcient at 10 m assuming that the observed wind speed is at 10 m in the neutral⃗ (zu )|). When the 10 m wind speed is already known, the neutral bulk transfer coeﬃcient isstability (U10 = |∆Ugiven as follows:p
CDN10 = 10−3 {ad + bd U10d },(14.102)where ad , bd , and pd are nondimensional numbers depending on the wind speed (Table 14.14).
2. Recalculate the roughness length using this neutral bulk transfer coeﬃcient at 10 m and correct the height for windspeed once again:{}−1/2
z0 = exp ln z10 − κ CDN10,(14.103)⃗ (zu )| ln(10/z0 )/ ln(zu /z0 ),U10 = |∆U

(14.104)

where z10 means z = 10 m.
3. Re-estimate the neutral bulk transfer coeﬃcient at 10 m using (14.102).
4. Operations 1 through 3 above are repeated twice or three times to determine CDN10 and U10 .
5. Convert the bulk transfer coeﬃcient to that in the neutral stability at the observed height of temperature and humidity(zh ):
κ2
CDNh = [(14.105)]2−1/2
κCDN10− ln(z10 /zh )CHNh,ENh =

1/2
κCDNh
1/2
κ CDN10/CHN10,EN10 + ln(zh /z10 )– 165 –.

(14.106)

14.9 Bulk transfer coeﬃcient
The neutral bulk transfer coeﬃcients at 10 m for evaporative flux and sensible heat flux (CEN10 and CHN10 respectively) are given as follows:
CEN10 = 10−3 {ae + beU10e + ce (U10 − 8) 2 },p

CHN10 = 10−3

(14.107)p{ah + bh U10h + ch (U10 − 8) 2 },(14.108)

where ae,h , be,h , ce,h , and pe,h are nondimensional numbers depending on wind speed (Table 14.14).
6. Estimate the bulk transfer coeﬃcient in which stability is considered using the neutral bulk transfer coeﬃcientobtained above.
Table14.14 Non-dimensional parameters used in calculating the bulk transfer coeﬃcient based on Kondo (1975)

U10(m · s−1 )0.3-2.2
2.2-5
5-8
8-25
25-50

ad

ae

ah

bd

be

bh

ce

ch

pd

pe

ph

0
0.771
0.867
1.2
0

0
0.969
1.18
1.196
1.68

0
0.927
1.15
1.17
1.652

1.08
0.08580.0667
0.025
0.073

1.23
0.05210.01
0.008−0.0161.185
0.05460.01
0.0075−0.01700
0−0.0004
00
0
0−0.00045
0

−0.15
1
1
1
1

−0.16
1
1
1
1

−0.1571
1
1
1

The bulk transfer coeﬃcient depending on the stability is estimated in the following operations. First, the stability of
the atmospheric boundary layer is defined as follows:s = s0where
s0 =⃗ (zh )| is approximated byand |∆U|s0 |,|s0 | + 0.01

(14.109)

Ts − θ a⃗ (zh )| 2 [1.0 + log10 (10/zh )]2 )|∆U.

⃗ (zh )| = |∆U⃗ (zu )| ln(zh /z0 )/ ln(zu /z0 ).
|∆U(14.110)(14.111)

The atmospheric boundary layer is unstable if s > 0 (Ts − θ a > 0) and stable if s < 0 (Ts − θ a < 0). Finally, the bulk
transfer coeﬃcient depending on the stability is given as follows: 0
CD = 
 CDNh {0.1 + 0.03s√+ 0.9 exp(4.8s)}
 CDNh (1.0 + 0.47 s)

s < −3.3−3.3 ≤ s < 00 ≤ s,(14.112) 0
CE = 
 CENh {0.1 + 0.03s√+ 0.9 exp(4.8s)}
 CENh (1.0 + 0.63 s)

s < −3.3−3.3 ≤ s < 00 ≤ s,(14.113) 0CHNh {0.1 + 0.03s + 0.9 exp(4.8s)}
CH = 
 C (1.0 + 0.63√ s) HNhs < −3.3−3.3 ≤ s < 00 ≤ s.

(14.114)14.9.4 Numerical implementation
Because the iteration loop of the bulk formulae is not vectorized, the count of iteration is set twice and the repeating
operation is explicitly written in the source programs.

14.9.5 Usage• BULKKONDO2 for Kondo (1975)• BULKNCAR for Large and Yeager (2004; 2009)• BULKITER for computing bulk-transfer coeﬃcient using the iterative method.
– 166 –

Chapter 14

Sea surface fluxes

14.10 Specification of input data
The attributes of external forcing data should be specified by namelist nml_force_data. Table 14.15 explains variables
that are contained in this namelist block.
Table14.15 Namelist nml_force_data related to surface forcing data.

variable name
name
txyu
l2d

units

logicalfile_data
file_data_grid
imfrc
jmfrc
num_elminterval
num_data_max
ifstartlrepeatlinterpilinearluniform
loffsetoffsetlfactorfactorldoubleiverbose
ldefinedsec

ifstart(6)logicallogicallogicallogicallogicallogicallogicaldescription
name of data object
grid point on which data should be placed
data is 2D or not
file that contains reference vales for surface restoring
forcingdata file name of grid
grid size of data in x direction
grid size of data in y direction
the number of elements contained in fileregular time interval of data
the number of record contained in the file[ymdhms] of the first record of the input filerepeat use of climatological data or notinterpolate horizontally or notinterpolation methoddata is horizontally uniform or notdata is oﬀset or not
oﬀsetting factor
data is multiplied or not
multiplicative factor
input data is double precision or notstandard output of progress
data object with this name is defined or not

usage.true. for surface data, default = .false.)
needed if linterp = .true.

−1 : monthly−999 : steady forcing
default = .false.
default = .false.
1 : linear, 2 : spline
default = .false.
default = .false.
default = .false.
default = .false.
1 : extensive, 0 : minimum
default = .true.

Format of data files that should be prepared is shown in the following. Note that one data file should contain only one
element.

– 167 –

14.11 Surface data objects
Format of surface forcing data (trcref(_surf)_conf%file_data)

integer(4), parameter :: num_data_max = 365*4, nu = 99
! "num_data_max" is the number of data
integer(4) :: imfrc, jmfrc! data size
character(128) :: file_data, file_data_gridreal(4) :: precip(imfrc,jmfrc,num_data_max)
real(8) :: alonf(imfrc), alatf(jmfrc)
logical :: linterp! main data
open (unit=nu,file=file_data,access=direct,recl=4*imfrc*jmfrc)
do n = 1, num_data_max
write(unit=nu,rec=n) precip(:,:,n)end doclose(nu)! longitude/latitude of the main data

if (linterp) then ! If input data is horizontally interpolated in the model.
open (unit=nu,file=file_grid)
write(nu) alonf, alatf
close(nu)end if

14.11 Surface data objects
External data either read from file or received from AGCM are stored in the data set object type type_surf_dbase(Table 14.16). This object contains a sub object meteor_alt of type type_surf_meteor_alt (Table 14.17) that storesthe measurement height of marine meteorological variables. The name (ReadData for Reanalysis or FromCGCM for
AGCM) of the object and the measurement height of marine meteorological variables must be specified by namelist
nml_srfflx_dbase (Table 14.18).
Air-sea fluxes computed on the basis of external data are stored in the data set object type type_surf_product_ao(Table 14.19). This object contains sub objects givn_ao and calc_ao of the object type type_surf_bulk_ao (Table
14.20). The object calc_ao stores elements related to the computation of fluxes based on the bulk formula and the object
givn_ao stores the corresponding elements received by other components, that is, atmospheric model. Similarly, air-ice
fluxes are stored in type_surf_product_ai (Table 14.21), which contains sub objects givn_ai and calc_ai of theobject type type_surf_bulk_ai (Table 14.22).
Table14.16 Contents of the object type_surf_dbase related to external surface forcing data.

variable namenamemeteor_alt
x_windy_windtemp_airtdew_airsphm_airs_windunits
type_surf_meteor_alt
cm sec−1 or dyn cm−2
cm sec−1 or dyn cm−2
◦C◦C1
cm sec−1

description
name of data object
Observation altitude of this data set
X-ward wind or wind stressY-ward wind or wind stresssurface air temperature
dew point temperature
specific humidity
scalar surface windContinued on next page

– 168 –

Chapter 14

Sea surface fluxes
Table 14.16 – continued from previous page
units
descriptionhPa
sea level pressure
erg s−1 cm−2
upward or downward shortwave, summed
erg s−1 cm−2
upward or downward shortwave, visible, beam
erg s−1 cm−2
upward or downward shortwave, visible, diﬀuse
erg s−1 cm−2
upward or downward shortwave, near Infra-red, beam
erg s−1 cm−2
upward or downward shortwave, near Infra-red, diﬀuse
erg s−1 cm−2
upward or downward longwave◦C
sea surface temperature
cm s−1
precipitation water flux
cm s−1river discharge water flux
−1−2
erg s cmriver discharge heat flux
cm s−1snow fall water flux
cm s−1iceberg discharge water flux
erg s−1 cm−2
iceberg discharge heat flux
1
sea ice fraction

variable nameslpress
sw_flxsw_flx_visb
sw_flx_visd
sw_flx_nirb
sw_flx_nird
lw_flxsst
precipriver_w
river_hsnowfalliceberg_w
iceberg_h
frac_ice

Table14.17 Contents of the object type_surf_meteor_alt that stores measurement height of meteorological elements of external data.

variable namealt_wind
alt_temp
alt_sphm

units
m
m
m

description
altitude of wind
altitude of temperature
altitude of specific humidity

Table14.18 Namelist nml_srfflx_dbase.

variable name
name

units

description
name of the external dataset

l_marine_meteor
alt_wind_meter
alt_temp_meter
alt_sphm_meterlogicalmmmread marine meteorological data or not
observed height of windobserved height of temperatureobserved height of specific humidity

usage
ReadData (Reanalysis) or FromCGCM (AGCM)(required)required
required if l_marine_meteor = .true.
required if l_marine_meteor = .true.
required if l_marine_meteor = .true.

Table14.19 Contents of the object type_surf_product_ao that stores air-sea fluxes based on type_surf_dbase.

variable namenamel_calc_wind_stressl_calc_bulk_fluxes
sw_nt_ao
sw_nt_visb_ao
sw_nt_visd_ao
sw_nt_nirb_ao
sw_nt_nird_ao
lw_nt_aotsfc_ocn
wind_speed
givn_aocalc_aounits
logicallogicalerg s−1 cm−2
erg s−1 cm−2
erg s−1 cm−2
erg s−1 cm−2
erg s−1 cm−2
erg s−1 cm−2◦C
cm sec−1
type_surf_bulk_ao
type_surf_bulk_ao

description
name of data object
calculate wind stress for this data set
calculate fluxes based on a bulk formula for this data set
net shortwave, summed
net shortwave, visible, beam
net shortwave, visible, diﬀuse
net shortwave, near Infra-red, beam
net shortwave, near Infra-red, diﬀuse
net longwave, air-ocean
sea surface (skin) temperaturescalar wind speed
given from other model or forcing data filecalculated using bulk formulae– 169 –

14.12 Technical details
Table14.20 Contents of the object type_surf_bulk_ao that stores elements related to computation of air-sea fluxes
based on bulk formula.

variable namenamex_rel_wind
y_rel_wind
x_stress
y_stress
frcvl
rel_windlatentsensible
evapo

units
m sec−1
m sec−1dyn cm−2
dyn cm−2
m sec−1
m sec−1
erg s−1 cm−2
erg s−1 cm−2
cm sec−1

description
name of data object
surface wind relative to surface current
surface wind relative to surface current
X-ward surface stressY-ward surface stressfriction velocity
surface wind relative to surface current
latent heat flux, air-ocean
sensible heat flux, air-ocean
evaporative water flux, air-ocean

Table14.21 Contents of the object type_surf_product_ai that stores air-ice fluxes based on type_surf_dbase.

variable namenamel_calc_wind_stressl_calc_bulk_fluxes
sw_nt_aisw_nt_insw_nt_visb_ai
sw_nt_visd_ai
sw_nt_nirb_ai
sw_nt_nird_ai
lw_nt_aitsfc_ice
wind_speed
givn_aicalc_aiunits
logicallogicalerg s cm−2
erg s cm−2
erg s cm−2
erg s cm−2
erg s cm−2
erg s cm−2
erg s cm−2◦Cm sec−1
type_surf_bulk_ai
type_surf_bulk_ai

description
name of data object
calculate wind stress for this data set
calculate fluxes based on a bulk formula for this data set
net shortwave, summed, absorbed at the ice surface
net shortwave, summed, penetrate into the ice interior
net shortwave, visible, beam
net shortwave, visible, diﬀuse
net shortwave, near Infra-red, beam
net shortwave, near Infra-red, diﬀuse
net longwave, air-ice
ice surface temperaturescalar wind speed
given from other model or forcing data filecalculated using bulk formulae

Table14.22 Contents of the object type_surf_bulk_ai that stores elements related to computation of air-ice fluxes
based on bulk formula.

variable namenamex_rel_wind
y_rel_wind
x_stress
y_stress
frcvl
rel_windlatentsensible
sublimctntmpcsntmpunits
m sec−1
m sec−1dyn cm−2
dyn cm−2
m sec−1
m sec−1
W m−2
W m−2
−1
m sec of sea waterm sec−1
m sec−1

description
name of data object
surface wind relative to ice drift
surface wind relative to ice drift
X-ward surface stressY-ward surface stressfriction velocity
surface wind relative to ice drift
latent heat flux, air-ice
sensible heat flux, air-ice
sublimation water flux, air-ice
bulk transfer coeﬃcient × scalar windbulk transfer coeﬃcient × scalar wind14.12 Technical details
14.12.1 Work flow in the main program
Work flow in the main program is as follows. Note that the sea ice model is part of the surface flux module (surfflux).
In the surface flux module, air-sea fluxes are calculated first, then they are partly modified by the presence of sea ice.

– 170 –

Chapter 14

Sea surface fluxes

ogcm__run (ogcm.F90)|+ LOOP_CGCM_SCUP|+ cgcm_scup_ctl__cpl_loop_start|||+ cgcm_scup__get_a2o(Update surface fluxes ’FromCGCM’)|||+ cgcm_scup__get_atmos_a2o (Update surface atmospheric state ’FromCGCM’)|+ LOOP_TIME_STEP|||+ part_1|||+ surfflux_ctl__set_surf_state
|
|
|
|
|
+ force__everytime(Update external forcing ’ReadData’)
|
|
|
|
|
+ solarangle__main(Compute solar insolation angle)|||+ surfflux_ctl__main|||+ surfflux__air_ocean(Determine Air-Sea flux,||
either ’ReadData’ or ’FromCGCM’)|||+ surfflux__seaice(Flux modification due to the presence of Sea Ice)|||
+ surfflux__oceanflux(Determine fluxes at the sea surface)|||+ solarangle__shrtwv_flux (Determine veritcal distribution|of solar insolation flux)|+ end LOOP_TIME_STEP|+ cgcm_scup_ctl__cpl_loop_end|||+ cgcm_scup__put_o2a(Transfer surface oceanic state to coupler)|+ end LOOP_CGCM_SCUP

14.12.2 Coupling with an atmospheric modela. Standard caseIn the coupled mode, the ocean model receives all kinds of surface fluxes from the atmospheric model and the surface
oceanic state such as sea surface temperature, surface oceanic current, and sea ice state, are transferred to the atmosphericmodel. The coupling interval is usually longer than the time step of the ocean model and the fluxes are kept fixed in theocean model during the coupling cycle. See Section 17.9.2 for details of flux adjustment for conserving heat and fresh
water fluxes by correcting errors caused by interpolation and sea ice evolution during the coupling loop.
Work flow is summarized as follows:
1. Getting fluxes from the atmospheric model at the beginning of a coupling cycle in cgcm_scup__get_a2o(cgcm_scup.F90).
2. Getting fluxes for this step from get_flux_a2o (get_fluxes.F90).
3. Sea surface fluxes are determined after solving the sea ice model
4. Solve time evolution of the oceanic state
5. Send oceanic state to the atmospheric model– 171 –

14.12 Technical details

b. Recalculating and replacing fluxes in the ocean model

In coupled mode (SCUPCGCM option) or in child mode of nesting (SUB option) in which fluxes are taken from the parent
model (NGETFLUX option which is often used when the parent model is coupled with the atmosphere), the received fluxes
may be replaced by the ones calculated on its own by using the surface atmospheric states taken from the parent component,
its surface oceanic states, and its bulk formula. User may also specify the grid points where fluxes are replaced. This
option is mainly intended to be applied to coastal region of a child model of nesting in which the fluxes received from theparent component may not fully reflect the fine scale surface state due to complex topography. Without an appropriate
feedback mechanism at the sea surface, a child model may sometimes get unstable. This runtime option is controlled bynamelist nml_cgcm_recalc and nml_cgcm_recalc_replace, which is explained in Tables 14.23 and 14.24.
Table14.23 Namelist nml_cgcm_recalc.

variable namel_recalc_flux

units
logicaldescription
Recalculating of fluxes

usage
default = .false.

Table14.24 Namelist nml_cgcm_recalc_replace.

variable namel_recalc_replace_all
file_recalc_replace

units
logicalcharacterdescription
Replacing fluxes for all grid points
File that contains mask that specify
grid points where fluxes are replaced.

usage
default = .false.
real(8) 2d array (1/0) written with direct access14.12.3 Properties of moist air
In MRI.COM, a set of formulae given by Gill (1982) is used to compute properties of moist air. Physical constants of sea
water is listed on Table 2.1 of Chapter 2.
a. Saturation specific humidity

We consider computing saturation specific humidity in an environment with the sea surface temperature t [◦C] and thepressure p [hPa].
The relation between the vapor pressure e [hPa] and specific humidity q is give by
e/p = q/(ϵ + (1 − ϵ )q),(14.115)

where ϵ is the molecular weight ratio between water vapor and air:
ϵ = mw /ma = 18.016/28.966 = 0.62197.

(14.116)This is solved for the specific humidity asq = ϵ e/(p − (1 − ϵ )e).

(14.117)

The relative humidity γ is the ratio of the mixing ratio of the mass of vapor to the mass of dry air to the saturated one.
The mixing ratio (r) of the mass of vapor to the mass of dry air is given by
r=

q
.
1−q

(14.118)

Thus the relative humidity is computed using the specific humidity q and the saturation specific humidity qs asγ=

q(1 − qs ).
qs (1 − q)(14.119)

The saturation vapor pressure esw [hPa] of pure water vapor over a plane water surface is given by
log10 esw (t) = (0.7859 + 0.03477 t)/(1 + 0.00412 t),
where t is temperature between ±40 ◦C.

– 172 –

(14.120)

Chapter 14

Sea surface fluxes′ of water vapor at saturation is not exactly e but is given by
In air, the partial pressure esw
sw

The value of f w is given by′esw= f w esw .

(14.121)f w = 1 + 10−6 p(4.5 + 0.0006 t 2 ),(14.122)

where p is the pressure (in units [hPa]).
The saturation vapor pressure over a salt solution is less than over fresh water. For sea water, the reduction is about 2 %′ ).
(a factor of 0.98 should be applied to esw
The saturation vapor pressure esi of pure water vapor over ice is given by
log10 esi (t) = log10 esw (t) + 0.00422 t.

(14.123)

The saturation partial pressure esi′ in moist air is f i times esi . Values of f i are given correct to 1 part in 104 by (14.122).
Thus(14.124)esi′ = f w esi .
b. Specific heat of air

7 (
8q )
R 1−q+
27ϵ= 1004.6 × (1 + 0.8735q) J kg−1 K−1(14.125)

L v = 2.5008 × 106 − 2.3 × 103 t J kg−1 .

(14.126)

cpa =c. Latent heat

The latent heat of vaporization is given by

The latent heat of sublimation is given by
L s = 2.839 × 106 − 3.6(t + 35) 2 J kg−1 .

(14.127)

d. Air density ( ρ a )The equation of state of moist air of temperature T [K] under the pressure P [Pa] is given by
ρa =P≡ P/RTv,
RT (1 − q + q/ϵ )(14.128)

whereis called the virtual temperature andTv ≡ T (1 − q + q/ϵ ) = T (1 + 0.6078q)(14.129)

R = 287.04 J kg−1 K−1(14.130)is the gas constant of dry air.

– 173 –

Chapter 15

Turbulence Closure Models
The surface boundary layer is made turbulent by wind injecting momentum, so vertical mixing may be induced even if thestratification is stable. The stratification near the surface may be made gravitationally unstable by the surface buoyancy
loss, so the vertical convection may be induced. Though these phenomena occur in small scales, they are important for thelarge scale oceanic processes. However, these are neither expressed by the fundamental equation of the general circulation
models∗ , nor resolved by large scale models. Therefore, general circulation models express the eﬀects of small scale
turbulence by making some closure assumptions called turbulence closure in which mixing is expressed by synoptic states(velocity and temperature (salinity) solution of the model).
There are mainly two approaches to the turbulence closure problem: statistical closure models and empirical approaches.
MRI.COM provides several options of statistical closure models, which are explained in this chapter. Note that an empiricalapproach of K-profile parameterization proposed by Large et al. (1994) is most popular in the ocean-climate modelingcommunity. However, the current version of MRI.COM does not take this approach. In a future version, an interface to theCommunity Vertical Mixing Project (CVMix; Griﬃes et al., 2015) will be implemented to test the empirical approaches.
Three statistical turbulence closure models are supported: So called Mellor-Yamada model (Mellor and Yamada 1982,
Mellor and Blumberg 2004) presented in Section 15.2, Noh and Kim (1999) model presented in Section 15.3, and a twoequation turbulence closure model using generic length-scale (GLS) equation by Umlauf and Burchard (2003) presented
in Section 15.4. Table 15.1 summarizes features of turbulence closure models.
Table15.1 Features and equation numbers of turbulence closure models explained in this chapter.

model option
prognosticated variables
surface boundary condition for TKE
formula for the length scalestability function

Mellor and Yamada
MELYAMq2 /2 (15.42)
specified (15.45)(15.49), (15.48)
Solution of (15.39)

Noh and Kim
NOHKIM
E(15.57)
flux (15.61)(15.75)(15.72), (15.73)

Umlauf and Burchard
GLS
E (15.83), Ψ (15.84)
flux (15.93)(15.78)(15.99), (15.100)

15.1 Statistical closure model
In the statistical closure models, the physical properties in the basic equations of motion for a Boussinesq fluid are separated
into averaged components and perturbed components (Reynolds decomposition), and then the equations are time averaged.
The expressions for averaged velocity U, averaged pressure P, and averaged potential temperature Θ are
∂Ui
= 0,
∂ xi
DU j
∂
1 ∂P
ρ
∂ [ 1 ( ∂Uk ∂U j ) ]+ ϵ jkl f k Ul =(−⟨uk u j ⟩) −
− gj
+
2ν
+
,
Dt
∂ xk
ρ0 ∂ x jρ0 ∂ x k
2 ∂xj
∂ xk
DΘ
∂
∂ ( ∂Θ )=κ,(−⟨uk θ⟩) +Dt∂ xk∂ xk ∂ xk(15.1)(15.2)(15.3)

where D( )/Dt ≡ Uk ∂( )/∂ x k + ∂( )/∂t, g j is the gravity vector, f k is the Coriolis vector, ϵ i jk is the alternating tensor,
ν is viscosity, and κ is diﬀusivity. Averaged quantities (resolved by the general circulation model) are represented bycapital letters, and turbulent components (unresolved by the general circulation model) are represented by lower-case∗

It could be expressed by relaxing the hydrostatic approximation but this is for future work.

– 175 –

15.2

Mellor and Yamada model

letters. The statistical averages of the turbulent components are represented by ⟨ ⟩. The equation for salinity is similar tothat for temperature (15.3). If density is calculated from temperature and salinity, the whole expression would becomecomplicated. Therefore, density is assumed to be a function only of temperature here (we set ρ = βθ, where β is thecoeﬃcient of thermal expansion). Though these equations basically correspond to the MRI.COM governing equations(2.16)-(2.20), they now have contributions from subgrid-scale processes (⟨uk u j ⟩ and ⟨uk θ⟩), which is the focus of thischapter.
The equations (15.1)-(15.3) show that the statistical averages of the turbulent components (⟨uk u j ⟩ and ⟨uk θ⟩), which
are called as the second order moments, are necessary to calculate the evolution of large scale quantities. However, they
are unknown and their evolution must be solved explicitly if we do not use a parameterization to presume them on thebasis of the large scale quantities. Following Kantha and Clayson (2000), the equations for the second order moments are
expressed as follows:∂⟨ui u j ⟩ ]D⟨ui u j ⟩∂ [1 [ ∂⟨pui ⟩ ∂⟨pu j ⟩ ]+⟨uk ui u j ⟩ − ν+++ f k (ϵ jkl ⟨ul ui ⟩ + ϵ ikl ⟨ul u j ⟩)
Dt
∂ xk
∂ xk
ρ0 ∂ x j∂ xi[⟨ p ( ∂ui ∂u j )⟩⟨ ∂ui ∂u j ⟩
∂U j
∂Ui ]
β
= −⟨uk ui ⟩− ⟨uk u j ⟩−(g j ⟨ui θ⟩ + gi ⟨u j θ⟩) +
+
− 2ν
,
∂ xk
∂ xk
ρ0
ρ0 ∂ x j ∂ x i∂ xk ∂ xk
D⟨u j θ⟩⟨ ∂θ ⟩⟨ ∂u j ⟩]∂ [1 ⟨ ∂p ⟩+⟨uk u j θ⟩ − κ u j−ν θ+θ
+ ϵ jkl f k ⟨ul θ⟩
Dt
∂ xk
∂ xk
∂ xk
ρ0 ∂ x j[⟨ ∂u j ∂θ ⟩∂U j ]∂Θβ= −⟨u j uk ⟩− ⟨uk θ⟩− g j ⟨θ 2 ⟩ − (κ + ν)
,
∂ xk
∂ xk
ρ0
∂ xk ∂ xk⟨ ∂θ ∂θ ⟩⟨ ∂θ 2 ⟩]D⟨θ 2 ⟩∂ [∂Θ+
⟨uk θ 2 ⟩ − κ= −2⟨uk θ⟩
− 2κ
.
Dt
∂ xk
∂ xk
∂ xk
∂ xk ∂ xk(15.4)(15.5)(15.6)

Thus, to solve the evolution of the second order moments, information about higher moments such as ⟨uuu⟩, ⟨uuθ⟩, ⟨uθ 2 ⟩is necessary. The system of equations is not closed, and assumption must be made at some level to close the problem.
The expression closed in the second order of the turbulent components is named second moment closure and is frequently
used in modeling turbulence. Many models have been proposed to parameterize the higher order correlation terms in (15.4)
through (15.6) in order to close the problem at the second moment. Those models diﬀer mainly in the manner how thepressure-strain correlators are parameterized (Burchard and Bolding, 2001). After applying independent parameterizations
for the higher order terms, the majority of models adopt several similar simplifications such as algebraization and boundary
layer approximation that result in the "stability functions" of their own. Stability functions are determined by local shear
and stability and relate the vertical gradient of large scale fields with vertical turbulent fluxes. Procedures adopted bymajor statistical closure models are summarized and compared by Burchard and Bolding (2001).

15.2 Mellor and Yamada model
Mellor and Yamada (1982) model is a classic model as well as a milestone in the eﬀorts of finding practical solutions formarine and atmospheric boundary layers. We show some details of the simplifying procedures made in this model as a
typical example. This is also intended to be an introduction to the problem of turbulence closure. However, it should be
noted that this is neither unique nor final solution.

15.2.1 Fundamental closure assumptions
Mellor and Yamada (1982) closed the system (15.1)-(15.6) by reducing the higher order terms as follows. Based on Rotta’s(1951a,1951b) hypothesis of energy redistribution, the covariances of pressure and velocity gradients are assumed to belinear functions of Reynolds stress:⟨ p ( ∂ui
ρ0 ∂ x j+( ∂Ui ∂U j )∂u j )⟩
δi j 2 )q (=−⟨ui u j ⟩ −q + C1 q2
+
,
∂ xi
3l 1
3
∂xj
∂ xi

(15.7)

where q2 ≡ ⟨ui2 ⟩, l 1 is the length scale, C1 is a non-dimensional constant, and δi j is Kronecker’s delta, which is unity fori = j and zero for i , j.
Using Kolmogorov’s hypothesis of local isotropy in small eddies, the energy dissipation is modeled as follows:2ν⟨ ∂ui ∂u j ⟩∂ xk ∂ xk=−

where Λ1 is the length scale.
– 176 –2 q3
δi j ,
3 Λ1(15.8)

Chapter 15

Turbulence Closure Models

The redistribution of temperature and the dissipation of heat are modeled in the same form as above:⟨ p ∂θ ⟩q= − ⟨u j θ⟩
ρ0 ∂ x j
3l 2(κ + ν)⟨ ∂u j ∂θ ⟩∂ xk ∂ xk= 0,(15.9)(15.10)where l 2 is the length scale. The dissipation of temperature variance is2κ⟨ ∂θ ∂θ ⟩q= −2 ⟨θ 2 ⟩,∂ xk ∂ xk
Λ2(15.11)

where Λ2 is the length scale.
In order to avoid the higher order problems, the turbulent velocity diﬀusion term and the other higher order terms aremodeled as follows:( ∂⟨ui u j ⟩ ∂⟨ui uk ⟩ ∂⟨u j uk ⟩ )3++,⟨uk ui u j ⟩ = lqSq
5
∂ xk
∂xj
∂ xi
( ∂⟨uk θ⟩ ∂⟨u j θ⟩ )⟨uk u j θ⟩ = − lqSuθ
+
,
∂xj
∂ xk
⟨uk θ 2 ⟩ = − lqSθ∂⟨θ 2 ⟩,
∂ xk

(15.12)(15.13)(15.14)

where Sq , Suθ , and Sθ are non-dimensional numbers and can be set as constants or functions of certain parameters. Other
relations are ⟨pθ⟩ = 0 and ⟨pui ⟩ = 0.
The essence of the Mellor-Yamada mixed layer model is that the above length scales are related linearly to each other:(l 1, Λ1, l 2, Λ2 ) = ( A1, B1, A2, B2 )l,(15.15)where l is the vertical scale of turbulence (also called the master length scale), and A1, B1, A2, B2 , and C1 are empirical constants and are determined from experiment data. Mellor and Yamada (1982) employ ( A1, B1, A2, B2, C1 ) =(0.92, 16.6, 0.74, 10.1, 0.08).

15.2.2 Algebraization, boundary layer approximation, and stability functions for the level 2.5 model
The turbulence model that solves the evolution of the statistically averaged values of the second-order turbulent components
based on the simplification described in the previous subsection is called the level-4 model.
The level-3 model solves the evolution of the turbulent kinetic energy (q2 /2) and the variance of potential temperature(⟨θ 2 ⟩) (in some cases, the covariance of potential temperature and salinity (⟨θs⟩) and the variance of salinity (⟨s2 ⟩)). The
other statistically averaged values are solved diagnostically through algebraic equations assuming them to be in the steadystate.
In the level-2.5 model, the variance of the potential temperature is also assumed to be in a statistically steady state (seeexpression (15.33) that appears later).
In the level-2 model, the turbulent kinetic energy is also assumed to be in a statistically steady state.
The level-2.5 model, which is employed as a surface boundary layer model by MRI.COM, is further simplified byapplying the following boundary layer approximations.
• Neglect the Coriolis term in the equations of motion for the turbulent components.
• Neglect the molecular viscosity and diﬀusivity.
• Use the hydrostatic assumption in the vertical component of the equation of motion.
• Consider only vertical diﬀerentiation (direction perpendicular to the boundary) in the spatial diﬀerentiation for theterm involving turbulent velocity.

– 177 –

15.2

Mellor and Yamada model

The equations for the large scale quantities, (15.2) and (15.3), become
DU
∂
1 ∂P
+
⟨uw⟩ = −+ f V,
Dt∂zρ0 ∂ xDV∂
1 ∂P+⟨vw⟩ = −− f U,
Dt∂zρ0 ∂ y
1 ∂P
ρ
0= −
−g ,
ρ0 ∂z
ρ0
∂
1 ∂R
DΘ
+
(⟨wθ⟩) =.
Dt∂zρ0 cp ∂z(15.16)(15.17)(15.18)(15.19)In (15.19), R is the heat flux due to penetration of shortwave radiation. The level-2.5 system consists of the time evolution
equation for turbulent kinetic energy and algebraic equations for other second-moment turbulent quantities.
The time evolution equation for the turbulent kinetic energy is
D ( q2 )∂ [∂ ( q2 ) ]−lqSq= Ps + Pb − ϵ,
Dt 2
∂z
∂z 2
where
Ps = −⟨wu⟩∂U∂V− ⟨wv⟩∂z∂z(15.20)(15.21)is the term for energy produced by the vertical shear of the averaged flow,
Pb = −g⟨w ρ⟩/ρ0(15.22)ϵ = q3 /Λ1(15.23)is the term for energy produced by buoyancy, andis the energy dissipation term. The constant Sq is given in the next subsection.
The algebraic equations for the statistically averaged values, which are expressed by other second-moment turbulent
quantities, are given below.
]
q2 l 1 [∂U∂V+−4⟨wu⟩+ 2⟨wv⟩− 2Pb ,
3
q
∂z
∂z
2
[
]
q
l1
∂U
∂V
⟨v 2 ⟩ =+2⟨wu⟩− 4⟨wv⟩− 2Pb ,
3
q
∂z
∂z
2
[
]
q
l1
∂U
∂V
⟨w 2 ⟩ =+2⟨wu⟩+ 2⟨wv⟩+ 4Pb ,
3q∂z∂z
3l 1 [∂V∂U ]⟨uv⟩ =−⟨uw⟩− ⟨vw⟩,q∂z∂z]
3l 1 [∂U⟨wu⟩ =− g⟨u ρ⟩ ,−(⟨w 2 ⟩ − C1 q2 )
q
∂z
[
]
3l 1
∂V
⟨vw⟩ =−(⟨w 2 ⟩ − C1 q2 )− g⟨v ρ⟩ ,
q
∂z
∂U ]
3l 2 [∂Θ− ⟨wθ⟩⟨uθ⟩ =
−⟨uw⟩
,
q
∂z
∂z
3l 2 [∂Θ∂V ]⟨vθ⟩ =−⟨vw⟩− ⟨wθ⟩,q∂z∂z]
3l 2 [∂Θ⟨wθ⟩ =−⟨w 2 ⟩− g⟨θ ρ⟩ ,
q
∂z
Λ
∂Θ
2
⟨θ 2 ⟩ = − ⟨wθ⟩.
q∂z
⟨u2 ⟩ =

(15.24)(15.25)(15.26)(15.27)(15.28)(15.29)(15.30)(15.31)(15.32)(15.33)

Some of the terms in these equations, which are used in (15.16)-(15.19), can be further deformed by substitution as– 178 –

Chapter 15

Turbulence Closure Models

follows:∂U,∂z∂V−⟨vw⟩ = K M,∂z∂Θ−⟨θw⟩ = K H,∂z
K M = lqSM ,−⟨uw⟩ = K M

K H = lqSH .

(15.34)(15.35)(15.36)(15.37)(15.38)This simplification means that the vertical turbulent fluxes are proportional to the gradient of the large scale fields. The
ultimate purpose of solving the mixed layer model is to determine the coeﬃcients of momentum and heat fluxes, K M andK H , using (15.37) and (15.38).
Assuming that the potential density is linearly related to the potential temperature (and salinity), the simultaneous
equations for SM and SH are derived as follows:
SM [6A1 A2 g M ] + SH [1 − 3A2 B2 gH − 12A1 A2 gH ] = A2,
SM [1 + 6A21 g M − 9A1 A2 gH ] − SH [12A21 gH + 9A1 A2 gH ] = A1 (1 − 3C1 ),(15.39)wherel 2 [( ∂U ) 2 ( ∂V ) 2 ]l2+= 2 S 2,
2
∂z
∂z
q
q
2
2
l g ∂ ρ̃lgH ≡ 2= 2 N 2,q ρ0 ∂zq

gM ≡(15.40)(15.41)and ∂ ρ̃/∂z is the vertical gradient of potential density. Note that SM and SH are stability functions of this closure model.
Using SM and SH , K M and K H are then obtained from (15.37) and (15.38) by determining q and l.

15.2.3 Solving for velocity scale (q) and length scale (l )The turbulent velocity q is obtained by solving the following expression that is modified from (15.20) using the above
results:[ ( ∂U ) 2 ( ∂V ) 2 ]∂ ( q2 )∂ [g∂ ( q2 ) ]
∂ ρ̃
−
KE
= KM
+
+
KH
− ϵ,
(15.42)
∂t 2
∂z
∂z 2
∂z
∂z
ρ0
∂z
where
K E = lqSq

(15.43)and advection terms are neglected. In MRI.COM, Sq is set proportional to SM (Sq ∝ SM ). We adopt the form
Sq = Sqc SM /SM n , where Sqc = 0.2 and SM n = 0.3927. With this choice, Sq = 0.2 when the stratification is neutral(gH = 0).
The sea surface boundary condition for the turbulent kinetic energy follows Mellor and Blumberg (2004):
Kq∂q2= 2αCB uτ3,∂z(15.44)

where αCB = 100 and uτ is the frictional velocity defined as uτ ≡ (τs /ρs ) 1/2 by using the surface stress (τs ) and the sea
surface density (ρs ). Mellor and Blumberg (2004) showed that this flux boundary condition is analytically converted tothe condition for q at the sea surface:q2 = (15.8αCB ) 2/3 uτ2 .
(15.45)On the other hand, the bottom boundary condition isq = 0.

(15.46)The vertical scale of the turbulence (master length scale, l) is estimated by many formulae such as a time evolution
equation (which is usually empirical and is not completely based on physics) and a diagnosis. MRI.COM uses diﬀerent
diagnosis equations for the surface layer influenced directly by surface wind wave eﬀects and for the internal region. The– 179 –

15.2

Mellor and Yamada modelsurface layer is defined by a depth of |z| < zw , where the roughness parameter zw due to surface wind waves is given by
Mellor and Blumberg (2004) as follows:u2z w = βw τ ,gβw = 2.0 × 105 .

(15.47)In this layer, the formula for l is
l = κzw,(15.48)where κ is the von Karman constant (κ = 0.4). In the internal region (|z| > zw ), the formula for l is
l=γ

∫ 0

′

|z |qdz

′

/∫ 0

zb

qdz ′,(15.49)zb

where γ = 0.2, and z b is the water depth. This is recognized as the averaged depth with the weight of the kinetic energy,
which is suﬃcient for the ocean boundary layer according to Mellor and Yamada (1982).

15.2.4 Implementation
This section briefly describes the solution procedure.
The mixed layer model (subroutine name mysl25 in my25.F90) is called as the last procedure of each time step thatproceeds from n to n+1. After the master length scale (l) for the present time step (n) is determined using (15.48) and(15.49), where the latter is actually computed in the previous time step, the turbulent kinetic energy (q2 /2) is solved using(15.42) with (15.23) under the boundary conditions of (15.45) and (15.46), where the forward finite diﬀerence (n → n+1)
is used in time. The implicit method is used for the vertical diﬀusion of the turbulent kinetic energy and energy dissipation
term, since these terms could become significantly large (see Section 19.5). The vertical viscosity and diﬀusivity for thetime step n+1 are estimated using q, l, and (15.37) to (15.39). These are used in the governing equations for the synoptic
scale field. The vertical scale of the turbulence (master length scale) based on the turbulence kinetic energy (15.49) is
calculated to prepare for the next time step.
The turbulent kinetic energy and the master length scale are defined at the bottom of the tracer cell (i, j, k). The specificexpression for the discretized form of the turbulent kinetic energy (E = q2 /2) equation, (15.42), is as follows:
Ekn+1 − Ekn
∆t

=

n+1
n+1
n+1
n+1
1 [ K E k− 12 (Ek−1 − Ek ) K E k+ 12 (Ek − Ek+1 ) ]−∆zk∆zk− 1∆zk+ 1
2+ KM k− KH k2(U n+11 − U n+11 )(Ũk− 1 − Ũk+ 1 )
k− 2

k+ 2

2

2

∆zk2
B n+11 − B n+11
k− 2

∆zk

k+ 2

+ KM k

(V n+11 − V n+11 )(Ṽk− 1 − Ṽk+ 1 )
k− 2

k+ 2

∆zk2

2

2

(15.50)− 2Ekn+1 qkn /B1 l kn,

where Ũ = (U n+1 + U n )/2, Ṽ = (V n+1 + V n )/2, and B is buoyancy (= − gρρ0 ). The discrete expression for shear production(the second and third terms on the r.h.s.) and the buoyancy sink (the fourth term on the r.h.s.) follows Burchard (2002),
which is consistent with the conservation law of the sum of mean and turbulent energy.
To summarize, the numerical operations proceed in the following order:
1. Calculate the master length scale for the present time step using (15.47) and (15.48) as well as (15.49) from theprevious time step.
2. Update the turbulent kinetic energy using (15.50), (15.45) and (15.46).
3. Solve the algebraic equation for SM and SH using (15.39).
4. Calculate the vertical viscosity and diﬀusivity for the next time step using (15.37), (15.38), and (15.43).
5. Calculate the master length scale using (15.49) in preparation for the next time step.

15.2.5 Usage
Mellor and Yamada model is invoked by MELYAM option on compilation. There is no tuning parameter at run time. Initialstate for vertical diﬀusion of turbulence kinetic energy (K E ), turbulence kinetic energy (q), and length scale (l) is readfrom restart file. Otherwise it is set as a state without turbulence. The state without turbulence is zero turbulence kinetic– 180 –

Chapter 15

Turbulence Closure Models

energy (q = 0), the mixing length of 1 meter (l = 1 m) and the vertical diﬀusion of turbulence kinetic energy is the constantbackground vertical viscosity given by visc_vert_bg_cm2ps in namelist nml_visc_vert_bg. How model is initializedis specified by namelist nml_melyam_run. Parameters are listed on Table 15.2.
Table15.2 namelist nml_melyam_run

variable namel_rst_melyam_in

units
logicaldescription.true. : Read restart files specified by nmlrs_my_avq,
nmlrs_my_q, and nmlrs_my_alo for the initial condition.
.false.: Start condition is that of no turbulence. See text
for details.

usage
default = l_rst_in of
nml_run_ini_state

15.3 Noh and Kim (1999) model
The mixed layer model proposed by Mellor and Yamada was originally developed for the atmospheric boundary layer, andits surface boundary is treated as a solid wall. When they applied this model to the ocean, they regarded that the turbulent
kinetic energy is injected into the ocean by the wind stress at the solid-wall sea surface. The model by Mellor and Yamada
could therefore be considered to insuﬃciently represent the oceanic turbulent mixed layer.
Noh and Kim (1999) presented a model that can resolve this insuﬃciency. The model is basically the same as Mellor
and Yamada level 2.5 scheme in that turbulent kinetic energy is prognosticated and the length scale is diagnosticallydetermined. It diﬀers in the stability function and the treatment of surface boundary condition.

15.3.1 Fundamental equation
The equations for the zonal and meridional components of the velocity, U, V , buoyancy B = −g∆ρ/ρ0 , and turbulent
energy E in the large scale fields are
DU
∂
1 ∂P
= − ⟨uw⟩ −+ f V,
Dt∂zρ0 ∂ xDV∂
1 ∂P= − ⟨vw⟩ −− f U,
Dt∂zρ0 ∂ yDB∂∂R= − ⟨bw⟩ +
,
Dt
∂z
∂z
)⟩
DE
∂ ⟨ ( p∂U∂V=−w
+ uu + vv + ww − ⟨uw⟩
− ⟨vw⟩+ ⟨bw⟩ − ϵ,
Dt
∂z
ρ0
∂z
∂z

(15.51)(15.52)(15.53)(15.54)

where R is the downward shortwave radiation and ∂ R/∂z is its convergence.
The turbulent flux is expressed by using the large scale fields (in capital letters) as follows:
DU
∂ (
∂U )
1 ∂P
=
KM
−
+ f V,Dt∂z∂zρ0 ∂ x
DV
∂ (
∂V )
1 ∂P
=
KM
−
− f U,Dt∂z∂zρ0 ∂ yDB∂ (∂B ) ∂ R
=
KH
−
,
Dt
∂z
∂z
∂z
∂ (
∂E )
∂U ∂U
∂V ∂V (
∂B )
DE
=
KE
+ KM
+ KM
− KH
− ϵ.
Dt
∂z
∂z
∂z ∂z
∂z ∂z
∂z

(15.55)(15.56)(15.57)(15.58)The surface boundary conditions are as follows:
∂U
τ
= ,
∂z
ρ0
∂B
KH
= Q0,
∂z
∂E
KE
= muτ3,∂z

KM– 181 –

(15.59)(15.60)(15.61)

15.3 Noh and Kim (1999) model
where m is a tuning parameter and m = 100 is recommended by Noh and Kim (1999). Unlike Mellor and Yamada model,
a flux boundary condition is used for turbulence kinetic energy. This is intended to express the direct input of turbulencekinetic energy due to wind waves. As such, in the presence of sea ice, the direct input of wave energy may be more
appropriate only through open water. The boundary condition may be optionally modified so thatKE∂E
= muτ3 (1 − a),∂z(15.62)where a is area fraction of sea ice. The no-flux condition is used for the bottom:
KE

∂E
= 0.
∂z

(15.63)15.3.2 Stability function
The central problem is how to determine the viscosity, diﬀusivity (K M , K H , K E ), and turbulent energy dissipation rate (ϵ).
In the Noh and Kim (1999) scheme, by using the typical velocity scale (q = (2E) 1/2 ) and the vertical length scale (l) of
the turbulence, they are obtained by the following formula:
K M = Sql,

(15.64)

K H = SB ql,(15.65)K E = SE ql,(15.66)ϵ = Cq3 l −1 .

(15.67)The constants (S, SB, SE , C) are obtained from experiments. For neutral stratification, it is assumed that S = 0.39,
Pr ≡ S/SB = 0.8, σ ≡ S/SE = 1.95, and C = 0.06. Hereafter, these values for neutral stratification are denoted as S0 ,Pr0 , σ0 , and C0 , respectively.
Regarding the influence of the stratification, that is, stability function, we assume that the vertical scale of turbulence islimited by the vertical scale of buoyancy l b = q/N (N 2 = ∂B/∂z). That is,where Rit is the turbulent Richardson numberK ∼ ql b ∼ qlRit −1/2,(15.68)

Rit = (Nl/q) 2 .

(15.69)Rit corresponds to gH of Mellor and Yamada model. See (15.41).
This means that when the stratification is strong (N is large, Rit is large, and K is small), the turbulent energy is nottransported downwards. It could also be considered that the local turbulent energy dissipation becomes large.
The following equation is used for S so that it satisfies (15.68) when Rit is large:
S/S0 = (1 + αRit ) −1/2,(15.70)where α is a tuning parameter. Noh and Kim (1999) recommend α ∼ 120.0, but α ∼ 5.0 is the default value of MRI.COM.
The eﬀect of stratification on the Prandtl number (Pr) is set following Noh et al. (2005):
Pr/Pr0 = (1 + βRit ) 1/2,

(15.71)

where β is a tuning parameter and 0.5 is used following Noh et al. (2005).
To summarize, the stability function for this model is expressed as follows:
S0(1 + αRit ) 1/2
S
S0
.
SB =
=
Pr Pr0 (1 + αRit ) 1/2 (1 + βRit ) 1/2
S=(15.72)(15.73)The eﬀect of stratification on the energy dissipation (C) is set as follows:
C/C0 = (1 + αRit ) 1/2 .
Note that this is independent of stratification for Mellor and Yamada and Umlauf and Burchard (2003) model.
– 182 –

(15.74)

Chapter 15

Turbulence Closure Models

In the case of unstable stratification (N 2 < 0), K M = K H = 1.0 m2 s−1 and K E is estimated from the turbulent velocity
scale and the vertical length scale in the model. This treatment is due to the diﬀerence between the time scales of thevertical convection and the development of turbulence.
Finally, the vertical scale of turbulence is diagnosed asl=κ(|z| + z0 ),(1 + κ(|z| + z0 )/h)(15.75)where z0 is the sea surface roughness (z0 = 1 m), z is the depth, and h is the mixed layer depth. The mixed layer depth (h)is determined as the depth at which the buoyancy frequency becomes the maximum in the vertical column. The verticalscale becomes longer as the mixed layer becomes deeper.

15.3.3 Implementation
Equation (15.58) is solved for the prognostic variable E in the subroutine nkoblm in nkoblm.F90 as the last procedure ofeach time step. The forward finite diﬀerence is used in the time evolution. The implicit method is used for the vertical
diﬀusion of the turbulent kinetic energy and energy dissipation term, since these terms could become significantly large(see Section 19.5). The new E is used to determine the coeﬃcients of viscosity and diﬀusivity for the next time step.
The turbulent kinetic energy and the master length scale are defined at the center of the tracer cell (i, j, k − 12 ). The
specific expression for the discretized form of the turbulent kinetic energy (E) equation is as follows:
E n+11 − E n 1
k− 2

k− 2

∆t

1 [
=
∆zk− 1K E k−1 (E n+13 − E n+11 )2

1+ K M k−1
2
1
+ K M k−1
2
1
− K H k−1
2

k− 2

k− 2

∆zk−1

−

K E k (E n+11 − E n+11 ) ]
k− 2

∆zk

(un+13 − un+11 )(un+13 − un 1 )
k− 2
k− 2
k− 2
k− 2
∆zk−1 ∆zk− 1

2(v n+13 − v n+11 )(v n+13 − v n 1 )
k− 2
k− 2
k− 2
k− 2
∆zk−1 ∆zk− 1
2

B n+13 − B n+11
k− 2

∆zk−1

k− 2

1
− KH k2

k+ 2

1+ KM k2
1+ KM k2

B n+11 − B n+11
k− 2

∆zk

k+ 2

(un+11 − un+11 )(un 1 − un+11 )
k− 2

k+ 2

k− 2

k+ 2

∆zk ∆zk− 1
2(v n+11 − v n+11 )(v n 1 − v n+11 )
k− 2
k+ 2
k− 2
k+ 2
∆zk ∆zk− 1(15.76)2n+1 n/l n .
− 2CEk−1q
k− 1 k− 12

2

2

The discrete expression for the shear production (the second through fifth terms on the r.h.s.) and the buoyancy sink(the sixth term on the r.h.s.) follows Burchard (2002), which is consistent with the conservation law of the sum of mean
and turbulent energy.
To summarize, the numerical operations proceed in the following order:
1. Update the master length scale using (15.75),
2. Update the turbulent kinetic energy using (15.76) under the boundary conditions (15.61) and (15.63)
3. Calculate the vertical viscosity and diﬀusivity for the next time step (15.64) and (15.65).

15.3.4 Usage
Noh and Kim (1999) model is invoked by NOHKIM option on compilation.
Initial state for vertical diﬀusion of turbulence kinetic energy (K E ), turbulence kinetic energy (E) is either read fromrestart file or set as a state without turbulence. The state without turbulence means that E is very small as 12 qmin 2 , whereqmin = 1 × 10−5 m s−1 is the minimum of turbulence velocity scale, and that K E is set the constant background verticalviscosity given by visc_vert_bg_cm2ps in namelist nml_visc_vert_bg. How model is initialized is specified bynamelist nml_nohkim_run. Parameters are listed on Table 15.4.
Tuning parameters may be specified at run time by namelist nml_nkoblm. Parameters are listed on Table 15.3.

– 183 –

15.4 Generic length scale model by Umlauf and Burchard (2003)
Table15.3 namelist nml_nkoblm

variable name
mfnk
alpnk
betankl_wave_damp_seaice

units
1
1
1
logicaldescription
m of Eq. (15.61)
α of Eq. (15.70)
β of Eq. (15.71)
Eq. (15.62) is used instead of (15.61)

usagedefault = 100.0
default = 5.0
default = 0.5
default = .false.

Table15.4 namelist nml_nohkim_run

variable namel_rst_nohkim_in

units
logicaldescription.true. : Read restart files specified by nmlrs_nk_avq
and nmlrs_nk_eb for the initial condition.
.false.: Start condition is that of no turbulence. See text
for details.

usage
default = l_rst_in of
nml_run_ini_state

15.4 Generic length scale model by Umlauf and Burchard (2003)†

The turbulent closure models of Mellor and Yamada level 2.5 (section 15.2) and Noh and Kim (section 15.3) asincorporated in MRI.COM have only one prognostic variable of turbulence (turbulent kinetic energy). These may beclassed as "one-equation" turbulent closure models. In these models, some formulae for length scale are given (e.g.,(15.49) and (15.75)). Therefore the turbulent length scale l is not a predicted but a diagnosed quantity.
Generic length scale model by Umlauf and Burchard (2003) is a generalized form of two-equation turbulence models.
It introduces a generic length-scale variable ψ as the second variable, expressed as
ψ = (cµ0 ) p E m l n .

(15.77)

With E and ψ given, the turbulent length scale can be computed aspl = (cµ0 ) − n E − n ψ n ,m

1(15.78)and the dissipation rate ϵ can be computed from the following relation,
ϵ = (cµ0 ) 3

E 3/2
.
l

(15.79)With appropriate values of the exponents p, m, n in (15.77), a number of well-known two-equation models (e.g., k-ϵ modeland k-ω model) can be identified as special case of the generic model.

15.4.1 Fundamental equation
The fundamental equations for U, V , B and E are basically common to those of Noh and Kim model. The equation forgeneric length scale variable ψ is added. Expressing the turbulent flux using the large scale fields with the assumption ofhorizontally homogeneous flow, the fundamental equations of the model is expressed as follows:
∂ (
∂U )
1 ∂P
DU
=
KM
−
+ f V,Dt∂z∂zρ0 ∂ x
DV
∂ (
∂V )
1 ∂P
=
KM
−
− f U,Dt∂z∂zρ0 ∂ yDB∂ (∂B ) ∂ R
=
KH
−
,
Dt
∂z
∂z
∂z
DE
∂ (
∂E )
=
KE
+ Ps + Pb − ϵ,Dt∂z∂z
∂ ( ∂ψ ) ψDψ=Kψ+ (cψ1 Ps + cψ3 Pb − cψ2 ϵ ),
Dt
∂z
∂z
E
† This model is implemented by Shiro Nishikawa as he was a visiting researcher at MRI.

– 184 –

(15.80)(15.81)(15.82)(15.83)(15.84)

Chapter 15

Turbulence Closure Models

where Ps and Pb are the shear production and the buoyancy production expressed according to
Ps = −⟨uw⟩∂u∂v− ⟨vw⟩= K M M 2,
∂z
∂z

and

(15.85)( ∂v ) 2∂B,
N2 =(15.86)∂z∂z∂z
are the shear frequency and the buoyancy frequency, respectively. The vertical turbulent diﬀusivities in horizontallyhomogeneous flows are expressed asM2 =( ∂u ) 2

Pb = ⟨bw⟩ = −K H N 2,+

K M = cµ E 1/2 l,(15.87)

K H = cµ′ E 1/2 l,(15.88)ψl/σk ,(15.89)Kψ = cµ E 1/2 l/σψ,(15.90)K E = cµ E

1/2ψwhere cµ and cµ′ are stability functions and σk and σψ are the constant Schmidt numbers. Here, cµ and cµ′ corresponds to
SM , and SH of the Mellor and Yamada model with a relation√√cµ = 2SM , cµ′ = 2SH .
(15.91)In the case of unstable stratification (N 2 < 0), K M = K H = 1.0 m2 s−1 is used in MRI.COM. This treatment is common
to Noh and Kim model (section 15.3).

15.4.2 Boundary conditions
At the surface, the shear-free boundary layers with injection of turbulent kinetic energy are assumed, in which
E = K (−z + z0 ) α,l = L(−z + z0 ),(15.92)are supposed, where K is the scale of turbulent kinetic energy, L and α are empirical constants given in Table 15.5, andz0 is the sea surface roughness (z0 = 1 m). K is determined by the injection of turbulent kinetic energy which is given by
the surface flux. We use flux forms for the surface boundary conditions (z = 0) of E and ψ as follows:∂E= ηuτ3,∂zcµ (cµ0 ) p
11∂ψFψ = Kψ=−(mα + n)K m+ 2 L n+1 (z0 ) (m+ 2 )α+n .
∂zσψ

FE = K E(15.93)(15.94)

As explained in the previous section for Noh and Kim (1999) model, injection of turbulence kinetic energy may be madeonly through the open water in the presence of sea ice.
FE = K E∂E= ηuτ3 (1 − a),∂z(15.95)where a is sea ice area fraction.
The parameter K can be determined from
ψ

σk

2
3

1
K = *−
FE + α ,
c
αL
z
, µ
- 0

(15.96)

where (15.89), (15.92) and (15.93) are used.
At the bottom, the logarithmic boundary layers are assumed, in which l = κ(z + H + z0b ) is supposed. The bottom
boundary conditions (z = −H) for E and ψ are given by flux forms as follows:
∂E
= 0,
∂z
0 p+1 n+11
∂ψ n(cµ ) κFψ = Kψ=E m+ 2 (z0b ) n,∂zσψFE = K E(15.97)(15.98)where z0b is the bottom roughness, which is set to be z0b = 1 m. Implementation of the bottom boundary condition isrealistic as compared to the Mellor and Yamada scheme and the Noh and Kim scheme.
– 185 –

15.4 Generic length scale model by Umlauf and Burchard (2003)

15.4.3 Model parameters
Umlauf and Burchard (2003) investigated the properties of the generic model in some fundamental flows and derived theψconstraints on the model parameters (cµ0 , cψ1 , cψ2 , cψ3 , σk , σψ , m and n). Table 15.5 is a recommended parameter set
for the generic model by Umlauf and Burchard (2003), based on their calibration. MRI.COM uses this as the default
parameters. Note that the parameter p (the factor (cµ0 ) p in (15.77)) is mathematically irrelevant, since (15.84) can be
multiplied by any constant without changing the solution (p = 2.0 is used as the default value).
Table15.5 A recommended parameter set for the generic model by Umlauf and Burchard (2003), where κ = 0.4,
− is based on the Algebraic Reynolds-Stress(cµ0 ) 2 = 0.3, and cψ1 = m are assumed. The value of the parameter cψ3
Model (ASM) of Canuto et al. (2001). In unstable situations (Pb > 0), a diﬀerent value of the parameter cψ3 needs to+ = 1.0 is adopted.
be used. In MRI.COM, cψ3

α
−2.0

L
0.20

m
1.00

ψ

σk
0.80

n
−0.67

σψ
1.07

cψ1
1.00

cψ2
1.22

−
cψ3
0.05

15.4.4 Stability functions
The version A of the Canuto et al. (2001) stability functions is expressed as follows:
0.1070 + 0.01741α N − 0.00012α M,
1 + 0.2555α N + 0.02872α M + 0.008677α 2N + 0.005222α N α M − 0.0000337α 2M
0.1120 + 0.004519α N + 0.00088α M
ĉµ′ = cµ′ (cµ0 ) 3 =,1 + 0.2555α N + 0.02872α M + 0.008677α 2N + 0.005222α N α M − 0.0000337α 2Mĉµ = cµ (cµ0 ) 3 =

where
αM =

E2 2
M ,
ϵ2

E2 2
N
ϵ2

αN =

(15.99)(15.100)(15.101)

are nondimensional parameters referred to as the shear number and the buoyancy number. Using (15.79), they are related
to the nondimensional parameter used by the Mellor and Yamada model (15.40) and (15.41) asgM =(cµ0 ) 6
2

α M,

gH =

(cµ0 ) 62

αN .

(15.102)

These are presented in Burchard and Bolding (2001)‡ . MRI.COM adopts these as the stability functions for the Umlauf
and Burchard (2003) model.

15.4.5 Implementation
Equations (15.83) and (15.84) are solved for the prognostic variable E and ψ in the subroutine gls_main in gls.F90 as
the last procedure of each time step. The forward finite diﬀerence is used in the time evolution. The implicit method isused for the vertical diﬀusion of the turbulent kinetic energy or generic variable and energy dissipation term, since theseterms could become significantly large (see Section 19.5). The new E and ψ are used to determine the coeﬃcients ofviscosity and diﬀusivity for the next time step.
The turbulent kinetic energy and the master length scale are defined at the center of the tracer cell (i, j, k − 21 ). The
specific expression for the discretized form of the turbulent kinetic energy (E) equation is as follows:
E n+11 − E n 1
k− 2

k− 2

∆t

1 [
=
∆zk− 1
2

K E k−1 (E n+13 − E n+11 )
k− 2

k− 2

∆zk−1

−

K E k (E n+11 − E n+11 ) ]
k− 2

1

n+1
n
n
+ Ps + Pb − (cµ0 ) 3 Ek−) 2 /l k−
1 (E
1.
k− 1
2

2

k+ 2

∆zk
(15.103)2′′′µ and cµ in Burchard and Bolding (2001) is diﬀerent than that of this chapter. ĉµ and ĉµ here correspond to cµ and cµof Burchard and Bolding (2001).

‡ Note that definition of c

– 186 –

Chapter 15

Turbulence Closure Models

The specific expression for the discretized form of the generic variable (ψ) equation is as follows:ψ n+11 − ψ n 1
k− 2

k− 2

∆t

1 [
=
∆zk− 1Kψ k−1 (ψ n+13 − ψ n+11 )
k− 2

k− 2

−

∆zk−1

2

Kψ k (ψ n+11 − ψ n+11 ) ]
k− 2

k+ 2

∆zk

ψn 1
k−
1
n
n+1
n
2
.
+ n 2 (cψ1 Ps + cψ3 Pb ) − cψ2 (cµ0 ) 3 ψk−
1 (E1 ) /l
k−
k− 21
E 1
2
2
k− 2

(15.104)

The shear and buoyancy production terms Ps and Pb are discretized as follows:1
Ps = K M k−1
2(un+13 − un+11 )(un+13 − un 1 )
k− 2

k− 2

k− 2

k− 2

∆zk−1 ∆zk− 1
2

1+ K M k−1
2
1
Pb = − K H k−1
2

1+ KM k2(v n+13 − v n+11 )(v n+13 − v n 1 )
k− 2

k− 2

k− 2

k− 2

∆zk−1 ∆zk− 1

2

B n+13 − B n+11
k− 2

k− 2

∆zk−1

1
− KH k2(un+11 − un+11 )(un 1 − un+11 )
k− 2

k+ 2

k− 2

∆zk

k− 2

k+ 2

∆zk ∆zk− 1
2

1+ KM k2

B n+11 − B n+11

k+ 2

(v n+11 − v n+11 )(v n 1 − v n+11 )
k− 2

k+ 2

k− 2

k+ 2

∆zk ∆zk− 1,(15.105)2.

(15.106)

The discrete expression for the shear production and the buoyancy production follows Burchard (2002), which is consistent
with the conservation law of the sum of mean and turbulent energy.
To summarize, the numerical operations proceed in the following order:
1. Update the turbulent kinetic energy and generic variable using (15.103), (15.104), (15.105) and (15.106) under theboundary conditions, (15.93), (15.94), (15.97) and (15.98).
2. Diagnose the turbulent length scale using (15.78),
3. Calculate the vertical viscosity and diﬀusivity using (15.88) and (15.88) for the next time step.

15.4.6 Limiters
Some kind of limiters for the turbulent variables may be needed to ensure numerical stability. MRI.COM supposes thefollowing conditions.
• Lower limiters of the turbulent kinetic energy and generic variable, Emin, ψmin , are introduced as basic limiters.
• If one of E and ψ becomes less than the limiter, turbulence is considered as stagnant. That is,
E = Emin,ψ = ψmin,l = l stag,

if either E ≤ Emin or ψ ≤ ψmin(15.107)• When turbulence is stagnant, the diagnosed viscosity and diﬀusivity (K M , K H ) should not exceed the backgroundvalues (K M bg, K H bg = 1.0 × 10−6 m2 s−1 ):K H = cµ′ (2Emin ) 1/2 l stag ≤ K H bg

K M = cµ (2Emin ) 1/2 l stag ≤ K M bg,(15.108)• Stagnant value of the turbulent length-scale (l stag ) is supposed to be the local vertical grid size (∆zlocal ) for numericalstability.
Table15.6

Default limiters for GLS model in MRI.COM.

1
2

(

Emin
10−6
0.1lstag)2

ψminl stagmin(cµ0 ) 2 E2/3

∆zlocallstag

15.4.7 Usage
Umlauf and Burchard (2003) model is invoked by GLS option on compilation.
Initial state for vertical diﬀusion of turbulence kinetic energy (K E ) and generic variable KΨ , turbulence kinetic energy(E), generic variable (Ψ), and length scale (l) is either read from restart file or set as the stagnant state defined in the– 187 –

15.4 Generic length scale model by Umlauf and Burchard (2003)
previous subsection. In the stagnant state, both of K E and KΨ are set as 1 × 10−6 m2 s. The initial state is specified bynamelist nml_gls_run. Parameters are listed on Table 15.7.
Tuning parameters may be specified at run time by namelist nml_gls. Parameters are listed on Table 15.8.
Table15.7 namelist nml_gls_run

variable namel_rst_gls_inunits
logicaldescription.true. : Read restart files specified by
nmlrs_gls_(avk, avq, eke, psi, alo) for the initial condition.
.false.: Start condition is that of no turbulence. See text for
details.

usage
default = l_rst_in of
nml_run_ini_state

Table15.8 namelist nml_gls

variable namemfnk
l_wave_damp_seaiceunits
1logicaldescription
η of Eq. (15.93)
Eq. (15.93) is used instead of (15.95)– 188 –

usage
default = 100.0
default = .false.

Chapter 16

Bottom Boundary Layer (BBL)
MRI.COM has an option of adopting a simple bottom boundary layer model (BBL option). This chapter describes theformulation and usage of this model.

16.1 General description
In general, neither dense overflows from the Nordic-Scotland-Greenland ridges to the Atlantic Ocean nor the dense
downslope flows from the continental slope around Antarctica can be properly reproduced by z-coordinate models. The
former becomes the core water of the North Atlantic Deep Water, and the latter becomes that of the Antarctic Bottom
Water. As a result, the abyssal waters in the modeled ocean tend to exhibit a warming bias. To mitigate these deficiencies,
MRI.COM incorporates a simple bottom boundary layer (BBL) model as introduced by Nakano and Suginohara (2002).
In this model, BBL is attached immediately below the bottom of the interior oceanic domain and conceptually treatedas a plane along the sea floor at the model’s lowest vertical level (Figure 16.1). The following processes relevant to theoverflow/downslope flows are (partly) incorporated into this BBL model:• The advection along the bottom topography.
• The pressure gradient terms when the dense water lies on slope.
• The eddy activity to create the flow crossing f /h contours.

16.2 Grid arrangement and exchanges
In z-coordinate models, the flow along the bottom topography is expressed as a sequence of horizontal and verticalmovements along a staircase-like topography. When the number of vertical grid points representing the bottom topography
is roughly the same as that of the horizontal grid points, the downslope flow can be reasonably represented even in zcoordinate models (Winton et al., 1998). In general, however, the number of the vertical grids used in ocean generalcirculation models is not large enough. Even if the number is large enough, the concentrations of tracers might besignificantly diﬀused during their movement owing to the horizontal mixing with the surrounding waters. The incorporationof BBL remedies these problems.
The BBL grid cells in MRI.COM are arranged as in Figure 16.1. General interior grid cells in MRI.COM are composedof U-cells, but it is intuitively easier to understand the grid arrangement of BBL as if they are located at T-cells (represented
by "•" points of Figure 16.1a and b). The T-cells in BBL are located just below the bottom of interior T-cells even if aninterior T-cell consists of only one quarter of a U-cell (Figure 16.1b). The U-cells in BBL are regarded as slanting cells
connecting neighboring T-points in BBL (Figure 16.1d).
For the purpose of calculating horizontal exchanges between BBL cells due to advection and subgrid-scale mixing, theprimary BBL is attached below the normal ocean grid (Figures 16.1a and c). Assuming that the number of the verticalgrids, km, is 50 before BBL option is applied, the number of vertical grids, km, becomes 51 (50 + 1) when BBL option isapplied. In addition, to easily express the vertical exchange between the BBL cells and the inner ocean cells, we place adummy BBL cell just below the bottom grid of each inner column and copy the temperature, salinity, and velocities fromthe primary BBL cell (Figures 16.1b and d).
Regarding vertical advection, primary vertical mass transport at the upper surface of BBL is defined at T-cell as in theinterior (there is not vertical transport at the bottom of BBL). Even when the horizontal area of the bottom of the interior
T-cell is smaller than that of BBL, all vertical mass transport at the top of BBL flows into or out of the bottom of theinterior cell (Figure 16.1b and d). Tracers immediately above and below are carried by this vertical transport. In the samemanner as explained in Chapter 5, the continuity equation for the U-cell is defined as an average of those of surrounding– 189 –

16.3

Pressure gradient terms

T-cells (5.21). At each of the T-points connecting the bottom of the interior and the BBL, velocity (momentum) at all the
U-cells in BBL that surround the T-point participates in the vertical momentum advection.
Regarding vertical mixing, tracer in BBL is exchanged with the one at the bottom of the interior as in vertical advection(the blue arrow in Figure 16.1b). On the other hand, velocity in BBL is exchanged with the one at the bottom of the watercolumn of the same U-point (the blue arrow in Figure 16.1d), whose quarter cells may not necessarily constitute bottom
T-cells that communicate with T-cells of the BBL.

(a)

(b)

(w*,kv)(u*,v*)(w*,kv)(u*,v*)
k=km

k=km

(c)

(d)

(w*,kv)(w*,kv)

k=km

k=km

Figure16.1 Grid arrangement and schematic of horizontal and vertical exchange of tracers. (a, b) Arrangement ofT-cell represented by "•" symbols. (c, d) Arrangement of U-cell represented by "×" symbols. The primary BBL cells
lies at the bottom of the main (interior) oceanic domain (a and c). Horizontal advection and subgrid-scale mixing iscomputed there (solid black arrows). In relation to the interior, the T-cells in BBL are located just below the bottom ofinterior T-cells even when an interior T-cell consists of only one quarter of a U-cell (Figure 16.1b). The U-cells in BBL
are regarded as slanting cells connecting neighboring T-points in BBL (dashed blue line in d). Vertical transport intoand out of BBL occurs according to the primary vertical velocity defined at T-points (solid red arrows). Diagonally
upward / downward advection of momentum may occur along the dashed red arrows in (d). Vertical mixing of tracers
and momentum occurs between the cells indicated blue arrow in (b) and (d), respectively.

16.3 Pressure gradient terms
a. Formulation

BBL of MRI.COM is attached at the bottom of interior T-cells. U-cells of BBL are regarded as slanting and connecting
the surrounding T-points. (Figure 16.1d). In other words, sea floor is considered to have gradient along U-cells.
The pressure perturbation term in eq. (2.56) is1
1
1∇s p ′ =∇p′ s=−H (x,y) −∂s p′∇H (x, y)
ρ0
ρ0
ρ0
g ρ′
1
∇p′ s=−H (x,y) +z s ∇H (x, y),=ρ0ρ0
and the geopotential term is

g ρ′
g ρ′
g ρ′
∇s z =∇z s=−H (x,y) −z s ∇H (x, y).
ρ0ρ0ρ0– 190 –

(16.1)(16.2)

Chapter 16

Bottom Boundary Layer (BBL)

Then, the pressure gradient for a U-cell in BBL is expressed as
1
g ρ′
1
g ρ′
∇s p ′ +∇s z =∇p′ s=−HBBL (x,y) +∇z s=−HBBL (x,y) .
ρ0
ρ0
ρ0
ρ0

(16.3)

b. Discretization

Horizontal gradient in (16.3) is evaluated along the bottom slope, not along the constant s-surface.
The depth of U-point in BBL (z = zBBL ) that appears in the geopotential term is separated into time-independent(−HBBL ) and time-dependent (z ′) part, so thatg ρ′g ρ′ ′g ρ′∇z s=−HBBL (x,y) = −∇HBBL +
∇z s=−HBBL (x,y) .
ρ0
ρ0
ρ0

(16.4)The pressure perturbation at T-point in BBL, whose vertical grid index is k = ktbtm, is computed by adding an increment
due the downward vertical integration along the T-point from the corner of the upper face of the U-cell at the bottom ofinterior, whose vertical grid index is k = kbtm − 1, to the T-cell in BBL (k = ktbtm). The finite diﬀerence form is given as
follows:
ktbtm−1∑(dzt)i, j,ktbtm+ 1
2( ρ′ )i, j,l+ 1 (dzt)i, j,l+ 1 + g( ρ′ )i, j,ktbtm+ 1.
(16.5)∆pi,′ j,ktbtm =22
22
l=kbtm−1
The gradient of pressure perturbation is computed as′′(∇pBBL )i+ 1 , j+ 1 =(∇r p )i+ 1 , j+ 1 ,kbtm−1 + x̂
2

2

2

′
′
∆p′
+∆p′
∆pi+1,
j+1,ktbtm +∆pi+1, j,ktbtm− i, j+1,ktbtm2 i, j,ktbtm
2∆x i+ 1 , j+ 1

2

2

+ ŷ

2

′
′
∆p′
∆pi+1,+∆p′
j+1,ktbtm +∆pi, j+1,ktbtm− i+1, j,ktbtm2 i, j,ktbtm
2∆yi+ 1 , j+ 1
2.

(16.6)

2

The geopotential term is computed as[ g ρ′
ρ0

∇zBBL

=−

g
ρ0

]

ρ′i+1, j+1 +ρ′i, j+1 HBBL i+1, j+1 −HBBL i, j+1
ρ′i+1, j +ρ′i, j HBBL i+1, j −HBBL i, j+2∆xi+ 1 , j
2∆xi+ 1 , j+1
{[
2

g
ρ0

2

2
+

+

i+ 12 , j+ 21x̂ρ′i, j+1 +ρ′i, j HBBL i, j+1 −HBBL i, j
ρ′i+1, j+1 +ρ′i+1, j HBBL i+1, j+1 −HBBL i+1, j+2∆y2∆yi+1, j+ 1[
i, j+ 1
2

2

2

′
′
′
′
ρ′i+1, j +ρ′i, j zi+1,
ρ′i+1, j+1 +ρ′i, j+1 zi+1,
j −z i, j
j+1 −zi, j+1+2∆xi+ 1 , j
2∆xi+ 1 , j+1
{[
2

2

2

]
x̂ +

]

ŷ

]}

′
′
′
′
ρ′i, j+1 +ρ′i, j zi,
ρ′i+1, j+1 +ρ′i+1, j zi+1,
j+1 −z i, j
j+1 −z i+1, j+2∆yi, j+ 1
2∆yi+1, j+ 1
[
2

2

2

]}
ŷ .
(16.7)In the source code, the depth of T-point hot is used for HBBL . To be exact, (hot − dzt_cnst/2) should be used.
However, because the initial thickness of the BBL (dzt_cnst) is spatially uniform in MRI.COM, this omission would notaﬀect the result.

16.4 Eddy eﬀects
In the rotational frame, the dense water along the continental slope flows along f /h without the eddy eﬀects. In this case,introducing the BBL model does not lead to a better representation of the overflow/downslope-flow in the Nordic Seas oron the continental shelf around Antarctica.
In the real world, eddy eﬀects create the cross f /h flow, resulting in the overflow/downslope-flows. Jiang and Garwood(1996) investigated the three-dimensional features of downslope flows using an eddy-resolving model and demonstrated
that the dense water descends roughly 45° left to the geostrophic contour (in the northern hemisphere) with vigorous eddy
activity. The observation of significant eddy activity south of the Denmark Strait is consistent with this result. In order to– 191 –

16.5 Usage
incorporate this eﬀect into the non-eddy resolving models, we apply Rayleigh drag whose coeﬃcient is nearly equal to the
Coriolis parameter, α ≃ f . In this case, the geostrophic balance is modified and written as follows:
1 ∂p
− αu,
ρ0 ∂ x1 ∂p
fu = −− αv,
ρ0 ∂ y−fv = −

(16.8)(16.9)

After some algebra we obtain
(
)
α
1 ∂p
− f v − u = − f v′ = −,fρ0 ∂ x)(1 ∂pα,
f u + v = f u′ = −fρ0 ∂ y(16.10)(16.11)

where v ′ = v − (α/ f )u and u ′ = u + (α/ f )v correspond to the geostrophic velocity for the pressure gradient. If we put
α = f , the direction of the flow is 45° to the right of the pressure gradient. In general, the horizontal pressure field isparallel to the topographic contour near the deep water formation area. Thus, incorporating the Rayleigh drag coeﬃcient,
α = f , leads to the dense water descending 45° to the left of the geostrophic contour.
Because this Rayleigh drag is thought to be caused by the eddy activity and is observed where the dense water descends
to the deep layer, the coeﬃcient of the Rayleigh drag should be parameterized as a function of the local velocity andtopography. However it is very diﬃcult to appropriately determine this function in coarse-resolution models. Accordingly,
in the default setting of BBL option in MRI.COM, the depth range where α = f is arbitrarily set above 2500 m in thenorthern Atlantic and above 4000 m around Antarctica to represent the observed dense overflow/downslope-flow. Below
those depths, α is set to zero. This setting may be explicitly specified by using namelist nml_bblrayfrc (Table 16.1).

16.5 Usage
In this section, we show how to use the BBL model in the MRI.COM.
• Add BBL option in the file configure.in.
• Set kbbl = 1 in the file configure.in. (kbbl is the number of BBL layers. Presently, only one layer (kbbl = 1)
is supported).
• Set the lowest layer of the array dz(km) as the thickness of BBL. Fifty to one hundred meters is recommended forthe thickness of BBL, which roughly corresponds to the observed thickness of the BBL layer near the Denmark
Strait.
• Append ho4bbl, exnnbbl to file_topo after ho4, exnn (Section 21.2.2). These two variables have the sameformat as ho4, exnn. Variable ho4bbl corresponds to the thickness of the BBL, and variable exnnbbl corresponds
to the number of BBL layers; only one layer is presently supported. Thus exnnbbl should be equal to or less thanone.
• Set the region where non-zero Rayleigh drag coeﬃcient is applied (nml_bblrayfrc; Table 16.1). The default
setting is above 2500 m in the North Atlantic and above 4000 m around the Southern Ocean.
If model integration starts from the 3-D distribution of temperature and salinity, the temperature and salinity in the BBL
cells are set to those in the lowermost inner cells.

Example of configure.in with BBL option
OPTIONS="BBL"
IMUT=184
JMUT=171
KM=48
KSGM=48
KBBL=1



The file that contains topographic information is read by the model at run time as follows (see also Section 21.2.2).

– 192 –

Chapter 16

Bottom Boundary Layer (BBL)

How topography data (file_topo) with BBL is read by model
integer(4) :: ho4(imut,jmut),exnn(imut,jmut)
integer(4) :: ho4bbl(imut,jmut),exnnbbl(imut,jmut)
integer(4),parameter :: lun = 10
open(lun, file=file_topo, form=unformatted)
read(lun) ho4, exnnread(lun) ho4bbl, exnnbbl ! This is for BBL.



Table16.1 Namelist nml_bblrayfrc for setting the lower limit above which Rayleigh damping is applied in BBL

variable
deprfcnunits
cmdeprfcscm

description
subjective lower limit of depth (for northern hemisphere) above which Rayleigh damping is applied
subjective lower limit of depth (for southern hemisphere) above which Rayleigh damping is applied

usage
default = 2500 m= 2500 × 102 cm
default = 4000 m= 4000 × 102 cm

16.6 Usage notes
16.6.1 Limit of the region where BBL model should be applied
Ideally, the BBL model should be applied universally and its behavior should change according to the local oceanic state.
In the real configuration, the BBL model is only eﬀective near the abyssal water formation areas. When used in otherareas, such as the near the equator, the BBL model does not improve the tracer and velocity fields. Furthermore, because
the BBL model connects the model cells along the topography, it inevitably induces a water mass exchange between thecold abyssal water and the warm shallow water. Thus, unphysical diapycnal diﬀusion could occur with the BBL model.
This eﬀect is not severe in high latitudes where the diﬀerence in temperature between the shallow continental shelves andthe deep layer is expected to be small, but it is extremely problematic for the cells in low latitudes. This problem is similar
to the problem for typical σ-layer models. To prevent this, it is recommended that the BBL model should be applied only
to the regions where the processes reproduced by the BBL model are thought to be important.

16.6.2 Limits of the BBL
Linear interpolation of the temperature and salinity along an extremely steep slope may cause problems. For the default
setting of MRI.COM, the BBL model is not applied in such places and isolated grids. This setting may be modified byusing namelist nml_bblgeo_limit (Table 16.2).
Table16.2 Namelist nml_bblgeo_limit for limiting the range where BBL scheme is applied

variable
hbblminunits
cm

bbldmaxcm

description
for sea floor depth shallower than hbbldmin, BBL
is not applied
if the sea floor is deeper than that of the neighboring point by bbldmax, the BBL is not applied

usage
default = 0 m
default = 10 000 m
10 000 × 102 cm

16.6.3 Notes for program coding and analyses
In each vertical column, the BBL exists both at the bottom cell (k = km) and at the lowest ocean cell (k = exn(i, j)). In
general, the lowest ocean cell of the T-point and U-point for the same horizontal indices might diﬀer. Thus, the treatmentof topographic mask array atexl and aexl is always very confusing. At the bottom cells (k = km) that corresponds theprimary BBL cells, we set atexl = 1 and aexl = 1 while we set atexl = 1 and aexl = 0 at the lowest ocean cells that
corresponds the dummy BBL cells. In taking global average, the treatment of atexl = 1 in the BBL model needs special
care to avoid double counting.

– 193 –

Chapter 17

Sea iceThis chapter describes the sea ice part. The sea ice part of MRI.COM treats formation, accretion, melting, and transfer
of sea ice and snow. Heat, water, salt, and momentum fluxes are exchanged with the ocean. Sea ice is categorized by its
thickness, but it has a single layer. Snow does not have heat capacity (so-called zero-layer). Thus, it might be regarded as
an intermediate complexity ice model.
This chapter is organized as follows. Section 17.1 outlines the model. The following sections describe details of
the solution procedure. According to the order of solving the equations, we deal with thermodynamics in Section 17.2,
remapping among thickness categories in Section 17.3, dynamics in Section 17.4, advection in Section 17.5, and ridging
in Section 17.6. Adjustment on the sea ice distribution is explained in Section 17.7. Discretization issues are described in
Section 17.8 and some technical issues are presented in Section 17.9. Finally, usage notes are presented in Section 17.10.

17.1 Outline
The sea ice part of an ocean-sea ice model gives sea surface boundary conditions to the oceanic part. Heat, fresh water,
salt, and momentum are exchanged at their interfaces. The sea ice part solves fractional area, heat content, thickness, andtheir transport of ice categorized by its thickness and dynamics of the grid-cell averaged ice pack.
The ice model of MRI.COM is based on the ice-ocean coupled model of Mellor and Kantha (1989). For processes thatare not explicitly discussed or included there, such as categorizing by thickness, ridging, and rheology, we adopt those of
the Los Alamos sea ice model (CICE) version 3.14 (Hunke and Lipscomb, 2006).
The fundamental property that defines the state of sea ice is the fractional area as a function of location (µ, ψ) andthickness (h I ). The equation for this distribution function (g(µ, ψ, h I )) is∂1 ( ∂(ghψ u I ) ∂(ghµ vI ) )∂g=−( f g) −
+
+ χ,
∂t
∂h I
h µ hψ
∂µ∂ψ(17.1)

where f is the thermodynamic growth rate of thickness, (u I , vI ) is the velocity vector of ice pack, and χ is the rate ofchange of distribution function caused by ridging. The growth rate of ice thickness is computed by solving thermodynamic
processes. Using this rate ( f ), thickness categories are remapped according to the first term on the r.h.s. of (17.1). To
compute the velocity of the ice pack (u I , vI ), we have to solve the momentum equation. On transporting the ice distribution(second and third terms on the r.h.s.), other conservative properties such as volume and energy are also transported. Using
the transported ice distribution function, the ridging process ( χ; fourth term on the r.h.s.) is solved. The formulation andsolving procedure of each process are presented in later sections.
We discretize the thickness in several categories. If an ice pack is divided into n categories separated at Hn with
H0 = 0 m, the fractional area of each category (an ) is defined as follows:
an =

∫ Hn
gdh.

(17.2)

Hn−1

Other major variables, ice and snow thickness, surface temperature, bottom temperature and salinity, and internal energyof ice, are defined for each category. Velocity is defined for an ice pack, the total ice mass in a grid cell. In the verticaldirection, both sea ice and snow have one layer. Sea ice has heat capacity, but snow does not. The heat capacity for sea ice
is due to brine and is represented by the temperature at the center of the sea ice. Figure 17.1 and Table 17.1 summarize
symbols used in this chapter and their variable names in the source code of MRI.COM. Note that the symbol T is used torepresent the temperature of any substance treated in this chapter. The unit is ◦C, though the use of an upper case letter(T) may recall absolute temperature.

– 195 –

17.1 Outline
Table17.1 Physical quantities used in the sea ice part (cf. Figure 17.1) and their variable names in the source code.

hI
hs
A
Ah I
Ahs
E1
Ah I E1
T3
T2
T1
T0 I (= T0 )T0 L
S0 I (= S0 )
S0 L
QI2
QIO
QAO
FTI
FTL
FS I
FS L
W
WAI
WAI
WIO
WAO
WROiceMeaning

Units

Array name foreach thickness
category

Array name foraverage or sum
over thickness
categories

ice thickness
snow thickness
area fraction (compactness)
grid cell average of ice thickness (volume per unit area)
grid cell average of snow thickness (volume per unit
area)
enthalpy of ice
grid cell average of enthalpy
skin temperature of upper surface
temperature at snow-ice interface
temperature of ice
skin temperature of lower surface
skin temperature of sea surface at open leadsskin salinity of lower surface
skin salinity of sea surface at open leads
heat flux in the upper half of ice (= Q S )heat flux on the ice side of the ice bottom
heat flux on the air side at open leads
heat flux on the ocean side of the ice bottom
heat flux on the ocean side at open leads
virtual salinity flux below sea ice driving the top layerof the ocean modelvirtual salinity flux in open water driving the top layerof the ocean model
fresh water flux driving the top layer of the ocean model(= AWIO + (1 − A)WAO + WRO + WFR )fresh water flux due to snow fall at the upper surface ofice
fresh water flux due to sublimation at the upper surfaceof icefresh water flux due to freezing and melting at the bottomof icefresh water flux due to freezing at open leads
fresh water flux due to melting of sea ice at the uppersurface of ice

m
m
1
m
m

hicen
hsnwn
aicen
hin
hsn

hiceo
hsnwo
a0iceohihsnw

J kg−1J kg−1 m
◦C
◦C
◦C
◦C
◦C
pss
pss
W m−2
W m−2
W m−2
W m−2
W m−2
pss m s−1

qicen
eicen
tsfcint2icent1icent0icen(1 :)t0icen(0)s0n(1 :)s0n(0)
fheatufheatn
fheat
ftio
ftao
-

tsfci
t0iceot0icel-pss m s−1

-

-

m s−1

-

wfluxim s−1-

snowfallm s−1

sublim

-

m s−1

wio

-

m s−1
m s−1

wao
-

wrsi

Continued on next page

– 196 –

Chapter 17

Sea ice
Table 17.1 – continued from previous page
Meaning
Units

WROsnow

fresh water flux due to melting of snow at the uppersurface of ice
fresh water flux due to formation of frazil ice
zonal component of ice pack velocity
meridional component of ice pack velocity

WFR
uI
vI

Array name foreach thickness
category

Array name foraverage or sum
over thickness
categories

m s−1

-

wrss

m s−1
m s−1
m s−1

-

wrso
uice
vice

shortwave

QAI

WAI
T3

snow

hS

QS
T2

Ss = 0.0

S0L
FSL

FTL

QI2

ice

WAO

QAO
T0L

E1

SI = 4.0

T1

hI
QIO

S0(=S0I)

T0 (= T0I)
FTI

FS I

T(1), S(1)

WFR

WRO
WIO

Figure17.1 Meaning of symbols and their locations. The left side is related to heat flux, and the right side is related
to fresh water flux. A column of ice is separated into the part that originated from sea water (thickness: h I and salinity:
SI = 4.0 pss) and the part that originated from snow (thickness: hs ). The former is further divided into upper andlower halves. Thus a column of ice has three vertical layers. The temperatures at the layer boundaries are T0 , T1 , T2 ,and T3 from the bottom. Heat fluxes within each layer are QIO , QI2 , and Q S from the bottom. The heat flux at the
air-ice interface is QAI and that at the ice-ocean interface is FTI . Sea ice is in fact categorized by thickness, and eachsymbol should have a suﬃx (n) of the category number. At open leads or open water, symbols have the suﬃx L. For
the definitions of fresh water fluxes, see Table 17.1.

17.2 Thermodynamic processes
Although the surface fluxes are positive downward (positive toward the ocean) in the ocean model, the sea ice part is codedsuch that fluxes are positive upward. In this section, we assume that the fluxes are positive upward.

17.2.1 Thermal energy of sea ice
In considering thermodynamics, thermal energy of sea ice should be defined. The basis of energy (i.e., zero energy) isdefined here as that of sea water at 0 ◦C. The thermal energy (enthalpy; E(T, r)) of sea ice that has temperature T (< 0)[◦C]and brine (salt water) fraction r is the negative of the energy needed to raise the temperature to 0 ◦C and melt all of it:
E(T, r) = r (CpoT ) + (1 − r)(−L F + CpiT ),(17.3)

where Cpo and Cpi are the specific heats of sea water and sea ice, and L F is the latent heat of melting/freezing. Thus
defined, the energy of sea ice is negative definite. The positive brine fraction represents the fact that the salt with salinity(SI ) in sea ice exists in a liquid state. If the ice temperature is T1 , the brine is assumed to have the same temperature, andits salinity is S = T1 /m, where m defines the freezing temperature as a function of salinity. Hence, the brine fraction ofsea ice is r = SI /S = mSI /T1 . As in Mellor and Kantha (1989), the specific heat of snow is not considered.
– 197 –

17.2 Thermodynamic processes
Once sea ice enthalpy E is known, sea ice temperature T1 is determined by solving the quadratic equation,
E=

(
mSI
mSI )
(CpoT1 ) + 1 −(−L F + CpiT1 ),T1
T1

which gives√−b − b2 − 4ac
T1 =
,
2a

(17.4)(17.5)

where
a = Cpi, b = −L F − E + mSI (Cpo − Cpi ), c = mSI L F .

(17.6)

Figure 17.2 shows that the other root is unphysical.

Figure17.2 Sea ice enthalpy E (17.4) as a function of sea ice temperature (T1 ). Units in J kg−1 .

17.2.2 Formation of new sea ice
a. From sea water with temperature below the freezing point

Sea ice is formed when sea surface temperature is below the freezing point. If the temperature of the first layer of theocean model is below the freezing point as a function of salinity, the temperature is set to the freezing point (Tfreeze ), andthe heat needed to raise the temperature is regarded as the release of latent heat and is used to form new ice.
The thickness of the new ice (h I ) is computed by assuming that the total thermal energy of the first layer of the ocean
model (whose layer thickness is ∆z 1 ) is conserved before and after the sea ice formation:2

ρo ∆z 1 [CpoT] = ρ I h I {r (CpoTfreeze ) + (1 − r)(−L F + CpiTfreeze )} + ρo (∆z 1 − ρ I h I /ρo )(CpoTfreeze ),22(17.7)

where r = mSI /Tfreeze . Using the above equation, we compute the thickness of the new ice:hI =ρo ∆z 1

Cpo (Tfreeze − T ).
ρ I (1 − r) L F + Tfreeze (Cpo − Cpi )2(17.8)

With the ice thickness known, the fractional area is determined by the following procedure:• For a grid cell without sea ice, the first thickness is set to hnew
= 10 cm and the fractional area is computed asI
new = h and A = 1.
A = h I /hnew
.
If
A
>
1
(i.e.,
h
>
10
cm),
h
I
I
I
I
• For a grid cell where sea ice already exists, h I is added to each category and open water.
– 198 –

Chapter 17

Sea ice

Note that this operation practically eliminates super-cooling in the ocean interior. Hence, the formation of frazil ice is notconsidered in this model.
The medium that contacts the sea ice diﬀers at the upper (air) and lower (sea water) surfaces. Even for a grid cellwith sea ice, not all the surface is covered by sea ice, i.e., there may be open water. We treat processes at each interface
separately. Heat flux and thermal energy are computed by solving the balance equation at each interface and each layer inFigure 17.1.
b. Input of iceberg

Input of iceberg (Ficeberg ) is given as a water mass flux per unit area (kg m−2 s−1 ). The thickness of the new ice (h I ) is
given byh I = Ficeberg ∆t/ρ I ,(17.9)
where ∆t is the unit time step. With the ice thickness known, the fractional area is determined by the same procedure used
for ice created from supercooled water as explained above.

17.2.3 Air-ice interface
a. Heat flux at the upper surface of ice (Q AI )The surface heat flux (QAI ) is expressed as follows:
QAI = QSI + QLI − (1 − α I )SW − LW + ϵ I σ(T3 + 273.15) 4,(17.10)where QSI is sensible heat flux, QLI is latent heat flux, SW is downward shortwave radiation flux, LW is downward
longwave radiation flux, α I is albedo, ϵ I is emissivity, and σ is the Stefan-Boltzmann constant. We will now examine
each component.
■ i. Shortwave: The downward shortwave radiation is represented by SW. The albedo of sea ice is α I , which is 0.82 forcold (T3 < −1 ◦C) snow, 0.73 for melting snow, and 0.64 for bare ice (while melting).
One might use a more sophisticated albedo scheme included in the Los Alamos sea ice model (CICE; Hunke andLipscomb, 2006) by choosing CALALBSI option. We briefly describe how the incoming shortwave radiation is treated bythis scheme. The downward shortwave radiation is treated at each interface as follows:• Among the net absorbed shortwave flux (= (1 − α I )SW), some fraction (i 0 ) penetrates into ice and the rest isabsorbed at the surface and used to warm the upper interface. See Table 17.2 for the specific value of i 0 .
• The part penetrating into the ice (= (1 − α I )i 0 SW) is attenuated according to Beer’s Law with the bulk extinction
coeﬃcient κ i = 1.4 m−1 . The attenuated part is used to warm the ice interior.
• The rest enters the ocean.
The albedos and penetration coeﬃcients of CICE are listed on Table 17.2. The property f snow is the snow fraction of
the upper surface of the ice, which is expressed as follows:
f snow =hs,
hs + hsnowpatch(17.11)

where hs is the snow thickness and hsnowpatch = 0.02 m.
If the ice thickness (h I ) is less than href = 0.5 m, the albedo of thin ice is computed as
αthinice = α o + β(αcoldice − α o ),
where
β=

arctan(ar h I ),arctan(ar href )ar = 4.0,(17.12)(17.13)and α o is the albedo of the ocean.
If the surface temperature T3 becomes −1 < T3 < 0 ◦C, the albedo of melting ice and snow is computed as
αmeltice = αthinice − γi (T3 + 1.0),(17.14)αmeltsnow = αcoldsnow − γs (T3 + 1.0),(17.15)– 199 –

17.2 Thermodynamic processes
where the condition αmeltice > α o is imposed. Using the snow fraction on the surface of the ice f snow , the total albedo iscomputed as
αi = αmeltice (1 − f snow ) + αmeltsnow f snow .
(17.16)
The albedos for visible and near infra-red wave lengths are computed separately. If the shortwave flux is given as thesum of all four components (direct and diﬀuse for visible and near infra-red wave lengths), a constant ratio (visible) : (nearinfra-red) = 0.575 : 0.425 is assumed, and the total albedo is computed as the weighted average.
Table17.2 Albedo and surface transparency of the albedo scheme of CICE.

albedo for cold snow αcoldsnow (T3 < −1 ◦C)
albedo for cold ice αcoldice (T3 < −1 ◦C, hi > 0.5 m)
reduction rate of albedo for melting ice γi (−1 ◦C < T3 < 0, hi < 0.5 m)
reduction rate of albedo for melting snow γs (−1 ◦C < T3 < 0, hi < 0.5 m)
fraction of transparent shortwave flux through the ice surface (i 0 )near infra-red(> 700 nm)
0.70
0.36
−0.075 /◦C−0.15 /◦C
0.0

visible(< 700 nm)
0.98
0.78
−0.075 /◦C−0.10 /◦C
0.7 × (1.0 − f snow )■ ii. Longwave: The downward longwave radiation from the atmosphere is represented by LW in (17.10). The black

body radiation from the ice surface is ϵ I σ(T3 + 273.15) 4 , where ϵ I is emissivity, and σ is the Stefan-Boltzmann constant.
Hereinafter, we use LW I = LW − ϵ I σ(T3 + 273.15) 4 as the net longwave radiation.
■ iii. Sensible heat flux:

The sensible heat flux (QSI in (17.10)) is computed using a bulk formula:
QSI = ρ a Cpa CHAI U10 (T3 − TA ),(17.17)

where ρ a is the density of air, Cpa is the specific heat of air, CHAI is the bulk transfer coeﬃcient for heat, U10 is the scalarwind speed at 10 m, TA is the surface air temperature, and T3 is the ice surface temperature (Figure 17.1).
■ iv. Latent heat flux:

The latent heat flux (QLI in (17.10)) is computed using a bulk formula:
QLI = ρ a L s CEAI U10 (qi − q A ),(17.18)where L s is the latent heat of sublimation, CEAI is the bulk transfer coeﬃcient for moisture, qi is the saturation humidity at
the ice surface temperature T3 , and q A is the specific humidity of air. Section 14.12.3 details a computing method for qi .
Fresh water loss due to sublimation is computed as
WAI = ρ a CEAI U10 (qi − q A )/ρo .

(17.19)■ v. Bulk coeﬃcient over sea ice: Over sea ice, it is common to set constant neutral bulk coeﬃcients at 10 m, as describedin Large and Yeager (2004). Then, bulk coeﬃcients and wind speed at each height are calculated on the basis of stability.
Large and Yeager (2004) uses
CDAI10 = CHAI10 = CEAI10 = 1.63 × 10−3,(17.20)

but the default in MRI.COM uses, following Mellor and Kantha (1989),
CDAI10 = 3.0 × 10−3

(17.21)

CHAI10 = CEAI10 = 1.5 × 10−3 .

(17.22)

These coeﬃcients are given by namelist nml_air_ice at run time. See Table 17.9.
The wind stress over sea ice can be calculated with TAUBULK option.
b. Heat flux in the snow (Q S )If we neglect the heat capacity of snow, the heat flux is constant within the snow layer and is computed as
QS =ks(T2 − T3 ),hswhere hs is the thickness of the snow layer and k s is the thermal conductivity of snow.
– 200 –

(17.23)

Chapter 17

Sea icec. Heat flux in the ice interior (Q I2, Q IO )In the upper half of the ice layer, the heat flux is computed as follows:
QI2 =

kI
(T1 − T2 ),h I /2(17.24)

where k I is the thermal conductivity of sea ice. If we neglect heat capacity of snow, Q S = QI2 . Using this relation, theinterface temperature is computed askskI
h s T3 + h I /2 T1
T2 = k
.
(17.25)kIsh s + h I /2
In the lower half of the ice layer, the heat flux is computed as follows:
QIO =

kI
(T0 − T1 ).
h I /2(17.26)d. Melting at the upper surface (WRO )The melting rate at the upper surface is computed as follows: First, the surface temperature (T3 ) is computed by equating
the fluxes at the upper surface (QAI = Q S ). If T3 is lower than the freezing temperature (0.0 ◦C for snow and mSI [◦C] forsea ice), melting does not occur. If T3 is higher than the freezing temperature, melting occurs. In this case, T3 is set to thefreezing temperature, and the interior heat flux just below the ice surface (Q S ) is recalculated. The imbalance between
QAI and Q S is used to melt snow or ice. The fresh water flux due to melting (by sign convention in this Chapter, this isalways negative) is computed as
WRO = (QAI − Q S )/( ρo L 3 )(17.27)
where L snow
= L F for snow melt and3L ice
3 ≡ [E(T3, 1) − E(T1, r 1 )](17.28)for ice melt. The brine fraction for sea ice is r 1 = mSI /T1 . The temperatures of the melted water are mSI [◦C] for sea ice
and 0 ◦C for snow.
If all pre-existing snow and ice is going to be melted, that is, (Q S − QAI )∆t > ρs hs L snow
+ ρ I h I L ice
3 , a new interior flux
3′
(Q S ) is computed as follows:′
Q S = QAI + ( ρs hs L snow
+ ρ I h I L ice(17.29)3 )/∆t.
3′

In this case, Q S is equated with QIO and treated as the sensible heat flux at the ice-ocean interface.
e. Notes• All precipitation on sea ice is assumed to be snow.
• Melted water is assumed to run oﬀ to the ocean immediately, that is, there is no melt pond.
f. Solution procedure

The solution procedures are basically the same with or without snow. To be exact, the interface fluxes should be computed
iteratively by adjusting surface temperature T3 until a balance is achieved. We adopt the semi-implicit method described
below. A situation without snow (hs = 0, T3 = T2 ) is considered. First, the surface temperature (T3 ) is computed byassuming that the fluxes on both sides are the same. Inserting (17.10) and (17.24) into QAI = QI2 with T3 → T3 + δT3 ,kI{T1 − (T3 + δT3 )} = QLI (T3 + δT3 ) + QSI (T3 + δT3 )h I /2− (1 − α I )(1 − i 0 )SW − LW + ϵ I σ{(T3 + δT3 ) + 273.15}4 .

(17.30)By expanding the specific heat, latent heat, and black body radiation in a Taylor series, we have,
∂QLI
kI
∂QSI
(T1 − (T3 + δT3 )) = QLI + QSI +δT3 +δT3 − (1 − α I )(1 − i 0 )SW
h I /2∂T3∂T3− LW I + 4ϵ I σ(T3 + 273.15) 3 δT3 .
– 201 –

(17.31)

17.2 Thermodynamic processes
Using this, we compute δT3 and add it to T3 to obtain a new temperature:δT3 =−QSI − QLI + (1 − α I )(1 − i 0 )SW + LW I + hkI I/2 (T1 − T3 )
∂QLI
∂QSI
kI
3
∂T3 + ∂T3 + 4ϵ I σ(T3 + 273.15) + h I /2,

(17.32)andT3new = T3old + δT3,

(17.33)

where∂QSI= ρ a Cpa CHAI U10,
∂T3
∂QLI
∂qi
= ρ a L s CEAI U10
.
∂T3
∂T3

(17.34)(17.35)Note that the dependency of L s on temperature (Section 14.12.3) is not considered in the partial diﬀerentiation with respect
to temperature. The specific form for the partial derivative of specific humidity (∂qi /∂T3 ) is presented in Section 17.11.1.
If the new surface temperature (T3new ) is below the freezing point, melting does not occur. If not, T3new is set to thefreezing temperature (= mSI ), and the heat flux in the ice interior is re-evaluated. The amount of melting (WRO ) is obtainedusing the imbalance:
QAI = − QSI − QLI + (1 − α I )(1 − i 0 )SW + LW I ,kI(T1 − T3new ),QI2 =h I /2L 3 = [E(T3new, 1) − E(T1, r 1 )],(QAI − QI2 )∆t= WRO ∆t.
∆h I =
ρI L3(17.36)

If there is snow, the above procedure is performed for the snow surface (T3new = 0 ◦C, L 3 = L F ). If all the snow melts
away, the residual heat (Eres ) is used to melt the ice:
Eres = (QI2 − QAI )∆t − hs ρs L F ,L 3 = [E(mSI , 1) − E(T1, r 1 )],∆h I = − Eres /ρ I L 3 = WRO ∆t.

(17.37)17.2.4 Heat balance in the ice interior
The thermal energy of the ice is aﬀected by vertical heat fluxes and horizontal heat transport due to advection. The
equation for the thermal energy (enthalpy) is written as follows:
ρI hI

[∂
∂t

E(T1, r 1 ) + u I i]∂
E(T1, r 1 ) = QIO − QI2 + [SW surface − SW bottom ],∂ xi(17.38)where
SW surface = (1 − α I )i 0 SW,(17.39)SW bottom = (1 − α I )i 0 SW × exp(−κ i h I ).

(17.40)

The above equation can be solved explicitly without causing serious problems when the time step is not too long.
If the enthalpy exceeds the upper limit, E(mSI , 1), QIO is adjusted so that the enthalpy gets the upper limit value at theend of the time step (∆t),
′

QIO =

ρ I h I [E(mSI , 1) − E n ]− [SW surface − SW bottom ] + QI2,
∆t′

(17.41)

where E n is the enthalpy at the time level n. QIO is used instead of QIO as the sensible heat flux at the ice-ocean interface.

– 202 –

Chapter 17

Sea ice17.2.5 Ice-ocean interface
Melting and freezing at the ice-ocean interface is computed using heat fluxes at the interface as depicted in Figure 17.1.
The ice-covered area and the open water are treated separately. The solution method slightly diﬀers from that of Mellor
and Kantha (1989).
In the ice-covered area, the heat flux on the ice side of the ice-ocean interface (QIO ) is computed according to (17.26).
In open water, the heat flux on the air side of the air-ocean interface (QAO ) is computed as in Chapter 14,
4
Qall
AO = Q SO (T ) + Q LO (T ) − (1 − α o )SW − LW + ϵ o σ(T + 273.15) .

(17.42)The temperature at the first level of the ocean model (T) is used. All the heat and fresh water fluxes are evaluated using thetemperature and salinity at the first level of the ocean model. By doing so, the equation to compute melting and freezing
rates becomes linear.
Here, shortwave radiation is assumed to pass through the skin layer without absorption and is excluded from theevaluation of the freezing rate in open water:
QAO = QSO + QLO − LW + ϵ o σ(T + 273.15) 4 .

(17.43)This operation causes the shortwave radiation to be absorbed in the ocean interior. In reality, the heat stored in the skinlayer in open water is used to melt ice laterally (edge melting). To include this eﬀect, a certain fraction (Ψ) of the bottom
melting (WIO ) may be used for the edge melting. The details are described in the last part of this section.
Melting and freezing occur owing to the imbalance between fluxes above (QIO , QAO ) and below (FTI , FTL ) the interface:
FTI = QIO − WIO ρo L o,
FTL = QAO − WAO ρo L o,

(17.44)(17.45)L o ≡ [E(T, 1) − E(T1, r 1 )].

(17.46)

where
T is the temperature at the first level of the ocean model. The heat flux that drives the first level of the ocean model isgiven by
FT = ( AQIO + (1 − A)QAO ) − WO ρo L o,
(17.47)
where
WO ≡ AWIO + (1 − A)WAO .

(17.48)

Following the formulation adopted by Mellor and Kantha (1989), we introduce skin layer temperature and salinity (T0 I ,T0 L , S0 I , and S0 L ) to solve for WIO and WAO , and thus FTI and FTL . To incorporate skin layer salinity in the systemof equations, we consider the problem in terms of virtual salt flux, in which the eﬀect of freshwater flux on salinity is
considered in terms of salt flux by keeping the water volume. The flux balance for salt below the interface is written as
follows:
FS I = WIO (SI − S),(17.49)FS L = WAO (SI − S).

(17.50)Here, unlike Mellor and Kantha (1989), the salinity at the first level of the ocean model (S) is used instead of the salinity
at the skin layer (S0 I , S0 L ). By doing so, the equations to solve for S0 I and S0 L become linear as shown below. It could
also be said that it is natural to use the first level salinity itself in evaluating the salt flux that drives the first level ofthe ocean model. Note that only fresh water fluxes that are relevant to freezing and melting at the ice-ocean interface
are included in the above equations. The restoration to climatological salinity and fresh water fluxes caused by surface
melting, precipitation, and evaporation are excluded in the above balance.
The virtual salinity flux caused by melting and freezing at the ocean surface that drives the first level of the ocean model
is given by
FS = ( AWIO + (1 − A)WAO )(SI − S).
(17.51)
Fluxes on the oceanic side of the interface (FTI , FTL , FS I , and FS L ) can also be obtained as the boundary conditions(z → 0) for the molecular boundary layer:
FTI /( ρo Cpo ) = − CTz (T0 I − T ),(17.52)

FTL /( ρo Cpo ) = − CTz (T0 L − T ),(17.53)– 203 –

17.2 Thermodynamic processes
and
FS I = − CSz (S0 I − S),(17.54)

FS L = − CSz (S0 L − S),(17.55)

where

uτ
,
−1
(Prt k ln(−z/z0 ) + BT )CTz =(17.56)

2 + τ 2 ) 1/4 ρ−1/2 is the friction velocity, k = 0.4 is von Karman’s constant, z is the roughness parameter,uτ ≡ (τIO
0
o
IO y
x
(τIO x , τIOy ) is the stress vector at the ocean-ice interface, andBT = b( z0 uτ ) 1/2
ν

Pr 2/3,(17.57)

with Pr ≡ ν/α t = 12.9. The specific values for other parameters are given in Section 17.11.2.
Parameters related to salinity are given by
CSz =

uτ
,
(Prt k −1 ln(−z/z0 ) + BS )and
BS = b( z0 uτ ) 1/2
ν

Sc2/3,(17.58)(17.59)where Sc ≡ ν/α b = 2432.
Following Mellor and Kantha (1989), roughness parameter z0 is computed as follows:
lnz0 = Alnz0 I + (1 − A)lnz0 L ,where
z0 I = 0.05

hI,
h I lim(17.60)h Ilim = 3.0 m,(17.61)ρo uτ2.
ρa g(17.62)andz0 L = 0.016

However, the roughness parameter below ice (17.61) is also used for open water (z0 L ) in MRI.COM.
The above equations are solved simultaneously to obtain melting and freezing at the upper surface of the ocean under
the following constraints:
WO = 0, if A = 0,
(17.63)andT0 I = mS0 I

and T0 L = mS0 L , if A > 0,

(17.64)

where m defines the freezing line as a function of salinity.
Solution procedure

We first solve for S0 I and S0 L using (17.44), (17.49), (17.52), (17.54), (17.45), (17.50), (17.53), and (17.55):
CSz S + ( ρo Cpo CTz T − QIO )(SI − S)/ρo L o,CSz + ρ0 Cpo CTz m(SI − S)/ρo L o
CS S + ( ρo Cpo CTz T − QAO )(SI − S)/ρo L o
S0 L = z.
CSz + ρo Cpo CTz m(SI − S)/ρo L o
S0 I =(17.65)(17.66)Using S0 I and S0 L , T0 I and T0 L are computed from (17.64), FTI and FTL are computed from (17.52) and (17.53), FS I and
FS L are computed from (17.54) and (17.55), and finally WIO and WAO are computed from (17.44) and (17.45). Since WAO
should be positive by definition (only freezing is allowed in open water), WAO is set to zero when WAO < 0. In this case,
S0 L = S from (17.50) and (17.55). Furthermore, FTL = Q AO in (17.45) and T0 L = T assuming that there is no ice eﬀect inopen water if freezing does not occur. Finally, the surface boundary conditions for the ocean model, the surface heat andvirtual salt fluxes, are computed from (17.47) and (17.51). If we allow variation of volume of the first layer of the ocean– 204 –

Chapter 17

Sea icemodel, virtual salt flux is not necessary. The fresh water flux (17.48) is used as a boundary condition, assuming that WO
has salinity of SI and temperature of the first layer of the ocean model.
If −ρo WIO ∆t/ρ I > h I , all ice would melt away. In this case, the amount of heat needed to melt all the ice is consumed,
and the residual is returned to the ocean. That is, the heat flux necessary to melt all the ice is removed from the ocean.
′Specifically, WIO is recomputed byρo ′h I = − WIO ∆t(17.67)
ρI
and FTI is obtained as′

FTI = QIO − WIO ρo L o .

(17.68)

If snow remains above the completely melted sea ice, the snow is also melted using ocean’s heat, which is treated as thelatent heat loss from the ocean (QLO ),QLO = ρs hs L F .
(17.69)When −ρo WIO ∆t/ρ I < h I , reduction of the fractional area is allowed by edge melting. The procedure is as follows:
The fraction used for edge melting is defined as Ψ, and −(1 − Ψ)WIO ρo ∆t/ρ I is used to reduce the thickness holding the′fractional area. The new thickness is h I and Ψ ρo WIO ∆t/ρ I is used to reduce compactness holding the thickness:′

A′ = A(1 + Ψ ρo WIO ∆t/ρ I h I ).

(17.70)

There seems to be no widely accepted parameterization scheme for edge melting. According to Steele (1992), bottom
and top melting are dominant processes, and thus Ψ ∼ 0.0 − 0.1 is usually used in MRI.COM.

17.2.6 Archimedes’ Principle
From Archimedes’ principle, the part of snow that is below freeboard absorbs sea water to become sea ice. The following
equality will be achieved at equilibrium:ρ I h I + ρ s hs = ρo h I .
(17.71)At the end of the time step, the above equality is checked, and if
ρ I h I + ρs hs> hI ,
ρ0(17.72)hnew=
Iρ I h I + ρs hs.
ρo(17.73)δhs = −(hnew
− hI ) ρII,ρs(17.74)then the new ice thickness is set as:

Since the change in snow thickness isthe new snow thickness is obtained as follows:hnew= hs + δhs .
s(17.75)Since the salinity of sea ice is SI = 4.0 pss, salt is removed from the first layer of the ocean model:
S new = S old − SI

δhs ρs.
∆z1 ρo(17.76)17.3 Remapping in thickness space
After the thermodynamic processes are solved, the resultant ice thickness in some thickness categories might not be within
the specified bound. Following the method adopted by CICE, we assume that there is a thickness distribution function ineach category and use it to redistribute the new thickness distribution into original categories.
This procedure corresponds to the first term on the r.h.s. of (17.1):∂g∂=−( f g).
∂t∂h I– 205 –

(17.77)

17.3

Remapping in thickness space

In practice, a thickness category is regarded as a Lagrange particle, and the category boundaries are displaced as aresult of thermodynamics. A linear thickness distribution function is assumed within each displaced category, and ice isremapped into the original categories using these functions.
First, boundaries of thickness categories are displaced. If the ice thickness in category n (hn ) changes from hnm to hnm+1(m is the time step index), the growth rate ( f n ) at thickness hn is represented as f n = (hnm+1 − hnm )/∆t. Using this, thegrowth rate (Fn ) at the upper category boundary Hn is obtained by linear interpolation:
Fn = f n +f n+1 − f n(Hn − hn ).
hn+1 − hn(17.78)

If the fractional area is zero in either category n or n + 1, Fn is set to the growth rate at the non-zero category. When thefractional area is zero on both categories, Fn = 0. The new category boundary after thermodynamics is obtained as
Hn∗ = Hn + Fn ∆t.

(17.79)∗ , H ∗ ] is determined. For simplicity, we
Next, the thickness distribution function g within the displaced category [Hn−1n∗∗write H L = Hn−1 and HR = Hn . Function g should satisfy the following equality for fractional area (an ) and volume (vn ):

∫ HR
HL
HR

∫

gdh = an,

(17.80)hgdh = vn .

(17.81)HL

We adopt a linear function of thickness for g. The thickness space is transformed to η = h − H L , and the thickness
distribution function is written as g = g1 η + g0 . These are substituted into (17.80) and (17.81) to yieldη 2R+ g0 η R = an,2
η3
η2
g1 R + g0 R = an η n,
3
2
g1

(17.82)(17.83)where η R = HR − H L and η n = hn − H L . These are algebraically solved for g0 and g1 as)
6an ( 2η R
−
η
,
n
3
η 2R
ηR )
12an (
g1 = 3 η n −
.
2
ηR

g0 =

(17.84)(17.85)The values of the thickness distribution function at category boundaries are given as follows:)
6an ( 2η R
−
η
,
n
3
η 2R
6an (
ηR )
g(η R ) = 2 η n −.
3
ηRg(0) =(17.86)(17.87)Equation (17.86) gives g(0) < 0 when the thickness is in the right third of the thickness range or η n > 2η R /3. Equation(17.87) gives g(η R ) < 0 when the thickness is in the left third of the thickness range or η n < η R /3. Since a negative
g is physically impossible, we redefine the range of the thickness distribution function. Specifically, when the thickness
is within the left third of the thickness range, a new right boundary is set at HC = 3hn − 2H L and g is set to zero for[HC , HR ]. In this case, η R = HC − H L in (17.84) and (17.85). When the thickness is within the right third of the thickness
range, a new left boundary is set at HC = 3hn − 2HR and g is set to zero for [H L, HC ]. In this case, η R = HR − HC andη n = hn − HC in (17.84) and (17.85).
Finally, we remap ice into the original categories using the above thickness distribution function. If Hn∗ > Hn , ice istransferred from category n to n + 1. The transferred area ∆an and volume ∆vn are
∆an =

∫ Hn∗
gdh
Hn

– 206 –

(17.88)

Chapter 17

Sea ice

and

∫ Hn∗

∆vn =

hgdh.

(17.89)Hn

If Hn∗ < Hn , ice is transferred from category n + 1 to n. The transferred area ∆an and volume ∆vn are
∆an =
and

∫ Hn
gdh

(17.90)hgdh.

(17.91)

Hn∗

∫ Hn

∆vn =

Hn∗

Snow and thermal energy are also transferred in proportion to the transferred volume. For example, ∆vsn = vsn (∆vin /vin )for snow and ∆ein = ein (∆vin /vin ) for thermal energy.
If ice is created in open water, the left boundary of category 1 (H0 ) is moved to F0 ∆t, where F0 is the growth rate inopen water. After area and volume are remapped in higher categories, ice area, volume, and energy are added to category1.
If ice is not created in open water, H0 remains zero, but the growth rate at the left boundary of category 1 is set toF0 = f 1 . If F0 < 0, the fractional area of category 1 thinner than ∆h0 = −F0 ∆t is added to open water area. In thisoperation, volume and energy are invariant. The area to be added to open water is
∆a0 =

∫ ∆h0
gdh.

(17.92)0

The right boundary of the thickest category N (H N ) is a function of its mean thickness h N . When h N is given, H N iscomputed as H N = 3h N − 2H N −1 . It is guaranteed that g(h) > 0 for H N −1 < h < H N and g(h) = 0 for H N < h.

17.4 Dynamics
17.4.1 Momentum equation for ice packThe momentum equation for an ice pack with mass ρ I Ah I is
1 ∂h∂
( Ah I u I ) − ρ I Ah I f vI = − ρ I Ah I g+ Fµ (σ) + A(τAI x + τIO x ),∂thµ ∂ µ∂1 ∂hρ I ( Ah I vI ) + ρ I Ah I f u I = − ρ I Ah I g+ Fψ (σ) + A(τAI y + τIOy ),
∂t
hψ ∂ψ
ρI

(17.93)(17.94)

where (u I , vI ) is the velocity vector, h is the sea surface height, (Fµ, Fψ ) is the ice’s internal stress (which is a function of−−→internal stress tensor (σ)), and −τ→AI and τIO are stresses exerted by the atmosphere and ocean.

17.4.2 Stresses at top and bottom
The stress at the top is wind stress:−
τ→AI = CDAI ρ a |U a − u I |[(U a − u I ) cos θ a + k × (U a − u I ) sin θ a ],(17.95)where Ua is the surface wind vector, CDAI is the bulk transfer coeﬃcient between air and ice, ρ a is the density of air, andθ a is the angle between the wind vector and the ice drift vector, which is set to zero in MRI.COM.
Stress at the bottom is ocean stress:−τ−→
IO = CDIO ρo |Uw − u I |[(Uw − u I ) cos θ o + k × (Uw − u I ) sin θ o ],(17.96)

where Uw is the velocity of the first level of the ocean model, CDIO is the bulk transfer coeﬃcient between the ice andocean, ρo is the density of sea water, and θ o is the angle between the ice drift vector and the surface velocity of the ocean,which is set to 25° in the northern and −25° in the southern hemisphere following Hunke and Dukowicz (2002).

– 207 –

17.4 Dynamics

17.4.3 Internal stressIn a highly concentrated icepack, the eﬀect of the internal stress is as large as the Coriolis eﬀect and the surface stresses.
The expression of the internal stress is derived by regarding the ice as continuous media. The elastic-plastic-viscous (EVP)
model by Hunke and Dukowicz (1997, 2002) is adopted for the constitutive law (the relation between stress and strainrate). The EVP model is a computationally eﬃcient modification of the viscous-plastic (VP) model (Hibler, 1979). In the
VP model, the internal stress could be very large when the concentration is high and strain rate is near zero, which makes
the explicit integration infeasible. An alternative, the implicit method, is usually adopted, but it is not suitable for parallel
computing. The EVP model treats the ice as an elastic medium and a large local force is released by elastic waves, which
would be damped within the time scale of the wind forcing.
The constitutive law of the EVP model is
1 ∂σi j1
η−ζP+
σi j +
σkk δi j +δi j = ϵ˙i j , i, j = 1, 2,
E ∂t
2η
4ηζ
4ζ

(17.97)

where ζ and η are viscous parameters, P represents ice strength, and E is an elastic parameter (mimics Young’s modulus).
In the VP model, tendency terms are zero.
The r.h.s. (ϵ˙i j ) is the strain rate tensor, expressed in Cartesian coordinates as:
ϵ˙i j =

1 ( ∂u I i ∂u I j )
+
.
2 ∂xj
∂ xi

(17.98)The divergence, tension, and shear of the strain rate are defined as follows:
DD = ϵ˙11 + ϵ˙22, DT = ϵ˙11 − ϵ˙22, DS = 2ϵ˙12 .

(17.99)The equation for the stress tensor for σ1 = σ11 + σ22 and σ2 = σ11 − σ22 is given by1 ∂σ1 σ1
P
+
+
= DD,
E ∂t
2ζ 2ζ
1 ∂σ2 σ2+= DT ,E ∂t2η
1 ∂σ12 σ12 1+= DS .
E ∂t2η
2(17.100)(17.101)(17.102)

In generalized orthogonal coordinates, divergence, tension, and shear of the strain rate are expressed by
1 [ ∂(hψ u I ) ∂(hµ vI ) ]+,h µ hψ∂µ∂ψhψ ∂ ( u I ) h µ ∂ ( v I )DT =−,h µ ∂ µ hψhψ ∂ψ hµh µ ∂ ( u I ) hψ ∂ ( v I )DS =+.
hψ ∂ψ hµh µ ∂ µ hψDD =(17.103)(17.104)(17.105)

The internal stress is obtained as the divergence of the internal stress tensor,)]
1 [ 1 ∂σ1
1 ∂(hψ σ2 )2 ∂ ( 2Fµ =++ 2
hµ σ12 ,2
2 hµ ∂ µ∂µ∂ψh µ hψ
h µ hψ

(17.106))]
1 [ 1 ∂σ1
1 ∂(hµ σ2 )2 ∂ ( 2− 2+
hψ σ12 .
2
2 hψ ∂ψhµ hψ ∂ψh µ hψ ∂ µ(17.107)

2

2

Fψ =

The viscous parameters are obtained from the ice strength and velocity as follows:P,2∆P
η= 2 ,
2e ∆
] 1/2
[
1
2
.
+ 2 (DT2 + DS2 )∆ = DDeζ=– 208 –

(17.108)(17.109)(17.110)

Chapter 17

Sea ice

The pressure (strength) of the ice is a function of ice concentration and thickness:
P = P∗ Ah I e exp[−c∗ (1 − A)],

(17.111)

where P∗ is the scaling factor for pressure, c∗ is a parameter that defines the dependency on concentration, and e is theaxis ratio of the elliptic yield curve (e = 2).
The elastic parameter E is given by
2Eo ρ I Ah I
min(∆x 2, ∆y 2 ),(17.112)E=∆t e2
where Eo is a tuning factor that satisfies 0 < Eo < 1, ∆t e is the time step for ice dynamics, and ∆x and ∆y are the zonal
and meridional grid widths.

17.4.4 Boundary conditions
Surface stresses on the ice are exerted for the fractional area of the ice within a grid cell. The ice concentration is multipliedby the wind and ocean stresses.
For the stress on the ocean, the ice-ocean stress is exerted for the ice-covered area, and the wind stress is exerted for the
open water area:( ∂U ∂V )A(1 − A)νV,= − (τIO x , τIOy ) +(τAO x , τAOy ).
(17.113)
k=0
∂z ∂z
ρo
ρo
Note that (τIO x , τIOy ) is reversed in sign because it is defined by (17.96) as the stress on the ice.

17.4.5 Solution procedure
Given the surface wind vector and the surface velocity of the ocean needed to compute surface stresses, the momentum
equations ((17.93) and (17.94)) and the equations for stress tensors ((17.100), (17.101), and (17.102)) are solved.
First, the stress tensor is computed using the equations for stress tensors, the momentum equation is then solved using
the stress tensor. Basically, the implicit method is used for prognostic variables for each equation. For example, stresstensor σ1 is solved for σ1m+1 as follows:
m+1
m
σ m+1
1 σ1 − σ1Pm+ 1 m + m = DD
.
E
∆t
2ζ
2ζ

(17.114)

Note that strain rate tensors and viscous parameters are updated every time step using a new velocity.
The momentum equations are solved using σ m+1 above:
ρ I Ah I

um+1
− um
1 ∂h
I
I
= ρ I Ah I f vIm+1 − ρ I Ah I g+ Fµ (σ m+1 ) + AτAI x∆thµ ∂ µ(17.115)

m+1+ Acw ρo |Uw − um
) cos θ o + (Vw − vIm+1 ) sin θ o ],I |[(Uw − u I

ρ I Ah I

vIm+1 − vIm
1 ∂h= − ρ I Ah I f um+1
− ρ I Ah I g+ Fψ (σ m+1 ) + AτAI y
I
∆t
hψ ∂ψ

(17.116)

m+1+ Acw ρo |Uw − um
) cos θ o − (Uw − um+1
) sin θ o ].
I |[(Vw − v I
I

Note that the surface velocity of the ocean (Uw, Vw ) is constant during the integration. The starting time level of the ocean
model is used, n − 1 for the leap-frog time step, and n for the Matsuno scheme. For the leap-frog time step of the ocean
model,
−
n−1
m
n−1
m+1
n−1
τ−→
) cos θ o + k × (Uw− um+1) sin θ o ].
(17.117)
IO = cw ρo |Uw − u I |[(Uw − u I
I
The time step of the ice dynamics is limited by the phase speed of the elastic wave. To damp the elastic waves during
the sub-cycle, the ice dynamics is sub-cycled several tens of steps during one ocean model time step.

– 209 –

17.5

Advection and Diﬀusion

17.5 Advection and Diﬀusion
Fractional area, snow volume, ice volume, ice energy, and ice surface temperature (optional; set flg_advec_tskin =.true. in mod_seaice_cat.F90) of each category are advected. A multidimensional positive definite advection transport
algorithm (MPDATA; Smolarkiewicz, 1984) is used. For all quantities which are advected, the harmonic-type diﬀusion
may be applied to remove noises.
The advection-diﬀusion equation for a property (α) is given by{}∂(hψ uα) ∂(hµ vα)∂α1++= D(α),(17.118)∂th µ hψ
∂µ∂ψ
where D represents diﬀusion operator. The specific representation for α is A for fractional area, Ahs for snow volume,
Ah I for ice volume, and Ah I E for ice energy.
In MPDATA, (17.118) is first solved to obtain a temporary value (α ∗ ) using the upstream scheme with a mid-point
velocity between time levels n and n + 1. Using this temporary value, an anti-diﬀusive velocity is computed as[]) ∂α ∗ 1∗111 ∂α
1 1 ( n+ 1
ũi+ 1 , j = ∗|u 2 |∆x − ∆t(un+ 2 ) 2− ∆tun+ 2 v n+ 2
2
α 2
∂x
2
∂y
1

1

1
1
∂u n+ 2 ∂v n+ 2 +− ∆tun+ 2 *
+
,
2
∂x
∂y
,
[
]
)
∗
∗
1
1 1 ( n+ 1
n+ 21 n+ 12 ∂αn+ 12 2 ∂α2|v
− ∆tu vṽi, j+ 1 = ∗|∆y − ∆t(v
)
2
α 2
∂y
2
∂x
1

(17.119)

1

1
1∂u n+ 2 ∂v n+ 2 +− ∆tv n+ 2 *
+
.
2
∂y
, ∂x
-

(17.120)

This velocity is used to compute a new value using the upstream scheme starting from the above temporary value. Since
MPDATA is positive definite, the new area and thickness should be positive. If the sum of the fractional area exceeds
one, the ridging scheme will adjust the fractional area. Since energy is negative definite, the sign is reversed just before
advection and returned to a negative value after the advection.
The specific form of the harmonic-type diﬀusion is represented as follows:
{
(
)
(
)}
1
∂ hψ κ H ∂α∂ hµ κ H ∂αD(α) =(17.121)+h µ hψ ∂ µhµ ∂ µ∂ψhψ ∂ψ∂(hψ Fµα ) ∂(hµ Fψα ) 


1 


=−
+
(17.122)h µ hψ  ∂ µ∂ψ where κ H is the horizontal diﬀusion coeﬃcients. Using the flux form (17.122), the diﬀusion term is advanced in time withthe advection term (at the "correction" step of the MPDATA scheme).

17.6 Ridging
As a result of advection, the sum of the fractional area might exceed one, especially where the velocity field is convergent.
In such a case, it is assumed that ridging occurs among ice to yield a sum equal to or less than one. Even if the sum is lessthan one, ridging or rafting might occur where the concentration is high.
The ridging scheme of MRI.COM follows that of CICE, which is briefly summarized in this section.
First, we determine a fractional area that undergoes ridging: a P (h) = b(h)g(h). The weighting function b(h) is chosensuch that the ridging occurs for thin ice:
G(h) )2 (1−
if G(h) < G∗
∗
G
G∗
=0
otherwise,b(h) =(17.123)(17.124)

where G(h) is the area of ice thinner than h and G∗ is an empirical constant with G∗ = 0.15. The participation function
for category n (a Pn ) is obtained by integrating a P (h) for a range [Hn−1, Hn ] asa Pn =(
G n−1 + G n )
2
(G
−
G
)
1
−
,
n
n−1
G∗
2G∗
– 210 –

(17.125)

Chapter 17

Sea icewhere dG = gdh is used. The property a Pn is the fractional contribution from the category n among the total area of ice
subject to ridging. The property G n is the area summed from category 0 to n. This equation is used for the category thatsatisfies G n < G∗ . If G n−1 < G∗ < G n , then G∗ is replaced by G n . If G n−1 > G∗ , then a Pn = 0. If the fractional area ofopen water exceeds G∗ (a0 > G∗ ), then ridging does not occur.
Ridging occurs such that the total area is reduced while conserving ice volume√ and energy. It is assumed that ice ofthickness hn is homogeneously distributed between Hmin = 2hn and Hmax = 2 H ∗ hn after ridging, where H ∗ = 25 m.
The thickness ratio before and after the ridging is k n = (Hmin + Hmax )/(2hn ). Therefore, when an area of category n is
reduced by ridging at a rate r n , the area of thicker categories is increased by r n /k n .
Among the new ridges, the fractional area that is distributed in category m is:
area
=
fm

HR − HL,
Hmax − Hmin

(17.126)

where H L = max(Hm−1, Hmin ) and HR = min(Hm, Hmax ). The fractional volume that is distributed in category m is:
vol
fm
=

(HR ) 2 − (H L ) 2.
(Hmax ) 2 − (Hmin ) 2(17.127)

The snow volume and ice energy are distributed by the same ratio as the ice volume.
The net area lost by ridging and open water closing is assumed to be a function of the strain rates. The net rate of area
loss of the ice pack (Rnet ) is given by
Rnet =

Cs(∆ − |DD |) − min(DD, 0),2(17.128)

where Cs is the fraction of shear dissipation energy that contributes to ridge building (0.5 is used in MRI.COM), DD is[]
2 + 1 (D 2 + D 2 ) 1/2 . These strain rates are computed by the dynamics scheme.
the divergence, and ∆ = DD
2
T
S
e
∑N
The total rate of area loss due to ridging, Rtot = n=0
r n , is related to the net rate as follows:[

Rnet = a P0 +

N
∑

(
1 )]
Rtot .
a Pn 1 −kn
n=1(17.129)

Since Rnet is computed from (17.128), Rtot is computed from (17.129). Thus, the area subjected to ridging from categoryn is computed as ar n = r n ∆t (r n = a Pn Rtot ). The area after ridging is ar n /k n and the volume after ridging is ar n hn .
Using these, the ice subjected to ridging is first removed from category n, and the ridged ice is then redistributed into eachcategory.
In practice, we require that ar n ≤ an . If A > 1 after ridging, Rnet is adjusted to yield A = 1, and the ridging procedure
is repeated.

17.7 Adjustment
As the final procedure in each time step, the following adjustment is applied to the sea ice distribution.
• If the thickness of sea ice of some category is out of its category bounds, the sea ice in that category is moved tothe appropriate category.
• If the thickness of sea ice in category 1 is less than 0.1 meter, the thickness is set to 0.1 meter and the area fractionis adjusted accordingly.
• If the fractional area that sea ice occupies is too small, the sea ice of that category is forcibly melted (See Table
17.8).

17.8 Discretization
17.8.1 Advection (MPDATA)In MPDATA, tracer (α in Figure 17.3) is updated following a three-step procedure.

– 211 –

17.8

Discretization

a. A temporary value is computed using an upstream scheme.

The tracer fluxes at the side boundaries of a T-cell are:
] ∆y
(
) [
n
n
n
n
n
n
n
n
,
Fx αi,n j , αi+1,
,
u
=
α
(u
+
|u
|)
+
α
(u
−
|u
|)
i, j i+ 1 , j
j i+ 1 , j
i+1, j i+ 1 , j
i+ 12 , ji+ 21 , j
2
2
2
2
[
]
)
(
∆x
Fy αi,n j , αi,n j+1, vi,n j+ 1 = αi,n j (vi,n j+ 1 + |vi,n j+ 1 |) + αi,n j+1 (vi,n j+ 1 − |vi,n j+ 1 |)
,
2
2
2
2
2
2

(17.130)

where
)
1(
ui+ 1 , j+ 1 + ui+ 1 , j− 1 ,
2
2
2
2
2
2
)
1(
vi, j+ 1 = vi+ 1 , j+ 1 + vi− 1 , j+ 1 .
2
2
2
2
2
2

ui+ 1 , j =(17.131)(17.132)

The zonal flux is defined at the closed circle and the meridional flux is defined at the closed square in Figure 17.3.
Using this, the temporary value (α ∗ ) is computed using an upstream scheme:(αi,∗ j − αi,n j )∆Si, j∆t((n+ 12 )
n+ 21 )nnn
= Fx αi−1,− Fx αi,n j , αi+1,
j , α i, j , u 1
j, u 1i− 2 , ji+ 2 , j(n+ 1 )
n+ 1 )+ Fy αi,n j−1, αi,n j , v 2 1 − Fy αi,n j , αi,n j+1, v 2 1 .
i, j−
i, j+
(

2

(17.133)

2

b. Compute the anti-diﬀusive transport velocity.

Using the temporary value computed in the first step, the anti-diﬀusive transport velocity is computed as follows:


)
∗
∗

1  1 ( n+ 21
1
n+ 12 2 ( ∂α )n+ 12 ( n+ 1 ∂α )ũi+ 1 , j =  ∗(2) |u 1 |∆x − ∆t(u 1 )− ∗(6) ∆tu 1 v 2
1
1 
i+
i+
,
j
,
j
,
j
,
j
,
j
i+
i+
i+
2
2 α 1
∂x
∂y
2
2 
2
2
2
α 1
 i+ 2 , j
i+ 2 , j1()∂u ∂v n+ 2
1
n+ 1,
− ∆tu 12+i+ 2 , j ∂ x
2∂ y i+ 12 , j


)
∗
∗

1  1 ( n+ 21
1
n+ 12 2 ( ∂α )n+ 12 ( n+ 1 ∂α )ṽi, j+ 1 =  ∗(2) |v 1 |∆y − ∆t(v 1 )−∆tv 1 u 2i, j+ 2i, j+ 2i, j+ 222 α 1∂ y i, j+ 21 α ∗(6) 1∂ x i, j+ 12  i, j+ 2
i, j+ 2() n+ 12
∂u ∂v
1
n+ 1
,
− ∆tv 2 1+
i, j+ 2 ∂ x
2∂ y i, j+ 1

(17.134)(17.135)

2

where
1
∗
α ∗(2)1 = (αi,∗ j + αi+1,j ),i+ 2 , j21
α ∗(2) 1 = (αi,∗ j + αi,∗ j+1 ),i, j+ 2
2
1
∗
∗
∗
∗
∗
α ∗(6)1 = (αi,∗ j−1 + αi+1,
j−1 + 2α i, j + 2α i+1, j + α i, j+1 + α i+1, j+1 ),i+ 2 , j
8
1 ∗
∗
∗
∗
∗
∗
α ∗(6) 1 = (αi−1,
j + α i−1, j+1 + 2α i, j + 2α i, j+1 + α i+1, j + α i+1, j+1 ),i, j+ 28( ∂α ∗ )

=

∗
∗
αi+1,
j − α i, j,
∂ x i+ 12 , j∆xαi,∗ j+1 − αi,∗ j( ∂α ∗ )=,∂ y i, j+ 12∆y– 212 –

(17.136)(17.137)(17.138)(17.139)(17.140)(17.141)

Chapter 17

Sea ice(∗∂α ∗ )
1 [ n+ 1 ( ∂α ∗ )n+ 12 ( ∂α )= v 211
1 + v
1i, j− 2 ∂ y i, j− 21∂ y i+ 2 , j 4 i, j+ 2 ∂ y i, j+ 2∗∗]
n+ 1 ( ∂α )n+ 21 ( ∂α )+ v 211 + v
1 ,1i, j+ 2 ∂ y i+1, j+ 2
i, j− 2 ∂ y i+1, j− 2∗(
[ n+ 1 ( ∂α ∗ )1 ( ∂α ∗ )1 ∂α )
1n++ u 12un+ 2= u 2
i− 2 , j ∂ x i− 12 , j∂ x i, j+ 12 4 i+ 12 , j ∂ x i+ 12 , j]( ∂α ∗ )( ∂α ∗ )n+ 1
n+ 1+ u 12,+ u 121
1
i+ 2 , j+1 ∂ x i+ 2 , j+1
i− 2 , j+1 ∂ x i− 2 , j+1
1

v n+ 2][∂v ) n+ 21
1 ( ∂u ∂v ) n+ 21 ( ∂u ∂v ) n+ 12
+
,
=
+
+
∂ x ∂ y i+ 12 , j 2 ∂ x ∂ y i, j
∂ x ∂ y i+1, j[]( ∂u ∂v ) n+ 21
1 ( ∂u ∂v ) n+ 21 ( ∂u ∂v ) n+ 12
+
+
+
+
,
=
∂ x ∂ y i, j+ 12 2 ∂ x ∂ y i, j
∂ x ∂ y i, j+1
( ∂u

and

(

+

(17.142)(17.143)(17.144)(17.145)( n))n
u 1 ∆y i+ 1 , j − un 1 ∆y i− 1 , j + v n 1 ∆x i, j+ 1 − v n 1 ∆x i, j− 1
i+ 2 , ji− 2 , ji, j+ 2i, j− 2
∂u ∂v
2
2
2
2
+
=
.
∂ x ∂ y i, j
∆Si, j(17.146)c. Update tracer using the anti-diﬀusive velocity starting from the temporary value.
∗(αi,n+1
j − α i, j )∆Si, j

∆t

)
(
)
(
∗
∗
∗
∗
= Fx αi−1,
j , α i, j , ũi− 12 , j − Fx α i, j , α i+1, j , ũi+ 12 , j()()+ Fy αi,∗ j−1, αi,∗ j , ṽi, j− 1 − Fy αi,∗ j , αi,∗ j+1, ṽi, j+ 1 .
2

Ui-1/2, j+3/2

α i+1, j+1

α i, j+1

Ui+3/2, j+1/2

Ui+1/2, j+1/2
α i, jUi-1/2, j-1/2(17.147)

Ui+3/2, j+3/2

Ui+1/2, j+3/2

Ui-1/2, j+1/2

2

α i+1, j
Ui+1/2, j-1/2

Ui+3/2, j-1/2

Figure17.3 Position for tracer (α) and velocity (U). Area and thickness are defined at tracer points. Zonal fluxes arecomputed at closed circles and meridional fluxes are computed at closed squares. The budget is computed for a unit
cell for α.

– 213 –

17.9 Technical Notes

17.8.2 Momentum equation
Specific forms of discretization for properties related to internal stress are given here.
The strain rate tensor (ϵ)˙ and stress tensor (σ) are defined at tracer points (Figure 17.4).
Components (divergence, tension, and shear) of the strain rate tensor are expressed in a discretized form as follows:∆y i− 1 , j( ∆y i+ 12 , j1
2(u I i+ 1 , j+ 1 + u I i+ 1 , j− 1 ) −(u I i− 1 , j+ 1 + u I i− 1 , j− 1 )
2
2
2
2
2
2
2
2
∆x i, j ∆yi, j22∆x i, j− 1
∆x i, j+ 1)
2
2+
(vI i+ 1 , j+ 1 + vI i− 1 , j+ 1 ) −(vI i+ 1 , j− 1 + vI i− 1 , j− 1 ) ,
2
2
2
2
2
2
2
2
2
2

(DD )i, j =(DT )i, j =u I i− 1 , j+ 1 ) ∆yi, j− 1 ( u I i+ 1 , j− 1
u I i− 1 , j− 1 ) ]1 [ ∆yi, j+ 21 ( u I i+ 12 , j+ 12
2
2
2
2
2
2
2
−
+
−
2 ∆x i, j+ 1 ∆y i+ 1 , j+ 1 ∆y i− 1 , j+ 1
∆x i, j− 1 ∆y i+ 1 , j− 1 ∆y i− 1 , j− 1
2

−

2

2

2 ∆yi+ 1 , j ∆x i+ 1 , j+ 1
2(DS )i, j =2

1 [ ∆x i+ 12 , j ( vI i+ 12 , j+ 12
2

−

2

2

2

vI i+ 1 , j− 1 )2

2∆x i+ 1 , j− 1
2

+

2

2

2

2

∆x i, j− 1 ( vI i− 1 , j+ 12

2

2

∆yi, j− 1 ∆x i− 1 , j+ 1
2

2

−

2

2

vI i− 1 , j− 1 ) ]2

2∆x i− 1 , j− 12,

2

vI i− 1 , j+ 1 ) ∆yi, j− 1 ( vI i+ 1 , j− 1
vI i− 1 , j− 1 ) ]1 [ ∆yi, j+ 12 ( vI i+ 12 , j+ 12
2
2
2
2
2
2
2
−
+
−
2 ∆x i, j+ 1 ∆y i+ 1 , j+ 1 ∆y i− 1 , j+ 1
∆x i, j− 1 ∆y i+ 1 , j− 1 ∆y i− 1 , j− 1
2

+

2

2

2

1 [ ∆x i+ 21 , j ( u I i+ 21 , j+ 12
2 ∆yi+ 1 , j ∆x i+ 1 , j+ 1
2

2

2

−

2

2

u I i+ 1 , j− 1 )2

2∆x i+ 1 , j− 1
2

+

2

2

2

2

∆x i, j− 1 ( u I i− 1 , j+ 12

2

2

∆yi, j− 1 ∆x i− 1 , j+ 1
2

2

2

−

2

u I i− 1 , j− 1 ) ]2

2∆x i− 1 , j− 1
2.

2

The internal stress is defined at velocity points and computed from stress tensor as follows:(Fµ )i+ 1 , j+ 1 =2

2

1 [ 1 ( (σ1 )i+1, j+1 + (σ1 )i+1, j − (σ1 )i, j+1 − (σ1 )i, j )2 2∆x i+ 1 , j+ 1
2

1(
+
2

∆y 2i+1, j+ 1 [(σ2 )i+1, j+1 + (σ2 )i+1, j ] − ∆y 2i, j+ 1 [(σ2 )i, j+1 + (σ2 )i, j ] )22∆y 2i+ 1 , j+ 1 ∆x i+ 1 , j+ 1
2

+

2

2

2

2

2

2
2
( ∆x i+ 12 , j+1 [(σ12 )i+1, j+1 + (σ12 )i, j+1 ] − ∆x i+ 12 , j [(σ12 )i+1, j + (σ12 )i, j ] )∆x 2i+ 1 , j+ 1 ∆y i+ 1 , j+ 1
2(Fψ )i+ 1 , j+ 1 =(17.148)

2

2

2

,

2

1 [ 1 ( (σ1 )i+1, j+1 + (σ1 )i, j+1 − (σ1 )i+1, j − (σ1 )i, j )2 2∆y i+ 1 , j+ 1
2

2

∆x
[(σ )
+ (σ2 )i, j+1 ] − ∆x i+ 1 , j [(σ2 )i+1, j + (σ2 )i, j ] )1 ( i+ 12 , j+1 2 i+1, j+1
2
2

−

2

2

∆x 2i+ 1 , j+ 1 ∆y i+ 1 , j+ 1
2+

(17.149)

2

2

2

2
2
( ∆y i+1, j+ 12 [(σ12 )i+1, j+1 + (σ12 )i+1, j ] − ∆y i, j+ 12 [(σ12 )i, j+1 + (σ12 )i, j ] )∆y 2i+ 1 , j+ 1 ∆x i+ 1 , j+ 1
2

17.9 Technical Notes
17.9.1 Source codesa. Program files

The sea ice part consists of the following programs.
– 214 –

2

2

2

.

Chapter 17

Sea iceUi-1/2, j+3/2

Ui+1/2, j+3/2

σi, j+1,εi, j+1
Ui-1/2, j+1/2

σi+1, j+1,εi+1, j+1
Ui+3/2, j+1/2

Ui+1/2, j+1/2
σi, j,εi, j

Ui-1/2, j-1/2

Ui+3/2, j+3/2

σi+1, j,εi+1, j
Ui+1/2, j-1/2

Ui+3/2, j-1/2

Figure17.4 Position of variables used by dynamics scheme.

ice_param.F90:

defines basic constants (This file is created by running the script configure on the basis of
the template ’ice_param.F90.in’. The number of thickness categories (NUM_ICECAT) mustbe specified in configure.in before running the script.)
ice_aggr_vars.F90: defines variables averaged over all categories for use in dynamics and ocean model
ice_cat_com.F90:
defines common variables and arrays
ice_cat_albedo.F90: computes surface albedo and fraction of shortwave penetrating into ice
ice_cat_bulk.F90:
computes surface fluxes using bulk formula
ice_cat_nest.F90:
contains subroutines relevant to nesting
ice_date.F90:
computes date and time
ice_dyn.F90:
computes ice dynamics
ice_flux.F90:
computes air-ice interface processes
ice_grid.F90:
sets grid cells (substitution from ocean model)
ice_hist.F90:
computes and outputs history
ice_main_cat.F90:
calls subroutines (main part of the ice model)
ice_mpdata.F90:
computes advection term using MPDATA
ice_remapv.F90:
remaps ice into thickness categories after thermodynamics
ice_restart.F90:
reads and writes restart
ice_ridge.F90:
computes ridging
ice_time.F90:
manages calendar
mod_seaice_cat.F90: computes thermodynamics and various adjustment processes

b. Structure

The ice model can be only used as a part of the ocean model. Among the subroutines included in the above programs, only
subroutines in ice_main_cat.F90 are called from surface flux routines of the ocean model. Relation with the OGCM is
as follows.
ogcm__ini (ogcm.F90)|+ surfflux_ctl__ini (surfflux_ctl.F90)|+ si_initialize (ice_main_cat.F90)... initializes the ice model by reading parameters and creating grid cells
ogcm__prepare (ogcm.F90)– 215 –

17.9 Technical Notes|+ surfflux_ctl__read_restart (surfflux_ctl.F90)|+ si_read_restart (ice_main_cat.F90)... reads restart and sets initial stateogcm__run (ogcm.F90)|+ LOOP_TIME_STEP|||+ part_1|||||+ surfflux_ctl__main (surfflux_ctl.F90)
|
|
|
|
|
+ surfflux_seaice (surfflux.F90)
|
|
|
|
|
+ simain (ice_main_cat.F90)||... the main program and calls subroutines of the ice model|||+ surfflux_ctl__write_restart (surfflux_ctl.F90)|||+ si_write_restart (ice_main_cat.F90)!
... writes restart|+ surfflux_ctl__run_end (surfflux_ctl.F90)|+ si_run_end (ice_main_cat.F90)... clears clock

Other things to be noted are:
• The sea ice model uses the forward scheme in time integration. The sea ice model is not called in the backward
part of the Matsuno (Euler backward) scheme.
• History of the basic state (integrated or averaged over all categories) might be monitored by the history routine of
the OGCM (specify namelist nmlhs_ice and nmlhs_iceuv).

17.9.2 Coupling with an atmospheric modela. General features

In coupled mode (SCUPCGCM), the boundary between the atmospheric component and the ocean-ice component is at the
air-ice(snow) boundary. The fluxes above the air-ice boundary are computed by the atmospheric component and passed
to the ocean-ice component via the coupler Scup (Yoshimura and Yukimoto, 2008). All the information needed by
the oceanic component is received by calling cgcm_scup__get_a2o (cgcm_scup.F90) from ogcm__run (ogcm.F90)
at the beginning of a coupling cycle. The information needed by the ice part is extracted by calling get_fluxi_a2o(get_fluxs.F90) from iaflux (ice_flux.F90). The main part of the ice is solved using surface fluxes and ice surface
temperature from the atmospheric component.
In the atmospheric component, temperature in the atmospheric boundary layer and at the ice surface (T3 (tsfcin)) arecomputed along with heat flux in snow layer (Q S = QI2 (fheatu)) using ice temperature (T1 (t1icen)), snow thickness(hs (hsnwn)), and ice thickness (h I (hicen)) given by the oceanic component (see Figure 17.1 and Table 17.1). The
properties needed by the atmospheric component are sent via cgcm_scup__put_o2a (cgcm_scup.F90) from ogcm__run(ogcm.F90) at the end of a coupling cycle.
To conserve heat and water in the coupled atmosphere-ocean system, globally integrated heat and fresh water flux mustmatch between the atmospheric and oceanic components. To achieve this, surface fluxes are adjusted in several steps,
which are explained in the following.

– 216 –

Chapter 17

Sea iceb. Correct errors due to interpolation
transFirst, to resolve errors that arise during the transformation process, a globally constant adjustment factor for heat flux qadj
is introduced so that
∫ (∑
ncat
)
trans
anini qn + qadj
dS = Qatm(17.150)
ocean,
n=0

where anini is the area fraction of sea ice for category n at the time of exchange, qn is the net heat flux for category n, andQatm
ocean is the net heat flux of the atmospheric model integrated over the ocean. The adjustment factor is given by[trans= Qatmqadj
ocean −

∫ ∑
ncat

]
anini qn dS /Socean,(17.151)n=0trans is added to the net longwave radiation flux.
where Socean is the global ocean area of the ocean model. qadj
trans
For fresh water, f adj is introduced so that

∫ (∑
ncat

evap

anini f n

)
trans
atm
+ f prcp + f rof + f adj
dS = Focean,(17.152)n=0
evap

where anini is the area fraction of sea ice for category n at the time of exchange, f n is the evaporation or sublimation for
atm is the net fresh water of the atmosphericcategory n, f prcp is the precipitation, and f rof is the continental run oﬀ, and Focean
model integrated over the ocean. The adjustment factor is given by
∫ (∑
ncat
) ]
[ atm
evap
trans
anini f n + f prcp + f rof dS /Socean,(17.153)f adj = Focean −n=0trans is added to either precipitation or evaporation according
where Socean is the global ocean area of the ocean model. f adj
to its sign.
This operation is done at subroutine adjust_fluxes_global of get_fluxs.F90.

c. Taking into account of the evolution of sea ice state

Second, as the ocean-sea ice system evolves during a coupling cycle, the area fraction of sea ice changes from the one at
the time of data exchange. Therefore, the global integral would be diﬀerent from the initial state. To resolve this diﬀerence,
evo is introduced so that
another adjustment factor qadj
∫ (∑
ncat

evo
anevo qn + qadj

)

dS =

∫ ∑
ncat

n=0

anini qn dS,(17.154)n=0

where anevo is the evolved area fraction of sea ice for category n. The adjustment factor is given as follows:
∫ ∑
ncat
ncat
[∫ ∑
]
evo
=
qadj
anini qn dS −anevo qn dS /Socean .
n=0(17.155)n=0trans + q evo is added to the longwave radiation flux.
In total, qadj
adj
LW
LW
trans
evo
qadj
= qatmn+ qadj+ qadjn(17.156)

evo is introduced so
For fresh water, because evaporation or sublimation may be aﬀected by the evolution of sea ice, f adj
that
∫ (∑
∫ ∑
ncat
ncat
)
evap
evap
evo
anevo f n + f adj
dS =anini f n dS,(17.157)n=0

n=0

where anevo is the evolved area fraction of sea ice for category n. The adjustment factor is given as follows:
∫ ∑
ncat
ncat
[∫ ∑
]
evap
evap
evo
anini f n dS −anevo f n dS /Socean .
f adj
=
n=0

n=0

evo is added to either evaporation or precipitation according to its sign.
f adjThis operation is done at subroutine adjust_fluxes_local of get_fluxs.F90.

– 217 –

(17.158)

17.9 Technical Notes

d. Correct errors due to solution method

Third, discrepancies due to solution methods adopted at the air-ice interface (Section 17.2.3) are corrected. To ensure astable model integration, the ice surface temperature is set as an average of those of the atmospheric component and theocean-ice component. This new temperature T3o is used to replace the upward longwave flux of the adjusted heat flux
AI ) that contains the net longwave flux (q LW ). Because the longwave flux q LW contains theabove the air-ice interface (qadj
adj
adj
′

LW is replaced with q LW in the following manner,
upward flux due to T3a of the atmospheric model, qadj
adj
′

LW
LW
− ϵ σ(T3a ) 4 + ϵ σ(T3o ) 4 .
qadj= qadj(17.159)′′

AI that contains q LW , theFor the surface where snow or ice is melting, the amount of melting is calculated using qadj
adj
discrepancy of the longwave radiation over melting surface should be stored for flux correction.
For the surface where melting does not occur, the interior flux (Q S or QI2 , they are equal), is obtained by equating′
AI
AI owing to the adjustment for the longwave radiation explained above. Furthermore,
qadj and Q S . Q S is diﬀerent than qadj′

AI and Q are not completely the same owing to the approximations used in the solution method explained in Section
qadj
S
17.2.3. This is not a serious problem for an ocean-only simulation. But this should be taken into account for the coupledmodel where exact surface heat flux conservation is required. Thus the diﬀerence between the adjusted air-ice flux and theAI − Q , should be stored.
interior flux, qadjS
To summarize, discrepancy of fluxes are stored in qdiﬀ in the following manner,
′

AI
AI
LW
LW
qdiﬀ = qadj
− qadj= qadj(T3a ) − qadj
(T3o ),
AI= qadj− Q S,

if surface is melting(17.160)

if surface is not melting.

(17.161)

stable , which
This is integrated over sea ice and divided by the oceanic area of the ocean model to give an oﬀset factor qadj
is added to the oceanic surface heat flux globally,

stable
qadj
=

∫ ∑
ncat

an (qdiﬀ )n dS/Socean .

(17.162)

n=1

Sampling is done at subroutine iaflux of ice_flux.F90 and flux adjustment is done at subroutine si_exit ofmod_seaice_cat.F90.

17.9.3 Nestinga. One-way nesting

In one-way nesting, it is recommended to set the boundary between coarse and fine resolution models (parent and childrespectively) along the line connecting tracer points. Setting velocity points as the boundary is awkward for several
reasons. For example, the velocity of the child model is zero at the boundary if sea ice of the parent model does not exist
around boundary. However, sea ice may exist inside the boundary in the child model in one-way nesting. Lacking thetransport removing sea ice from the child model through the boundary, sea ice may accumulate at tracer cells adjacent tothe boundary. A serious discontinuity may result. If tracer points are set as the boundary, this kind of accumulation will
not occur. However, conservation does not hold.
b. Two-way nesting

In two-way nesting, it is necessary to set velocity points as the boundary to impose global conservation including sea ice.
By reflecting the child model state to the parent model, the problem seen in one-way nesting, an inconsistent distribution ofsea ice around the boundary, may be avoided. However, owing to the application of the flux adjustment for conservation,
the positive definiteness of area fraction and volume of sea ice and snow, or negative definiteness of sea ice enthalpy are
not guaranteed after the advection equation is integrated.
This problem is avoided by the following treatment on the transport fluxes. First, the child model flux is adjusted tomatch the parent model flux (Chapter 18). Second, if the original flux calculated by the child model is going out of childmodel’s main region, the flux received from the parent model is replaced by this outgoing flux. Third, the diﬀerence
between the outgoing flux and the parent model flux is sent to the parent model. Fourth, if the value just inside theboundary of the child model is predicted to be negative, the boundary flux is adjusted so that the value does not get– 218 –

Chapter 17

Sea icenegative (a certain amount of quantity is taken from the parent). The adjustment factor is sent to the parent model. Fifth,in the parent model, the flux corrections received from the child model are summed over the corresponding parent gridand added to the original flux of the parent model.
Another problem arises for the interior temperature of sea ice. Owing to the discrepancy between the thermal energy
and the volume of the ice pack after advection, the interior temperature of sea ice layer may get erroneous. Therefore, onboth sides of the parent-child boundary, interior temperature of sea ice is calculated as the average of the top and bottom
surfaces of the sea ice,
T2 + T0
adj.
(17.163)
T1 =2
The diﬀerence between the predicted and adjusted ice energy is integrated along the parent-child boundary (Γ) as
Qdiﬀ =

∫ ∑
ncat
Γ n=1

[
]
adj adj
ρ I an hn E(T1 , r 1 ) − E(T1, r 1 ) dS.

(17.164)

This is reflected as a correction factor for the ocean surface heat flux of each model, which is computed asthermalcore
qadj= Qdiﬀ /Socean,(17.165)

core is the area of the main region of the parent or child model.
where Socean

17.10 Usage notes
17.10.1 Compilation
The information needed to compile the sea ice model with the OGCM should be given to configure.in. At least, ICE
option should be specified in the OPTIONS line and the number of categories of the sea ice thickness should be specified as
NUM_ICECAT, for example, a line NUM_ICECAT = 5 should be inserted somewhere in configure.in. The model options
related to the sea ice model are listed on Table 17.3.
Table17.3 Model options related to the sea ice model

option name
ICE
SIDYN

description
Sea ice model is included to the OGCM
Dynamics of sea ice is solved

CALALBSIHISTICECATAn albedo scheme of CICE is used
Monitor time evolution of basic state variables

ICEFULLMONIT

Activate extensive monitor

usage
otherwise, sea ice drifts with a third of the surface
ocean velocity.
otherwise constant
Use NAMELIST.name_model.MONITOR to specify theoutput group and interval. name_model is the nameof the model specified by configure.in
Use NAMELIST.name_model.MONITOR to specify theoutput group and interval

17.10.2 Job parameters (namelist)
The runtime job parameters (namelist) that should be given by NAMELIST.name_model are listed on Tables 17.4 to 17.11.
Job parameters related to monitoring are specified in the next subsection.
a. Thickness category

The bounds of thickness categories, whose number should be specified by NUM_ICECAT in configure.in before compilation, should be given at run time to namelist nml_seaice_thickness_category, which is explained in Table
17.4.

– 219 –

17.10 Usage notes
Table17.4 Namelist of thickness category of the sea ice model (nml_seaice_thickness_category)variable namehbound(0 : ncat)
lsicat_volchk

dimensionm
logicaldescription
category boundary
flag for checking mass conservation

usage (default value)
requireddefault = .false.

b. Time

The parameters related to time should be given by nml_seaice_time_conf and nml_seaice_time_calendar, which are
explained in Table 17.5. Note that the specification of nml_seaice_time_conf is not mandatory, necessary information
will be taken from the ocean model.
Table17.5 Namelist related to time of the sea ice model

variable namenstep_seaice
ibyri
ibmni
ibdyi
ibhri
ibmii
ibsci
l_force_leap

group
nml_seaice_time_conf
nml_seaice_time_conf
nml_seaice_time_conf
nml_seaice_time_conf
nml_seaice_time_conf
nml_seaice_time_conf
nml_seaice_time_conf
nml_seaice_calendar

l_leapnml_seaice_calendar

description
time steps to be proceedstart year of this run
start month of this run
start day of this run
start hour of this run
start minute of this run
start second of this run
forcibly specify leap year atrun time with l_leap
specification of whether thisyear is leap or not (valid whenl_force_leap = .true.)

usage (default value)same as OGCM
same as OGCM
same as OGCM
same as OGCM
same as OGCM
same as OGCM
same as OGCM
.false.
.false.

c. Dynamics

The parameters related to dynamics of sea ice (SIDYN option) should be given to namelist nml_seaice_dyn, which islisted on Table 17.6
Table17.6 Namelist related to dynamics of the sea icel (nml_seaice_dyn)

variable namedt_seaice_dyn_sec

dimension
secdescription
time step interval for dynamics (in seconds)

damp_vicem sec−1

damp_vmax

m sec−1

Rayleigh damping coeﬃcient for rapid velocity
minimum velocity for damping

usage (default value)
about a tenth of the baroclinic
time step of the ocean model
0.0
1.0

d. Diﬀusion

Coeﬃcient for the harmonic-type horizontal diﬀusion should be given to namelist nml_seaice_diff, which is explainedin Table 17.7.
Table17.7 Namelist of diﬀusion of the sea ice model (nml_seaice_diff)variable name
diff_seaice_m2ps

dimension
m2 sec−1

description
horizontal diﬀusion coeﬃcient

usage (default value)
requirede. Minimum fractional area

Minimum of fractional area applied to all thickness categories may be given at run time to namelist nml_seaice_model,
which is explained in Table 17.8.

– 220 –

Chapter 17

Sea ice
Table17.8 Namelist of minimum fractional area of the sea ice model (nml_seaice_model)

variable nameconcentration_min

dimension
1

description
minimum fractional area

usage (default value)
1 × 10−4

f. Air-ice transfer

The bulk transfer coeﬃcient at the air - sea ice interface should be given to nml_air_ice, which is listed on Table 17.9.
Table17.9 Namelist of air-ice bulk transfer coeﬃcient (nml_air_ice)variable name
chaiw
chait
chaie

dimension1
1
1

description
air-ice exchange coeﬃcient for windair-ice exchange coeﬃcient for sensible heat
air-ice exchange coeﬃcient for evaporation

usage (default value)
3.0 × 10−3
1.5 × 10−3
1.5 × 10−3

g. Restart

How to handle restart files is specified by namelists nml_seaice_run, nml_seaice_rst_in, and nml_seaice_rst_out,
which is explained in Table 17.10
Table17.10 Namelist related to restart of the sea ice model

variable namel_rst_seaice_in

group
nml_seaice_run

description

file_icecat_restart_in

nml_seaice_rst_in

nwrt_rst_ic

nml_seaice_rst_out

file_icecat_restart_out_temp

nml_seaice_rst_out

restart file name of categorized ice for input
the interval of time stepfor snap shot output
core part of the file namefor snap shot

usage (default value).true. : the initial stateis read from restart file.false.(default) : startfrom the state without sea
ice
required
required
requiredh. Albedo

When CALALBSI option is selected, an albedo scheme of CICE is used. The parameters of this scheme should be given bynamelist nml_albedo_seaice, which is listed on Table 17.11.
Table17.11 Namelist for the albedo of sea ice used by CALALBSI option (nml_albedo_seaice)variable namealb_ice_visible_t0
alb_ice_nearIR_t0
alb_snw_visible_t0
alb_snw_nearIR_t0
alb_ice_visible_dec_ratio
alb_ice_nearIR_dec_ratio
alb_snw_visible_dec_ratio
alb_snw_nearIR_dec_ratio

dimension(◦C) −1(◦C) −1(◦C) −1(◦C) −1

description
visible ice albedo for thicker ice
near infrared ice albedo for thicker ice
visible, cold snow albedo
near infrared, cold snow albedo
visible ice albedo declination rate
near infrared ice albedo declination rate
visible snow albedo declination rate
near infrared ice albedo declination rate

– 221 –

usage (default value)
0.78
0.36
0.98
0.70
0.075
0.075
0.10
0.15
Continued on next page

17.10 Usage notes

variable namehi_refatan_reftsfci_t0
tsfci_t1
fsnow_patch

Table 17.11 – continued from previous page
dimension
descriptionmthe maximum ice thickness to which connection function is used
the base value of the tangent hyperbolic connection function◦C
the temperature at which ice albedo isequated to that of ocean◦Cthe temperature at which ice albedo isstarted to decline to that of the ocean
meter
thickness of snow patch on melting bare ice

usage (default value)
0.50
4.0
0.0
−1.0
0.02

17.10.3 Monitoring
To monitor or sample the state of sea ice, HISTICECAT option must be specified. Extensive monitoring is activated byfurther specifying ICEFULLMONIT option.
The general behavior should be specified by the nml_seaice_budget and nml_seaice_hst in NAMELIST.name_model,
which is explained in Table 17.12.
Table17.12 Namelist of monitor of the sea ice model

variable namenstep_si_budget_intervalgroup
nml_seaice_budget

undef_value

nml_seaice_hst

undef_for_land

nml_seaice_hst

description
the interval of time step bywhich water budget is written
undefined value to fill ice freegrid point
undefined value to fill land
grid points

usage
0 : none, −1 : monthly
real(8)real(8)Sampling variables are grouped according to the processes they are involved, the grid points they are defined, and thepriorities given by Sea Ice Model Intercomparison Project (SIMIP) (Notz et al., 2016).
The groups are listed on Table 17.13. Users specify nml_history blocks in file NAMELIST.name_model.MONITOR in
the following manner.
&nml_history
namefile_baseinterval_step
suffix&end

Specification of NAMELIST.name_model.MONITORforseaicemonitoring= ’sea ice basic state vars at t-point’,= ’result/hs_ice_t_sta1’,= 12,= ’minute’,
Table17.13 List of groups for sea ice monitoringgroup namesea ice basic state vars at t-point
sea ice state vars at t-point
tendencies of sea-ice mass and area fraction at t-point
heat and freshwater fluxes over sea ice fraction at t-point
sea ice state vars of each category at t-point
sea ice basic dynamics at u-point

usage note"mean" is forced"mean" is forced"mean" is forced"mean" is forced"mean" is forced"mean" is forced
Continued on next page– 222 –

Chapter 17

Sea ice

Table 17.13 – continued from previous pagegroup nameusage notesea ice dynamics at u-point"mean" is forced
sea ice dynamics at t-point"mean" is forced
sea ice vars at x-point
sea ice vars at y-point
sea ice integrated measures
sea ice dynamics at u-point priority 3
valid if ICEFULLMONIT, "mean" is forced
sea ice dynamics at t-point priority 3
valid if ICEFULLMONIT, "mean" is forced
sea ice fluxes of each category at t-point
valid if ICEFULLMONIT
heat and freshwater fluxes over open leads at t-point
valid if ICEFULLMONIT

17.11 Appendix
17.11.1 Partial derivative of specific humidity with respect to temperature
In MRI.COM surface temperature of ice is computed using semi-implicit method, where an expression for the partial
derivative of the saturation specific humidity with respect to temperature (t) is needed as in (17.35).
Properties of moist air used by MRI.COM is based on Gill (1982) and replicated in Section 14.12.3. Using (14.117),
the saturation specific humidity (qi ) is given asqi = ϵ esi′ /(ps − (1 − ϵ )esi′ ).

(17.166)

where saturation partial pressure of moist air over ice (esi′ ) is given by (14.123) and (14.124). Using (17.166),
∂esi′
ϵ ps
∂qi
=
,
′
2
∂t
{ps − (1 − ϵ )esi } ∂t(17.167)

where ∂esi′ /∂t is expressed by setting f w = 1 as∂esi′
= ln 10 · 10g(t) · g ′ (t),∂t(17.168)

where g(t) = (0.7859 + 0.03477t)/(1 + 0.00412t) + 0.00422t and g ′ (t) = ∂g(t)/∂t, where t is temperature at the uppersurface of sea ice in ◦C.

17.11.2 Physical constant, parameters
Since the ice part is coded in SI units, constants and parameters are written in SI units.
a. Thermodynamics

Table17.14 Physical constants and parameters relevant to sea ice thermodynamics

parameter
Thermal ice conductivity
Thermal snow conductivity
Specific heat of sea waterSpecific heat of air
Specific heat of icevaluek I = 2.04 J m−1 s−1 K−1
k s = 0.31 J m−1 s−1 K−1
Cpo = 3990 J kg−1 K−1Equation (14.125)
Cpi = 2093 J kg−1 K−1– 223 –

variable name in MRI.COM
cki
cks
cp0
cpair
cpi
Continued on next page

17.11
Table 17.14 – continued from previous pagevalue
Specific heat of snow
Cps = 0.0 J kg−1 K−1
Stefan Boltzmann constantσ = 5.67 × 10−8 W m−2 K−4
Albedo of open ocean surface
α o = 0.1 (default)
Albedo of ice
α I = 0.6 (default)
Albedo of snow
α s = 0.75 (default)
Emissivity of ocean surface
ϵ o = 1.0
Emissivity of ice surface
ϵ I = 1.0
Emissivity of snow surface
ϵ s = 1.0
neutral bulk transfer coeﬃcient for heat
CHAI = 1.5 × 10−3 (default)
neutral bulk transfer coeﬃcient for moisture
CEAI = 1.5 × 10−3 (default)
Latent heat of fusion
L F = 3.347 × 105 J kg−1
Latent heat of sublimation
Equation (14.127)
Constants for fusion phasem = −0.0543 ◦C/pss
equation: T f = mS + nz
n = −0.000 759 ◦C m−1
Ice roughness parameterz0 I = 0.05h I /3
Salinity of sea ice
SI = 4.0 pss
von Karman’s constantk = 0.4
Thickness/compactness diﬀusion of iceκ H = 1.0 × 103 m2 s−1
Seawater kinematic viscosityν = 1.8 × 10−6 m2 s−1
Seawater heat diﬀusivity
α t = 1.39 × 10−7 m2 s−1
Seawater salinity diﬀusivity
α b = 6.8 × 10−10 m2 s−1
Turbulent Prandtl number
Prt = 0.85
b in eqs (17.57),(17.59)
b = 3.14
parameter

Appendixvariable name in MRI.COM
—
stbl
albw
albi
albs
eew
eei
ees
chaih
chaiw
alf
rlth
xmxm
xnxn
z0
si
xk
akh
anu
at
as
prt
ab

b. Dynamics

Table17.15 Physical constants and parameters relevant to sea ice dynamics

parameter
Density of sea water
Density of air
Density of sea ice
Reference snow density (ratio between snow andwater)
e-folding constant for ice pressurepressure scaling factor
drag coeﬃcient (air-ice)drag coeﬃcient (ice-ocean)yield curve axis ratio
scaling factor for Young’s modulus
water turning angle
air turning anglevalueρo = 1036 kg m−3
ρ a , Equation (14.128)
ρ I = 900 kg m−3
ρs /ρo= 330 kg m−3 /1036 kg m−3c∗ = 20.0
P∗ = 2.75 × 104 N m−2
CDAI = 3.0 × 10−3 (default)
CDIO = 5.5 × 10−3
e = 2.0
Eo = 0.25
θ o = ±25° (positive/negative in thenorthern/southern hemisphere)θ a = 0° (not considered)– 224 –

variable name in MRI.COM
ro
roair
rice
rdsw
cstar
prsrefchaiw
cdrgiwelips
eyoungwiangl–Chapter 18

Nesting
In MRI.COM, embedding of a fine-resolution regional model within a coarse-resolution model can be realized by nesting
method. The available methods range from a serial execution of a coarse-resolution (parent) model writing a boundary data
and a fine-resolution (child) model reading it (oﬀ-line one-way nesting) to a parallel execution of both models exchanging
data in both directions (on-line two-way nesting), as summarized in Table 18.1.
This chapter explains facilities available in MRI.COM for nesting. How to construct and run a set of coarse- and fineresolution models in nesting are also explained. Usage of nesting in the sea ice model and the tidal scheme is described in
Section 17.9.3 and 6.5.3, respectively.

18.1 Feature
In a set of nested grid models, a fine-resolution (child) model is embedded in a coarse-resolution (parent) model. In
one-way nesting, values at the side boundary of the child model are given by the parent model. The side boundary data
may be transferred both oﬀ-line and on-line. In oﬀ-line mode, the data needed to calculate side boundary values are
written to files by first running the parent model, and the child model is executed reading these data and calculating the sideboundary values by interpolation. In on-line mode, parent and child models are run in parallel. Simple coupler (Scup),
which is originally developed by Yoshimura and Yukimoto (2008) for the communication among components of the MRI
Earth System Model, is used to transfer data.
In two-way nesting, in addition to the transfer of the side boundary data from parent to child model, the result of thechild model is reflected to the parent model in the embedded region. Data are exchanged on-line and Scup is used for datatransfer.
The boundary between parent and child models may be formed by connecting either the tracer points or the velocity
points of the Arakawa B-grid arrangement (Figure 18.1). Table 18.1 shows the side boundaries suited for the available
nesting methods. From experiences, we recommend setting the tracer points as the side boundary for one-way nesting(imposing Dirichlet boundary condition for the sea surface height and tracer equations) and the velocity points as the sideboundary for two-way nesting (imposing Neumann boundary condition for the sea surface height and tracer equations).
In one-way nesting, the child model is quite stable when the side boundary is set to the tracer points. In two-way nesting,
imposing the conservation on water volume and tracer content for the set of parent and child models is more straightforward
when the side boundary is set to the velocity (flux) points.
To be able to treat both side boundary conditions with a single topographic configuration, the domain of a child model
is constructed by taking tracer points as the boundary of the main region (Figure 18.1a). Inside the boundary, the same
topography as the parent model should be given for at least one parent grid cell. By doing so, we may use the velocity
points located inside the boundary by a half parent grid width as the parent-child boundary for two-way nesting (Figure
18.1b). The model domain needs additional two rows or columns of velocity cells outside the main region along each sideboundary (hereinafter referred to as halo; the region shaded by dark green in Figure 18.1a).
Whether the side boundary of the child model is the tracer points or the velocity points, three columns or rows of theparent model are suﬃcient for a child model to fill both halo and boundary regions. Here, the boundary region refers to a
Table18.1 Nesting methods available in MRI.COM and the recommended side boundary of a child model in eachmethod. It should be noted that the region and the topography of a child model must be made as if tracer-points areused as the side boundary.

model run
one-way nest
two-way nest

on-line (coupler)
tracer point
velocity point– 225 –

oﬀ-line (file I/O)
tracer point
unavailable18.2 Grid and Topography
region inside the boundary of the main region by one grid of the parent model. For example, T1 − T3 and U1 − U3 in Figure18.1 are suﬃcient for filling the southern boundary (except for one-to-one nesting). In oﬀ-line one-way nesting, the parent
model outputs the three columns or rows to files, and the child model reads them. In on-line nesting, the mapping tablebetween parent and child grids should be prepared for the Scup library. The Scup library exchanges only necessary data
among MPI processes of diﬀerent models.
In this way, prognostic variables at the side boundary of the child model are directly replaced by those from the parent
model. This is usually called the clamped method (Cailleau et al., 2008). This method does not guarantee conservation of
tracers in contrast to the method where fluxes are imposed at the boundary, but the integration is quite stable.
(a)

(b)

U3

U3

T3
T2
T1

T3
U2

T2

U1

T1

×：tracer

U2
U1●：velocity

×：tracer●：velocity

Figure18.1 Relation between parent (large symbols) and child (small symbols) grid points. The region shaded withblue (green) is the main region of the child (parent) model. The boundary of the main region is indicated by the redlines. In (a), the tracer points, and in (b), the velocity points, are set as the boundary between the two models. The
domain of the child model should be constructed to include the dark green area, which is expanded outward by two
velocity cells from the tracer point boundary (red line in (a)). This expanded region is used as halos for the main region.
Variables in the halo region are obtained by using values of the parent model. Thus, for the southern boundary, threerows (T1 - T3 and U1 - U3 ) are sent to the child model from the parent model in both case (a) and (b). In two-way nesting,variables in the main region of the child model shown in (b) are sent to the parent model, except on the boundary line
(red line).

Multistage nesting is available in MRI.COM, and on-line nesting and oﬀ-line nesting can be used together. This
flexibility makes various kinds of experiments possible, e.g., a double on-line nesting experiment using global, NorthPacific, and Japanese coastal models, or an oﬀ-line one-way nesting experiment of a Japanese coastal model reading theside boundary values from an on-line nesting experiment for a set of global and North-Pacific models.

18.2 Grid and Topography
18.2.1 Parent (coarse-resolution) model
There is no particular issue in constructing the parent model. Because the grid size ratio between the child and parent
models must be odd, the grid size of the child model may be taken into account in determining the grid size of the parent
model. A grid size ratio of 1:3 or 1:5 is recommended.

18.2.2 Child (fine-resolution) modela. Horizontal grid size

The same grid size as the parent model should be used in the vertical direction. The child model is embedded in thehorizontal directions. The distance between the velocity points of the parent model (i.e., T-cell) is divided equally in thechild model.
– 226 –

Chapter 18

Nesting

The domain of a child model is constructed by taking the tracer point as the boundary of the main region and adding
the halo region, even if the velocity-point boundary will be used. Here, the halo region is two rows or columns of velocity
cells outside the boundaries as shown in Figure 18.1a. There should be always two marginal velocity cells outside themain region at the western and southern boundaries, even if the western or southern boundary is filled by land and does
not receive data from the parent model. However, if the eastern or northern boundary is filled by land and does not receive
boundary data, one marginal velocity cell will be enough.
How to divide parent cells"dxt" of the parent model (U-point distance, T-point at its center) should be
divided equally in the child model.
There should be two marginal U-points in the child model(one cell will do if the eastern or northern boundary is filled by land)***** parent (coarse-resolution) model *****"ifst" (variable name is sub_region_ifirst) is the western end grid point numberof the parent model to be used for interpolation.
boundary|(i=ifst)(i=ifst)(i=ifst+1) (i=ifst+1)<------- dxt -------> U <------- dxt -------> U||-----------+-----------|-----------+-----------|---|
T <------- dxu -------> T <------- dxu -------> T(i=ifst)(i=ifst)(i=ifst+1) (i=ifst+1)|***** child (fine-resolution) model *****(nesting ratio is 1:3)|
(1) (1) (2) (2) (3) (3)<-dxt->U<-dxt->U<-dxt->U<-dxt->U|---+---|---+---|---+-T<-dxu->T<-dxu->T<-dxu->T(1) (1) (2) (2) (3) (3)b. Topography of the child model

To achieve flux conservation, the child model should have the same topography as the parent model around the sideboundary of the child model. It is recommended that the topography of the child model should have the same topography
as the parent model for two parent velocity cells inward and one parent velocity cell outward relative to the boundary of
the main region constructed by connecting tracer points (Figure 18.2).
c. Weighting ratio between parent and child models

For smoothness, prognostic values of the child model around the side boundary may be given as a weighted average of thesolutions of the parent and child models. For variable ϕ at a grid point with index (i, j),p

ϕchild= αi j ϕi j + (1 − αi j )ϕicj ,ij(18.1)

where ϕchild is the value for the child model, ϕ p is the parent model solution interpolated on the child grid, ϕc is the childmodel solution, and α is a spatially dependent weight between parent and child solutions.
Although this weight is introduced for a smooth transition from parent to child model solution, we recommend that theweight for the parent model to be unity in the halo and at the boundary of the main region and zero elsewhere. Instead,this weight is used to specify the boundary between the main regions of parent and child models. When the boundary isat the tracer points, the weight for the parent solution should be unity in the dark green region of Figure 18.1a. When theboundary is at the velocity points, the weight for the parent solution should be unity in the dark green region of Figure
18.1b. Weight must be prepared separately for the tracer and velocity points. (Incidentally, a radiation boundary condition
may be imposed by elaborating weight values.)– 227 –

18.3

×：tracerIntermodel transfer●：velocity

Figure18.2 Condition on the topography of the child model around the side boundary. The topography of the childmodel should have the same topography as the parent model for two parent velocity cells inward (dark blue) and one
parent velocity cell outward (light blue) relative to the boundary of the main region constructed by connecting tracer
points (red rectangle).

18.3 Intermodel transfer
Communication between models is both downscaling and upscaling in two-way nesting, whereas only downscaling in
one-way nesting.

18.3.1 Downscaling
In the downscaling from parent to child model, the method of data transfer and interpolation is diﬀerent between tracer
point and velocity point.
a. Tracer points

It is straightforward to do a linear interpolation for variables on tracer points. Figure 18.3 shows the relative positionsof tracer points of parent and child models. Because topography is common between parent and child models aroundthe side boundary, all child tracer points in an ocean velocity cell of the parent model (bounded by the red rectangle) areocean points. Values on these child tracer points are computed by using values at the four corner tracer points of theocean velocity cell of the parent model (⊚ in Figure 18.3). All of the four corner points are always available for bilinear
interpolation to compute the tracer values of the child model. Thus, assuming that parent and child grid points coincide
at (i p, j p ) and (i c, jc ), the tracer value at (i c + m, jc + n), where (0 ≤ m ≤ M) and (0 ≤ n ≤ N ), and M and N are thezonal and meridional nesting ratios, is computed as follows:
p

p

p

p

ϕic +m, jc +n = (1 − w x )(1 − wy )ϕi p , j p + w x (1 − wy )ϕi p +1, j p + (1 − w x )wy ϕi p , j p +1 + w x wy ϕi p +1, j p +1,(18.2)

where w x = (x itc +m − x it p )/(x it p +1 − x it p ) and wy = (y tjc +n − y tj p )/(y tj p +1 − x tj p ), x t and y t represent the tracer point inzonal and meridional coordinates, respectively.
b. Velocity points

Figure 18.4 shows the relative positions of velocity points of parent and child models. Unlike the tracer points, all of thefour parent velocity points surrounding a child velocity point are not necessarily ocean points. When all parent grid points
are ocean, bi-linear interpolation can be applied.
A problem arises for velocity points of the child model between an ocean point and a land point of the parent model(shaded by blue in Figure 18.5a), because only data at a single point is available for interpolation. The same problem also
– 228 –

Chapter 18

Nesting

Tic , jc+N

Tic+1, jc+N

Tic+M-1, jc+N

Tip, jp+1

Tic+M, jc+N

Tip+1, jp+1

Tic , jc+N-1

Tic+1, jc+N-1

Tic+M-1, jc+N-1

Tic+M, jc+N-1

Uip , jp
Tic -1, jc +1

Tic , jc +1

Tic+1, jc +1

Tic+M-1, jc+1

Tic+M, jc+1

Tic -1, jc

Tic, jc

Tic+1, jc

Tic+M-1, jc

Tic+M, jc

Tip , jp
Tic -1, jc -1

Tip+1, jp
Tic, jc -1

Tic+1, jc -1

Figure18.3 Relative positions of tracer points of parent and child models (⊚ and ◦, respectively), coinciding at (i p, j p )and (i c, jc ). The red and blue frames indicate U- and T-boxes in the parent model, respectively. M and N are the zonal
and meridional nesting ratios.

arises for a velocity cell that has a partial velocity cell at its neighbor. The velocity for the partially blocked part (shadedby purple in Figure 18.5a) should be treated separately from the upper unblocked part.
In this case, the child model velocity perpendicular to the coast line (uc ) is assumed to be zero at the coast and linearlyinterpolated from the nearest ocean grid point of the parent model (Figure 18.5b). The component tangent to the coast line
(v c ) is set to the same value at the nearest ocean grid point of the parent model (Figure 18.5c). By doing so, the interpolated
velocity field yields nearly the same transport along the coast line as the parent model. To be specific, when the parent
model has a partially shaved bottom cell at (i p − 1, j p, k), velocities of the child model between x u (i p − 1) ≤ x u ≤ x u (i p )are computed as follows:pp

uc (i) =w x ui p + (1 − w x )ui p −1,(18.3)ppv c (i) =w x vi p + (1 − w x )vi p −1,

uc (i) =v c (i) =∆z i p −1,k∆z i p ,k∆z i p −1,k∆z i p ,kp

if x u (i) < x t (i p )p∆z i p ,k − ∆z i p −1,kp∆z i p ,k∆z i p ,k − ∆z i p −1,k(w x ui p + (1 − w x )ui p −1 ) +p(w x vi p + (1 − w x )vi p −1 ) +∆z i p ,kp

wui p ,p

vi p ,(18.4)(18.5)

if x t (i p ) < x u (i)(18.6)

where w x = (x u (i) − x u (i p − 1))/(x u (i p ) − x u (i p − 1)) and w = (x u (i) − x t (i p ))/(x u (i p ) − x t (i p )).
To apply this method to the barotropic transport, the barotropic transport is first divided by the height of the watercolumn to obtain the vertically uniform barotropic velocity. This velocity is given to all the velocity points in the watercolumn and then the same method as used for the 3-D velocity field is applied.

– 229 –

18.3

Uic , jc+N

Uic+1, jc+N

Uic+M-1, jc+N

Uip , jp+1

Uic -1, jc+1

Uip+1, jp+1
Uic+1, jc+N-1

Uic+M-1, jc+N-1 Uic+M, jc+N-1

Uic , jc+1

Uic+1, jc+1

Uic+M-1, jc+1

Uic+1, jc

Uic+M-1, jc

Uic , jc -1

Uic+M, jc+1

Uic+M, jc
Uip+1, jp

Uip , jp
Uic -1, jc -1

Uic+M, jc+N

Uic , jc+N-1

Uic , jc

Uic -1, jc

Intermodel transfer

Uic+1, jc -1

Figure18.4 Same as Figure 18.3 except for velocity point.

18.3.2 Upscaling
In two-way nesting, values of the parent model in the child model’s main region are filled with those based on the childmodel. The region to receive feedback in the parent model is the main region of the child model excluding the boundaryline (inside the red line of Figure 18.1b). MRI.COM has two options for upscaling, replacement and nudging.
a. Replacement

One option is to simply replace variables of the parent model by those of the child model. Ideally, fine scale features thatcannot be resolved by the parent model should be removed in upscaling. In MRI.COM, simple averaging within a parent
grid is used (averaging over blue rectangle in Figures 18.3 and 18.4), though more sophisticated filters have been proposed(e.g., Debreu et al., 2012). Because applying a full averaging for the entire embedded region is costly in terms of datatransfer between models, a full averaging is only applied to the 2 to 3 grid points inside the parent-child boundary,p

ϕi p , j p =1 ∑ c
ϕ , i c − i m ≤ i ≤ i c + i m and jc − jn ≤ j ≤ jc + jn,
M × N i, j i, j

(18.7)

where i m = (M − 1)/2 and jn = (N − 1)/2. For the region further inside the boundary, values on the child grid points thatcoincide with the parent grid points may be used without filtering,p

ϕi p , j p = ϕicc , jc ,x t (i p ) = x t (i c ) and y t ( j p ) = y t ( jc ).

(18.8)

This treatment does not cause a problem because this inner region away from the boundary does not aﬀect the temporal
evolution of the physical field in the main region of the parent model.

– 230 –

Chapter 18(a)Nesting
ip-1
ic -M

×：tracer / normal ﬂux

ip
ic

(b)

ip-1
ic -M

●：velocity

×：ﬂux

ip
ic

●：velocity

(c)

ip-1
ic -M

×：ﬂux

ip
ic

●：velocity

Figure18.5 Interpolation of the parent model velocity to the child grid with bottom topography. (a) For velocity pointsof the child model between an ocean point and a land point (shaded with blue or purple), only one data is available forinterpolation. Examples of (b) normal and (c) tangential component of velocity relative to the coast line (red arrows).
For an ocean velocity cell next to a partial cell, the cell is divided into a part blocked by a wall and a part unblocked(green point). Linear interpolation is used for an unblocked part and special treatment is applied to the blocked part
(purple arrows). The velocity is obtained as a weighted average of the two parts.

b. Nudging

Nudging is another option to reflect the child result to the parent model. In MRI.COM, tracer values of the child modelp
may be used as reference values (ϕC ) to which parent tracer fields (ϕ p ) are nudged,∂ϕ p1p= − (ϕ p − ϕC ),∂tγ(18.9)

where γ is a restoring time scale usually taken to be very small, about 0.1 day. Only tracer fields are corrected in the parent
model. Neither velocity nor sea surface height is modified. Our assumption lying behind this treatment is that reflecting
only slowly evolving quasi-geostrophic baroclinic fields is enough for a parent model in many applications.

18.4 Conservation
Conservation of sea water volume and scalar quantities are generally preferable. This is especially so for the case ofclimate applications with long-term integration. To achieve this in nesting, MRI.COM has options to adjust side boundaryand surface fluxes in the child model, or source / sink due to nudging in the parent model.

18.4.1 Side boundaries
Adjustment of the flux at the side boundary is performed as follows. Consider an evolution equation for a quantity ϕ in a
simple flux form:∂ϕ= −∇ · f,(18.10)∂twhere f = ( f x, f y ), which are due to either advective or diﬀusive fluxes. Integration (18.10) over each model’s main
region (Figure 18.6) gives
∫
∫
p
∂ΦΩ−ω
∂ϕ
≡
dS = −
f p · ndl
(18.11)
∂t
∂t
Ω−ω
∂Ω
for parent model’s main region and

c
∂Φω
≡
∂t

∫

∂ϕ
dS = −
ω ∂t– 231 –∫
∂ωf c · ndl

(18.12)

18.4

Conservation

for child model’s main region. If integral values of the flux normal to the boundary (that is, r.h.s.) cancel between parent
and child models, the sum of integral values of the tendency of ϕ in each model’s main region is conserved,
p

∂ΦΩ−ω

+

∂t

c
∂Φω
= 0.
∂t

(18.13)

However, this conservation does not always hold owing to discretization. Some adjustment is needed.
In MRI.COM, conservation of water volume and tracers is achieved by adjusting boundary fluxes computed in eachmodel. Diﬀerent approaches are taken for advective and diﬀusive fluxes.
For advective fluxes, fluxes of the child model are adjusted to match those of the parent model. Fluxes for tracer cells of
the child model ( f yc ) are area-integrated along the side boundary section over the width of a tracer cell of the parent model(Figure 18.7a),∑(18.14)f yc (i) × ∆s c (i), i c − i m ≤ i ≤ i c + i m,
Fyc (i p ) =iwhere ∆s c is the area of the side boundary of a tracer cell of the child model and i m = (M − 1)/2. This is compared withp
the flux for the tracer cell of the parent model ( f y ) (Figure 18.7b),pp

Fy (i p ) = f y (i p ) × ∆s p (i p ),(18.15)where ∆s p is the area of the side boundary of a tracer cell of the parent model. On the basis of this comparison, fluxes of
the child model are adjusted using a quantity f yc adj (i p ) given byp

f yc adj (i p ) = (Fy (i p ) − Fyc (i p ))/∆S c (i p ),where
∆S c (i p ) =∑∆s c (i), i c − i m ≤ i ≤ i c + i m .

(18.16)(18.17)i

The adjusted flux f yc mod is given by
f yc mod (i) = f yc (i) + f yc adj (i p ), i c − i m ≤ i ≤ i c + i m,

(18.18)so that the flux integral in the child model equals the parent model flux for parent’s tracer cell.
For diﬀusive fluxes, the sum of fluxes in the child model ( f xc ) may be given to the parent model instead of adjusting thechild model flux to the parent model flux as in the advective fluxes (Figure 18.8),∑pp
Fx mod ( j p ) = f x mod ( j p )∆s p ( j p ) =f xc ( j) × ∆s c ( j), jc − jn ≤ j ≤ jc + jn .
(18.19)jp

p

Fx mod ( j p ) replaces the flux calculated on the parent grid Fx ( j p ) at the boundary.
The flux adjustment should be applied to any horizontal flux of volume and tracers. Currently, this is implemented in
the following fluxes:• Volume flux in the 3-D continuity equation• Advective volume flux in the 2-D continuity equation (both barotropic and baroclinic time step)• Diﬀusive volume flux in the 2-D continuity equation (both barotropic and baroclinic time step)• Tracer advection flux by Second Order Moment and MPDATA schemes• Tracer diﬀusion flux by horizontal diﬀusion• Tracer flux by isoneutral transport scheme (isoneutral diﬀusion plus Gent-McWilliams parameterization)• Sea ice advection plus diﬀusion flux (area fraction, volume, thermal energy) with some special treatment. See
Chapter 17 for details.

18.4.2 Surface fluxes
In order to impose conservation on some properties in a set of nested models, surface fluxes must be also taken into
account.
In MRI.COM, you may impose conservation condition for the following cases by adjusting the net surface flux.
• Volume of sea water in an ocean-only mode (WADJ option).
– 232 –

Chapter 18

NestingΩωΓFigure18.6 Schematic for considering conservation in a system of coarse and fine grid models. The coarse grid(parent) model region is denoted by Ω and the fine grid (child) model region is denoted by ω. The boundary between
models is denoted by Γ. These notations follow Figure 1 of Debreu et al. (2012).

(a)

(b)

ic

ip

(c)

+

+

diﬀerence divided by+total cross sectional area

Figure18.7 Schematic explaining flux adjustment for a child model. In this zonal-vertical section, tracer points
coincide at i p and i c . (a) Meridional fluxes are computed in the child model using the values interpolated from theparent model. Flux is integrated over the corresponding parent grid and this is compared with that of the parent model(b). (c) Diﬀerence between the integrated fluxes is divided by the integration area (unit parent grid area) to give flux
correction for the child model (pink).

• Salt content in the ocean - sea ice system in any mode (l_sss_rst_cnsv = .true. in namelist nml_sss_restore)• Heat content and volume of sea water in a coupled mode (SCUPCGCM and NPUTFLUX options for parent and NGETFLUX
option for child)
For example, to conserve sea water volume in the system (WADJ option), a globally constant adjustment factor f w adj isused so that∫∫p
( f w + f w adj )dS +( f wc + f w adj )dS = 0,
(18.20)ωΩ−ωpwhere f w and f wc represent total surface water fluxes of parent and child models, respectively. The adjustment factor isgiven byf w adj = −

(

∫

∫
p

f w dS +Ω−ωω)
f wc dS /(SΩ−ω + Sω ),(18.21)∫∫
where SΩ−ω = Ω−ω dS and Sω = ω dS.
To avoid a drift of total salt content of the system (l_sss_rst_cnsv = .true. in namelist nml_sss_restore) in thecase of surface salinity restoring, a globally constant adjustment factor f s adj is used so that∫∫p
( f s + f s adj )dS +( f sc + f s adj )dS = 0,
(18.22)Ω−ωω– 233 –

18.4

Conservation

parent

child

adv

jp jc
diﬀ

×：tracer●：velocity

Figure18.8 Computation of parent model’s zonal diﬀusive flux across the boundary, coinciding at j p and jc in termsof the tracer point. In this case, the sum of diﬀusive fluxes of the child model (green) is used as the diﬀusive flux forthe parent model at j p . This is contrasted with the advective flux (blue) where the parent model flux is used to adjust
the child model flux.

pwhere f s and f sc represent surface salinity flux due to restoring in parent and child models, respectively. The adjustment
factor is given by
∫
∫
(
)
p
f s adj = −f s dS +f sc dS /(SΩ−ω + Sω ).
(18.23)ωΩ−ω

When a set of nested grid models is coupled with an atmospheric model (SCUPCGCM and NPUTFLUX options for parent
and NGETFLUX option for child), the sum of sea surface heat flux of the ocean model must equal that of the atmosphericmodel. To achieve this, a globally constant adjustment factor f q adj is used so that
∫

∫
Ω−ω

p
( f q + f q adj )dS +∫ω( f qc + f q adj )dS =f qa dS,(18.24)Ωpwhere f q , f qc , and f qa represent net surface heat flux in parent oceanic, child oceanic, and atmospheric models, respectively.
The adjustment factor is given by
∫
[∫
(∫
)]
p
f q adj =f qa dS −f q dS +f qc dS /(SΩ−ω + Sω ).
(18.25)ΩωΩ−ω

f q adj is used to oﬀset the net longwave flux in the ocean models.
These adjustments are applicable only to on-line two-way nesting by using pre-communicator of Scup.

18.4.3 Nudging
To conserve the total amount of tracer of the parent model when a tracer of the parent model is nudged toward that of theϕchild model, an additional flux adjustment is needed. A globally constant adjustment factor Sadj is added to the right handside of (18.9) so that∫1
ϕp(18.26)[− (ϕ p − ϕC ) + Sadj ]dV = 0.
γωThe adjustment factor is given byϕ

Sadj =

∫

1
ω γ

p

(ϕ p − ϕC )dV /Vω,

– 234 –

(18.27)

Chapter 18

Nesting∫
where Vω = ω dV . By doing so, nudging only changes the distribution of the tracer in the parent model withoutaﬀecting its total amount. MRI.COM oﬀers this adjustment as an option (flg_feedback_cnsv_ndg = .true. in namelist
nml_nest_par).

18.5 Sponge region
Appearance of noises and discontinuity around the parent-child boundary is almost unavoidable. It is a common practice
to set a sponge region around the boundary to reduce them. Several methods are available for MRI.COM.

18.5.1 Giving 2-D distribution of diﬀusion and viscosity coeﬃcient
A simple choice would be to give 2-D distribution of time-invariant coeﬃcient for diﬀusion and viscosity operators thathave been selected for each model. For example, when a common operator is selected for both parent and child models,the coeﬃcients of the child model should be the same as those of the parent model around the side boundary to obtain
similar results there. The coeﬃcients in the boundary region should be connected smoothly with the small coeﬃcients in
the interior.
Among the mixing operators available in MRI.COM, the additional harmonic viscosity is most eﬀective for getting
a smooth solution around the parent/child boundary. However, this scheme should be used with caution because theharmonic viscosity may even damp resolved processes of the parent model. These additional diﬀusivity and viscosity areavailable, whether nesting is one-way or two-way.

18.5.2 Scheme specialized for two-way nesting
MRI.COM has some schemes specialized for two-way nesting in order to make the model stable.
a. Parent model

A parent model can use an additional Laplacian operator for both velocity and tracer. This operator may be applied to thevelocity points on the boundary (red line in Figure 18.9) and to the tracer point (blue line) just outside the feedback region
where parent-model values are upscaled (replaced) by child-model values (light blue region). Laplacian is calculated in
the direction perpendicular to the boundary.

×：tracer●：velocity

Figure18.9 Same as Figure 18.1b except for indicating of sponge regions for a parent model. Laplacian diﬀusionand viscosity operators are applied in the blue and red bands, respectively.

– 235 –

18.6 Usageb. Child model

For a child model, Laplacian operator is applied to the diﬀerence between the child value and the parent value mapped on
the child grid in the sponge region following Section 2.5 of Debreu et al. (2012),∂ϕc= ∇ · κ∇(ϕc − ϕcP ),∂t(18.28)where ϕc is the child model value, ϕcP is the parent value mapped on the child grid, and κ is viscosity/diﬀusion coeﬃcient.
The operator is prepared for velocity and tracers. The width of the sponge region is two parent velocity cells from theboundary constructed by connecting tracer cells (region shaded with purple in Figure 18.10).

×：tracer●：velocity

Figure18.10 Same as Figure 18.1b except for indicating of sponge region for child model. Laplacian diﬀusion/viscosity operator is applied to the diﬀerence between child and parent values in the region shaded with purple.

18.6 Usage
18.6.1 Oﬄine one-way nesting (OFFNESTPAR, OFFNESTSUB option)a. Compilation

The model option for the parent model is OFFNESTPAR (OFFline-NESTing PARent), and that for the child model is
OFFNESTSUB. These model options and the model name should be specified in configure.in.
b. Runtime specification for the parent modelAt run time, namelists nml_parentmodel_grid (Table 18.2) and nml_parentmodel_package (Table 18.3) must be
specified. They are read by parent/parent_common.F90 on execution. See docs/README.Namelist for details.
Following files are output from the parent model to be read by the child model.
• Three rows or columns of data for each side boundary of specified packages. For package (package_name),’uv’, ’barotropic’, ’active_tracer’, ’passive_tracer’, ’vmixcoef’, ’nohkim’, ’my25’, ’gls’, ’ice_thermodynamics’,’ice_dynamics’, and ’tide’ are available.
• Land-sea index of the parent model (file_parent_sea_index_out).
• Latitude and longitude of the parent model (file_parent_grid_out).
The daily output of side boundary data will work in running the child model.

– 236 –

Chapter 18

Nesting
Table18.2 Namelist nml_parentmodel_grid for the oﬀ-line parent model

variable namesub_region_ifirst

dx_ratio_w

description
the western end point in the parent model to be
used to interpolate data (T-point)
the eastern end point in the parent model to be
used to interpolate data (T-point)
the southern end point in the parent model to be
used to interpolate data (T-point)
the northern end point in the parent model to be
used to interpolate data (T-point)
refinement ratio at west boundary

dx_ratio_erefinement ratio at east boundary

dy_ratio_srefinement ratio at south boundary

dy_ratio_nrefinement ratio at north boundary

file_parent_sea_index_out
file_parent_grid_out

land-sea index of the parent model
latitude-longitude of the parent model

sub_region_ilast
sub_region_jfirst
sub_region_jlast

usage

odd number, zero for no nestingalong this side
odd number, zero for no nestingalong this side
odd number, zero for no nestingalong this side
odd number, zero for no nestingalong this side

Table18.3 Namelist nml_parentmodel_package for the oﬀ-line parent model

variable namepackage_namefile_root_namebndr_dt_sec
nrec_firstdescription
root name of the output boundary data files
time interval [sec] of output
the first record number of the output files

l_write_initial_state

usage

must be multiple of model time step
= 1 : create new file
> 1 : append to old file.true. : output initial state.false. : do not output initial statec. Runtime specification for the child modelAt run time, namelists nml_submodel_grid (Table 18.4), nml_submodel_bnd_cnd (Table 18.5), and nml_submodel_package(Table 18.6) must be specified. They are read by sub/sub_common.F90 on execution. See docs/README.Namelist for
details.
To run the child model, prepare the following two data files:• file_sub_wgt_t: The contribution ratio between parent and child models around the boundary (T-point),• file_sub_wgt_u: The contribution ratio between parent and child models around the boundary (U-point).
In addition, produce the following files by running the parent model:• file_root_name in nml_submodel_package: Three rows or columns of the necessary elements at each sideboundary,• file_parent_sea_index: Land-sea index of the parent model,• file_parent_grid: Latitude and longitude of the parent model.
Note the following when preparing barotropic boundary data. Since time filtering is used to feedback the result of thebarotropic equations to the baroclinic mode, the barotropic equations are integrated over the end of the baroclinic time.
Thus, the child model needs barotropic data longer than the original output from the parent model. To fulfill this need,prepare the barotropic data file to include one additional time record. For example, in a historical (yearly) run, the second
record of the following year of the parent model should be appended to the present year data. If the additional record is
not prepared, MRI.COM integrates the barotropic equation by repeating the final record just as needed.

– 237 –

18.6 Usage
Table18.4 Namelist nml_submodel_grid for the oﬀ-line child model

variable nameparent_grid_im

dx_ratio_w

description
Total number of grid points in X directionof the parent model
Total number of grid points in Y directionof the parent model
the western end point in the parent model
to be used to interpolate data (T-point)
the eastern end point in the parent model tobe used to interpolate data (T-point)
the southern end point in the parent model
to be used to interpolate data (T-point)
the northern end point in the parent model
to be used to interpolate data (T-point)
refinement ratio at west boundary

dx_ratio_erefinement ratio at east boundary

dy_ratio_srefinement ratio at south boundary

dy_ratio_nrefinement ratio at north boundary

file_parent_grid
file_parent_sea_index

latitude-longitude of the parent model
land-sea index of the parent model

parent_grid_jm
sub_region_ifirst
sub_region_ilast
sub_region_jfirst
sub_region_jlast

usage

odd number, zero for no nesting along thisside
odd number, zero for no nesting along thisside
odd number, zero for no nesting along thisside
odd number, zero for no nesting along thisside

Table18.5 Namelist nml_submodel_bnd_cnd for the oﬀ-line child model

variable namefile_sub_wgt_t
file_sub_wgt_u

description
filename of weighing factor for parent/child model aroundthe boundary (T-points)
filename of weighing factor for parent/child model aroundthe boundary (U-points)

usage

Table18.6 Namelist nml_submodel_package for the oﬀ-line child model

variable namepackage_namefile_root_namebndr_dt_sec
bndr_first_date
num_bndr_record

description

usage

root name of the output boundary data files
time interval [sec] of the input data
integer array(6) indicating date and time (YMDMHS) of
the first record
the last record number of the input files

18.6.2 On-line nesting (SCUPNEST option)a. Compilation

For on-line mode, SCUPNEST option should be specified. Also, the name of the model should be explicitly specified suchas NAME_MODEL = modelname. The model option for the parent model is PARENT, and that for the child model is SUB.
In ocean-only mode, surface fluxes are calculated in each model, preferably on the basis of common atmospheric state.
When the parent model is coupled with an atmospheric model (SCUPCGCM), the parent model receives surface fluxes
from the atmospheric model. The surface flux may be sent from parent to child model. In the parent (child) model,
NPUTFLUX (NGETFLUX) option must be selected.
These model options and the model name should be specified in configure.in.

– 238 –

Chapter 18

Nestingb. Runtime specification: NAMELIST_SCUP

In on-line mode, parent and child models are run at the same time and Scup (simple coupler) by Yoshimura and Yukimoto(2008) is used to exchange data. User should tell the coupler how data are exchanged between parent and child models
via a namelist file NAMELIST_SCUP. A template of NAMELIST_SCUP is available from MRI.COM execution environment(MXE). Mapping tables for data transfer used by Scup can be created in MXE. User may comment out unnecessary dataexchange according to the model and runtime options chosen for a specific experiment. An example of NAMELIST_SCUP
is listed in the following. In this example, the model name of the parent model is GLOBAL and that of the child model is
NP01.


An example of NAMELIST_SCUP# pre-communicator## (parent -> child)&nam_scup_pre model_put=’GLOBAL’, model_get=’NP01’, type=’REAL8’ /&nam_scup_pre var_put=’ALONTC’, var_get=’ALONTC’, dst_get=’ALL’ /&nam_scup_pre var_put=’ALATTC’, var_get=’ALATTC’, dst_get=’ALL’ /&nam_scup_pre var_put=’ALONUC’, var_get=’ALONUC’, dst_get=’ALL’ /&nam_scup_pre var_put=’ALATUC’, var_get=’ALATUC’, dst_get=’ALL’ /# activate if l_sss_rst_cnsv = .true.
&nam_scup_pre var_put=’SFLXADJ’, var_get=’SFLXADJ’, dst_get=’ALL’ /## (child -> parent)&nam_scup_pre model_put=’NP01’, model_get=’GLOBAL’, type=’REAL8’ /# activate if l_sss_rst_cnsv = .true. and flg_trcflux_cnsv=.true.,&nam_scup_pre var_put=’SFLXSUB’, var_get=’SFLXSUB’, dst_get=’ALL’ /## main-communicator## (parent -> child) side boundary prognosticated variables (ALWAYS activate)&nam_scup model_put=’GLOBAL’, model_get=’NP01’ ,comp_put =’BAROCLIP_GLOBAL’ , comp_get =’BAROCLIS_NP01’ ,grid_put =’BCLI3DP_GLOBAL_T’, grid_get =’BCLI3DS_NP01_T’ ,fl_remap=’../data-np/rmp_tw_p2ct_3d.d’ /&nam_scup
var_put=’TRC_SIDE01’, var_get=’TRC_SIDE01’, intvl=-1, lag=0, flag=’SNP’/&nam_scup
var_put=’TRC_SIDE02’, var_get=’TRC_SIDE02’, intvl=-1, lag=0, flag=’SNP’/## (child -> parent) for replace (activate IF flg_feedback_replace=.true.)&nam_scup model_put=’NP01’, model_get=’GLOBAL’ ,comp_put =’BAROCLIS_NP01’, comp_get =’BAROCLIP_GLOBAL’,
grid_put =’BC3DS_NP01_TC’, grid_get =’BC3DP_GLOBAL_TA’,
fl_remap=’../data-np/rmp_c2pt_merge_3d.d’ /&nam_scup
var_put=’TRC_REPL01’, var_get=’TRC_REPL01’, intvl=-1, lag=0, flag=’SNP’/&nam_scup
var_put=’TRC_REPL02’, var_get=’TRC_REPL02’, intvl=-1, lag=0, flag=’SNP’/

c. Runtime specification for parent modelAt run time, namelist nml_nest_par (Table 18.7) must be specified. They are read by nest_scup_par.F90 on execution.
See docs/README.Namelist for details. If sponge layer is used for the parent model (flg_sponge = .true.), the twodimensional distribution of the diﬀusion and viscosity coeﬃcient in the sponge layer is read from a file whose name isgiven by namelist nml_par_diff (Table 18.8) and nml_par_visc (Table 18.9).

– 239 –

18.6 Usage
Table18.7 Namelist nml_nest_par for the on-line parent model

variable nameflg_carbon
flg_verbose
flg_sponge
flg_sponge_sub
flg_feedback_nudge
flg_feedback_replace
flg_volflux_cnsv

units
logicallogicallogicallogicallogicallogicallogicalflg_iceflux_cnsv

logicalflg_send_sshhdif_c2p
flg_trcflux_cnsv

logicallogicalflg_send_trchdif_p2clogicalflg_send_trchdif_c2plogicalflg_send_trcidif_p2clogicalflg_send_trcidif_c2plogicalchfbc_sub_day
scup_norecv
nest_depthday
1integer

num_sub_models
file_upscale

integercharacterdescription
Send surface CO2 flux or not
Output details of the processing
Sponge region is set just outside the boundary
Send data to child model’s sponge region
Use nudging of tracers as two-way feedback
Use replacing of all fields as two-way feedback
Impose conservation of volume flux at u(v)star
for child model
Impose conservation of sea ice flux at u(v)star
for child model
Send SSH diﬀusion flux from child to parent
Impose conservation of tracer flux at u(v)star
for child model
Send tracer lateral diﬀusion flux from parent tochild
Send tracer lateral diﬀusion flux from child toparentSend tracer isopycnal diﬀusion flux from parent
to parent
Send tracer isopycnal diﬀusion flux from childto parent
restoring time of received tracer in day
undefined value of received tracer given by scup
nesting is operated from surface to nest_depth
level
the number of child (sub) models
Name of the file that contains mask of the upscale region

usage

default = .false.
default = .false.
default = .false.
default = .false.
default = .false.
default = .false.
default = .false., choose
either p2c or c2p or neither
default = .false., choose
either p2c or c2p or neither
default = .false., choose
either p2c or c2p or neither
default = .false., choose
either p2c or c2p or neither

default = 1

Table18.8 Namelist nml_par_diff for the on-line parent model

variable namefile_sponge_par_diff

description
2D distribution of horizontal diﬀusivity justoutside the boundary

usage

Table18.9 Namelist nml_par_visc for the on-line parent model

variable namefile_sponge_par_visc

description
2D distribution of horizontal viscocity just outside the boundary

usage

d. Runtime specification for child modelAt run time, namelist nml_submodel_grid (Table 18.10) and nml_nest_sub (Table 18.11) must be specified. They areread by nest_scup_sub.F90 on execution. See docs/README.Namelist for details.
The two dimensional distribution of the diﬀusion and viscosity coeﬃcient in the sponge layer is optionally
read from a file whose name is given by namelist nml_sub_diff (Table 18.12), nml_sub_visc (Table 18.13) (ifflg_sponge_sub = .true.), nml_sub_isopyc (Table 18.14), and nml_sub_ssh_diff (Table 18.15).

– 240 –

Chapter 18

Nesting
Table18.10 Namelist nml_submodel_grid for the on-line child model

variable nameparent_grid_im

dx_ratio_w

description
Total number of grid points in X direction of theparent model
Total number of grid points in Y direction of theparent model
the western end point in the parent model to be
used to interpolate data (T-point)
the eastern end point in the parent model to be used
to interpolate data (T-point)
the southern end point in the parent model to be
used to interpolate data (T-point)
the northern end point in the parent model to be
used to interpolate data (T-point)
refinement ratio at west boundary

dx_ratio_erefinement ratio at east boundary

dy_ratio_srefinement ratio at south boundary

dy_ratio_nrefinement ratio at north boundary

parent_grid_jm
sub_region_ifirst
sub_region_ilast
sub_region_jfirst
sub_region_jlast

l_parent_tripolarusage

odd number, zero for no nesting along
this side
odd number, zero for no nesting along
this side
odd number, zero for no nesting along
this side
odd number, zero for no nesting along
this side.true. : The parent model uses
TRIPOLAR grid

Table18.11 Namelist nml_nest_sub for the on-line child model

variable namefile_sub_wgt_t

unitscharacter

file_sub_wgt_ucharacter

file_sub_wgt_x

character

file_sub_wgt_ycharacter

flg_carbon
flg_verbose
flg_sponge_sub
flg_feedback_nudge
flg_feedback_replace
flg_volflux_cnsv

logicallogicallogicallogicallogicallogicalflg_iceflux_cnsv

logicalflg_send_sshhdif_c2p
flg_trcflux_cnsv

logicallogicaldescription
filename of weighing factor for parent/childmodel around the boundary (T-points)
filename of weighing factor for parent/childmodel around the boundary (U-points)
filename of weighing factor for parent/childmodel around the boundary (Ustar-points)
filename of weighing factor for parent/childmodel around the boundary (Vstar-points)
Send surface CO2 flux or not
Output details of the processing
Sponge region is set just inside the boundary
Use nudging of tracers as two-way feedback
Use replacing of all fields as two-way feedback
Impose conservation of volume flux at u(v)star
for the child model
Impose conservation of sea ice flux at u(v)star
for the child model
Send SSH diﬀusion flux from child to parent
Impose conservation of tracer flux at u(v)star
for the child model

usage

default = .false.
default = .false.
default = .false.
default = .false.
default = .false.
default = .false.
Continued on next page– 241 –

18.7 Program structure

variable nameflg_send_trchdif_p2c
flg_send_trchdif_c2p
flg_send_trcidif_p2c
flg_send_trcidif_c2p
nest_depth

Table 18.11 – continued from previous page
units
description
logicalSend tracer lateral diﬀusion flux from parent tochildlogicalSend tracer lateral diﬀusion flux from child toparentlogicalSend tracer isopycnal diﬀusion flux from parent
to parent
logicalSend tracer isopycnal diﬀusion flux from childto parent
integer
Nesting is operated from surface to nest_depth
level

usage
default = .false., choose
either p2c or c2p or neither
default = .false., choose
either p2c or c2p or neither
default = .false., choose
either p2c or c2p or neither
default = .false., choose
either p2c or c2p or neither

Table18.12 Namelist nml_sub_diff for the on-line child model (tracer_nest.F90)

variable namefile_sponge_sub_diff

description
3D distribution of horizontal diﬀusivity justinside the boundary

usage
valid if flg_sponge_sub = .true.

Table18.13 Namelist nml_sub_visc for the on-line child model (clinic_nest.F90)

variable namefile_sponge_sub_visc

description
3D distribution of horizontal diﬀusivity justinside the boundary

usage
valid if flg_sponge_sub = .true.

Table18.14 Namelist nml_sub_isopyc for the on-line child model (ipcoef.F90)

variable namel_isopyc_mask_file
file_isopyc_mask_sub

description.true./.false 2D distribution of isopycnal diﬀusion mask is read from file
2D mask of isopycnal and GM diﬀusionusage

diﬀusion coeﬃcients are determined as in the normal case

Table18.15 Namelist nml_sub_ssh_diff for the on-line child model (surface_integ.F90)

variable namel_ssh_diff_file
file_ssh_diff_sub

description.true./.false 2D-distribution of SSH diﬀusion isread from file
2D distribution of horizontal diﬀusivity of SSH

18.7 Program structure
The program structure of an oﬀ-line parent model is as follows.
ogcm__prepare|+- parent_ctl__ini|+- parent_ctl__prepare

ogcm__run|+- parent_ctl__write|+- parent_ctl__fin_run(initialize the parent model)(output boundary data of the initial state)(output boundary data at specified interval)(close files)– 242 –

usage

Chapter 18

Nesting

The program structure of an oﬀ-line child model is as follows.
ogcm__ini|+- sub_ctl__ini|+- sub_ctl__prepare(initialize the child model)(open boundary files, etc.)

ogcm__run|+- part_1|||+- surface_ctl__predict|||+- surface_predict
|
|
|
|
|
+- surfce_integ__predict
|
|
|
|
|
+- DO LOOP_N
|
|
|
|
|
|
|
+- surface_sub__input (replace barotropic boundary data)
|
|
|
|
|
+- surface_sub__input (replace time-filtered data)|||+- surface_bcli_predict_h (l_global_local_cnsv_ssh = .true.)|||+- surface_sub__input (replace time-filtered data again)|+- part_2|+- sub_ctl__input_side(read/receive baroclinic boundary data|
after "tracer_ctl__implicit_z" and before "strat_adjust_ctl__main" )|+- sub_ctl__input_side_vd (read/receive viscosity/diffusivity data
after mixed layer model)– 243 –

Part VI

MiscellaneousChapter 19

Basics of the finite diﬀerence method
This chapter describes the basics of finite diﬀerence methods for solving diﬀerential equations. The general principles ofthe finite diﬀerencing methods are introduced using the diﬀusion equation as an example in Section 19.1. Sections 19.2 and19.3 describe applying finite diﬀerence methods of time and space derivatives in diﬀerential equations. Considerations infinite-diﬀerence methods for advection-diﬀusion equations are discussed in Section 19.4. An implicit method for solving
the diﬀusion equation is described in Section 19.5. Section 19.6 summarizes discretization schemes used by MRI.COM.
Durran (1999) treats the basics of finite diﬀerence methods for solving diﬀerential equations of advection and diﬀusionin geophysical fluid dynamics.

19.1 Diﬀusion equation
As an example, consider an initial-boundary value problem expressed by a one-dimensional diﬀusion equation (heatconductive equation),∂T∂ 2T= κ 2.
(19.1)∂t∂x
Given T (x, 0) = f (x) as the initial distribution and T (0, t) = T (L, t) = 0 as the boundary condition, the analytical solutionis∞∑2
T (x, t) =f m e−κkm t sin(k m x),(19.2)
m=0

where
fm =

2
L

∫ L
f (x) sin(k m x)dx,

km =

0

mπ
.
L

(19.3)

Next, consider the finite diﬀerence method to get the solution numerically. In the finite diﬀerence method, grids are set
with a finite increment in space and time, and each term in the equation is evaluated at each grid using Tjn = T (x j , t n ). For
example,nn
Tj+1− 2Tjn + Tj−1
Tjn+1 − Tjn=κ,(19.4)∆t∆x 2
where ∆t = t n+1 − t n and ∆x = x j+1 − x j .
Distribution at the new time level T n+1 can then be calculated if T n is known. This finite diﬀerence equation is identicalto the original diﬀerential equation (19.1) in the limit ∆t → 0, ∆x → 0 (consistency).
If the initial distribution is assumed to be f (x) = T0 sin k1 x, the solution of the finite diﬀerence equation (19.4) fort = t n is
Tjn = λ nT0 sin k1 x j ,(19.5)
where
λ =1−

2κ∆t
∆x 2

(1 − cos k1 ∆x).

(19.6)In order to suppress oscillation and divergence of the solution (stability), 0 < λ < 1 is necessary and ∆x and ∆tmust be set to satisfy this condition. This solution is identical to the analytical solution in the limit of ∆t → 0, ∆x → 0(convergence).
To summarize, the finite diﬀerence method that satisfies consistency, stability, and convergence is the necessary condition
for an accurate solution.

– 247 –

19.2 Finite diﬀerence expressions for time derivatives

19.2 Finite diﬀerence expressions for time derivatives
The following four finite diﬀerence expressions are employed for the time derivatives in MRI.COM:
T n+1 − T n= F (T n )∆t
T n+1 − T n= F (T n+1 )backward :∆t
T ∗n+1 − T n
T n+1 − T n
Matsuno := F (T n ),= F (T ∗n+1 )∆t∆t
T n+1 − T n−1
leap-frog := F (T n ).
2∆tforward :(19.7)(19.8)(19.9)(19.10)The scheme used in the previous section is the forward scheme. The forward, backward, and Matsuno schemes use thevalues at two time levels and are accurate to O(∆t), while the leap-frog scheme uses three time levels and is accurate toO(∆t 2 ). Basically, the leap-frog scheme is employed in MRI.COM because of its higher order accuracy.
However, the leap-frog scheme cannot be applied to the diﬀusion equation. A solution by the finite diﬀerence method
using the leap-frog scheme is given by
Tjn = (Ta λ na + Tb λ bn ) sin k1 x j ,(19.11)√−α + α 2 + 4
λa = −
,
2

where

√
−α − α 2 + 4
4κ∆t(1 − cos k1 ∆x)).
λb = −(α ≡2∆x 2(19.12)

Because λ b < −1 for arbitrary values of α, the divergent mode with oscillation is always included (computational mode).
In order to avoid this computational mode, the forward scheme is employed for diﬀusion and viscosity terms in MRI.COM.
When the diﬀusion and viscosity coeﬃcients are very large as in the surface mixed layer, the time step has to be unusuallysmall for the stability of the forward scheme according to (19.6). In such a case, the backward scheme is used for vertical
diﬀusion and viscosity (implicit method; see Section 19.5). Though the time integration at each point can proceed without
referring to the result of other points by the forward, leap-frog, and Matsuno schemes, it must be done by solving combined
linear equations in the backward scheme (see Section 19.5).
The Matsuno scheme is useful for suppressing the computational mode in the leap-frog scheme. By defaults, theMatsuno scheme is used once per twelve steps of the leap-frog scheme in MRI.COM. This interval can be changed at run
time using a namelist parameter (nstep_matsuno_interval) of namelist nml_time_step (Table 21.6). It should be
noted that the Matsuno scheme needs twice as many numerical operations as the forward and leap-frog schemes.

19.3 Finite diﬀerence expression for space derivatives
Let us consider a one-dimensional advection equation,∂T∂T= −u ,∂t∂x(19.13)T (x, t) = T (x − ut, 0).

(19.14)

where u is a constant velocity. The solution is

Using the leap-frog scheme for time diﬀerencing, the finite diﬀerence equation can be written as follows:
Tjn+1 − Tjn−1
2∆t

= −u

n
n
Tj+
1 −T
j− 1
2

2

∆x

,

(19.15)where T n 1 and T n 1 are the values at the left and right faces of the grid cell for x j , i.e., values at x j− 1 and x j+ 1 . The point
j− 2

j+ 2

2

2

x j− 1 is defined as the central point between x j and x j−1 . Because the transport of T at the boundary that enters a grid2cell is identical to that leaving the adjacent grid cell, the total T in the whole system is conserved in this finite diﬀerence
equation.
– 248 –

Chapter 19

Basics of the finite diﬀerence method

There are several methods to decide T n 1 using a value at a single or multiple grid points. The following are two simple
and frequently used formulations,

j− 2

n
n
n
n
upstream finite diﬀerence : Tj−
Tj−
1 = T j−1 (u > 0),
1 = T j (u < 0),2(19.16)2ncentral finite diﬀerence : Tj−
1 =

n
+ Tjn
Tj−1

2

2

.

(19.17)

The former is accurate to O(∆x), and the latter is accurate to O(∆x 2 ).
In central finite diﬀerencing, the expression for (19.15) is
Tjn+1 − Tjn−1

= −u

2∆t

n
n
Tj+1
− Tj−12∆x.

(19.18)Assuming the solution to be T (x, t) = τ(t)e−ik x ,τ n+1 = τ n−1 + 2iατ n, where α ≡u∆tsin k∆x.
∆x(19.19)It is stable (neutral) if |α| ≤ 1. To be stable for any wave number,
u∆t
≤1
∆x

(19.20)must be satisfied (CFL condition). However, if τ n = τ 0 e−in∆θ ,∆θ = − sin−1 [µ sin k∆x](where µ ≡

u∆t
).
∆x

(19.21)

Expanding the r.h.s. by a Taylor expansion we obtain1(µ sin k∆x) 36µ(k∆x) 3 µ3 (k∆x) 3−≃ − µk∆x +
6
6
{
}
2
(k∆x)
= − µk∆x 1 −(1 − µ2 ) .
6∆θ ≃ − µ sin k∆x −(19.22)This means that the phase of the solution from this finite diﬀerence scheme is delayed relative to that of analytical solution,
depending on its wavenumber (numerical dispersion). Therefore, a distribution with maxima and minima that do notexist in the initial distribution arises. However, this method is popularly used since the kinetic energy is conserved by
employing the central diﬀerence in the advection term in the equation of motion. Moreover, the "Arakawa method," which
can nearly conserve the enstrophy (squared vorticity) for horizontally non-divergent flows, is adopted in MRI.COM byusing the central diﬀerence. This topic is treated in Chapter 7.
Using the upstream finite diﬀerence, the finite diﬀerence equation (19.15) is
Tjn+1 − Tjn−1−un
Tjn − Tj−1.

(19.23)∂T u∆x ∂ 2T+
+ O(∆x 2 ).
∂x
2 ∂ x2(19.24)2∆tExpanding the r.h.s. by a Taylor expansion we obtain

= −u

∆x

The second term has the diﬀusion (heat conductive) form (which disappears in the central finite diﬀerencing). Actually, theinitial distribution diﬀuses when the advection equation is solved by the upstream finite diﬀerence (numerical diﬀusion).
The third order schemes (QUICK, QUICKEST, and UTOPIA) can be used in MRI.COM to suppress the numericaldispersion and diﬀusion somewhat in the advection calculation for tracers, but not completely. The grid boundary value
is set in QUICK as
n
n
−Tj−2
+ 6Tj−1+ 3Tjn2n(u > 0), Tj−
1 =

n
n
3Tj−1
+ 6Tjn − Tj+1

(u < 0).
(19.25)8
8
2The QUICKEST method uses the time averaged value at the grid boundary as the tracer value to be transported, considering
the change of the value there by advection during one time step. UTOPIA is a multi-dimensional extension of QUICKEST.
The details of these schemes are described in Chapter 8.
n
Tj−
1 =

– 249 –

19.4 Finite diﬀerencing of advection-diﬀusion equation

19.4 Finite diﬀerencing of advection-diﬀusion equation
According to the above restriction, when the advection-diﬀusion equation (Eqs. (8.1) and (9.1)) is expressed in finite
diﬀerence form using the leap-frog scheme, it is necessary to use present (previous) time level for advection (diﬀusion)
term. The following finite diﬀerence equation is then employed:
T n+1 − T n−1= −A(T n ) + D(T n−1 ),2∆t(19.26)

where A and D are advection and diﬀusion operators, respectively.
When the vertical diﬀusion term is very large, D(T n+1 ) is used instead of D(T n−1 ). This formula is an implicit scheme
and is described in the next section.

19.5 Implicit method for vertical diﬀusion equation
Turbulent mixing is parameterized by using high vertical diﬀusivity and viscosity determined by boundary layer models,
which was treated in Chapter 15. The time step must be set very small to keep the calculation stable when viscosity anddiﬀusivity are very high, since the time tendency becomes very large due to the rapid mixing. To avoid this problem, theimplicit method uses the advanced (mixed) state for evaluating viscosity and diﬀusivity, unlike the normal explicit method
where previous or present values are used.
Expressing the present time step as n and the time step before and after as n ± 1, the finite-diﬀerence method is applied
to the advection-diﬀusion equation using the leap-frog scheme. The diﬀusion term is written separately using (n − 1) stepfor the horizontal direction and (n + 1) step for the vertical direction,(T n+1 − T n−1 )= −A(T n ) + D H (T n−1 ) + DV (T n+1 ).
2∆t(19.27)

Putting all the terms involving T n+1 on the l.h.s.,
T n+1 − 2∆tDV (T n+1 ) = T n−1 + 2∆t(−A(T n ) + D H (T n−1 ))(19.28)

is obtained, which is an algebraic equation for T n+1 . The finite diﬀerence form is rewritten specifically as
Tkn+1 − 2∆t

)
1 (
n+1
n+1
κ k− 1 (Tk−1− Tkn+1 )/∆zk− 1 − κ k+ 1 (Tkn+1 − Tk+1)/∆zk+ 1
2
2
2
2
∆zk
n−1
n
= Tk + 2∆t(−A(Tk ) + D H (Tkn−1 )).

By puttinga=2∆t κ k− 1

2

∆zk ∆zk− 1,

b = 1 + a + c,c=22∆t κ k+ 1
2

∆zk ∆zk+ 1,(19.29)(19.30)2

we get,n+1
n+1= Tkn−1 + 2∆t(−A(Tkn ) + D H (Tkn−1 )).
+ bTkn+1 − cTk+1−aTk−1(19.31)

Setting Fk ≡ −A(Tkn ) + D H (Tkn−1 ), this is expressed in the matrix form as
b
*. −a
..
..
..
..
..
.
,

−c
b
−a

−c
b
..
.

−c
..
.

−a

..

.

b
−a

−c
b
−a

T n+1
T n−1 + 2∆tF1+/+/ *. T1n+1 +/ *. T1n−1 + 2∆tF
2/
2// .. 2n+1 // ..
// .. T3n−1 + 2∆tF3 ///// .. T3// .
..
// .. .. // = ..
. / .
.
//
// .. n+1
/ . n−1// .. TKM−2 // .. TKM−2 + 2∆tFKM−2 //−c / .. T n+1 // .. T n−1 + 2∆tFKM−1 //
KM−1

n+1
b - , TKM-

The l.h.s. has the form of the tri-diagonal matrix.

– 250 –

KM−1

n−1
, TKM + 2∆tFKM -(19.32)

Chapter 19

Basics of the finite diﬀerence method

19.5.1 A solution of tri-diagonal matrix
In general, simultaneous linear equations for n variables with tri-diagonal matrix coeﬃcients
B1
*. A
.. 2
..
..

C1
B2
..
.

C2
..
.
An−1

,

..

.

Bn−1
An

X1
D1
+/ *. X +/ *. D +/// .. .2 // .. . 2 //// .. .. // = .. .. ///.
/ .
/Cn−1 / . X n−1 / . Dn−1 /Bn - , X n - , Dn -(19.33)are solved using the Thomas method, which is modified from LU decomposition,
P1 = C1 /B1(19.34)

Q1 = D1 /B1Ck
Pk =Bk − Ak Pk−1
Dk − Ak Q k−1
Qk =Bk − Ak Pk−1
Xn = Qn(19.35)

Xk = Q k − Pk Xk+1

(2 ≤ k ≤ n − 1)

(19.36)(2 ≤ k ≤ n)

(19.37)(19.38)(1 ≤ k ≤ n − 1).

(19.39)19.6 Summary of schemes used by MRI.COM
MRI.COM employs centered diﬀerence schemes for spatial discretization except for the tracer advection terms.
Figure 19.1 shows the schematic of the relation between components of the governing equations in terms of time
integration. The situation depicted here is the advancement in time from n to n + 1, represented by the thick red arrows.
The main components employ the leap-frog scheme, inserting Matsuno (Euler-backward) scheme with a specified interval.
Sea ice and turbulence closure employs forward scheme.
Colored arrows represent data transfer among the components in advancing in time using the leap-frog time (n → n + 1).
(a) State at the starting time (n − 1) is used for computing surface fluxes. (b) Surface current at the starting time and SST
and SSS of the present time is given to the sea ice component. (c) Flux from the sea ice component is averaged withthat of the previous time step for use in the ocean component (d) for conservation. (e) Horizontal velocity field of thepresent time is used to compute vertical velocity and tracer transport. (f) Diagnosed vertical velocity is used immediately
in the advection schemes of momentum and tracers. (g) Temperature and salinity at the present time is used to computethe pressure gradient term. (h) Vertically integrated forcing term computed by the baroclinic component is used by thebarotropic component. (i) The barotropic transport is returned to the baroclinic component and used immediately as thevertically averaged velocity for computing the total velocity. For turbulent closure scheme, states at both present (j) andnew (k) time level are used to estimate shear production and buoyancy loss of turbulent kinetic energy. (l) Turbulent
kinetic energy at the present time is used to compute vertical mixing coeﬃcient. Blue arrows represent the transfer of
extant data obtained by the previous time steps. Green arrows represent the immediate use of the new data computed inthis time step. These relations constrain the order of operations within a time step. The source side of the green arrows
must be calculated earlier than the recipient side.

– 251 –

19.6 Summary of schemes used by MRI.COM

Result of this time step is used immediately
Advance of variablesat the time step nResults of previous time steps are used

n-1

n

n+1

n+2

Sea ice(c)(c)

Surface flux
(τ,Ft,Fs,Fw)

(d)

(a)

(a)

(b)

(b)

Contintuity(w)(f)(f) (e) (e)Baroclinic
momentum(u,v)
Barotropic
momentum
(U,V)

(i)

(h)

(e)

(e)

(g)

Surfaceheight(η)(j) (k)Tracers(θ,s)(j)

Turbulence
closure
(q)

(k)

(l)

Figure19.1 Schematic of the relation between components of the governing equations in terms of time integration.
The situation depicted here is the advancement in time from n to n + 1, which is represented by the thick red arrows.
Blue arrows represent the transfer of extant data obtained by the previous time steps. Green arrows represent theimmediate use of data computed at this time step. These relations constrain the order of operations within a time step.
See text for the detailed explanation.

– 252 –

Chapter 20

Generalized orthogonal curvilinear coordinate
gridsThis chapter introduces generalized orthogonal curvilinear coordinates and presents related calculus.

20.1 Outline
A finite volume ocean model on geographic coordinates does not have any problem concerning the South Pole because
it does not calculate around the South Pole. However, serious problems arise around the North Pole where the meridian
concentrates to one point in the ocean. First, it is necessary to calculate the temporal evolution of the physical quantity ina special way only there, because the relations between U-cells that surround the North Pole and the northernmost T-cellare topologically peculiar. Next, even if a cell doesn’t touch the North Pole, its zonal lattice interval is extremely small
near the North Pole. Therefore, a short time step for integration is required owing to the limitation of the CFL condition.
This limitation is reflected directly in the increased calculation time required. Moreover, when the zonal grid intervals in
low latitudes and the Arctic region are extremely diﬀerent, the arguments about accuracies of numerical schemes and theparameters for diﬀusion and viscosity operators generally cannot be applied uniformly to a model domain.
The following can be considered to avoid such problems concerning the North Pole. 1) Creating a huge island including
the North Pole. The finite-diﬀerence calculation in the island is abandoned, and the lateral boundary values are restored
to the climatology. 2) Shifting the singular points of the model to a continent or a huge island by changing the model’s
horizontal grid system. The MRI.COM scheme adopts the latter approach, which is outlined in this section.
Because the MRI.COM code is based on generalized orthogonal coordinates, the geographical latitude (ϕ) and longitude(λ) are not of a great concern for the calculus in the model. However, it is necessary to know the land and sea distribution,
sea depth, scale factor, and the Coriolis parameter given as a function of λ and ϕ at every grid point of the model prior
to the calculation. We describe the method of generating the orthogonal coordinate grid system in Section 20.2. Using a
conformal transformation in the general sense, the functions that describe the relation between model coordinates (µ, ψ)and geographic coordinates (λ, ϕ), λ(µ, ψ), ϕ(µ, ψ), µ(λ, ϕ), and ψ(λ, ϕ), are obtained.
Because an atmospheric boundary condition is given in many cases at grid points in geographic coordinates, it isnecessary to prepare tables for converting the surface atmospheric temperature, the wind stress, and so on. To convert avector quantity, we must remember that the direction of the µ contour diﬀers from that of the λ contour (meridian). The
diﬀerence is described in Section 20.3. We can use the functions, (λ, ϕ) ⇐⇒ (µ, ψ), to convert a scalar quantity as well
as sea depth and the Coriolis parameter. The total flux that the ocean receives from the atmosphere should be equal to thetotal flux that the atmosphere gives to the ocean. The method for conserving the total flux is explored in Section 20.4. The
vector operation in generalized orthogonal coordinates is concisely described in Section 20.5.

20.2 Generation of orthogonal coordinate system using conformal mapping
We designate the plane that touches the sphere at the North Pole as SN . A polar stereographic projection is a conformaltransformation in the general sense, so that an orthogonal coordinate system on the sphere is mapped onto an orthogonal
coordinate system on SN and the orthogonality is preserved on the reverse transformation (Figure 20.1). Moreover, if SN isassumed to be a complex plane, various conformal transformations can be defined on it. Therefore, applying (i) the polarstereographic projection, (ii) a conformal transformation on the complex plane SN , and (iii) the reverse polar stereographic
projection to a geographic coordinate grid point (λ, ϕ) on the sphere, an orthogonal coordinate grid point on the spherecan be obtained (Bentzen et al., 1999).
The functions µ(λ, ϕ) and ψ(λ, ϕ) are obtained by the following procedure:– 253 –

20.2

Generation of orthogonal coordinate system using conformal mapping

Figure20.1 Schematic illustration of a Polar stereographic projection (a conformal transformation in the general sense
between the sphere and SN ).

1. From a point (λ, ϕ) on the sphere to a point z on SN (polar stereographic projection). Defining colatitude
ϕ ′ = π/2 − ϕ,( ′)ϕz = taneiλ,(20.1)
2
where the origin of SN corresponds to ϕ ′ = 0 (ϕ = π/2), and the positive part of the real axis corresponds to λ = 0.
2. Conformal transformation MC on SN :
ζ = MC (z).
(20.2)
3. From a point ζ on SN to a point (µ, ψ) on the sphere (reverse polar stereographic projection).
µ = arg(ζ ),(20.3)′

ψ = 2 arctan |ζ |,′ψ = π/2 − ψ .
Functions λ(µ, ψ) and ϕ(µ, ψ) are obtained by reversing the above procedure.
Defining ψ ′ = π/2 − ψ,( ′)ψ
ζ = taneiµ,2(20.4)(20.5)(20.6)

z = MC−1 (ζ ),(20.7)

λ = arg(z),(20.8)and′ϕ = 2 arctan |z|,′ϕ = π/2 − ϕ .

(20.9)(20.10)Thus, when a model coordinate grid point, (µ0 + ∆µ × (i − 1), ψ0 + ∆ψ × ( j − 1)) is given, we know the geographic positionof the point, Coriolis parameter, etc., at once.
Bentzen et al. (1999) used the linear fraction conversion as a conformal transformation on SN . That is,ζ=MC (z)=– 254 –(z − a)(b − c),(c − a)(b − z)

(20.11)

Chapter 20

Generalized orthogonal curvilinear coordinate grids

where the three complex numbers a, b, and c expressed by()
ϕ a′ iλ aa = tane ,2(b = tan

ϕb′

)

2

(

e

iλ b

,

)
ϕc′ iλcc = tane ,2(20.12)

correspond to the three geographic coordinate grid points (λ a, ϕ a ), (λ b, ϕb ), and (λ c, ϕc ), which are mapped to the model
coordinate grid points, (µ, ψ) = (0, π/2), (0, −π/2), (0, 0), respectively. Therefore, the singular point (µ, ψ) = (0, π/2)
in the model calculation can be put on Greenland, by setting (λ a, ϕ a ) at 75◦ N and 40◦ W. If TRIPOLAR or JOT option is
specified instead of SPHERICAL, then two singular points: (µ, ψ) = (0, π/2) and (0, −π/2) can be put on arbitrary land
locations by suitably setting (λ a, ϕ a ) and (λ b, ϕb ), which are model parameters (north_pole_lon, north_pole_lat,
south_pole_lon, and south_pole_lat in degree) that should be specified in namelist nml_poles (Table 3.2).
When TRIPOLAR option is specified, the parameters are set to ϕ a = ϕb = 64°N, λ a = 80°E, and λ b = 100°W. The
transformed grids are used for the region north of 64°N, and geographic coordinates are used for the region south of 64°N.
This tripolar coordinate system can express the Arctic Sea with a higher resolution than the Southern Ocean. The adoptionof geographic coordinates south of 64°N enables us to do the assimilation and analysis with relative ease (Figure 20.2).
When JOT option is specified, the Joukowski conversion is used as a conformal transformation on SN . That is,ψ ′2
z = MC−1 (ζ ) = *ζ + 0 + eiµ0 .
ζ ,(20.13)This Joukowski conversion maps the area outside the circle with a radius of ψ0′ centered at the origin of the ζ-plane to thewhole domain of the z-plane, and rotates it by µ0 . The left panel of Figure 20.2 presents an example where the coordinate
system is created by setting ψ0′ to 20° and µ0 to 80°. Because there is no discontinuity of grid spacing in this coordinate
system, the singular points can be put at various positions. For instance, the singular point on the North American side
can be put on the Labrador Peninsula or in Greenland.

Figure20.2 Model coordinate grid arrangement in the Arctic sea. Left: Grid system made through the Joukowski
conversion (JOT). Right: Combination of the coordinate systems made through the linear fraction conversion andconventional geographic coordinates (TRIPOLAR).

Functions λ(µ, ψ) and ϕ(µ, ψ) are defined as subroutine mp2lp, and functions µ(λ, ϕ) and ψ(λ, ϕ) are defined as subroutine lp2mp. Module programs trnsfrm.{spherical, moebius, tripolar, jot}.F90 contain these internal subroutines.
These functions, especially mp2lp, are frequently used when the topography and the surface boundary condition are madebefore starting the main integration of model.

20.3 Rotation of vector
A vector expressed in geographic coordinates (λ, ϕ) should be rotated when observed from model coordinates (µ, ψ).
Taking advantage of the local orthogonality of coordinate axes, the angle (α) is obtained as an angle at which the meridian– 255 –

20.4

Mapping a quantity from geographic coordinates to transformed coordinates(λ = λ 0 ) of geographic coordinates intersects that of the model coordinates (µ = µ0 ) at a certain point. First, we set
z = f (ζ ),f ′ (ζ )=z = x + iy,
∂y
∂x
+i
∂u
∂u

ζ = u + iv,
=

∂y
∂x
−i .
∂v
∂v

(20.14)(20.15)

At a certain point z0 = f (ζ0 ) on geographic coordinates, the angle θ at which a curve v = v0 meets a straight line y = y0
is given by
[ ]
[ / ]
∂y
∂y ∂x
tan θ ==,
∂ x v0∂u ∂u v0
then (see Figure 20.3),

θ = arg( f ′ (ζ0 )).

(20.16)

Figure20.3 A meridian (red) in geographic coordinates (λ, ϕ) (left) and a meridian (blue) in model coordinates (µ, ψ) (right).

Assuming λ = arg(z) and µ = arg(ζ ), at point ζ0 the straight line (v = v0 ) meets the straight line (µ = µ0 ) at the angle−µ0 (µ0 → v0 ), and at point z0 the meridian (λ = λ 0 ) meets the curve (v = v0 ) at the angle λ 0 − θ (v0 → λ 0 ). The
meridian (λ = λ 0 ) in λ-ϕ coordinates meets the line (µ = µ0 ) in µ-ψ coordinates at angle α given as follows:
α = − µ0 + λ 0 − θ= λ 0 − µ0 − arg( f ′ (ζ0 )).

(20.17)

Subroutine rot_mp2lp defined in trnsfrm.{spherical, moebius, tripolar, jot}.F90 returns (cos α, sin α) at a
specified grid point of the model. A wind stress vector (τx, τy ) in geographic coordinates should appear in the model
ocean described in the µ-ψ coordinate system as (τx cos α − τy sin α, τx sin α + τy cos α).

20.4 Mapping a quantity from geographic coordinates to transformed coordinates
We consider a method to receive a quantity G I,J given at the geographic coordinate grids (I, J) as the quantity H M, N
at the model coordinate grids (M, N ) (Figure 20.4). The quantities are wind stress components after the vector rotation,
precipitation per unit area, sea surface atmospheric temperature, and so on. In addition, the average depth at a model grid
point can also be calculated by the following method because bottom topography (depths of sea floor) is usually given in
geographic coordinates.
– 256 –

Chapter 20

Generalized orthogonal curvilinear coordinate grids

Figure20.4 Grids (I, J) and (M, N ) subdivided into finer grids (i, j) and (m, n).

Grids (I, J) and (M, N ) are suitably subdivided into finer grids (i, j) and (m, n). We call these filter grids. A quantity
Gi,′ j is assumed to be homogeneously distributed in the geographic filter grids (i, j) covered by grid (I, J),
Gi,′ j = G I,J .
′Assume the quantity at a model filter grid Hm,nis equal to that at the nearest geographic filter grid,′′Hm,n= Gi(m,n),j (m,n) .

The quantity at model grid (M, N ) is obtained as the area-weighted average:
H M, N =1∑

AH M, N m,n′
AH ′ m,n Hm,n,(20.18)where AH M, N is the area of model grid and AH ′ m,n is the area of model filter grid.
When the grid intervals of geographic filter grid (i, j) and model filter grid (m, n) are extremely small, the total quantity(flux) received on the model grids (M, N ) is equal to the total quantity (flux) given by the geographic grids (I, J). The
relation between the quantity in the geographic grids and that in the model grids is defined by weight w,∑
H M, N =w(M, N, I, J)G I,J .
(20.19)I,J

How is the quantity converted in an actual calculation in the model?
1. When the strict conservation of quantity (flux) is necessary:
Fresh water is not permitted to be generated or vanish at the surface boundary in a run using an atmosphere-ocean
coupled model, for instance. In this case, w(M, N, I, J) is prepared beforehand, and the flux is passed from theatmosphere through equation (20.19) to the ocean. The resolution of the filter grid need not be extremely fine,provided that every geographic filter grid is linked to more-than-zero model filter grids and∑∑∑∑
AG I,J =A′ G′ i, j =A′ H ′ m,n =AH M, N .
I,J

i, j

m,n

M, N

2. When conservation need not be guaranteed:
When the ocean model is driven by the surface boundary condition based on atmospheric re-analysis data, theamount of fresh water entering the sea as precipitation and river discharge is not equal to that drawn from the ocean– 257 –

20.5 Vector operation and diﬀerentiation in generalized orthogonal coordinates
through evaporation and sublimation. Therefore, the global sea surface height rises or descends during years ofintegration. It is not very important to pursue complete conservation of fresh-water under this condition. In such acase, the flux at a model grid point can be prepared beforehand using equation (20.19), to avoid the time-consuming
flux conversions in the model calculation.

20.5 Vector operation and diﬀerentiation in generalized orthogonal coordinates
To formulate the model equations, we have to know the vector operation and diﬀerentiation in generalized orthogonal
coordinates. Some basic formulae used in formulating primitive equations are presented here.
The line element vector δx at a certain point (µ, ψ, r) in an arbitrary general orthogonal coordinate system is expressedasδx = hµ δ µeµ + hψ δψeψ + hr δrer ,(20.20)where basis vectors eµ , eψ , and er are mutually orthogonal unit vectors, and hµ , hψ , and hr are scale factors.
Defining
eµ ∂
eψ ∂
er ∂
∇=
+
+
,
hµ ∂ µ hψ ∂ψ hr ∂r
the gradient of scalar A(µ, ψ, r) is∇A =eµ ∂ A eψ ∂ A er ∂ A++,
hµ ∂ µ hψ ∂ψ hr ∂r(20.21)(20.22)and the divergence of vector A = Aµ eµ + Aψ eψ + Ar er is∇·A=1h µ hψ hr

The r component of curlA is[]∂(hψ hr Aµ ) ∂(hr hµ Aψ ) ∂(hµ hψ Ar )
+
+
.
∂µ
∂ψ
∂r

[
]
∂(hψ Aψ ) ∂(hµ Aµ )1−.
h µ hψ
∂µ∂ψ(20.23)(20.24)The calculation of velocity advection includes (a · ∇)A, where a is an arbitrary vector (a = aµ eµ + aψ eψ + ar er ).
The µ component of (a · ∇)A is
(
)
(
)
Aψ
∂hµ
∂hψ
∂hµ
Ar
∂hr
a · ∇Aµ +
aµ
− aψ
+
aµ
− ar
.
(20.25)h µ hψ
∂ψ∂µhr h µ∂r∂µ
The second and third terms are so-called "metric" terms in the equation of motion in spherical coordinates.
These expressions in spherical coordinates (λ, ϕ, r) are shown next. Defining longitude λ, latitude ϕ, and radius of theearth r, scale factors are hλ = r cos ϕ, hϕ = r, and hr = 1.
Velocity vector v is
v = ueλ + veϕ + wer ,(20.26)
where eλ , eϕ , and er are the eastward, northward, and upward unit vectors, respectively, and (u, v, w) = (r cos ϕ λ̇, r ϕ̇, ṙ).
The gradient of scalar function A(λ, ϕ, r) is,∇A =eλ ∂ A e ϕ ∂ A
∂A
+
+ er
,
r cos ϕ ∂λr ∂ϕ∂r

(20.27)

∇=

eϕ ∂
∂
eλ
∂
+
+ er .
r cos ϕ ∂λr ∂ϕ∂r

(20.28)where

For vector A = Aλ eλ + Aϕ eϕ + Ar er , the divergence is
∇·A=

[
]
1
∂ Aλ ∂(cos ϕAϕ )∂(r 2 Ar )+.
+r cos ϕ ∂λ
∂ϕr 2 ∂r

(20.29)[]∂ Aϕ ∂(cos ϕAλ )1.
−r cos ϕ ∂λ
∂ϕ(20.30)and the r component of curlA is[curlA]r =– 258 –

Chapter 20

Generalized orthogonal curvilinear coordinate grids

The λ component of (a · ∇)A is[(a · ∇)A]λ = a · ∇Aλ −Aϕ aλ tan ϕ Ar aλ+.
rr(20.31)The Coriolis force in generalized orthogonal coordinates (µ, ψ, r) is given as2Ω × v = (2Ωψ w − 2Ωr v)eµ + (2Ωr u − 2Ωµ w)eψ + (2Ωµ v − 2Ωψ u)er ,(20.32)where Ω = Ωµ eµ + Ωψ eψ + Ωr er is the rotation vector of the Earth, and v = ueµ + veψ + wer is the velocity vector. We
designate f µ = 2Ωµ , f ψ = 2Ωψ , and f = f r = 2Ωr in Chapter 2. The rotation vector of the Earth is (Ωλ, Ωϕ, Ωr ) =(0, Ω cos ϕ, Ω sin ϕ) in geographic coordinates (λ, ϕ, r).

– 259 –

Chapter 21

User’s Guide
This chapter briefly explains the procedures needed to run MRI.COM. The description in this chapter is based on MRI.COM
version 4.4 (MRICOM-4_4_beta01-20170118) and some contents presented in this chapter may not be used for the latest
version. It is recommended that users refer to README.First, README.Options, README.Namelist, README.Monitor,
and README.Restart in the docs directory when setting up a model.
The minimal information to prepare, run, and post-process is presented in this chapter in the following order:• Model setup: User defined parameter files and compilation (Section 21.1 and Table 21.1).
• Input data: Grid spacing, topography, and surface forcing etc., to be read at run time (Section 21.2 and Table
21.2).
• Restart file: Explanation of restart files (Section 21.3).
• Execution: Explanation of the runtime parameters that control time-integration (Section 21.4).
• Post process: A description of monitor files (Section 21.5).
Note that cgs units are employed to express physical values in the model.
We are developing a comprehensive package of tools "MRI.COM eXecution Environment (MXE)" that aggregatesprograms for preprocessing, execution, postprocessing, and analysis of MRI.COM experiments. This is briefly explainedin Section 21.6

21.1 Model setup
21.1.1 Model configuration file (configure.in)This section describes the procedures necessary for setting up the model and compiling its programs. First, prepare
configure.in that contains the information about model options and grid size. This is needed for compilation. Following
is an example of configure.in.
An example configure.in for Global tripolar 1° × 0.5° grid model
DEFAULT_OPTIONS="IDEALAGE ICE SIDYN CALALBSI SMAGOR VIS9P DIFAJS GLS VVDIMPSOMADVEC ISOPYCNAL HFLUX TAUBULK WFLUX RUNOFF SFLUXR LWDOWN BULKNCAR BULKITERCYCLIC ZSTAR BBL TRIPOLAR PARALLEL"#
NAME_MODEL=’GLOBAL’
IMUT=364
JMUT=368
KM=51
KSGM=51
KBBL=1
NPARTX=8
NPARTY=4
NUM_ICECAT=5
NUMTRC_P=1

– 261 –



21.2 Preparation of input data files for execution
Additional parameters are required for some particular model options. Those are listed in Table 21.1 (see also
README.Options).
Table21.1 Model parameters to be set in configure.in

option namealways required

variable name
IMUT, JMUT, KM
NAME_MODELKSGM
NSFMRGNBBLPARALLELKBBL
NPARTX, NPARTY

passive tracers
ICE

NUMTRC_P
NUM_ICECAT

description
zonal/meridional/vertical grid number
name of the model (default = "tmOGCM")
The number of variable layers near the sea surface. KSGM must equal KM for
ZSTAR option. See Chapter 6.
the number of side-boundary ghost cells to reduce the communication costin parallel computation (see Ishizaki and Ishikawa, 2006)
the number layers of bottom boundary layer model; must be 1
the number of zonally/meridionally partitioned region for a computation using parallel processors: the number of parallel processes should be NPARTX× NPARTY
only when any passive tracer is calculated (NUMTRC_P ≥ 1)
the number of thickness categories of sea ice

21.1.2 Compilation of the modelA standard compiling script is prepared as compile.sh in the src directory. The part depending on the system (OS,Fortran compiler, and compiler option) are found in machines directory.
To compile the programs, execute compile.sh.
The function of compile.sh is to create param.F90,
icecat/ice_param.F90, and Makefile from configure.in, param.F90.in, icecat/ice_param.F90.in, andMakefile.in respectively, by running configure and to execute the command make to create the executable file ogcm.
The environment variables for compilation are set in configure using the options prescribed in configure.in. If
configure.in is newer than param.F90, the parameter values defined in configure.in replace those in param.F90.into create new param.F90.
The program files that should be compiled are automatically selected according to the descriptions of the relationships in
Makefile, but users should be careful since it might not be perfect. The compilation should be carried out after executing./compile.sh clean when any compile option in configure.in is rewritten.

21.2 Preparation of input data files for execution
According to user’s specification of model options and job parameters, the data files listed on Table 21.2 should be preparedalong with the files that must be always prepared. See Section 21.3 for how to handle restart files.
Table21.2 Main input data files and their related program files. Here, name_model represents the specific name given
as NAME_MODEL in configure.in.

subject
runtime job parametersspecification about monitoring
variable horizontal grid spacing
variable vertical grid spacing
grid cell area and line elements
topographyreference data for tracers
restoring coeﬃcient for tracers
surface forcing
surface forcing grid
restart files

file name specified in(NAMELIST.name_model)
NAMELIST.name_model
NAMELIST.name_model.MONITOR
file_dxdy_tbox_deg (nml_horz_grid)
file_dz_cm (nml_vert_grid)
file_scale (nml_grid_scale)
file_topo (nml_topo)
trcref(_surf)_conf%file_data
rstcoef(_surf)_conf%file_data
file_data (nml_force_data)
file_data_grid (nml_force_data)
nmlrs_element, element being elements– 262 –

read fromeach package
history.F90 etc.
gridm.F90
gridm.F90
gridm.F90
topo.F90
restore_cond.F90
restore_cond.F90
force.F90
force.F90
history.F90

Chapter 21

User’s Guide

21.2.1 Grid spacing and cell area
Details on how to specify grid information and how to prepare the necessary data is given in Section 3.6. The grid spacing
data file should be prepared for each of the horizontal (file_dxdy_tbox_deg) or vertical (file_dz_cm) directions when
variable grid spacing is used for that direction. The units are in degrees for the horizontal and in cm for the vertical.
When the model grid points are defined on the basis of general orthogonal coordinates, the quarter cell area and line
elements should be prepared. The units are in cgs. It is read from the file file_scale.
When spherical coordinates are used (SPHERICAL option), e.g., the grids are defined on geographical latitude andlongitude, the grid information is analytically calculated in gridm.F90, and the file file_scale is not necessary.

21.2.2 Topography
Land-sea distribution and sea-floor topography should be given by the topography data file file_topo. The topographic
data consist of the 4-byte integer array HO4(imut, jmut) that contains the sea floor depths of the velocity grid points (incm) and the 4-byte integer array EXNN(imut, jmut) that contains its corresponding vertical level. They should be written
unformatted and sequentially as follows:Format of topographic data (file_topo)
integer(4) :: ho4(imut,jmut),exnn(imut,jmut)
open(unit=nu,file=file_topo)
write(unit=nu) ho4, exnn
An example of the topography for global 1◦ × 0.5◦ model is shown in Figure 21.1. In creating a model topography,
especially for a low-resolution model, the user should be careful that the important gateways for the ocean circulation be
kept open and that the land blocking the ocean circulation be kept closed.

Figure21.1 Example of ocean model topography (global 1◦ × 0.5◦ grid model).

21.2.3 Reference data and restoring coeﬃcient for tracers
For each tracer, the integration may be started from an initial state based on the reference data and the tracer values may berestored to the reference data during the integration. To do this, tracer reference values and restoring coeﬃcients should
be prepared. See Chapter 13 for how to prepare these data.

– 263 –

21.3 Restart file

21.2.4 Surface forcing data
See Chapter 14 for how to prepare surface forcing data. The surface forcing data are read at a uniform time interval. A
leap year is set according to the calendar subroutine. Climatological data may be used repeatedly.
The following data files should be prepared according to the chosen model options. Each file is opened only once at the
beginning of run time and thus should contain all the data needed for that run.
Table21.3 Surface data to be read from namelist nml_force_data.

name

units
dyn cm−2
dyn cm−2
cm s−1cm s−1erg s−1 cm−2 =10−3 W m−2
erg s−1 cm−2

X-ward wind stressY-ward wind stress
X-ward wind speed
Y-ward wind speed
Downward shortwave radiation

U-windV-windU-windV-windShortWave

Downward longwave radiation

LongWave

Sea surface temperature
Surface air temperature
Surface air specific humidity
Scalar wind speed
Sea level pressure
precipitation
river discharge
Sea ice area fraction

SST
TempAir
SphAirScalarWind
SeaLevelPressure
Precipitation
RiverDischargeRate
IceConcentrationClimatology

◦C
◦C

1
cm s−1hPa
g s−1 cm−2
g s−1 cm−2
1

usage
if not TAUBULK
if not TAUBULK
if TAUBULK
if TAUBULKNet longwave by default(SST is required).
Downward if LWDOWN
unnecessary if LWDOWN

unnecessary if TAUBULK
also available for SLP
WFLUX
RUNOFFICECLIM

21.3 Restart file
A set of restart files provides an instantaneous state necessary to resume a model integration. This section explains how tohandle restart files. Table 21.4 lists restart variables of MRI.COM as well as their namelist that specifies their restart fileattributes.
Table21.4 Restart variables and their namelist block that specifies their restart file attributes.

variable nametemperaturesalinityx velocity
y velocity
sea surface heightx barotropic transport
y barotropic transport
x transport due to SSH diﬀusion
y transport due to SSH diﬀusion
horizontally averaged density and pressure used
for the equation of state

namelist block
nmlrs_t
nmlrs_s
nmlrs_u
nmlrs_v
nmlrs_ssh
nmlrs_uml
nmlrs_vml
nmlrs_ssh_dflx_x
nmlrs_ssh_dflx_y
nmlrs_density

model options

required if CALPP
Continued on next page

– 264 –

Chapter 21

User’s Guide

Table 21.4 – continued from previous pagevariable namenamelist block
vertical diﬀusivity coeﬃcient
nmlrs_vmix_avd
vertical viscosity coeﬃcient
nmlrs_vmix_avm
passive tracers
nmlrs_ptrc
x tidal transport
nmlrs_tide_um
y tidal transport
nmlrs_tide_vm
tidal sea surface height
nmlrs_tide_ssh
vertical diﬀusion coeﬃcient of turbulence kinetic nmlrs_my_avq
energyturbulence velocity scale
nmlrs_my_q
turbulence length scale
nmlrs_my_alo
vertical diﬀusion coeﬃcient of turbulence kinetic nmlrs_nk_avq
energyturbulence kinetic energy
nmlrs_nk_eb
vertical diﬀusion coeﬃcient of turbulence kinetic nmlrs_gls_avk
energyvertical diﬀusion coeﬃcient of generic variable
nmlrs_gls_avp
turbulence kinetic energy
nmlrs_gls_eke
generic variable
nmlrs_gls_psi
turbulent length scale
nmlrs_gls_alo
moments of second order moment advection nml_somadv
schemesea ice initial state
sea ice final state

nml_seaice_rst_in
nml_seaice_rst_out

model options
VVDIMP
VVDIMPNUMTRC_P > 0
TIDE
TIDE
TIDE
MELYAMMELYAMMELYAM
NOHKIM
NOHKIM
GLS
GLS
GLS
GLS
GLS
SOMADVEC andadv_scheme%name = ”som”
in nml_tracer_data
for any tracer
ICE
ICE

21.3.1 Restart for the ocean model
For restart variables of the main part of ocean model, the attributes of their restart files are given by the namelist blocks
that start with "nmlrs_". The element names that follow are t, s, u, v, ssh, etc. as listed on Table 21.4. Requiredelements depend on model options. See docs/README.Restart for more information. Namelists must be written in
NAMELIST.name_model (default) as follows:An example namelist for temperature restart files&nmlrs_t
fname = ’result/rs_t’,/
where fname specifies the basename of the restart files. This information is shared by both input and output files. In theabove example, the name of a restart file for temperature is result/rs_t.YYYYMMDDHHMMSS. A suﬃx indicating the date
and time of data is always added to the basename.
For passive tracers, namelist nmlrs_ptrc should be repeated as many times as the number of passive tracers(=numtrc_p) with a right order. The order is determined at tracer_vars.F90. An example for a set of biogeochemical
tracers is shown in Section 11.6.
Restart files are saved in a Fortran direct-access format, which can be read by a following program.
Program to read a restart file of the ocean modelcharacter(14):: date = ’20010101000000’
real(8):: d(imut,jmut,km)
integer(4),parameter :: nu = 10 ! device number!- for 0:00z1JAN2001

open(nu, file=’result/rs_t.’//date, form=’unformatted’, && access=’direct’,recl=imut*jmut*km*8)
read(nu,rec=1) dclose(nu)– 265 –

21.3 Restart file
The file endian is that of the computer where the model runs. There are support tools to visualize restart files inthe directory, tools/MK_GRADS. Please follow the instructions given by tools/MK_GRADS/00README.txt. The MXE
package (Section 21.6) also has tools for visualization in the directory, postp/.

21.3.2 Restart for the SOM advection scheme
When the SOM tracer advection scheme is used (SOMADVEC), the model may read and write additional 9 restart files of the
2nd order moments for each tracer. Their names are BASENAME_SS_NN.YYYYMMDDHHMMSS, where BASENAME indicates acharacter string specified separately for input and out files by file_som_in and file_som_out in namelist nml_somadv,
SS elements of the moments (9 types), NN a tracer number (01 for temperature, 02 for salinity, numbers after 03 for othertracers), and the suﬃx for the date and time. The format of each file is the same as the tracer restart files. See Section 8.6
for the SOM scheme, and docs/README.Namelist for other items of nml_somadv.

21.3.3 Restart for the sea ice model
To handle restart files of the sea ice model, dedicated namelist blocks must be specified separately for input and output asfollows:An example namelist for sea ice restart input&nml_seaice_rst_in
file_icecat_restart_in = ’result/rs_ice’,/

An example namelist for sea ice restart output

&nml_seaice_rst_out
file_icecat_restart_out_temp = ’result/rs_ice’,nwrt_rst_ic= 480, !- set same as nstep_seaice/where file_icecat_restart_in and file_icecat_restart_out_temp specify the basenames of input and output
files, and nwrt_rst_ic specifies the time step interval to write instantaneous states (usually, set it the same as the time
step number of the integration). See docs/README.Namelist for other items. (Before MRI.COM ver.4.2, one must set
l_file_name_calendar = .true. in nml_seaice_time_conf to add a suﬃx of the date and time to the file name inthe same manner as the ocean model. After ver.4.4, the suﬃx is always added. )The restart file can be read by the following program.

– 266 –

Chapter 21

User’s GuideProgram to read restart for the sea ice model

integer(4), parameter :: ncat = 5 ! number of thickness categories! ice concentration
real(8) :: aicen (1:imut,1:jmut,0:ncat), a0iceo(imut,jmut)! ice thickness
real(8) :: hicen (1:imut,1:jmut,0:ncat), hiceo (imut,jmut)! averaged sea-ice thickness
real(8) :: hin(1:imut,1:jmut,0:ncat), hi(imut,jmut)! snow depthreal(8) :: hsnwn (1:imut,1:jmut,0:ncat), hsnwo (imut,jmut)! averaged snow thickness
real(8) :: hsn(1:imut,1:jmut,0:ncat), hsnw (imut,jmut)! ice surface temperaturereal(8) :: tsfcin(1:imut,1:jmut,0:ncat), tsfci (imut,jmut)! ice temperaturereal(8) :: t1icen(1:imut,1:jmut,0:ncat)! sea surface skin temperaturereal(8) :: t0icen(1:imut,1:jmut,0:ncat)! sea surface skin salinity
real(8) :: s0n(1:imut,1:jmut,0:ncat)! skin temperature beneath the sea ice
real(8) :: t0iceo(imut,jmut)! skin temperature in the open leadsreal(8) :: t0icel(imut,jmut)! stress tensor
real(8) :: sigma1(imut,jmut), sigma2(imut,jmut), sigma3(imut,jmut)
integer(4),parameter :: nu = 10 ! device number
integer(4):: i = 0, m
open(nu, file=’result/rs_ice’, form=’unformatted’, && access=’direct’, recl=imut*jmut*8)
do m = 0, ncat ; i = i + 1 ; read(nu,rec=i) aicen(1:imut,1:jmut,m) ; enddo
do m = 0, ncat ; i = i + 1 ; read(nu,rec=i) hin (1:imut,1:jmut,m) ; enddo
do m = 0, ncat ; i = i + 1 ; read(nu,rec=i) hsn (1:imut,1:jmut,m) ; enddo
do m = 0, ncat ; i = i + 1 ; read(nu,rec=i) hicen(1:imut,1:jmut,m) ; enddo
do m = 0, ncat ; i = i + 1 ; read(nu,rec=i) hsnwn(1:imut,1:jmut,m) ; enddo
do m = 0, ncat ; i = i + 1 ; read(nu,rec=i) tsfcin(1:imut,1:jmut,m) ; enddo
do m = 0, ncat ; i = i + 1 ; read(nu,rec=i) t1icen(1:imut,1:jmut,m) ; enddo
do m = 0, ncat ; i = i + 1 ; read(nu,rec=i) t0icen(1:imut,1:jmut,m) ; enddo
do m = 0, ncat ; i = i + 1 ; read(nu,rec=i) s0n(1:imut,1:jmut,m) ; end do
i = i + 1 ; read(nu,rec=i) a0iceo(1:imut,1:jmut)
i = i + 1 ; read(nu,rec=i) hi(1:imut,1:jmut)
i = i + 1 ; read(nu,rec=i) hsnw(1:imut,1:jmut)
i = i + 1 ; read(nu,rec=i) hiceo(1:imut,1:jmut)
i = i + 1 ; read(nu,rec=i) hsnwo(1:imut,1:jmut)
i = i + 1 ; read(nu,rec=i) uice(1:imut,1:jmut)
i = i + 1 ; read(nu,rec=i) vice(1:imut,1:jmut)
i = i + 1 ; read(nu,rec=i) sigma1(1:imut,1:jmut)
i = i + 1 ; read(nu,rec=i) sigma2(1:imut,1:jmut)
i = i + 1 ; read(nu,rec=i) sigma3(1:imut,1:jmut)
close(nu)– 267 –

21.4 Execution

21.3.4 Control of input and output at run timea. Input

Input of restart files at run time is controlled by namelist listed on Table 21.5. By setting l_rst_in = .true. in
nml_run_ini_state, all necessary elements are read from restart files. However, the specification by l_rst_in may beoverridden for some elements by specifying namelists dedicated to those elements. Note that model tries to read restart
files for all passive tracers, thus namelist nmlrs_ptrc must be always specified appropriately.
Table21.5 Namelist blocks that control input of restart files.

namelist block
nml_run_ini_state

nml_barotropic_run
nml_baroclinic_run
nml_tracer_run
nml_vmix_run
nml_tide_run
nml_melyam_run
nml_nohkim_run
nml_gls_run
nml_density_run
nml_seaice_run
nml_tracer_data
nml_tracer_data

variable namel_rst_in

Usage.true. : read restart files and resume integration.false. (default): initial state is set internallyNo motion (u,v,ssh=0).
Initial condition for temperature and salinity are
deteremined by nml_tracer_run
l_rst_barotropic_in
default = l_rst_in
l_rst_baroclinic_in
default = l_rst_in
l_rst_tracer_in
default = l_rst_in
applicable to only temperature and salinity
l_rst_vmix_in
default = l_rst_in
l_rst_tide_in
default = l_rst_in
l_rst_melyam_in
default = l_rst_in
l_rst_nohkim_in
default = l_rst_in
l_rst_gls_in
default = l_rst_in
l_rst_density_in
default = l_rst_in
l_rst_seaice_in
default = .false.
adv_scheme%lrstin_som = .true. to read restart of SOM scheme
adv_scheme%lrstout_som = .true. to write restart of SOM schemeb. Output

Model always try to write restart files of the oceanic part according to the specification by the namelists listed on Table
21.4. Users have to specify those namelists appropriately to obtain restart files and terminate the model normally. Output
from the SOM scheme may be suppressed if adv_scheme%lrstout_som = .false. in nml_tracer_data.

21.4 Execution
To run a model, a shell script that handles input/output files, executes the compiled binary ogcm, and post-processes isusually prepared. The namelist parameters that control the time integration and the namelist file that specify sampling
have to be given.
• NAMELIST.name_model controls the job and gives parameters to sub-packages. See docs/README.Namelist for
details.
• NAMELIST.name_model.MONITOR specifies sampling. See docs/README.Monitor for details.
Parameters used by sub-packages are explained in corresponding chapters. Runtime parameters relevant to timeintegration are listed on Tables 21.6 through 21.11.

– 268 –

Chapter 21

User’s Guide
Table21.6 Namelist nml_time_step.

variable namedt_secunits
secnstep_matsuno_interval

integeralpha_bryan_1984

real(8)l_monitor_time

logicaldescription
unit time interval for equation of motion
and tracer
time step interval with which Matsuno(Euler-backward) time-integration scheme
is usedacceleration parameter α of Bryan (1984).
See Section 2.3.1.
time step monitor is written to standard output ()usage
requireddefault = 12 steps
default = 1
default = .true.

Table21.7 Namelist nml_run_period.

variable namenstep_total

units
integer

description
total number of time step of this run

usage
required

Table21.8 Namelist nml_exp_start that specifies start time of the whole experiment.

variable name
year
month
day
hour
minutesecondunits
integer(4)integer(4)integer(4)integer(4)integer(4)integer(4)description

usage
default = −999
default = 1
default = 1
default = 0
default = 0
default = 0

year
month
day
hour
minutesecondTable21.9 Namelist nml_run_ini that specifies start time of this run.

variable name
year
month
day
hour
minutesecondunits
integer(4)integer(4)integer(4)integer(4)integer(4)integer(4)description

usage
default = 1
default = 1
default = 1
default = 0
default = 0
default = 0

year
month
day
hour
minutesecondTable21.10 Namelist nml_calendar that specifies treatment of leap year.

variable namel_force_leap

unitslogicall_leaplogicaldescription
Use l_leap to decide whether the current year
is leap or not.
the current year is a leap year or not.

usage
default = .false., the realistic calendar isfollowed
default = .false.

Table21.11 Namelist nml_stdout that specifies standard output (program log).

variable namel_stdout2file
file_base_stdout

units
logicall_debug

logicaldescription
Standard output is written to separate files for MPI processes
file name is name_model-file_base_stdout-stdout.XXXX(XXXX: mpi process number)debug mode

– 269 –

usage
default = .false.
default = .false.

21.5

Monitoring

21.5 Monitoring
This section summarizes outputs of the mean state (history) data used for monitoring and analyses.

21.5.1 History of the ocean and ice models
In MRI.COM version 4, specification of history outputs has been significantly changed and commonized in the oceanand sea ice models. Users prepare a namelist file named NAMELIST.name_model.MONITOR dedicated to history outputs(name_model can be specified by NAME_MODEL in configure.in, default=OGCM), and write the following namelist blocks,
nml_history, in it as needed,An example namelist for mean temperature output&nml_history
name= ’temperature’
file_base= ’result/hs_t’
interval_step = 48
[suffix= ’day’]/where name specifies the variable name to be output、file_base the basename of the history files, interval_step theoutput interval in terms of the integration time step. An option item of suffix specifies the depth of calendar date and timeused in the file suﬃx. In the above example, the time-mean temperature is written to the file, result/hs_t.YYYYMMDD, atevery 48-th time step.
History files are saved in a Fortran direct-access format, which can be read by a following program.
Program to read ocean model history datacharacter(8):: date = ’20010101’ !- for 1JAN2001
real(4):: d(imut,jmut,km)
integer(4),parameter :: nu = 10 ! device number
open(nu, file=’result/hs_t.’//date, form=’unformatted’, && access=’direct’, recl=imut*jmut*km*4)
read(nu,rec=1) dclose(nu)
A GrADS control file to visualize data is also made by default (result/hs_t.ctl in the above example).
The state variables that can be monitored by namelist nml_history are listed in docs/README.Monitor. There aremany options in nml_history, such as netCDF output, snapshot output, averaging in the model region, output in a
specified sub region, addition of an oﬀset value, multiplication by a factor, and so on. See docs/README.Monitor for theavailable options.
History outputs of the sea ice model are also specified by nml_history in the same manner as the ocean model. Statevariables that can be monitored are listed in the sea ice section of docs/README.Monitor. The format of output files
is also common, except that a missing value for grids of ocean without sea ice should be diﬀerent from the one for land
grids. By default, the value of 0.e0 is used for the former, while -9.99e33 for the latter. (These values can be changed by
nml_seaice_hst written in NAMELIST.name_model.)21.6 MRI.COM eXecution Environment (MXE)
We have been developing a package "MRI.COM eXecution Environment (MXE)" that aggregates programs for preprocessing, execution, postprocessing, and analysis of MRI.COM experiments. By using prepared tools, users can relatively
easily perform complex work of model building and various analyses. In fact, this package is also used as a common basis
for developing various ocean models at the Meteorological Research Institute.
The MXE package includes the following tools.
• Preprocessing tools for creating experiment setup files (directory prep)• Directory template for running MRI.COM (exp)• Postprocessing tools for visualizing output files (postp)• Various analysis tools (anl)– 270 –

Chapter 21

User’s Guide• A Fortran library that provides basic subroutines such as grid and topography information (lib)The main programs are written in Fortran and sample shell scripts are also included for execution. Several tools in the
package have been confirmed by unit testing. This package is managed independently from MRI.COM, but it is provided
to users as an accompanying material. Since it is often updated like MRI.COM, it is recommended to use the latest version.
For details on how to use MXE, see the file README-MXE.md in the top directory (in Japanese).

21.7 Appendix
21.7.1 Model options
The model options are listed on Table 21.12. Only major options are listed here. Description about those related tobio-geochemical models can be found in Chapter 11. The description of all options for the latest version can be foundin src/README.Options. Though an expression like OGCM_PARALLEL is used in the source program, OGCM_ is omitted
when users specify options in configure.in.
Table21.12 Description of Model Options

Model option
BBL
BIHARMONICBULKKONDO2BULKNCAR

BULKITERCALALBSICALPPCARBONCBNHSTRUNCHLMA94CYCLICDIFAJSF2003
FSVISCGMANISOTROPGMTAPER

Description
uses the bottom boundary layer model
uses biharmonic operator for both horizontal viscosity and diﬀusion(*) If ISOPYCNAL is also selected, the biharmonic form is used only for viscosity and not for diﬀusion.
Kondo (1975) is used for the surface flux bulk formula.
Large and Yeager (2004; 2009) is used for the surface flux bulk formula. This option corresponds to
the COREs.
(*) BULKKONDO2, BULKNCAR is available only for HFLUX case.
Bulk transfer coeﬃcient is calculated using iterative method if the observed wind speed is not at
10 m.
(*) use with BULKKONDO2 and BULKNCAR
Sea-ice albedo is calculated using sea-ice conditions according to Los-Alamos model instead of using
a constant value
considers the time variation of pressure for the equation of state
bio-geochemical process is included(*) numtrc_p= 4 for Obata-Kitamura model; numtrc_p= 8 for NPZD model
atmospheric pCO2 is given from file(*) use with CARBON
shortwave penetration scheme with chlorophyll concentration by Morel and Antoine (1994)(*) use with NPZD and SOLARANGLE
uses zonally cyclic condition
sets large vertical diﬀusion coeﬃcient (1.0 m2 s−1 ) between unstable points instead of convective
adjustment
Program uses Fortran 2003 features
calculates viscosity explicitly in the barotropic momentum equation
Anisotropic horizontal variation of thickness diﬀusion is used(*) use with ISOPYCNALTaper GM vector stream function near the sea surface(*) cannot be used with SLIMIT, GMANISOTROP, AFC
Continued on next page– 271 –

21.7 Appendix

Model option
GMVAR
GLS
HFLUX
ICE
ICECLIMISOPYCNALISOTAPERLWDOWNMELYAM
MPDATAADVECNOHKIM
NOMATSUNONPZDOFFNESTPAR
OFFNESTSUB
PARALLEL
PARENT
RUNOFFSFLUXRSFLUXWSIDYN
SLIMITSLPSMAGHDSMAGORSOLARANGLE
SOMADVECSPHERICALSUBTAUBULK
TDEW
TIDE
TRCBIHARM

Table 21.12 – continued from previous page
Description
Horizontal thickness diﬀusion is allowed to vary in horizontal
Generic length scale model of Umlauf and Burchard (2003)
calculates sea surface heat flux using bulk formulasea ice is included
reading climatological sea-ice fractional area from fileuses isopycnal diﬀusion and Gent-McWilliams’ parameterization for eddy
induced tracer transport velocity (thickness diﬀusion)
Taper isopycnal diﬀusion coeﬃcient (ISOPYCNAL)
Longwave radiation data include only downward component instead of default net radiation
uses Mellor and Yamada Level 2.5 for mixed layer model
uses MPDATA for tracer advectionuses Noh’s mixed layer model
uses forward finite diﬀerence instead of Matsuno scheme
NPZD process is included(*) use with CARBON
Used as the lower-resolution part of an oﬀ-line one-way nesting calculation
Used as the higher-resolution part of an oﬀ-line one-way nesting calculation
parallel calculation using MPI. The number of zonally and meridionally partitioned regions shouldbe specified as NPARTX and NPARTY, respectively.
executed as low resolution model of the nesting calculation
uses river runoﬀ data(*) available only for WFLUX
SSS is restored to the climatological sea surface salinity as the salinity flux
calculates salinity flux converting from the surface freshwater flux(*) available only for WFLUX
sea-ice dynamics model with EVP rheology(*) available only for ICE
Tapers thickness diﬀusion near the sea surface
Sea surface is elevated/depressed according to surface atmospheric pressure
uses the Smagorinsky viscosity coeﬃcient multiplied by a constant ratio as the horizontal diﬀusion
coeﬃcient(*) available only for SMAGOR
uses the Smagorinsky parameterization for horizontal viscosity
solar insolation angle is considered in calculating shortwave penetration
uses second order moment advection by Prather (1986)
calculates scale factor semi-analytically for spherical coordinates
executed as a high resolution model of the nesting calculation
calculates the wind stress using bulk formulae by reading wind speed over the ocean
reads dew-point temperature and converts to specific humidity(*) available only for HFLUX
Tide producing forcing is activateduses biharmonic operator for horizontal diﬀusion(*) Should not be used with ISOPYCNAL
Continued on next page– 272 –

Chapter 21

User’s Guide

Model option
TRIPOLARUTZQADVECVISANISO
VIS9P
VISBIHARM
VMBG3D
VVDIMP
WADJ
WFLUX
ZSTAR
MOVE
SCUP
SCUPCGCM
SCUPNEST

Table 21.12 – continued from previous page
Description
Tripolar system is used to construct model grids of a global model(*) Cannot be used with SPHERICAL"UTOPIA" and "QUICKEST" scheme can be used for horizontal and vertical tracer advection withultimate limiter
Anisotropic viscosity coeﬃcients are used(*) use with VIS9P
calculates the viscosity using adjacent 9 grid points
uses biharmonic operator for both horizontal viscosity
reads 3-D vertical viscosity and diﬀusion coeﬃcients from a filecalculates the vertical diﬀusion/viscosity by implicit method(*) it is automatically loaded if any mixed layer model is used or ISOPYCNAL option is selectedadjusts sea surface freshwater flux every time step to keep its global sum to be zero(*) available only for WFLUX
uses the sea surface freshwater flux to force the model
zstar (z*) vertical coordinate(*) KSGM = KM
used as ocean module for data assimilation (MOVE) systemuse simple coupler (SCUP) libraryused as an ocean module for a coupled model using scup for communication
on-line nesting(*) use with SCUP

– 273 –

Acknowledgments
The authors of this manual made new contributions to the source code development of MRI.COM after publication of
the previous version in 2010. They would like to express their sincere gratitude to the following persons who wereinvolved in the development of the previous versions on which MRI.COM version 4 has been built: The developmentof MRI.COM was first proposed by Yoshiteru Kitamura. The early stage of its development was led by Ichiro Ishikawa,whose contribution was indispensable for the existence of MRI.COM. The thermodynamic process of sea ice is basedon what developed by Tatsuo Motoi. During the course of the development, significant contributions were also made by
Tamaki Yasuda, Atsushi Obata, and Seiji Yukimoto.
The authors owe special thanks to Hiroyasu Hasumi of the Atmosphere and Ocean Research Institute (AORI), theUniversity of Tokyo for his valuable suggestions. Application of the EVP rheology to the sea ice model, of σ-coordinate
to the free surface formulation, and of high-accuracy advection schemes to the tracer equations would not have beensuccessful without his support. The continuous collaborative relation with the development group of COCO, which is also
maintained by Hasumi-san, has been of great help to keep MRI.COM at a high level in international standards.
The experiences of coupling with other component models, such as atmospheric models and data assimilation schemes,
greatly helped improve this model. The authors are grateful to the developers of those models. Among them, various
comments made by Yosuke Fujii during his development of an adjoint code of this model were indispensable for refining
the model codes.
Thanks are extended to Shunji Konaga, Tatsushi Tokioka, Masahiro Endoh, Yoshihiro Kimura, Noriya Yoshioka for
their numerous contributions to the original models that MRI.COM is based on.
Introduction of state-of-the-art technologies of typesetting using LATEX was contributed by Shigeru Hayashi.
Most figures of this manual were drawn by Yukiko Suda.
The people listed above and all other members and former members of the Oceanographic Research Department (atthe time) and the Oceanography and Geochemistry Research Department of MRI contributed to the development andrefinement of this model. Specifically, earlier versions of this manual (for version 4) were proofread by some members of
this department, which was of great help for the improvement of those earlier versions.
Since the first version of this manual was published in Japanese, some authors had been requested or encouraged to
publish an English version on many occasions to introduce this model at international scientific conferences and workshops.
This fact strongly motivated us to write the later versions in English. Continuous encouragement and numerous suggestions
by Stephen Griﬃes at Geophysical Fluid Dynamics Lab., NOAA, USA and Gurvan Madec at LODYC, Institute Pierre
Simon Laplace, France, during panel meetings of the working group for ocean model development (WGOMD), now theocean model developing panel (OMDP), of CLIVAR are gratefully acknowledged.

– 275 –

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IndexA
AFC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117, 271
B
BBL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189, 192, 262, 271
BIHARMONIC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80, 271
BULKITER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163, 165, 166, 271
BULKKONDO2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156, 165, 166, 271
BULKNCAR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156, 166, 271
C
CALALBSI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199, 219, 221, 271
CALPP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37, 264, 271
CARBON . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133, 271, 272
CBNHSTRUN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133, 271
CFC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
CHLCLIM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
CHLMA94 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133, 153, 154, 271
configure.in28, 29, 133, 163, 165, 192, 215, 219, 236, 238, 261, 262, 270
CYCLIC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
D
DIFAJS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120, 271
F
F2003 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
FSVISC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61, 271
G
GLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175, 187, 265, 272
GMANISOTROP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116, 117, 118, 271
GMTAPER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115, 117, 271
GMVAR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113, 117, 118, 272
H
HFLUX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151, 152, 156, 159, 271, 272
HISTICECAT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219, 222
I
ICE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219, 262, 265, 272
ICECLIM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264, 272
ICEFULLMONIT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219, 222, 223
IDEALAGE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
IMUT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29, 262, 263
ISOPYCNAL . . . . . . . . . . . . . . . . . . 108, 109, 112, 120, 139, 271, 272, 273
ISOTAPER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111, 112, 272
J
JMUT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29, 262, 263
JOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28, 29, 30, 31, 255
K
KBBL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192, 262
KM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29, 189, 192, 193, 262, 273
KSGM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262, 273
L
LWDOWN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151, 152, 155, 264, 272
M
MELYAM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175, 180, 265, 272
MOVE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
MPDATAADVEC . . . . . . . . . . . . . . . . . . . . 85, 86, 105, 139, 140, 141, 272

N
NAME_MODEL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238, 262, 270
NAMELIST_SCUP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
NGETFLUX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172, 233, 234, 238
nml_air_ice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200, 221
nml_albedo_ocean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153, 154
nml_albedo_seaice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
nml_baroclinic_run . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81, 268
nml_baroclinic_visc_horz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77, 80
nml_barotropic_diﬀ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
nml_barotropic_model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
nml_barotropic_run . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62, 268
nml_barotropic_visc_horz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
nml_bblgeo_limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
nml_bblrayfrc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
nml_bioNPZD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
nml_bottom_friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
nml_bulk_wind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
nml_calendar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
nml_calpp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38, 39
nml_cgcm_recalc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
nml_cgcm_recalc_replace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
nml_density_run . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39, 268
nml_diﬀ_vert_bg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
nml_emissivity_sea_water . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
nml_exp_start . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
nml_force_data . . . . . . . . . . . . . . . . . 61, 138, 151, 152, 158, 167, 262, 264
nml_force_windsetup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150, 151
nml_frac_nearir . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
nml_gls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
nml_gls_run . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188, 268
nml_gm_anisotrop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
nml_gm_transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116, 118
nml_gmanisotrop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116, 117
nml_gmvar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
nml_gmvar_danabasoglu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115, 118
nml_gmvar_eden . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
nml_gmvar_select . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113, 114, 115, 117
nml_gmvar_visbeck . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
nml_grid_scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30, 262
nml_grid_size_change_mix_coefs . . . . . . . . . . . . . . . . . . . . . . 109, 113, 117
nml_history . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222, 270
nml_horz_grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29, 30, 262
nml_hvisc_add . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
nml_isopy_slope_clip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110, 112, 115
nml_melyam_run . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181, 268
nml_mkflux. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .158, 160
nml_nest_par . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235, 239, 240
nml_nest_sub . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240, 241
nml_nkoblm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183, 184
nml_nohkim_run . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183, 184, 268
nml_par_diﬀ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239, 240
nml_par_visc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239, 240
nml_parentmodel_grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236, 237
nml_parentmodel_package . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236, 237
nml_poles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28, 29, 255
nml_run_ini . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
nml_run_ini_state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39, 140, 184, 188, 268
nml_run_period . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
nml_seaice_budget . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
nml_seaice_calendar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
nml_seaice_diﬀ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
nml_seaice_dyn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
nml_seaice_hst . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222, 270– 283 –

nml_seaice_model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220, 221
nml_seaice_rst_in . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221, 265
nml_seaice_rst_out . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
nml_seaice_run . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221, 268
nml_seaice_thickness_category . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219, 220
nml_seaice_time_calendar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
nml_seaice_time_conf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220, 266
nml_smagor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77, 80
nml_solarangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
nml_somadv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103, 265, 266
nml_srﬄx_dbase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163, 165, 168, 169
nml_sss_restore . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160, 233
nml_stdout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
nml_sub_diﬀ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240, 242
nml_sub_isopyc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240, 242
nml_sub_ssh_diﬀ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240, 242
nml_sub_visc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240, 242
nml_submodel_bnd_cnd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237, 238
nml_submodel_grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237, 238, 240, 241
nml_submodel_package . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237, 238
nml_submodel_tide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
nml_tide_forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
nml_tide_model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
nml_tide_run . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61, 268
nml_time_step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248, 269
nml_topo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262
nml_tracer_adv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
nml_tracer_data . . . . . . . . . . . 137, 138, 140, 141, 143, 144, 145, 150, 268
nml_tracer_diﬀ_horz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
nml_tracer_diﬀ_isopy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .112, 116
nml_tracer_diﬀ_isopy_taper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111, 112
nml_tracer_idealage_start . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
nml_tracer_run . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140, 268
nml_vert_grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29, 30, 262
nml_visaniso . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76, 80
nml_visc_vert_bg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81, 181, 183
nml_vmbg3d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
nml_vmix_river . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
nml_vmix_run . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268
nml_waterflux_adjust . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158, 159
nmlrs_density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39, 264
nmlrs_gls_alo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188, 265
nmlrs_gls_avk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188, 265
nmlrs_gls_avp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
nmlrs_gls_avq . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
nmlrs_gls_eke . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188, 265
nmlrs_gls_psi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188, 265
nmlrs_my_alo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181, 265
nmlrs_my_avq . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181, 265
nmlrs_my_q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181, 265
nmlrs_nk_avq . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184, 265
nmlrs_nk_eb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
nmlrs_ptrc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133, 137, 138, 265
nmlrs_s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140, 264
nmlrs_ssh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264
nmlrs_ssh_dflx_x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57, 264
nmlrs_ssh_dflx_y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57, 264
nmlrs_t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140, 264
nmlrs_tide_ssh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
nmlrs_tide_um . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
nmlrs_u . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264
nmlrs_uml . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264
nmlrs_v . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264
nmlrs_vmix_avd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
nmlrs_vmix_avm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
nmlrs_vml . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264
NOHKIM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175, 183, 265, 272
NOMATSUNO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272
NPARTX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262, 272
NPARTY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262, 272
NPUTFLUX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233, 234, 238
NPZD. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .133, 134, 153, 154, 271, 272

NSFMRGN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262
NUM_ICECAT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215, 219, 262
NUMTRC_P . . . . . . . . . . . . . . . . . . . . . . . . . . . 133, 137, 138, 262, 265, 271

O
O2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
OBT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
OFFNESTPAR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236, 272
OFFNESTSUB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236, 272
OPTIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28, 219
P
PARALLEL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262, 272
PARENT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238, 272
Q
QUICKADVEC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85, 86, 139, 140
R
RUNOFF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120, 158, 264, 272
S
SCUP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
SCUPCGCM . . . . . . . . . . . . . . . . . . . . . . . . . . . 172, 216, 233, 234, 238, 273
SCUPNEST . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238, 273
SF6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .138
SFLUXR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129, 160, 272
SFLUXW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129, 156, 158, 160, 272
SIDYN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219, 220, 272
SLIMIT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115, 117, 271, 272
SLP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50, 61, 152, 264, 272
SMAGHD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80, 272
SMAGOR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63, 75, 77, 80, 272
SOLARANGLE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152, 153, 154, 271, 272
SOMADVEC . . . . . . . . . . . . . . . 85, 86, 103, 139, 140, 141, 265, 266, 272
SPHERICAL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28, 30, 255, 263, 272, 273
STABLERUN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
SUB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172, 238, 272
T
TAUBULK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150, 152, 200, 264, 272
TDEW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272
TIDE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57, 61, 265, 272
TRCBIHARM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108, 109, 117, 272
TRIPOLAR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28, 29, 30, 31, 255, 273
TURBULENCE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
U
UTZQADVEC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85, 86, 91, 139, 273
V
VIS9P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80, 81, 273
VISANISO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75, 76, 80, 273
VISBIHARM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63, 75, 76, 273
VMBG3D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119, 273
VVDIMP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119, 139, 265, 273
W
WADJ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158, 232, 233, 273
WFLUX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158, 159, 160, 264, 272, 273
Z
ZSTAR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262, 273– 284 –TECHNICAL REPORTS OF THE METEOROLOGICAL RESEARCH INSTITUTE No.80 2017

気象研究所技術報告一覧表
第１号

バックグラウンド大気汚染の測定法の開発（地球規模大気汚染特別研究班，1978）
Development of Monitoring Techniques for Global Background Air Pollution. (MRI Special Research Group on Global
Atmospheric Pollution, 1978)
第２号
主要活火山の地殻変動並びに地熱状態の調査研究（地震火山研究部，1979）
Investigation of Ground Movement and Geothermal State of Main Active Volcanoes in Japan. (Seismology andVolcanology Research Division, 1979)
第３号
筑波研究学園都市に新設された気象観測用鉄塔施設（花房龍男, 藤谷徳之助, 伴野登, 魚津博，1979）
On the Meteorological Tower and Its Observational System at Tsukuba Science City. (T. Hanafusa, T. Fujitani, N. Banno,and H. Uozu, 1979)
第４号
海底地震常時観測システムの開発（地震火山研究部，1980）
Permanent Ocean－Bottom Seismograph Observation System. (Seismology and Volcanology Research Division, 1980）
第５号
本州南方海域水温図－400m（又は 500m）深と 1,000m 深－（1934－1943 年及び 1954－1980 年）（海洋研究部，
1981）
Horizontal Distribution of Temperature in 400m (or 500m) and 1,000m Depth in Sea South of Honshu, Japan and Western－North Pacific Ocean from 1934 to 1943 and from 1954 to 1980. (Oceanographical Research Division, 1981)
第６号
成層圏オゾンの破壊につながる大気成分及び紫外日射の観測（高層物理研究部，1982）
Observations of the Atmospheric Constituents Related to the Stratospheric ozon Depletion and the Ultraviolet Radiation.
(Upper Atmosphere Physics Research Division, 1982)
第７号
83 型強震計の開発（地震火山研究部，1983）
Strong－Motion Seismograph Model 83 for the Japan Meteorological Agency Network. (Seismology and Volcanology
Research Division, 1983)
第８号
大気中における雪片の融解現象に関する研究（物理気象研究部，1984）
The Study of Melting of Snowflakes in the Atmosphere. (Physical Meteorology Research Division, 1984)
第９号
御前崎南方沖における海底水圧観測（地震火山研究部・海洋研究部，1984）
Bottom Pressure Observation South off Omaezaki, Central Honsyu. (Seismology and Volcanology Research Division andOceanographical Research Division, 1984)
第 10 号 日本付近の低気圧の統計（予報研究部，1984）
Statistics on Cyclones around Japan. (Forecast Research Division, 1984)
第 11 号 局地風と大気汚染質の輸送に関する研究（応用気象研究部, 1984）
Observations and Numerical Experiments on Local Circulation and Medium－Range Transport of Air Pollutions.
(Applied Meteorology Research Division, 1984)
第 12 号 火山活動監視手法に関する研究（地震火山研究部，1984）
Investigation on the Techniques for Volcanic Activity Surveillance. (Seismology and Volcanology Research Division,
1984)
第 13 号 気象研究所大気大循環モデル－Ⅰ（MRI・GCM－Ⅰ）（予報研究部，1984）
A Description of the MRI Atmospheric General Circulation Model (The MRI・GCM－Ⅰ). (Forecast Research Division,
1984)
第 14 号 台風の構造の変化と移動に関する研究－台風 7916 の一生－（台風研究部，1985）
A Study on the Changes of the Three - Dimensional Structure and the Movement Speed of the Typhoon through its Life
Time. (Typhoon Research Division, 1985)
第 15 号 波浪推算モデル MRI と MRI－Ⅱの相互比較研究－計算結果図集－（海洋気象研究部，1985）
An Intercomparison Study between the Wave Models MRI and MRI － Ⅱ － A Compilation of Results －(Oceanographical Research Division, 1985)
第 16 号 地震予知に関する実験的及び理論的研究（地震火山研究部，1985）
Study on Earthquake Prediction by Geophysical Method. (Seismology and Volcanology Research Division, 1985)
第 17 号 北半球地上月平均気温偏差図（予報研究部，1986）
Maps of Monthly Mean Surface Temperature Anomalies over the Northern Hemisphere for 1891－1981. (Forecast
Research Division, 1986)
第 18 号 中層大気の研究（高層物理研究部, 気象衛星研究部, 予報研究部, 地磁気観測所，1986）
Studies of the Middle Atmosphere. (Upper Atmosphere Physics Research Division, Meteorological Satellite Research
Division, Forecast Research Division, MRI and the Magnetic Observatory, 1986)
第 19 号 ドップラーレーダによる気象・海象の研究（気象衛星研究部・台風研究部・予報研究部・応用気象研究部・海洋研究部，1986）
Studies on Meteorological and Sea Surface Phenomena by Doppler Radar. (Meteorological Satellite Research Division,
Typhoon Research Division, Forecast Research Division, Applied Meteorology Research Division, and OceanographicalResearch Division, 1986)
第 20 号 気象研究所対流圏大気大循環モデル（MRI・GCM－Ⅰ）による 12 年間分の積分（予報研究部，1986）
Mean Statistics of the Tropospheric MRI・GCM－Ⅰbased on 12－year Integration. (Forecast Research Division, 1986)
第 21 号 宇宙線中間子強度 1983－1986（高層物理研究部，1987）
Multi－Directional Cosmic Ray Meson Intensity 1983－1986. (Upper Atmosphere Physics Research Division, 1987)

TECHNICAL REPORTS OF THE METEOROLOGICAL RESEARCH INSTITUTE No.80 2017

第 22 号

静止気象衛星「ひまわり」画像の噴火噴煙データに基づく噴火活動の解析に関する研究（地震火山研究部，1987）
Study on Analysis of Volcanic Eruptions based on Eruption Cloud Image Data obtained by the Geostationary
Meteorological satellite (GMS). (Seismology and Volcanology Research Division, 1987)
第 23 号 オホーツク海海洋気候図（篠原吉雄, 四竃信行，1988）
Marine Climatological Atlas of the sea of Okhotsk. (Y. Shinohara and N. Shikama, 1988)
第 24 号 海洋大循環モデルを用いた風の応力異常に対する太平洋の応答実験（海洋研究部，1989）
Response Experiment of Pacific Ocean to Anomalous Wind Stress with Ocean General Circulation Model.
(Oceanographical Research Division, 1989)
第 25 号 太平洋における海洋諸要素の季節平均分布（海洋研究部，1989）
Seasonal Mean Distribution of Sea Properties in the Pacific. (Oceanographical Research Division, 1989)
第 26 号 地震前兆現象のデータベース（地震火山研究部, 1990）
Database of Earthquake Precursors. (Seismology and Volcanology Research Division, 1990)
第 27 号 沖縄地方における梅雨期の降水システムの特性（台風研究部，1991）
Characteristics of Precipitation Systems During the Baiu Season in the Okinawa Area. (Typhoon Research Division, 1991)
第 28 号 気象研究所・予報研究部で開発された非静水圧モデル（猪川元興・斉藤和雄，1991）
Description of a Nonhydrostatic Model Developed at the Forecast Research Department of the MRI. (M. Ikawa and K.
Saito, 1991)
第 29 号 雲の放射過程に関する総合的研究（気候研究部・物理気象研究部・応用気象研究部・気象衛星・観測システム
研究部・台風研究部，1992）
A Synthetic Study on Cloud－Radiation Processes. (Climate Research Department, Physical Meteorology Research
Department, Applied Meteorology Research Department, Meteorological Satellite and Observation System Research
Department, and Typhoon Research Department, 1992)
第 30 号 大気と海洋・地表とのエネルギー交換過程に関する研究（三上正男・遠藤昌宏・新野 宏・山崎孝治，1992）
Studies of Energy Exchange Processes between the Ocean－Ground Surface and Atmosphere. (M. Mikami, M. Endoh, H.
Niino, and K. Yamazaki, 1992)
第 31 号 降水日の出現頻度からみた日本の季節推移－30 年間の日降水量資料に基づく統計－（秋山孝子，1993）
Seasonal Transition in Japan, as Revealed by Appearance Frequency of Precipitating-Days. －Statistics of Daily
Precipitation Data During 30 Years－(T. Akiyama, 1993)
第 32 号 直下型地震予知に関する観測的研究（地震火山研究部，1994）
Observational Study on the Prediction of Disastrous Intraplate Earthquakes. (Seismology and Volcanology Research
Department, 1994)
第 33 号 各種気象観測機器による比較観測（気象衛星・観測システム研究部，1994）
Intercomparisons of Meteorological Observation Instruments. (Meteorological Satellite and Observation System Research
Department, 1994)
第 34 号 硫黄酸化物の長距離輸送モデルと東アジア地域への適用（応用気象研究部，1995）
The Long－Range Transport Model of Sulfur Oxides and Its Application to the East Asian Region. (Applied Meteorology
Research Department, 1995)
第 35 号 ウインドプロファイラーによる気象の観測法の研究（気象衛星・観測システム研究部，1995）
Studies on Wind Profiler Techniques for the Measurements of Winds. (Meteorological Satellite and Observation System
Research Department, 1995)
第 36 号 降水・落下塵中の人工放射性核種の分析法及びその地球化学的研究（地球化学研究部，1996）
Geochemical Studies and Analytical Methods of Anthropogenic Radionuclides in Fallout Samples. (Geochemical
Research Department, 1996)
第 37 号 大気と海洋の地球化学的研究（1995 年及び 1996 年）（地球化学研究部，1998）
Geochemical Study of the Atmosphere and Ocean in 1995 and 1996. (Geochemical Research Department, 1998)
第 38 号 鉛直２次元非線形問題（金久博忠，1999）
Vertically 2-dmensional Nonlinear Problem (H. Kanehisa, 1999)
第 39 号 客観的予報技術の研究（予報研究部，2000）
Study on the Objective Forecasting Techniques（Forecast Research Department, 2000）
第 40 号 南関東地域における応力場と地震活動予測に関する研究（地震火山研究部，2000）
Study on Stress Field and Forecast of Seismic Activity in the Kanto Region（Seismology and Volcanology Research
Department, 2000）
第 41 号 電量滴定法による海水中の全炭酸濃度の高精度分析および大気中の二酸化炭素と海水中の全炭酸の放射性炭素
同位体比の測定（石井雅男・吉川久幸・松枝秀和，2000）
Coulometric Precise Analysis of Total Inorganic Carbon in Seawater and Measurements of Radiocarbon for the Carbon
Dioxide in the Atmosphere and for the Total Inorganic Carbon in Seawater（I.Masao, H.Y.Inoue and H.Matsueda, 2000）
第 42 号 気象研究所／数値予報課統一非静力学モデル（斉藤和雄・加藤輝之・永戸久喜・室井ちあし，2001）
Documentation of the Meteorological Research Institute / Numerical Prediction Division Unified Nonhydrostatic Model(Kazuo Saito, Teruyuki Kato, Hisaki Eito and Chiashi Muroi，2001)
第 43 号 大気および海水中のクロロフルオロカーボン類の精密測定と気象研究所クロロフルオロカーボン類標準ガスの確立（時枝隆之・井上(吉川)久幸，2004）
Precise measurements of atmospheric and oceanic chlorofluorocarbons and MRI chlorofluorocarbons calibration scale

TECHNICAL REPORTS OF THE METEOROLOGICAL RESEARCH INSTITUTE No.80 2017

(Takayuki Tokieda and Hisayuki Y. Inoue，2004)
PostScript コードを生成する描画ツール"PLOTPS"マニュアル（加藤輝之，2004）
Documentation of "PLOTPS": Outputting Tools for PostScript Code (Teruyuki Kato，2004)
第 45 号 気象庁及び気象研究所における二酸化炭素の長期観測に使用された標準ガスのスケールとその安定性の再評価に関する調査・研究（松枝秀和・須田一人・西岡佐喜子・平野礼朗・澤 庸介・坪井一寛・堤 之智・神谷ひとみ・根本和宏・長井秀樹・吉田雅司・岩野園城・山本 治・森下秀昭・鎌田匡俊・和田 晃，2004）
Re-evaluation for scale and stability of CO 2 standard gases used as long-term observations at the Japan Meteorological
Agency and the Meteorological Research Institute (Hidekazu Matsueda, Kazuto Suda, Sakiko Nishioka, Toshirou Hirano,
Yousuke, Sawa, Kazuhiro Tuboi, Tsutumi, Hitomi Kamiya, Kazuhiro Nemoto, Hideki Nagai, Masashi Yoshida, Sonoki
Iwano, Osamu Yamamoto, Hideaki Morishita, Kamata, Akira Wada，2004)
第 46 号 地震発生過程の詳細なモデリングによる東海地震発生の推定精度向上に関する研究（地震火山研究部，2005）
A Study to Improve Accuracy of Forecasting the Tokai Earthquake by Modeling the Generation Processes (Seismology
and Volcanology Research Department, 2005)
第 47 号 気象研究所共用海洋モデル（MRI.COM）解説（海洋研究部，2005）
Meteorological Research Institute Community Ocean Model (MRI.COM) Manual (Oceanographical Research Department,
2005)
第 48 号 日本海降雪雲の降水機構と人工調節の可能性に関する研究（物理気象研究部・予報研究部，2005）
Study of Precipitation Mechanisms in Snow Clouds over the Sea of Japan and Feasibility of Their Modification by
Seeding (Physical Meteorology Research Department, Forecast Research Department, 2005)
第 49 号 2004 年日本上陸台風の概要と環境場（台風研究部, 2006）
Summary of Landfalling Typhoons in Japan, 2004 (Typhoon Research Department, 2006)
第 50 号 栄養塩測定用海水組成標準の 2003 年国際共同実験報告（青山道夫，2006）
2003 Intercomparison Exercise for Reference Material for Nutrients in Seawater in a Seawater Matrix (Michio Aoyama,
2006)
第 51 号 大気および海水中の超微量六フッ化硫黄(SF 6 )の測定手法の高度化と SF 6 標準ガスの長期安定性の評価（時枝隆之、石井雅男、斉藤 秀、緑川 貴, 2007）
Highly developed precise analysis of atmospheric and oceanic sulfur hexafluoride (SF 6 ) and evaluation of SF 6 standard
gas stability (Takayuki Tokieda, Masao Ishii, Shu Saito and Takashi Midorikawa, 2007)
第 52 号 地球温暖化による東北地方の気候変化に関する研究（仙台管区気象台, 環境・応用気象研究部, 2008）
Study of Climate Change over Tohoku District due to Global Warming (Sendai District Meteorological Observatory,
Atmospheric Environment and Applied Meteorology Research Department, 2008)
第 53 号 火山活動評価手法の開発研究（地震火山研究部, 2008）
Studies on Evaluation Method of Volcanic Activity (Seismology and Volcanology Research Department, 2008)
第 54 号 日本における活性炭冷却捕集およびガスクロ分離による気体計数システムによる 85Kr の測定システムの構築および 1995 年から 2006 年の測定結果（青山道夫, 藤井憲治, 廣瀬勝己, 五十嵐康人, 磯貝啓介, 新田 済, Hartmut
Sartorius, Clemens Schlosser, Wolfgang Weiss, 2008）
Establishment of a cold charcoal trap-gas chromatography-gas counting system for 85Kr measurements in Japan and resultsfrom 1995 to 2006 (Michio Aoyama, Kenji Fujii, Katsumi Hirose, Yasuhito Igarashi, Keisuke Isogai, Wataru Nitta,
Hartmut Sartorius, Clemens Schlosser, Wolfgang Weiss, 2008)
第 55 号 長期係留による 4 種類の流速計観測結果の比較（中野俊也, 石崎 廣, 四竈信行, 2008）
Comparison of Data from Four Current Meters Obtained by Long-Term Deep-Sea Moorings (Toshiya Nakano, Hiroshi
Ishizaki and Nobuyuki Shikama, 2008)
第 56 号 CMIP3 マルチモデルアンサンブル平均を利用した将来の海面水温・海氷分布の推定（水田 亮, 足立恭将, 行本
誠史, 楠 昌司, 2008）
Estimation of the Future Distribution of Sea Surface Temperature and Sea Ice Using the CMIP3 Multi-model Ensemble
Mean (Ryo Mizuta, Yukimasa Adachi, Seiji Yukimoto and Shoji Kusunoki, 2008)
第 57 号 閉流路中のフローセルを用いた分光光度法自動分析装置による海水の高精度 pH T 測定（斉藤 秀, 石井雅男, 緑
川 貴, 井上（吉川）久幸, 2008）
Precise Spectrophotometric Measurement of Seawater pH T with an Automated Apparatus using a Flow Cell in a Closed
Circuit（Shu Saito, Masao Ishii, Takashi Midorikawa and Hisayuki Y. Inoue, 2008）
第 58 号 栄養塩測定用海水組成標準の 2006 年国際共同実験報告（青山道夫,J. Barwell-Clarke, S. Becker, M. Blum, Braga E.S.,
S. C. Coverly, E. Czobik, I. Dahllöf, M. Dai, G. O Donnell, C. Engelke, Gwo-Ching Gong, Gi-Hoon Hong, D. J. Hydes,
Ming-Ming Jin, 葛西広海, R. Kerouel, 清本容子, M. Knockaert, N. Kress, K. A. Krogslund, 熊谷正光, S. Leterme,
Yarong Li, 増田真次, 宮尾 孝, T. Moutin, 村田昌彦, 永井直樹, G. Nausch, A. Nybakk, M. K. Ngirchechol, 小川浩史,
J. van Ooijen, 太田秀和, J. Pan, C. Payne, O. Pierre-Duplessix, M. Pujo-Pay, T. Raabe, 齊藤一浩, 佐藤憲一郎, C.
Schmidt, M. Schuett, T. M. Shammon, J. Sun, T. Tanhua, L. White, E.M.S. Woodward, P. Worsfold, P. Yeats, 芳村 毅, A.
Youénou, Jia-Zhong Zhang, 2008）
2006 Inter-laboratory Comparison Study for Reference Material for Nutrients in Seawater（M. Aoyama, J. Barwell-Clarke,
S. Becker, M. Blum, Braga E. S., S. C. Coverly, E. Czobik, I. Dahllöf, M. H. Dai, G. O. Donnell, C. Engelke, G. C. Gong,
Gi-Hoon Hong, D. J. Hydes, M. M. Jin, H. Kasai, R. Kerouel, Y. Kiyomono, M. Knockaert, N. Kress, K. A. Krogslund, M.
Kumagai, S. Leterme, Yarong Li, S. Masuda, T. Miyao, T. Moutin, A. Murata, N. Nagai, G. Nausch, M. K. Ngirchechol, A.
Nybakk, H. Ogawa, J. van Ooijen, H. Ota, J. M. Pan, C. Payne, O. Pierre-Duplessix, M. Pujo-Pay, T. Raabe, K. Saito, K.
第 44 号

TECHNICAL REPORTS OF THE METEOROLOGICAL RESEARCH INSTITUTE No.80 2017

Sato, C. Schmidt, M. Schuett, T. M. Shammon, J. Sun, T. Tanhua, L. White, E.M.S. Woodward, P. Worsfold, P. Yeats, T.
Yoshimura, A. Youénou, J. Z. Zhang, 2008）
第 59 号 気象研究所共用海洋モデル(MRI.COM)第 3 版解説（辻野博之, 本井達夫, 石川一郎, 平原幹俊, 中野英之, 山中吾郎, 安田珠幾, 石崎廣（気象研究所海洋研究部）, 2010）
Reference manual for the Meteorological Research Institute Community Ocean Model (MRI.COM) Version 3（Hiroyuki
Tsujino, Tatsuo Motoi, Ichiro Ishikawa, Mikitoshi Hirabara, Hideyuki Nakano, Goro Yamanaka, Tamaki Yasuda, andHiroshi Ishizaki (Oceanographic Research Department), 2010）
第 60 号 栄養塩測定用海水組成標準の 2008 年国際共同実験報告（青山道夫, Carol Anstey, Janet Barwell-Clarke, François
Baurand, Susan Becker, Marguerite Blum, Stephen C. Coverly, Edward Czobik, Florence D’amico, Ingela Dahllöf,
Minhan Dai, Judy Dobson, Magali Duval, Clemens Engelke, Gwo-Ching Gong, Olivier Grosso, 平山篤史, 井上博敬,
石田雄三, David J. Hydes, 葛西広海, Roger Kerouel, Marc Knockaert, Nurit Kress, Katherine A. Krogslund, 熊谷正光,
Sophie C. Leterme, Claire Mahaffey, 光田均, Pascal Morin, Thierry Moutin, Dominique Munaron, 村田昌彦, GüntherNausch, 小川浩史, Jan van Ooijen, Jianming Pan, Georges Paradis, Chris Payne, Olivier Pierre-Duplessix, Gary Prove,
Patrick Raimbault, Malcolm Rose, 齊藤一浩, 斉藤宏明, 佐藤憲一郎, Cristopher Schmidt, Monika Schütt, Theresa M.
Shammon, Solveig Olafsdottir, Jun Sun, Toste Tanhua, Sieglinde Weigelt-Krenz, Linda White, E. Malcolm. S. Woodward,
Paul Worsfold, 芳村毅, Agnès Youénou, Jia-Zhong Zhang, 2010）2008 Inter-laboratory Comparison Study of a Reference Material for Nutrients in Seawater（青山道夫, Carol Anstey, Janet
Barwell-Clarke, François Baurand, Susan Becker, Marguerite Blum, Stephen C. Coverly, Edward Czobik, Florence D’amico, Ingela Dahllöf, Minhan Dai, Judy Dobson, Magali Duval, Clemens Engelke, Gwo-Ching Gong, Olivier Grosso, 平
山篤史, 井上博敬, 石田雄三, David J. Hydes, 葛西広海, Roger Kerouel, Marc Knockaert, Nurit Kress, Katherine A.
Krogslund, 熊谷正光, Sophie C. Leterme, Claire Mahaffey, 光田均, Pascal Morin, Thierry Moutin, Dominique Munaron,
村 田 昌 彦 , Günther Nausch, 小 川 浩 史 , Jan van Ooijen, Jianming Pan, Georges Paradis, Chris Payne, Olivier
Pierre-Duplessix, Gary Prove, Patrick Raimbault, Malcolm Rose, 齊藤一浩, 斉藤宏明, 佐藤憲一郎, Cristopher
Schmidt, Monika Schütt, Theresa M. Shammon, Solveig Olafsdottir, Jun Sun, Toste Tanhua, Sieglinde Weigelt-Krenz,
Linda White, E. Malcolm. S. Woodward, Paul Worsfold, 芳村毅, Agnès Youénou, Jia-Zhong Zhang, 2010）
第 61 号 強雨をもたらす線状降水帯の形成機構等の解明及び降水強度・移動速度の予測に関する研究（大阪管区気象台･
彦根地方気象台･京都地方気象台･奈良地方気象台･和歌山地方気象台･神戸海洋気象台･松江地方気象台･鳥取地
方気象台･舞鶴海洋気象台･広島地方気象台･徳島地方気象台･予報研究部, 2010）
Studies on formation process of line-shaped rainfall systems and predictability of rainfall intensity and moving speed（Osaka District Meteorological Observatory, Hikone Local Meteorological Observatory, Kyoto Local MeteorologicalObservatory, Nara Local Meteorological Observatory, Wakayama Local Meteorological Observatory, Kobe Marine
Observatory, Matsue Local Meteorological Observatory, Tottori Local Meteorological Observatory, Maizuru Marine
Observatory, Hiroshima Local Meteorological Observatory, Tokushima Local Meteorological Observatory AND Forecast
Research Department, 2010）
第 62 号 WWRP 北京オリンピック 2008 予報実証/研究開発プロジェクト（齊藤和雄, 國井勝, 原昌弘, 瀬古弘, 原旅人, 山
口宗彦, 三好建正, 黄偉健, 2010）
WWRP Beijing Olympics 2008 Forecast Demonstration/Research and Development Project (B08FDP/RDP)（Kazuo Saito,
Masaru Kunii, Masahiro Hara, Hiromu Seko, Tabito Hara, Munehiko Yamaguchi, Takemasa Miyoshi and Wai-kin Wong,2010）
第 63 号 東海地震の予測精度向上及び東南海・南海地震の発生準備過程の研究（地震火山研究部, 2011）
Improvement in prediction accuracy for the Tokai earthquake and research of the preparation process of the Tonankai and
the Nankai earthquakes (Seismology and Volcanology Research Department, 2011)
第 64 号 気象研究所地球システムモデル第 1 版（MRI-ESM1）―モデルの記述―（行本誠史, 吉村裕正, 保坂征宏, 坂見智法, 辻野博之, 平原幹俊, 田中泰宙, 出牛真, 小畑淳, 中野英之, 足立恭将, 新藤永樹, 籔将吉, 尾瀬智昭, 鬼頭昭雄, 2011）
Meteorological Research Institute-Earth System Model Version 1 (MRI-ESM1) — Model Description — (Seiji Yukimoto,
Hiromasa Yoshimura, Masahiro Hosaka, Tomonori Sakami, Hiroyuki Tsujino, Mikitoshi Hirabara, Taichu Y. Tanaka,
Makoto Deushi, Atsushi Obata, Hideyuki Nakano, Yukimasa Adachi, Eiki Shindo, Shoukichi Yabu, Tomoaki Ose and Akio
Kitoh, 2011)
第 65 号 東南アジア地域の気象災害軽減国際共同研究（斉藤和雄, 黒田徹, 林修吾, 瀬古弘, 國井勝, 小司禎教, 上野充,
川畑拓矢, 余田成男, 大塚成徳, Nurjanna Joko Trilaksono, 許智揚, 古関俊也, Le Duc, Kieu Thi Xin, 黄偉健,
Krushna Chandra Gouda, 2011）
International Research for Prevention and Mitigation of Meteorological Disasters in Southeast Asia (Kazuo Saito, Tohru
Kuroda, Syugo Hayashi, Hiromu Seko, Masaru Kunii, Yoshinori Shoji, Mitsuru Ueno, Takuya Kawabata, Shigeo Yoden,
Shigenori Otsuka, Nurjanna Joko Trilaksono, Tieh-Yong Koh, Syunya Koseki, Le Duc, Kieu Thi Xin, Wai-Kin Wong andKrushna Chandra Gouda, 2011)
第 66 号 太平洋における大気－海洋間二酸化炭素フラックス推定手法（杉本裕之, 平石直孝, 石井雅男, 緑川貴, 2012）
A method for estimating the sea-air CO2 flux in the Pacific Ocean (Hiroyuki Sugimoto, Naotaka Hiraishi, Masao Ishii and
Takashi Midorikawa, 2012)
第 67 号 太平洋における大気－海洋間二酸化炭素フラックス推定手法（坪井一寛, 松枝秀和, 澤庸介, 丹羽洋介, 中村雅
道, 久保池大輔, 岩坪昇平, 齊藤和幸, 花宮義和, 辻健太郎, 大森英裕, 西秀紘, 2012）

TECHNICAL REPORTS OF THE METEOROLOGICAL RESEARCH INSTITUTE No.80 2017

Development of a flask sampling and its high-precision measuring system for greenhouse gases observations using a cargo
aircraft C-130H (Kazuhiro Tsuboi, Hidekazu Matsueda, Yousuke Sawa, Yosuke Niwa Masamichi Nakamura, Daisuke
Kuboike, Shohei Iwatsubo, Kazuyuki Saito Yoshikazu Hanamiya, Kentaro Tsuji, Hidehiro Ohmori, Hidehiro Nishi, 2012)
第 68 号 国際シンポジウム 電子顕微鏡を用いたエアロゾル研究（五十嵐康人, Weijun Li, Peter.R.Buseck, 岡田菊雄, 張代洲, 足立光司, 藤谷雄二, 嶋寺光, 五藤大輔, 三井千珠, 野島雅, 大島長, 松井仁志, 石元裕史, 松木篤, Pradeep
Khatri, 中山智喜, 向井将平, 大石乾詞, 間山憲仁, 坂本哲夫, 直江寛明, 財前祐二, 塩流水洋樹, 田中泰宙, 梶野
瑞王, 2013）
International Symposium on Aerosol Studies Explored by Electron Microscopy (Yasuhito Igarashi, Weijun Li, Peter. R.
Buseck, Kikuo Okada, Daizhou Zhang, Kouji Adachi, Yuji Fujitani, Hikari Shimadera, Daisuke Goto, Chizu Mitsui,
Masashi Nojima, Naga Oshima, Hitoshi Matsui, Hiroshi Ishimoto, Atsushi Matsuki, Pradeep Khatri, Tomoki Nakayama,
Shohei Mukai, Kenji Ohishi, Norihito Mayama, Tetsuo Sakamoto, Hiroaki Naoe, Yuji Zaizen, Hiroki Shiozuru, Taichu Y.
Tanaka and Mizuo Kajino, 2013)
第 69 号 マグマ活動の定量的把握技術の開発とそれに基づく火山活動度判定の高度化に関する研究（地震火山研究部,
2013）
Development of Quantitative Detection Techniques of Magma Activity and Improvement of Evaluation of Volcanic
Activity Level (Seismology and Volcanology Research Department, MRI, 2013)
第 70 号 平成 23 年（2011 年）東北地方太平洋沖地震による津波高の現地調査報告（林豊，前田憲二，対馬弘晃，岡田正
實，木村一洋，岩切一宏, 2013）
Reports on Field Surveys of Tsunami Heights from the 2011 off the Pacific Coast of Tohoku Earthquake (Yutaka Hayashi,
Kenji Maeda, Hiroaki Tsushima, Masami Okada, Kazuhiro Kimura and Kazuhiro Iwakiri, 2013)
第 71 号 気候研究のための気象研究所アンサンブル予測システムの概要とその応用（藪将吉，水田亮，吉村裕正，黒田友二，向川均, 2014）
Meteorological Research Institute Ensemble Prediction System (MRI-EPS) for climate research - Outline and its
applications – (Shoukichi Yabu, Ryo Mizuta, Hiromasa Yoshimura, Yuhji Kuroda, and Hitoshi Mukougawa, 2014)
第 72 号 日本各地域の繰り返し相似地震発生状況に関する研究（地震火山研究部，地震火山部，気象大学校，札幌管区
気象台，仙台管区気象台，大阪管区気象台，福岡管区気象台，沖縄気象台, 2014）
Survey of moderate repeating earthquakes in Japan (Seismology and Volcanology Research Department of MRI,
Seismology and Volcanology Department, Meteorological College, Sapporo Regional Headquarters, Sendai Regional
Headquarters, Osaka Regional Headquarters, Fukuoka Regional Headquarters, and Okinawa Regional Headquarters, 2014)
第 73 号 気象研究所非静力学地域気候モデルによる日本付近の将来気候変化予測について（佐々木秀孝，村田昭彦，川瀬宏明，花房瑞樹，野坂真也，大泉三津夫，水田亮，青栁曉典，志藤文武，石原幸司, 2015）
Projection of Future Climate Change around Japan by using MRI Non-hydrostatic Regional Climate Model (Hidetaka
Sasaki, Akihiko Murata, Hiroaki Kawase, Mizuki Hanafusa, Masaya Nosaka, Mitsuo Oh’izumi, Ryou Mizuta, Toshinori
Aoyagi, Fumitake Shido, and Koji Ishihara, 2015)
第 74 号 新型自己浮上式海底水圧計の開発（平田賢治, 山崎明, 対馬弘晃, 2015）
Development of a new pop-up ocean-bottom pressure gauge (Kenji Hirata, Akira Yamazaki, and Hiroaki Tsushima, 2015)
第 75 号 2012 年・2013 年に日本に接近・上陸した台風の概要と特性（北畠尚子, 小山亮, 嶋田宇大, 櫻木智明, 沢田雅洋,
2015）
Summary and Characteristics of Approaching and Landfalling Tropical Cyclones in Japan in 2012 and 2013 (Naoko
Kitabatake, Ryo Oyama, Udai Shimada, Tomoaki Sakuragi and Masahiro Sawada, 2015)
第 76 号 ＷＭＯ福島第一原発事故に関する気象解析技術タスクチーム活動と気象研究所の大気拡散モデリング（斉藤和雄, 新堀敏基, R. Draxler, 原旅人, 豊田英司, 本田有機, 永田和彦, 藤田司, 坂本雅巳, 加藤輝之, 梶野瑞王, 関山
剛, 田中泰宙, 眞木貴史, 寺田宏明, 茅野政道, 岩崎俊樹, M.C. Hort, S.J. Leadbetter, G. Wotawa, D. Arnold, C.
Maurer, A. Malo, R. Servranckx, P. Chen, 2015）
Contribution of JMA to the WMO Technical Task Team on Meteorological Analyses for Fukushima Daiichi Nuclear
Power Plant Accident and Relevant Atmospheric Transport Modelling at MRI(K. Saito, T. Shimbori, R. Draxler, T. Hara,
E. Toyoda, Y. Honda, K. Nagata, T. Fujita, M. Sakamoto, T. Kato, M. Kajino, T.T. Sekiyama, T.Y. Tanaka, T. Maki, H.
Terada, M. Chino, T. Iwasaki, M.C. Hort, S.J. Leadbetter, G. Wotawa, D. Arnold, C. Maurer, A. Malo, R. Servranckx and P.
Chen, 2015)
第 77 号 海溝沿い巨大地震の地震像の即時的把握に関する研究（地震津波研究部, 2017）
Research on rapid estimation of the parameters for large earthquakes along trenches (Seismology and Tsunami Research
Department, 2017)
第 78 号 2013-2015 年西之島噴火のモニタリングに関する研究（高木朗充, 長岡優, 福井敬一, 安藤忍, 木村一洋, 土山博昭, 2017）
Studies on Monitoring of the 2013 – 2015 Nishinoshima Eruption (Akimichi Takagi, Yutaka Nagaoka, Keiichi Fukui,
Shinobu Ando, Kazuhiro Kimura, and Hiroaki Tsuchiyama, 2017)
第 79 号 2012 年から 2016 年に実施された温室効果ガス観測に関する相互比較実験（iceGGO）（坪井一寛、中澤高清、松枝秀和、町田敏暢、青木周司、森本真司、後藤大輔、下坂琢哉、加藤健次、青木伸行、渡邉卓朗、向井人史、
遠嶋康徳、勝又啓一、村山昌平、石戸谷重之、藤谷徳之助、小出寛、髙橋正臣、川﨑照夫、滝沢厚詩、澤庸介, 2017）
InterComparison Experiments for Greenhouse Gases Observation (iceGGO) in 2012–2016 (A K. Tsuboi, T. Nakazawa, H.
Matsueda, T. Machida, S. Aoki, S. Morimoto, D. Goto, T. Shimosaka, K. Kato, N. Aoki, T. Watanabe, H. Mukai, Y.
Tohjima, K. Katsumata, S. Murayama, S. Ishidoya, T. Fujitani, H. Koide, M. Takahashi, T. Kawasaki, A. Takizawa and Y.
Sawa, 2017)

気

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研

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1 9 4 6（昭和 21）年
所
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研究総務官 ： 齊
研究調整官 ： 竹

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気象衛星・観測システム研究部
地 震 津 波 研 究 部
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海 洋 ・ 地 球 化 学 研 究 部

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部 長 ： 理 博
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気 象研 究所 技術 報告 は 、1978（ 昭 和 53）年の 初刊 以来、気 象 研究 所が 必要 の都 度発 行 する 刊行 物で あり 、気 象 研 究 所 の 研 究 計 画 に 基づ き 実 施 し た 研 究 に 関 す る手 法 、 デ ー タ 、 結 果 等 に つい て の ま と め 、 ま た は 、す で に公 表し た研 究論 文類を と りま とめ 総合 的報 告とし た もの を掲 載す る。

本 紙 に 掲 載 さ れ た 報 告 の著 作 権 は 気 象 研 究 所 に 帰 属す る 。 本 紙 に 掲 載 さ れ た 報告 を 引 用 す る 場 合 は 、出所 を 明 示 す れ ば 気 象 研 究 所の 許 諾 を 必 要 と し な い 。 本紙 に 掲 載 さ れ た 報 告 の 全 部又 は 一 部 を 複 製 、 転 載 、
翻 訳 、 あ る い は そ の 他 に 利用 す る 場 合 は 気 象 研 究 所 の許 諾 を 得 な け れ ば な ら な い。 個 人 が 研 究 、 学 習 、 教
育 に 使用 する 場合 は、 出所を 明 示す れば 気象 研究 所の許 諾 を必 要と しな い。

気

象

研

究

所

技 術 報 告第 80 号

平成 29 年 3 月

編 集 兼
発 行 者

気

ISSN 2189-8871

発行

象 研 究 所

〒305-0052 茨城県つくば市長峰１－１
TEL(029)853-8535