R EFERENCE M ANUAL FOR THE
M ETEOROLOGICAL R ESEARCH I NSTITUTEC OMMUNITY O CEAN M ODEL(MRI.COM)V ERSION 3
気象研究所共用海洋モデル(MRI.COM)
第３版解説
H IROYUKI T SUJINO , TATSUO M OTOI , I CHIRO I SHIKAWA ,M IKITOSHI H IRABARA , H IDEYUKI NAKANO , G ORO YAMANAKA ,TAMAKI YASUDA , AND H IROSHI I SHIZAKI(O CEANOGRAPHIC R ESEARCH D EPARTMENT )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）

Preface
It has long been recognized that the role of the ocean in the earth’s climate system is conclusively importantin such issues as the climate warming, the long-term variability in the air-sea coupled system, meteorologicalextreme phenomena, and so forth. In these situations, modeling of the ocean has become an indispensable methodof studying the climate variability and predicting its future state as well as studying the mechanisms of the oceanic
variability itself.
The Oceanographic 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 modelingactivities are maintained 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 basedon 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), and”Development of ocean environmental forecasting methods” (started from fiscal year 2009). The present publication is the first English version of the MRI.COM manual and has been improved from the Japanese version
published in 2005 by including newly developed parts.
The ocean modeling study in MRI began in the late 1970s for investigating the variability of the Kuroshio southof Japan. First, an ocean model with the primitive-equation system developed by former Prof. K. Takano in
UCLA, USA, was introduced. Another ocean model was then introduced slightly later in 1981. It was similar to
the former but developed by an ocean research group in the University of Tokyo. Since that time, the two ocean
models with different codes have been 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 beenused 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 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 andseasonal 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 initiatedbased on the two ocean models used so far to achieve efficiency in model improvement and management and tointegrate their merits. In designing the new model, the main frame of the former model and the various physicaloptions of the latter model were transferred to the new model, and many newly developed physical processes andschemes were added.
The first Japanese version of the MRI.COM manual was published in 2005 and the model has been continuously
updated through further improvements in physical processes and addition of new processes. One of the mostipronounced improvements is the introduction of the chemical and biogeochemical processes associated with theoceanic carbon cycle. This is continuing even now and will be finished in a few years.
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 and has already achieved many satisfactory results.
Based on our experiences, we believe MRI.COM is one of the best ocean models in the world. We thank the present
and past participants in the model development for their great deal of efforts and help. We hope MRI.COM andthe present manual will contribute to research work in the fields of climatology, oceanography, and environmental
sciences in domestic and foreign institutions as well as to the research and operational work in MRI and JMA.

Hiroshi Ishizaki
Director
Oceanographic Research DepartmentiiЭ૨
地球規模の気候温暖化、大気海洋結合系の長期変動、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）において継続されている。本技術報告は、2005 年に出版された日本語版の解説書の内容を改訂し、日本語版出版後新たに付加された開発項目に関する解説を加えた、最初の英語による解説書である。
海洋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システム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。
最初の日本語による解説書が 2005 年に出版された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。特筆すべきは、海洋炭素循環に関連して、化学過程および生物地球化学過程が付加されたことである。生物地球化学過程の開発は現在も継続中で今後数年で完了すると思われる。

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MRI.COM システム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ある。その経験から、本モデルシステムは世界に幾つかある他の海洋モデルシステム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、本モデルシステム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。

海洋研究部長石崎 廣

iv

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). As the name suggests, it has been used for studying large
scale oceanic phenomena and as the oceanic part of the coupled climate models developed at MRI.
The current version of MRI.COM is version 3. 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 to be 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 ofusers 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 eventuallybecame the prototype of this manual.
Version 3 was intended for use as an oceanic component of the Earth System Model of MRI (MRI-ESM1;
Yukimoto et al., 2010). One of the reasons for creating a new version was that the definition of vertical gridarrangement was modified during the development. MRI plans to participate in the phase five of the Coupled
Model Intercomparison Project (CMIP5) using MRI-ESM1, and its results on the future projection are expected to
be used by a wide range of communities, so we decided to prepare a detailed description of its oceanic component
in English.
Note that the purpose of this manual is to present a detailed description of a particular model system. The
mathematical expressions 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 meanthat they are the complete reflections of physical, mathematical, and numerical integrity. Every method is subject
to possible sophistication. We welcome critical comments and suggestions from any reader or user, which we
believe are necessary for further improvement.
For a more general or detailed description of OGCMs, please refer to textbooks by Griffies (2004) and Kantha
and Clayson (2000). The former thoroughly describes the fundamentals of OGCMs, and the latter concisely
summarizes the modeling of various oceanic processes such as tide and sea ice.

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.

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Part I describes the model configuration. Governing equations are derived in Chapter 2, and the spatial gridarrangement and definition of continuity equations for unit grid cells are presented in Chapter 3.
Part II describes the dynamical core. The method of solving the barotropic and the baroclinic part of the momentum equation are presented in Chapters 4 and 5, respectively. The method of solving the advection-diffusion
equation for tracers (temperature and salinity) is presented in Chapter 6.
Part III describes additional processes. Surface mixed-layer models are presented in Chapter 7, surface fluxes
in Chapter 8, sea ice in Chapter 9, bottom boundary layer parameterization in Chapter 10, and biogeochemical
models are presented in Chapter 11.
Part IV contains miscellaneous topics. Basics of the finite difference method are presented in Chapter 12, somehigh-accuracy advection schemes are presented in Chapter 13, general orthogonal curvilinear coordinates andrelated calculus are introduced in Chapter 14, how to construct a pair of nested-grid models is presented in Chapter
15, and user’s guide to construct and run a model is presented in Chapter 16.
Each chapter is almost independent from other chapters. Thus the readers might be able to understand thecontents of each 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 to express staggered grid arrangements. They do not necessarily correspond to the array indices inprogram codes.

References
Griffies, S. M., 2004: Fundamentals of ocean climate models, Princeton University Press, 518pp.
Ishikawa, I., H. Tsujino, M. Hirabara, H. Nakano, T. Yasuda, and H. Ishizaki, 2005: Meteorological ResearchInstitute Community Ocean Model (MRI.COM) Manual, Technical Reports of the Meteorological ResearchInstitute, No.47, 189pp.
Kantha, L., and C. Clayson, 2000: Numerical models of ocean and oceanic processes, International Geophysics
Series, Vol. 66, 940pp.
Yukimoto, S., and coauthors, 2010: Meteorological Research Institute-Earth System Model v1 (MRI-ESM1) Model Description -, Technical Reports of the Meteorological Research Institute, No.64, in press.

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。海洋大循環モデルの名の通り、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。

MRI.COM の最新バージョンは３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 年台前半）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 （石川他、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（本バージョン）は、気象研地球システムモデル（MRI-ESM1; Yukimoto et al., 2010） の海
洋部分への供用を主な目的としたものである。新バージョン作成理由のひとつは、開発段階で、鉛直格子点の定義位置を変更したことである。MRI-ESM1 は第５次結合モデル相互比較プロジェクト（CMIP5）への参加を予定しており、同モデルによる将来予測結果などは、幅広い分野の人々に使用されることが見込まれるため、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。
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。
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 andClayson (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。

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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。空間格子配置と、単位格子に対する連続方程式の差分式の定義を第 3 章で行う。
第 II 部ではモデルの核心部分の解説を行う。運動方程式の順圧成分と傾圧成分の解法を第 4 章と第 5 章
でそれぞれ述べる。トレーサー（水温と塩分）に対する移流拡散方程式の解法を第 6 章に述べる。
第 III 部では付加的物理プロセスの解説を行う。幾つかの海面混合層モデルを第 7 章で、海面フラックスの取り扱いを第 8 章で、海氷モデルを第 9 章で、海底境界層モデルを第 10 章で、幾つかの生物地球化学モ
デルを第 11 章で解説する。
第 IV 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 12 章で、幾つかの高精度トレーサー移流スキームについて第 13 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 14 章で、入れ子モデルの作成と使用法について第 15 章で解説する。最後にモデルの作成と実行方法の解説を第 16 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が必要である。

References
Griffies, S. M., 2004: Fundamentals of ocean climate models, Princeton University Press, 518pp.
石川一郎・辻野博之・平原幹俊・中野英之・安田珠幾・石崎廣，2005: 気象研究所共用海洋モデル (MRI.COM)
解説，気象研究所技術報告第 47 号，189pp.

Kantha, L., and C. Clayson, 2000: Numerical models of ocean and oceanic processes, International Geophysics
Series, Vol. 66, 940pp.
Yukimoto, S., and coauthors, 2010: Meteorological Research Institute-Earth System Model v1 (MRI-ESM1) Model Description -, Technical Reports of the Meteorological Research Institute, No.64, in press.

viii

Contents
Chapter 1

OGCMs and MRI.COM

1

1.1
1.2

What do OGCMs cover? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Classification of OGCMs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1
1

1.2.1

1.3

Z-coordinate models (z-models) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.2 Sigma-coordinate models (σ -models) . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.3 Isopycnal-coordinate models (ρ -models) . . . . . . . . . . . . . . . . . . . . . . . . .
About MRI.COM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2
3
3
3

1.4

Future of OGCMs and MRI.COM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5Part I

Configuration

Chapter 2
2.1

2.2

2.3

7

Governing Equations

9

Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 Coordinate System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.2 Momentum Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.3 Continuity equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.4 Temperature and salinity equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.5 Equation of state of sea water . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.6 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.7 Acceleration method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9
9
9
11
11
13
13
15
16
16

2.2.2

Momentum equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17

2.2.3

Continuity equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20

2.2.4 Temperature and salinity equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.5 Equation of state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20
21
24

Physical constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

24

2.3.1
Chapter 3

Spatial grid arrangement and definition of continuity equation

27

Horizontal grid arrangement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Vertical grid arrangement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Indices and symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27
27
28

3.4

Continuity equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29

3.5

Calculation of area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

32

General orthogonal coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Geographic coordinate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

32
33

3.1
3.2
3.3

3.5.1
3.5.2

ix

Part II

Main Processes

37

Chapter 4
Equations of motion (barotropic component)
4.1 Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39
40

4.2

Time integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

40

4.3

Prognostics of physical properties in the uppermost layer . . . . . . . . . . . . . . . . . . . . .
4.3.1 Standard scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.2 Locally conserved scheme (option FSMOM) . . . . . . . . . . . . . . . . . . . . . . . .
Introduction of σ -coordinates near the sea surface . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.1 Formulation of σ -layer model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43
43
44
45

4.4

4.4.2
4.4.3

Governing equations in the σ -coordinates . . . . . . . . . . . . . . . . . . . . . . . . .
Redistribution of tracers among the σ -layers . . . . . . . . . . . . . . . . . . . . . . . .

45
46
47

Chapter 5
Equations of motion (baroclinic component)
5.1 Advection terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49
49

5.1.1 Vertical mass fluxes and its momentum advection . . . . . . . . . . . . . . . . . . . . .
5.1.2 Horizontal mass flux and its momentum advection . . . . . . . . . . . . . . . . . . . . .
Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.1 Horizontal viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.2 Horizontal anisotropic viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.3 Smagorinsky parameterization for horizontal viscosity . . . . . . . . . . . . . . . . . .

49
53
59
59
60
60

Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Vertical viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Bottom friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61
62
64

Chapter 6
Temperature and salinity equations
6.1 Flux form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Advection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3 Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

65
65
65
69

6.3.1
6.3.2

Vertical diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Harmonic horizontal diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69
70

6.3.3

Biharmonic horizontal diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

70

6.3.4
6.3.5

Isopycnal diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Gent and McWilliams parameterization . . . . . . . . . . . . . . . . . . . . . . . . . .

71
72

6.3.6

Anisotropic Gent-McWilliams scheme . . . . . . . . . . . . . . . . . . . . . . . . . . .

73

Convective adjustment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

74

Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Numerical procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

74
77

5.2

5.2.4
5.2.5
5.2.6

6.4

6.4.1
6.4.2

Part III

Additional Processes

81

Chapter 7

Mixed layer model

83

7.1

Mellor and Yamada’s Turbulence Closure Model . . . . . . . . . . . . . . . . . . . . . . . . .
7.1.1 Turbulence Closure Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

x

83
83

Level 2.5 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

85
88

Turbulent mixed layer model by Noh and Kim (1999) . . . . . . . . . . . . . . . . . . . . . . .
7.2.1 Fundamental equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

89
89

7.2.2 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
K Profile Parameterization (KPP) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

91
91

7.3.1
7.3.2

Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Monin-Obukhov similarity law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

91
92

7.3.3

Coefficients of vertical viscosity and diffusivity . . . . . . . . . . . . . . . . . . . . . .

93

7.3.4
7.3.5

Coefficients of vertical viscosity and diffusion at the base of the mixed layer . . . . . . .
Thickness of the mixed layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

94
95

7.3.6
7.3.7

Mixing due to shear instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Nonlocal Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

95
95

7.1.2
7.1.3
7.2

7.3

Chapter 8
8.1

8.2

8.3

Sea surface fluxes

8.2.1 Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.2.2 Salinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Heat flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.3.1 Shortwave radiation flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.3.2 Shortwave radiation flux based on chlorophyll concentration . . . . . . . . . . . . . . .
8.3.3
8.3.4

8.4

8.7

8.8
8.9

100
101
102
102
103

Longwave radiation flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
Latent and sensible heat fluxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

Freshwater flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
8.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
8.4.2

8.5
8.6

99

Momentum flux (surface stress) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
8.1.1 Input of wind stress data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
8.1.2 Calculating wind stress using a bulk formula . . . . . . . . . . . . . . . . . . . . . . . 100
Sea surface forcing for temperature and salinity . . . . . . . . . . . . . . . . . . . . . . . . . . 100

Calculating freshwater flux

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

Equivalent surface temperature and salinity fluxes for constant first layer volume . . . . . . . . . 106
Bulk transfer coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
8.6.1

Formulation of bulk formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

8.6.2
8.6.3

Kondo (1975) BULKKONDO2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
Large and Yeager (2004) BULKNCAR . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

8.6.4

Kara et al. (2002) BULKKARA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

8.6.5

Bulk coefficient over sea ice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

Work flow in MRI.COM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
8.7.1 Momentum flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
8.7.2 Temperature (heat) flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
8.7.3 Salinity and fresh water flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
8.9.1

Unit of constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

8.9.2

Unit of variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
xiChapter 9
Sea ice119
9.1 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
9.2

Thermodynamic processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
9.2.1 Formation of new sea ice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
9.2.2
9.2.3

Air-ice interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
Heat balance in the ice interior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

9.2.4
9.2.5

Ice-ocean interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
Archimedes’ Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

9.3

Remapping in thickness space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

9.4

Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
9.4.1 Momentum equation for ice pack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
9.4.2
9.4.3

Stresses at top and bottom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
Internal stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

9.4.4
9.4.5

Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
Solution procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

9.5

Advection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

9.6
9.7

Ridging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

9.8

9.7.1 Advection (MPDATA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
9.7.2 Momentum equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
Technical issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
9.8.1
9.8.2
9.8.3

9.9

Source codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
Coupling with an atmospheric model . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
Job parameters (namelist) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
9.9.1 Saturation water vapor pressure and latent heat . . . . . . . . . . . . . . . . . . . . . . 144
9.9.2 Physical constant, parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

Chapter 10

Bottom Boundary Layer (BBL)

147

10.1 General description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
10.2 Grid arrangement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
10.3 Pressure gradient terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
10.4 Eddy effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
10.5 Usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
10.6 Usage notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
10.6.1 Limit of the area where BBL model should be applied . . . . . . . . . . . . . . . . . . . 151
10.6.2 Limits of the BBL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
10.6.3 Notes for the program code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
Chapter 11 Biogeochemical model
153
11.1 Inorganic carbon cycle and biological model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
11.2 Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
11.3 Carbon cycle component . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
11.3.1 Air-sea gas exchange fluxes at the sea surface (Jg ) . . . . . . . . . . . . . . . . . . . . . 155

xii

11.3.2 Dilution and concentration effects of evaporation and precipitation on DIC and Alk . . . . 157
11.4 Obata and Kitamura model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
11.5 NPZD model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
11.5.1 Description of each term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
11.5.2 Primary Production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
11.5.3 Variation of DIC and Alk due to biological activity . . . . . . . . . . . . . . . . . . . . 161
11.6 Usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
11.7 Program structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

Part IV

Miscellaneous

169

Chapter 12 Basics of the finite difference method
17112.1 Diffusion equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
12.2 Finite difference expressions for time derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . 172
12.3 Finite difference expression for space derivatives . . . . . . . . . . . . . . . . . . . . . . . . .
12.4 Finite differencing of advection-diffusion equation . . . . . . . . . . . . . . . . . . . . . . . . .
12.5 Implicit method for vertical diffusion equation . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.5.1 A solution of tri-diagonal matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

172
174
174
175

Chapter 13 Tracer advection schemes
177
13.1 QUICKEST for vertical advection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
13.2 UTOPIA for horizontal advection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
13.3 Second Order Moment (SOM) scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
13.3.1 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
13.3.2 Calculating SOM advection in MRI.COM . . . . . . . . . . . . . . . . . . . . . . . . . 188
Chapter 14

Generalized orthogonal curvilinear coordinate grids

191

14.1 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
14.2 Generation of orthogonal coordinate system using conformal mapping . . . . . . . . . . . . . . 192
14.3 Rotation of vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
14.4 Mapping a quantity from geographic coordinates to transformed coordinates . . . . . . . . . . . 195
14.5 Vector operation and differentiation in a general orthogonal coordinate system . . . . . . . . . . 197
Chapter 15

Nesting199

15.1 Feature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
15.2 Low-resolution model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
15.3 High-resolution model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
15.3.1 Required data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
15.3.2 Creating data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
15.4 Usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
15.4.1 Compilation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
15.4.2 Running the models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
15.5 Program structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

xiii

Chapter 16 User’s Guide
213
16.1 Model setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
16.1.1 Files needed for compilation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
16.1.2 Compilation of the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
16.2 Preparation of input data files for execution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
16.2.1 Topographic and grid spacing data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
16.2.2 Climatological data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
16.2.3 Nudging (body forcing) data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
16.2.4 Atmospheric forcing data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
16.3 Execution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
16.4 Structure of output files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
16.4.1 Snapshot (restart) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
16.4.2 Averaged value (history) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234
16.5 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
16.5.1 Model options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

xiv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 are 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 areeither unresolved or neglected might be incorporated in some forms of sub-grid scale (SGS) parameterizations. In
the future, the coverage of the OGCMs will be extended to directly resolve the smaller space-scale and the shortertime-scale phenomena such as tides and vertical convection, but they are not the primary targets of the current
OGCMs. The current OGCMs could cover phenomena from thermohaline circulations to mesoscale eddies. However, it is almost impossible to conduct 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, evenwith 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 severalhundred-year integration of a high resolution model that resolves mesoscale eddies using enormous resources (e.g.,
the Earth Simulator), but such a resource is not available to everyone.

1.2

Classification of OGCMs

Most OGCMs used by ocean research scientists and by operational centers for forecasting climate and oceanicstates numerically 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 fortemperature and salinity, the equation of state of sea water, and the mass conservation equation, collectively calledprimitive 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 that attempt to simulate realistic
oceanic states adopt the finite difference method to discretize the equations. The spectral approach widely usedby an atmospheric model 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 models.
Ocean models 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: z-coordinate models or z-models
adopt depth as the vertical coordinate, σ -coordinate or terrain following models adopt fractional depth between
the sea surface and the sea floor as the vertical coordinate, and ρ -coordinate or isopycnal models adopt 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’sdisadvantages by using advantages of other classes. Readers are referred to the book by Griffies (2004) for more–

1–

Chapter 1

OGCMs and MRI.COM

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Figure 1.1. Various oceanic phenomena in terms of their space and time scales and coverage of the ocean generalcirculation model. The figure for oceanic phenomena is adopted from von Storch and Zwiers (2001).
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 theGeophysical Fluid Dynamics Laboratory (GFDL) in the 1960’s. 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 isexpressed as a function of depth by zero-order approximation, which makes implementating the equation of statestraightforward. Implementation of bottom topography and drawing of results are also straightforward. The modelsof this class are most widely used in the community because of their versatility. Such models were first used ascomponents of coupled atmosphere-ocean models.
The descendant of the GFDL model is called the Modular Ocean Model (MOM; Griffies et al., 2004) andis the most widely used. Most climate centers participating in climate model intercomparison projects use the zcoordinate models; MRI.COM also adopts z-coordinates. 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
the coast and the sea floor tend to be poorly reproduced.
• Numerical inaccuracy in the tracer transport algorithm immediately leads to spurious diapycnal mixing of– 2 –1.3. About MRI.COM
the transported 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’s disadvantages are not completely overcome; these substitutes have their own difficulties.

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 σ -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
the shoreward 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 theinterior. Thus, the character of a water mass is well maintained in the ocean interior. Since many theoreticalstudies 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 and Boudra, 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
climate change 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 of these disadvantages.

1.3 About MRI.COM
MRI.COM is a z-coordinate model. 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 points, i.e., the lines connecting
the tracer points. This arrangement is suitable for the discrete expressions for the side boundary conditions forvelocity, and transport through a narrow passage can be achieved with a single grid cell.
–

3–Chapter 1

OGCMs and MRI.COM

The program structure of MRI.COM is presented along with the typical computational cost of each process inFigure 1.2.
subroutine name [program ﬁle name]exchange withatmosphere

initialization[stmdlp.F90, rdjobp.F90, rdbndt.F90
glatlon.F90, rdinit.F90, calcoe.F90, etc.]

chap 16

cost
2%
11%

surface ﬂuxes

bulk formulasea icechap 8chap 9mkﬂux [mkﬂux.F90]
bulk [bulk.F90]
simain [ice̲main̲cat.F90]

1%continuity equation(determine vertical transport)

chap 3

cont [cont.F90]

19%equation of motion(baroclinic component)

chap 5clinic [clinic.F90]

8%equation of motion(barotropic component)

chap 4surfce [surfce̲ctl.F90][surfce̲integ.F90]

34%

advection of tracers

chap 6
chap 13

tracer [tracer.F90][trcadv.F90, som̲adv.F90]

17%

diﬀusion of tracers(isopycnal and diapycnal)

chap 6
chap 12

trcimp [trcimp.F90]
ipcoef [ipcoef.F90]
ipycmix[ipycmix.F90]

1.5%calculation of density
stability check and adjustment

chap 2chap 6difajs [difajs.F90]
cnvajs [cnvajs.F90][stable.F90, stbden.F90]

1%

preparation for the next time step

5%

mixed layer model
determine vertical mixing coeﬀ.

chap 7mysl25 [my25.F90]
nkoblm [nkoblm.F90]

0.5%

output to ﬁleswith speciﬁed intervals

chap 16

writdt [writdt.F90]

rewrit [rewrit.F90]
sfc̲rewrit [surface̲ctl.F90]

termination

Figure 1.2. Program Structure of MRI.COM

Though the program code should ideally use MKS units, MRI.COM basically uses cgs units for historicalreasons. The sea ice model, the mixed-layer model, and some surface bulk formulae are coded in MKS units andconverted into cgs units before their outputs are used by the main part. Details are described in each chapter.

– 4 –1.4. Future of OGCMs and MRI.COM

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 scientificand 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 isshifting from 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, ever higher resolution simulations will be achieved. The result will be more strongly affectedby how sub-grid scale processes are parameterized and thus sub-grid 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 data assimilation schemes. Having an interface to universal 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.

References
Arakawa, A., 1972: Design of the UCLA general circulation model, Numerical Simulation Weather and Climate,
Tech. Rep. No. 7, Dep. of Meteorology, University of California, Los Angeles, 116 pp.
Bleck, R., and D. B. Boudra, 1986: Wind-driven spin up in eddy-resolving ocean models formulated in isopycnic
and isobaric coordinates, J. Geophys. Res., 91, 7611-7621.
Bleck, R., G. Halliwell, A. Wallcraft, S. Carroll, K. Kelly, K. Rushing, 2002: Hybrid Coordinate Ocean Model(HYCOM) User’s Manual,available online at http://hycom.rsmas.miami.edu/hycom-model/documentation.html.
Mellor, G. L., 2004: Users guide for a three-dimensional, primitive equation, numerical ocean model, Prog. in
Atmos. and Ocean. Sci, Princeton University, 53pp.,
available online at http://www.aos.princeton.edu/WWWPUBLIC/htdocs.pom/PubOnLine/POL.html.
Griffies, S. M., 2004: Fundamentals of ocean climate models, Princeton University Press, 518pp.
Griffies, S. M., M. J. Harrison, R. C. Pacanowski, and A. Rosati, 2004: A Technical Guide to MOM4, GFDL
Ocean Group Technical Report No. 5, NOAA/Geophysical Fluid Dynamics Laboratory, Version prepared on
March 3, 2004,
available online at http://www.gfdl.noaa.gov.
Hasumi, H., 2006: CCSR Ocean Component Model (COCO) version 4.0, CCSR report, 25, 103pp.
Shchepetkin, A. F., and J. C. McWilliams, 2003: A method for computing horizontal pressure-gradient force inan oceanic model with a non-aligned vertical coordinate, J. Geophys. Res., 108, 3090-3124.
Shchepetkin, A. F., and J. C. McWilliams, 2005: The Regional Ocean Modeling System: A split-explicit, freesurface, topography-following-coordinate oceanic model, Ocean Modell., 9, 347-404.
von Storch, H., and F. Zwiers, 2004: Statistical Analysis in climate research, Cambridge university press, 484pp.

–

5

–

Part I

Configuration

–

7

–

Chapter 2

Governing Equations

In this chapter, the governing equations for the general ocean circulation are formulated. These equations areusually called primitive equations. The numerical methods to solve these equations are detailed in later chapters.

2.1

Formulation

2.1.1

Coordinate System

The governing equations of an ocean general circulation model need to be formulated on a sphere. Originally,
spherical coordinates were adopted, and the equations are discretized on a geographical (latitude-longitude-depth)
grid. A problem arises for a global model because the North Pole is singular in spherical coordinates. Since thezonal grid widths within five latitudinal degrees from the Pole become less than a tenth of those in middle to low
latitudes, a short time step is required owing to the limitation of the Courant-Friedrichs-Lewy (CFL) condition.∗
This could hinder long-term integration.
One simple means to remove this North Pole singularity is to shift both poles to land. In this case, one could use
the spherical coordinate model without major modification by just adjusting the Coriolis parameter. Unfortunately,
there are not many pairs of points on land that are symmetric about the Earth’s center.† One would like to choosea pair that provides the largest zonal grid size for the oceanic grid point having the smallest zonal grid size. The
land mass around the shifted pole should be as large as possible or the oceanic grid point nearest to the shiftedpole should be as far from the pole as possible. However, even the most suitable pair with poles on Greenland andAntarctica (near the Ross Sea) places the nearest oceanic grid point to the shifted pole only five degrees away. 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 ofspherical coordinates. Users may choose the coordinate system most suitable to their purposes. For example, theresolution of a target region can be intentionally enhanced by placing a pole of the distorted grid near the target
region. Of course, a regional model without the North Pole could be arranged on geographical coordinates since thespherical coordinates are one form of the generalized orthogonal coordinates. In our model, the model equations
are formulated on generalized orthogonal coordinates. Chapter 14 summarizes the concepts and calculus related
to the generalized orthogonal coordinates.

2.1.2

Momentum Equation

Most ocean general circulation models use the Boussinesq approximation, where the density of sea water isreplaced by a reference density (ρ0 ) for all terms except for the buoyancy term. On a general orthogonal coordinate
system ((μ , ψ , r)) whose unit vectors are eμ , eψ , and er , the momentum equation for velocity v = ueμ + veψ + wer ,∗ The time step, Δt, needs to satisfy vΔt/Δx ≤ 1, where v is velocity and Δx 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.

–

9

–

Chapter 2

Governing Equations

where u = hμ μ̇ , v = hψ ψ̇ , w = hr ṙ, is represented by




∂ hμ
∂ hψ
∂ hμ
∂u
∂ hr
1 ∂P
v
w
−
u−
v −
w + Vμ ,(2.1)+ v · ∇u + fψ w − f v = −u−∂t
ρ0 hμ ∂ μ hμ hψ ∂ ψ
∂μ
hr hμ
∂r
∂μ




∂ hψ
∂ hψ
∂ hμ
∂v
∂ hr
1 ∂P
w
u
−
w −
v−
u + Vψ ,(2.2)+ v · ∇v + f u − f μ w = −v−∂tρ0 hψ ∂ ψ hψ hr ∂ r∂ψhμ hψ ∂ μ
∂ψ




∂ hμ
∂w
ρ
∂ hr
∂ hr
∂ hr
v
1 ∂P
u
w−
w−
+ v · ∇w + f μ v − fψ u = −−g −u −v + Vr , (2.3)∂tρ0 h r ∂ r
ρ0 hr hμ ∂ μ∂rhψ hr ∂ ψ∂rwhere P is pressure, V is viscosity, and g is the acceleration due to gravity. The radial distance from the Earth’s
center is represented by r and the gravity is in the negative direction of r.
The Coriolis force is represented by2Ω × v = (2Ωψ w − 2Ωr v)eμ + (2Ωr u − 2Ωμ w)eψ + (2Ωμ v − 2Ωψ u)er ,(2.4)

where Ω = Ωμ eμ + Ωψ eψ + Ωr er is the rotation vector of the Earth. We designate f μ = 2Ωμ , fψ = 2Ωψ , andf = fr = 2Ωr .
The above expression is the most general form under the Boussinesq approximation. This form is used toformulate a non-hydrostatic model and a quasi-hydrostatic model.
Further approximations are adopted for the standard model calculations. Since the vertical scale of motion ofa water particle is far smaller than the Earth’s radius (a), the radial distance r as a scalar quantity is replacedby the Earth’s radius a. The new vertical coordinate (z) is the distance (positive upward) from the geoid (seasurface height in the state of rest) and ∂ /∂ r is replaced by ∂ /∂ z. To conserve angular momentum under the above
approximations, we neglect the metric terms that involve w for the horizontal components, all the metric terms forthe vertical components, and the Coriolis terms that do not involve f (Phillips, 1966).
Hydrostatic approximation is adopted for the vertical momentum equation:
0=−∂P− ρ g.
∂z

The resultant momentum equation described below is discretized by the default version of MRI.COM:


∂ hμ
∂ hψ
∂u
v
1 ∂P
− A (u) −u−v + Vu ,− fv = −∂t
ρ0 h μ ∂ μhμ hψ ∂ ψ
∂μ


∂ hψ
∂ hμ
∂v
1 ∂P
u
− A (v) −v−
u + Vv .
+ fu = −∂t
ρ0 h ψ ∂ ψ
hμ hψ ∂ μ∂ψThe term A in (2.6) and (2.7) represents advection. For a scalar quantity α , A is defined by∂ (hψ uα ) ∂ (hμ vα )∂ (wα )1
A (α ) =
+
+
.
hμ hψ
∂μ
∂ψ
∂z

(2.5)

(2.6)
(2.7)

(2.8)

The viscosity term is represented by V , which is calculated separately in the lateral and vertical directions. For
lateral viscosity, a harmonic (default) or biharmonic scheme is used. The harmonic scheme is used for verticalviscosity. The specific form for harmonic viscosity is as follows: 1 ∂  ∂  ∂u1 ∂  2,hψ νH DT + 2
h2μ νH DS +νV2hψ hμ ∂ μhμ hψ ∂ ψ∂z∂z
 1 ∂  ∂  ∂v1 ∂  2
Vv = 2hψ νH DS − 2h2μ νH DT +νV,hψ hμ ∂ μhμ hψ ∂ ψ
∂z
∂z

Vu =

– 10 –(2.9)(2.10)

2.1. Formulation
where νH is the horizontal viscosity coefficient, νV is the vertical viscosity coefficient, DT is the horizontal tension
and DS is the horizontal shear:∂  u ∂  v − hμ,hμ ∂ μ hψhψ ∂ ψ h μ∂ u∂  v DS = hμ+ hψ.
hψ ∂ ψ hμhμ ∂ μ hψDT = hψ(2.11)(2.12)

The above representation of the viscous term was derived by Bryan (1969) and is consistent with Smagorinsky(1963). Note that in the latter the coefficients also varies according to DT and DS (cf. Section 5.2.1). If we takehμ = 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.
When the biharmonic scheme is selected for lateral viscosity, the above harmonic operation is repeated twice.
The friction terms Vu and Vv obtained by the first procedure are sign-reversed and substituted for u and v in (2.11)
and (2.12) in the second procedure.

2.1.3

Continuity equation

Mass conservation is represented by the continuity equation for an incompressible fluid:
A (1) = 0.

2.1.4

(2.13)

Temperature and salinity equation

The advection-diffusion equation is solved for temperature (potential temperature) and salinity:∂T(2.14)= −A (T ) + D(T ) + ST ,∂t
∂S
(2.15)= −A (S) + D(S) + SS ,∂twhere D is the diffusion operator and Sτ represents internal sources and sinks for tracer τ caused by nudging,convective adjustment (Section 6.4), short wave absorption (Section 8.3.1), etc.
There are several forms for the diffusion operator. By default, the diffusion operator mixes a tracer in eachdirection of the model coordinates with the harmonic scheme. For horizontal diffusion, the biharmonic scheme canbe used instead of the harmonic scheme. Using (14.22) and (14.23), the harmonic-type diffusivity is represented
as follows:





hψ κ H ∂ T
h μ κH ∂ T∂∂∂  ∂T 
1(2.16)++
κV
D(T ) =
hμ hψ ∂ μhμ ∂ μ∂ψhψ ∂ ψ∂z∂z∂ hψ FμT ∂ hμ FψT
∂ FT
1
+ z ,
=
+
(2.17)
hμ hψ
∂μ
∂ψ
∂z





hψ κ H ∂ S
h μ κH ∂ S∂∂∂  ∂S1(2.18)++
κV
D(S) =
hμ hψ ∂ μhμ ∂ μ∂ψhψ ∂ ψ∂z∂z∂ hψ FμS ∂ hμ FψS
∂ FS
1
+ z ,
=
+
(2.19)
hμ hψ
∂μ
∂ψ
∂z
where κH and κV are the horizontal and vertical diffusion coefficients and the diffusive fluxes are represented by
F. When the biharmonic-type is selected for horizontal diffusion, the above Laplacian operation is repeated twice(the diffusion coefficient has a negative value).
– 11 –

Chapter 2

Governing Equations

In the real ocean, transport and mixing would occur along neutral (isopycnal) surfaces. Thus horizontal mixing
along the constant depth surface is generally inappropriate since neutral surfaces are generally slanting relative
to the constant depth surface. To represent excursion and mixing of water masses along neutral surfaces, neutral
physics schemes are developed and widely used in place of the horizontal diffusion scheme presented above. Whenthe neutral physics schemes are selected, the advection-diffusion equation for any scalar quantity (τ ) is expressed
as follows (Gent and McWilliams, 1990):∂∂
Dττ ∇H · (−κT ∇H ρ /ρz ) = D(τ ) + Sτ ,+ ∇H · τ (κT ∇H ρ /ρz ) +
Dt
∂z
∂z

(2.20)

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

⎛

ρy2 + ρz2
1⎜
K= 2
⎝ −ρx ρy
ρx + ρy2 + ρz2−ρx ρz−ρx ρyρx2 + ρz2⎞
−ρx ρz⎟−ρy ρz ⎠−ρy ρzρx2 + ρy2⎛

(2.21)(2.22)⎞−(ρx /ρz )(ρy /ρz )−ρx /ρz1⎜⎟=
1 + (ρx /ρz )2−ρy /ρz⎝ −(ρx /ρz )(ρy /ρz )⎠,1 + (ρx /ρz )2 + (ρy /ρz )222−ρx /ρz−ρy /ρz(ρx /ρz ) + (ρy /ρz )1 + (ρy /ρz )2(2.23)(Redi, 1982). In the above, the Cartesian notation is used for simplicity. The subscript x represents ∂ /(hμ ∂ μ )and y represents ∂ /(hψ ∂ ψ ). The isopycnal diffusion coefficient is represented by κI . Diapycnal diffusion is notconsidered here.
The second and third terms on the l.h.s. of (2.20) have the form of advection terms with a transport velocity
vector (uT , vT , wT ):∂  1 ∂ρ ∂ρ ,
(2.24)uT ≡κT/∂zhμ ∂ μ ∂ z∂ 
1 ∂ρ ∂ρ 
vT ≡
κT
/
,
(2.25)∂zhψ ∂ ψ ∂ z∂  hμ ∂ ρ ∂ ρ 
1  ∂  hψ ∂ ρ ∂ ρ 
wT ≡ −
κT
/
κT
/
+
,
(2.26)
hμ hψ ∂ μhμ ∂ μ ∂ z∂ψhψ ∂ ψ ∂ z(Gent and McWilliams, 1990). This velocity can be understood as the advection caused by the thickness diffusionof an isopycnal layer, where κT is thickness diffusivity.
Note that these could be rewritten as
G (τ ) = ∇ · (κT A∇τ )
with

⎛
⎜
A=⎝

0

0

0
ρx / ρz0ρy /ρz⎞−ρx /ρz⎟−ρy /ρz ⎠ .

(2.27)(2.28)

0Comparing with (2.23), we notice that the isopycnal diffusion and the thickness diffusion terms are combined toyield a concise form (Griffies, 1998) and (2.20) can be rewritten as:Dτ= ∇ · {(κI K − κT A)∇τ } + Sτ .
Dt

–

12

–

(2.29)

2.1. Formulation

2.1.5

Equation of state of sea water

The in situ density of sea water is a function of (potential) temperature, salinity, and pressure:ρ = ρ (T, S, P).

(2.30)The equation of state is usually given as a polynomial fit to the available measurements. A detailed description ofthis will be presented later (Section 2.2.5).

2.1.6 Boundary conditions
a. Momentum equation
Upper surface (z = 0):
At the sea surface, momentum flux enters the ocean in the form of wind stress (or stress from sea ice where sea
ice exists):νV(τx , τy )∂ (u, v) =.
∂ z z=0ρ0(2.31)

Note that z is defined positive upward (the surface wind stress is positive into the ocean).
In the model algorithm, this is expressed as a body force to the first level velocity,uv, Fsurf)=(Fsurf(τx , τy ),
ρ0 Δz 1(2.32)

2

where Δz 1 is the thickness of the first layer, and τx and τy are zonal and meridional stress, respectively.
2

Surface fresh water flux is assumed to have zero velocity.
Bottom (z = −H):
Bottom friction (τxb in zonal and τyb in meridional direction) exists at the sea floor.
For the bottom level velocity (u, v), the stress caused by bottom friction is proportional to the magnitude of thevelocity and has an angle (θ0 + π ) to the velocity vector (Weatherly, 1972):
(τxb , τyb ) = −Cbtm ρ0 u2 + v2 (u cos θ0 − v sin θ0 , v cos θ0 + u sin θ0 ),(2.33)
where Cbtm is a dimensionless drag coefficient with a value 1.225 × 10−3 and θ0 = ±10◦ . The sign of θ0 is positivein the northern hemisphere and negative in the southern hemisphere.
Side wall:
No slip condition is applied (u = v = 0).

b. Temperature and Salinity
Upper surface (z = 0):
At the sea surface, heat and fresh water are exchanged with air and sea ice. Salt is also exchanged below the sea
ice. All these exchanges are expressed as surface fluxes and become surface boundary conditions.

–

13

–

Chapter 2

Governing Equations

The surface boundary conditions for temperature and salinity are expressed as follows:∂ T κV= FzT ,
(2.34)
∂ z z=0∂ S κV = FzS ,
(2.35)∂ z z=0
where flux is defined positive downward (positive into the ocean).
Fresh water flux is explicitly incorporated into the sea level equations ((2.44) and (2.66)). It could also exert
a force on temperature and salinity. The water transported through the air-sea interface (WAO ) and ice bottom(WIbot ) is assumed to have the first level temperature (T1 ). The water transported from the ice surface (WIsurf > 0)is assumed to have the freezing point temperature (Tfreeze ). The water exchanged with ice is assumed to have low
salinity (SI = 4.0 [psu]). Note that the freezing point temperature (Tfreeze ) is given by mSI , where m = −0.0543 [K/ psu].
The expression for the surface fluxes is given byQ+ (WAO +WIbot ) · T1 +WIsurf · Tfreeze ,ρ0 C p

FzT

=

FzS

= (WIbot +WIsurf ) · SI ,(2.36)(2.37)

where Q is heat flux, and Cp is the specific heat of sea water.
To avoid an unexpected rising or falling trend of sea level during a long-term integration, fresh water flux might
be converted to salt flux. In this case, the surface flux is given by
Q−WIsurf · (T1 − Tfreeze ),ρ0 C p

FzT

=

FzS

= −WAO · S1 − (WIbot +WIsurf ) · (S1 − SI ),(2.38)(2.39)

where S1 is the first level salinity (see Chapter 8 for derivation). Note that fresh water flux should not be added tothe sea level in this case.
In a long-term run driven by surface flux, temperature and salinity might exhibit unexpected drift. To avoid this,surface temperature and salinity might be restored to observational or climatological values (T ∗ , S∗ ):
FzT
FzS

1
= − (T − T ∗ )Δz 1 ,2γt1= − (S − S∗ )Δz 1 ,2γs(2.40)(2.41)

where Δz 1 is the surface layer thickness, and γt and γs are restoring time for temperature and salinity in units of2

seconds.
Bottom (z = −H):At the sea floor, the adiabatic boundary condition is applied:

∂T
∂S
= 0,
= 0.
∂z
∂z

(2.42)Side wall:
For any tracer, the adiabatic condition is applied at the side wall:

∂T
∂S
= 0,
= 0,
(2.43)∂n∂n
where n denotes the direction perpendicular to the wall. River discharge is expressed as a part of the surface fresh
water flux.
–

14

–

2.1. Formulationc. Continuity equation
Upper surface (z = 0):
At the sea surface, vertical velocity is generated because a water parcel moves following the freely moving sea
surface:
∂η
dη
1 ∂η
1 ∂η
w=
+v
− (P − E + R),(2.44)− (P − E + R) =+udt∂thμ ∂ μhψ ∂ ψ
where P is precipitation, E is evaporation, and R is river discharge.
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

(2.45)

d. Mixing at the surface boundary layer
Near the sea surface, strong vertical mixing could 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 model withtypical horizontal and vertical resolutions. These processes are parameterized as enhanced vertical viscosity anddiffusivity. The user may specify either a high vertical viscosity and diffusivity a priori or use a surface boundary layer model. MRI.COM supports three surface boundary layer models: Mellor and Blumberg (2004), Noh
and Kim (1999), and Large et al. (1994). In the surface boundary layer models, vertical viscosity and diffusivity
change in time and are calculated every time step. See Chapter 7 for details.

2.1.7 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]). A several-thousand-year integration will not be a workable exercise as long as we are restricted bythis criteria in determining the time step. Bryan (1984) proposed a method to accelerate the ocean’s convergenceto equilibrium. In this method, the phase speed of waves is reduced by modifying the governing equations, and athermally 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 fortracers) to increase inertia and to reduce specific heat. When a steady state is reached in these equations, we
expect that the same balance 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



∂ hμ
∂ hψ
u−
v + Vu ,
∂ψ
∂μ


∂ hμ
∂ hψ
∂v
u
1 ∂P
α
− A (v) +u−
v + Vv .
+ fu = −∂t
ρ0 h ψ ∂ ψ
hμ hψ ∂ ψ
∂μ

∂u
v
1 ∂P
α
− A (u) −− fv = −∂t
ρ0 h μ ∂ μ
hμ hψ

–

15

–

(2.46)(2.47)

Chapter 2

Governing Equations

The modified temperature and salinity equations are given by

∂T
∂t
∂S
γ
∂t

γ

= −A (T ) + D(T ) + ST ,(2.48)= −A (S) + D(S) + SS .

(2.49)

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 ).
α2α(2.50)

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:ω = −β k α (k2 + l 2 ) +f 2 −1.
gHn(2.51)

Again, a large α yields a low phase speed.
In standard practice, a value from several tens to a few hundreds 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.

2.2 Numerical Methods
2.2.1 Discretization
To solve the primitive equations formulated in the previous section, the equations are projected on a discrete
lattice and then advanced for a discrete time interval using solution algorithms.
Since MRI.COM basically adopts z-coordinate, a fixed 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 gridcell. This approach is called a finite volume approach or sometimes a flux form expression. The finite volume
approach realizes a smooth transition from z-coordinates in the interior to σ -coordinates near the sea surface asdetailed in Chapter 4.
A vertically integrated expression for the primitive equations is useful for describing the solution procedure.
These are called semi-discrete equations. The body force and the metric term will be simply multiplied by the grid
width. The material derivative and the sub-grid scale flux need some attention.
The material derivative of any quantity α ,∂ (hψ uα ) ∂ (hμ vα )∂α
∂ (wα )
Dα
1
+
+
=
+
Dt
∂t
hμ hψ
∂μ
∂ψ
∂z

(2.52)§ But note that the upper several levels are allowed to move with the undulating sea surface like the σ -coordinate models. See Chapter 4 fordetails.

–

16–

2.2. Numerical Methods
is vertically integrated over a (k + 12 )-th grid cell bounded by zk and zk+1 to give


 zk
 zk
 zk
∂ (hψ uα ) ∂ (hμ vα )∂α
1∂ (wα )
dz +
+
dz +
dz
∂μ
∂ψ
∂z
zk+1 ∂ t
zk+1 hμ hψ
zk+1







∂  zk∂  zk∂  zk1
α dz +hψ uα dz +hμ vα dz=
∂ t zk+1hμ hψ ∂ μ zk+1
∂ ψ zk+1

∂z
)
∂
z
)
∂
z
v(z
u(z
k
k
k
k
k
+
− w(zk ) α (zk )−+∂thμ ∂ μhψ ∂ ψ
∂z
u(zk+1 ) ∂ zk+1 v(zk+1 ) ∂ zk+1
k+1++
− w(zk+1 ) α (zk+1 ).
+
∂t
hμ
∂μ
hψ
∂ψ
The first line on the r.h.s. is expressed in a semi-discrete form as





∂
∂ 
∂ 
1
Δzα
,
hψ Δzuαhμ Δzvα++∂thμ hψ ∂ μ
∂ψ
k+ 12
k+ 12
k+ 12

(2.53)(2.54)

where any quantity is assumed to have a uniform distribution within a grid cell.
For an interior grid cell, the last two lines reduce to the difference between vertical advective fluxes,
w(zk )α (zk ) − w(zk+1 )α (zk+1 ).

(2.55)For the sea surface (k = 0; z0 = η ) and the bottom (k = N; zN = −H), kinematic conditions (2.44) and (2.45) areused to give−(P − E + R)α (0) − w(z1 )α (z1 )(2.56)
at the surface andwzN−1 α (zN−1 )at the bottom.
Similarly, the vertical integral of the divergence of the sub-grid scale fluxes gives


 zk
 zk
∂ (hψ Fμ ) ∂ (hμ Fψ )
1
∂ Fz
dz +
+
dz
∂μ
∂ψ
zk+1 hμ hψzk+1 ∂ z




∂ 
∂ 
1
Δz hψ Fμ
Δz
h
=
+
F
μ
ψ
hμ hψ ∂ μ
∂ψ
k+ 12
k+ 12
 F (z) ∂ z F (z) ∂ z
  F (z) ∂ z F (z) ∂ z

μ
ψ
μ
ψ
+
− Fz (z) ++− Fz (z).
−hμ ∂ μhψ ∂ ψ
hμ ∂ μhψ ∂ ψ
k
k+1(2.57)(2.58)The quantityα= (P − E + R)α (0) +Fsurf F (z ) ∂ z
μ

0

hμ

0

∂μ

+


Fψ (z0 ) ∂ z0
− Fz (z0 ) (≡ −Fzα 0 )hψ ∂ ψ

(2.59)

taken from (2.56) and (2.58) could 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 (2.59) is the tracer transport by the
fresh water flux, and the second term is the microstructure flux calculated by sub-grid scale (SGS) parameterizaα = −F α ).
tions such as bulk formula. We generalize this flux as a vertical flux (i.e., Fsurf
z 0

2.2.2 Momentum equation
By default, the momentum equation with hydrostatic and Boussinesq approximation is solved. Equations are(2.6) and (2.7). To integrate these equations in time, the instantaneous vector field and pressure should be known.
– 17 –

Chapter 2

Governing Equations

The vector field of the previous time level is used. The pressure field might be obtained by integrating the hydrostatic equation vertically:P(μ , ψ , z,t) = Ps (μ , ψ ,t) + g 0zρ (μ , ψ , z )dz .

(2.60)

This equation indicates that the surface pressure Ps (μ , ψ ,t) should be known.
There is no problem if the surface height η is known (Ps (μ , ψ ,t) = ρ0 gη ). To obtain the surface height, weshould solve vertically integrated equations of motion. Since the external gravity waves caused by the rise and fallof the sea level have high phase speeds, a short time step is required to satisfy the CFL condition. However, whena target phenomenon has a longer time scale, the external gravity waves are usually not important. One might want
to separate or remove this wave since its phase speed is two orders of magnitude greater than that of other waves.
Historically, external gravity waves are removed from the model by prohibiting the vertical movement of the sea
surface (rigid-lid approximation). In this case, the vertically integrated equations lead to the vorticity equation in aform of the Poisson equation and are solved iteratively. The surface pressure is diagnostically obtained as pressure
to push the lid.
When the sea surface is allowed to move vertically, the problem of external gravity waves can be resolved byseparating the barotropic mode from the baroclinic mode. We can achieve a long time step for the baroclinic modeby reflecting a temporally averaged state of the barotropic mode that is sub-cycled with a short time step.
Since the free surface option is more suitable for parallel computation, only the free surface option is now
officially supported by MRI.COM.

a. Barotropic mode
If we put
U=

 η
−H

N

udz = ∑ uk− 1 Δzk− 1 ,
k=1

2

2

V=

 η
−H

N

vdz = ∑ vk− 1 Δzk− 1 ,2

k=1(2.61)

2then the vertically summed semi-discrete momentum equations are∂Ug(η + H) ∂ η+ X,− fV = −
∂t
hμ
∂μ
∂V
g(η + H) ∂ η+Y,+ fU = −
∂t
hψ
∂ψ
where
X

=

−∇H ·

 N
∑ (Δz(u, v)u)k− 1 − ∑

 N

k=1
N

1 1
ρ
0 hμ
k=1

−∑
( ≡

z

k− 21



∂ hμ
∂ hψ
u−
v
∂ψ
∂μ

k− 12

Δzk− 12Nuugρμ dz Δzk− 1 + ∑ (ΔzVHu )k− 1 + Fsurf
Δz 1 + Fbottom
ΔzN− 1
2

2

k=1

2

2

∑ Fμ ),= −∇H ·

(2.64) N+∑(Δz(u,
v)v)
1
∑
k−

 N

k=1
N

1 1
k=1 ρ0 hψ

−∑
( ≡

 0



(2.63)

N

k=1

Y

2

v
h
μ hψ
k=1

(2.62)

2

 0
z

k− 12

u
k=1 hμ hψ





∂ hμ
∂ hψ
u−
v
∂ψ
∂μ

k− 12

Δzk− 1
2

Nv
v
gρψ dz Δzk− 1 + ∑ (ΔzVHv )k− 1 + Fsurf
Δz 1 + Fbottom
ΔzN− 1
2

k=1

2

2

2

N

∑ Fψ ).

(2.65)

k=1– 18 –

2.2. Numerical Methods
The vertically integrated continuity equation is given by∂η
1  ∂ (hψ U) ∂ (hμ V ) = (P − E + R).
+
+
∂t
hμ hψ
∂μ
∂ψ

(2.66)

We solve these equations for U, V , and η . See Chapter 4 for details.

b. Baroclinic mode
To solve the baroclinic mode, we could use Ps (μ , ψ ,t) that can be diagnostically obtained using barotropic
equations (2.62), (2.63), and (2.66). However, we could avoid the procedure of obtaining Ps (μ , ψ ,t) by using themethod described below.
Velocity is decomposed into a barotropic component and a baroclinic component as follows:u= û + ū,(2.67)v= v̂ + v̄,(2.68)

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 surface pressure gradientterm is dropped:uk− 1 Δz
2

(new)
(old)
− uk− 1 Δz 1
k− 12
k− 2
2

Δt

(new)
(old)
vk− 1 Δz 1 − vk− 1 Δz 1
k− 2

2

where



Fμ = −∇H · Δz(u, v)u 0(2.69)= − f uΔzk− 1 + Fψ ,(2.70)

2

k− 2

2

Δt

= f vΔzk− 1 + Fμ ,

2



v
hμ hψ

−
1

k− 2



∂ hμ
∂ hψ
u−
v
∂ψ
∂μ

k− 12

Δzk− 12

1 1gρμ dz Δzk− 1 + (ΔzVHu )k− 1 − Fzu k + Fzu k+1 ,22
ρ0 hμ zk− 1
2




∂ hμ
∂ hψ
u
Fψ = −∇H · Δz(u, v)v+u−v
Δzk− 12hμ hψ ∂ ψ
∂μ
k− 12
k− 12
 0
1 1
−
gρψ dz Δzk− 1 + (ΔzVHv )k− 1 − Fzv k + Fzv k+1 .
2
2
ρ0 hψ zk− 1−(2.71)(2.72)

2

Summing over the whole water column gives
(new)



(old)

∑Nk=1 (uk− 1 Δzk− 1 ) − ∑Nk=1 (uk− 1 Δzk− 1 )
2

2

2

2

Δt

k=1

(new)



(old)

∑Nk=1 (vk− 1 Δzk− 1 ) − ∑Nk=1 (vk− 1 Δzk− 1 )
2

2

2

N

2

Δt

(old)
+
k− 12

= f ∑ vk− 1 Δz
2N

= − f ∑ uk− 1 Δz
k=1N

∑ Fμ ,(2.73)

k=1

(old)
+
k− 12

N

∑ Fψ .

(2.74)

η (old) + H ∂ Ps
(old)
+
,
1
k− 2
ρ0 h μ ∂ μ(2.75)

η (old) + H ∂ Ps
(old)
+
.
1
k− 2
ρ0 h ψ ∂ ψ(2.76)

2

k=1

From (2.62), (2.63), (2.64), (2.65), we haveN

∑ Fμ =

(new)

(new)

2

2

k=1
N

∑ Fψ =

k=1(old)∑Nk=1 (uk− 1 Δzk− 1 ) − ∑Nk=1 (uk− 1 Δzk− 1 )
(new)

(new)

2

2

2

Δt

2

N

− f ∑ vk− 1 Δz
k=1

(old)

∑Nk=1 (vk− 1 Δzk− 1 ) − ∑Nk=1 (vk− 1 Δzk− 1 )
2

2

Δt

N

+ f ∑ uk− 1 Δz
k=1

–

19

–

2

2

Chapter 2

Governing Equations

From the above equations, we have
(new)

(new)



∑Nk=1 (uk− 1 − uk− 1 )Δzk− 1
2

2

Δt

(new)

2

Δt

η (old) + H ∂ Ps,
ρ0 h μ ∂ μ(2.77)=

η (old) + H ∂ Ps.
ρ0 h ψ ∂ ψ(2.78)(new)∑Nk=1 (vk− 1 − vk− 1 )Δzk− 1
2

=

2

2

Thus we obtain
(new)
k− 12

(u

z

(new)
k− 12

(v



+ uk− 1 − u
2

z



z



k− 12

Δt
z



+ vk− 1 − v
2

)Δz

k− 12

(new)
(old)
− uk− 1 Δz 1
k− 12
k− 2
2

η (old) + H Δzk− 12 ∂ Ps
+ Fμ
= f vΔzk− 1 − (new)2η+ H ρ0 h μ ∂ μ

(new)
(old)
− vk− 1 Δz 1
k− 12
k− 2
2

η (old) + H Δzk− 12 ∂ Ps
+ Fψ ,= − f uΔzk− 1 − (new)2+ H ρ0 hψ ∂ ψ
η

)Δz

Δt

(new)

(2.79)(new)(2.80)zwhere (...) denotes the thickness weighted vertical average.
z

z

z

z

Since we could regard u(new) + u − ū = u and v(new) + v − v̄ = v as the real updated velocity, the baroclinic
component is expressed as û = u − ū and v̂ = v − v̄ .
To summarize, we first solve for (u , v ) using (2.69) and (2.70), and then compute the baroclinic component byû = u − ū and v̂ = v − v̄ . 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 from top tobottom. By using a flux form, the surface boundary condition (2.44) could be naturally included. The vertical
integration for the k-th vertical level is performed as follows:∂ (hψ Δzk− 1 uk− 1 ) ∂ (hμ Δzk− 1 vk− 1 )
1
2
2
2
2
wk = wk−1 +,(2.81)
+
hμ hψ
∂μ
∂ψ
where Δzk− 1 is the width of the (k − 12 )-th layer with Δz 1 = Δz 1 const + η and
2

2

w0 =

2.2.4

2

∂η
1  ∂ (hψ U) ∂ (hμ V ) .
+− (P − E + R) = −
∂t
hψ hμ
∂μ
∂ψ

(2.82)

Temperature and salinity equation

We solve for potential temperature instead of in situ temperature, because we want to express vertical mixing asa simple mixing of water masses. If we use in situ temperature, the temperature change caused by the change ofpressure accompanied by vertical excursion of a water parcel should be calculated before mixing.

– 20 –

2.2. Numerical Methods
a. A semi-discrete expression
The equation for potential temperature and salinity is an advection-diffusion equation (2.14) and (2.15) (or(2.20)). Its semi-discrete expression is as follows:∂− (wT )k−1 + (wT )k(Tk− 1 Δzk− 1 ) = − ∇H · Δzhψ uT, Δzhμ vT
2
2
∂t
k− 1
 2

− ∇H · Δzhψ FμT , Δzhμ FψT− FzT k−1 + FzT k + ST Δzk− 1 ,1(2.83)


∂− (wS)k−1 + (wS)k(Sk− 1 Δzk− 1 ) = − ∇H · Δzhψ uS, Δzhμ vS
2
2
∂t
k− 1
 2

− ∇H · Δzhψ FμS , Δzhμ FψS− FzS k−1 + FzS k + SS Δzk− 1 .
1(2.84)

k− 2

2

k− 2

2

Several options for discretizing each term on the r.h.s. are detailed in Chapter 6.

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, which is explained in Section 6.4.
One might also choose to mix tracers by setting the local vertical diffusion coefficient to a large value such as
10000 [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 12.5.

2.2.5

Equation of state

The in situ density is needed to calculate the pressure gradient term in the momentum equation. As indicated in(2.30), the equation of state is a function of pressure, temperature and salinity. Here we present the specific formof the equation of state.

a. Basics of the equation of state
The standard equation of state provided by UNESCO (1981) is a function of in situ temperature, salinity, andpressure. Note that in situ temperature is used, not potential temperature. Density (ρw ) of pure water (S = 0) under
sea level pressure is given as a function of temperature (T ):ρw (T ) = 999.842594 + 6.793952 × 10−2 T − 9.095290 × 10−3 T 2(2.85)+ 1.001685 × 10−4 T 3 − 1.120083 × 10−6 T 4 + 6.536332 × 10−9 T 5 .
Density at the sea surface (ρ0 = ρ (T, S, 0)) is expressed using sea surface temperature and salinity:

ρ0 = ρw(2.86)+ (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 )S3

+ (−5.72466 × 10−3 + 1.0227 × 10−4 T − 1.6546 × 10−6 T 2 ) S 2+ 4.8314 × 10−4 S2 .

–

21

–

Chapter 2

Governing Equations

Density in the interior is calculated using the secant bulk modulus K(S, T, P). The pure water value Kw is givenbyKw = 19652.21 + 1.484206 × 102 T − 2.327105T 2−2 3(2.87)−5 4+ 1.360477 × 10 T − 5.155288 × 10 T .
The value at the sea surface 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

(2.88)

32+ 1.6483 × 10 T − 5.3009 × 10 T ) S ,and the value at pressure P is given by
K = K0(2.89)
−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
3+ P (1.91075 × 10−4 ) S 2+ 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.
Density is computed using the following equations,ρ = ρ0 /(1 − P/K)

(2.90)σ = ρ − 1000.0.

(2.91)

andSince potential temperature (θ ) is the prognostic variable, an equation of state should be given as a function ofpotential temperature, salinity, and pressure.
To do this, potential temperature should be converted to in situ temperature. The conversion equation is obtained
as follows using the adiabatic lapse rate Γ(T, S, P):
T (θ0 , S, P) = θ0 +

 P
P0Γ(T, S, P )dP .

(2.92)A polynomial for the adiabatic lapse rate Γ(T, S, P) is given by UNESCO:
Γ(T, S, P) = a0 + a1 T + a2 T 2 + a3 T 3+ (b0 + b1 T )(S − 35)+ {c0 + c1 T + c2 T 2 + c3 T 3 + (d0 + d1 T )(S − 35)} P+ (e0 + e1 T + e2 T 2 ) P2 ,

–

22

–

(2.93)

2.2. Numerical Methodswhere
a0 = +3.5803 × 10−5 ,c2 = +8.7330 × 10−12 ,

−6

−14

a1 = +8.5258 × 10 ,c3 = −5.4481 × 10

a2 = −6.8360 × 10−8 ,d0 = −1.1351 × 10−10 ,a3 = −6.6228 × 10−10 ,d1 = +2.7759 × 10−12 ,b0 = +1.8932 × 10−6 ,e0 = −4.6206 × 10−13 ,b1 = −4.2393 × 10−8 ,e1 = +1.8676 × 10−14 ,c0 = +1.8741 × 10−8 ,e2 = −2.1687 × 10−16 ,(2.94),

c1 = −6.7795 × 10−10 .
With the converted in situ temperature, the equation of state is used to calculate density using salinity andpressure.

b. An equation of state used by MRI.COM
An equation of state of MRI.COM is in the same polynomial form as UNESCO, but has a modified set ofparameters. The parameters are determined by the least square fit for a realistic range of potential temperature andsalinity. We follow the method of Ishizaki (1994), but prescribe a different range (−2 ≤ θ ≤ 40 [◦ C], 0 ≤ S ≤ 42[psu], and 0 ≤ P ≤ 1000 [bar]).
We first calculate density at the sea surface (potential density or σθ ) using equations (2.85) and (2.86) without
any modification.
The pressure dependent part, or specific volume K(θ , S, P) is given byK(θ , S, P) = e1 (P) + e2 (P)θ + e3 (P)θ 2 + e4 (P)θ 3 + e5 (P)θ 4(2.95)+ S( f1 (P) + f2 (P)θ + f3 (P)θ + f4 (P)θ )2

3

3+ S 2 ( f5 (P) + f6 (P)θ + f7 (P)θ 2 ),where
e1 (P) = ec1 + (gc1 + hc1 P)P,f1 (P) = fc1 + (gc5 + hc4 P)P,e2 (P) = ec2 + (gc2 + hc2 P)P,f2 (P) = fc2 + (gc6 + hc5 P)P,

e3 (P) = ec3 + (gc3 + hc3 P)P,f3 (P) = fc3 + (gc7 + hc6 P)P,

e4 (P) = ec4 + gc4 P,
e5 (P) = ec5 ,(2.96)

f4 (P) = fc4 ,f5 (P) = fc5 + gc8 P,
f6 (P) = fc6 ,f7 (P) = fc7 .

The set of coefficients in the above equation is computed using a least square fit as follows. Given 43×43×101
combinations of the above range of potential temperature, salinity, and pressure, in situ temperature is first computed using (2.92). Density is then calculated by the UNESCO equations using in situ temperature and salinity.
The above coefficients are determined using these data of density, potential temperature, salinity, and pressure by
the least square method. They are given as follows.
– 23 –

Chapter 2

Governing Equations
ec1
ec2
ec3

19659.35
144.5863−1.722523

fc1
fc2
fc3

52.85624−3.128126 × 10−1
6.456036 × 10−3

ec4
ec5

1.019238 × 10−2
−4.768276 × 10−5

fc4
fc5

−5.370396 × 10−5
3.884013 × 10−1

fc6
fc7

9.116446 × 10−3
−4.628163−4

gc1
gc2

3.185918
2.189412 × 10−2

hc1
hc2

2.111102 × 10−4
−1.196438 × 10−5

gc3
gc4

−2.823685 × 10−4
1.715739 × 10−6

hc3
hc4

1.364330 × 10−7
−2.048755 × 10−6

gc5
gc6
gc7

6.703377 × 10−3
−1.839953 × 10−4
1.912264 × 10−7

hc5
hc6

6.375979 × 10−85.240967 × 10−10gc8

1.477291 × 10−4

2.3 Appendix
2.3.1 Physical constants
Here we list basic physical constants used for MRI.COM. These are defined in param.F90. Physical constants
used to calculate surface fluxes and sea ice processes are listed in Chapters 8 and 9, respectively.
value

variable name in MRI.COM

radius of the Earthacceleration due to gravity

6375.0 × 105 cm

RADIUS

980.1 cm2 · s−1

angular velocity of the Earth’s rotation

π /43082.0 radian · s−1
273.16 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 )

GRAV
OMEGA

the absolute temperature of 0 ◦ C
the average density of sea water
the specific heat of sea water

TAB
RO
CP

References
Bryan, K., 1969: A numerical method for the study of the circulation of the world ocean, J. Comput. Phys., 4,347-376.
Bryan, K., 1984: Accelerating the convergence to equilibrium of ocean-climate models, J. Phys. Oceanogr., 14,666-673.
Gent, P. R., and J. C. McWilliams, 1990: Isopycnal mixing in ocean circulation models, J. Phys. Oceanogr., 20,
150-155.
Griffies, S. M., 1998: The Gent-McWilliams skew flux, J. Phys. Oceanogr., 28, 831-841.
– 24 –

2.3. Appendix
Ishizaki, H., 1994: A Simulation of the abyssal circulation in the North Pacific Ocean. Part II: Theoretical
Rationale, J. Phys. Oceanogr., 24, 1941-1954.
Large, W. G., J. C. McWilliams, and S. C. Doney, 1994: Oceanic vertical mixing: a review and a model with anonlocal boundary layer parameterization, Rev. Geophys., 32, 363-403.
Mellor, G. L., and A. Blumberg, 2004: Wave breaking and ocean surface layer thermal response, J. Phys. Oceanogr.,
34, 693-698.
Noh, Y., and H.-J. Kim, 1999: Simulations of temperature and turbulence structure of the oceanic boundary layer
with the improved near-surface process, J. Geophys. Res., 104, 15,621-15,634.
Phillips, N., 1966: The equation of motion for a shallow rotating atmosphere and the “Traditional approximation”,
J. Atmos. Sci., 23, 626–628.
Redi, M. H., 1982: Oceanic isopycnal mixing by coordinate rotation, J. Phys. Oceanogr., 12, 1154–1158.
Smagorinsky, J., 1963: General circulation experiments with the primitive equations: I. The basic experiment,
Mon. Weather Rev., 91, 99–164.
UNESCO, 1981: Tenth report of the Joint Panel on Oceanographic Tables and Standards, Sidney, B. C., September 1980, Unesco Technical papers in marine science, 36, 25pp.
Weatherly, G. L., 1972: A study of the bottom boundary layer of the Florida current, J. Phys. Oceanogr., 2, 54–72.

–

25

–

Chapter 3

Spatial grid arrangement anddefinition of continuity equation

The model ocean domain is defined as a three-dimensional aggregate of rectangular grid cells limited by latitudinal
circles, longitudinal lines, and horizontal surfaces with fixed depths. Just above the bottom, vertical thickness ofthe cell can be locally changed. The horizontal and vertical lengths of the cells are regarded as the horizontal andvertical grid sizes, respectively. In MRI.COM, the grid size can be varied spatially in each direction (variable gridsize).

3.1

Horizontal grid arrangement

Figure 3.1(a) illustrates the horizontal grid arrangement. 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 3.1(a)); the tracer point as the T-point;
and the grid cell centered on the T-point, the T-cell (Figure 3.3). The T-cells are staggered from the U-cells bya half grid size and consists of partial cells along the coast lines (Figure 3.3). The coast lines are defined by the
periphery of the U-cells, i.e., the lines connecting the T-points. This type of horizontal grid arrangement is called
Arakawa’s B-grid arrangement (Arakawa, 1972).
Although Arakawa’s B-grid arrangement is also used in MOM (NOAA-GFDL, USA) and COCO (CCSR,
U. Tokyo, Japan), the primary cell is the T-cell in those models and the coast lines are defined by the lines connecting the U-points (Figure 3.1(b)).
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
3.1(c), 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.

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 3.2(a), tracers (◦) and velocity (×) are defined at just the mid-depth level of thecell, and the vertical mass fluxes W (, 2) are defined at the boundary of the cell. There are two kinds of W , one
for the T-cell (W T ; ) and another for the U-cell (W U ; 2), W U being obtained by an averaging operation on W T .
Their horizontal locations are the T-points and the U-points depicted in Figure 3.1(a).
In order to express the gentle bottom slope as smoothly as possible, the thickness of the deepest U-cell at eachhorizontal location is variable, with a limitation that it must exceed 10 percent of the nominal thickness of the layer
to avoid violating the vertical CFL condition (Figure 3.2(b)). Otherwise, as presented in Figure 3.2(c), the gentle
bottom slope is expressed by wide, flat bottoms and cliffs here and there with height of vertical grid size Δz. In
–

27

–

Chapter 3

Spatial grid arrangement and definition of continuity equation

Figure 3.1. Horizontal grid arrangement. (a) MRI.COM (◦ :T, S, η , ×:u, v, H), (b) MOM and COCO (◦ :T, S, η , H,
×:u, v), (c) Variable grid size in MRI.COM
these regions, the vertical velocity is concentrated at the cliffs, resulting in relatively strong fictitious horizontal
currents there.

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 distance corresponding to an increment of Δμ in the zonal direction of the generalized orthogonal coordinate
is expressed as follows:Δx ≡ hμ Δμ ,(3.1)

where hμ is the scaling factor. The meridional distance is defined similarly:Δy ≡ hψ Δψ .

(3.2)

The vertical distance is expressed by Δz. For a discretized grid cell, the horizontal area is expressed by ΔS and thevolume is expressed by ΔV .
The subscript indices expressing the horizontal grid position in the finite difference expression of the equations
are usually integers for the T-points, i.e., (i, j) and are increased by a half for the U-points (i + 12 , j + 12 ). 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 thevertical mass flux 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 3.2(a)). In some cases, the T-point and U-point levels may be numbered as k, witha notice.

– 28 –

3.4. Continuity equation
(a)
k-1
k-1/2
k

○䇭T,S
×䇭U,V
T
䂦䇭W
U
□䇭W

k+1/2
k+1
k+3/2

(b)

(c)

Figure 3.2. Vertical grid arrangement. (a)Vertical grid arrangement. (b) Smooth bottom slope. (c) Stair-like bottom
slope.

3.4

Continuity equation

The mass (volume) fluxes, which are fundamental for estimating the advection of momentum and tracers, arecalculated based on the finite difference expression of the continuity equation.
The finite difference expression of the continuity equation differs for the T-cell and U-cell (Figure 3.3). In
MRI.COM, the mass continuity for the T-cell is fundamental 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).
The finite difference expression of the continuity equation for the T-cell is given as follows, based on the mass
fluxes passing through each side of the grid cell (Figure 3.3):

MCi,T j

T
T
T
T
T
T
≡ Ui−
1 , j −Ui+ 1 , j +Vi, j− 1 −Vi, j+ 1 +Wi, j,k+1 −Wi, j,k

= 0,

2

2

2

2

(3.3)

where
T
∗
T
∗
Ui+
1 , j = ui+ 1 , j Δyi+ 1 , j Δz, Vi, j+ 1 = vi, j+ 1 Δxi, j+ 1 Δz,222u∗i+ 1 , j=v∗i, j+ 1
2

=

2

2

2

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

2

(3.4)

(3.5)

The finite difference analog of the continuity for the partial T-cell along the coastline (Figure 3.3(b)) is defined
as follows:
1
1
(3.6)
Vi,Tj+ 1 = v∗i, j+ 1 Δxi, j+ 1 Δz, Vi,Tj− 1 = v∗i, j− 1 Δxi, j− 1 Δz,
2
2
2
2
2
2
2
2
–

29

–

Chapter 3

Spatial grid arrangement and definition of continuity equation
(a)

(b)

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

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

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

i+1/2, j+1/2

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, ji, j

u*i+1/2, jvmi, ju*i-1/2, jumi, 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-1Figure 3.3. Horizontal arrangement of variables for the continuity equation. (a) Relationship between T-cell andU-cell (standard form). (b),(c) Relationship between T-cell and U-cell near the coast. (d) Diagonal square grid celland mass fluxes.
v∗i, j+ 1 = vi+ 1 , j+ 1 ,22v∗i, j− 1 = vi+ 1 , j− 1 ,

2

2

2

u∗i− 1 , j = 0.

2

(3.7)

2

For the corner part of land as shown in Figure 3.3(c), it is given as follows:
1 ∗
1
T
Ui−
ui− 1 , j Δyi− 1 , j Δz, Vi,Tj− 1 = v∗i, j− 1 Δxi, j− 1 Δz,
1,j =
2
2
2
2
2
2
2
2

(3.8)

u∗i− 1 , j = ui− 1 , j+ 1 , v∗i, j− 1 = vi+ 1 , j− 1 .
2

2

2

2

2

(3.9)

2

The boundary condition for Wi,Tj is as follows:Wi,Tj = Ui− 1 , j Δyi− 1 , j −Ui+ 1 , j Δyi+ 1 , j + Vi, j− 1 Δxi, j− 1 −Vi, j+ 1 Δxi, j+ 1 /ΔSi, j
2

2

2

2

2

2

2

2

at the surface (where U = ∑Nk=1 uk− 1 Δzk− 1 and V = ∑Nk=1 vk− 1 Δzk− 1 ), and2222
Wi,Tj = 0
at the bottom.
On the other hand, the finite difference expression of the continuity equation for the U-cell (i + 12 , j + 12 ) isdefined as follows (Figure 3.3(a, b, and c)):UMCi+
1 , j+ 1 ,k+ 1
2
2
2

≡

MCi,T j,k+ 1
2

Ni, j,k+ 1

= 0,

2

+

T
MCi+1,
j,k+ 1
2

Ni+1, j,k+ 1
2+

MCi,T j+1,k+ 1
2

Ni, j+1,k+ 1
2+

T
MCi+1,
j+1,k+ 1
2

Ni+1, j+1,k+ 1
2(3.10)– 30 –

3.4. Continuity equation
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
2

for T-cells away from land (Figure 3.3(a)), but N < 4 for the partial T-cells along coast lines (Figure 3.3(b,c)).
This equation means that the 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 asfollows:
UMCi+
1 , j+ 1
2

2

1
T
T
T
(MCi,T j + MCi+1,
j + MCi, j+1 + MCi+1, j+1 )
4
= 0.
≡

(3.11)

This can be rewritten as
1 U
1 U
1 U
U
U
(Ui, j + Ui,Uj+1 ) − (Ui+1,
j + Ui+1, j+1 ) + (Vi, j + Vi+1, j )
2
2
2
1
U
U
U
− (Vi,Uj+1 + Vi+1,
j+1 ) = Wi+ 1 , j+ 1 ,k −Wi+ 1 , j+ 1 ,k+1 .
2
2
2
2
2
Terms Ui,Uj , Vi,Uj , and W U 1

i+ 2 , j+ 12 ,k(3.12)are defined as follows:
Ui,Uj = umi, j Δyi, j Δz, Vi,Uj = vmi, j Δxi, j Δz,umi, j=vmi, j=1 ∗(u 1 + u∗i− 1 , j ),2 i+ 2 , j2
1 ∗∗(v 1 + vi, j− 1 ),2 i, j+ 2
2(3.13)(3.14)

1 T
U
TT
T(W + Wi+1,(3.15)
Wi+
1 , j+ 1 ,k =j,k + Wi, j+1,k + Wi+1, j+1,k ).
4 i, j,k2
2
Thus, for the standard form of the U-cell continuity, the following equations are derived from (3.4), (3.13), and(3.14):
Ui,Uj

=

Vi,Uj

=

1 T
T
(U 1 + Ui−
1 , j ),
2 i+ 2 , j
2
1 TT(V 1 + Vi, j− 1 ).
2 i, j+ 2
2(3.16)All the above relationships hold for the cases with variable grid sizes (Figure 3.1(c)).
The l.h.s. of the standard form of the continuity equation (3.12) expresses the convergence of mass fluxes along
the horizontal coordinate axes and it is completed as far as the continuity equation is concerned. However, whenthe mass continuity is used to calculate the momentum advection, the l.h.s. of (3.12) is rewritten as follows to
express the convergence of the diagonal mass fluxes to the coordinate axes, and is used together with its originalform (3.12):
1 U
1 U
1 U
U
U
(Ui, j + Vi,Uj ) − (Ui+1,
j+1 + Vi+1, j+1 ) + (Ui, j+1 −Vi, j+1 )
2
2
2
1 U
U
U
U
− (Ui+1,
j −Vi+1, j ) = Wi+ 1 , j+ 1 ,k −Wi+ 1 , j+ 1 ,k+1 .
2
2
2
2
2

(3.17)Let us explain the meaning taking the first term on the l.h.s. of (3.17)
Ui,Uj + Vi,Uj =


1 T
T
T
(Ui− 1 , j + Vi,Tj− 1 ) + (Ui+
1 , j + Vi, j+ 1 )
2
2
2
2
2
–

31

–

(3.18)

Chapter 3

Spatial grid arrangement and definition of continuity equationT
as an example, where (3.16) is used. If the flow is horizontally nondivergent, the horizontal mass fluxes Ui−
1,j2

and Vi,Tj− 1 in the first term on the r.h.s. are expressed by streamfunction at two pairs of U-points, (i − 12 , j + 12 ) and2(i − 12 , j − 12 ), and (i − 12 , 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 − 12 ) (Figure 3.3(d)). The
second term on the r.h.s. expresses the same quantity, though the route is different. Thus, multiplying by a factorof two, the l.h.s. of (3.17) means the horizontal mass convergence in the diagonal square defined by four U-points(i − 12 , j + 12 ), (i + 12 , j − 12 ), (i + 32 , j + 12 ), and (i + 12 , j + 32 ). Multiplying by a factor of 12 , the l.h.s. of (3.17) itself
means the horizontal mass convergence in the U-cell (i + 12 , j + 12 ), whose area is a half of that of the diagonalsquare.
Taking the l.h.s. of (3.12) as Ai+ 1 , j+ 1 and that of (3.17) as Bi+ 1 , j+ 1 , the standard form of the continuity equation22
2
2
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+
1 , j+ 1 ,k −Wi+ 1 , j+ 1 ,k+1 ,(3.19)

α + β = 1.

(3.20)

2

2

2

2

2

2

2

2

where
As shown later in Chapter 5, (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).

3.5 Calculation of area
3.5.1 General orthogonal coordinates
When equations are solved in MRI.COM, the temporal variations of physical quantities are calculated as abudget of their fluxes through the boundaries of the U-cells or T-cells. In those situations, it is necessary to know
the area and volume of the grid cells. These are numerically calculated as follows.
The longitude and latitude (λ , ψ ) of grid points on the sphere are defined by user as a function of the model
coordinate (μ , ψ , a)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 + 12 ), ψ ( j))(variable name in the model: dx bl) is approximated numerically as follows taking μ1 = μ (i), μ2 = μ (i + 12 ), andψ1 = 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

.

(3.21)

m=1

Here, L[λ1 , φ1 , λ2 , φ2 ] is the distance between the two points (λ1 , φ1 ) and (λ2 , φ2 ) on the sphere along a great circle
and δ μ = (μ2 − μ1 )/M (divided by M ∼ 20 between μ1 and μ2 ).
Similarly, a quarter grid area (a bl) surrounded by four points (μ (i), ψ ( j)), (μ (i + 12 ), ψ ( j)), (μ (i + 12 ), ψ ( j +1112 )), and ( μ (i), ψ ( j + 2 )) is, taking ψ2 = ψ ( j + 2 ) and δ ψ = (ψ2 − ψ1 )/N (divided by N ∼ 20 between ψ1 and– 32 –

3.5. Calculation of areaψ2 ), calculated as:
N

M





1



1
φ μ1 + (m − 1)δ μ , ψ1 + (n − )δ ψ ,2
1
φ μ1 + mδ μ , ψ1 + (n − )δ ψ
2

∑ ∑ L λ μ1 + (m − 1)δ μ , ψ1 + (n − 2 )δ ψ ,n=1 m=11
λ μ1 + mδ μ , ψ1 + (n − )δ ψ ,
2
×


1
L λ μ1 + (m − )δ μ , ψ1 + (n − 1)δ ψ ,2
1
λ μ1 + (m − )δ μ , ψ1 + nδ ψ ,2

1
φ μ1 + (m − )δ μ , ψ1 + (n − 1)δ ψ ,2
1
φ μ1 + (m − )δ μ , ψ1 + nδ ψ .
2(3.22)As depicted in Figure 3.4, a bli, j is the area of the lower left quarter of the central U-point. Those for the lower
right a bri, j , upper left a tli, j , and upper right a tri, j quarters are obtained similarly.
The unit area centered on U-point (areaui, j ) is then expressed as:
areaui, j = a bli, j + a bri, j + a tli, j + a tri, j ,(3.23)and the area centered on T-point (areati, j ) as
areati, j = a bli, j + a bri−1, j + a tli, j−1 + a tri−1, j−1 .

(3.24)

Following the conventions for indexing introduced in Section 3.3, the above equations are expressed in later
chapters as follows:areaui+ 1 , j+ 1 = a bli+ 1 , j+ 1 + a bri+ 1 , j+ 1 + a tli+ 1 , j+ 1 + a tri+ 1 , j+ 1 ,(3.25)

areati, j = a bli+ 1 , j+ 1 + a bri− 1 , j+ 1 + a tli+ 1 , j− 1 + a tri− 1 , j− 1 .

(3.26)

2

2

2
2

3.5.2

2

2

2

2

2
2

2
2

2
2

2
2

2
2

Geographic coordinate

Let us examine the situation of T-cell quarterly divided (Figure 3.4). The area of the northeastern quarter (anhft,
the same as that of the northwestern quarter) is obtained by the latitudinal integration of the thick line in Figure
3.4, where Δφ = φ ( j + 12 ) − φ ( j − 12 ).
Using the latitude of T-point φ ( j), the zonal width of the grid unit for T-points Δλ = λ (i + 12 ) − λ (i − 12 ), andthe Earth’s radius a, the length of the thick line along the latitude circle (Δs) is expressed as:
Δs = aΔλcos φ .
2

(3.27)

Integrating this in the latitudinal direction, we obtain the following.
anhfti, j

=
=

=
=
=

 φ + ΔφΔφa2 Δλ φ + 2a2 ΔλΔsad φ =cos φ d φ =
2
2
φ
φ


Δφ
Δφ
a2 Δλ cos φ +
sin
4
4


φ
Δφ
Δ
Δφ
sin
a2 Δλ cos φ cos− sin φ sin
4
4
4


φ
φ
φ
Δ
Δ
Δ
a2 Δλ cos φ cos
1 − tan φ tan
sin
4
4
4


2
a
Δφ
Δφ
1 − tan φ tan.
Δλ cos φ sin
22
42– 33 –

Δφ
sin φ +− sin φ2

(3.28)

Chapter 3

Spatial grid arrangement and definition of continuity equation(a)dy_tli, j
dx_tli, j

dx_tri-1, j
dy_bri-1, j

dy_bli, j

dy_tri, j

Ui, j

dx_tri, j

dy_bri, j

Ti, j
dx_bri-1, j
dy_tri-1, j-1

dx_bri, j

dx_bli, j
dy_tli, j-1

dy_tri, j-1

dx_tli, j-1

dx_tri-1, j-1(b)anhfti, j

a_tri, ja_tli, jUi, janhfti-1, j

a_bli, jTi, ja_bri, jӠsashfti, j

ashfti-1, j

Figure 3.4. Variables that define a grid unit. (a) distance. (b) area. Grid indices (i, j) follow array indices inprogram codes.

– 34 –

3.5. Calculation of area
Similarly, the area of the southeastern quarter of the T-cell (variable name in the model: ashft, the same as thatof the southwestern quarter) is expressed as:ashfti, j =a2ΔφΔλ cos φ sin
22


1 + tan φ tan

Δφ
4


.

(3.29)At the north and the south poles, where φ = ±90◦ , we obtain the following by changing (3.28) and (3.29) to thefollowing forms.


a2
Δφ
Δφ
Δλ sincos φ − sin φ tan(3.30)
anhfti, j =
2
2
4


a2
Δφ
Δφ
ashfti, j =Δλ sincos φ + sin φ tan.
(3.31)
2
2
4
At the north pole:anhfti, j= 0(3.32)
Δφ
Δφ
tan
.
2
4

(3.33)a2ΔφΔφΔλ sin
tan
2
2
4
= 0.

(3.34)

a2

ashfti, j=anhfti, j=2

Δλ sinAt the south pole:ashfti, j(3.35)

In our modela bli, j = anhfti, j ,a bri, j = anhfti+1, j ,a tli, j = ashfti, j+1 , a tri, j = ashfti+1, j+1 ,and the areas of the grid cells centered on the U-points and T-points are calculated by (3.23) and (3.24).

References
Arakawa, A., 1972: Design of the UCLA general circulation model, Numerical Simulation Weather and Climate,
Tech. Rep. No. 7, Dep. of Meteorology, University of California, Los Angeles, 116 pp.
Webb, D. J., 1995: The vertical advection of momentum in Bryan-Cox-Semtner ocean general circulation models,J. Phys. Oceanogr., 25, 3186–3195.

–

35

–

Part II

Main Processes

–

37

–

Chapter 4

Equations of motion (barotropic
component)

Historically, iterative methods have been used to solve the barotropic part of the momentum equations by applying
the rigid-lid approximation. The number of iterations to get convergence of the solution is of order N, where N is
the larger number of grid points in the horizontal direction. Thus the number of iterations increases as the numberof grid points increases. This means that the iterative process could occupy a large part of the total CPU time. Thisis a severe burden and should be remedied.
An alternative is to replace the rigid lid with a free surface. The number of short barotropic time steps in eachlong baroclinic time step is roughly the ratio of linear 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 more suitable forparallel computation than the iterative methods. Thus, for fine-resolution models, the free-surface formulations
have numerous advantages over the iterative methods.
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. However, this free-surface
model does not work when the sea surface is below the bottom of the uppermost layer and the thickness of thislayer vanishes. This occurs in world ocean models, because the difference between the maximum and minimumof the sea surface height becomes several meters.†
To remedy this problem, the σ -coordinates are introduced for the free-surface formulation near the sea surface.
In this method, the thicknesses of the several upper layers (z ≤ −HB ) vary as the sea surface height does (Figure4.1). These layers are referred to as the σ -layers. The σ -coordinates are written as

σ≡

z−η
.
HB + η

(4.1)

This extension of the free-surface formulation is described in Section 4.4.
The free-surface formulation was adopted in the Bryan-Cox-Semtner numerical ocean general circulation models by Killworth et al (1991). The σ -coordinates were adopted in the free-surface formulation by Hasumi (2006).
In this chapter, we explain the free-surface formulation of MRI.COM adopting these two methods.

∗ A North Pacific Model with 1/4◦ × 1/6◦ resolution has 742 grid points in the zonal direction.

† 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] (the Ross Sea).

–

39

–

Chapter 4

Equations of motion (barotropic component)z=ηz = -HB

Figure 4.1. Schematic of the near surface σ -layers.

4.1 Governing equations
As described in Chapter 2, the prognostic variables in the free-surface model are the surface elevation (η ) andthe vertically integrated velocity (U and V ). The momentum equation is given as

∂U
− fV
∂t
∂V
+ fU
∂t

g(η + H) ∂ η
+ X,
hμ
∂μ
g(η + H) ∂ η
+Y,
= −
hψ
∂ψ
= −

where
X

= −∇H ·N 

∑ Δz(u, v)u k− 12

k=1

N

1 1
k=1 ρ0 hμ

−∑
Y

=

−∇H ·



N 

 0
z

k− 21

N

1 1
ρ
0 hψ
k=1

−∑

z

k− 12

v
−∑
k=1 hμ hψ





∂ hμ
∂ hψ
u−
v
∂ψ
∂μ

k− 12

Δzk− 1
2

N

2



 0

N

(4.3)

u
u
gρμ dz Δzk− 1 + ∑ (ΔzVHu )k− 1 + Fsurf
Δz 1 + Fbottom
ΔzN − 1∑ Δz(u, v)v k− 12

k=1



(4.2)



k=1

N

u
+∑
h
μ hψ
k=1

2



2

2



∂ hμ
∂ hψ
u−
v
∂ψ
∂μ

k− 12

Δzk− 1
2

Nv
v
gρψ dz Δzk− 1 + ∑ (ΔzVHv )k− 1 + Fsurf
Δz 1 + Fbottom
ΔzN − 1 .

The continuity equation is

∂η
1
+
∂t
hμ hψ

(4.4)

2



2

k=1

∂ (hψ U) ∂ (hμ V )
+
∂μ
∂ψ

2

2

(4.5)


= (P − E + R),(4.6)

where P is precipitation (positive downward), E is evaporation (positive upward) and R is the river discharge rate
(positive into the ocean).
Figure 4.2 illustrates the grid arrangement of the free-surface formulation. The variable η is defined at T-points,
and the variables U and V are defined at U-points. Forcing terms (X and Y ) in Eqs. (4.2) and (4.3) are calculated
in the subroutine for the baroclinic component and defined at U-points.

4.2 Time integration
Figure 4.3 presents schematics of the time integration of the barotropic modes in the free-surface formulation.
When the time integration of the baroclinic mode is performed from step n (t = tn ) to step n + 1 (t = tn+1 , Δt =– 40 –

4.2. Time integrationΔxη 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, jη i-1, jΔyη i+1, j

U 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

Figure 4.2. Grid arrangement of the free-surface formulation
tn+1 − tn ), the corresponding time integration of the barotropic mode is carried out from step n to step n + 2 using
the barotropic time interval Δttr and the vertically integrated values (X,Y ) at t = tn calculated in the program thatsolves the baroclinic mode. The time-averaged value of the barotropic mode over the two baroclinic time intervals
between t = tn and t = tn+2 is used to calculate the total velocity at t = tn+1 .
The “Euler forward-backward” scheme is a stable and economical numerical scheme for linear gravity wave
equations without advection terms (Meisinger and Arakawa, 1976), and this is adopted for the governing equationsin the free-surface formulation of MRI.COM. This scheme is more stable than the leap-frog scheme. The time step
can be doubled for 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 then used 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, thecomputer burden increases considerably because this scheme calculates the equations twice for one time step. In
MRI.COM, stable solutions are efficiently obtained by using this Euler forward-backward scheme because the time
filter is applied for the barotropic mode.‡
The finite-difference expression of the continuity equation (Eq. 4.6) is(ηi, j − ηi, j )
Δttr

+

1
ψ
μ
(δμ hψ U )i, j + (δψ hμ V )i, j = (P − E + R)i, j ,(hμ hψ )i, j(4.7)

where the subscripts are labeled on the basis of T-grid points. The variables ηi, j and wi, j are located at the T-gridpoints, and the variables Ui+ 1 , j+ 1 and Vi+ 1 , j+ 1 are located at U-grid points. (They are located at (i + 12 , j + 12 ) on22
22
the basis of T-grid points; see Figure 3.2).
The finite-difference operator is defined as follows:δ μ Ai ≡μ

Ai ≡Ai+ 1 − Ai− 1
2

Δ μi

2

2

Ai+ 1 + Ai− 12, δμ Ai+ 1 ≡2

2

Ai+1 − Ai,
Δμi+ 12μ, Ai+ 1 ≡2

Ai+1 + Ai.
2‡ An option (FSEB) in MRI.COM uses the Euler backward scheme for the barotropic equations.

–

41

–

(4.8)

Chapter 4

Equations of motion (barotropic component)

The same applies to ψ . In the program codes, the above variables are multiplied by the area of each model cell at
the T-grid point (ΔST ):(ηi, j −ηi, j ) · ΔST i, j
(4.9)
 



ψ
ψ
μ
μ
= Δttr · (P − E + R)i, j · ΔST i, j − Δyi+ 1 , jU i+ 1 , j − Δyi− 1 , jU i− 1 , j − Δxi, j+ 1 V i, j+ 1 − Δxi, j− 1 V i, j− 1 .
2

2

2

2

2

2

2

2

Each operator is defined the same way as the previous one. This equation is used to obtain the new surfaceelevation, ηi, j .
After obtaining ηi, j , the momentum equations, Eqs. (4.2) and (4.3), are solved. A longer time step can be used
when the semi-implicit scheme is applied for the Coriolis terms in the momentum equations. Their finite-difference
forms are

(Ui+ 1 , j+ 1 −Ui+ 1 , j+ 1 )
2

2

2

2

Δttr



−

f (Vi+ 1 , j+ 1 + Vi+ 1 , j+ 1 )
2

2

2

2

2



=−

μ ,ψ

g(Hi+ 1 , j+ 1 + η i+ 1 , j+ 1 )
2

2

2

2

(hμ )i+ 1 , j+ 12ψδμ η  i+ 1 , j+ 1 + Xi+ 1 , j+ 1
2

2

2

2

2

(4.10)
(Vi+ 1 , j+ 1 −Vi+ 1 , j+ 1 )
2

2

2

2

Δttr



+

f (Ui+ 1 , j+ 1 + Ui+ 1 , j+ 1 )
2

2

2

2

2



=−

μ ,ψ

g(Hi+ 1 , j+ 1 + η i+ 1 , j+ 1 )222(hψ )i+ 1 , j+ 1
22μδψ η  i+ 1 , j+ 1 + Yi+ 1 , j+ 1 .
2

2

2

2

2

(4.11)


Next, we solve these equations for Ui+ 1 , j+ 1 and Vi+ 1 , j+ 1 . Let the r.h.s. of the above equations be GX and GY .
22
2
2
Multiplying both sides by Δttr , we havef Δttr (Vi+ 1 , j+ 1 + Vi+ 1 , j+ 1 ) = Δttr GXi+ 1 , j+ 1 ,
2
2
2
2
2
2
2

f Δttr (Ui+ 1 , j+ 1 + Ui+ 1 , j+ 1 ) = Δttr GYi+ 1 , j+ 1 ,(Vi+ 1 , j+ 1 −Vi+ 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

2

(4.12)(4.13)

leading to
f Δttr 
V 1 1
2 i+ 2 , j+ 2
f Δttr 
Vi+ 1 , j+ 1 +Ui+ 1 , j+ 1
2
2
2
2
2


Ui+ 1 , j+ 1 −2

2

f ΔttrV 1 1 + Δttr GXi+ 1 , j+ 1 ,2
2
2 i+ 2 , j+ 2
f Δttr= Vi+ 1 , j+ 1 −Ui+ 1 , j+ 1 + Δttr GYi+ 1 , j+ 1 .
2
2
2
2
2
2
2

= Ui+ 1 , j+ 1 +2

2(4.14)(4.15)

Letting the r.h.s. of the above equations be RX and RY , we could simplify the equations as follows:

Ui+ 1 , j+ 1
2

2



Vi+ 1 , j+ 1
2

2

=
=



 f Δttr 2 
f Δttr1+,
RXi+ 1 , j+ 1 +RYi+ 1 , j+ 1
2
2
2
2
2
2



 f Δttr 2
f Δttr1+.
RYi+ 1 , j+ 1 −RXi+ 1 , j+ 1
2
2
2
2
2
2

(4.16)(4.17)

We obtain η , U, and V at t = tn+1 by averaging them between [tn ,tn+2 ]. We can see from Figure 4.3 that there
are 2 × Δt/Δttr barotropic time steps between the two baroclinic time intervals [tn ,tn+2 ]. Thus the value at t = tn+1
is the average of 2 × Δt/Δttr + 1 barotropic steps centered at t = tn+1 . It is not always necessary to calculate
the governing equations for the barotropic modes until t = tn+2 to obtain the value at t = tn+1 . We can use thevalue at t = tn+1 without any time averaging when they are calculated until t = tn+1 . This method requires lesscomputational cost than the above method. We apply the time-filter method because we empirically know that thismethod is more computationally stable.
The degree of the time-filter procedure can be specified by the namelist parameter ntflt, which specifies the
additional time steps to be calculated past the time t = tn+1 . When ntflt=Δt/Δttr , the averaged barotropic mode
– 42 –

4.3. Prognostics of physical properties in the uppermost layer
at t = tn+1 is evaluated by calculating the additional Δt/Δttr barotropic modes, and then averaging 2 × Δt/Δttr + 1
barotropic steps. This is the default setting and is applied when ntflt=-1 or namelist ntflt is not described in
the input namelist files. If ntflt= 0, time-filtering is not applied. If ntflt satisfies 0 < ntflt < Δt/Δttr , thenthe averaged barotropic mode is evaluated by the average of 2 × ntflt+1 steps centered at t = tn+1 . We usually
recommend the default setting.

n+1nbaroclinic

n+2

Δt

depth-integrated
properties

averagedbarotropic
m-1

Δttr

m

m+1

Figure 4.3. Schematic figure of the time integration of the barotropic mode and its time-filtering procedure
Subroutine SURFCE diagnoses the volume of the uppermost layer and the mass flux due to the surface elevations
using the prognostic variables η , U, and V . The mass flux due to the surface elevations (labeled ws in the programcodes) is defined at T-points and is obtained by vertically integrating the continuity equation without precipitationand evaporation,
wsi, j= wi, j · ΔST i, j


ψ
ψ
μ
μ
= −Δt · Δyi+ 1 , jU i+ 1 , j − Δyi− 1 , jU i− 1 , j + Δxi, j+ 1 V i, j+ 1 − Δxi, j− 1 V i, j− 1 .
2

2

2

2

2

2

2

(4.18)

2

The relationship between the continuity equation and the vertical velocity at the surface is explained later.

4.3

Prognostics of physical properties in the uppermost layer

4.3.1

Standard scheme

The flux form is used in MRI.COM for calculating the advection and diffusion (viscosity) of momentum, temperature, and salinity. The prognostic variables at the next time step are obtained as the convergence of these fluxes
divided by the corresponding volume. In the free-surface formulation, the volume in the uppermost layer changes
with time. This requires many careful and special procedures for calculating the prognostic variables in the uppermost layer. In MRI.COM, the prognostic variables themselves and (the prognostic variables) × (their volume) are
saved for the prognostic procedures, which are labeled as uv1a, vv1a, and trcv1a in the program codes. In the
prognostic procedures of subroutines clinic and tracer for momentums and tracers in the uppermost layer, theirvalues times their volume at the new time step are temporarily stored in uv1a, vv1a, and trcv1a. The prognostic
value at the new time step is obtained by dividing them by the volume of the new time step, volt1, which is
calculated at the end of the subroutine surfce. In this procedure, the global integrals of the prognostic variables
are conserved. However, they are not conserved locally (see the next subsection). The following is the calculationof temperature (T ) in the uppermost layer when the leap-frog time is used.
trcv1b = ΔVt old · Told
↓
–

43

–

Chapter 4

Equations of motion (barotropic component)
trcv1a = trcv1b + Convergence of the fluxes↓

Tnew = trcv1a/ΔVt new↓

trcv1 = trcv1a , trcv1b = trcv1
In MRI.COM, we also use the flux form for body forcing such as wind-forcing and restoration of temperatureand salinity to the prescribed values in the uppermost layer.
The body force for the uppermost layer becomes ∂ u ∂ v 
1 (τμ , τψ )
= ... +,,
∂ t ∂ t k= 12
ρ0 Δz 1 + η(4.19)

2

where (τμ , τψ ) are the wind stress at the surface (momentum flux), and Δz 1 is the standard thickness of the upper2
most layer. Thus, the uppermost layer is more accelerated when this layer is thinner than the standard value.
When the restoring condition is applied at the surface, the corresponding temperature and salinity fluxes are
FzT∂ T 
1= ... +,
(4.20)
FzT = − (T − T ∗ )Δz 1 ,2γt∂ t k= 1
Δz 1 + η22FzS∂ S 
1
FzS = − (S − S∗ )Δz 1 ,= ... +.
(4.21)
2γs∂ t k= 1
Δz 1 + η22

Thus, the temperature and salinity are more strongly restored to the prescribed values when this layer is thinnerthan the standard value.

4.3.2

Locally conserved scheme (option FSMOM)

For the standard free-surface formulations stated above, the relation between the surface elevation, convergenceof the barotropic flow, and P − E + R (precipitation minus evaporation) isη n+1 − η n−1+ ∇ ·U n = P − E + R.
2Δt

(4.22)

Therefore, the mass is not locally conserved even if the r.h.s. (P − E + R) is zero. However, the mass is globallyconserved owing to the flux form. Thus, starting from the uniform salinity distribution, we have small deviationfrom it even if P − E + R is not applied.§ This inconsistency occurs because η n+1 is evaluated as the average ofthe barotropic steps from tn to tn+2 , while U n is evaluated as the average of the barotropic steps from tn−1 to tn+1 .
To remedy this problem, the following procedure can be used in MRI.COM as option FSMOM. In this option,η̂ n+1 is calculated in addition to η n+1 so thatη̂ n+1 − η n−1+ ∇ ·U n = P − E + R
2Δt(4.23)

is satisfied. The temperature and salinity in the uppermost layer are obtained based on η̂ n+1 , and local conservation
is maintained. However, global conservation is not maintained in this case. Thus, we do not usually use this option.

§ If the times of the barotropic and baroclinic steps are the same, and time-filtering for the free-surface is not applied (ntflt=0), then theuniform salinity distribution is maintained. This is not practical for simulations but is useful for checking program codes.

– 44 –

4.4. Introduction of σ -coordinates near the sea surface

4.4 Introduction of σ -coordinates near the sea surface
This section introduces the σ -coordinates in the upper layers as an extension of the free-surface formulation.¶ Introducing the σ -coordinates does not require significant modification in the program codes because the barotropic
equations are calculated based on the flux form.
In MRI.COM, the σ -coordinates are automatically introduced when option FREESURFACE is used. The numberof the σ -layer, ksgm, must be specified in the model configuration file configure.in and should be less than orequal to the total number of vertical grid levels km. The bottom topography should not appear for 1 ≤ k ≤ ksgm.

4.4.1

Formulation of σ -layer model

There is an arbitrariness in the definition of the σ -coordinates. We follow the notation of the Princeton Ocean
Model (POM; Mellor 2004). Let the surface elevation z = η (μ , ψ ,t). A vertical coordinate (σ -coordinate) that isnormalized by the thickness between z = −H (< 0) and z = η (μ , ψ ,t) is defined as

σ=

z−η
H +η

(−1 < σ < 0).

(4.24)

See Figure 4.1. Thus, the values of σ range from σ = 0 at z = η to σ = −1 at z = −H. Hereinafter, we define thethickness of the σ -layer as D, i.e., D ≡ H + η . The quantity Dd σ = (η + H )d σ corresponds to the real verticalgrid size dzu in the program code of MRI.COM. (Note that dzu varies with time.) For the general σ -coordinates,
the depth (H) is also a function of horizontal position, but here H is assumed to be constant.
Next we derive the governing equations in the σ -coordinates. The equations expressed in terms of the original
Cartesian coordinates z(μ ∗ , ψ ∗ , z∗ ,t ∗ ) are transformed into sigma coordinates σ (μ , ψ , σ ,t), whereμ = μ ∗, ψ = ψ ∗, σ =z∗ − η
, t = t ∗.
H +η(4.25)The partial derivatives transform according to

∂ α∗
∂ μ∗
∂ α∗
∂ ψ∗
∂ α∗
∂ z∗
∂ α∗
∂ t∗

=
=
=
=

∂α 1+σ ∂η ∂α−,∂μD ∂μ ∂σ∂α 1+σ ∂η ∂α−,∂ψD ∂ψ ∂σ1 ∂α,D ∂σ∂α 1+σ ∂α ∂η−.
∂tD ∂σ ∂t(4.26)(4.27)(4.28)(4.29)The material derivatives transform according to∗dα ∗∂ α ∗ u∗ ∂ α ∗ v∗ ∂ α ∗
∗ ∂α
=
+
+
w
+
∗
∗
∗
∗
dt ∗
∂t
hμ ∂ μhψ ∂ ψ
∂z


∗
∂α
u ∂α
v ∂α w ∂α 1+σ ∂α u ∂ηv ∂η ∂η
.
=
+
+
+
−
+
+
∂t
hμ ∂ μ hψ ∂ ψD ∂σ
D ∂ σ hμ ∂ μ hψ ∂ ψ∂tHere, we define ω as

ω≡w∗ 1 + σ
dσ
−
=
dt ∗
D
D



v ∂η ∂ηu ∂η
++h μ ∂ μ hψ ∂ ψ∂t¶ The introduction of the σ -layer, and the expression of equations are based on Hasumi (2006).

–

45

–

(4.30)

.

(4.31)

Chapter 4

Equations of motion (barotropic component)

The above equation then becomes

∂α
u ∂α
v ∂α
∂α
dα
dα ∗
=
+
+ω
≡
+
.
dt ∗
∂t
hμ ∂ μ hψ ∂ ψ∂σdt(4.32)

Thus, they take nearly the same form as in z-coordinates.
In the program code, we do not directly calculate ω , but diagnose Dω and Dd σ from the equation

ω

∂α
∂α
= Dω
.
∂σ
D∂ σ

(4.33)

They are stored in the model code as variables wlwl and dzu. (Their dimensions are the same as those in thez-coordinates, and their magnitudes are comparable.)

4.4.2

Governing equations in the σ -coordinatesa. Continuity equations
The continuity equation in the σ -layer is∂ (hψ u) ∂ (hμ v)
1∂ω 1 u ∂ηv ∂η ∂η
+
+
+
+
+
= 0.
hμ hψ
∂μ
∂ψ
∂ σ D hμ ∂ μ hψ ∂ ψ∂t(4.34)In the flux form, this becomes simpler.

∂η
1
+
∂t
hμ hψ




∂ (hψ uD) ∂ (hμ vD)
∂ω
+D
+
= 0.
∂μ
∂ψ
∂σ

(4.35)In the program code, this equation is used to diagnose Dω , which is used as vertical velocity w (wlwl) in theσ -layer.

b. Advection terms
Advection terms in the flux form A (α D) are as follows:∂ (hψ uα D) ∂ (hμ vα D)
1∂ (ωα )A (α D) =+.
+Dh μ hψ∂μ∂ψ∂σ
In the program code, the convergence of the advective flux per unit area is calculated as follows:∂ (hψ uα Dd σ ) ∂ (hμ vα Dd σ )1A (α Dd σ ) =
+
hμ hψ
∂μ
∂ψ

(4.36)(4.37)+ (α Dω )upper − (α Dω )lower .
c. Pressure gradient term
The hydrostatic equation is1 ∂p= −ρ g.
D ∂σWhen the atmospheric pressure at the surface is neglected, the pressure in the σ -coordinates isp(σ ) = g 0
σDρ d σ  = gD
–

46

–

 0
σ

ρ dσ .

(4.38)(4.39)

4.4. Introduction of σ -coordinates near the sea surface
After some manipulations, the following expression for the pressure gradient in the σ -coordinates is obtained:
 0
∂ p∗
∂ p 1+σ ∂η ∂ p∂ρ σ ∂η ∂ρ  
∂η
−
= gD
−
,
∗ =
 dσ + ρ g∂μ∂μ
D ∂μ ∂σ
∂
μ
D
∂
μ
∂μ
∂
σ
σ

0  ∂ρ
∂ p∗∂ p 1+σ ∂η ∂ pσ ∂η ∂ρ  ∂ηdσ + ρ g
−
= gD
−
.
∗ =
∂ψ
∂ψ
D ∂ψ ∂σ∂ψ D ∂ψ ∂σ∂ψσ(4.40)

The first term on the r.h.s. corresponds to the baroclinic part of the pressure gradient. The second term on the
r.h.s. is the barotropic part of the pressure gradient. Density ρ in the last term on the r.h.s. is replaced by thereference density ρ0 .

d. Time integration of velocities and tracers
Horizontal viscosity and diffusion are assumed to be parallel to the σ -layer in the σ -layer model. In MRI.COM,the volume integral of a tracer with variable name trcl, trcv1 (≡ trcl ∗ tvol1), is stored to calculate trcl in
the next step. Introduction of the σ -layer corresponds to the direct extension of trcv1 in the uppermost layer toall σ -layers. The following equation expresses the tracer equations per unit area:∂ (hψ uT D) ∂ (hμ vT D)
∂
∂
1
−
+
[ω T D] + DD (T ).
[T D] = −(4.41)
∂t
hμ hψ
∂μ
∂ψ
∂σ

4.4.3 Redistribution of tracers among the σ -layers
The volume change due to freshwater flux from the atmosphere is distributed among all the σ -layers, so that theratio of the thicknesses for each σ -layer remains unchanged. However, in reality freshwater flux like rain should
affect only the uppermost layer. In addition, the uppermost layer gets excess heat flux because the heat flux due
to this freshwater flux is only applied to the uppermost layer. In order to fix this problem, tracers are redistributed
among the σ -layers at the beginning of the subroutine tracer.
See the schematic diagram in Figure 4.4. Let us consider the k-th layer in the σ -layer (1 ≤ k ≤ ksigma ). Let
surface height, temperature, and salinity before this redistribution be η n+1(old) , T n+1(old) , and Sn+1(old) . The
surface height due to the freshwater flux isη n+1 = η n+1(old) + δ η ,(4.42)

where δ η = 2Δt (P − E + R) during the leap-frog time integration.
The following thicknesses are redistributed at the lower boundary and the upper boundary of the k-th layer.
k−1

δ zTk = 1 − ∑ Δσl δ η (1 < k ≤ ksigma ), and δ z1T = 0,

δ zBk =

l=1
k

(4.43)

1 − ∑ Δσl δ η (1 ≤ k < ksigma ), and δ zBksigma = 0,(4.44)

l=1

where zBk , and zTk are the redistributed thickness at the lower end and the upper end of the k-th layer, respectively(δ zTk = δ zBk−1 ).
The redistribution of tracer (S) among the σ -coordinate layer depends on the sign of δ η :When δ η > 0 (Figure 4.4a),n+1(old)

Skn+1 (η n+1 + H )Δσk = Skn+1(old)(η n+1(old) + H )Δσk − Sk
– 47 –

n+1(old)

δ zBk + Sk−1

δ zTk .

(4.45)

Chapter 4

Equations of motion (barotropic component)

When δ η < 0 (Figure 4.4b),n+1(old)

Skn+1 (η n+1 + H )Δσk = Skn+1(old)(η n+1(old) + H )Δσk − Sk+1

n+1(old)

δ zBk + Sk

δ zTk .

(4.46)In the program, trcv1b is substituted with trcv1a at the beginning of the subroutine tracer, then trcv1a is
modified using either (4.45) or (4.46).
(b)

(a)

Sk-1

Sk-1
Sk-1

Sk-1
T

T

-δzk

δzk
Sk

Sk
Sk

Sk
B
k

B
k

-δz

δz
Sk+1

Sk+1

Sk+1

Sk+1

Figure 4.4. Schematic diagram of the redistribution of tracer among the σ -coordinate layers. When fresh water is(a) added, and (b) removed from the sea surface.

References
Hasumi, H., 2006: CCSR Ocean Component Model (COCO) version 4.0, CCSR report, 25, 103pp.
Killworth, P. D., D. Stainforth, D. J. Webb, and S. M. Paterson, 1991: The Development of a Free-Surface
Bryan-Cox-Semtner Ocean Model, J. Phys. Oceanogr., 31, 1333–1348.
Meisinger, F., and A. Arakawa 1976: Numerical methods used in atmospheric models, GARP Publications Series,17, 65pp.
Mellor, G. L., 2004: Users guide for a three-dimensional, primitive equation, numerical ocean model, Prog. in
Atmos. and Ocean. Sci, Princeton University, 53pp.,
available online at http://www.aos.princeton.edu/WWWPUBLIC/htdocs.pom/PubOnLine/POL.html.

–

48

–

Chapter 5

Equations of motion (baroclinic
component)

This chapter explains the advection terms (Section 5.1) and the viscosity terms (Section 5.2) of momentum.
One of the unique characteristics of MRI.COM’s momentum advection terms is that there are exchanges ofmomentum between U-cells that share only a corner without a common side boundary. 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 away from land are conserved in calculating the momentum advection forhorizontally non-divergent flows. The description of momentum advection in Section 5.1 is based on Ishizaki and
Motoi (1999).
The discrete expressions for the viscosity terms in the momentum equations are based on the generalized orthogonal coordinates. A harmonic operator is used as the default assuming a no-slip condition on the land-sea
boundaries. A biharmonic operator (option VISBIHARM) and a parameterization of viscosity as a function of thevelocity gradients (option SMAGOR) could also be used.

5.1 Advection terms
Chapter 3 demonstrated that the mass fluxes used for calculating momentum advection are those for the masscontinuity of the U-cell and are obtained by an averaging operation (3.10) of those for the T-cell mass continuity(3.3) to (3.9). This is the preliminaries for constructing the general mass flux form over an arbitrary bottom andcoastal 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 to be 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 theT- and U-points a half vertical grid size lower is expressed by k + 12 (Figure 3.2(a)).

5.1.1

Vertical mass fluxes and its momentum advection

According to the definition (3.3) and (3.10) in Chapter 3, the vertical mass flux at the upper surface, level k, ofU
the U-cell (i + 12 . j + 12 , k + 12 ), W i+ 1 , j+ 1 ,k , is defined by surrounding W T as
2

U

W i+ 1 , j+ 1 ,k =2

2

2

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

+

T
Wi+1,
j,k

Ni+1, j,k+ 1+

2

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

+

T
Wi+1,
j+1,k

Ni+1, j+1,k+ 1

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

–

49

–

2

,

(5.1)

Chapter 5

Equations of motion (baroclinic component)

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 U1

i+ 2 , j+ 12 ,kis defined asUWi+
1 , j+ 1 ,k =2

2

Wi,Tj,kNi, j,k− 1

+

2

T
Wi+1,
j,k

Ni+1, j,k− 1+

2

Wi,Tj+1,kNi, j+1,k− 1
2

+

T
Wi+1,
j+1,k

Ni+1, j+1,k− 1

.

(5.2)

2

U

Since W T are continuous at the boundary of vertically adjacent T-cells, W i+ 1 , j+ 1 ,k and W U 1 1 seem to bei+ 2 , j+ 2 ,k2
2
discontinuous at the boundary when N are vertically different, for example, Ni, j,k+ 1 < Ni, j,k− 1 , over the bottom
22

relief. 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 (Figure 5.1(a)). Assume
a barotropic current flows over the topography. The U-points are just intermediate between T-points. This indicatesthe mass continuity for T-cells. The barotropic flow comes from the left and barotropy is conserved in shallow
regions. Figure 5.1(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 mean value of neighboring fluxes for T-cells. Just on
the bottom slope, we must introduce a flux that flows along the slope to ensure mass continuity. The lowermost
U-cells at the slope have nonzero vertical flux at the bottom. The barotropy of the flow and the distribution ofvertical velocity are thereby kept reasonable for U-cell fluxes.
(a)

(b)

U, V

U, V

U, V

U, V

3

3

4.5

9

U, V

U, V

U, V

3

3.75

6.75

4.5

1.5
3

3

3

0.75
3

4.5

3

3
3

U, V
9

3
2.25

3.75
1.5

3

3

3

1.5

Figure 5.1. (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 forflows over bottom topography that varies two-dimensionally. For simplicity, we consider a two-layer case without
losing generality. First, we consider three examples of bottom relief, 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 thelower layer (cell dl ) (Figure 5.2). We use suffixes l and u to designate the lower and the upper layer. The central
T-point and T-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– 50 –

5.1. Advection terms
consider how this T-cell mass 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 layer, WT is shared by three cells, al , bl , and cl , so the contribution of W T to eachW 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 . HereW 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, and the remaining W T /12 (= W T /3 − W T /4) flows to cell du through theinterface. Gathering these diagonal fluxes from the 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 cellsof the flux, if the centered difference scheme is used, which is necessary to conserve the total kinetic energy.

UPPER

LOWER

du

cu
䃁

cl

dl
䃁

䃁

T

W
Al

Au
䃁

䃁

䃁

au

bu

䃁
T

al

bl

T

T

1/4W

T

䃁

䃁 1/3W
T

1/3W 䋭 1/4W = 1/12W

Figure 5.2. First example of land-sea patterns, in which all four upper cells are sea cells, with three sea cells andone 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 aresea cells in the upper layer (Figure 5.3). In the lower layer, W T is shared only by bl but in the upper layer, it isshared 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 three cells 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 areland cells and the others are sea cells (Figure 5.4). In the lower layer, W T is shared by two cells (al , and bl ) while
it is shared by three 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).

–

51

–

Chapter 5

Equations of motion (baroclinic component)

UPPER

LOWER

du

cu
䃁

cl

䃁

dl
T

Au

W

Au
䃁

䃁

䃁

au

bu

䃁

T

1/4W

T

al

bl

䃁
T

T

W

T

1/3( W 䋭 1/4W ) = 1/4W

Figure 5.3. Second example of land-sea patterns, in which all four upper cells are sea cells, with one sea cell andthree land cells in the lower layer.
c. Generalization
The relationship between the land-sea distribution and the vertically and diagonally upward fluxes stated aboveis 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 the number in the upper layer (1 ≤ Nl < Nu ≤ 4). Each cell in the lower layer carries W T /Nl , andW T /Nu of it goes vertically upward. The remaining
W T /Nl −W T /Nu = W T (Nu − Nl )/(Nl Nu )(5.3)

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 is
W T (Nu − Nl )/(Nl Nu ) × 1/(Nu − Nl ) = W T /(Nl Nu ).
(5.4)
The number of such fluxes coming to the cell du is Nl , so their total is
W T /(Nl Nu ) × Nl = W T /Nu .

(5.5)

Based on these discussions we understand the difference between (5.1) and (5.2).
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 isa 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 ) andal W T (ual + udu )/(2Nl Nu ),(5.6)

where ual and udu are the velocity at cells al and du , respectively. Purely vertical mass flux and momentum flux
from cell al to cell au are expressed as
al W T /Nuand

al W T (ual + uau )/(2Nu ),– 52 –(5.7)

5.1. Advection terms

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 1/2WT

1/2W 䋭 1/3W = 1/6W

T

Figure 5.4. Third example of land-sea patterns, in which one of the upper cells is a land cell, with two land andtwo sea cells in the lower layer.
respectively, where uau is the velocity at cell au . Similar formulations apply to cells bl and cl .
Mass and momentum fluxes for W T at other T-points around cell du should be calculated similarly to complete
vertically and diagonally upward momentum advections around cell du . When Nu = Nl , diagonally upward fluxes
need not be considered and only vertical fluxes (5.7) apply.

5.1.2

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. To do this, westart with the generalization of the T-cell mass continuity (3.3)-(3.9). Assuming that ei+ 1 , j+ 1 is a land-sea index
22TT(unity for sea and zero for land) for U-cell (i + 12 , j + 12 ), the general formulae for Ui+
1 , j and Vi, j+ 1 are given as
2

2

1
T
∗
Ui+
1 , j = (ei+ 1 , j− 1 + ei+ 1 , j+ 1 )ui+ 1 , j Δyi+ 1 , j Δz
2
2
2
2
2
2
2
2
and
1
Vi,Tj+ 1 = (ei− 1 , j+ 1 + ei+ 1 , j+ 1 )v∗i, j+ 1 Δxi, j+ 1 Δz
2
2
2
2
2
2
2
2

(5.8)

Here we neglect the vertical subscript (k + 12 ).
Substituting these formulae into the T-cell mass continuity (3.3), the X (zonal) component of the mass continuity

–

53

–

Chapter 5

Equations of motion (baroclinic component)

for U-cell (i + 12 , j + 12 ) (3.10), XMCUi+ 1 , j+ 1 , multiplied by its own land-sea signature ei+ 1 , j+ 1 , is expressed as
2

2

2

2

Δyi+ 1 , j+ 1
2
2
XMCUi+ 1 , j+ 1 = ei+ 1 , j+ 1
Δz
2
2
2
2
2
 1
[(e 1 1 + ei− 1 , j+ 1 )u∗i− 1 , j − (ei+ 1 , j− 1 + ei+ 1 , j+ 1 )u∗i+ 1 , j ]
×
2
2
2
2
2
2
Ni, j i− 2 , j− 2221+[(e 1 1 + ei+ 1 , j+ 1 )u∗i+ 1 , j − (ei+ 3 , j− 1 + ei+ 3 , j+ 1 )u∗i+ 3 , j ]
2
2
2
2
2
2
Ni+1, j i+ 2 , j− 2221+[(e 1 1 + ei− 1 , j+ 3 )u∗i− 1 , j+1 − (ei+ 1 , j+ 1 + ei+ 1 , j+ 3 )u∗i+ 1 , j+1 ]
2
2
2
2
2
2
Ni, j+1 i− 2 , j+ 2
2
2

1
+
[(ei+ 1 , j+ 1 + ei+ 1 , j+ 3 )u∗i+ 1 , j+1 − (ei+ 3 , j+ 1 + ei+ 3 , j+ 3 )u∗i+ 3 , j+1 ]
2
2
2
2
2
2
2
2
Ni+1, j+1
22
 1
Δyi+ 1 , j+ 1
2
2
= ei+ 1 , j+ 1
(e 1 1 + ei− 1 , j+ 1 )u∗i− 1 , j
Δz ×
2
2
2
2
2
Ni, j i− 2 , j− 2
2
 1

1
+ −
+
(ei+ 1 , j− 1 + ei+ 1 , j+ 1 )u∗i+ 1 , j
2
2
2
2
Ni, j Ni+1, j2
1−(e 3 1 + ei+ 3 , j+ 1 )u∗i+ 3 , j2
2
Ni+1, j i+ 2 , j− 221+
(e 1 1 + ei− 1 , j+ 3 )u∗i− 1 , j+1
2
2
Ni, j+1 i− 2 , j+ 2
2


1
1
+ −
+
(ei+ 1 , j+ 1 + ei+ 1 , j+ 3 )u∗i+ 1 , j+1
2
2
2
2
Ni, j+1 Ni+1, j+12
1−(e 3 1 + ei+ 3 , j+ 3 )u∗i+ 3 , j+1 .
2
2
Ni+1, j+1 i+ 2 , j+ 2
2

(5.9)

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

(5.10)

2

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
(5.11)and−

1
Ni, j+1+

1
Ni+1, j+1

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

1
Ni, j+1222(ei− 1 , j+ 1 + ei− 1 , j+ 3 ) −
2

2

2

2

1
Ni+1, j+1

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

2

2

2

(5.12)– 54 –

5.1. Advection terms
Thus, based on (3.13) and (3.14),
XMCUi+ 1 , j+ 1 = ei+ 1 , j+ 1
2

2

2

Δyi+ 1 , j+ 1
2

2

2

2



1
(e 1 1 + ei− 1 , j+ 1 )(u∗i− 1 , j + u∗i+ 1 , j )22Ni, j i− 2 , j− 2
2
2

Δz

1

(e 3 1 + ei+ 3 , j+ 1 )(u∗i+ 1 , j + u∗i+ 3 , j )22
Ni+1, j i+ 2 , j− 2221+
(e 1 1 + ei− 1 , j+ 3 )(u∗i− 1 , j+1 + u∗i+ 1 , j+1 )2
2
Ni, j+1 i− 2 , j+ 2
2
2

1
−
(ei+ 3 , j+ 1 + ei+ 3 , j+ 3 )(u∗i+ 1 , j+1 + u∗i+ 3 , j+1 )2
2
2
2
Ni+1, j+1
2
2
 1

1
= ei+ 1 , j+ 1 ei− 1 , j+ 1
Ui,Uj +
UU
2
2
2
2 Ni, jNi, j+1 i, j+1
 1

1
1 U
U
U
− ei+ 3 , j+ 1
Ui+1,
+
U
+ei− 1 , j− 1
Ui, j
j
i+1,
j+1
2
2 Ni+1, j
2
2 Ni, jNi+1, j+1
1
1
1U.
− ei+ 3 , j+ 3
Ui+1,
Ui,Uj+1 − ei+ 3 , j− 1
UU
j+1 + ei− 12 , j+ 32
22 Ni+1, j+1
2
2
Ni, j+1
Ni+1, j i+1, j−(5.13)

Adding the Y (meridional) component to the above formula, we obtain the horizontal part of the U-cell masscontinuity as follows:

HMCUi+ 1 , j+ 1 = ei+ 1 , j+ 1
2

2

2

2

× ei− 1 , j+ 1

 1



Ui,Uj+1 − ei+ 3 , j+ 1

1

Ui,Uj +

1

U
Ui+1,
j+

1

U
Ui+1,
j+1

2 Ni, j2
2 Ni+1, j
Ni, j+1
Ni+1, j+1
 1



1
1
1
U
U
− ei+ 1 , j+ 3
+ ei+ 1 , j− 1
Vi,Uj +
Vi+1,
Vi,Uj+1 +
Vi+1,
j
j+1
2
2 Ni, j2
2 Ni, j+1
Ni+1, jNi+1, j+1
1
1
UU+ ei− 1 , j− 1
(Ui,Uj + Vi,Uj ) − ei+ 3 , j+ 3
(Ui+1,
j+1 + Vi+1, j+1 )2
2 Ni, j2
2 Ni+1, j+1
1
1UU+ ei− 1 , j+ 3
(Ui,Uj+1 −Vi,Uj+1 ) − ei+ 3 , j− 1
(Ui+1,
j −Vi+1, j ) .
2
2 Ni, j+1
2
2 Ni+1, j2

(5.14)

Assuming mass fluxes ME , MN , MNE , MSE as follows:ME
MN

i, j+ 21i+ 21 , j

MNEi, jMSEi, j= ei+ 1 , j+ 1 ei− 1 , j+ 1

 1

Ui,Uj+1 ,Ni, j+11U,= ei+ 1 , j+ 1 ei+ 1 , j− 1
Vi,Uj +
Vi+1,
j
2
2
2
2 Ni, jNi+1, j1
= ei+ 1 , j+ 1 ei− 1 , j− 1
(Ui,Uj + Vi,Uj ),22
2
2 Ni, j1= ei− 1 , j+ 1 ei+ 1 , j− 1
(Ui,Uj −Vi,Uj ),22
2
2 Ni, j
2

2

2

2

Ni, j
 1

1

Ui,Uj +

(5.15)then,
HMCUi+ 1 , j+ 1
2

2

= ME

i, j+ 21− ME

i+1, j+ 21+ MN

i+ 21 , j

− MN

i+ 12 , j+1+MNEi, j − MNEi+1, j+1 + MSEi, j+1 − MSEi+1, j .

(5.16)

Here, ME and MN are axis-parallel mass fluxes, and MNE and MSE are horizontally diagonal ones (Figure 5.5).

–

55

–

Chapter 5

Equations of motion (baroclinic component)

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, ji, j
MNE i, j

Figure 5.5. Distribution of generalized mass fluxes for U-cell (i + 1/2, j + 1/2)
If we derive the formula for the standard case from (5.14) (all of N are 4),
1 1 U
1(Ui, j +UiU, j+1 ) − (UiU+1, j +UiU+1, j+1 )
2 2
2
2
1 U
1
U
+ (Vi, j +Vi, j+1 ) − (ViU+1, j +ViU+1, j+1 )
2
2
1 1 U
1 U
U
(U +Vi, j ) − (Ui+1, j+1 +ViU+1, j+1 )+2 2 i, j2
1 U
1U+ (Ui, j+1 −Vi, j+1 ) − (UiU+1, j −ViU+1, j ) .
2
2

HMCUi+ 1 , j+ 1 =2(5.17)

This expression means that the horizontal mass flux convergence is a mean of those of the axis-parallel mass fluxes(3.12) and of the diagonal ones (3.17). 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(δ v/δ x)2 and (δ u/δ y)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 ) to average the convergences of the axis-parallel and the horizontally diagonal mass fluxes, as long as α + β = 1, as seen in (5.17). In
MRI.COM α = 2/3 and β = 1/3 are chosen for the standard case so that the momentum advection terms lead to– 56 –

5.1. Advection terms
the generalized Arakawa scheme. In this case the zonal momentum advection is expressed as follows:
2 1
1(u 1 1 + ui+ 1 , j+ 1 ) (Ui,Uj + Ui,Uj+1 )2
2 2
3 2 i− 2 , j+ 2
1 U
1U− (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 U
1U− (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 U
1
U
− (ui+ 1 , j+ 1 + ui+ 3 , j− 1 ) (Ui+1,
j −Vi+1, j )
2
2
2
2 2
2

CADi+ 1 , j+ 1 (u) =2

2(5.18)

This scheme under Arakawa’s B-grid arrangement conserves the quasi-enstrophies ((∂ u/∂ y)2 and (∂ v/∂ x)2 ) in ahorizontally non-divergent flow.
To merge the generalized Arakawa scheme for the standard case into the general form of the horizontal massflux expressed in Figure 5.5 and related momentum flux, let us examine the axis-parallel and horizontally diagonalmass flux associated with Ui,Uj . Look at Figure 5.2b, where letters a, b, c, and d designate the land-sea index andnames of U-cells. Cell d is assumed to be a sea cell (d = 1). We analyze two kinds of mass fluxes associatedwith Ui,Uj under different combinations of a, b, and c (eight cases), as indicated in the first column in Table 5.1.
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 (5.14). Column(C) indicates the coefficient of Ui,Uj in the horizontally diagonal mass flux of (5.14).
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 summing the product of A and C. That is,
c1

=

8

1

∑ An Bn = 6 c(ab − a − b+3)

n=1

and
c2

=

8

1

∑ An Cn = 6 b(3−a − c).

(5.19)

n=1

Then, the axis-parallel and the horizontally diagonal flux of zonal momentum (u) related with Ui,Uj , multiplied by
the land-sea index d, ared(uc + ud )c1Ui,Uj
2d(ub + ud )c2Ui,Uj
2

1
1(uc + ud ) cd(ab − a − b+3)Ui,Uj ,
2
6
and
1
1
=
(ub + ud ) bd(3−a − c)Ui,Uj ,2
6=respectively.

–

57

–

(5.20)

Chapter 5

Equations of motion (baroclinic component)

Table 5.1. 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),
for eight combinations of indices a, b, and c in Figure 5.2b. Cell d is assumed to be a sea cell, and the momentum
advection by means of Ui,Uj into and from cell d is generalized.
B
Coefficient of Ui,UjC
Coefficient of Ui,Uj

CASEn

Land-sea index
a b c

A

(axisi-parallel)+(horizontally-diagonal)
×

1

1 1 1

abc

1/3

1/6

2

1 1 0

ab(1−c)

0

1/3

3

1 0 1

ab(1−b)c

1/3

0

4

0 1 1

(1−a)bc

1/3

1/3

5

1 0 0

−

0

0

6

0 1 0

(1−a)b(1−c)

0

1/2

7

0 0 1

(1−a)(1−b)c

1/2

0

8

0 0 0

−

0

0

The resultant momentum fluxes are as follows:FE

i, j+ 21FN

i+ 21 , j1
(u 1 1 + ui+ 1 , j+ 1 )ME 1 ,i, j+ 2222 i− 2 , j+ 2
1
(u 1 1 + ui+ 1 , j+ 1 )MN 1 ,i+ 2 , j22
2 i+ 2 , j− 2
1
(u 1 1 + ui− 1 , j− 1 )MNEi, j ,22
2 i+ 2 , j+ 2
1
(u 1 1 + ui+ 1 , j− 1 )MSEi, j ,22
2 i− 2 , j+ 2

(u) =
(u) =

FNEi, j (u) =FSEi, j (u) =(5.21)

where
ME

=

MN

=

MNEi, j=MSEi, j=i, j+ 12i+ 21 , j

1
(CXNi, j Ui,Uj + CXSi, j+1 Ui,Uj+1 ),6
1U(CY Ei, j Vi,Uj + CYWi+1, j Vi+1,
j ),6
1
CNEi, j (Ui,Uj + Vi,Uj ),6
1
CSEi, j (Ui,Uj −Vi,Uj ),6(5.22)

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),

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),

CY Ei, 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),

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),

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

– 58 –

2

2

2

2

2

2

2

2

2
2

2
2
2
2

2
2
2
2

2
2
2
2

2
2

(5.23)

5.2. Viscosity
Finally, convergence of the horizontal momentum fluxes is written as
CADi+ 1 , j+ 1 (u) = FE22i, j+ 21(u) − FE

i+1, j+ 21(u) + FN

i+ 21 , j(u) − FN

i+ 21 , j+1(u)+FNEi, j (u) − FNEi+1, j+1 (u) + FSEi, j+1 (u) − FSEi+1, j (u).

5.2

(5.24)

Viscosity

For horizontal viscosity, the harmonic (default) or biharmonic (option VISBIHARM) scheme can be selected.
Anisotropy of viscosity with respect to the flow direction can be applied (option VISANISO) when harmonic viscosity is chosen. The viscosity coefficient is a constant by default but can be a function of local velocity gradientsand grid-size (option SMAGOR).
For vertical viscosity, the harmonic scheme is used and the local coefficient is the larger of a background constantand that calculated through a turbulence closure scheme. A parameterization of bottom friction (Weatherly, 1972)
is adopted at the lowest layer.

5.2.1

Horizontal viscosity

Horizontal tension DT and horizontal shear DS are defined as follows.
 
 
∂
∂
u
v
− hμ
,
DT = hψhμ ∂ μ hψhψ ∂ ψ hμ
 
 
∂
∂
v
u
DS = hψ.
+ hμhμ ∂ μ hψhψ ∂ ψ hμ
The friction terms are(5.25)(5.26)Vu =1 ∂  2  1 ∂  2 
h σT + 2
h σS ,
h2ψ hμ ∂ μ ψh μ hψ ∂ ψ μ(5.27)Vv =1 ∂  2  1 ∂  2 h σS − 2h σT ,
h2ψ hμ ∂ μ ψ
hμ hψ ∂ ψ μ(5.28)

where νH is the horizontal viscosity coefficient, σT = νH DT , and σS = νH DS .
In the geographical coordinate system, where (μ , ψ ) = (λ , φ ), hλ = a cos φ , and hφ = a, tension and shear areDT =1 ∂u 1 ∂v v−
− tan φ ,a cos φ ∂ λ a ∂ φ a(5.29)DS =1 ∂v 1 ∂u u+
+ tan φ .
a cos φ ∂ λ a ∂ φ a(5.30)

Vu =

1 ∂
∂
1
2 tan φσT +σS − σS,a cos φ ∂ λa ∂φa(5.31)Vv =1
2 tan φ∂1 ∂σS −σT + σT,a cos φ ∂ λa ∂φa(5.32)

The friction terms in this case arewhere the third term on the r.h.s. is called the metric term.
When the VISBIHARM option is selected, the above operation is repeated twice using a viscosity coefficient νBH .
The friction terms Vu and Vv given by (5.27) and (5.28) are sign-reversed and substituted as u and v in equations(5.25) and (5.26). A biharmonic scheme dissipates noise only on scales near the grid size. This scale selectivity– 59 –

Chapter 5

Equations of motion (baroclinic component)

allows the explicitly represented eddies to survive without unphysical damping in eddy-resolving models, although
we must note that a 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 meso-scale eddies.

5.2.2 Horizontal anisotropic viscosity
The viscosity in an ocean general circulation model seeks to attenuate numerical noise rather than parameterizing the sub-grid scale momentum transport. The momentum advection scheme should conserve the total kinetic
energy in the general 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 numerical dispersion. In eddy-resolving models, the current velocity and the numerical noise
are greater than in eddy-less models. A biharmonic viscosity scheme has been widely used to reduce numericalnoise while maintaining the eddy structure.
Smith and McWilliams (2003) proposed a method of making a harmonic viscosity scheme anisotropic in anarbitrary direction. Setting σT and σS in equations (5.27) and (5.28) to
 


 


1
2 − n2
σT
(
ν
+
ν
)
0
n
n
−2n
D
μ
ψ
T
0
1
μ
ψ
2
,
(5.33)=+ (ν0 − ν1 )nμ nψσS
ν1n2μ − n2ψ 2nμ nψDS0where n̂ = (nμ , nψ ) is a unit vector in an arbitrary direction and ν0 (ν1 ) is the viscosity coefficient parallel (perpendicular) to n̂. When option VISANISO 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 are maintained.
The following is a note on usage. The ratio ν1 /ν0 is read from namelist: nmlvisaniso (variable name: cc0;
default value is 0.2). The ratio at the lateral boundary is also read from the namelist (cc1; default is 0.5). Whenthe variable flgvisequator is set as a positive number in the namelist, the ratio ν1 /ν0 is tapered linearly fromcc0 at the latitude flgvisequator (in degrees) to zero at the equator. The ratio is not tapered when a negative
number is set, and the default value of flgvisequator is −1.

5.2.3 Smagorinsky parameterization for horizontal viscosity
To give the necessary but minimum viscosity to reduce numerical noise, the viscosity coefficient is made proportional to the local deformation rate (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|:T −1 = |D| =D2T + D2S ,(5.34)

the viscosity coefficients are set as follows.


2

νH

=

CΔmin
π

νBH

=

Δ2min
νH ,
8

|D|,

(5.35)(5.36)

where C (cscl) is a dimensionless scaling parameter set by considering numerical stability and Δmin is the smallerof the zonal and meridional grid-spacings.
–

60

–

5.2. Viscosity
The parameter C should be selected to satisfy the following conditions.
• Restriction of grid Reynolds number:ΔminνH > U,
2• Restriction on the width of the lateral boundary layer:(5.37)

νH > β Δ3min ,(5.38)• CFL condition:

Δ2min,(5.39)
2Δtwhere β = d f /dy is the meridional gradient of the Coriolis parameter. Scaling the deformation rate |D| by U/Δmin√
gives the condition for stability: C > π / 2 ≈ 2.2 from (5.37) (Griffies and Hallberg, 2000).

νH <

5.2.4

Discretization

Using the notations

δμ Ai, j≡δψ Ai, j≡

Ai+ 1 , j − Ai− 1 , j
2

2ΔμAi, j+ 1 − Ai, j− 1,

δi Ai, j2,Δψ≡ Ai+ 1 , j − Ai− 1 , j ,δ j Ai, j≡ Ai, j+ 1 − Ai, j− 1 ,

2

2

2

2

2

and
Ai, j
Ai, j

μ

≡

ψ

≡

1
(A 1 + Ai+ 1 , j ),2
2 i− 2 , j
1(A 1 + Ai, j+ 1 ),2
2 i, j− 2(5.40)

deformation rates are discretized as follows:
DT i, jDSi, j==hψ i, jδμhμ i, jhψ i, jδμhμ i, j




u
hψ
v
hψ



ψ

i, j



hμ i, j−δψhψ i, jψ+i, jhμ i, jδψhψ i, j




v
hμ
u
hμ



μ

,
i, j



μ

.

(5.41)

i, j

Viscosity forces are discretized as follows:
Fx i+ 1 , j+ 1
2

2

=
×

1

×
ΔVi+ 1 , j+ 1
2
2
⎤
⎡




1
1
ψ
μ
⎦,
⎣
δi ΔyΔzh2ψ νH DT+ 2
δ j ΔxΔzh2μ νH DS
i+ 12 , j+ 12
i+ 12 , j+ 12
h2ψ i+ 1 , j+ 1
hμ i+ 1 , j+ 1
2

Fy i+ 1 , j+ 1
2

2

=
×

2

2

2

1

×
(5.42)
ΔVi+ 1 , j+ 1
2
2
⎤
⎡




1
1
ψ
μ
⎦.
⎣
δi ΔyΔzh2ψ νH DS− 2
δ j ΔxΔzh2μ νH DT
i+ 12 , j+ 12
i+ 12 , j+ 12
h2ψ i+ 1 , j+ 1
hμ i+ 1 , j+ 1
2

2

2

–

61

–

2

Chapter 5

Equations of motion (baroclinic component)

When the grid point (i − 12 , j + 12 ) is defined as a (vertically partial) land (Figure 5.6a), the velocity gradients at
the wall are calculated as follows.




∂u
∂x
∂v
∂x




=

i, j+ 12=i, j+ 12

ui+ 1 , j+ 1
2

Δxi−j

2

vi+ 1 , j+ 1
2

Δxi−j

2

,
,

(5.43)

where Δxi−j is the length between the points (i, j + 12 ) and (i + 12 , j + 12 ). The contribution of this wall to the forceis:
FxW
i+ 1 , j+ 1
2

2

=

−

ui+ 1 , j+ 1
1
2
2
2
Δy
Δz̃
h
ν
,
1
1
1
1
ψ
−
ΔVi+ 1 , j+ 1 hψ 2i+ 1 , j+ 1 i, j+ 2 i, j+ 2 i− 2 , j Hi− 2 , j Δxi+
1 , j+ 1
2

FyW
i+ 1 , j+ 1
2

2

=

2

2

2

2

2

vi+ 1 , j+ 1
1
2
2
2
−
Δy
Δz̃
h
ν
,
1
1
1
1
ψ
−
ΔVi+ 1 , j+ 1 hψ 2i+ 1 , j+ 1 i, j+ 2 i, j+ 2 i− 2 , j Hi− 2 , j Δxi+
1 , j+ 1
2

2

2

2

2

(5.44)

2

where Δz̃i, j+ 1 is the wall height.
2

5.2.5 Vertical viscosity
Only the harmonic scheme is considered. The vertical momentum flux is assumed to be proportional to thevertical gradient 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− 1 − uk+ 1∂u2
2= −νvk
,
− νv
∂z k
Δzk
where Δzk = (Δzk− 1 + Δzk+ 1 )/2, and Δzk+ 1 is the thickness of the U-cell (dzu). Similarly, the momentum flux in2
2
2
the lower part of the U-cell is calculated as follows:
uk+ 1 − uk+ 3∂u2
2= −νvk+1,− νv∂ z k+1
Δzk+1
where νvk+1 is set to zero if the (k + 32 )th level is the solid Earth. The bottom friction is calculated independently(see the next subsection). Also note that the variations of the grid thickness at the bottom and near the sea surface
are not considered when evaluating fluxes for simplicity.
To calculate viscosity, the divergence of the momentum flux is first calculated. The expression for the verticalviscosity term is ∂u   ∂u 
νv ∂ z k − νv ∂ z k+1 νvk (uk− 12 − uk+ 12 ) νvk+1 (uk+ 12 − uk+ 32 )∂  ∂u
νv
=
=
−
%
% 1
∂z
∂ z k+ 12
Δzk ΔzΔzk+1 %
Δz 1
Δz 1
k+ 2

k+ 2

(5.45)

k+ 2% that is,where the variation of the grid thickness for the U-cell is now taken into account and is represented by Δz,& 1 (Figure 5.6(b)). Note that the first term on the r.h.s. of equation (5.45) is set to zero in% 1 = Δz 1 − Δz
Δz
k+ 2

k+ 2

k+ 2

calculating the viscosity term for the vertical level of 12 (k = 0).

– 62 –

5.2. Viscosityau i-1/2, j+1/2

u i+1/2, j+1/2
㼺y i+1/2, j㼺x i, j+1/2

u i+1/2, j-1/2
㼺z i

u i-1/2-u i-1/2~
㼺

u i+1/2zi㼺z i+1
㼺z~ i+1

u i+3/2~

u k+1/2

㼺z k+1/2

㼺z k+1/2

~
㼺z
k+3/2

㼺z k+1

u k+1/2

u k-1/2

㼺z k+3/2

u k+1/2

㼺z k

u k-1/2

㼺z k+1/2b

Figure 5.6. (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 + 32 ) has a sea bed. Right: The U-cell (k + 12 ) has a sea bed.

–

63

–

Chapter 5

Equations of motion (baroclinic component)

5.2.6 Bottom friction
When a U-cell in the (k + 12 )th layer contains solid earth (Figure 5.6(b) right), the stress from the lower boundary(τxb , τyb ) is calculated following Weatherly (1972). The specific expression is as follows.


τxb
τyb




=

−ρ0Cbtmu 2 1 + v2 1
k+ 2

k+ 2

cos θ0sin θ0− sin θ0cos θ0



uk+ 1
2

vk+ 1


,

2

where Cbtm is a dimensionless constant. Viscous stress at the lower boundary has a magnitude proportional to thesquare of the flow speed at the U-cell and an angle (θ0 + π ) relative to the flow direction.
In MRI.COM,
Cbtm = 1.225 × 10−3
θ0 = ±π /18 [rad](≡ 10[◦ ]),where θ0 is positive (negative) in the northern (southern) hemisphere. The variables are designated in the model
as Cbtm = abtm, cos θ0 = bcs, and sin(±θ0 ) = isgn ∗ bsn, where isgn = 1 in the northern hemisphere andisgn = −1 in the southern hemisphere.

References
Delhez, E. J. M., and E. Deleersnijder, 2007: Overshootings and spurious oscillations caused by biharmonic
mixing., Ocean Modell., 17, 183–198.
Griffies, S., and R. Hallberg, 2000: Biharmonic friction with a Smagorinsky-like viscosity for use in large-scale
eddy-permitting ocean models, Mon. Weather Rev., 128, 2935–2946.
Ishizaki, H., and T. Motoi, 1999: Reevaluation of the Takano-Oonishi scheme for momentum advection on bottom
relief in ocean models, J. Atmos. Oceanic Tech., 16, 1994–2010.
Smagorinsky, J., 1963: General circulation experiments with the primitive equations: I. The basic experiment,
Mon. Weather Rev., 91, 99–164.
Smith, R. D., and J. C. McWilliams, 2003: Anisotropic horizontal viscosity for ocean models., Ocean Modell., 5,
129–156.
Weatherly, G. L., 1972: A study of the bottom boundary layer of the Florida current, J. Phys. Oceanogr., 2, 54–72.

–

64

–

Chapter 6

Temperature and salinity equations

This chapter formulates a basic part of the temperature and salinity prediction procedure, except for ocean surface
processes (related to mixed layer and surface fluxes). First, the finite difference expression of the flux form is
explained (Section 6.1). The formulation of each component of the flux (advection and diffusion) is then given(Sections 6.2 and 6.3). The advection scheme mainly focuses on the Quadratic Upstream Interpolation for Convective Kinematics (QUICK; Leonard, 1979) scheme. Other advection schemes, QUICK with Estimated Streaming
Terms (QUICKEST; Leonard, 1979; option QUICKEST), the Uniformly Third-Order Polynomial Interpolation Algorism (UTOPIA; Leonard et al., 1993; option UTOPIA), and the Second Order Moment (SOM; Prather, 1986;
option SOMADVEC) are explained in Chapter 13. Finally, a convective adjustment scheme is explained (Section
6.4).

6.1

Flux form

The equations governing the time change of temperature and salinity (differential form) are presented in equations (2.14) to (2.29) of Chapter 2. The finite difference forms of (2.14) and (2.15) are given by calculating fluxes
through each cell face, and setting their divergence and convergence to be the time change at the grid cell (Figure6.1) as follows:
ΔVi,n+1= T n−1 1 ΔV n−1 1
Ti,n+1j,k+ 1j,k+ 1i, j,k+i, j,k+222

2+ 2 Δt {FX i− 1 , j,k+ 1 − FX i+ 1 , j,k+ 1 + FY i, j− 1 ,k+ 1 − FY i, j+ 1 ,k+ 1 + FZ i, j,k+1 − FZ i, j,k }, (6.1)
2

2

2

2

2

2

2

2

where T is temperature or salinity, ΔV is the volume of the grid cell, and FX, FY, and FZ represent (flux) × (areaof the grid boundary). The fluxes include contributions by advection and diffusion. Note that ΔV from k = 1 tok = ksgm varies with time (Chapter 4).
The volume of the grid cell is written as follows:
ΔVi, j,k = a tri− 1 , j− 1 Δzi− 1 , j− 1 ,k + a tli+ 1 , j− 1 Δzi+ 1 , j− 1 ,k
2

2

2

2

2

2

2

2

+ a bri− 1 , j+ 1 Δzi− 1 , j+ 1 ,k + a bli+ 1 , j+ 1 Δzi+ 1 , j+ 1 ,k ,
2

2

2

2

2

2

2

2

(6.2)

where Δz is the thickness of the corresponding velocity cell (in which fluctuations of the surface height are considered). The horizontal area of the left-lower quarter of the T-cells at (i, j) (corresponding to the southwestern part ingeographical coordinates) is represented by a tri− 1 , j− 1 . Similarly, a tli+ 1 , j− 1 is the right-lower (southeastern),2222a bri− 1 , j+ 1 is the left-upper (northwestern), and a bli+ 1 , j+ 1 is the right-upper (northeastern) quarter of a T-cell2222
at (i, j). See also Section 3.5.

6.2

Advection

There are several choices for tracer advection schemes. For horizontal advection, the Quadratic Upstream Interpolation for Convective Kinematics (QUICK; Leonard, 1979), the Uniformly Third-Order Polynomial Interpo–

65–

Chapter 6

Temperature and salinity equations

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
Figure 6.1. Grid arrangement around TS-Box (Upper: Views from the upper, Below: views from the horizontal).
Fluxes represented by an arrow are calculated.

lation Algorithm (UTOPIA; Leonard et al., 1993), and the Second Order Moment (SOM; Prather, 1986) schemes
are available. For vertical advection, QUICK with Estimated Streaming Terms (QUICKEST; Leonard, 1979), isavailable in addition to QUICK and SOM. When SOM is selected, it should be used for both horizontal and vertical directions. Refer to Leonard (1979) for a comparison between QUICK and QUICKEST. Leonard et al. (1993)
provides a detailed explanation of the extension from QUICKEST to UTOPIA. The procedure for calculating
UTOPIA is based on Leonard et al. (1994). This section explains the QUICK scheme. Explanations for QUICKEST, UTOPIA, and SOM are given in Chapter 13.
Fluxes due to advection are given as follows:
FXAi+ 1 , j,k+ 1
2

T
= Ui+
1 , j,k+ 1 Ti+ 1 , j,k+ 1 ,(6.3)

FYAi, j+ 1 ,k+ 1= Vi,Tj+ 1 ,k+ 1 Ti, j+ 1 ,k+ 1 ,(6.4)= Wi,Tj,k+1 Ti, j,k+1 ,

(6.5)

2

2

2

FZAi, j,k+1

2

2

2

2

2

2

2

2

where horizontal volume transport U T and V T are defined as follows, using (3.4) to (3.9),
TUi+
1 , j,k+ 1 =22

Δyi+ 1 , j 2
ui+ 1 , j− 1 ,k+ 1 Δzi+ 1 , j− 1 ,k+ 1 + ui+ 1 , j+ 1 ,k+ 1 Δzi+ 1 , j+ 1 ,k+ 1 ,
2
2
2
2
2
2
2
2
2
2
2
2
2
– 66 –(6.6)

6.2. Advection
Vi,Tj+ 1 ,k+ 1 =22

Δxi, j+ 1 2
vi− 1 , j+ 1 ,k+ 1 Δzi− 1 , j+ 1 ,k+ 1 + vi+ 1 , j+ 1 ,k+ 1 Δzi+ 1 , j+ 1 ,k+ 1 .
2
2
2
2
2
2
2
2
2
2
2
2
2

(6.7)

Vertical volume transport W T is then obtained by diagnostically solving (3.3). 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+1=

wi, j,k+1 × areati, j,k+ 1 ,

(6.8)

2

where
= a tri− 1 , j− 1 × eui− 1 , j− 1 ,k+ 1 + a tli+ 1 , j− 1 × eui+ 1 , j− 1 ,k+ 1

areati, j,k+ 1
2

2

2

2

2

2

2

2

2

2

2

+ a bri− 1 , j+ 1 × eui− 1 , j+ 1 ,k+ 1 + a bli+ 1 , j+ 1 × eui+ 1 , j+ 1 ,k+ 1 .
2

2

2

2

2

2

2

2

2

2

(6.9)

An array eu 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 wayof determining grid boundary values of tracer Ti+ 1 , j,k+ 1 , Ti, j+ 1 ,k+ 1 , and Ti, j,k+1 . The centered finite difference22
22
scheme uses the average between two neighboring points of temperature and salinity as the grid boundary value.
In the QUICK scheme, the grid boundary value is interpolated by a quadratic function, using three points, with oneof them added from the upstream side (Figure 6.2).
T
Ti+1

quadratic function thatpasses three points

Ti
Ti-1

Ti+1/2
Ui+1/2

i-1

i+1

i

X

T-box

Figure 6.2. Schematic for interpolation: Ti indicates the representative temperature currently calculated at T-point,
and Ti+ 1 is the grid boundary value. In the QUICK scheme, Ti+ 1 is interpolated by a quadratic function that passes
22
Ti and the neighboring T-point values, Ti−1 and Ti+1 .

Originally, grid boundary values in the QUICK scheme are given as follows:
Ti+ 1 , j,k+ 1=

Ti, j+ 1 ,k+ 1=

Ti, j,k+1

=

2

2

2

2

Δxi Ti+1, j,k+ 1 + Δxi+1 Ti, j,k+ 1
2

2

Δxi+1 + Δxi
Δy j Ti, j+1,k+ 1 + Δy j+1 Ti, j,k+ 1−

Δxi+1 Δxi
ci+ 1 , j,k+ 1 ,2
2
4

Δy j+1 Δy j
di, j+ 1 ,k+ 1 ,2
2
Δy j+1 + Δy j
4
Δzk+ 1 Ti, j+1,k+ 3 + Δzk+ 3 Ti, j,k+ 1 Δzk+ 1 Δzk+ 3
2
2
2
2
2
2
−
ei, j,k+1 ,Δzk+ 3 + Δzk+ 1
4
2

2

2

–

2

67

–

−

(6.10)(6.11)(6.12)

Chapter 6

Temperature and salinity equations

where c, d, and e are defined depending on the direction of the mass flux as follows:2

Δxi δx δx Ti, j,k+ 1=

ci+ 1 , j,k+ 1
2

2(= c p ),x
2Δxi
Δxi+1 δx δx Ti+1, j,k+ 1
2(= cm ),x2Δxi+1Δy j δy δy Ti, j,k+ 1
2(= d p ),y2Δy j
Δy j+1 δy δy Ti, j+1,k+ 1
2(= dm ),y
2Δy j+1
Δzk+ 3 δz δz Ti, j,k+ 3
2
2(= e p ),z
2Δzi, j,k+ 3
2
Δzk+ 1 δz δz Ti, j,k+ 1
2
2(= em ),z
2Δzi, j,k+ 1==

di, j+ 1 ,k+ 1
2

2

=
=

ei, j,k+1=

T
if Ui+
1 , j,k+ 1 > 0,2

2

T
if Ui+
1 , j,k+ 1 < 0,2

2

if Vi,Tj+ 1 ,k+ 1 > 0,22

(6.13)

if Vi,Tj+ 1 ,k+ 1 < 0,2

2

if Wi,Tj,k+1 > 0,
if Wi,Tj,k+1 < 0.

2

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

Ai+ 1 − Ai− 12, δx Ai+ 1 ≡

2

Δxi

2

Ai+1 − Ai,
Δxi+ 1
2

Ai+ 1 + Ai− 1

Ai+1 + Ai
x2, Ai+ 1 ≡.
(6.14)
2
2
2
Letting c p , d p , and e p represent their values for positive velocity at the grid boundary and cm , dm , and em represent
their values for negative velocity at grid boundary and takingxAi ≡we obtainT
FXAi+ 1 , j,k+ 1 =Ui+
1 , j,k+ 1
2

2

2

2

ca

= cm + c p(6.15)

cd= cm − c p(6.16)da= dm + d p(6.17)dd= dm − d p(6.18)ea= em + e p(6.19)

ed= em − e p ,(6.20)

Δxi, j Ti+1, j,k+ 1 + Δxi+1, j Ti, j,k+ 1

Δxi+1, j Δxi, j2−
ca i+ 1 , j,k+ 1
2
2
8

2

Δxi+1, j + Δxi, j2

Δxi+1, j Δxi, j
T+ |Ui+
cd i+ 1 , j,k+ 1 ,1 , j,k+ 1 |
2
2
8
2
2

Δyi, j Ti, j+1,k+ 1 + Δyi, j+1 Ti, j,k+ 1 Δyi, j+1 Δyi, jT
22
FYAi, j+ 1 ,k+ 1 =Vi, j+ 1 ,k+ 1−
da i, j+ 1 ,k+ 12
2
22Δyi, j+1 + Δyi, j822Δyi, j+1 Δyi, j+ |Vi,Tj+ 1 ,k+ 1 |dd i, j+ 1 ,k+ 1 ,
2
2
8
2 2

Δzi, j,k+ 1 Ti, j,k+ 3 + Δzi, j,k+ 3 Ti, j,k+ 1 Δzi, j,k+ 3 Δzi, j,k+ 1
T
2
2
2
2
2
2
FZAi, j,k+1 =Wi, j,k+1−
eai, j,k+1
Δzi, j,k+ 3 + Δzi, j,k+ 1
8
2+ |Wi,Tj,k+ 1 |Δzi, j,k+ 3 Δzi, j,k+ 1
2

8

2

2

(6.21)(6.22)

2

ed i, j,k+1 .

(6.23)

Equation (6.21) can be rewritten as
FXAi+ 1 , j,k+ 1
2

2

T
Ui+
1 , j,k+ 1 T̃i+ 1 , j,k+ 1 + AQ2
2
2
2

– 68 –∂ 3 Ti+ 1 , j,k+ 1
2

∂ x3

2

,

(6.24)

6.3. Diffusion
where T̃i+ 1 , j,k+ 1 is the value of T at the grid boundary interpolated by the cubic polynomial, and22T
AQ = |Ui+
1 , j,k+ 1 |2

Δxi+1 Δxi+ 1 Δxi
2

8

2

.

(6.25)

Although the time integration for advection is done by the leap-frog scheme, the second term on the r.h.s. of(6.24) 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 > 0 and the T-point at (i, j, k + 32 ) is below
the bottom. The upstream-side weighting ratio is given by the user as the namelist parameter vupp.

6.3

Diffusion

Historically, a harmonic diffusion operator is applied in each direction of the model coordinates to express
mixing of tracers. In the real ocean, transport and mixing would occur 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 the horizontal direction, while the harmonic scheme continues to be used for vertical diffusion.
Three types of horizontal diffusion, harmonic horizontal diffusion (default), biharmonic horizontal diffusion (option TRCBIHARM), and isopycnal diffusion (option ISOPYCNAL), 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) is used with it. This is realized by merely setting two mixing coefficients for the isopycnal
diffusion tensor (Griffies, 1998). An anisotropic GM scheme (Smith and Gent, 2004, option GMANISOTROP), which
gives greater diffusivity only in the direction of the current vector, is also available.
The following is a guide to selecting a horizontal diffusion scheme. Biharmonic diffusion is appropriate for ahigh resolution model that can resolve eddies because it is more scale-selective than harmonic diffusion and does
not unnecessarily suppress disturbances in resolved scales. However, biharmonic diffusion is not recommended
in eddy-less models because this would result in numerical instability. Harmonic horizontal diffusion is not recommended because this scheme would cause unrealistic cross-isopycnal (diapycnal) mixing as mentioned above.
Instead, both isopycnal diffusion and the GM scheme should be used. Isopycnal diffusion mixes tracers along neutral surfaces. The GM scheme represents eddy-induced transports in isopycnal layers, mimicking transport caused
by baroclinic instability. Using an anisotropic GM scheme can maintain the meso-scale eddy structures and swift
currents by restricting the direction of diffusion, and thus may be usable even for a high resolution model.

6.3.1

Vertical diffusion

Vertical diffusion assumes that vertical diffusion flux is proportional to the vertical gradient of temperature andsalinity. By default, vertical diffusivity is given as a function of depth. Its profile is stored in the one dimensional
array vdbg(1 : km) (e.g., Tsujino et al., 2000). A three dimensional distribution can be set by selecting option
VMBG3D 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.
In addition to these static profiles, the following processes give vertical diffusivity coefficients for every modeltime step.
–

69

–

Chapter 6

Temperature and salinity equations• Surface mixed layer models.
• Vertical component of isopycnal diffusion,• Enhanced diffusivity where the stratification is unstable (option DIFAJS).
• Enhanced diffusivity around rivermouths to avoid negative salinity (by setting flg enhance vm rivmouthof namelist nrivermouth to be .true.). This scheme is needed when positive definiteness is not guaranteed
by a tracer advection algorithm.
The vertical diffusion for “this” time step is taken as the largest of the above estimations. When one or more of theabove mixing schemes is employed, a backward (implicit) scheme is used in the time integration (Section 12.5)
because high diffusivity is expected. Otherwise, a forward scheme is used.
The finite difference form is as follows:
FZDi, j,k+1 = −κz areati, j,k+1 δz Ti, j,k+1 ,whereδz Ti, j,k+1 ≡(6.26)Ti, j,k+ 1 − Ti, j,k+ 3
2

2

Δzk+1

(6.27)

Noted that, for simplicity, the change of the grid thickness at the bottom and fluctuations of the surface height are
not considered in the grid distance Δzk+1 when calculating the gradient.

6.3.2 Harmonic horizontal diffusion
Harmonic horizontal diffusion assumes that diffusion fluxes are proportional to the gradient of temperature andsalinity. The finite differences of the fluxes are as follows:y= −κH Δyi+ 1 , j Δzi+ 1 , j,k+ 1 δx Ti+ 1 , j,k+ 1 ,FXDi+ 1 , j,k+ 1
2

2

2

2

2

2

x

= −κH Δxi, j+ 1 Δzi, j+ 1 ,k+ 1 δy Ti, j+ 1 ,k+ 1 ,FYDi, j+ 1 ,k+ 1
2

2

2

2

2

2

2

2

(6.28)(6.29)

whereδx Ti+ 1 , j,k+ 1
2

2

Ti+1, j,k+ 1 − Ti, j,k+ 1≡

2

Δxi+ 1 , j

2,(6.30).

(6.31)

2

δy Ti, j+ 1 ,k+ 1
2

2

Ti, j+1,k+ 1 − Ti, j,k+ 1

≡

2

2

Δyi, j+ 1
2

6.3.3 Biharmonic horizontal diffusion
Biharmonic horizontal diffusion assumes that diffusion flux is proportional to the gradient of the Laplacian oftemperature and salinity. The finite difference of the flux is as follows:y= κb Δyi+ 1 , j Δzi+ 1 , j,k+ 1 δx ∇2 Ti+ 1 , j,k+ 1 ,FXDi+ 1 , j,k+ 1
2

2

2

2

2

2

(6.32)= κb Δxi, j+ 1 Δzi, j+ 1 ,k+ 1 δy ∇2 Ti, j+ 1 ,k+ 1 ,(6.33)

Δxi, j Δyi, jyx(δx Δzi, j,k+ 1 δx Ti, j,k+ 1 + δy Δzi, j,k+ 1 δy Ti, j,k+ 1 ).
2
2
2
2
ΔVi, j,k+ 1(6.34)

2∇2 Ti, j,k+ 1 =2x

FYDi, j+ 1 ,k+ 1
where

2

2

2

2

– 70 –

2

2

2

2

6.3. Diffusion

6.3.4

Isopycnal diffusion

In isopycnal diffusion, diffusion flux is expressed by high diffusivity along the isopycnal surface κI , low diffusivity across the isopycnal surface κD , and their product with the gradient of temperature and salinity in eachdirection. Using diffusion tensor K, each flux component is written as F m (T ) = −K mn ∂n T , and then⎛K=

1 + Sy2 + ε Sx2

κI ⎜⎝ (ε − 1)Sx Sy
1 + S2(1 − ε )Sx(ε − 1)Sx Sy
1 + Sx2 + ε Sy2⎞(1 − ε )Sx⎟(1 − ε )Sy ⎠ ,(1 − ε )Sy

ε + S2(6.35)

where
⎞
∂ρ
∂ρ
⎜ ∂x
∂y ⎟
⎟
S = (Sx , Sy , 0) = ⎜⎝− ∂ ρ , − ∂ ρ , 0⎠ ,∂z∂z
S = | S |,
κD
ε =
κI
⎛

(6.36)(6.37)(6.38)(Redi, 1982).
The finite difference is based on Cox (1987) except for the small isopycnal slope approximation. The finite
difference form of three components of the gradient of temperature and salinity in calculating the east-west component of flux FXDi+ 1 , j,k is as follows:2(δx T )i+ 1 , j,k+ 1= δx Ti+ 1 , j,k+ 1 ,(δy T )i+ 1 , j,k+ 1xy= δy Ti+ 1 , j,k+ 1 ,(δz T )i+ 1 , j,k+ 1

xz

2

2

2

2

2

2

2

2

(6.39)(6.40)

2= δz Ti+ 1 , j,k+ 1 .

(δx T )i, j+ 1 ,k+ 1= δx Ti, j+ 1 ,k+ 1 ,(6.42)(δy T )i, j+ 1 ,k+ 1= δy Ti, j+ 1 ,k+ 1 ,(6.43)(δz T )i, j+ 1 ,k+ 1

yz

2

2

2

(6.41)

Similarly, the north-south component is given by
2
2
2

2
2
2

xy

2

2

2

2

= δz Ti, j+ 1 ,k+ 1 ,2

2(6.44)and the vertical component is given byxz(δx T )i, j,k+1= δx Ti, j,k+1 ,(δy T )i, j,k+1

yz= δy Ti, j,k+1 ,(6.46)(δz T )i, j,k+1= δz Ti, j,k+1(6.47)(6.45)(Figure 6.3).
The density gradient in the calculation of each component of the diffusion tensor can be obtained by replacing
T in the above equation with ρ . However, density is calculated at the reference level k + 12 for the east-west andnorth-south components, and at the reference level k + 1 for the vertical component.
In addition, the upper bound on the isopycnal slope Smax is set because a nearly vertical isopycnal slope and theresultant low horizontal diffusivity could cause numerical instability. If |S| > Smax , ∂z ρ in all flux components is
replaced so as to satisfy |S| = Smax .
– 71 –

Chapter 6

Temperature and salinity equations

Ti, j+1

Ti+1, j+1

Ti, j

Ti+1, jTi, j-1

Ti+1, j-1

Figure 6.3. The way of calculating the gradient at the circle (i + 12 , j, k + 12 ) 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.

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 two properties, the down-gradient orientation of the
diffusive fluxes along the neutral directions and the zero isoneutral diffusive flux of locally referenced density, are
not guaranteed because of the nature of the finite difference method and the non-linearity of the equation of state.
Griffies et al. (1998) proposed a remedy, but this remains to be implemented in MRI.COM.

6.3.5 Gent and McWilliams parameterization
The Gent and McWilliams (1990) parameterization represents transports of temperature and salinity due todisturbances smaller than the grid 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 and isohaline 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 ∂n T )(6.48)

Diffusion tensor J mn is expressed as the sum of the symmetric component K mn = (J mn + J nm )/2 and the antisymmetric component Amn = (J mn − J nm )/2. Isopycnal diffusion has the form of a symmetric diffusion tensor.

– 72 –

6.3. Diffusion
m = −Amn ∂ T is as follows:
Convergence of a skew flux caused by the anti-symmetric component Fskewn

RA (T ) =∂m (Amn ∂n T )= ∂m (Amn )∂n T= ∂n (∂m Amn T ),(6.49)

where Amn ∂m ∂n T = 0 and ∂m ∂n Amn = 0 are used. If we set a virtual velocity un∗ ≡ −∂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 RA (T ) = −u∗ · ∇T since u∗ is divergence free by definition.
The Gent and McWilliams parameterization is given byu∗ = − ∂h∂(κT ∇ρ h),∂ρ∂ρ(6.50)

where h is the depth of the neutral surface (ρ = const). This velocity is expressed in the depth coordinate asu∗ = (−∂z (κT Sx ), −∂z (κT Sy ), ∇h · (κT S))(6.51)(Gent et al., 1995).
Generally, the corresponding anti-symmetric tensor A can not be uniquely determined. Here, from u1∗ = −∂y A21 −∂z A31 = −∂z (κT Sx ), we choose A21 = 0, A31 = κT Sx . Likewise, we adopt A12 = 0, A32 = κT Sy . A specific form of
A for the Gent and McWilliams parameterization is given as follows:⎞⎛
00−κT Sx⎟⎜(6.52)A=⎝ 00−κT Sy ⎠ .
κT Sx

κT Sy

0

The flux due to advection can be expressed using a vector streamfunction, Ψ = (−κT Sy , κT Sx , 0), which produces u∗ in (6.51):
Fadv = u∗ T = T (∇ × Ψ).
The skew diffusive expression for the flux using tensor A in (6.52) 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 otherwords, the Gent and McWilliams parameterization is realized by only adding A to the tensor of the isopycnal
diffusion K (Griffies, 1998).

6.3.6 Anisotropic Gent-McWilliams scheme
Using unit vector n̂ = (nx , ny ) in an arbitrary direction, the two-dimensional anisotropic diffusion tensor is

defined as follows:
K2 =

L
M

M
N




=

κA n2x + κB n2y
κB nx nyκB nx nyκB n2x + κA n2y,(6.53)

where κA is the diffusivity in the n̂ direction, and κB is that in the direction normal to n̂. This is applied to theanti-symmetric tensor in the Gent-McWilliams scheme, and the following expression is obtained (Smith and Gent,
2004),⎛⎞
00−LSx − MSy⎜⎟(6.54)A = ⎝
0
0−MSx − NSy ⎠ .
LSx + MSyMSx + NSy
– 73 –

0Chapter 6

Temperature and salinity equations

In the choice of option GMANISTROP, n̂ is set in the direction of the local horizontal velocity. The valueof κA is read from the namelist njobpi (variable name aitd). The ratio of κB /κA is read from the namelist
nmlgmanisotrop (variable name cscl isotrop). The default value of cscl isotrop is set to 1/2.

6.4 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 neighboringlevels (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 causedby the numerical calculation error and smoothes the distribution.
In general, there are three numerical schemes for convective adjustment. In the simplest one, adjustment isdone for a pair of two neighboring levels, and then for a pair of another two neighboring levels. By repeating thisprocedure, 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 is done 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 the mixed layer or isopycnal diffusion schemes and thus
needs an implicit method to solve it. In MRI.COM, this scheme is invoked by specifying the option DIFAJS. The
vertical diffusivity between the unstable grid points is set to 10000 cm2 s−1 . In the third scheme, the unstablepart is first neutralized. The stability at the top and bottom of the neutralized column is then reexamined, If theunstable condition remains, the part including the already-neutralized column is re-neutralized. This procedurecontinues 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).

6.4.1 Algorithm
In order to minimize the judgment process (”IF” statement) and replace it by arithmetic calculation, this scheme
defines two integer indices, αk and βk , at the layer boundaries, and six real variables TU k , TLk , SU k , SLk , VU k , andVLk , (k = 1, KM −1), in addition to the vertical rows of temperature, salinity, and density Tk , Sk , Rk , (k = 12 , KM − 12 )(KM is the number of levels; see Figure 6.4). The level at the vertical boundary of a T-cell corresponds to the integer
k. The index αk indicates an unstable part within a water column: αk = 1 if it is unstable at the level between k − 12and k + 12 , and αk = 0 if it is neutral or stable. The index βk memorizes the mixed part: βk = 1 at the boundary
where it is neutral as a result of mixing, and βk = 0 elsewhere. Variables TU k , SU k , and VU k and TLk , SLk , andVLk are temperature, salinity and volume accumulated by multiplying α above the level k and below the level k,respectively, and are expressed by the following recursive relation.

–

74

–

6.4. Convective adjustment

VU 1= α1V 1 ,VU 2

= α2 (V1+ 1 + α1V 1 ) = α2 (V1+ 1 + VU 1 ),

2

2

2

2

· · ·,= αk (Vk− 1 + VU k−1 ),

VU k

2

· · ·,= αKM−1 (VKM−1− 1 + VU KM−2 ),VU KM−1(6.55)2andVLKM−1= αKM−1VKM− 1 ,VLKM−2= αKM−2 (VKM−1− 1 + αKM−1VKM− 1 ) = αKM−2 (VKM−1− 1 + VLKM−1 ),

2

2

2

2

· · ·,VLk= αk (Vk+ 1 + VLk+1 ),2· · ·,
VL1= α1 (V1+ 1 + VL2 ),(6.56)

2

where Vk+ 1 denotes a volume of the cell at the level k + 12 . In a similar way, other quantities are expressed as2

follows:
TU 1 = α1 T 1 V 1 ,2

TU k = αk (Tk− 1 Vk− 1 + TU k−1 ),SU 1 = α1 S 1 V 1 ,SU k = αk (Sk− 1 Vk− 1 + SU k−1 ),

2

2

2

2

2

2

2

TLKM−1 = αKM−1 TKM− 1 VKM− 1 , T Lk = αk (Tk+ 1 Vk+ 1 + TLk+1 ),
2

2

2

2

SLKM−1 = αKM−1 SKM− 1 VKM− 1 , SLk = αk (Sk+ 1 Vk+ 1 + SLk+1 ),
2

2

2

2

(6.57)

where Tk+ 1 and Sk+ 1 are temperature and salinity at the level k + 12 .
22
According to this definition, if αk = 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

2

2

indicating a volume and accumulated temperature and salinity in an unstable part and
TU k + TLk,
2
2
VU k + VLk
SU k + SLk,
SM k− 1 ,k+ 1 =2
2
VU k + VLk

TM k− 1 ,k+ 1 =(6.58)

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 andsalinity. For example, let αk−1 = αk = 1 and αk−2 = αk+1 = 0,

–

75

–

Chapter 6

Temperature and salinity equations

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

BottomFigure 6.4. Reference vertical grid points in Section 6.4

VU k−1 + VLk−1= VU k + VLk= Vk−1− 1 +Vk− 1 +Vk+ 1 ,2

TU k−1 + TLk−1

2

2= TU k + TLk= Tk−1− 1 Vk−1− 1 + Tk− 1 Vk− 1 + Tk+ 1 Vk+ 1 ,2

SU k−1 + SLk−1

2

2

2

2

2

= SU k + SLk= Sk−1− 1 Vk−1− 1 + Sk− 1 Vk− 1 + Sk+ 1 Vk+ 1 ,
2

2

2

2

2

2

(6.59)

andTU k−1 + TLk−1
TU k + TLk=,
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– 76 –(6.60)

6.4. Convective adjustment
These are averages of the three layer, k − 1 − 12 , k − 12 , and k + 12 .

6.4.2

Numerical procedure[1] Density is calculated at the intermediate depth between adjacent levels using a pair of temperature andsalinity and is 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 (6.55) to (6.57), VU, TU, SU, VL, TL, and SL are calculated using α n for a water columnthat includes an unstable part.
[3] The vertical mean TM and SM are calculated for the unstable part using equation (6.58) and substituted forthe original 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. Thisis presented by the following:(6.61)
βkn = αkn + βkn−1 (1 − αkn ).
[5] The static stability is judged only for βkn = 0. Let αkn+1 = 1 if statically unstable, and 0 otherwise. If there isno unstable part, the procedure for that water column is completed.
[6] For a water column which still includes an unstable part, modification for αkn+1 is done using βkn by thefollowing. After the procedure [2], any instability will be found only at the bottom of the part that is neutral as aresult of prior mixing. In that case, the neutral part must be treated as an unstable part, that is, αkn+1 = 1. On theother hand, no more procedure is needed if the upper and lower end is stable, giving αkn+1 = 0. This is done by arecursive formula going down and up in the following.
(n+1)

γ1 = α1
(n+1)

(n+1)

+ (1 − αk(n+1)= γk + (1 − γk )βk αk+1 ,, γ k = αk

αKM−1 = γKM−1 , αk

(n+1)

(n)

)βk γk−1
(n)

(n+1)

(6.62)

where γ is a work variable, but may be treated as α itself in a FORTRAN program. Then, the procedure goes backto [2].
Table 6.1 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 αkn+1 using βkn based on equation (6.62), as
described in [6]. Note that βk0 = 0, though there is no description in the table.

– 77 –

Chapter 6

n

1

VU

Table 6.1. Example of the convective adjustment procedure
VL
VU+VL
TU+TL
V 1 + V 1 1 + V2 1

T 1 V 1 + T1 1 V1 1 + T2 1 V2 1

1

V2 1

V 1 + V 1 1 + V2 1

T 1 V 1 + T1 1 V1 1 + T2 1 V2 1

1

0

0

0
T4 1 V4 1 + T5 1 V5 1

0
1

1

1

V1

2

1

1

V 1 + V1 1

3

0

0

0

0

0

4
5

0
1

0
1

0
V4 1

0
V5 1
2

0
V 4 1 + V5 1

1

0

1

V1

V1 1 + V2 1 + V3 12

V 1 + V1 1 + V 2 1 + V3 1

V 2 1 + V3 1

3

0
1

1
1

2
2

2

2

2

V 1 + V1 1

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

V 1 + V1 1 + V 2 1 + V3 1

V 1 + V 1 1 + V2 1

V3 1

V 1 + V1 1 + V 2 1 + V3 1

2

2

2

2

β

V1 1 + V2 1

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

2

3

∑ Tk+ 12 Vk+ 12

1∑ Tk+ 12 Vk+ 12

1∑ Tk+ 12 Vk+ 12

1

k=0
3
k=0
3
k=0

4

0

0

0

0

0

0

0

5

0

0

0

0

0

0

1

1

0

1

V1

V1 1 + V2 1 + V3 1 + V4 1 + V5 1

∑ Tk+ 12 Vk+ 12

1∑ Tk+ 12 Vk+ 12

1∑ Tk+ 12 Vk+ 12

1∑ Tk+ 12 Vk+ 12

1∑ Tk+ 12 Vk+ 12

1

2

3

α

k

2

2

Temperature and salinity equations

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

V 4 1 + V5 1

V 1 + V 1 1 + V2 1 + V 3 1 + V4 1

V5 1

2

2

2

2

2

2

2

2

2

2

2

V3 1 + V4 1 + V5 1

V 1 + V 1 1 + V2 1 + V 3 1
22

V 2 1 + V3 1 + V 4 1 + V5 1

V 1 + V 1 1 + V2 1
2

2

2

2

2

5

∑ Vk+ 12

5

k=0
5

k=0
5

k=0
5

k=0
5

k=0
5

k=0
5

k=0
5

k=0
5

k=0

k=0

∑ Vk+ 12
∑ Vk+ 12
∑ Vk+ 12
∑ Vk+ 12

– 78 –

6.4. Convective adjustment

References
Cox, M. D., 1987: Isopycnal diffusion in a z-coordinate ocean model, Ocean Modell., 74, 1-5.
Gent, P. R., and J. C. McWilliams, 1990: Isopycnal mixing in ocean circulation models, J. Phys. Oceanogr., 20,
150-155.
Gent, P. R., J. Willebrand, T. J. McDougall, and J. C. McWilliams, 1995: Parameterizing eddy-induced tracer
transports in ocean circulation models, J. Phys. Oceanogr., 25, 463–474.
Griffies, S. M., 1998: The Gent-McWilliams skew flux, J. Phys. Oceanogr., 28, 831-841.
Griffies, S. M., A. Gnanadesikan, R. C. Pacanowski, V. D. Larichev, J. K. Dukowicz, and R. D. Smith., 1998:
Isoneutral diffusion in a z-coordinate ocean model., J. Phys. Oceanogr., 28, 805-830.
Holland, W. R., J. C. Chow, and F. O. Bryan, 1998: Application of a third-order upwind scheme in the NCAR
ocean model., J. Climate, 11, 1487–1493.
Ishizaki, H., 1997: A massive treatment scheme of complete convection for ocean models, unpublished manuscript.
Leonard, B. P., 1979: A stable and accurate convective modeling procedure based upon quadratic upstream
interpolation, J. Comput. Methods Appl. Mech. Eng., 19, 59-98.
Leonard, B. P., M. K. MacVean, and A. P. Lock, 1993: Positivity-Preserving Numerical Schemes for Multidimensional Advection, NASA Tech. Memo., 106055, ICOMP-93-05, 62pp.
Leonard, B. P., M. K. MacVean, and A. P. Lock, 1994: The flux integral method for multidimensional convection
and diffusion, NASA Tech. Memo., 106679, ICOMP-94-13, 27pp.
Prather, M. J., 1986: Numerical advection by conservation of second-order moments, J. Geophys. Res., 91, 66716681.
Redi, M. H., 1982: Oceanic isopycnal mixing by coordinate rotation, J. Phys. Oceanogr., 12, 1154–1158.
Smith, R. D., and P. R. Gent, 2004: Anisotropic GM parameterization for ocean models, J. Phys. Oceanogr., 34,
2541–2564.
St. Laurent, L. C., H. L. Simmons, and S. R. Jayne, 2002: Estimating tidally driven mixing in the deep ocean,Geophys. Res. Lett., 29,, 2106,doi:10.1029/2002GL015633.
Tsujino, H., H. Hasumi, and N. Suginohara, 2000: Deep pacific circulation controlled by vertical diffusivity at
the lower thermocline depth, J. Phys. Oceanogr., 30, 2853–2865.

–

79

–

Part III

Additional Processes

–

81

–

Chapter 7

Mixed layer model

The surface boundary layer is made turbulent by wind and other factors injecting momentum, so vertical mixing
could be induced even if the stratification is stable. However, this phenomenon is not expressed by the fundamental
equation of the model.∗ Therefore, general circulation models express the effects of turbulent phenomena by using
synoptic states (variables of velocity and temperature in the model). In MRI.COM, the vertical viscosity anddiffusivity are set to large values in the uppermost layer or calculated at each time step by using mixed layer
models. Three mixed layer models are supported: the turbulence closure model by Mellor and Yamada (1982) and
Mellor and Blumberg (2004) introduced in Section 7.1, the turbulence closure model by Noh and Kim (1999) in
Section 7.2, and K-profile parameterization by Large et al. (1994) in Section 7.3. Table 7.1 lists the notation of thevertical viscosity and diffusivity as well as physical properties that are predicted or diagnosed in each mixed layer
model.
Table 7.1. Physical properties predicted and diagnosed in each mixed layer model and their notations used for
explanation in each section
Physical property (name of variable in MRI.COM)

Mellor and Yamada

Noh and Kim

KPP

coefficient of diffusion (avdsl)

KH

KH

Kx

coefficient of viscosity (avm)turbulent velocity (q: only Mellor-Yamada)turbulent kinetic energy (eb: only Noh and Kim)

KM
q
2
q /2

KM
E

Kx
-

vertical turbulent scale (alo: except for KPP)

l

l

-

7.1

Mellor and Yamada’s Turbulence Closure Model

7.1.1

Turbulence Closure Model

The physical properties in the basic equations of motion for a Boussinesq fluid are separated into averaged
components and perturbed components, 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
∂
ρ
∂
1  ∂ Uk ∂ U j 
1 ∂P+ ε jkl fkUl =2ν
,(− uk u j ) −
−gj +
+
Dt
∂ xk
ρ0 ∂ x jρ0 ∂ xk
2 ∂xj
∂ xk
∂
∂  ∂Θ DΘ,(− uk θ ) +
κ
=
Dt
∂ xk
∂ xk
∂ xk
∗ It could be expressed by relaxing the hydrostatic approximation but this is for future work.

–

83

–

(7.1)
(7.2)
(7.3)

Chapter 7

Mixed layer model

where D( )/Dt ≡ Uk ∂ ( )/∂ xk + ∂ ( )/∂ t, g j is the gravity vector, fk is the Coriolis vector, εi jk is the alternating
tensor, ν is viscosity, and κ is diffusivity. Averaged quantities (resolved by the general circulation model) are represented by capital letters, and turbulent components (unresolved by the general circulation model) are represented
by lower-case letters. The statistical averages of the turbulent components are represented by . The equation forsalinity is similar to that for temperature (7.3). If density is calculated from temperature and salinity, the wholeexpression would become complicated. Therefore, density is assumed to be a function only of temperature here
(we set ρ = β θ , where β is the coefficient of thermal expansion).
The evolution of large scale physical quantities could be obtained by calculating the statistically averaged turbulent component ( ) expressed above for each time step, but the calculation is very complicated and unknown
higher order terms arise when evolution equations for statistically averaged turbulent components are derived (e.g.,
to calculate the covariance of turbulent velocities, the equation for a turbulent velocity component is multipliedby another component of turbulent velocity and then statistically averaged). Thus, the expression is not closed,
although it should be closed at a certain level (usually called closure). The expression closed in the second orderof the turbulent component is named second moment closure and is popular in modeling turbulence. Following
Kantha and Clayson (2000), the second moment closure is expressed as∂ ui u j 
1 ∂ pui  ∂ pu j ++ fk (ε jkl ul ui  + εikl ul u j )+∂ xk
ρ0 ∂ x j∂ xi'∂u ∂u (' p ∂u∂Uj∂ u j (
∂ Ui
β
j
i
i
= − uk ui − (g j ui θ  + gi u j θ ) +− 2ν, (7.4)− uk u j 
+
∂ xk
∂ xk
ρ0
ρ0 ∂ x j ∂ xi
∂ xk ∂ xk) ∂θ *) ∂uj *
D u jθ ∂1 ) ∂p*uk u j θ  − κ u j−ν θ++ ε jkl fk ul θ 
θ
+
Dt
∂ xk
∂ xk
∂ xk
ρ0 ∂ x j'∂u ∂θ (∂Uj∂Θβj− g j θ 2  − (κ + ν ),(7.5)− uk θ = − u j uk 
∂ xk
∂ xk
ρ0
∂ xk ∂ xk' ∂θ ∂θ ()∂θ2 *
D θ 2∂∂Θuk θ 2  − κ.
(7.6)= −2 uk θ 
− 2κ
+
Dt
∂ xk
∂ xk
∂ xk
∂ xk ∂ xk

D ui u j 
∂
+
Dt
∂ xk

uk ui u j  − ν

Mellor and Yamada (1982) reduced the higher order terms as follows. It should be noted that it is not unique.
Based on Rotta’s (1951a,b) hypothesis of energy redistribution, the covariances of pressure and velocity gradients are assumed to be linear functions of Reynolds stress:' p ∂ui

ρ0 ∂ x j+

 ∂U∂ u j (δi j 
∂Uj 
q 
i
=−
ui u j  − q2 +C1 q2
,
+
∂ xi
3l1
3
∂xj
∂ xi

(7.7)

where q2 ≡ u2i , l1 is the length scale, C1 is a non-dimensional constant, and δi j is Kronecker’s delta, which isunity for i = j and zero for i = j.
Using Kolmogolov’s hypothesis of local isotropy in small eddies the energy dissipation is modeled as follows,2ν'∂u ∂u (ji∂ xk ∂ xk=−

2 q3
δi j ,
3 Λ1(7.8)

where Λ1 is the length scale.
The redistribution of temperature and the dissipation of heat are modeled as follows in the same form as above,' p ∂θ (qu jθ 
3l2(7.9)'∂u ∂θ (j= 0,(κ + ν )∂ xk ∂ xk(7.10)

ρ0 ∂ x j

–

=−

84

–

7.1. Mellor and Yamada’s Turbulence Closure Model
where l2 is the length scale. The dissipation of temperature variance is2κ' ∂θ ∂θ (∂ xk ∂ xk

= −2

q 2
θ ,
Λ2

(7.11)

where Λ2 is the length scale.
In order to avoid the higher order problems, the turbulent velocity diffusion term and the other higher orderterms are modeled as follows:∂ u u  ∂ u u  ∂ u u 
3
i j
j k
i k
lqSq
,
(7.12)++
uk ui u j  =
5
∂ xk
∂xj
∂ xi
∂ u θ ∂ u θjk,(7.13)+
uk u j θ  = −lqSuθ∂xj∂ xk
uk θ 2  =−lqSθ∂ θ 2
,∂ xk(7.14)

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 eachother:(7.15)(l1 , Λ1 , l2 , Λ2 ) = (A1 , B1 , A2 , B2 )l,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).

7.1.2 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 ( θ s) and the varianceof salinity ( s2 )). The other statistically averaged values are solved diagnostically through algebraic equations
assuming them to be in the steady state.
In the level-2.5 model, the variance of the potential temperature is also assumed to be in a statistically steadystate (see expression (7.33) 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 is employed as the surface boundary layer model by MRI.COM and 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 the term involving turbulent velocity.

–

85

–

Chapter 7

Mixed layer model

The equations for the large scale physical quantity become

∂
DU
1 ∂P
+
uw = −
+ fV,
Dt∂z
ρ0 ∂ x∂DV
1 ∂P+vw = −
− fU,
Dt∂z
ρ0 ∂ y
ρ
1 ∂P
0 = −
−g ,
ρ0 ∂ zρ0DΘ ∂+ ( wθ ) = 0.
Dt∂z

(7.16)(7.17)(7.18)(7.19)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∂  q2 D  q2  ∂−lqSq= Ps + Pb − ε ,
Dt 2
∂z
∂z 2

(7.20)

where

∂U
∂V
− wv
∂z
∂z
is the term for energy produced by the vertical shear of the averaged flow,
Ps = − wu

Pb = −g wρ /ρ0

(7.21)(7.22)

is the term for energy produced by buoyancy, andε = q3 /Λ1(7.23)

is the energy dissipation term.
The algebraic equations for the statistically averaged values, which are expressed by other second-momentturbulent quantities, are given below.

∂U
∂V
q2 l1
−4 wu
++ 2 wv
− 2Pb ,
3
q
∂z
∂z
∂U
∂V
q2 l1
2 wu
v2  =+− 4 wv
− 2Pb ,
3
q
∂z
∂z
∂U
∂V
q2 l1
2 wu
w2  =++ 2 wv
+ 4Pb ,
3
q
∂z
∂z
∂V
∂U
3l1
− uw
,
uv =
− vw
q
∂z
∂z
3l1
∂U
wu =
−( w2  −C1 q2 )− g uρ  ,
q
∂z
∂V
3l1
vw =
−( w2  −C1 q2 )− g vρ  ,
q
∂z
∂Θ
∂U
3l2
− uw
,
uθ  =− wθ 
q
∂z
∂z
∂Θ
∂V
3l2
vθ  =− vw
,− wθ 
q
∂z
∂z
∂Θ
3l2
wθ  =− w2 
− g θ ρ ,
q
∂z
∂Θ
Λ2
θ 2 = −
wθ 
.
q
∂z
u2  =– 86 –(7.24)(7.25)(7.26)(7.27)(7.28)(7.29)(7.30)(7.31)(7.32)(7.33)

7.1. Mellor and Yamada’s Turbulence Closure Model
Some of the terms in these equations can be further simplified as follows:

∂U
,
∂z
∂V
,
− vw = KM∂z∂Θ− θ w = KH,∂z− uw = KM

(7.34)(7.35)(7.36)

KM = lqSM ,(7.37)

KH = lqSH .

(7.38)

This simplification means that the vertical turbulent fluxes are proportional to the gradient of the large scalevalues. The ultimate purpose of solving the mixed layer model is to determine the coefficients of momentum andheat fluxes, KM and KH , using (7.37) and (7.38).
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 GM ] + SH [1 − 3A2 B2 GH − 12A1 A2 GH ] = A2 ,SM [1 + 6A21 GM − 9A1 A2 GH ] − SH [12A21 GH + 9A1 A2 GH ] = A1 (1 − 3C1 ),(7.39)

where
GM

≡

GH

≡

l 2  ∂ U 2  ∂ V 2
,
+
q2
∂z
∂z
l 2 g ∂ ρ̃,
q 2 ρ0 ∂ z(7.40)(7.41)and ∂ ρ̃ /∂ z is the vertical gradient of potential density.
Using SM and SH , KM and KH are then obtained from (7.37) and (7.38) by determining q and l.
The turbulent velocity q is obtained by solving the following expression that is modified from (7.20) using theabove results: ∂ U 2  ∂ V 2
∂  q2  ∂∂  q2 
∂ ρ̃
g
−
KE
= KM
+ KH
+
(7.42)
− ε,
∂t 2
∂z
∂z 2
∂z
∂z
ρ0
∂z
where
KE = lqSq(7.43)

and advection terms are neglected. In MRI.COM, Sq is set proportional to SM (Sq ∝ SM ). We adopt the form
Sq = Sqc SM /SMn , where Sqc = 0.2 and SMn = 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 u3τ ,∂z(7.44)

where αCB = 100 and uτ is the frictional velocity defined as uτ ≡ (τs /ρs )1/2 by using the surface stress (τs ) andthe sea surface density (ρs ). This flux boundary condition is analytically converted to the condition for q at the sea
surface:q2 = (15.8αCB )2/3 uτ2 .

–

87

–

(7.45)

Chapter 7

Mixed layer model

The vertical scale of the turbulence (master length scale) 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. The formula used in
MRI.COM is a diagnosis for cases without surface wind wave effects:
l=γ

 0
zb

|z |qdz / 0zb

qdz ,(7.46)

where γ = 0.2, and zb is the depth of the bottom. This is recognized as the averaged depth with the weightof the kinetic energy, which is sufficient for the ocean boundary layer according to Mellor and Yamada (1982).
Roughness parameter zw due to surface wind waves is given by Mellor and Blumberg (2004) as follows:u2zw = βw τ , βw = 2.0 × 105 .
g(7.47)

l = max(κ zw , κ z),(7.48)For a depth of |z| < zw ,where κ is the von Karman constant (κ = 0.4). For depths exceeding zw , (7.46) is employed.

7.1.3

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 stepthat proceeds from n to n+1. After the master length scale (l) for the present time step (n) is determined using(7.46), (7.47), and (7.48), the turbulent kinetic energy (q2 /2) is solved using (7.42), (7.15), (7.23), and (7.45),
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 12.5). The vertical viscosity and diffusivity for the time step n+1 are estimated using q, l, and(7.37) to (7.39). The vertical scale of the turbulence (master length scale) based on (7.46) 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 grid cell (i, j, k).
The specific expression for the discretized form of the turbulent kinetic energy (E = q2 /2) equation is as follows:
n+1
n+1
n+1
n+1
Ekn+1 − Ekn
1 KE k− 12 (Ek−1 − Ek ) KE k+ 12 (Ek − Ek+1 )
−
=
Δt
Δzk
Δzk− 1
Δzk+ 1
2

2(U n+11 −U n+11 )(Ũk− 1 − Ũk+ 1 )(V n+11 −V n+11 )(Ṽk− 1 − Ṽk+ 1 )
k− 2
k+ 2
k− 2
k+ 2
2
2
2
2
+ KM k
+
K
Mk
Δz2k
Δz2k
Bn+11 − Bn+11
k− 2
k+ 2
− KH k− 2Ekn+1 qnk /B1 lkn ,Δzk(7.49)

where ũ = (un+1 + un )/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), whichis 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 (7.46), (7.47), and (7.48).
2. Update the turbulent kinetic energy using (7.49).
– 88 –

7.2. Turbulent mixed layer model by Noh and Kim (1999)
3. Solve the algebraic equation for SM and SH using (7.39).
4. Calculate the vertical viscosity and diffusivity for the next time step using (7.37), (7.38), and (7.43).
5. Calculate the master length scale using (7.46) for the next time step.

7.2

Turbulent mixed layer model by Noh and Kim (1999)

The mixed layer model proposed by Mellor and Yamada was originally developed for the atmospheric boundary
layer, and its 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 asthat by Mellor and Yamada and is classified as a second moment closure model.

7.2.1

Fundamental equation

The equations for the zonal and meridional components of the velocity, U, V , buoyancy B = −gΔρ /ρo , andturbulent energy E in the large scale fields are

∂
DU
1 ∂P
=−
uw −
+ fV,
Dt
∂z
ρ0 ∂ x
∂
DV
1 ∂P
=−
vw −
− fU,
Dt
∂z
ρ0 ∂ y
∂
∂R
DB
=−
bw +
,
Dt
∂z
∂z

*
∂ )
∂U
∂V
DE
p
+ uu + vv + ww − uw
=−
w
− vw
+ bw − ε ,
Dt
∂z
ρ0
∂z
∂z

(7.50)(7.51)(7.52)(7.53)

where R is the downward short wave radiation and ∂ R/∂ z is its convergence.
The turbulent flux is expressed by using the large scale fields (in capital letters) as follows,∂  ∂U  1 ∂ P
DU
=
KM
−
+ fV,
(7.54)Dt∂z∂zρ0 ∂ x∂  ∂V  1 ∂ P
DV
=
KM
−
− fU,
(7.55)Dt∂z∂zρ0 ∂ y∂  ∂B ∂R
DB
KH
−
=
,
(7.56)Dt∂z∂z∂z∂  ∂E ∂U ∂U∂V ∂V  ∂ B 
DE
KE
+ KM
− ε.
(7.57)
=
+ KM
− KH
Dt
∂z
∂z
∂z ∂z
∂z ∂z
∂z
The central problem is how to determine the viscosity, diffusivity (KM , KH , KE ), and turbulent energy dissipation
rate (ε ). By using the typical velocity scale (q = (2E)1/2 ) and the vertical length scale (l) of the turbulence, we
assume the following
KM

=

Sql,

(7.58)KH=

SB ql,(7.59)

KE=

SE ql,(7.60)

ε= Cq3 l −1 .

(7.61)– 89 –

Chapter 7

Mixed layer model

The constants (S, SB , SE ,C) are obtained from experiments and it is assumed that S = S0 = 0.39, Pr = Pr0 = S/SB =0.8, σ = S/SE = 1.95, and C = C0 = 0.06 for neutral stratification.
As for the influence of the stratification, we assume that the vertical scale of turbulence is limited by the verticalscale of buoyancy lb = q/N (N 2 = ∂ B/∂ z). That is,K ∼ qlb ∼ qlRit −1/2 ,(7.62)

Rit = (Nl/q)2 .

(7.63)

where Rit is the turbulent Richardson numberThis means that when the stratification is strong (N is large, Rit is large, and K is small), the turbulent energy is
not transported downwards since the internal waves are induced by the injected turbulent energy, resulting in theirhorizontal propagation. It could also be considered that the local turbulent energy dissipation becomes large.
The following equation is used for S so that it satisfies (7.62) when Rit is large:
S/S0 = (1 + α Rit )−1/2 ,(7.64)

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 energy dissipation (C) is set as follows:
C/C0 = (1 + α Rit )1/2 .

(7.65)The effect of stratification on the Prandtl number (Pr) is set following Noh et al. (2005):
Pr/Pr0 = (1 + β Rit )1/2 ,(7.66)

where β is a tuning parameter and 0.5 is used following Noh et al. (2005).
The vertical scale of turbulence is given byl=κ (|z| + z0 ),(1 + κ (|z| + z0 )/h)(7.67)

where z0 is the sea surface roughness (z0 = 1[m]), z is the depth, and h is the mixed layer depth. The vertical scale
becomes longer as the mixed layer becomes deeper.
The boundary conditions are as follows:

∂U
∂z
∂B
KH
∂z
∂E
KE
∂z

KM

τ
,
ρ0

(7.68)= Q0 ,(7.69)= mu3∗ ,(7.70)=

where m is a tuning parameter and m = 100 is recommended by Noh and Kim (1999).
In the case of unstable stratification (N 2 < 0), KM = KH = 1.0[m2 s−1 ] and KE is estimated from the turbulent
velocity scale and the vertical length scale in the model. This treatment is due to the difference between the timescales of the vertical convection and the development of turbulence.

–

90

–

7.3. K Profile Parameterization (KPP)

7.2.2

Implementation

Equation (7.57) is solved for the prognostic variable E in the subroutine nkoblm in nkoblm.F90 as the lastprocedure 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 and energy dissipation term, since these terms could
become significantly large (see Section 12.5). The new E is used to determine the coefficients of viscosity anddiffusivity for the next time step.
The turbulent kinetic energy and the master length scale are defined at the center of the tracer grid cell (i, j, k − 12 ).
The specific expression for the discretized form of the turbulent kinetic energy (E) equation is as follows:n
E n+11 − Ek+
1
k+ 2

Δt

2

=

1

KE k (E n+11 − E n+11 )Δzk+ 1

Δzk

k− 2

2

k+ 2

−

KE k+1 (E n+11 − E n+13 )
k+ 2

Δzk+1

k+ 2

(un+11 − un+11 )(un+11 − unk+ 1 ) 1(un+11 − un+13 )(unk+ 1 − un+13 )
1
k− 2
k+ 2
k− 2
k+ 2
k+ 2
k+ 2
2
2
+ KM k+ KM k+1
2
Δzk Δzk+ 1
2
Δzk+1 Δzk+ 1
2

1+ KM k22(vn+11 − vn+11 )(vn+11 − vnk+ 1 )
k− 2

k+ 2

k− 2

Δzk Δzk+ 1

2

2

1+ KM k+12(vn+11 − vn+13 )(vnk+ 1 − vn+13 )
k+ 2

k+ 2

2

Δzk+1 Δzk+ 1

k+ 2(7.71)

2

Bn+11 − Bn+11 1
Bn+11 − Bn+13
1
k− 2
k+ 2
k+ 2
k+ 2
n
− KH k+1− 2CE n+11 qnk+ 1 /lk+
− KH k
1.
k+ 2
2
Δzk
2
Δzk+1
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 thesum of mean and turbulent energy.
To summarize, the numerical operations proceed in the following order:
1. Update the master length scale using (7.67),
2. Update the turbulent kinetic energy using (7.71),
3. Calculate the vertical viscosity and diffusivity for the next time step.

7.3

K Profile Parameterization (KPP)

7.3.1

OutlineThe K-profile parameterization (KPP) (Figure 7.1, Equation 7.79) is a method to determine the coefficients ofvertical viscosity and diffusivity. First, the turbulent vertical velocity scale in the mixed layer (the boundary layer)
is determined following the Monin-Obukhov similarity law near the boundary. This scale is then multiplied by themixed layer thickness and the non-dimensional profile function that are separately obtained to produce the verticalviscosity and diffusivity. The coefficient νx below the mixed layer is set to different values from those within
the mixed layer and connected continuously. In this way, KPP differs from a series of turbulent closure model
represented in the previous subsections. The KPP scheme is an application of the nonlocal K-profile model (Treonand Mahrt, 1986) used in the atmospheric model to the ocean model by Large et al. (1994). The KPP subroutinein MRI.COM is based on that of the NCEP ocean model (NCOM). The paramterization for mixing due to salt
fingering is not adopted and a vertically one-dimensional background profile is given at vdbg.F90 by user (seeChapter 16).
– 91 –

Chapter 7

Mixed layer model

The time tendency of an averaged value X due to the turbulent eddy is expressed by the vertical divergence of
turbulent flux wx,∂t X = −∂z wx,(7.72)
where X represents the time averaged components of the velocity component U, V , temperature T , salinity S,
buoyancy B, and so on, and x represents the turbulent components of the velocity component u, v, temperaturet, salinity s, buoyancy b, and so on. Furthermore, w is the vertical velocity due to the turbulent eddy (positive
upward). Hereinafter, the momentum component is expressed as m and the scalar property as s in some cases.
In the KPP scheme, the turbulent flux within the mixed layer is expressed by the vertical gradient term of X andthe nonlocal transport term.† That is(7.73)
wx = −Kx (∂z X − γx ).
The vertical viscosity and diffusivity coefficients Kx and the nonlocal transport γx are calculated in the following
order by the KPP subroutine of MRI.COM.
• Calculation of the sea surface fluxes (momentum and buoyancy) wx0 • Calculation of the stability scale L• Calculation of the mixed layer thickness h• Calculation of the non-dimensional universal function φx• Calculation of the turbulent vertical velocity scale wx• Calculation of the vertical viscosity and diffusivity coefficient within the mixed layer Kx• Connection of the coefficient in the mixed layer to the coefficient νx below the mixed layer base Kx∗• Calculation of the vertical viscosity and diffusivity coefficient due to shear instability νxs• Adopt the larger value of Kx∗ and νxs for the vertical viscosity and diffusivity coefficient• Calculation of the nonlocal transport γx7.3.2 Monin-Obukhov similarity law
The Monin-Obukhov similarity law applies to the boundary layer near the sea surface. In this boundary layer,
only the distance from the sea surface d(= −z) and the sea surface flux wx0  are important. From these parameters,
the following three basic turbulent parameters are determined,• Frictional velocity:u∗2 = ( wu0 2 + wv0 2 )1/2 = |τ0 |/ρ0 ,(7.74)• Turbulent scale of scalar (temperature and salinity):s∗ = − ws0 /u∗ ,(7.75)† Nonlocal means the phenomenon in which the material is transported upwards (that is opposite to the gradient of the averaged fields) due
to the gradient induced by the turbulent component ∂ x/∂ z even though the gradient of the averaged fields ∂ X/∂ z is locally positive. See Figure
7.1(a).

– 92 –

7.3. K Profile Parameterization (KPP)
(a)

(b)

buoyancy(solid) and its ﬂux (dotted)

diﬀusion coeﬃcient

mixed layer(solid), interior(dotted)

mixed layer

0

-dk
-he
-h

0

0

Figure 7.1. Schematic diagram of KPP. (a) Relative buoyancy (solid line) and buoyancy flux (dotted line) profiles.
Upward buoyancy flux (non-local transport; Section 7.3.7) could occur in the boundary layer even if the stratification is neutral. Also indicated are the entrainment depth (he ) and mixed layer base (h). (b) The boundary layer
diffusivity profile Kx (solid line; equation (7.79), and the interior diffusivity profile (dotted line). The diffusivity
at depth (dk ) immediately above the mixed layer base (h) is set to the value marked by 2, and the boundary layer
profile and the interior profile are connected by the dashed line (see Section 7.3.4).

• Monin-Obukhov stability scale:
L = u∗3 /(κ B f ),(7.76)

where τ0 is the sea surface wind stress, ρ0 is density, κ = 0.4 is the von Karman constant, and B f is the surface
buoyancy flux (positive downward and thus negative means unstable). It should be noted that the direction of
B f is opposite to that within the ocean. Note also that the fluxes need not be constant in the boundary layer(d < ε h [ε  1, in general ε ∼ 0.1]), but are affected by the sea surface flux wx0  and derived properties u∗ , s∗ ,L. In this case, the non-dimensional profiles of velocity and scalar are defined as functions of stability parameterζ = d/L:κd∂z (U 2 +V 2 )1/2 ,u∗κdφs (ζ ) = ∗ ∂z S.
sφm (ζ ) =(7.77)(7.78)

These functions are determined empirically based on observations.

7.3.3

Coefficients of vertical viscosity and diffusivity

The profile of vertical viscosity and diffusivity Kx in the mixed layer is defined as the product of the turbulent vertical velocity scale wx and the non-dimensional vertical shape function G(σ ). Since the mixing becomes
more effective due to the turbulent eddies as the mixed layer becomes thicker, the coefficient Kx is set linearly
proportional to h:Kx (σ ) = hwx (σ )G(σ ),– 93 –(7.79)

Chapter 7

Mixed layer model

where σ = d/h (fractional depth within the mixed layer) is the non-dimensional vertical coordinate. The verticalshape G(σ ) is approximated by a third order polynomial (O’Brien, 1970),
G(σ ) = a0 + a1 σ + a2 σ 2 + a3 σ 3 .

(7.80)

Since the turbulent eddy does not cross the sea surface, Kx = 0 at σ = 0. Thus, a0 = 0 in equation (7.80).
The Monin-Obukhov similarity law is applied in the boundary layer (σ < ε [= 0.1]). Assuming that the turbulent
flux wx is linear (Lumley and Panofski, 1964; Tennekes, 1973), we havewx (σ )(a1 + a2 σ ) =κ u∗ wx(d)
φx (ζ ) wx0 (7.81)

from (7.73) with γx = 0, (7.78), and (7.79). Equation (7.81) holds if
wx (σ ) =κ u∗.
φx (ζ )

(7.82)

To avoid wx becoming too large, wx is set to be constant below the boundary layer (σ = ε (∼ 0.1)) under theunstable condition (ζ (= d/L) < 0),κ u∗
φx (ε h/L)κ u∗
wx (σ ) =φx (σ h/L)

wx (σ ) =ε < σ < 1 ζ < 0,otherwise.

(7.83)The non-dimensional profile function φx is determined as a function of the stability parameter ζ (= d/L) basedon experiments. They are determined so that wx could be scaled by κ u∗ for a neutrally stable case (h/L ≤ 0), andso that wx could be larger (smaller) than κ u∗ for an unstable (stable) case. Large et al. (1994) employ the following
expressions.

φm
φm
φm
φs
φs

= φs = 1 + 5ζ= (1 − 16ζ )−1/4= (am − cm ζ )−1/3=(1 − 16ζ )−1/2= (as − cs ζ )−1/3

0 ≤ ζ,

ζm ≤ ζ < 0,
ζ < ζm ,ζs ≤ ζ < 0,
ζ < ζs .

(7.84)

Here, (ζs , cs , as , ζm , cm , am ) = (−1.0, 98.96, −28.86, −0.2, 8.38, 1.26).
Assuming a linear profile for the turbulent flux wx (flux decreases linearly from the sea surface value), equation(7.81) iswx(σ )/ wx0  = 1 − βr σ /ε = a1 + a2 σ .

(7.85)

From this, we have a1 = 1 and a2 = −βr /ε . Using the condition at the mixed layer base, G(1) = ∂σ G(1) = 0, andassuming ε = 0.1, we have a2 = −2, a3 = 1, and βr = 0.2.

7.3.4 Coefficients of vertical viscosity and diffusion at the base of the mixed layer
In the KPP scheme, viscosity and diffusion coefficients in the mixed layer (Kx ) are connected to the backgroundprofile (νx ) at the base of the mixed layer.
The coefficient at the boundary (level k) is determined by using the following equations:

δ = (h − dk− 1 )/(dk+ 1 − dk− 1 )2

22

dk− 1 < h < dk+ 1 ,2

2(7.86)

Kx∗ = (1 − δ )2 Kx (dk− 1 ) + δ 2 νx (dk )dk− 1 < h ≤ dk ,(7.87)

Kx∗ = (1 − δ )2 Kx (dk− 1 ) + δ 2 Kx (dk )dk < h ≤ dk+ 1 ,(7.88)222

Λx = (1 − δ )νx (dk− 1 ) + δ Kx∗ .

2(7.89)

2– 94 –

7.3. K Profile Parameterization (KPP)
The vertical viscosity and diffusion coefficients at depth dk that corresponds to the boundary between the mixed
layer base (h) and the interior region is presented by Λx (2 in Figure 7.1(b)). The coefficient Kx∗ is set to yield largecoefficients there. When dk− 1 < h < dk (when the mixed layer is shallower than dk ), Kx (dk ) that appears in (7.88)
2
is not defined and is replaced by the background value νx (dk ).

7.3.5

Thickness of the mixed layer

The thickness h of the mixed layer is estimated from the vertical profiles of buoyancy B(d) and velocity V (d).
Here, it is determined as the depth where the bulk Richardson number referred to the sea surface,
Rib (d) =(Br − B(d))d,|Vr − V (d)|2 +Vt2 (d)(7.90)

equals the reference value Ric (0.3 in MRI.COM), where Br and Vr are the buoyancy and the velocity in theuppermost layer. The quantity Vt /d is called the turbulent velocity shear and is expressed as follows:
Vt2 (d) =Cv (−βT )1/2
(cs ε )−1/2 dNws ,Ric κ 2

(7.91)

where Cv = 1.8 and βT = −0.2. The turbulent velocity shear is introduced to achieve entrainment at the mixed
layer base under strong stratification.

7.3.6

Mixing due to shear instability

In the stratified interior, mixing could occur locally due to shear instabilities. The tendency for shear instability
is measured by the local gradient Richardson number,
Rig =

N2
(∂zU)2 + (∂zV )2.

(7.92)

When Rig is below a critical value Ri0 , the vertical velocity shear overcomes the stabilizing effect of the buoyancy
gradient. This process is parameterized using a vertical mixing coefficient νxs given by the following formula andis the same for viscosity and diffusivity.

νxs /ν 0 = 1 Rig < 0νxs /ν 0 = [1 − (Rig /Ri0 )2 ] p1

0 < Rig < Ri0νxs /ν 0 = 0 Ri0 < Rig(7.93)(7.94)(7.95)

Here, ν 0 = 50 × 10−4 [m2 s−1 ], Ri0 = 0.7, and p1 = 3.
The final vertical mixing coefficient for “this” time step is taken as the larger of this shear instability mixing
coefficient and the one obtained by connecting boundary layer and background interior (Section 7.3.4).

7.3.7

Nonlocal Transport

If the stratification is locally stable or neutral, the turbulent buoyancy flux should be downward or zero according
to the r.h.s. of (7.73 [γx = 0]). However, an upward turbulent buoyancy flux exists if the sea surface buoyancy flux
is unstable (upward; Figure 7.1(a)). This is called a nonlocal transport (or counter gradient transport).

–

95

–

Chapter 7

Mixed layer model

In the mixed layer, the turbulence exhibits a nonlocal character, and the local buoyancy flux depends on theboundary layer parameters such as the sea surface flux wb and the thickness of the mixed layer h in addition to
the local gradient. Furthermore, nonlocal transport has a non-zero value only for the scalar under unstable forcing(Deardroff, 1972). Here, using the parameterization of Mailhôt and Benoit (1982), the nonlocal transport γs for thescalar under the unstable forcing is calculated by

γs = C∗

ws0 
,
w∗ h

(7.96)

where C∗ = 10. These are summarized as follows:

γx
γm

=

0

=

0

ζ ≥ 0,
ζ < 0,γs= Cs

ζ < 0,γθ

ws0 
ws (σ )hwθ0  + wθR = Cs
ws (σ )h(7.97)

ζ < 0,

where
Cs = C∗ κ (cs κε )1/3 ,(7.98)

and wθR  expresses the absorption of short wave radiation, which will be detailed in Chapter 8.

References
Burchard, H., 2002: Energy-conserving discretization of turbulent shear and buoyancy production, Ocean Modell., 4, 347-361.
Deardroff, J. W., 1972: Theoretical expression for the counter-gradient vertical heat flux, J. Geophys. Res., 77,
5900-5904.
Kantha, L. H., and C. A. Clayson, 2000: Small Scale Processes in geophysical Fluid Flows, Academic Press,
888pp.
Large, W. G., J. C. McWilliams, and S. C. Doney, 1994: Oceanic vertical mixing: a review and a model with anonlocal boundary layer parameterization, Rev. Geophys., 32, 363-403.
Lumley, J. A., and H. A. Panofsky, 1964: The structure of the atmospheric turbulence, 239pp., John Wiley, NewYork.
Mailhôt, J., and R. Benoit, 1982: A finite-element model of the atmospheric boundary layer suitable for use withnumerical weather prediction models, J. Atmos. Sci., 39, 2249-2266.
Mellor, G. L., and A. Blumberg, 2004: Wave breaking and ocean surface layer thermal response, J. Phys. Oceanogr.,
34, 693-698.
Mellor, G. L., and T. Yamada, 1982: Development of a turbulence closure model for geophysical fluid problems,
Rev. Geophys. Space Phys., 20, 851-875.
Noh, Y., Y.-J. Kang, T. Matsuura, and S. Iizuka, 2005: Effect of the Prandtl number in the parameterization of vertical mixing in an OGCM of the tropical Pacific, Geophys. Res. Lett., 32, L23609, doi:10.1029/2005GL024540.

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7.3. K Profile Parameterization (KPP)
Noh, Y., and H.-J. Kim, 1999: Simulations of temperature and turbulence structure of the oceanic boundary layer
with the improved near-surface process, J. Geophys. Res., 104, 15,621-15,634.
O’Brien, J. J., 1970: A note on the vertical structure of the eddy exchange coefficient in the planetary boundary
layer, J. Atmos. Sci., 27, 1213-1215.
Rotta, J. C., 1951a: Statistische Theotie nichthomogener Turbulenz, Z. Phys., 129, 547-572.
Rotta, J. C., 1951b: Statistische Theotie nichthomogener Turbulenz, Z. Phys., 131, 51-77.
Tennekes, H., 1973: A model for the dynamics of the inversion above a convective boundary layer, J. Atmos. Sci.,
30, 558-567.
Treon, I. B., and L. Mahrt, 1986: A simple model of the atmospheric boundary layer; sensitivity to surface
evaporation, Boundary Layer Meteorol., 37, 129-148.

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Chapter 8

Sea surface fluxes

Sea surface momentum, heat, and fresh water fluxes are needed to drive an ocean model. This chapter explains
how those fluxes are treated in MRI.COM. The treatment is designed for the stand-alone simulation driven byatmospheric reanalysis data or observed data and simulation using a coupled atmosphere-ocean model.
By default, a stand-alone simulation is driven by momentum fluxes and temperature and salinity fluxes derived
by restoring the model sea surface temperature and salinity to the observed ones. It would be natural to use
sea surface momentum, heat and fresh water fluxes based on observation when a realistic simulation is intended.
However, the simulated sea surface temperature and salinity do not always agree with the observed ones due to thefollowing reasons.
1. Sea surface heat and fresh water fluxes derived from observations include large errors.
2. The ocean stand-alone simulations do not include feedback mechanisms from the ocean to the atmosphere.
3. Since ocean models have dynamical errors, they do not necessarily capture all of the phenomena.
Thus, it is not common in a stand-alone simulation to drive the model by fluxes without a feedback process, such as
flux data derived from atmospheric reanalysis data. Rather, it is common to calculate fluxes using a bulk formulawith sea surface meteorological elements derived from atmospheric reanalysis data.
For a simulation using a coupled atmosphere-ocean model, all surface fluxes are calculated by the atmosphericcomponent and fluxes on the oceanic grid points are given by a flux coupler. Users are referred to Yukimoto etal. (2010) for the treatment of fluxes in a coupled model.
Section 8.1 describes momentum flux. The surface forcing terms for temperature and salinity are presented in
Section 8.2, and their components, (heat flux (Section 8.3), fresh water flux (Section 8.4), and the equivalent fluxes
under invariable first layer volume condition (Section 8.5)) are detailed in the succeeding sections. Several options
are available for the choice of the bulk formula to calculate momentum, latent and sensible heat, and evaporative
fluxes. Each bulk formula is detailed in Section 8.6 with description about a general formulation for the bulk
transfer coefficient. Finally, the work flow in MRI.COM is summarized in Section 8.7.

8.1

Momentum flux (surface stress)

Surface forcing to the momentum equation, or surface momentum flux into the ocean, is in the form of windstress (or stress from sea ice below sea ice), and is expressed as a body force to the first level velocity in the model
algorithm(Fx , Fy ) =(τx , τy ),
ρ0 Δz 12where Δz 1 is the thickness of the first layer and τx and τy are zonal and meridional stress.
2

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(8.1)

Chapter 8

Sea surface fluxes

8.1.1 Input of wind stress data
By default, wind stresses are given as external data and are read from a file. (This input file is always required,
and thus a file filled with zeros is necessary when wind stress is not applied.) The input data should be in units of[dyn · cm−2 ]. The order of operations is as follows:• Wind stress data are read from a file in force.F90 in the following format:
read(unit = iwind) τx , τy .
• Time-interpolated in mkflux.F90 to produce surface stress.
• A part of the surface stress might be modified by the stress from sea ice.
• Surface stress is applied to the first level horizontal velocity in clinic.F90 as follows:∂1(τx , τy ).
(u1 , v1 ) = ... +∂t
ρ0 Δz 1

(8.2)

2

In the last of the above operations, (u1 , v1 ) is horizontal velocity at the first level and Δz 1 is the thickness of the2

first layer.

8.1.2 Calculating wind stress using a bulk formula
If the model option TAUBULK is specified, the wind stress is calculated based on a bulk formula. In this option,
zonal and meridional wind velocities at 10 m in units of [cm · s−1 ] should be prepared as external data, instead ofthe default wind stress data. Each component of wind stress is calculated with a bulk transfer coefficient CD (seeSection 8.6) as follows:τ = ρaCD |Ua − Us |(Ua − Us ),(8.3)

where ρa is the density of air, Ua is the wind vector at 10 m height, and Us is the first level velocity.

8.2

Sea surface forcing for temperature and salinity

8.2.1

Temperature

The contribution of surface forcing (FzT ) to the first layer temperature is expressed as
FzT∂ T 
=...
+,1∂ t k= 2
Δz 1

(8.4)

2

where Δz 1 is the thickness of the first layer of the T-cell. The term (FzT ) consists of the net sea surface heat flux2

gained by the ocean (QNET ), heat transport due to fresh water flux, and restoring of SST (To ) to a specific value(T ∗ ):
QNET
1
FzT =
+ (WAO +WIbot ) · To +WIsurf · Tfreeze − (To − T ∗ )Δz 1 .
(8.5)
2
ρ0 C pγt
The heat flux term is converted to heat transport using the reference density (ρ0 ) and specific heat (C p ) of sea water.
The water transported through the air-sea interface (WAO ) and ice bottom (WIbot ) is assumed to have the first level
temperature (To ). The water transported from the ice surface (WIsurf > 0) is assumed to have the freezing point
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8.2. Sea surface forcing for temperature and salinity
temperature (Tfreeze ). The fourth term of the r.h.s. is a restoring term to the observed SST. The parameter γt is a
restoring time (in units of seconds) and should be specified in the namelist njobbdy as rtmsc in units of days.
The restoring term is converted to heat transport by multiplying the thickness of the first layer of T-cell (Δz 1 ).
2
By default, surface temperature forcing consists only of the restoring term. By specifying the option HFLUX,
sea surface heat fluxes are calculated using the observed atmospheric variables. The fresh water flux terms areincluded when the option WFLUX is specified.
If the option SFLUXW is specified, fresh water flux is not added to the ocean and the expression for the heattransport term is modified as follows:
FzT =

QNET
1
−WIsurf · (To − Tfreeze ) − (To − T ∗ )Δz 1 .
2
ρ0 C pγt(8.6)

See Section 8.5 for the derivation.
Net sea surface heat flux QNET is given by
QNET = QSH + QLO + QLA + QSN + Qice ,(8.7)

where QSH is the net shortwave radiation flux, QLO is the net longwave radiation flux, QLA is the latent heat flux,
and QSN is the sensible heat flux. The last term on the r.h.s., Qice , is due to exchange of heat with sea ice and will
be detailed in Chapter 9.
Downward heat flux is defined as positive. Three of the first four components of the heat flux, QLO , QLA , and
QSN , depend on sea surface temperature, and thus they are calculated using the model SST.

8.2.2

Salinity

The contribution of surface forcing (FzS ) to the first layer salinity is expressed asFzS∂ S 
,1 = ... +
k=
2
∂t
Δz 1

(8.8)

2

where Δz 1 is the thickness of the first layer of the T-cell.
2
The explicit surface forcing (FzS ) consists of salt transport due to formation and melting of sea ice and restoringof SSS (So ) to a specific value (S∗ ):FzS = (WIbot +WIsurf ) · SI −1
(So − S∗ )Δz 1 .
2
γs

(8.9)

The water exchanged with ice (WIbot + WIsurf ) is assumed to have low salinity (SI = 4.0 [psu]). The parameter γs
is a restoring time (in units of seconds) and should be specified in the namelist njobbdy as rtmsc in units of days
(the same as temperature). The restoring term is converted to salinity transport by multiplying the thickness of thefirst layer of the T-cell (Δz 1 ). The fresh water flux modifies the volume but not the salt content of the surface layer,
2changing the salinity of the surface layer (see also Chapter 4).
By default, surface salinity forcing consists only of the restoring term. When the option WFLUX is specified, theeffect of the fresh water flux on salinity could be included.
If the option SFLUXW 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 · So − (WIbot +WIsurf ) · (So − SI ) −where So is the first level salinity. See Section 8.5 for the derivation.
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101

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1
(So − S∗ )Δz 1 ,2
γs(8.10)

Chapter 8

Sea surface fluxes

8.3 Heat flux
8.3.1 Shortwave radiation flux
By default, the downward shortwave radiation (Qdown
SH ) is read as external data. A part of the insolating shortwave
down
radiation (αo QSH ) is reflected at the sea surface, and the remainder penetrates into the ocean interior as the net
shortwave radiation flux (QSH (0) = (1 − αo )Qdown
SH ), where αo is the albedo at the ocean surface. More than
50% of the insolating radiation (near-infrared band) is absorbed within a depth of 1 m below the sea surface, but
the remainder (visible and ultraviolet bands) penetrates further into the ocean interior and affects the subsurface
temperature structure. According to Paulson and Simpson (1977), the shortwave radiation flux penetrating into theocean interior is given by
QSH (z) = QSH (0)[R exp(z/ζ1 ) + (1 − R) exp(z/ζ2 )],(8.11)

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). Vertical convergence of the penetrating shortwave radiation energy ∂ QSH (z)/∂ z isconverted to heat within each layer.
Using option SOLARANGLE enables us to represent the shortwave radiation flux including the effect of the insolation angle. This scheme is based on Ishizaki and Yamanaka (2010). In this case, if the depth (z) on the r.h.s. of
Equation (8.11) is replaced by the penetrating distance from the sea surface, the shortwave radiation is given by
QSH (z) = QSH (0)[R exp(z/ζ1 sin θw ) + (1 − R) exp(z/ζ2 sin θw )],(8.12)

where θw is the penetrating angle in the ocean interior. When option SOLARANGLE is specified in coupled models,Equation (8.12) is replaced by
QSH (z) = QSHb (z) + QSHd (z),(8.13)

where QSHb (z) and QSHd (z) are the shortwave radiation fluxes due to direct solar radiation and diffuse solar radiation. Those fluxes are expressed as follows:
QSHb (z) = QSHb (0)[Rb exp(z/ζ1 sin θw ) + (1 − Rb ) exp(z/ζ2 sin θw )],(8.14)

QSHd (z) = QSHd (0)[Rd exp(z/ζ1 ) + (1 − Rd ) exp(z/ζ2 )],(8.15)

where Rb and Rd are the ratio of near-infrared radiation to total radiation for direct solar radiation and diffuse solar
radiation. QSHb (0) and QSHd (0) are the net direct and diffuse solar radiation fluxes at the sea surface.
There are three options for the sea surface albedo. The first option (albedo choice = 1 in the namelist
njobalb) is a constant value, which should be specified as alb in the namelist njobalb. The second option(albedo choice = 2) is based on Large and Yeager (2008) and is given by

αo = 0.069 − 0.011 cos(2φ ),(8.16)

where φ is latitude. The third option (albedo choice = 3) is based on Baker and Li (1995) and is given by

αo = 0.06 + 0.0421x2 + 0.128x3 − 0.04x4 + (3.12
0.074x+)x5 ,5.68 +U 1.0 + 3.0U

where x = 1 − sin θa (θa is a height angle of the Sun), and U is the surface wind speed [m · s−1 ].

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102

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(8.17)

8.3. Heat flux

8.3.2

Shortwave radiation flux based on chlorophyll concentration

Recent studies indicate that solar radiation absorption and local heating within the upper ocean are strongly
influenced by the chlorophyll concentration. Using both the CHLMA94 and NPZD options enables us to use theshortwave penetration model with the chlorophyll concentration (Morel and Antoine, 1994). In this scheme, theshortwave radiation flux penetrating into the ocean interior is given by
QSH (z) = QSH (0)[R exp(−z/0.267 sin θw ) + (1 − R){V1 exp(−z/ζ1 ) +V2 exp(−z/ζ2 )}],

(8.18)

The first exponential is for the infrared waveband (> 750nm), which is not influenced by biological materials. The
second and third exponentials are for the ultraviolet and visible bands (< 750nm). 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 + 0.132C2 + 0.038C3 − 0.017C4 − 0.007C5 ,(8.19)

V2 = 0.679 − 0.008C − 0.132C2 − 0.038C3 + 0.017C4 + 0.007C5 ,(8.20)

5

ζ1 = 1.540 − 0.197C + 0.166C − 0.252C − 0.055C + 0.042C ,(8.21)

ζ2 = 7.925 − 6.644C + 3.662C2 − 1.815C3 − 0.218C4 + 0.502C5 ,(8.22)

2

3

4

where C = log10 (chl). It is noted that V1 +V2 = 1.
When the option SOLARANGLE is added to the options mentioned above, the shortwave radiation is slightly
modified by
QSH (z) = QSH (0)[R exp(−z/0.267 sin θw ) + (1 − R){V1 exp(−z/ζ1 sin θw ) +V2 exp(−z/ζ2 sin θw )}].

(8.23)

In coupled models, Equation (8.23) is replaced by
QSH (z) = QSHb (z) + QSHd (z),(8.24)

where QSHb (z) and QSHd (z) are the shortwave radiation fluxes due to direct solar radiation and that due to diffuse
solar radiation. Those fluxes are expressed as follows:
QSHb (z) = QSHb (0)[Rb exp(−z/0.267 sin θw ) + (1 − Rb ){V1 exp(−z/ζ1 sin θw ) +V2 exp(−z/ζ2 sin θw )}], (8.25)
QSHd (z) = QSHd (0)[Rd exp(−z/0.267) + (1 − Rd ){V1 exp(−z/ζ1 ) +V2 exp(−z/ζ2 )}],

(8.26)

where Rb and Rd are the ratios of the near-infrared part to the total radiation for direct and diffuse solar radiation,
respectively.

8.3.3 Longwave radiation flux
Longwave radiation is calculated by removing the upward radiation due to the observed SST ToOBS (externaldata) from the net longwave radiation QOBS
LO (external data) and by adding the upward radiation due to the model
SST.
OBS
QLO = QOBS
+ 273.16)4 − (To + 273.16)4 ).
LO + em σ ((To(8.27)

Here, em = 0.97 is the emissivity for sea water, and σ = 5.67 × 10−5 [erg·s−1 ·cm−2 ·K−4 ] is the Stefan-Boltzmann
constant.
Using option LWDOWN enables us to use the downward longwave radiation as external data. In this case, dummy
data should be prepared for the SST data as external data.
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103

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Chapter 8

Sea surface fluxes

8.3.4 Latent and sensible heat fluxes
The bulk method is used to calculate latent and sensible heat fluxes. In the bulk method, latent heat flux QLA andsensible heat flux QSN are calculated using bulk transfer coefficients CE and CH (see Section 8.6).
QLA = −ρa LCE U10 (qs − qa )(8.28)QSN = −ρaC paCH U10 (To − Ta ).

(8.29)

andHere, ρa is the atmospheric density, L is the latent heat for evaporation, qa is the specific humidity at the sea
surface, qs is the saturated specific humidity of the sea surface temperature, Ta is the sea surface air temperature,
and U10 is the scalar wind speed. The quantity Cpa is the specific heat for the atmosphere.
MRI.COM can provide three calculation methods for bulk transfer coefficients: Kondo (1975), Large and Yeager(2004), and Kara et al. (2002). In the following, we explain the elements necessary for calculating these bulk
transfer coefficients and calculation of the latent and sensible heat fluxes.
(a) Kondo (1975)ᲢBULKKONDO2Უ
The bulk formula based on Kondo (1975) uses sea surface pressure Ps [hPa], sea surface air temperature Ta [◦ C],
sea surface specific humidity qa [g·g−1 ], and scalar wind speed U10 [cm·s−1 ] as input data.
Elements necessary for calculating the latent and sensible heat fluxes are obtained as follows:
1. Saturated water vapor pressure es [hPa] for sea surface temperature (To [◦ C])
es = 0.98 × 6.1078 × 107.5To /(237.3+To ) ,(8.30)

2. Saturated specific humidity for sea surface temperature qs [g·g−1 ]qs =0.62197es,
Ps − 0.378es(8.31)

3. Latent heat of evaporation L [erg·g−1 ]L = 4.186 × 107 (594.9 − 0.5To ),(8.32)

ρa = 1.205 × 10−3 .

(8.33)

4. Atmospheric density ρa [g·cm−3 ](b) Large and Yeager (2004)ᲢBULKNCARᲣThe bulk formula based on Large and Yeager (2004) uses the same input data and elements as those of Kondo(1975).
(c) Kara et al. (2002)ᲢBULKKARAᲣThe bulk formula based on Kara et al. (2002) uses four elements: sea surface pressure Ps [hPa], sea surface airtemperature Ta [◦ C], sea surface dew point temperature Td [◦ C], and scalar wind speed U10 [cm·s−1 ].
Elements necessary for calculating the latent and sensible heat fluxes are obtained as follows:
1. Water vapor pressure at the sea surface ea [hPa]ea = 6.1121 exp[(18.729 − Td /227.3)Td /(Td + 257.87)],
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104

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(8.34)

8.4. Freshwater flux
2. Saturated water vapor pressure es [hPa] for sea surface temperature (To [◦ C])
es = 0.9815 × 6.1121 exp[(18.729 − To /227.3)To /(To + 257.87)],(8.35)

3. Specific humidity at the sea surface qa [g·g−1 ]qa =0.62197ea,
Ps − 0.378ea(8.36)

4. Latent heat of evaporation L[erg·g−1 ]L = 1010 (2.501 − 0.00237To ),(8.37)

ρa = 103 Ps /[Rg (Ta + 273.16)(1.0 + 0.61qa )].

(8.38)

5. Air density ρa [g·cm−3 ]Here, Rg is the gas constant for dry air.
The saturated specific humidity for sea surface temperature qs [g·g−1 ] is given by Equation (8.31), which is equivalent to Kondo (1975).

8.4 Freshwater flux
8.4.1 Introduction
Freshwater flux through the sea surface is caused mainly by precipitation (P), evaporation (E), river discharge(R), and formation-melting of sea ice (I). The fresh water fluxes are included in the model when option WFLUX
is specified. In this case, precipitation and river discharge data should be prepared as input data. A free surface
model can deal with freshwater flux explicitly and calculate the salinity change using a volume change of the firstlayer.
Even with the WFLUX option, the salinity restoring term could be added to the water flux by specifying the option
SFLUXR to avoid unexpected salinity drift. The salinity restoring is converted to the equivalent fresh water flux inthis case. It is possible to transform all freshwater fluxes to salinity flux by specifying option SFLUXW, in order toprevent the thickness of the first layer (nominally several meters) from becoming zero or unrealistically large due
to excess precipitation or evaporation.

8.4.2 Calculating freshwater flux
Freshwater flux through the sea surface F W [cm·s−1 ] is given by
F W = P − E + R + I,

(8.39)

where precipitation P is given as external data, and evaporation E is calculated in the model,
E = ρaCE U10 (qs − qa ) = −QLA /L.

(8.40)

River discharge R is given as external data when RUNOFF is selected. Freshwater flux I due to the formation andmelting of the sea ice is added when the sea ice model is used (ICE) (see Chapter 9 for the sea ice model). We get
F W = P − E when only WFLUX is selected.
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Chapter 8

Sea surface fluxes

Since F W is not related directly to the sea surface salinity, the model sea surface salinity might be far from theobserved 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 is available at present.
By converting the restoring term to the fresh water flux, the expression for F W becomes
FW = P − E + R + I +1 So − S∗.
γs S0(8.41)In this case, the salinity flux should be
FzS = (WIbot +WIsurf ) · SI ,(8.42)

where WIbot is the water transported from the ice bottom, WIsurf is the water transported through the ice surface,and SI is the salinity of the water exchanged with ice.
The global mean of F W should not necessarily be zero, and thus the volume of the first layer averaged globally
may continue to increase or decrease. To avoid this, the integral of the globally averaged freshwater flux is set tozero by selecting option WADJ,
W= FW − FW ,FADJ

(8.43)

where F W is the global mean of F W (= P − E(+R) + γs (So − S∗ )/S0 ).

8.5 Equivalent surface temperature and salinity fluxes for constant firstlayer volume
Precipitation, evaporation, and river discharge (WAO = P − E + R) are assumed to have the first level temperature(To ) and zero salinity. The water transported through the ice bottom (WIbot ) is also assumed to have the firstlevel temperature. The water transported from the ice surface (WIsurf > 0) is assumed to have the freezing pointtemperature (Tfreeze ). The water exchanged with ice is assumed to have low salinity (SI = 4.0 [psu]). Note that thefreezing point temperature (Tfreeze ) is given by mSI , where m = −0.0543 [K / psu].
If fresh water flux is simply added to the first layer volume, surface temperature and salinity fluxes due to the
fresh water flux are given by the following.
T
FWF= (WAO +WIbot ) · To +WIsurf · Tfreeze ,(8.44)S
FWF= (WIbot +WIsurf ) · SI .

(8.45)

If the surface fresh water flux is not added to the first layer to avoid an unexpected sea level rise or fall during along-term integration, equivalent temperature and salinity fluxes should be imposed.
The conservation equations for the first layer heat and salinity are considered. Temperature and salinity are
assumed to evolve from the old values (Told , Sold ) to the new values (Tnew , Snew ) due to fresh water flux.
If a volume change is allowed,
Vnew Tnew= Vold Told + (WAO Told +WIbot Told +WIsurf Tfreeze ) · ΔA · Δt,(8.46)

Vnew Snew= Vold Sold + (WIbot SI +WIsurf SI ) · ΔA · Δt,(8.47)

Vnew= Vold + (WAO +WIbot +WIsurf ) · ΔA · Δt,(8.48)

where ΔA is the horizontal area.
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8.6. Bulk transfer coefficient
If a volume change is not allowed,
Vold Tnew

T= Vold Told + FWF· ΔA · Δt,(8.49)

Vold Snew

S
Vold Sold + FWF· ΔA · Δt,(8.50)=

T and F S are temperature and salinity flux for the constant volume case.
where FWF
WF
Removing Tnew , Snew , and Vnew from the above equations, we have,

Vold {Vold Told + (WAO Told +WIbot Told +WIsurf Tfreeze ) · ΔA · Δt}
T· ΔA · Δt),= {Vold + (WAO +WIbot +WIsurf ) · ΔA · Δt}(Vold Told + FWF(8.51)

Vold {Vold Sold + (WIbot SI +WIsurf SI ) · ΔA · Δt}
S· ΔA · Δt).
= {Vold + (WAO +WIbot +WIsurf ) · ΔA · Δt}(Vold Sold + FWF(8.52)

The fluxes for the constant volume case are given by
T
FWF
S
FWF

Vold
WIsurf · (Told − Tfreeze ),Vold +W · ΔA · Δt
Vold= −{WAO · Sold + (WIbot +WIsurf ) · (Sold − SI )},Vold +W · ΔA · Δt
= −(8.53)(8.54)

where W = WAO +WIbot +WIsurf .
If |Vold |  |W · ΔA · Δt|, we have,

8.6

T
FWF

= −WIsurf · (Told − Tfreeze ),(8.55)S
FWF= −WAO · Sold − (WIbot +WIsurf ) · (Sold − SI ).

(8.56)

Bulk transfer coefficientThis section briefly describes how to calculate sea surface fluxes using a bulk formula. For details, refer toKantha and Clayson (2000) and Large and Yeager (2004).

8.6.1

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 require specific interactions between the atmosphere and the ocean, and attributes all unknown processes to bulk transfer
coefficients.
Momentum (τ ), sensible heat (HS ), latent heat (HL ), and water vapor (E) fluxes are written in terms of turbulent
components as follows:τ= −ρa wu = ρa u2∗ ,(8.57)HS= −ρa c p wθ = ρa c p u∗ θ∗ ,(8.58)HL= −ρa LE wq = ρa LE u∗ q∗ ,(8.59)E= −ρa wq = ρa u∗ q∗ = HL /LE ,(8.60)

where a bar over a variable denotes a covariance between the turbulent component of vertical velocity and theturbulent component of each physical property, ρa is air density, u∗ is friction velocity, θ∗ ≡ HS /(ρa c p u∗ ) is the
–

107

–

Chapter 8

Sea surface fluxes

temperature scale in the boundary layer, q∗ ≡ HL /(ρa LE u∗ ) = E/(ρa u∗ ) is the scale of specific humidity in theboundary layer, c p is the specific heat of air, and LE is the latent heat of water vapor evaporation.
Sea surface fluxes are also represented using bulk formulae as follows:τ= ρaCDh (Ua −Us )2 ,(8.61)HS= −ρa c p |Ua −Us |CHh (Ts − Ta ),(8.62)HL= −ρa LE |Ua −Us |CEh (qs − qa ),(8.63)

E= ρa |Ua −Us |CEh (qs − qa ),(8.64)

where Ua is the wind velocity at height of zh , and Us is the velocity component in the Ua direction at the sea surface.
The subscript “a” means a value of each physical property at z = zh , and the subscript “s” means the value justabove the sea surface.
Parameter CDh is called a drag coefficient. CHh and CEh are transfer coefficients for heat and water vapor andare 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 byobserved turbulent fluxes, using the following Equations (8.65), (8.66), and (8.67) and the similarity law of MoninObukhov mentioned below,
CDh
CHh
CEh

u2∗
,
(Ua −Us )2
HS /ρa c p
u∗ θ∗
= −
=
,
|Ua −Us |(Ts − Ta ) |Ua −Us |(Ts − Ta )
E/ρa
u∗ q∗
=
=
.
|Ua −Us |(qs − qa ) |Ua −Us |(qs − qa )=(8.65)(8.66)(8.67)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 vertical profiles when they are scaled with the stability parameter and sea surface fluxes. Vertical profilesof wind velocity, air temperature, and humidity can be written as follows:κz ∂Uu∗ ∂ zκz ∂ Tθ∗ ∂ zκz ∂ qq∗ ∂ z

z
,
L
z
,
= φH
L
z
,
= φE
L

= φM

(8.68)(8.69)(8.70)

where κ = 0.4 is the von Karman constant, and L is the Monin-Obukhov length scale
L=

u3 Tv
u3∗
=− ∗
.
κ Qb
κ g wθv(8.71)In (8.71), Qb = −g wθv /Tv is the buoyancy flux, Tv is the virtual temperature (Tv = T (1 + 0.608 q)), and wθv is thenet heat flux including the water vapor flux:wθv = wθ (1 + 0.608 q) + 0.608 T wq.

(8.72)In (8.68) to (8.70), ζ = z/L is the Monin-Obukhov similarity variable, and φM,H,E is a nondimensional function
for wind velocity, air temperature, and specific humidity. The nondimensional function is assumed to be a mathematically simple function.
– 108

–

8.6. Bulk transfer coefficient
Integrating Equations (8.68), (8.69), and (8.70) vertically, we have the following.
+,zu∗ln − ΨM (ζ )U(z) −Us =
κ
z0
+
,
θ∗
z
ln
T (z) − Ts =− ΨH (ζ )
κ
z0T
+
,
z
q∗
ln
− ΨE (ζ ) ,q(z) − qs =
κ
z0E
where
ΨM,H,E (ζ ) =

 ζ



0

dζ
1 − φM,H,E (ζ ) ζ

(8.73)(8.74)(8.75)

.

(8.76)In (8.73) to (8.75), z0 , z0T , and z0E are roughness lengths for each physical property. When the stability of theboundary layer is already known, bulk transfer coefficients can be estimated using these roughness lengths,

κ2

CDh = +,2 ,zhln − ΨM (ζh )
z0

κ2
,+
,.
CHh,Eh = +zhzhln − ΨM (ζh ) ln
− ΨH,E (ζh )z0
z0T,0E

(8.77)(8.78)

If neutrally stable (ζ = 0), the bulk transfer coefficient is a function of the roughness length only and is expressed
as follows:
CDNh = +

κ2

, ,
zh 2
ln
z0N

CHNh,ENh

=

=

κ2
+
,+
,
zh
zh
ln
ln
z0N
z0TN,0EN
CDNh+,
1 1/2 z0TN,0EN
1 − CDNh lnκ
z0N(8.79)(8.80)(8.81)

1/2

=

+
ln

κ CDNh,.
zhz0TN,0EN(8.82)

Normally, these neutral bulk transfer coefficients are estimated at a height of 10 m. Non-neutral bulk transfer
coefficients at an arbitrary height (zh ) are connected with the neutral and stable bulk transfer coefficients at a heightof 10 m by eliminating the roughness length as follows:
CDN10

,2 ,
1
zh
−1/2
1 + CDN10CDN10 ln− ΨM (ζh )κz10

CDh = +


CDh 1/2
CHN10,EN10
C
DN10
, ,
CHh,Eh = +
1
zh
−1/2
1 + CHN10,EN10CDN10 ln− ΨH,E (ζh )κz10(8.83)



– 109

–

(8.84)

Chapter 8

Sea surface fluxes

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
coefficients and fluxes because height can be adjusted using the similarity law. This requires an iteration of thecalculation 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 and stable caseand use a bulk coefficient at 10 m in the neutral case,• Convert a bulk coefficient at 10 m into one with a height and a stability at which atmospheric elements areobserved.
• Convert air temperature and specific humidity into those at a height where wind speeds are observed, andconvert the bulk coefficient at 10 m and neutral case into one with a height and a stability at which windspeeds are observed.

8.6.2 Kondo (1975) BULKKONDO2When atmospheric elements are observed at various heights, latent and sensible heat fluxes are estimated ata height where air temperature and humidity are observed. Momentum fluxes are estimated at the same height.
In this case, wind speed and bulk transfer coefficients should be corrected to ones at the height where the airtemperature and humidity are observed.
In the program of bulk.F90, you may specify BULKITER in configure.in to do this iteration. The observedheights are specified as altu (wind), altt (temperature), and altq (humidity) in the namelist njbkondo.
The following procedure is executed:
1. Estimate the neutral bulk transfer coefficient at 10 m assuming that the observed wind speed is at 10 m in
the neutral case. When the 10 m wind speed is already known, the neutral bulk transfer coefficient is given
as follows:
CDN10 = 10−3 {ad + 10−2 bd U10d },p(8.85)

where ad , bd , and pd are nondimensional numbers depending on the wind speed (Table 8.1).
2. Recalculate the roughness length using this neutral bulk transfer coefficient at 10 m and correct the height
for wind speed once again:



−1/2
z0 = exp ln z10 − κ CDN10 ,(8.86)

u10 = u(z) ln(10/z0 )/ ln(z/z0 ).

(8.87)

3. Re-estimate the neutral bulk transfer coefficient at 10 m.
4. Convert the bulk transfer coefficient to that in the neutral case at the observed height:
CDNh =

–

κ2
2
−1/2
κ CDN10 − ln(z10 /zh )110–(8.88)

8.6. Bulk transfer coefficient
1/2

CHNh,ENh =κ CDNh1/2

κ CDN10 /CHN10,EN10 + ln(zh /z10 ),(8.89)

where z10 means z = 10[m].
5. Estimate the bulk transfer coefficient in which stability is considered using the neutral bulk transfer coefficient obtained above.
The neutral bulk transfer coefficient at 10 m is given as follows:
CDN10

=

CEN10

=

CNN10

=

10−3 {ad + 10−2 bd U10d },p−3(8.90)−2pe
10 {ae + 10 beU10+ ce (10−2U10 − 8)2 },p
10−3 {ah + 10−2 bhU10h + ch (10−2U10 − 8)2 },(8.91)(8.92)

where CDN10 , CEN10 , and CHN10 are bulk transfer coefficients for neutral atmospheric stability. and ae,h , be,h , ce,h ,and pe,h are nondimensional numbers depending on wind speed (Table 8.1).
Table 8.1. Non-dimensional parameters used in calculating the bulk transfer coefficient based on Kondo (1975)

U10

ad

ae

ah

bd

be

bh

ce

ch

pd

pe

ph

0.3-2.2
2.2-5

0
0.771

0
0.969

0
0.927

1.08
0.08581.23
0.0521

1.185
0.0546

0
0

0
0

-0.15
1

-0.16
1

-0.157
1

5-8
8-25

0.867
1.2

1.18
1.196

1.15
1.17

0.0667
0.025

0.01
0.008

0.01
0.00750-0.0004

0-0.00045

1
1

1
1

1
1

25-50

0

1.68

1.652

0.073

-0.016-0.017

0

0

1

1

1

(m · s−1 )The bulk transfer coefficient depending on the stability is estimated in the following operations. First, thestability of the atmospheric boundary layer is defined as follows:s = s0|s0 |,|s0 | + 0.01

(8.93)s0 =To − Ta.
2
10−4U10(8.94)

where

The atmospheric boundary layer is unstable if s > 0 (To − Ta > 0) and stable if s < 0 (To − Ta < 0). Finally, thebulk transfer coefficient depending on the stability is given as follows:⎧⎪s < −3.3⎨ 0
CD =CDNh {0.1 + 0.03s + 0.9 exp(4.8s)} −3.3 ≤ s < 0⎪√⎩0 ≤ s,
CDNh (1.0 + 0.47 s)⎧⎪
s < −3.3⎨ 0
CE =CENh {0.1 + 0.03s + 0.9 exp(4.8s)} −3.3 ≤ s < 0⎪√⎩0 ≤ s,
CENh (1.0 + 0.63 s)⎧⎪
s < −3.3⎨ 0
CH =CHNh {0.1 + 0.03s + 0.9 exp(4.8s)} −3.3 ≤ s < 0⎪√⎩0 ≤ s.
CHNh (1.0 + 0.63 s)
–

111

–

(8.95)(8.96)(8.97)

Chapter 8

Sea surface fluxes

8.6.3 Large and Yeager (2004) BULKNCAR
In the bulk formula based on Large and Yeager (2004), each flux is estimated at a height where wind speed
is observed, by transforming air temperature and humidity from the height where they are observed (zθ and zq ,respectively) to the height where wind speed is observed. To do this iteration in the program bulk.F90, specify
the option BULKITER in configure.in. The observed heights are specified as altu (wind) altt (temperature),
and altq (humidity) in the namelist njbncar.
The bulk transfer coefficient at 10 m in the neutral case is given as follows:
2.70
U10N
+ 0.142 +
,
U10N
13.09

103CHN10,EN10 = 18.0 CDN10 , stable ζ > 0,
103CHN10,EN10 = 32.7 CDN10 , unstable ζ ≤ 0,
103CDN10 =(8.98)(8.99)(8.100)Each physical property is estimated at a height where wind speed is observed in the following operations. First,
calculate the virtual temperature θv as follows:θv = θ (zθ )(1 + 0.608 q(zq ))(8.101)

Next, calculate the bulk transfer coefficient at 10 m in the neutral and stable cases assuming that the first guess
for the 10 m wind speed in the neutral and stable cases is U10N = U(zu ) (Equations (8.98) to (8.100)). The first
guesses of the scales for the friction velocity, air temperature, and specific humidity are estimated assuming thatthese bulk coefficients are at the observed height and stability,
1

τ
u∗ =
= CDN10 U(zu ),(8.102)
ρa
CHN10
HS
θ∗ =
=
(θ (zθ ) − Ts ),(8.103)
ρa c p u ∗
CDN10
E
CEN10
=
(q(zq ) − qsat (Ts )),(8.104)q∗ =ρa u ∗
CDN10
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, ζθ = zθ /L, andζq = 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,,+κ gz θ∗q∗.
(8.105)ζ= 2+u∗ θv (q(zq ) + 0.608−1 )The integral of the non-dimensional profile function is expressed asΨM (ζ ) = ΨH (ζ ) = −5ζ ,(8.106)

if it is stable (ζ ≥ 0), andΨM (ζ ) = 2 ln

1+X
2





+ ln


1 + X2,
ΨH (ζ ) = 2 ln
2– 112

1 + X22− 2 tan−1 (X) +π,2(8.107)(8.108)–8.6. Bulk transfer coefficient
if it is unstable (ζ < 0). In the above,
X = (1 − 16ζ )1/4 .

(8.109)Using these values, convert the wind speed to that at 10 m in the neutral and stable cases, and convert thetemperature and specific humidity to those at a height where the wind speed is observed,

,−1
+
CDN10
zu
ln
− ΨM (ζu ),U10N = U(zu ) 1 +
κ
z10
+
,
θ∗
zθ
ln + ΨH (ζu ) − ΨH (ζθ ) ,θ (zu ) = θ (zθ ) −
κ
zu
,
+
z
q∗
q
q(zu ) = q(zq ) −ln + ΨH (ζu ) − ΨH (ζq ) ,κzu(8.110)(8.111)(8.112)(8.113)where z10 means z = 10[m]. Estimate the bulk coefficient at 10 m in the neutral and stable cases using U10N , andthen obtain the bulk coefficient at a height (zu ) where wind speed is observed,
CDu

=

CHu

=

CEu

=

CDN10
+

,2 ,
1
zu
−1/2
1 + CDN10CDN10 ln− ΨM (ζu )
κ
z10


CDu 1/2
CHN10
CDN10
, ,
+

1
zu
−1/2
1 + CHN10CDN10 ln− ΨH (ζu )
κ
z10


CDu 1/2
CEN10
CDN10
+
, .

1
zu
−1/2
1 + CEN10CDN10 ln− ΨH (ζu )κz10(8.114)(8.115)(8.116)

Repeat the procedures to calculate the bulk coefficients using these bulk coefficients with temperature and specifichumidity at z = zu , and recalculate the scales for virtual temperature, friction velocity, temperature, and specifichumidity (Equations of (8.102), (8.103), and (8.104)).

8.6.4

Kara et al. (2002) BULKKARA

In the bulk formula based on Kara et al. (2002), bulk transfer coefficients CD (momentum), CE (latent heat),and CH (sensible heat) are calculated using a polynomial fitting as follows to reduce the calculation cost due toiteration:2),
C0D = 10−3 (0.692 + 0.071U10 − 0.00070U10−3

C1D = 102(0.083 − 0.0054U10 + 0.000093U10),

CD = C0D +C1D (To − Ta ),−3

C0H = 10

(8.117)(8.118)(8.119)2(0.8195 + 0.0506U10 − 0.0009U10),(8.120)2),
C1H = 10−3 (−0.0154 + 0.5698/U10 − 0.6743/U10(8.121)

CE = C0H +C1H (To − Ta ),(8.122)

CH = 0.95CE ,(8.123)–113–

Chapter 8

Sea surface fluxes

where 2.5[m · s−1 ] ≤ U10 ≤ 32.5[m · s−1 ].
Flux correction factors may be specified by setting close budgets, corr factor n, and corr factor s innamelist njbkara.

8.6.5 Bulk coefficient over sea ice
A similar treatment is necessary even over sea ice, but it is common to replace the neutral and stable bulk
coefficients at 10 m with constants, as described in Large and Yeager (2004). Then, bulk coefficients and windspeed at each height are calculated based on the stability.
Large and Yeager (2004) uses
CDN10 = CHN10 = CEN10 = 1.63 × 10−3 ,(8.124)but the default in MRI.COM uses, following Mellor and Kantha (1989),
CDN10 = 3.0 × 10−3

(8.125)

CHN10 = CEN10 = 1.5 × 10−3 .

(8.126)

The wind stress could be separately calculated over sea ice under the option TAUBULK. Users should specify
flg strsi = .true. in ice main cat.F90.

8.7 Work flow in MRI.COM
8.7.1 Momentum flux
• Use of bulk formula (TAUBULK)
1. Input of external data (velocity vectors at 10 m height) in force.F90.
(a) Time interpolation of external data in mkflux.F90.
(b) Calculation of wind stress (τx , τy ) (Equation (8.3)) in bulk.F90 (called from mkflux.F90).
(c) Calaculation of ice-water stress and total surface stress based on area-weighted average in ice dyn.F90(called from the sea ice part (ICE), see Chapter 9).
2. Update velocity for the first layer (Equation (8.2)) in clinic.F90.
• Use of external data (default)
1. Input of external data in force.F90.
2. Time interpolation of external data in mkflux.F90.
3. Calaculation of ice-water stress and total surface stress based on area-weighted average in ice dyn.F90(called from the sea ice part (ICE), see Chapter 9).
4. Update velocity for the first layer (Equation (8.2)) in clinic.F90.
• Coupled model
1. Getting fluxes from the atmospheric model at the beginning of a coupling cycle in cgcm scup get a2o(cgcm scup.F90).
– 114

–

8.7. Work flow in MRI.COM
2. Getting fluxes for this step from get flux a2o (get fluxes.F90).
3. Calaculation of ice-water stress and total surface stress based on area-weighted average in ice dyn.F90(called from the sea ice part (ICE), see Chapter 9).
4. Update velocity for the first layer (Equation (8.2)) in clinic.F90.

8.7.2

Temperature (heat) flux

• Heat flux type (HFLUX)
1. Input of external data in force.F90
OBS
OBS(a) read(unit = ihflx) QOBS
SH , QLO , To(b) read(unit = ibulk) Ta , qa (Td , if TDEW), U10 , Ps
2. Calculation of FT in mkflux.F90(a) Time interpolation of external data.
(b) Calculation of QSH (Equation (8.11)).
(c) Calculation of QLO （Equation (8.27)).
(d) Calculation of QLA and QSN (Equation (8.28)-(8.29)) in bulk.F90 (called from mkflux.F90).
(e) Preparation for QSH and (QLO + QLA + QSN ).
(f) Heat flux between sea ice and sea water (Qice ) in ice main cat.F90 (called from the sea ice part(ICE), see Chapter 9).
T (Equation (8.44) or (8.55)).
(g) Calculation of FWF

3. Update first layer temperature using F T (Equation (8.4)) in tracer.F90.
– Update temperature at the depth deeper than the first layer using shortwave absorption (Equation(8.11)).
• Restoring sea surface temperature type (default)
1. Calculation of F T (the last term on the r.h.s. of Equation (8.5)) in mkflux.F90.
2. Update the first layer temperature using F T (Equation (8.4)) in tracer.F90.
• Coupled model
1. Getting fluxes from the atmospheric model at the beginning of a coupling cycle in cgcm scup get a2o(cgcm scup.F90).
2. Calculation of FT in mkflux.F90(a) Getting fluxes for this step from get flux a2o (get fluxes.F90).
(b) Preparation for QSH and (QLO + QLA + QSN ).
(c) Heat flux between sea ice and sea water (Qice ) in ice main cat.F90 (called from the sea ice part(ICE), see Chapter 9).
T (Equation (8.44) or (8.55)).
(d) Calculation of FWF

3. Update first layer temperature using F T (Equation (8.4)) in tracer.F90.
– Update temperature at the depth deeper than the first layer using shortwave absorption (Equation(8.11)).
–

115–

Chapter 8

Sea surface fluxes

8.7.3 Salinity and fresh water flux
• Freshwater flux type WFLUX
1. Input of external data in force.F90:(a) read(ipcpr) P(b) read(irnof) R, when RUNOFF
2. Calculation of F W and F S in mkflux.F90:(a) Time interpolation of external data.
(b) Calculation of E (Equation (8.40)) in bulk.F90 (called from mkflux.F90).
(c) Freshwater and salinity fluxes between sea ice and sea water I and F S (Equation (8.42)) in ice main cat.F90(called from the sea ice part (ICE), see Chapter 9).
(d) Restoring term for sea surface salinity and conversion to F W (Equation (8.41)).
W for the option WADJ (Equation (8.43)).
(e) Calculation of FADJ(f) Set F W = 0 and reevaluate F S for option SFLUXW (Equation (8.10)).
3. Update salinity of the first layer:
W ) in surface integ.F90.
(a) Update the first layer volume using F W (or FADJ(b) Update the first layer salinity using F S (Equation (8.8)) in tracer.F90.
• Restoring sea surface salinity (default)
1. Calculation of F S (the last term on the r.h.s. of Equation (8.9)) in mkflux.F90
2. Update the first layer salinity (Equation (8.8)) in tracer.F90• Coupled model
1. Getting fluxes from the atmospheric model at the beginning of a coupling cycle in cgcm scup get a2o(cgcm scup.F90).
2. Calculation of F W and F S in mkflux.F90:(a) Getting fluxes for this step from get flux a2o (get fluxes.F90).
(b) Freshwater and salinity fluxes between sea ice and sea water I and F S (Equation (8.42)) in ice main cat.F90(called from the sea ice part (ICE), see Chapter 9).
(c) Calculation of restoring term for sea surface salinity and conversion to F W (Equation (8.41)).
W for the option WADJ (Equation (8.43)).
(d) Calculation of FADJ(e) Set F W = 0 and reevaluate F S for the option SFLUXW (Equation (8.10)).
3. Update salinity of the first layer:
W ) in surface integ.F90.
(a) Update the first layer volume using F W (or FADJ(b) Update the first layer salinity using F S (Equation (8.8)) in tracer.F90.

– 116

–

8.8. Remarks

8.8

Remarks

Although recent satellite observations enable us to obtain sea surface fluxes with high resolution in space andtime, even higher accuracy is necessary for practical use. For example, a bias of several W·m−2 in heat flux
greatly affects 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 basedon various methods have been expanded globally. In the future, a high-resolution model and a new scheme foradvection 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 unlikelythat the accuracy of a bulk coefficient would be improved). Hence, it should be kept in mind that sea surface fluxes
presently involve large uncertainties.

8.9

Appendix

8.9.1

Unit of constants
Symbol and Name

ρo
ρa
Cp
C pa
αo
em
σ
Rg

8.9.2

sea water density
air density
sea water specific heat
air specific heatsea surface albedo
sea water emissivity
Stefan-Boltzmann constantgas constant for dry air

Unit

NumeralModel(CGS)

1.0
1.205 × 10−3

SI×g·cm−3×103 Kg·m−3

×g·cm−3×103 Kg·m−3

3.99
1.00467

107 erg·g−1 ·K−1
107 erg·g−1 ·K−1

103 J·Kg−1 ·K−1
103 J·Kg−1 ·K−1

alb
0.97
5.67

10−5 erg·s−1 ·cm−2 ·K−4

10−8 J·s−1 ·m−2 ·K−4

2.871

106 erg·g−1 ·K−1

102 J·Kg−1 ·K−1

Unit of variables
Unit

Symbol and NameQ∗∗

Model(CGS)×g·s−3

Sea surface heat flux

SI
×10−3 W·m−2

Freshwater flux

cm·s−110−3 J·s−1 ·m−2
10−2 m·s−1

τ∗
Δz 1

Momentum flux (wind stress)
the first layer thickness

dyn · cm−2
cm10−1 N · m−2
10−2 mT∗

temperature

◦C

◦C

Ps
U10

sea surface pressure
scalar wind speedhPa (not in CGS)
cm·s−1hPa
10−2 m·s−1

ρ∗
q∗
e∗
L

densityspecific humidity
water vapor pressure
evaporation latent heatg·cm−3g·g−1
hPa (not in CGS)

103 Kg·m−3
Kg·Kg−1hPa

erg·g−1

J·Kg−1

FW

2

–

117

–

Chapter 8

Sea surface fluxes

References
Baker, H. W., and Z. Li, 1995: Improved simulation of clear-sky shortwave radiative transfer in the CCC-GCM.,
J. Climate, 8, 2213-2223.
Ishizaki, H., and G. Yamanaka, 2010: Impact of explicit sun altitude in solar radiation on an ocean model simulation, Ocean Modell., xxx, in press.
Jerlov, N. G., 1976: Marine Optics, 231pp., Elsevier.
Kantha, L. H., and C. A. Clayson, 2000: Small Scale Processes in Geophysical Fluid Flows., 888pp., Academic
Press.
Kara, A. B., P. A. Rochford, and H. E. Hurlburt, 2002: Air-sea flux estimates and the 1997-1998 ENSO event.,
Boundary-Layer Meteorol., 103, 439-458.
Kondo, J., 1975: Air-sea bulk transfer coefficients in diabatic conditions., Boundary-Layer Meteorol., 9, 91-112.
Large, W. G., and S. G., Yeager, 2008: The global climatology of an interannually varying air-sea flux data set.,
Climate Dynamics, DOI 10.1007/s00382-008-0441-3, 24pp.
Large, W. G., and S. G., Yeager, 2004: Diurnal to decadal global forcing for ocean and sea-ice models: the data
sets and flux climatologies., NCAR Technical Note: NCAR/TN-460+STR. CGD Division of the National
Center for Atmospheric Research.
Mellor, G. L., and L. Kantha, 1989: An ice-ocean coupled model, J. Geophys. Res., 94, 10937-10954.
Morel, A., and D. Antoine, 1994: Heating rate within the upper ocean in relation to its bio-optical state, J. Phys. Oceanogr.,
24, 1652-1665.
Paulson, C. A., and J. J. Simpson, 1977: Irradiance measurements in the upper ocean, J. Phys. Oceanogr., 7,952-956.
WGASF, 2000: Intercomparison and validation of ocean-atmosphere energy flux fields., Final report of the Joint
WCRP/SCOR Working Group on Air-Sea Fluxes (WGASF), pp.306..
Yukimoto, S., and coauthors, 2010: Meteorological Research Institute-Earth System Model v1 (MRI-ESM1) Model Description -, Technical Reports of the Meteorological Research Institute, No.64, in press.

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Chapter 9

Sea iceThis chapter describes the sea ice part. The sea ice part of MRI.COM treats formation, accretion, melting, andtransfer of sea ice and snow. Heat, water, salt, and momentum fluxes are exchanged with the ocean. Sea ice iscategorized 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 9.1 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
9.2, remapping among thickness categories in Section 9.3, dynamics in Section 9.4, advection in Section 9.5, andridging in Section 9.6. Discretization issues are described in Section 9.7, and finally some technical issues arepresented in Section 9.8.

9.1

Outline

The sea ice part of an ocean model gives surface boundary conditions. Heat, fresh water, salt, and momentum
are exchanged at their interfaces. The sea ice part solves fractional area, heat content, thickness, and their transportof 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 that are not explicitly discussed nor included there, such as categorizing by thickness, ridging, and rheology,
we adopt those of the Los Alamos sea ice model (CICE; Hunke and Lipscomb, 2006).
The fundamental property that defines the state of sea ice is the fractional area as a function of location (x, y)and thickness (hI ). The equation for this distribution function (g(x, y, hI )) is∂g∂
1  ∂ (ghψ uI ) ∂ (ghμ vI ) + χ,( f g) −
+
=
∂t
∂ hI
h μ hψ
∂μ
∂ψ

(9.1)

where f is the thermodynamic growth rate of thickness, (uI , vI ) is the velocity vector of ice pack, and χ is therate of change 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. To compute the velocity of the ice pack (uI , 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 and solving 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
Hn−1

gdh.

(9.2)

Other major variables, ice and snow thickness, surface temperature, bottom temperature and salinity, and internalenergy of ice, are defined for each category. Velocity is defined for an ice pack, the total ice mass in a grid cell. In
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Chapter 9

Sea icethe vertical direction, both ice and snow have one layer with the heat capacity for sea ice but without heat capacity
for snow. The heat capacity for sea ice is due to brine and is represented by the temperature at the center of the ice.
It is assumed that sea ice has the same energy (temperature) throughout the whole layer. Figure 9.1 and Table 9.1
summarize symbols used in this chapter and their variable names in the source code of MRI.COM.
Table 9.1. Physical quantities used in the sea ice part (cf. Figure 9.1) and their variable names in the source code.
Meaning
Variable name Variable name of averagehIhs

ice thickness
snow thickness

hicen
hsnwn

hiceo
hsnwo

A
AhI

area fraction (compactness)average ice thickness (volume)

aicen
hin

a0iceo
hi

Ahs
T3
T1

average snow thickness (volume)
skin temperature of upper surface
temperature of ice

hsn
tsfcint1icen

hsnw
tsfci
-

T0
T0L
S0
S0L

skin temperature of lower surface
skin temperature of sea surface at open leadsskin salinity of lower surface

t0icent0icens0nt0iceo (under the sea ice)
t0icel (open leads)
s0skin salinity of sea surface at open leads
heat flux on the ice side of the ice bottom

s0nfheatns0l-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

fheat
ftio
ftao

-

QIO
QAO
FT I
FT L
FSI , FSL

salinity flux driving first layerof the ocean model-sfluxi = FSI + FSL-

WAI

fresh water flux driving first layerof the ocean model
fresh water flux due to snow fall

wfluxi= WIO +WAO +WRO +WFR
snowfallWAI

at the upper surface of ice
fresh water flux due to sublimationW

sublim-WIO

at the upper surface of ice
fresh water flux due to freezing and melting
at the bottom of ice

wio

-

WAO
WRO

fresh water flux due to freezing at open leads
fresh water flux due to melting

wao-

wrss, wrsi (snow, ice)

WFR
uI

at the upper surface of ice
fresh water flux due to formation of frazil ice
zonal component of ice pack velocity

-

wrso
uice

vI

meridional component of ice pack velocity

-

vice

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9.2. Thermodynamic processes

short wave

QAI

WAI
T3

snow

hS

QS
T2

Ss = 0.0

S0L
FSL

FTL

QI2

ice

WAO

QAO

T0L

Si = 4.0

T1

hI
QIO

S0

T0
FTI

FSI

T(1), S(1)

WFR

WRO
WIO

Figure 9.1. Meaning of symbols and their locations. The left side is related to heat flux, and the right side is related
to fresh water flux. Sea ice is separated into the part that originated from sea water (thickness: hI and salinity: SI= 4.0 [psu]) and the part that originated from snow (thickness: hs ). The former is further divided into upper andlower halves. Sea ice thus 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 QS from the bottom. The heat flux at the air-ice
interface is QAI and that at the ice-ocean interface is FT I . 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 9.1.

9.2

Thermodynamic processes

In considering thermodynamics, thermal energy of sea ice should be defined. The basis of energy (i.e., zeroenergy) is defined 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(C po T ) + (1 − r)(−LF +C pi T ),(9.3)
where C po and C pi are the specific heats of sea water and sea ice, and LF is the latent heat of melting/freezing. Thus
defined, the energy of sea ice is negative definite. The brine 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, and its
salinity is S = T1 /m, where m defines the freezing temperature as a function of salinity. Hence, the brine fractionof sea ice is r = SI /S = mSI /T1 . As in Mellor and Kantha (1989), the specific heat of snow is not considered.
Although the surface fluxes are positive downward (positive toward the ocean) in the ocean model, the sea ice
part is coded such that fluxes are positive upward. In this section, we assume that the fluxes are positive upward.

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Chapter 9

Sea ice

9.2.1 Formation of new sea ice
a. From a sea water below the freezing point temperature
Sea ice is formed when sea surface temperature is below the freezing point. If the temperature of the first layerof the ocean model is below the freezing point as a function of salinity, the temperature is set to the freezing point(Tfreeze ), and the 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 (hI ) is computed by assuming that the total thermal energy of the first layer of theocean model (whose layer thickness is Δz 1 ) is conserved before and after the sea ice formation:2

ρo Δz1 [C po T ] = ρI hI {r(C po Tfreeze ) + (1 − r)(−LF +C pi Tfreeze )} + ρo (Δz − ρI hI /ρo )(Cpo Tfreeze ),(9.4)

where r = mSI /Tfreeze . Using the above equation, we compute the thickness of the new ice:hI =C po (Tfreeze − T )ρo Δz.
ρI (1 − r) LF + Tfreeze (C po −C pi )(9.5)

With the ice thickness known, the fractional area is determined by the following procedure:= 10[cm] and the fractional area is computed• For a grid cell without sea ice, the first thickness is set to hnewI
new = h and A = 1.
.
If
A
>
1
(i.e.,h>10[cm]),
has A = hI /hnew
I
I
I
I• For a grid cell where sea ice already exists, hI is added to each category and open water.
Note that this operation practically eliminates super-cooling in the ocean interior. Hence, the formation of frazil
ice is not considered in this model.
The medium that contacts the sea ice differs at the upper (air) and lower (sea water) surfaces. Even for an ocean
grid cell with 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 eachinterface and each layer in Figure 9.1.

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(hI ) is give by(9.6)
hI = Ficeberg Δt/ρI ,where Δt is the unit time step. With the ice thickness known, the fractional area is determined by the sameprocedure used for ice created from supercooled water as explained above.

9.2.2

Air-ice interface

a. Heat flux at the upper surface of ice (QAI )The surface heat flux (QAI ) is expressed as follows:
QAI = QSI + QLI − (1 − αI )SW − LW + εI σ (T3 + 273.16)4 .
We will now examine each component.
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122

–

(9.7)

9.2. Thermodynamic processesi) Short wave
The downward short wave radiation is represented by SW. The albedo of sea ice is αI , which is 0.82 for cold
(T3 < −1◦ C) snow, 0.73 for melting snow, and 0.64 for bare ice (while melting).
One might use the more sophisticated albedo scheme of the Los Alamos sea ice model (CICE; Hunke andLipscomb, 2006) by choosing model option CALALBSI. We briefly describe how the incoming short wave radiationis treated by CICE. The downward short wave radiation is treated at each interface as follows:• Among the net absorbed shortwave flux (= (1 − αI )SW), some fraction (i0 ) penetrates into ice and the rest
is absorbed at the surface and used to warm the upper interface. See Table 9.2 for the specific value of i0 .
• The part penetrating into the ice (= (1 − αI )i0 SW ) is attenuated according to Beer’s Law with the bulk
extinction coefficient κi = 1.4m−1 . The attenuated part is used to warm the ice interior.
• The rest is absorbed into the ocean.
The albedos and penetration coefficients of CICE are listed on Table 9.2. The property fsnow is the snow fractionof the upper surface of the ice, which is expressed as follows:
fsnow =hs,
hs + hsnowpatch(9.8)

where hs is the snow thickness and hsnowpatch = 0.02m.
If the ice thickness (hI ) is less than href = 0.5[m], the albedo of thin ice is computed as

αthinice = αo + β (αcoldice − αo ),(9.9)

arctan(ar hI ), ar = 4.0,
arctan(ar href )(9.10)

whereβ=

and αo 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),(9.11)

αmeltsnow= αcoldsnow − γs (T3 + 1.0),(9.12)

where the condition αmeltice > αo is imposed. Using the snow fraction on the surface of the ice fsnow , the total
albedo is computed as

αi = αmeltice (1 − fsnow ) + αmeltsnow fsnow .

(9.13)

The albedos for visible and near infra-red wave lengths are computed separately. If the short wave flux is given
as the sum of all four components (direct and diffuse for visible and near infra-red wave lengths), a constant ratio(visible) : (near infra-red) = 0.575 : 0.425 is assumed, and the total albedo is computed as the weighted average.
ii) Long wave
The downward long wave radiation from the atmosphere is represented by LW in (9.7). The black body radiation from the sea surface is εI σ (T3 + 273.16)4 , where εI is emissivity, and σ is the Stefan-Boltzmann constant.
Hereinafter, we use LW I = LW − εI σ (T3 + 273.16)4 as the net longwave radiation.
iii) Sensible heat flux
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123

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Chapter 9

Sea ice

Table 9.2. Albedo and surface transparency of the albedo scheme of CICE.
near infra-red

visible(> 700nm)(< 700nm)

0.70
0.36

0.98
0.78

reduction rate of albedo for melting snow γs (−1◦ C < T3 < 0, hi < 0.5m)−0.075/◦ C−0.15/◦ C−0.075/◦ C−0.10/◦ C

fraction of transparent short wave flux through the ice surface (i0 )0.0

0.7 × (1.0 − fsnow )albedo for cold snow αcoldsnow (T3 < −1◦ C)
albedo for cold ice αcoldice (T3 < −1◦ C, hi > 0.5m)
reduction rate of albedo for melting ice γi (−1◦ C < T3 < 0, hi < 0.5m)

The sensible heat flux (QSI in (9.7)) is computed using a bulk formula:
QSI = ρaC paCHAI U10 (T3 − TA ),(9.14)

where ρa is the density of air, C pa is the specific heat of air, CHAI is the bulk transfer coefficient at the air-ice
interface (Section 8.6), U10 is the scalar wind speed at 10 [m], TA is the surface air temperature, and T3 is the ice
surface temperature (Figure 9.1).
iv) Latent heat flux
The latent heat flux (QLI in (9.7)) is computed using a bulk formula:
QLI = ρa LsCHAI U10 (qi − qA ),(9.15)

where Ls is the latent heat of sublimation, qi is the saturation humidity at the ice surface temperature T3 , and qA is
the specific humidity of air. Section 9.9.1 details a computing method for qi .
Fresh water loss due to sublimation is computed as
WAI = ρaCHAI U10 (qi − qA )/ρo .

(9.16)

b. Heat flux in the snow (QS )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 ),hs(9.17)

where hs is the thickness of the snow layer and ks is the thermal conductivity of snow.
c. Heat flux in the ice interior (QI2 , QIO )In the upper half of the ice layer, the heat flux is computed as follows:
QI2 =

kI
(T1 − T2 ),hI /2(9.18)

where kI is the thermal conductivity of sea ice. If we neglect heat capacity of snow, QS = QI2 . Using this relation,
the interface temperature is computed as
T2 =

k1
ks
hs T3 + hI /2 T1.
k1
ks
hs + hI /2

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–

(9.19)

9.2. Thermodynamic processes
In the lower half of the ice layer, the heat flux is computed as follows:
QIO =

kI
(T0 − T1 ).
hI /2(9.20)

d. Melting at the upper surface (WRO )The melting rate at the upper surface is computed as follows: First, the surface temperature (T3 ) is computedby equating the fluxes at the upper surface (QAI = QS ). If T3 is lower than the freezing temperature (0.0◦ C forsnow and mSI [◦ C] for sea ice), melting does not occur. If T3 is higher than the freezing temperature, melting
occurs. In this case, T3 is set to the freezing temperature, and the interior heat flux just below the ice surface QS is
recalculated. The imbalance between QAI and QS is used to melt snow or ice. A melting rate is computed as
WRO = (QAI − QS )/(ρo L3 )(9.21)

L3 ≡ [E(T3 , 1) − E(T1 , r1 )](9.22)

where L3 = LF for snow melt andfor ice melt. The brine fraction for sea ice is r1 = mSI /T1 . The temperatures of the melted water are mSI [◦ C] forsea ice and 0 [◦ C] for snow.

e. Notes• All precipitation on sea ice is assumed to be snow.
• Melted water is assumed to run off to the ocean.
• If all pre-existing ice is melted, the residual surface heat flux QAI is added to the ice-ocean flux QIO .

f. Procedure
The solution procedures are basically the same with or without snow. To be exact, the interface fluxes should becomputed 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 by assuming that the fluxes on both sides are the same. Inserting (9.7) and (9.18) into QAI = QI2
with T3 → T3 + δ T3 ,kI
{T1 − (T3 + δ T3 )}hI /2= QLI (T3 + δ T3 ) + QSI (T3 + δ T3 )−(1 − αI )(1 − i0 )SW − LW + εI σ {(T3 + δ T3 ) + 273.16}4 .

(9.23)

By expanding the specific heat, latent heat, and black body radiation in a Taylor series, we have,
kI
∂ QLI
∂ QSI
δ T3 +
δ T3 − (1 − αI )(1 − i0 )SW(T1 − (T3 + δ T3 )) = QLI + QSI +
hI /2
∂ T3
∂ T3
−LWI + 4εI σ (T3 + 273.16)3 δ T3 .

(9.24)

Using this, we compute δ T3 and add it to T3 to obtain a new temperature:δ T3 =

I(T1 − T3 )−QSI − QLI + (1 − αI )(1 − i0 )SW + LWI + hIk/2

∂ QSI
∂ QLI
kI
3
∂ T3 + ∂ T3 + 4εI σ (T3 + 273.16) + hI /2

–

125

–

,

(9.25)

Chapter 9

Sea iceandT3new = T3old + δ T3 ,(9.26)

where

∂ QSI
∂ T3
∂ QLI
∂ T3

= ρaC paCHAI U10 ,= ρa LsCHAI U10

∂ qi
.
∂ T3

(9.27)(9.28)

Note that the dependency of Ls on temperature (Section 9.9.1) is not considered in the partial differentiation withrespect to temperature. The specific form for the partial derivative of specific humidity (∂ qi /∂ T3 ) is presented in
Section 9.9.1.
If the new surface temperature (T3new ) is below the freezing point, melting does not occur. If not, T3new is set tothe freezing temperature (= mSI ), and the heat flux in the ice interior is re-evaluated. The amount of melting (WRO )is obtained using the imbalance:
QAI
QI2
L3
ΔhI

= −QSI − QLI + (1 − αI )(1 − i0 )SW + LWI ,kI=(T1 − T3new ),hI /2= [E(T3new , 1) − E(T1 , r1 )],(QAI − QI2 )Δt== WRO Δt.
ρi L 3(9.29)

If there is snow, the above procedure is performed for the snow surface (T3new = 0[◦ C], L3 = LF ). If all the snow
melts away, the residual heat (Eres ) is used to melt the ice:

9.2.3

Eres

= (QI2 − QAI )Δt − hs ρs LF ,L3

= [E(mSI , 1) − E(T1 , r1 )],ΔhI=−Eres /ρi L3 = WRO Δt.

(9.30)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

∂
∂
E(T1 , r1 ) = QIO − QI2 + [SWsurface − SWbottom ],E(T1 , r1 ) + uI i∂t
∂ xi

(9.31)

where
SWsurface= (1 − αI )i0 SW,(9.32)

SWbottom= (1 − αI )i0 SW × exp(−κi hI ).

(9.33)

The above equation can be solved explicitly without causing serious problems when the time step is not too long.

9.2.4 Ice-ocean interface
Melting and freezing at the ice-ocean interface is computed using heat fluxes at the interface as depicted in
Figure 9.1. The ice-covered area and the open water are treated separately. The solution method slightly differsfrom that of Mellor and Kantha (1989).
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9.2. Thermodynamic processes
In open water, the heat flux on the air side of the air-ocean interface (QAO ) is computed as in Chapter 8,
4
Qall
AO = QSO (T ) + QLO (T ) − (1 − αo )SW − LW + εo σ (T + 273.16) .

(9.34)The temperature at the first level of the ocean model (T ) is used. All the heat and fresh water fluxes are evaluatedusing the temperature and salinity at the first level of the ocean model. By doing so, the equation to compute
melting and freezing rates becomes linear.
Here, short wave radiation is assumed to pass through the skin layer without absorption and is excluded fromthe evaluation of the freezing rate in open water:
QAO = QSO + QLO − LW + εo σ (T + 273.16)4 .

(9.35)

This operation causes the short wave radiation to be absorbed in the ocean interior. In reality, the heat stored in theskin layer in open water is used to melt ice laterally (edge melting). To include this effect, some fraction (Ψ) of thebottom melting (WIO ) may be used for the edge melting. The details are described in the last part of this section.
In the ice-covered area, the heat flux on the ice side of the ice-ocean interface (QIO ) is computed according to(9.20).
Melting and freezing occur due to the imbalance between fluxes above and below the interface:
FTI=

QIO −WIO ρo Lo ,(9.36)FTL= QAO −WAO ρo Lo ,(9.37)

where
Lo ≡ [E(T0 , 1) − E(T1 , r1 )](= LF ).

(9.38)The heat flux that drives the first level of the ocean model is given by
FT = (AQIO + (1 − A)QAO ) −WO ρo Lo ,(9.39)

WO ≡ AWIO + (1 − A)WAO .

(9.40)

where

The flux balance for salt is written as follows:
FSI= WIO (SI − S),(9.41)FSL= WAO (SI − S).

(9.42)

Unlike Mellor and Kantha (1989), the salinity at the first level of the ocean model (S) is used instead of thesalinity at the skin layer (S0 ). By doing so, the equation to solve for S0 becomes linear. 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 of the ocean model.
Note that only fresh water fluxes that are relevant to freezing and melting at the ice-ocean interface are includedin 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 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).

(9.43)

Note that the effect of melting at the upper surface of the ice could also be included in the driving salt flux forthe ocean model:
FS  = −A{WROice (SI − S) −WROsnow S}.
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(9.44)

Chapter 9

Sea iceHowever, fluxes on the oceanic side of the interface (FTI , FTL , FSI , and FSL ) could be obtained as the boundary
conditions (z → 0) for the molecular boundary layer:
FTI /(ρoC po ) = −CTz (T0I − T ),(9.45)FTL /(ρoC po ) = −CTz (T0L − T ),(9.46)andFSI= −CSz (S0I − S),(9.47)FSL= −CSz (S0L − S),(9.48)uτ,−1(Prt k ln(−z/z0 ) + BT )(9.49)

where
CTz =
−1/2

2 + τ 2 )1/4 ρis the friction velocity, k = 0.4 is von Karman’s constant, z0 is the roughness parameter,uτ ≡ (τIOo
IOyx(τIOx , τIOy ) is the stress vector at the ocean-ice interface, and z u 1/2
0 τ
BT = b
Pr2/3 ,
(9.50)
νwith Pr ≡ ν /αt = 12.9. The specific values for other parameters are given in Section 9.9.2.
Parameters related to salinity are
CSz =

uτ
,
−1
(Prt k ln(−z/z0 ) + BS )and
BS = b

 z u 1/2

where Sc ≡ ν /αb = 2432.

0 τνSc2/3 ,

(9.51)(9.52)

Roughness parameter z0 is computed as follows:
lnz0 = Alnz0I + (1 − A)lnz0L ,where
z0I = 0.05

hI
, hIlim = 3.0 [m],hIlim

(9.53)(9.54)andρo u2τ
.
(9.55)
ρa g
The roughness parameter below ice (9.54) is also used for open water in MRI.COM.
The above equations are solved simultaneously to obtain melting and freezing temperatures at the upper surfaceof the ocean under the following constraint:z0L = 0.016

WO = 0T0 = mS0

A=0
A > 0,(9.56)

where m defines the freezing line as a function of salinity.
Solving procedure
We first solve for S0I and S0L using (9.36), (9.41), (9.45), (9.47), (9.37), (9.42), (9.46), and (9.48):
S0I

=

S0L

=

CSz S + (ρoC poCTz T − QIO )(SI − S)/ρo Lo,
CSz + ρ0C poCTz m(SI − S)/ρo Lo
CSz S + (ρoC poCTz T − QAO )(SI − S)/ρo Lo.
CSz + ρoC poCTz m(SI − S)/ρo Lo
–

128

–

(9.57)(9.58)

9.2. Thermodynamic processes
Using S0I and S0L , T0I and T0L are computed from (9.56), FTI and FTL are computed from (9.45) and (9.46), FSI
and FSL are computed from (9.47) and (9.48), and finally WIO and WAO are computed from (9.36) and (9.37). Since
WAO is positive (only freezing is allowed in open water), WAO = 0 when WAO < 0. In this case, S0L = S from (9.42)
and (9.48). Furthermore, FTL = QAO in (9.37) and T0L = T assuming that there is no ice effect in open water iffreezing does not occur. Finally, the surface boundary conditions for heat and salt fluxes are computed from (9.39)
and (9.43).
z → 0:∂T= FT ,∂z∂S−κV= FS ,∂z−κV(9.59)(9.60)

where κV is the vertical diffusivity for the surface level of the ocean model.
If −WIO Δt > hI , 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 theocean. Specifically, WIO is computed by
hI = −ρo
WIO ΔtρI(9.61)and FTI is obtained as
FTI = QIO −WIO ρo Lo .

(9.62)

When −WIO Δt < hI , 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 Δt is used to reduce the thickness holding thefractional area. The new thickness is hI and ΨWIO Δt is used to reduce compactness holding the thickness:A = A(1 + ΨWIO Δt/hI ).

(9.63)

It does not seem that there is a widely accepted parameterization scheme for edge melting. According to Steele(1992), bottom and top melting are dominant processes, and thus Ψ ∼ 0.1 is usually used in MRI.COM.

9.2.5

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:

ρI hI + ρs hs = ρo hI .

(9.64)At the end of the time step, the above equality is checked, and if

ρI hI + ρs hs> hI ,
ρ0(9.65)=hnew
I

ρI hI + ρs hs.
ρo(9.66)

δ hs = −− hI )ρI
(hnew
I
,
ρs

(9.67)

then the new ice thickness is set as:

Since the change in snow thickness isthe new snow thickness is obtained as follows,= hs + δ hs .
hnew
s
– 129

–

(9.68)

Chapter 9

Sea ice

Since the salinity of sea ice is SI = 4.0 [psu], salt is removed from the first layer of the ocean model:
Snew = Sold − SI9.3

δ hs ρs.
Δz1 ρo(9.69)

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 in each 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 (9.1):∂g∂( f g).
=∂t∂ hI(9.70)

In practice, a thickness category is regarded as a Lagrange particle, and the category boundaries are displaced
as a result of thermodynamics. A linear thickness distribution function is assumed within each displaced category,
and ice is remapped into the original categories using these functions.
First, boundaries of thickness categories are displaced. If the ice thickness in category n (hn ) changes from hm
n to(m is the time step index), the growth rate ( fn ) at thickness hn is represented as fn = (hm+1
− hm
hm+1
n
n
n )/Δt. Using
this, the growth rate at the upper category boundary Hn is obtained by linear interpolation:
Fn = fn +fn+1 − fn(Hn − hn ).
hn+1 − hn(9.71)

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 the fractional area is zero on both categories, Fn = 0. The new category boundary after thermodynamics is
obtained as
Hn∗ = Hn + Fn Δt.

(9.72)∗ , H ∗ ] is determined. For simplicNext, the thickness distribution function g within the displaced category [Hn−1n∗and HR = Hn∗ . Function g should satisfy the following equality for fractional area andity, we write HL = Hn−1
volume:

 HR
HL
 HR
HL

gdh

= an ,(9.73)

hgdh=(9.74)

vn .

We adopt a linear function of thickness for g. The thickness space is transformed to η = h−HL , and the thickness
distribution function is written as g = g1 η + g0 . These are substituted into (9.73) and (9.74) to yield

ηR2+ g0 ηR2
η3
η2
g 1 R + g0 R
32
g1

= an ,(9.75)= an ηn ,(9.76)

where ηR = HR − HL and ηn = hn − HL . These are algebraically solved for g0 and g1 as
g0

=

g1

=


6an  2ηR
η
−
n ,
3
ηR2
ηR 
12an 
.
η
−
n
2
ηR3
– 130

–

(9.77)(9.78)

9.3. Remapping in thickness space
The values of the thickness distribution function at category boundaries are given as follows
6an  2ηR
− ηn ,
2
3
ηR

ηR 
6an
g(ηR ) = 2 ηn −.
3
ηRg(0) =(9.79)(9.80)

Equation (9.79) gives g(0) < 0 when the thickness is in the right third of the thickness range or ηn > 2ηR /3.
Equation (9.80) gives g(ηR ) < 0 when the thickness is in the left third of the thickness range or ηn < ηR /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 − HL and g is
set to zero for [HC , HR ]. In this case, ηR = HC − HL in (9.77) and (9.78). When the thickness is within the rightthird of the thickness range, a new left boundary is set at HC = 3hn − 2HR and g is set to zero for [HL , HC ]. In thiscase, ηR = HR − HC and ηn = hn − HC in (9.77) and (9.78).
Finally, we remap ice into the original categories using the above thickness distribution function. If Hn∗ > Hn ,ice is transferred from category n to n + 1. The transferred area Δan and volume Δvn are
Δan =
and
Δvn =

 H∗
n

gdh

(9.81)

hgdh.

(9.82)

Hn

 H∗
n

Hn

If Hn∗ < Hn , ice is transferred from category n + 1 to n. The transferred area Δan and volume Δvn are
Δan =
and
Δvn =

 Hn

gdh

(9.83)

hgdh.

(9.84)

Hn∗

 Hn
Hn∗

Snow and thermal energy are also transferred in proportion to the transferred volume. For example, Δvsn =vsn (Δvin /vin ) for snow and Δein = ein (Δvin /vin ) for thermal energy.
If ice is created in open water, the left boundary of category 1 (H0 ) is moved to F0 Δt, where F0 is the growth rate
in open water. After area and volume are remapped in higher categories, ice area, volume, and energy are added tocategory 1.
If ice is not created in open water, H0 remains zero, but the growth rate at the left boundary of category 1 is set
to F0 = f1 . If F0 < 0, the fractional area of category 1 thinner than Δh0 = −F0 Δt is added to open water area. In
this operation, volume and energy are invariant. The area to be added to open water is
Δa0 =

 Δh0

gdh.

(9.85)

0

The right boundary of the thickest category N (HN ) is a function of its mean thickness hN . When hN is given, HN
is computed as HN = 3hN − 2HN−1 . It is guaranteed that g(h) > 0 for HN−1 < h < HN and g(h) = 0 for HN < h.

–

131

–

Chapter 9

Sea ice

9.4 Dynamics
9.4.1 Momentum equation for ice packThe momentum equation for an ice pack with mass ρI AhI is∂(AhI uI ) − ρI AhI f vI∂t∂
ρI (AhI vI ) + ρI AhI f uI
∂t
ρI

1 ∂h
+ Fμ (σ ) + A(τAIx + τIOx ),hμ ∂ μ1 ∂h= −ρI AhI g+ Fψ (σ ) + A(τAIy + τIOy ),hψ ∂ ψ
= −ρI AhI g(9.86)(9.87)

where (uI , vI ) is the velocity vector, h is the sea surface height, (Fμ , Fψ ) is the ice’s internal stress (which is afunction of internal stress tensor (σ )), and τAI and τIO are stresses exerted by the atmosphere and ocean.

9.4.2

Stresses at top and bottom

The stress at the top is wind stress:

τAI = ca ρa |Ua − uI |[(Ua − uI ) cos θa + k × (Ua − uI ) sin θa ],(9.88)

where Ua is the surface wind vector, ca is the bulk transfer coefficient between air and ice, ρa is the density of air,and θa is the angle between the wind vector and the ice drift vector, which is set to zero.
Stress at the bottom is ocean stress:

τIO = cw ρo |Uw − uI |[(Uw − uI ) cos θo + k × (Uw − uI ) sin θo ],(9.89)

where Uw is the velocity of the first level of the ocean model, cw is the bulk transfer coefficient between the ice andocean, ρo is the density of sea water, and θo is the angle between the ice drift vector and the surface velocity of theocean, which is set to 25◦ in the northern and -25◦ in the southern hemisphere.

9.4.3

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 elasticplastic-viscous (EVP) model by Hunke and Ducowicz (1997, 2002) is adopted for the constitutive law (the relation
between stress and strain rate). 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 highand strain rate is near zero, which makes the explicit integration infeasible. An alternative, the implicit method, isusually adopted, but it is not suitable for parallel computing. The EVP model treats the ice as an elastic media anda 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

η −ζ
P
1
1 ∂ σi j+
σi j +
σkk δi j + δi j = ε̇i j , i, j = 1, 2,
E ∂t
2η
4ηζ
4ζ

(9.90)

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.

– 132

–

9.4. Dynamics
The r.h.s. (ε̇i j ) is the strain rate tensor, expressed in Cartesian coordinate as:ε̇i j =

1  ∂ uI i ∂ uI j 
.
+
2 ∂xj
∂ xi

(9.91)The divergence, tension, and shear of the strain rate are defined as follows:
DD = ε̇11 + ε̇22 , DT = ε̇11 − ε̇22 , DS = 2ε̇12 .

(9.92)The equation for the stress tensor for σ1 = σ11 + σ22 and σ2 = σ11 − σ22 is given by1 ∂ σ 1 σ1
P
+
+
E ∂t
2ζ 2ζ
1 ∂ σ 2 σ2
+
E ∂t
2η
1 ∂ σ12 σ12
+
E ∂t
2η

= DD ,(9.93)= DT ,(9.94)

1
DS .
2(9.95)=

In generalized orthogonal coordinates, divergence, tension, and shear of the strain rate are expressed by
DD

=

DT

=

DS

=

∂ (hψ u) ∂ (hμ v)
1
,
+
hμ hψ
∂μ
∂ψ
hψ ∂  u  hμ ∂  v −,
hμ ∂ μ hψhψ ∂ ψ hμhμ ∂  u  hψ ∂  v +.
hψ ∂ ψ hμhμ ∂ μ hψ(9.96)(9.97)(9.98)The internal stress is obtained as the divergence of the internal stress tensor,
Fμ

=

3
1 2 1 ∂ σ1
1 ∂ (h2ψ σ2 )2 ∂  2
hμ σ12 ,++ 2
2
2 hμ ∂ μhμ hψ ∂ μh μ hψ ∂ ψ(9.99)Fψ=

3
1 2 1 ∂ σ1
1 ∂ (h2μ σ2 )2 ∂  2hψ σ12 .
− 2+
22 hψ ∂ ψhμ hψ ∂ ψh μ hψ ∂ μ(9.100)(9.101)The viscous parameters are obtained from the concentration and velocity as follows:

ζ

=

η

=

Δ

=

P
,
2Δ
P
,
2e2 Δ
D2D +

(9.102)(9.103)
1/2
1 2(DT + D2S ).
2e(9.104)The pressure of the ice is a function of ice concentration and thickness:
P = P∗ AhI e exp[−c∗ (1 − A)],

(9.105)where P∗ is the scaling factor for pressure, c∗ is a parameter that defines the dependency on concentration, and e isthe axis ratio of the elliptic yield curve (e = 2).
The elastic parameter E is given byE=

2Eo ρI AhImin(Δx2 , Δy2 ),Δte2(9.106)

where Eo is a tuning factor that satisfies 0 < Eo < 1, Δte is the time step for ice dynamics, and Δx2 and Δy2 are thezonal and meridional grid widths.
–

133

–

Chapter 9

Sea ice

9.4.4 Boundary conditions
Surface stresses on the ice are exerted for the fractional area of the ice within a grid cell. The ice concentration
is multiplied by 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 non-ice area:νV ∂ U ∂ V 
(1 − A)A(τAOx , τAOy ).
= − (τIOx , τIOy ) +,∂ z ∂ z k=0ρoρo(9.107)Note that (τIOx , τIOy ) is reversed in sign because it is defined by (9.89) as the stress on the ice.

9.4.5

Solution procedure

Given the surface wind vector and the surface velocity of the ocean needed to compute surface stresses, themomentum equations ((9.86) and (9.87)) and the equations for stress tensors ((9.93), (9.94), and (9.95)) 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, stress tensor σ1 is solved for σ1m+1 as follows:
1 σ1m+1 − σ1m σ1m+1P+ m = Dm
+
D.
m
E
Δt
2ζ
2ζ

(9.108)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:um+1 − umIρI AhI IΔt− ρI AhI g= ρI AhI f vm+1I1 ∂h
+ Fμ (σ m+1 ) + AτAIx
hμ ∂ μ(9.109)m+1) cos θo + (Vw − vm+1
) sin θo ],+Acw ρo |Uw − um
I |[(Uw − uII

vm+1 − vm
IρI AhI IΔt=−ρI AhI f um+1
− ρI AhI g
I

1 ∂h+ Fψ (σ m+1 ) + AτAIyhψ ∂ ψ

(9.110)m+1) cos θo − (Uw − um+1
) sin θo ].
+Acw ρo |Uw − um
I |[(Vw − vII

Note that the surface velocity of the ocean (Uw ,Vw ) is constant during the integration. The starting time level ofthe 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,
m+1
m+1
m+1
τIO = cw ρo |Un−1|[(Un−1) cos θo + k × (Un−1) sin θo ].
w − uI
w − uI
w − uI(9.111)

The time step of the ice dynamics is limited by the phase speed of the elastic wave. To damp the elastic waves
during the subcycle, the ice dynamics is subcycled several tens of steps during one ocean model time step.

9.5 Advection
Fractional area, snow volume, ice volume, ice energy, and ice surface temperature (optional; set flg advec tskin
to be .true. in mod seaice cat.F90) of each category are advected. A multidimensional positive definite advection transport algorithm (MPDATA; Smolarkiewicz, 1984) is used.

– 134

–

9.6. Ridging
The advection equation for a property (α ) is given by∂α+ A (α ) = 0,∂t

(9.112)where A is an advection operator defined as
1
A (α ) =
hμ hψ




∂ (hψ uα ) ∂ (hμ vα )
.
+
∂μ
∂ψ

(9.113)The specific representation for α is A for fractional area, Ahs for snow volume, AhI for ice volume, and AhI E
for ice energy.
In MPDATA, (9.112) is first solved to obtain a temporary value 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
,
+

∗
∗
1 1  n+ 1
1
n+ 12 2 ∂ αn+ 12 n+ 12 ∂ α
2|u|Δx − Δt(u
)
v
− Δtu
ũi+ 1 , j =
2
α∗ 2
∂x
2
∂y


1
1
1
∂ u n+ 2 ∂ v n+ 2
1,
(9.114)− Δtun+ 2
+
2
∂x
∂y
,
+
 ∂ α∗ 1∗1
1 1  n+ 1
n+ 12 n+ 12 ∂ α
2 |Δy − Δt(vn+ 2 )2|vvṽi, j+ 1 =
Δtu
−
2
α∗ 2
∂y
2
∂x


n+ 12
n+ 12
1
∂u
∂v
1
.
(9.115)− Δtvn+ 2+2∂x∂yThis 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 isreversed just before advection and returned to a negative value after the advection.

9.6

Ridging

As a result of advection, the sum of the fractional area might exceed one, especially where the velocity field isconvergent. 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 less than 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: aP (h) = b(h)g(h). The weighting function b(h) is
chosen such that the ridging occurs for thin ice:
G(h) 
2 
1− ∗if G(h) < G∗∗GG
= 0 otherwise,b(h) =(9.116)(9.117)

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 (aPn ) is obtained by integrating for a range [Hn−1 , Hn ] as
aPn =


2
Gn−1 − Gn 
.
(G
−
G
)
1
−
n
n−1
G∗
2G∗

(9.118)The property aPn is the fractional contribution from the category n among the total area of ice subject to ridging.
The property Gn is the area summed from category 0 to n. This equation is used for the category that satisfies
–

135

–

Chapter 9

Sea iceGn < G∗ . If Gn−1 < G∗ < Gn , then G∗ is replaced by Gn . If Gn−1 > G∗ , then aPn = 0. If 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 of thickness hn is homogeneously distributed between Hmin = 2hn and Hmax = 2 H ∗ hn after ridging, whereH ∗ = 25 [m]. The thickness ratio before and after the ridging is kn = (Hmin + Hmax )/(2hn ). Therefore, when an
area of category n is reduced by ridging at a rate rn , the area of thicker categories is increased by rn /kn .
Among the new ridges, the fractional area that is distributed in category m is:fmarea =HR − HL,
Hmax − Hmin

(9.119)where HL = max(Hm−1 , Hmin ) and HR = min(Hm , Hmax ). The fractional volume that is distributed in category mis:(HR )2 − (HL )2.
(9.120)fmvol =
(Hmax )2 − (Hmin )2
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 rateof area loss of the ice pack (Rnet ) is given by
Rnet =Cs(Δ − |DD |) − min(DD , 0),2(9.121)

where Cs is the fraction of shear dissipation energy that contributes to ridge building (0.5 is used in MRI.COM),
DD is the divergence, and Δ = D2D + e12 (D2T + D2S )1/2. These strain rates are computed by the dynamics scheme.

The total rate of area loss due to ridging, Rtot = ∑Nn=0 rn , is related to the net rate as follows:N
1
Rnet = aP0 + ∑ aPn 1 −Rtot .
kn
n=1(9.122)Since Rnet is computed from (9.121), Rtot is computed from (9.122). Thus, the area subjected to ridging fromcategory n is computed as arn = rn Δt (rn = aPn Rtot ). The area after ridging is arn /kn and the volume after ridging
is arn hn . Using these, the ice subjected to ridging is first removed from category n, and the ridged ice is thenredistributed into each category.
In practice, we require that arn ≤ an . If A > 1 after ridging, Rnet is adjusted to yield A = 1, and the ridging
procedure is repeated.

9.7 Discretization
9.7.1 Advection (MPDATA)In MPDATA, tracer (α in Figure 9.2) is updated following a three-step procedure.
1. A temporary value is computed using an upstream scheme.
The tracer fluxes at the side boundaries of the unit grid cell where the tracer is defined are:nnFx (αi,n j , αi+1,
j , ui+ 1 , j ) =2

Fy (αi,n j , αi,n j+1 , vni, j+ 1 )
2

=

Δy
,
2
2
2
2
2
Δx
αi,n j (vni, j+ 1 + |vni, j+ 1 |) + αi,n j+1 (vni, j+ 1 − |vni, j+ 1 |)
,
2
2
2
2
2
n
n
n
αi,n j (uni+ 1 , j + |uni+ 1 , j |) + αi+1,
j (ui+ 1 , j − |ui+ 1 , j |)

– 136

–

(9.123)

9.7. Discretization
where


1
ui+ 1 , j+ 1 + ui+ 1 , j− 1 ,(9.124)
2
2
2
2
2
2


1
v 1 1 + vi− 1 , j+ 1 .
vi, j+ 1 =(9.125)2222 i+ 2 , j+ 2
The zonal flux is defined at the closed circle and the meridional flux is defined at the closed square in Figure 9.2.
Using this, the temporary value (α ∗ ) is computed using an upstream scheme:ui+ 1 , j =(αi,∗ j − αi,n j )ΔSi, jΔtnn= Fx (αi−1,
j , αi, j , un+ 12

i− 12 , j

+ Fy (αi,n j−1 , αi,n j , vn) − Fx (αi,n j , αi+1,
j, u

n+ 12

i, j− 12n+ 12

i+ 12 , j

) − Fy (αi,n j , αi,n j+1 , v

)

n+ 12

i, j+ 12).

(9.126)

2. 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
1
1
∂
α
∂
α
1⎣ 1
1
n+
n+
n+
⎦
|u 12 |Δx − Δt(u 12 )2ũi+ 1 , j =−Δtu 12 vn+ 2
i+ 2 , ji+ 2 , ji+ 2 , j2
2 α ∗(2)∂ x i+ 12 , j α ∗(6)∂ y i+ 21 , j
i+ 12 , j1 ∂uṽi, j+ 1
2

i+ 12 , j

∂ v n+ 1

1
n+2,(9.127)− Δtu 12+i+ 2 , j ∂ x
2∂ y i+ 12 , j⎡⎤ ∗ 

∗
1
∂α
∂
α
1⎣ 1
1
n+ 12
n+ 12 2
n+ 12⎦
|v 1 |Δy − Δt(v 1 )=−Δtv 1 un+ 2i, j+ 2i, j+ 2i, j+ 22 α ∗(2)∂ y i, j+ 12 α ∗(6)
∂ x i, j+ 12
i, j+ 12

1 n+ 1− Δtv 2 1
2 i, j+ 2



∂u ∂v
+
∂x ∂y

i, j+ 12

n+ 1

2i, j+ 12,(9.128)where∗(2)
i+ 12 , j=∗(2)i, j+ 12=∗(6)
i+ 12 , j=∗(6)i, j+ 12

=

α
α
α
α

1 ∗
∗
(α + αi+1,j ),2 i, j1 ∗(α + αi,∗ j+1 ),2 i, j
1 ∗
∗
∗
∗
∗
∗
+ αi+1,(αj−1 + 2αi, j + 2αi+1, j + αi, j+1 + αi+1, j+1 ),6 i, j−1
1 ∗
∗
∗
∗
∗
∗
+ αi−1,(αj+1 + 2αi, j + 2αi, j+1 + αi+1, j + αi+1, j+1 ),6 i−1, j ∂ α∗ ∂xi+ 12 , j∂yi+ 12 , j

 ∂ α∗ 



1

vn+ 2



1

un+ 2

∂ α∗ 
∂ y i+ 12 , j∂ α∗ 
∂ x i, j+ 12

=

=

=
=

∗
∗
αi+1,
j − αi, j

Δx∗αi, j+1 − αi,∗ j
Δy

137

–

(9.130)(9.131)(9.132),(9.133),(9.134) ∂ α∗ 
1 n+ 12  ∂ α ∗ 
n+ 12
v 1
+
v
1
i, j− 2
4 i, j+ 2 ∂ y i, j+ 12∂ y i, j− 12 ∂ α∗  ∂ α∗ 1
1n+n++v 2 1+ v 21,i, j+ 2i, j− 2∂ y i+1, j+ 12∂ y i+1, j− 12 ∂ α∗ 
1 n+ 12  ∂ α ∗ 
n+ 1
u 1+ u 12
1
i− 2 , j
4 i+ 2 , j ∂ x i+ 2 , j∂ x i− 12 , j ∂ α∗  ∂ α∗ 
n+ 1
n+ 1
,
+u 12
+ u 12
1
i+ 2 , j+1
i− 2 , j+1∂ x i+ 2 , j+1∂ x i− 12 , j+1–(9.129)(9.135)(9.136)

Chapter 9

Sea ice


∂u ∂v
+
∂x ∂y

n
=


 n
ui+ 1 , j Δyi+ 1 , j − uni− 1 , j Δyi− 1 , j + vni, j+ 1 Δxi, j+ 1 − vni, j− 1 Δxi, j− 1
2

2

2

2

ΔSi, ji, j∂ v n+ 12+∂ x ∂ y i+ 12 , j ∂ u ∂ v n+ 1
2+∂ x ∂ y i, j+ 12

∂u

=
=

2

2

2

2

,

+
,
1  ∂ u ∂ v n+ 21  ∂ u ∂ v n+ 12
,
+
+
+
2
∂ x ∂ y i, j
∂ x ∂ y i+1, j+,1 ∂ u ∂ v n+ 21  ∂ u ∂ v n+ 12
.
+
+
+
2
∂ x ∂ y i, j
∂ x ∂ y i, j+1(9.137)(9.138)(9.139)

3. Update tracer using the anti-diffusive velocity starting from the temporary value.
∗(αi,n+1
j − αi, j )ΔSi, j

Δt

∗
∗
∗
∗
= Fx (αi−1,
j , αi, j , ũi− 1 , j ) − Fx (αi, j , αi+1, j , ũi+ 1 , j )22+ Fy (αi,∗ j−1 , αi,∗ j , ṽi, j− 1 ) − Fy (αi,∗ j , αi,∗ j+1 , ṽi, j+ 1 ).
2
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

Ui+3/2, j+3/2

Ui+1/2, j+3/2

Ui-1/2, j+1/2(9.140)α i+1, j
Ui+3/2, j-1/2

Ui+1/2, j-1/2

Figure 9.2. Position for tracer (α ) and velocity (U). Area and thickness are defined at tracer points. Zonal fluxes
are computed at closed circles and meridional fluxes are computed at closed squares. The budget is computed fora unit grid cell for α .

9.7.2 Momentum equation
Specific forms of discretization for properties related to internal stress are given here.
The strain rate tensor (ε̇ ) and stress tensor (σ ) are defined at tracer points (Figure 9.3).
Components (divergence, tension, and shear) of the strain rate tensor are expressed in a discretized form as
follows:(DD )i, j= Δyi+ 1 , jΔyi− 1 , j
1
2
2(uI i+ 1 , j+ 1 + uI i+ 1 , j− 1 ) −(uI i− 1 , j+ 1 + uI i− 1 , j− 1 )
2
2
2
2
2
2
2
2
Δxi, j Δyi, j22Δxi, j+ 1
Δxi, 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

– 138

–

9.7. Discretization(DT )i, juI i− 1 , j+ 1  Δyi, j− 1  uI i+ 1 , j− 1
uI i− 1 , j− 1 
1 Δyi, j+ 12  uI i+ 12 , j+ 12
2
2
2
2
2
2
2
+
−
−
2 Δxi, j+ 1 Δyi+ 1 , j+ 1 Δyi− 1 , j+ 1
Δxi, j− 1 Δyi+ 1 , j− 1 Δyi− 1 , j− 1

=

2

2

2

2

2

2

2

2

2

2

vI i+ 1 , j− 1  Δxi, j− 1  vI i− 1 , j+ 1
vI i− 1 , j− 1 
1 Δxi+ 12 , j  vI i+ 12 , j+ 12
2
2
2
2
2
2
2
+
,
−
−
−
2 Δyi+ 1 , j Δxi+ 1 , j+ 1 Δxi+ 1 , j− 1
Δyi, j− 1 Δxi− 1 , j+ 1 Δxi− 1 , j− 1
2(DS )i, j

2

2

2

2

2

2

2

2

2

vI i− 1 , j+ 1  Δyi, j− 1  vI i+ 1 , j− 1
vI i− 1 , j− 1 
1 Δyi, j+ 12  vI i+ 12 , j+ 12
2
2
2
2
2
2
2
+
−
−
2 Δxi, j+ 1 Δyi+ 1 , j+ 1 Δyi− 1 , j+ 1
Δxi, j− 1 Δyi+ 1 , j− 1 Δyi− 1 , j− 1

=

2

2

2

2

2

2

2

2

2

2

uI i+ 1 , j− 1  Δxi, j− 1  uI i− 1 , j+ 1
uI i− 1 , j− 1 
1 Δxi+ 12 , j  uI i+ 12 , j+ 12
2
2
2
2
2
2
2
+
.
−
−
+
2 Δyi+ 1 , j Δxi+ 1 , j+ 1 Δxi+ 1 , j− 1
Δyi, j− 1 Δxi− 1 , j+ 1 Δxi− 1 , j− 1
2

2

2

2

2

2

2

2

2

2

The internal stress is defined at velocity points and computed from stress tensor as follows:(Fμ )i+ 1 , j+ 122=

1 2 1  (σ1 )i+1, j+1 + (σ1 )i+1, j − (σ1 )i, j+1 − (σ1 )i, j 
2 2
Δxi+ 1 , j+ 1
2

(9.141)2

2
2
1  Δyi+1, j+ 12 [(σ2 )i+1, j+1 + (σ2 )i+1, j ] − Δyi, j+ 21 [(σ2 )i, j+1 + (σ2 )i, j ] +2
Δy2 1 1 Δxi+ 1 , j+ 1
i+ 2 , j+ 222

 Δx2i+ 1 , j+1 [(σ12 )i+1, j+1 + (σ12 )i, j+1 ] − Δx2i+ 1 , j [(σ12 )i+1, j + (σ12 )i, j ] 22,+
Δx2i+ 1 , j+ 1 Δyi+ 1 , j+ 1
22
22(Fψ )i+ 1 , j+ 122=

1 2 1  (σ1 )i+1, j+1 + (σ1 )i, j+1 − (σ1 )i+1, j − (σ1 )i, j 
2 2
Δyi+ 1 , j+ 1
2

−

2

2
2
1  Δxi+ 1 , j+1 [(σ2 )i+1, j+1 + (σ2 )i, j+1 ] − Δxi+ 1 , j [(σ2 )i+1, j + (σ2 )i, j ] 22

Δx2i+ 1 , j+ 1 Δyi+ 1 , j+ 1

2

2

2

2

2

 Δy2i+1, j+ 1 [(σ12 )i+1, j+1 + (σ12 )i+1, j ] − Δy2i, j+ 1 [(σ12 )i, j+1 + (σ12 )i, j ] 22.
+Δy2i+ 1 , j+ 1 Δxi+ 1 , j+ 1
2
2
2
2

Ui-1/2, j+3/2

Ui+3/2, 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+1, j,εi+1, j

σi, j,εi, j
Ui-1/2, j-1/2

Ui+3/2, j-1/2

Ui+1/2, j-1/2

Figure 9.3. Position of variables used by dynamics scheme.

–

139

–

(9.142)

Chapter 9

Sea ice

9.8 Technical issues
9.8.1 Source codes
The sea ice part consists of the following programs.
ice bulk.F90: computes fluxes at the air-ice interface
ice cat albedo.F90: computes surface albedo and fraction of short wave penetrating into ice
ice cat com.F90: defines variables and sets parameters
ice cat bulk.F90: computes surface fluxes using bulk formulaice 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 restartice ridge.F90: computes ridging
ice time.F90: manages calendar
mod seaice cat.F90: computes thermodynamics and various adjustment processes
The ice model could only be used as a part of the ocean model. Among the subroutines included in the above programs, only si initialize and simain in ice main cat.F90 are called from the ocean model. si initialize
initializes the ice model by reading parameters, creating grid cells, and reading restart, which is called from thesubroutine ogcm ini of ogcm.F90. simain is the main program and calls subroutines of the ice model, which iscalled from the subroutine mkflux of mkflux.F90.
To use the ice part, the ocean model options ICE, SIDYN, ICECAT, should be selected. If SIDYN is not selected,
dynamics is not solved and ice drifts with a third of the surface ocean velocity. By selecting CALALBSI, the albedo
scheme of CICE is used. Otherwise, the constant albedo is used.
Other things to be noted are:
• The ice model uses the forward scheme in time integration. The ice model is not called in the backward partof the Matsuno (Euler backward) scheme.
• The output of history and restart for the category integrated/averaged state is automatically done at the sametime as the ocean model for separate files (Chapter 16).
– 140

–

9.8. Technical issues

9.8.2

Coupling with an atmospheric model

In a coupled mode, 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 andpassed to the ocean-ice component via the coupler scup (Yoshimura and Yukimoto, 2008). To use all the heat
fluxes given by the atmospheric component, fluxes are globally adjusted at each subcycle time step of the oceaniccomponent to account for changes in ice area during the subcycle. See Yukimoto et al. (2010) for details.
In the atmospheric component, temperature in the atmospheric boundary layer and at the ice surface (T3 (tsfcin))are computed along with heat flux in snow layer (QS = QI2 (fheatu)) using ice temperature (T1 (t1icen)), snow
thickness (hs (hsnwn)) and ice thickness (hI (hicen)) given by the oceanic component.
All the informations needed by the oceanic component are received via cgcm scup get a2o in ogcm.F90 at
the beginning of the coupling cycle. The information needed by the ice part is extracted via get fluxi a2oin ice flux.F90. The main part of the ice is solved using surface fluxes and ice surface temperature (optional) from the atmospheric component. The properties needed by the atmospheric component are sent via
cgcm scup put o2a in ogcm.F90 at the end of the coupling cycle.

9.8.3

Job parameters (namelist)

The run-time job parameters (namelist) are listed on Table 9.3.
Table 9.3. namelist for the ice model
variable namegroupdescription

usage

file ice restart in

infli

restart file of average for input

used only in OGCM

file ice restart out temp

outflir

restart file of average for output

used only in OGCM

file ice hist temp

outflihhistory file of average for output

used only in OGCM

irstrtnjobpsi

set initial state

nstepinjobpsitime steps to be proceed

1:read restart, 0: start without ice
same as OGCM if not speci-

the interval of time step by which

fied
0 for no output

int bgticenjobpsiwater budget is written
ibyri
ibmni
ibdyi

njobpsinjobpsinjobpsistart year of this series of integra-

same as OGCM if not speci-

tion

fied

start month of this series of integra-

same as OGCM if not speci-

tion

fied

start day of this series of integration

same as OGCM if not specified

adtdi

njobidyntime step interval for dynamics (inabout a tenth of the baro-

minutes)clinic time step of the ocean
model

hbound(0:ncat)

njobpscat

category boundary

lsicat volchk

njobpscat

flag for checking mass conservation.false. by default

num hint ic

nhsticint

the number of time step intervals of

not exceed 3

history output of categorized ice

–

141

–

Chapter 9

Sea icevariable namegroupdescription

usage

maxnum hist ic

nhsticfile

nwrt hist ic

nhsticfile

maximum histories allowed to be
written to one filethe interval of time step for history

given for each num hint ic(same for below)
the number of time steps or

nhsticfile

outputwestern end for history output-1 for monthly output

imin hist ic
imax hist ic
jmin hist ic
jmax hist ic

nhsticfile
nhsticfile
nhsticfile

eastern end for history output
southern end for history output
northern end for history output

file ice hist ic temp

nhsticfile

core part of the file name for history

file icecat restart in

inflicrestart file name of categorized ice
for input

num rst ic

outflic

the number of files allowed to be
outputmaxnum rst ic

outflic

nwrt rst ic

outflic

the maximum snap shots allowed tobe written to one filethe interval of time step for snap

outflic

shot output
core part of the file name for snap

file icecat restart out temp

the number of time steps or-1 for monthly output

shot
alb ice visible t0
alb ice nearIR t0

njobalbsi
njobalbsi

visible ice albedo for thicker ice
near infrared ice albedo for thicker

0.78
0.36

alb snw visible t0
alb snw nearIR t0

njobalbsi
njobalbsi

ice
visible, cold snow albedo
near infrared, cold snow albedo

0.98
0.70

alb ice visible dec ratio
alb ice nearIR dec ratio

njobalbsi
njobalbsi

visible ice albedo declination rate
near infrared ice albedo declination

0.075 [◦ C−1 ]0.075 [◦ C−1 ]njobalbsi
njobalbsi

ratevisible snow albedo declination rate
near infrared ice albedo declination

0.10 [◦ C−1 ]0.15 [◦ C−1 ]hi refnjobalbsi

rate
the maximum ice thickness to

0.50 [m]atan refnjobalbsi

which connection function is used
the base value of the tangent hyper-

4.0

alb snw visible dec ratio
alb snw nearIR dec ratio

bolic connection function
tsfci t0njobalbsi

the temperature at which ice albedo

0.0 [◦ C]

is equated to that of ocean
tsfci t1

njobalbsi

the temperature at which ice albedo
is started to decline to that of the
ocean

– 142

–

-1.0 [◦ C]

9.8. Technical issues
variable namegroupdescription

usage

fsnow patch

njobalbsi

area fraction of snow on melting
bare ice

0.02

–

143

–

Chapter 9

Sea ice

9.9 Appendix
9.9.1 Saturation water vapor pressure and latent heat
The latent heat of sublimation and the saturation specific humidity is computed according to Appendix 4 ofGill (1982) as follows.
The saturation vapor pressure ew (in units [hPa]) of pure water vapor over a plane water surface is given by
log10 ew (t) = (0.7859 + 0.03477t)/(1 + 0.00412t).

(9.143)



In air, the partial pressure ew of water vapor at saturation is not exactly ew but is given by

ew = fw ew .

(9.144)

fw = 1 + 10−6 p(4.5 + 0.0006t 2 ),(9.145)The value of fw is given by

where p is the pressure (in units [hPa]).
The saturation vapor pressure ei of pure water vapor over ice is given by,

log10 ei (t) = log10 ew (t) + 0.00422t.
(9.146)

The current version does not include this correction. Thus, ei = ew .
The saturation specific humidity q is by definition:
ei /ps = q/(ε + (1 − ε )q),(9.147)

where ε is the molecular weight ratio between water vapor and air:

ε = mw /ma = 18.016/28.966 = 0.62197,(9.148)and ps is sea level pressure (units in [hPa]).
Using this, the saturation specific humidity is given as

q = ε ei /(ps − (1 − ε )ei ).

(9.149)

Ls = 2.839 × 106 − 3.6(T3 + 35)2 Jkg−1 .

(9.150)The latent heat of sublimation is given by

Surface temperature of ice is computed using semi-implicit method, where an expression for the partial derivativeof the saturation specific humidity by temperature is needed as in (9.28).
First, using equation (9.149),



∂ ei
∂ qi
ε ps
=
,
 2
∂t
{ps − (1 − ε )ei } ∂ t(9.151)

where ∂ ei /∂ t is expressed by setting fw = 1 as


∂ ei

= ln 10 · 10g(t) · g (t),∂twhere g(t) = (0.7859 + 0.03477t)/(1 + 0.00412t) and g (t) = ∂ g (t)/∂ t.
–

144

–

(9.152)

9.9. Appendix

9.9.2

Physical constant, parameters

Since the ice part is coded in SI units, constants and parameters are written in SI units.

Thermodynamics
parameter

value

variable name in MRI.COM

Thermal ice conductivity
Thermal snow conductivity
Specific heat of sea water= 2.04 J m−1 s−1 K−1

kI
ks = 0.31 J m−1 s−1 K−1
Cpo = 3990 J kg−1 K−1

CKI
CKS
CP0

Specific heat of air
Specific heat of ice

Cpa = 1004.67 J kg−1 K−1
C pI = 2093 J kg−1 K−1

CPAIR
CPI

Specific heat of snow
Stefan Boltzmann constant
Albedo of open ocean surface

Cps = 0.0 J kg−1 K−1
σ = 5.67 × 10−8 W m−2 K−4
αo = 0.1—
STBL

Albedo of ice
Albedo of snow

αI = 0.6
αs = 0.75
εo = 0.97
εI = 1.0
εs = 1.0
CHAI = 1.5 × 10−3
LF = 3.347 × 105 J kg−1
equation (9.150)
m = −0.0543 K/ppt
n = −0.000759 K m−1
zoI = 0.05hI /3
SI = 4.0 psu
k = 0.4
κH = 1.0 × 103 m2 s−1
North : 3.0, South : 3.0
ν = 1.8 × 10−6 m2 s−1
αt = 1.39 × 10−7 m2 s−1
αb = 6.8 × 10−10 m2 s−1
Prt = 0.85
b = 3.14

Emissivity of ocean surface
Emissivity of ice surface
Emissivity of snow surface
bulk transfer coefficient
Latent heat of fusion
Latent heat of sublimation
constants for fusion phaseequation: T f = mS + nz
Ice roughness parameter
Salinity of sea ice
von Karman’s constantThickness/compactness diffusion of ice
Scaling factor for κH
Seawater kinematic viscosity
Seawater heat diffusivity
Seawater salinity diffusivity
Turbulent Prandtl number
b in eqs (9.50),(9.52)

–

145

–

ALBW
ALBI
ALBS
EEW
EEI
EES
CHAI
ALF
RLTH
XMXM
XNXN
Z0
SI
XK
AKH
FKHDN, FKHDS
ANU
AT
AS
PRT
AB

Chapter 9

Sea iceDynamics
parameter

value

Reference water density
Reference air density
Reference ice density
Reference snow density (ratio between

ρo = 1036 kg m−3
ρa = 1.205 kg m−3
ρI = 900 kg m−3
ρs = 330 kg m−3

ROAIR
RICE

snow and water)e-folding constant for ice pressurec∗ = 20.0

CSTAR

pressure scaling factor
drag coefficient (air-ice)P∗ = 2.75 × 104 N m−2

PRSREF
CDRGAI

drag coefficient (ice-ocean)yield curve axis ratio
scaling factor for Young’s modulus
water turning angle

cw = 5.5 × 10−3
e = 2.0
Eo = 0.25
θo = ±25◦ (positive/negative in the

air turning angle

northern/southern hemisphere)θacavariable name in MRI.COM= 1.5 × 10−3

RO

RDSW

CDRGIWELIPSEYOUNGWIANGL–

References
Gill, A. E., 1982: Atmosphere-Ocean Dynamics, Academic Press, 662pp.
Hibler, W. D., III, 1979: A dynamic thermodynamic sea ice model, J. Phys. Oceanogr., 9, 815–846.
Hunke, E. C, and J. K. Dukowicz, 2002: The Elastic-Viscous-Plastic Sea Ice Dynamics Model in General Orthogonal Curvilinear Coordinates on a Sphere – Incorporation of Metric Terms, Mon. Weather Rev., 130,
1848–1865.
Hunke, E. C, and J. K. Dukowicz, 1997: An Elastic-Viscous-Plastic Model for Sea Ice Dynamics, J. Phys. Oceanogr.,
94, 1849–1867.
Hunke, E. C, and W. H. Lipscomb, 2006: CICE: the Los Alamos Sea Ice Model Documentation and Software
User’s Manual, available at
http://climate.lanl.gov/source/projects/climate/Models/CICE/index.shtml, 59pp.
Mellor, G. L., and L. Kantha, 1989: An Ice-Ocean Coupled Model, J. Geophys. Res., 94, 10,937–10,954.
Smolarkiewicz, P. K., 1984: A fully multidimensional positive definite advection transport algorism with small
implicit diffusion, J. Comput. Phys., 54, 325–362.
Steele, M., 1992: Sea ice melting and floe geometry in a simple ice-ocean model, J. Geohys. Res., 97, 17,729–
17,738.
Yoshimura, H., and S. Yukimoto, 2008: Development of simple couplar (Scup) for earth sysyem modeling,
Pap. Meteor. Geophys., 59, 19-29, doi:10.2467/mripapers.59.19.
Yukimoto, S., and coauthors, 2010: Meteorological Research Institute-Earth System Model v1 (MRI-ESM1),
Technical Reports of the Meteorological Research Institute, No.64, in press.

– 146

–

Chapter 10

Bottom Boundary Layer (BBL)

MRI.COM has the option of adopting a simple bottom boundary layer model (the option BBL). This chapter
describe the formulation and usage of this model.

10.1

General description

In general, z-coordinate models cannot properly reproduce either dense overflows from the Nordic-ScotlandGreenland ridges to the Atlantic oceans or the dense downslope flows from the continental slope around Antarctica.
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 world ocean tend to exhibit a warming bias. To mitigate
these deficiencies, MRI.COM incorporates a simple bottom boundary layer (BBL) model, used in Nakano and
Suginohara (2002). This BBL model lies at the bottom of the normal ocean grid along the bottom topography. The
following components of the overflow/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.

10.2

Grid arrangement

In z-coordinate models, the flow along the bottom topography is expressed as a sequence of horizontal andvertical movements along a staircase-like topography. When the number of vertical grid points representing thebottom topography is roughly the same as that of the horizontal grid points, the downslope flow can be reasonably
represented even in the z-coordinate model (Winton et al., 1998). In general, however, the number of the verticalgrids used in ocean general circulation models is not large enough. Even if the number is large enough, theconcentrations of tracers might be significantly diffused during their movement owing to the horizontal diffusion
with the surrounding waters. The incorporation of BBL remedies these problems.
The BBL grid cells in MRI.COM are arranged as in Figure 10.1. The grid cells in MRI.COM are composed of
U-grid cells, but it is intuitively easier to understand the grid arrangement of BBL as if they are located at T-gridcells. This is because the depth where the pressure gradient is evaluated is defined as the average depth between
the neighboring BBL cells. (In Figure 10.1a, the pressure gradient used to calculate the velocity between the BBLs
with different depths is evaluated at the averaged depth between these BBLs.) Assume that the number of thevertical grids, km, is 50 before the BBL option is applied. When the BBL option is applied, the number of verticalgrids, km, becomes 51 (50 + 1). The master BBL cells are set at the lowest level of each array regardless of theactual depths of the BBL cells. This arrangement is suitable for calculating the horizontal advection between the
BBL cells. In addition, to easily express the interaction between the BBL cells and the inner ocean cells, we place
–

147

–

Chapter 10

Bottom Boundary Layer (BBL)

=⇒

(a)

(b)

Figure 10.1. (a) Arrangement of BBL(T-box). (b) Schematic arrangement that is identical to (a).

a dummy BBL cell in each vertical column just below the bottom grid of the inner cell and copy the temperature,salinity, and velocities from the master BBL cell.

10.3 Pressure gradient terms
(a)
<<

ǹ$

ǹ+

<* Z

7
<* Z

ǹ$

Figure 10.2. Evaluation of horizontal pressure gradient.

Consider the case in which water in the inner grids is uniform, and there is a dense layer along the bottom slope.
From the physical point of view, the horizontal pressure gradient should move the dense water downward along theslope in the non-rotational frame. However, in the original MRI.COM code without the BBL option, the pressure
gradient is zero in the lowermost inner cells along the bottom slope. In the BBL cells, the horizontal pressure
gradient is calculated as follows when the slope is smooth:∂ p 
∂ p 
∂ p ∂H,=+∂ x z=const∂ x z=H(x) ∂ z ∂ x∂ p 
∂H=.
− g ρ |z=H(x)∂x∂x(10.1)(10.2)z=H(x)In the above case, the density of the water in the inner cells is uniform (ρ = ρI ), and there is a dense layer (ρ = ρB )

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10.4. Eddy effects
along the bottom slope. The horizontal pressure gradient becomes∂ [pz=z0 + gH(x)ρI ]∂ p ∂H∂H− g ρB=− g ρB,∂ x z=H(x)
∂x
∂x
∂x

∂ p 
∂H=− g(ρB − ρI )
,

∂x
∂x

(10.3)(10.4)

z=z0

where pz=z0 is the the pressure at z = z0 in Figure 10.2. The second term on the r.h.s. of Eq. (10.4) represents theeffect of the slope. The larger the difference in the density or the steeper the slope of the topography, the larger thehorizontal pressure gradient becomes. When there is no difference in the density between the inner cells and BBL
cells, the pressure gradient is zero regardless of the slope of the bottom topography.

10.4

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 or on the continental shelf around Antarctica.
In the real world, eddy effects create the cross f /h flow, resulting in the overflow/downslope-flows. Jian andGarwood (1996) investigated the three-dimensional features of downslope flows using an eddy-resolving modeland 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 isconsistent with this result. In order to incorporate this effect into the non-eddy resolving models, we apply Rayleighf . In this case, the geostrophic balance isdrag whose coefficient is nearly equal to the Coriolis parameter, α
modified and written as follows:∂p− α u,∂x∂p
fu = −− α v,
∂y

−fv

=

−

(10.5)(10.6)

After some algebra we obtain



α
∂p
− f v − u = − f v = − ,
f
∂x


α
∂p
f u + v = f u = − ,f∂y(10.7)(10.8)

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 is parallel to the topographic contour near the deep water formation area. Thus, incorporating the Rayleighdrag 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 and topography. However it is very difficult to appropriately determine this function in coarse-resolution
models. Accordingly, in the default setting of the BBL option in MRI.COM, the depth range where α = f isarbitrarily set above 2500 m in the northern Atlantic and above 4000 m around Antarctica to represent the observed
dense overflow/downslope-flow. Below those depths, α is set to zero.

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Chapter 10

Bottom Boundary Layer (BBL)

10.5 Usage
In this section, we show how to use the BBL model in the MRI.COM.
• Add the 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 available).
• Set the lowest layer of dz.F90 as the depth of BBL. Fifty to one hundred meters is recommended for thedepth of BBL, which roughly corresponds to the observed thickness of the BBL layer near the Denmark
Strait.
• Add ho4bbl, exnnbbl to file topo after ho4, exnn. These two variables have the same format as ho4, exnn.
Variable ho4bbl corresponds to the thickness of the BBL, and variable exnnbbl corresponds to the numberof BBL layers; only one layer is presently available. Thus exnnbbl should be equal to or less than one.
• Set the area where the Rayleigh drag coefficient is applied. The default setting is 2500m in the North Atlantic
and 4000 m around the Southern Ocean.
In the default setting, the temperature and salinity in the BBL cells are set to those in the cells k=km. When theoption INILEV is used, the temperature and salinity in the BBL cells are set to those in the lowermost inner cells.

Example of configure.in with the BBL option
DEFAULT_OPTIONS="OMIP FREESURFACE UTOPIA ULTIMATE ZQUICKEST ZULTIMATECYCLIC ISOPYCNAL MELYAM HFLUX WFLUX RUNOFF CLMFRC HIST HISTFLUX ICE
SIDYN INILEV BBL"
IMUT=184
JMUT=171
KM=48
KSGM=1
KBBL=1
SLAT0=-84.D0
SLON0=0.D0
DXTDGC=2.D0
DYTDGC=1.D0
ITMSC=0
ITMSCB=0
ISRSTB=NSDAYNPARTA=4
NPLAT=75.D0
NPLON=320.D0



The above example includes BBL in DEFAULT OPTIONS and KBBL = 1.

A sample Fortran program of the topography data follows.

A Fortran program to create topography data with BBL

integer(4) :: ho4(imut,jmut),exnn(imut,jmut)
integer(4) :: ho4bbl(imut,jmut),exnnbbl(imut,jmut)
write(inidt) ho4, exnn
write(inidt) ho4bbl,exnnbbl





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10.6. Usage notes

10.6

Usage notes

10.6.1

Limit of the area where BBL model should be applied

Ideally, the BBL model should be applied to the globe, and its behavior should change due to temperature,salinity, velocity, and topography. In the real configuration, the BBL model is only effective near the abyssal water
formation areas. When used in other areas, such as the near the equator, the BBL model does not improve thetracer and velocity fields. Furthermore, because the BBL model connects the model cells along the topography,
it inevitably induces a water mass exchange between the cold 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 and the 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 typicalσ -layer models. To prevent this, we choose to apply the BBL model in areas where the BBL model is important.

10.6.2 Limits of the BBL
Linear interpolation of the temperature and salinity along an extremely steep slope may cause problems. For thedefault setting of MRI.COM, the BBL model is not applied in such places and isolated grids.

10.6.3 Notes for the program code
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, thetreatment of atexl and aexl is always very confusing. At the bottom cell (k=km), we set atexl=1 and aexl=1
while we set atexl=1 and aexl=0 at the lowest ocean cells. In the program energy.F90, the treatment of atexl=1
in the BBL model needs special care to avoid double counts.

Reference
Jiang, L., and R. W. Garwood Jr., 1996: Three-dimensional simulations of overflows on continental slopes,
J. Phys. Oceanogr., 26, 1214-1233.
Nakano, H., and N. Suginohara, 2002: Effects of bottom boundary layer parameterization on reproducing deep
and bottom waters in a world ocean model, J. Phys. Oceanogr., 32, 1209–1227.
Winton, M., R. Hallberg, and A. Gnanadesikan, 1998: Simulation of density-driven frictional flow in z-coordinate
ocean models, J. Phys. Oceanogr., 28, 2163–2174.

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Chapter 11

Biogeochemical model

There are several options for Biogeochemical models in MRI.COM. These biogeochemical models have beendeveloped for both ocean-only and coupled ocean-atmosphere-vegetation carbon cycle studies. They feature an
explicit representation of a marine ecosystem, which is assumed to be limited by light, temperature, and nutrientsavailability. 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
inorganic carbon-cycle component, the partial pressure of CO2 at the sea surface (pCO2 ) is diagnosed from thevalues of dissolved inorganic carbon (DIC) and Alkalinity (Alk) at the sea surface, which should be calculated in
the ecosystem component. The difference in pCO2 between the atmosphere and ocean determines uptake or releaseof CO2 from the ocean to the atmosphere 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 fairaccuracy. The ecosystem component deals with various biological activities, and gives sources and sinks of thenutrients, DIC, Alk, and dissolved oxygen through these activities. Our knowledge of these activities is far fromcomplete, and they are difficult to estimate even in state-of-the-art models.
There are many biological models and methods for calculating the ecosystem components. One of the simplest
biological models 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, short wave radiation, and nutrients.
A Nutrient-Phytoplankton-Zooplankton-Detritus (NPZD) model is more complex than the above model, but stilla simple biological model. The NPZD model has four prognostic variables (nutrient, phytoplankton, zooplankton,
and detritus). 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 becomesthe 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 computerresources.
To simulate the carbon cycle in the ocean, some biological process should be calculated in the ecosystem component to obtain DIC at the sea surface. However, the carbon cycle component is not always necessary when ourinterests are to simulate 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 their ecosystem components in these studies are quite simple. Biogeochemical models
adopted in MRI.COM are classified in this category. The latter studies usually use complex biological models such

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Chapter 11

Biogeochemical model

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 in hopes of better simulation of carbon cycle.
The carbon cycle component follows the OCMIP protocols (Orr et al., 1999) whose authority is recognized in thecommunity. MRI.COM has several options for the ecosystem component. At present, MRI.COM can incorporate
the Obata and Kitamura model (Obata and Kitamura, 2003) or an NPZD model based on Oschlies (2001), and theycannot be used without the carbon cycle component. The biogeochemical model of MRI.COM is largely based on
Schmittner (2008) when an NPZD model is adopted as an ecosystem component.
Units in MRI.COM are cgs, but in these biogeochemical subroutines, we use MKS units for the sake of future
development. We use mol/m3 for the units of nutrients. The unit of mmol/m3 is used in some models such as
Oschlies (2001). 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 theecosystem component, the governing equations are as follows. When Obata and Kitamura model is used insteadof the NPZD model, the first four biogeochemical compartments (DIC, Alk, PO4 , and O2 ) are used.

∂ [DIC]∂t∂ [Alk]∂t∂ [PO4 ]∂t∂ [O2 ]∂t
∂ [NO3 ]∂t∂ [PhyPl]∂t∂ [ZooPl]∂t∂ [Detri]∂t= A ([DIC]) + D([DIC]) + Sb([DIC]) + Jv([DIC]) + Jg([DIC]),(11.1)= A ([Alk]) + D([Alk]) + Sb([Alk]) + Jv([Alk]),(11.2)A ([PO4 ]) + D([PO4 ]) + Sb([PO4 ]),(11.3)= A ([O2 ]) + D([O2 ]) + Sb([O2 ]) + Jg([O2 ]),(11.4)= A ([NO3 ]) + D([NO3 ]) + Sb([NO3 ]),(11.5)=A ([PhyPl]) + D([PhyPl]) + Sb([PhyPl]),(11.6)=A ([ZooPl]) + D([ZooPl]) + Sb([ZooPl]),(11.7)== A ([Detri]) + D([Detri]) + Sb([Detri]),(11.8)

where A () is advection, D() is diffusion, and Sb() is source minus sink due to the biogeochemical activities. The
square brackets mean dissolved concentration in mol/m3 of the substance within them. The terms represented byJg() and Jv() are the air-sea gas fluxes at the sea surface, and they appear only in the uppermost layer. The termJg() is calculated based on the OCMIP 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
the volume at the surface layers due to evaporation and precipitation.

11.3

Carbon cycle component

To estimate Jg and Jv, we follow the protocol of OCMIP, which is described in detail in Najjar and Orr (1998),
and Orr (1999). The program to calculate them is based on the subroutine downloaded from the OCMIP website.
We have modified this subroutine so that it can be used in the vector oriented calculation of the MRI.COM code.
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11.3. Carbon cycle component

11.3.1Air-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 onlyin the uppermost layer. When these fluxes are expressed as Fg([DIC]) and Fg([O2 ]), Jg([DIC]) and Jg([O2 ]) are
given as follows:
Jg([DIC]) =Jg([O2 ]) =Fg([DIC]),Δz1
Fg([O2 ]),Δz1(11.9)(11.10)

where
2
Fg([DIC]) = KCO
w ∗ ([CO2 ]sat − [CO2 ]surf ),2
Fg([O2 ]) = KOw ∗ ([O2 ]sat − [O2 ]surf ),(11.11)(11.12)[CO2 ]sat= α ∗ pCO2 atm ∗ P/Po ,(11.13)[O2 ]sat=(11.14)C[O2 ]sato ∗ P/Po .

Here a standard gas transfer formulation is adopted. Next, we elaborate on the above equations.

Piston velocity
O2
2
Parameters KCO
w and Kw are the air-sea gas exchange transfer (Piston) velocity and are diagnosed as follows.
2
KCO
w
2
KO
w

2
= (1 − A)[Xconv aU10](ScCO2 /660)−1/2 ,(11.15)2(1 − A)[Xconv aU10](ScO2 /660)−1/2 ,(11.16)=

where• A is the fraction of the sea surface covered with ice,
• U10 is 10m scalar wind speed,• a is the coefficient of 0.337, consistent with a piston velocity in cm/hr, and specified in the OCMIP protocol,• Xconv = 1/(3.6 × 105 ), is a constant factor to convert the piston velocity from [cm/hr] to [m/s],• ScCO2 and ScO2 are the Schmidt numbers for CO2 and O2 . They are computed using the formulation of
Wannikhof (1992) for CO2 and that of Keeling et al. (1998) for O2 .

Computing CO2 and O2 concentrations at the surface
The concentration of CO2 is computed as follows.
• [CO2 ]surf is diagnosed every step from [DIC],[Alk], temperature, and salinity at the surface.
• α C is the solubility of CO2 , which is diagnosed from sea surface temperature and salinity.
• [O2 ]sato is the saturation oxygen concentration before the variations in total pressure are taken into accountand is diagnosed from the sea surface temperature and salinity.
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Chapter 11

Biogeochemical model• pCO2 atm is specified. (In the OCMIP climatology, it is set to 280ppm.)• P is the sea surface pressure in units of [hPa] and Po = 1013.25 [hPa] is the standard surface atmosphericpressure.

Diagnosis of [CO2 ] at the surface
Diagnosis of [CO2 ] at the surface is the most complex of the above calculations and has the heaviest computational burden. 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 concentrationof 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:
2−[DIC] = [CO2 ] + [H2CO3 ] + [HCO−3 ] + [CO3 ]= [CO∗2 ]2−+ [HCO−3 ] + [CO3 ].

(11.17)(11.18)In the OCMIP protocol, the following equations are solved to obtain [CO∗2 ].
The equilibrium expressions for dissociation are:K1 =[H + ][HCO−3]∗[CO2 ]KB =K1P =[H + ][H2 PO−
4][H3 PO4 ][H + ][CO2−3 ],−[HCO3 ]

(11.19)[H + ][B(OH)−4],[B(OH)3 ]

K2P =
KSi =

K2 =

[H + ][HPO2−4 ]−[H2 PO4 ](11.20)K3P =[H + ][PO3−4 ],2−[HPO4 ]

[H + ][SiO(OH)−3],[Si(OH)4 ](11.21)(11.22)

KW = [H + ][OH − ],(11.23)[H + ]F [SO2−
4 ]
KS =
,
−
[HSO4 ]

(11.24)[H + ]F [F − ],[HF](11.25)andKF =where [H+ ] is the hydrogen ion concentration in seawater and [H+ ]F is the free hydrogen ion concentration. Thereis another 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 +
KS KF


ST
and
[H + ]T = [H + ]F 1 +.
KS(11.26)(11.27)

There are three pH scales corresponding to these three hydrogen ion concentrations.
The equilibrium constants Kx 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 , which is referenced to the free pH scale.
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11.3. Carbon cycle component
The total dissolved inorganic carbon, boron, phosphate, silicate, sulfate, and fluoride are expressed as follows:
2−[DIC] = [CO∗2 ] + [HCO−3 ] + [CO3 ],(11.28)[BT ] = [B(OH)3 ] + [B(OH)−4 ],(11.29)2−
3−[PT ] = [H3 PO4 ] + [H2 PO−
4 ] + [HPO4 ] + [PO4 ],(11.30)[SiT ] = [Si(OH)4 ] + [Si(OH)−3 ],(11.31)

2−[ST ] = [HSO−
4 ] + [SO4 ],(11.32)[FT ] = [HF] + [F − ].

(11.33)andAlkalinity used in this calculation is defined as follows:
2−
−
−
2−
3−
−
[Alk] = [HCO−3 ] + 2[CO3 ] + [B(OH)4 ] + [OH ] + [HPO4 ] + 2[PO4 ] + [SiO(OH)3 ](11.34)−[H + ]F − [HSO−4 ] − [HF] − [H3 PO4 ].
These expressions exclude the effect 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 can be solved using the Newton-Raphson method. The concentration [SiT ] is not predicted in the biogeochemical model adopted in MRI.COM but rather is specified as a typical value of 7.68375×10−3 mol/m3 . (Thesensitivity to [SiT ] is much less than that to other variables).

11.3.2

Dilution and concentration effects of evaporation and precipitation on DIC andAlk

The dilution and concentration effects of evaporation and precipitation significantly impact the concentrationsof some chemical species in seawater. This is particularly true for DIC and Alk, which have large backgroundconcentrations compared with their spatial variability. MRI.COM uses a free surface, so the impact of evaporationand precipitation is straightforward to model unless the option SFLUXW or SFLUXR is used. In these options, salinity
flux is diagnosed and applied instead 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)/Δz,(11.35)where S(i, j, 1) and Δz are the salinity and thickness of the uppermost layer. Note that the variable sflux(i, j) is
not the salinity flux but the time change rate of salinity due to the flux even though the spelling brings up the imageof 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),(11.36)Jv(Alk(i, j, 1)) = sflux(i, j)/S(i, j, 1) ∗ Alk(i, j, 1).

(11.37)

Strictly speaking, air-sea fluxes of fresh water impact other species. However, these modifications are not usuallyapplied because their spatial variabilities are significantly greater than those of DIC and Alk,
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Biogeochemical model

In the OCMIP protocol, the global averaged salinity Sg is used instead of S(i, j, 1) in equations(11.36,11.37).
In addition, globally integrated Jv(DIC) and Jv(Alk) are set to 0. In MRI.COM, these modifications are not thedefault 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 theMichaelis-Menten kinetics (Dugdale, 1967), phosphorus in the new production exported to depth (ExprodP) is
parameterized as rL[PO4]2 /([PO4] + k), where r is a proportional factor (r = 0.9[yr−1 ]), L is the insolation normalized by the annual mean insolation on the equator, and k is the half-saturation constant (k = 0.4[molkg−1 ]). 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 (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 theexport production occurs is fixed at 60 m. The vertical distribution of POM and calcite vertical flux below a depthof 100 m is 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 computed from the formula of Weiss (1970). Source and sink terms of Sb()
representing the above processes are as follows.

Sb([DIC]) = 106 ∗ RemiP + SolnCa − 106 ∗ ExprodP(11.38)Sb([Alk]) = 2 ∗ SolnCa + 16 ∗ ExprodP − 16 ∗ RemiP(11.39)Sb([PO4 ]) = RemiP − ExprodP(11.40)Sb([O2 ]) = − 138 ∗ Sb([PO4 ])(11.41)

11.5 NPZD model
The NPZD model used in MRI.COM is constructed on the assumptions that the biological composition ratio isnearly constant (Redfield ratio) and that the concentration of the biology can be estimated by nitrogen or phosphorus. The prognostic variables of nitrogen (NO3 ), phytoplankton (PhyPl), zooplankton (ZooPl), and detritus (Detri)
are normalized in terms of nitrogen 1 mol/m3 . For example, [PhyPl] represents the concentration of phytoplankton estimated in terms of nitrogen in one cubic meter (N mol /m3 ). The increase and decrease of carbon can be
diagnosed by multiplying by Rcn (ratio of C:N).
Source and sink terms Sb() calculated in the NPZD model are as follows. Those for DIC and Alk, Sb(DIC) and

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11.5. NPZD model
Sb(Alk), used for calculating the carbon cycle, are described later in this section.
Sb([PhyPl]) = Priprod − MortP1 − MortP2 − GrP2Z(11.42)

Sb([ZooPl]) = assim ∗ GrP2Z − Excrtn − MortZ(11.43)

Sb([Detri]) = [(1 − assim) ∗ GrP2Z + MortP2 + MortZ] − RemiD − wdetri∂ Detri∂z(11.44)

Sb([NO3 ]) = MortP1 + Excrtn + RmeiD − PriProd(11.45)

Sb([PO4 ]) = R pn ∗ Sb([NO3 ])(11.46)

Sb([O2 ]) = − Ron ∗ Rnp ∗ Sb([PO4 ])(11.47)

There is no input from the atmosphere such as nitrogen fixation in the above equations, so the sum of these fiveequations becomes zero at each grid point except for the term for detritus sinking (−wdetri ∂ Detri∂ z ). The termfor detritus sinking expresses the biological pump, whose role is to remove nutrients from the upper layers andtransport them into the deep ocean where the plankton cannot use the nutrients. When vertically integrated, thesum of each grid is 0 even though this sinking term is included. The nutrients are transported horizontally throughphysical processes such as advection and diffusion.
In general, the nitrate limit is more severe than the phosphate limit 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 elaborateon 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 = φP ∗ [PhyPl]
The conversion of mortality phytoplankton directly into nutrients. This term was introduced by Oschlies(2001) to increase 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, thisis given as G(P) = g ∗ ε ∗ [PhyPl]2 /(g + ε ∗ [PhyPl]2 ). Among the grazing, the ratio assim is used for thegrowth of zooplankton, 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 ).
• MortZ = φZ ∗ [ZooPl]
The conversion from the zooplankton to detritus (mortality of zooplankton).
• RemiD = φD ∗ [Detri]Remineralization of detritus. This is converted to nutrients through the activity of bacteria.

–

159

–

Chapter 11

Biogeochemical model
2TKOCT[2TQFWEVKQP

㧺

2
[

OQTVCNKV[

P

OQ

KQ

TV

GV

CN

ET

KV

GZ

TGOKPGTCNK\CVKQP&

ITC\KPI<GZRQTVOQTVCNKV[

Figure 11.1. 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 inseveral ways. Here we adopt an expression with a minimum function:
J(I, N, P) = min (JI , JN , JP ) ,(11.48)where JI denotes the purely light-limited growth rate, and JN and JP are nutrient-limited growth rates that arefunctions of nitrate or phosphate.
The light-limited growth is calculated as follows:
JI =Jmax α I.
2[Jmax + (α I)2 ]1/2

(11.49)Here, Jmax is the light-saturated growth, which depends on temperature based on Eppley (1972) as
Jmax = a · bcT ,(11.50)where a = 0.6day−1 , b = 1.066, and c = 1(◦ C)−1 . Note that the default values in MRI.COM are based on Schmittner (2008) and differ from these values (see Table 11.1). Equation (11.49) 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(11.51)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.
 z̃I = Iz=0 PAR exp −kw z̃ − kePdz ,(11.52)0where 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 theangle of incidence in water. The angle of incidence θ is a function of the latitude φ and declination δ .
– 160

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11.5. NPZD model
For the nutrient-limited growth rate (JN and JP ), we adopt the Optimal Uptake (OU) equation instead of theclassic Michaelis-Menten (MM) equation. For the classic MM equation, the nitrate-limited growth rate is expressedasJmax N
,(11.53)
KN + N
where KN is a half-saturation constant for NO3 uptake. In contrast, the Optimal Uptake (OU) equation for a nitrateis expressed as follows:
V0 N
JN = JOU =,(11.54)N + 2 VA00 N + VA00
JN = JMM =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 that phytoplankton alters the number of its surface uptake sites (or ion channels), which determines theencounter timescale, versus internal enzymes, which assimilate the nutrients once encountered.
We set parameters VO and AO so that the rates of uptake, JMM and JOU , are equal at N = KN . In addition, we fix
the ratio V0 /A0 = αOU , where αOU is determined from fitting the data. This requires
1
2

αOU
Jmax .
(11.55)V0 = 0.5 1 +KNFinally, we obtain

V0 N.
(11.56)N + 2 αOU N + αOU
We use αOU = 0.19, which is determined from the fitting of log KN vs log N in the wide range of N by Smith et al.
√

JOU =

(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(11.57)
Sb([DIC]) = Sb([PO4 ])Rcp − Sb([CaCO3 ]),Sb([Alk]) = −Sb([NO3 ]) − 2 · Sb([CaCO3 ]).

(11.58)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 is much 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 ]).

(11.59)Following Schmittner et al. (2008), the source term (Pr([CaCO3 ])) of calcium carbonate is determined by the
production of detritus as follows:
Pr([CaCO3 ]) = [(1 − assim) ∗ [GrP2Z] + [MortP2] + [MortZ]] RCaCO3 /POC RC:N ,(11.60)where assim, GrP2Z, MortP2, and MortZ are as described above. The sink term (Di([CaCO3 ])) of calcium carbonate is parameterized as
Di([CaCO3 ]) =

Pr([CaCO3 ])dz ·
–

161

–


d 
exp (−z/DCaCO3 ) ,dz(11.61)

Chapter 11

Biogeochemical model

which expresses an instantaneous sinking with an e-holding depth of DCaCO3 =3500m. In this equation, z is positive
downward. This depth of 3500m was estimated by Yamanaka and Tajika (1996) to reproduce the observed nutrient
profile. This 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 theratio RCaCO3 /POC = 0.035 used by Schmittner et al. (2008), the resultant global mean Rain ratio should be roughly
consistent with the recently estimated range (0.07 to 0.11) based on various observations.

11.6 Usage
There are three options, CARBON, NPZD, and CBNHSTRUN. Options NPZD and CBNHSTRUN require CARBON.
• When option NPZD is not used in configure.in, the simple biological model of Obata and Kitamura (2003)
is applied as the ecosystem component.
– numtrc p=4 should be specified in configure.in. Using a different number causes the program tostop.
– When used in the NEC super computer, the compile option of FFLAGS = −pvctlexpand = 6 should
be used to activate the in-line expansion. If this option is not used, the model works but quite slowly
due to the low efficiency in the vectorization. The configurations in restart and history file are set innamelist inflpt, outfpt, outfph:
namelist /inflpt/ fn_ptrc_in(1:numtrc_p+np_inout)

namelist /outfpt/ fn_ptrc_out (1:numtrc_p+np_inout)
namelist /outfph/ fn_ptrc_hist (1:numtrc_p+np_hist)
np_inout = 2 np_hist = 5 (when NPZD is used, this number is 6)
fn_ptrc_in (1) : restart file for dic(input)(2) : restart file for alk(3) : restart file for po4(input)(input)(4) : restart file for o2(input)(5) : restart file for pco2o(input)(6) : restart file for pco2a(input)fn_ptrc_out (1) : restart file for dic(2) : restart file for alk(output)(output)(3) : restart file for po4(output)(4) : restart file for o2(output)(5) : restart file for pco2o(output)(6) : restart file for pco2a
fn_ptrc_hist(1) : history file for dic(output)(2) : history file for alk(3) : history file for po4(4) : history file for o2(5) : history file for pco2o(6) : history file for pco2a(7) : history file for oaco2
– 162

–

11.6. Usage(8) : history file for gswd

np inout and np hist is automatically set in ogfile.F90.
• When CBNHSTRUN is used, the partial pressure of atmospheric CO2 , pCO2a, should be applied as additional
atmospheric forcing, and the namelist inflpco2a that indicates file name should be specified. The input
intervals and the file size should be the same as other atmospheric forcing files for heat flux (see Chapter
16).
namelist /inflpco2a/ file_pco2a_ref
file_pcof2a_ref: Partial pressure of atmospheric CO2 (ppt)• When CBNHSTRUN is not used in the ocean-only model, the ocean interacts with the one-box model that
contains the uniform partial pressure of atmospheric CO2 . In this case, namelist njobco2io should bespecified as follows.
namelist /njobco2io/ file_atmco2_in, file_atmco2_out
file_atmco2_in : Partial pressure of atmospheric CO2 (ppt) in the one-box model at the start of theintegration
file_atmco2_out: Partial pressure of atmospheric CO2 (ppt) in the one-box model at the end of theintegration• When NPZD is used, an NPZD model is used as the ecosystem component.
– numtrc p=8 should be described in configure.in. When another value is specified, the model stops.
– The compile option related to the in-line expansion should appear as FFLAGS = −pvctlexpand = 10.
– When the model option CHLMA94 is used, the chlorophyll concentration is considered to calculate theshortwave penetration following Morel and Antoine (1994). See section 8.3.2.
– The settings of restart and history files are specified in namelist inflpt, outfpt, outfph as follows:
namelist /inflpt/ fn_ptrc_in(1:numtrc_p+np_inout)

namelist /outfpt/ fn_ptrc_out(1:numtrc_p+np_inout)

namelist /outfph/ fn_ptrc_hist (1:numtrc_p+np_hist)
np_inout = 2
np_hist = 6
fn_ptrc_in (1) : restart file for dic(input)(2) : restart file for alk(input)(3) : restart file for po4(4) : restart file for o2(5) : restart file for no3(input)(input)(input)(6) : restart file for PhyPl(input)(7) : restart file for ZooPl(input)(8) : restart file for Detri(9) : restart file for pco2o(input)(input)(10) : restart file for pco2a(input)fn_ptrc_out (1) : restart file for dic
–

163

–

(output)

Chapter 11

Biogeochemical model(2) : restart file for alk(3) : restart file for po4(4) : restart file for o2(output)(output)(output)(5) : restart file for no3(6) : restart file for PhyPl(output)(output)(7) : restart file for ZooPl(output)(8) : restart file for Detri(output)(9) : restart file for pco2o(output)(10) : restart file for pco2a
fn_ptrc_hist(1) : history file for dic(2) : history file for alk(output)(3) : history file for po4(4) : history file for o2(5) : history file for no3(6) : history file for PhyPl(7) : history file for ZooPl(8) : history file for Detri(9) : history file for pco2o(10) : history file for pco2a(11) : history file for oaco2(12) : history file for eprdc(13) : history file for gswd
(14) : history file for pprdc
The parameters of NPZD are set in namelist nbioNPZD. The default values are based on Schmittner etal. (2008) and listed on Table 11.1. If the parameters of Oschlies (2001) are used, the high nutrient-low
chlorophyl (HNLC) region in the North Pacific is not appropriately expressed. This may be because theparameters 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 thebiological parameter is specified by using the unit [day]. In the model, the time unit is converted to seconds,[s].
namelist /nbioNPZD/ && alphabio,abio,bbio,cbio,dkcbio,dkwbio,rk1bioNO3,rk1bioPO4,&& gbio,epsbio,phiphy,phiphyq,a_npz,phizoo,d_npz,remina, && w_detr,fac_wdetr, && c_mrtn, Rcn, Ron,Rnp, dp_euph, dp_mrtn, dp_eprdc, Rcaco3poc, Dcaco3

11.7 Program structure
ogcm__ini|+-- rdinit
– 164

–

11.7. Program structure|+-- ptrc_init|+-- obgc_init|+-- (obgcinit0)|+-- cbn_readdt|+-- co2calc
ogcm__run|+-- part_1|||+-- cbn_flx

|
|
|

|
|
|

|
|

|
|

|
|
|

|
|
+-- tracer|||+-- part_2||+-- o2flux|+-- co2flux|+-- co2calc|+-- bio_calc||+-- cbn_rewrit|+-- writdt|+-- cbn_writdt

Reference
Dugdale, R. C., 1967: Nutrient limitation in the sea: dynamics, identification, and significance, Limnol. Oceanogr.,
12, 685–695.
Eppley, R., 1972: Temperature and phytoplankton growth in the sea, Fish. Bull., 70, 1063–1085.
Keeling, R. F., B. B. Stephens, R. G. Najjar, S. C. Doney, D. Archer and M. Heimann, 1998: Seasonal variations

–

165

–

Chapter 11

Biogeochemical modelin the atmospheric O2 /N2 ratio in relation to the air-sea exchange of O2 , Global Biogeochem. Cycles, 12,141-164.
Kishi, M. J., H. Motono, M. Kashiwai and A. Tsuda, 2001: An ecological-physical coupled model with ontogenetic vertical migration of zooplankton in the northwestern Pacific, J. Oceanogr., 57, 499-507.
Morel, A., and D. Antoine, 1994: Heating rate within the upper ocean in relation to its bio-optical state, J. Phys. Oceanogr.,
24, 1652-1665.
Najjar, R., J. C. Orr, 1998: Design of OCMIP-2 simulations of chlorofluorocarbons, the solubility pump andcommon biogeochemistry, Internal OCMIP Report, LSCE/CEA Saclay, 25 pp, Gif-sur-Yvette, France.
Obata, A., and Y. Kitamura, 2003: Interannual variability of the sea-air exchange of CO2 from 1961 to 1998 simulated with a global ocean circulation-biogeochemistry model, J. Geophys. Res., 108, 3377,doi:10.1029/2001
JC001088,2003.
Orr, J., R. Najjar, C. Sabine, and F. Joos, 1999: Abiotic-HOWTO, Internal OCMIP Report, LSCE/CEA Saclay,
25 pp, Gif-sur-Yvette, France.
Oschlies, A., 2001: Model-derived estimates of new production: New results point toward lower values, DeepSea Res. II, 48, 2173–2197.
Redfield, A. C., B. H. Ketchum, and F. A. Richards, 1963: The influence of organisms on the composition of sea
water, in The sea, vol. 2, edited by M. N. Hill, pp. 26-77, Wiley-Intersci., New York..
Schmittner, A., A. Oschlies, H. Matthews, and E. Galbraith, 2008: Future changes in climate, ocean circulation,
ecosystems, and biogeochemical cycling simulated for a business-as-usual CO2 emission scenario until year
4000 AD, Global Biogeochem. Cycles, 22, GB1013,doi:10.1029/2007GB002953.
Schmittner, A., A. Oschlies, H. Matthews, and E. Galbraith, 2009: Correction to “Future changes in climate,ocean circulation, ecosystems, and biogeochemical cycling simulated for a business-as-usual CO2 emission
scenario until year 4000 AD”, ,, Global Biogeochem. Cycles, 23. GB3005,doi:10.1029/2009GB003577
Smith, L., Y. Yamanaka, M. Pahlow, and A. Oschlies, 2009: Optimal uptake kinetics: physiological acclimation
explains the pattern of nitrate uptake by phytoplankton in the ocean, Mar., Ecol. Prog. Ser., , in press.
Wanninkhof, R., 1992: Relationship between wind speed and gas exchange over the ocean, J. Geophys. Res., 97,
7373-7382.
Weiss, R. F., 1970: The solubility of nitrogen, oxygen and argon in water and seawater, Deep-Sea Res., 17,
721–735.
Yamanaka, Y., and E. Tajika, 1996: The role of the vertical fluxes of particulate organic matter and calcite
in the oceanic carbon cycle: Studies using an ocean biogeochemical general circulation model, Global
Biogeochem. Cycles, 10, 361-382.

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11.7. Program structure

Table 11.1. Parameters used for the NPZD ecosystem component (NPZD).
description
unit

namelist namealphabio
abio
bbio

:
:
:

cbio

:

Initial slope of P-I curve
Maximum growth rate parameter
Maximum growth rate = a * b ** (c * T)[(W m−2 )−1 day−1 ][day−1 ]default value# 0.1d0# 0.2d0# 1.066d0# 1.d0PARbio:dkcbio:dkwbio:rk1bioNO3 :Photosynthetically active radiation
Light attenuation due to phytoplankton
Light attenuation in the water
Half-saturation constant for NO3 uptake

rk1bioPO4 :alpha_ou :Half-saturation constant for PO4 uptake
Fitting constant for Optical Uptake kinetics[mol m−3 ]# 0.0d0# 0.19d0

gbio

Maximum grazing rate[day−1 ]# 1.575d0# 1.6d6# 0.014d0:[m−1 (mol m−3 )−1 ][m−1 ][mol m−3 ]# 0.43d0# 0.03d3# 0.04d0# 0.7d-3

epsbiophiphy::Prey capture rate
Specific mortality/recycling rate[(mol m−3 )−2 day−1 ]phiphyqa_npz
phizoo:::Quadratic mortality rate
Assimilation efficiency
Quadratic mortality of zooplankton[(mol m−3 )−1 day−1 ]# 0.05d3# 0.925d0

d_npz
remina::Excretion
Remineralization rate[(mol m−3 )−1 day−1 ][day−1 ][day−1 ]# 0.34d3# 0.01d0# 0.048d0[m day−1 ]# 2.0d0# 3.d0w_detr:
fac_wdetr ::::c_mrtn:[s−1 ]

Sinking velocity
Arbitrary parameter for numerical stability.
When the concentration of detritus in the n+1 st levelis higher than fac wdetri times that in the n th level,w detri is set to 0 between n and n+1 level.
Dimensionless scaling factor for Martin et al (1987)# 0.858d0:

Phi(z) = Phi(zo) * (z/dp mrtn)**(-c mrtn)

Rcn
Ron
Rnp

:
:
:

Molar elemental ratio (C/N)
Molar elemental ratio (O2 /N)
Molar elemental ratio (N/P)

dp_euphdp_mrtn::

Maximum depth of euphotic zone
Characteristic depth of martin curve

[m]
[m]

# 150.d0# 400.d0

dp_eprdc:[m]# 126.d0

Rcaco3poc :Dcaco3:The depth where the bio-export is diagnosed
This value should be less than dp mrtn.
CaCO3 over nonphotosynthetical POC production ratio
CaCO3 remineralization e-folding depth[m]# 3500.d0

shwv_intv :Interval for calculating the irradiance and light-limited[min]# 10.d0::# 7.d0# 10.d0# 16.d0

growth rate.
This must be a divisor of the time step for tracer.

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167

–

# 0.05d0

Part IV

Miscellaneous

–

169

–

Chapter 12

Basics of the finite difference method

This chapter describes the basics of finite difference methods for solving differential equations. The general principles of the finite differencing methods are introduced using the diffusion equation as an example in Section 12.1.
Sections 12.2 and 12.3 describe applying finite difference methods of time and space derivatives in differential
equations. Considerations in finite-difference methods for advection-diffusion equations are discussed in Section
12.4. An implicit method for solving the diffusion equation is described in Section 12.5. Durran (1999) treats thebasics of finite difference methods for solving differential equations of advection and diffusion in geophysical fluid
dynamics.

12.1

Diffusion equation

As an example, consider an initial-boundary value problem expressed by a one-dimensional diffusion equation(heat conductive equation),∂T∂ 2T(12.1)=κ 2.
∂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 analyticalsolution is∞2(12.2)T (x,t) = ∑ fm e−κ km t sin(km x),
m=0

where



2 L
mπ
f (x) sin(km x)dx, km =.
(12.3)L 0
L
Next, consider the finite difference method to get the solution numerically. In the finite difference method, grids
fm =are set with a finite increment in space and time, and each term in the equation is evaluated at each grid using
T jn = T (x j ,tn ). For example,nn
T jn+1 − T jn
T j+1
− 2T jn + T j−1,(12.4)=κΔt
Δx2
where Δt = tn+1 − tn and Δx = 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
identical to the original differential equation (12.1) in the limit Δt → 0, Δx → 0 (consistency).
If the initial distribution is assumed to be f (x) = T0 sin k1 x, the solution of the finite difference equation (12.4)
for t = tn is(12.5)
T jn = λ n T0 sin k1 x j ,
where

2κ Δt
(1 − cos k1 Δx).
(12.6)
Δx2
In order to suppress oscillation and divergence of the solution (stability), 0 < λ < 1 is necessary and Δx and Δt

λ = 1−

must be set to satisfy this condition. This solution is identical to the analytical solution in the limit of Δt → 0, Δx →0 (convergence).
– 171

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Chapter 12

Basics of the finite difference method

To summarize, the finite difference method that satisfies consistency, stability, and convergence is the necessary
condition for an accurate solution.

12.2 Finite difference expressions for time derivatives
The following four finite difference expressions are employed for the time derivatives in MRI.COM:

forward :backward :Matsuno :leap-frog :T n+1 − T n= F(T n )Δt
T n+1 − T n= F(T n+1 )Δt
T ∗ n+1 − T n
T n+1 − T n= F(T n ),= F(T ∗ n+1 )ΔtΔt
T n+1 − T n−1= F(T n ).
2Δt(12.7)(12.8)(12.9)(12.10)The scheme used in the previous section is the forward scheme. The forward, backward, and Matsuno schemes
use the values at two time levels and are accurate to O(Δt), while the leap-frog scheme uses three time levels
and is accurate to O(Δt 2 ). Basically, the leap-frog scheme is employed in MRI.COM because of its higher orderaccuracy.
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
T jn = (Ta λan + Tb λbn ) sin k1 x j ,
where



α2 + 4, λb = −

−α −



4κ Δt
(1 − cos k1 Δx)).
(12.12)
(α ≡
2
2
Δx2
Because λb < −1 for arbitrary values of α , the divergent mode with oscillation is always included (computationalmode). In order to avoid this computational mode, the forward scheme is employed for diffusion and viscosity

λa = −−α +(12.11)

α2 + 4

terms in MRI.COM.
When the diffusion and viscosity coefficients are very large as in the surface mixed layer, the time step has to beunusually small for the stability of the forward scheme according to (12.6). In such a case, the backward scheme
is used for vertical diffusion and viscosity (implicit method; see Section 12.5). Though the time integration at eachpoint 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 12.5).
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 (matsuno int of the namelist group njobp). It should be noted that the
Matsuno scheme needs twice as many numerical operations as the forward and leap-frog schemes.

12.3 Finite difference expression for space derivatives
Let us consider a one-dimensional advection equation,

∂T
∂T
= −u
,
∂t
∂x
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–

(12.13)

12.3. Finite difference expression for space derivatives
where u is a constant velocity. The solution is
T (x,t) = T (x − ut, 0).

(12.14)Using the leap-frog scheme for time differencing, the finite difference equation can be written as follows:
T jn+1 − T jn−1
2Δt

= −u

n
n
T j+
1 − T j− 1
2

2

Δx

,

(12.15)nnwhere T j−
1 and T j+ 1 are the values at the left and right (in the same way for up-down and front-rear grid cells)
22
faces of the grid cell for x j , i.e., values at x j− 1 and x j+ 1 . The point x j− 1 is defined as the central point between x j222

and x j−1 . Because the transport of T at the boundary that enters a grid cell is identical to that leaving the adjacentgrid cell, the total T in the whole system is conserved in this finite difference equation.
n
There are several methods to decide T j−
1 using a value at a single or multiple grid points. The following are
two simple and popular formulations,2

upstream finite difference :

n
n
n
n
T j−
1 = T j−1 (u > 0), T j− 1 = T j (u < 0),2

central finite difference :

n
T j−
1 =

(12.16)2n
T j−1+ T jn2

2.

(12.17)

The former is accurate to O(Δx), and the latter is accurate to O(Δx2 ).
In central finite differencing, the expression for (12.15) isT jn+1 − T jn−1
2Δt

= −u

n
n
− T j−1
T j+1
.
2Δx

(12.18)Assuming the solution to be T (x,t) = τ (t)e−ikx ,τ n+1 = τ n−1 + 2iατ n , where α ≡uΔtsin kΔx.
Δx

It is stable (neutral) if |α | ≤ 1. To be stable for any wave number, uΔt 
 Δx  ≤ 1(12.19)(12.20)must be satisfied (CFL condition). However, if τ n = τ 0 e−inΔθ ,Δθ = − sin−1 [μ sin kΔx](where μ ≡

uΔt
).
Δx

(12.21)

Expanding the r.h.s. by a Taylor expansion we obtainΔθ1−μ sin kΔx − (μ sin kΔx)36μ (kΔx)3 μ 3 (kΔx)3−μ kΔx +−66(kΔx)2= −μ kΔx 1 −(1 − μ 2 ) .
6(12.22)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 andminima that do not exist 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,
–

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Chapter 12

Basics of the finite difference method

the “Arakawa method,” which can nearly conserve the enstrophy (squared vorticity) for horizontally non-divergent
flows, is adopted in MRI.COM by using the central difference. This topic is treated in Chapter 5.
Using the upstream finite difference, the finite difference equation (12.15) isT jn+1 − T jn−1n
T jn − T j−1.
Δx(12.23)∂ T uΔx ∂ 2 T+ O(Δx2 ).
+∂x
2 ∂ x2(12.24)2Δt= −uExpanding the r.h.s. by a Taylor expansion we obtain:−u

The second term has the diffusion (heat conductive) form (which disappears in the central finite differencing).
Actually, the initial 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 numerical dispersion and diffusion somewhat in the advection calculation for tracers, but not completely. The gridboundary value is set in QUICK as
n
T j−
1 =
2

n
n
n
n
−T j−2
3T j−1+ 6T j−1+ 3T jn+ 6T jn − T j+1
n=(u > 0), T j−
(u < 0).
1
882(12.25)The 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 13.

12.4 Finite differencing of advection-diffusion equation
According to the above restriction, when advection-diffusion equations (2.14) and (2.15) are 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
2Δt
Sn+1 − Sn−1
2Δt

= −A (T n ) + D(T n−1 )(12.26)= −A (Sn ) + D(Sn−1 ).

(12.27)

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.

12.5 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 7. The time step must be set very small to keep the calculation stable when
viscosity and diffusivity are very high, since the time tendency becomes very large due to the rapid mixing. To
avoid this problem, the implicit 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 isapplied to the temperature equation using the leap-frog scheme. The diffusion term is written separately using the
– 174

–

12.5. Implicit method for vertical diffusion equation(n − 1) step for the horizontal direction and the (n + 1) step for the vertical direction,(T n+1 − T n−1 )/2Δt = −A (T n ) + DH (T n−1 ) + DV (T n+1 ).

(12.28)

Putting all the terms involving T n+1 on the l.h.s.,
T n+1 − 2ΔtDV (T n+1 ) = T n−1 + 2Δt(−A (T n ) + DH (T n−1 ))(12.29)is obtained, which is an algebraic equation for T n+1 . The finite difference form is rewritten specifically as
Tkn+1 − 2Δt


1 
n+1
n+1
κk− 1 (Tk−1− Tkn+1 )/Δzk− 1 − κk+ 1 (Tkn+1 − Tk+1)/Δzk+ 1
2
2
2
2
Δzk

(12.30)= Tkn−1 + 2Δt(−A (Tkn ) + DH (Tkn−1 )).
By puttinga=2Δt κk− 1
2

Δzk Δzk− 1, b = 1 + a + c, c =2

2Δt κk+ 1
2

Δzk Δzk+ 1,(12.31)

2

we get,n+1
n+1+ bTkn+1 − cTk+1= Tkn−1 + 2Δt(−A (Tkn ) + DH (Tkn−1 )).
−aTk−1(12.32)

Setting Fk ≡ −A (Tkn ) + DH (Tkn−1 ), this is expressed in the matrix form as
⎛

b
⎜
⎜ −a
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝

−c
b
−a

⎞⎛
−c
b
..
.

−c
..
.
−a

..

.

b
−a

T1n+1

⎞

⎛

T1n−1 + 2ΔtF1⎞⎟ ⎜ n+1 ⎟ ⎜⎟⎟ ⎜ T2⎟ ⎜ T2n−1 + 2ΔtF2 ⎟⎟⎜⎟ ⎜⎟⎟ ⎜ T n+1 ⎟ ⎜ T n−1 + 2ΔtF
⎟
3
⎟⎜ 3
⎟ ⎜
⎟
3
⎟⎜ . ⎟ ⎜⎟..
⎟⎜ . ⎟ = ⎜⎟.
.
⎟⎜ . ⎟ ⎜⎟⎟ ⎜ n+1 ⎟ ⎜ n−1
⎟
⎜
⎜
⎟
⎟
⎟
−c
⎟ ⎜ TKM−2 ⎟ ⎜ TKM−2 + 2ΔtFKM−2 ⎟⎟ ⎜ n+1 ⎟ ⎜ n−1⎟
b −c ⎠ ⎝ TKM−1 ⎠ ⎝ TKM−1 + 2ΔtFKM−1 ⎠
n−1
n+1
−a b
TKM
TKM
+ 2ΔtFKM(12.33)The l.h.s. has the form of the tri-diagonal matrix.

12.5.1A solution of tri-diagonal matrix

In general, simultaneous linear equations for n variables with tri-diagonal matrix coefficients
⎛

B1

⎜
⎜ A2
⎜
⎜
⎜
⎜
⎜
⎝

⎞⎛

C1
B2
..
.

C2
..
.
An−1

..

.

Bn−1
An

X1

⎞

⎛

D1

⎟⎜
⎟ ⎜
⎟
⎟ ⎜ X2 ⎟ ⎜ D2 ⎟⎟⎜⎟ ⎜⎟⎟ ⎜ .. ⎟ ⎜ .. ⎟⎟⎜ . ⎟ = ⎜ . ⎟
⎟⎜
⎟ ⎜
⎟
⎟⎜
⎟ ⎜
⎟
Cn−1 ⎠ ⎝ Xn−1 ⎠ ⎝ Dn−1 ⎠
Bn
Xn
Dn

are solved using the Thomas method, which is modified from LU decomposition,

–

⎞

175

–

(12.34)

Chapter 12

Basics of the finite difference method

P1= C1 /B1(12.35)Q1(12.36)Xn= D1 /B1
Ck=
Bk − Ak Pk−1
Dk − Ak Qk−1=
Bk − Ak Pk−1
= Qn

Xk

= Qk − Pk Xk+1

Pk
Qk

(2 ≤ k ≤ n − 1)

(12.37)(2 ≤ k ≤ n)

(12.38)(12.39)(1 ≤ k ≤ n − 1).

(12.40)

References
Durran, D. R., 1999: Numerical methods for wave equations in geophysical fluid dynamics, Springer-Verlag,
465pp.

– 176–

Chapter 13

Tracer advection schemes

The default tracer advection scheme of MRI.COM is the Quadratic Upstream Interpolation for Convective Kinematics (QUICK; Leonard, 1979) as described in Chapter 6. Other options with higher accuracy are described inthis chapter.
Section 13.1 describes the QUICK with Estimated Streaming Terms (QUICKEST; Leonard, 1979) for verticaladvection (option QUICKEST). Section 13.2 describes the Uniformly Third-Order Polynomial Interpolation Algorithm (UTOPIA; Leonard et al., 1993) for horizontal advection (option UTOPIA). The UTOPIA scheme is a
two-dimensional generalization of the QUICKEST scheme. For these schemes, a flux limiter that prevents unrealistic extrema should be used (Leonard et al., 1994). A flux limiter for the QUICKEST scheme is used when option
ZULTIMATE is specified. A flux limiter for the UTOPIA scheme is used when option ULTIMATE is specified.
The above schemes seek to improve the accuracy by refining the finite-difference expression at the cell faces.
There is another approach that seeks to improve the accuracy by considering the distribution within the cell. The
second order moment (SOM; Prather, 1986) scheme takes this approach and is available in MRI.COM through theoption SOMADVEC (Section 13.3).

13.1

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

(13.1)

where w is a constant. Although the velocities are not constants in the real three dimensional ocean, we assume aconstant velocity 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) and bottom (k + 1) faces of the verticalcells. The following relation holds for the vertical grid spacings:
Δzk =Δzk+ 1 + Δzk− 1
2

2

2

.

(13.2)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

–

177

–

Chapter 13

Tracer advection schemes

second order interpolation are calculated first. A Taylor expansion of T about point zk gives

Tk− 3
2

Tk− 1
2

Tk+ 1
2

Tk+ 3
2

= c0 + c1= c0 + c1= c0 − c1Δzk− 3
2+ Δzk− 1

Δzk− 1Δz2k− 1

2

+ c2

2

+ c2

2

2

2

4



Δzk− 32

2

2+ Δzk− 12+ O(Δz3 ),+ O(Δz3 ),(13.4)

Δz2k+ 1

Δzk+ 12+ c2+ O(Δz3 ),
2
4



2
Δzk+ 3Δzk+ 32
2= c0 − c1+ O(Δz3 ).
+ Δzk+ 1 + c2
+ Δzk+ 1
2
2
2
2
2(13.3)(13.5)(13.6)

Coefficients c0 , c1 , and c2 can be solved using three of the four equations (13.3), (13.4), (13.5), and (13.6). The
three upstream-side equations are chosen. When w > 0 (w < 0), equations (13.4), (13.5), and (13.6) ((13.3), (13.4),and (13.5)) are used. The solution is as follows.
c0

=

c1

=

c2

Tk− 1 Δzk+ 1 + Tk+ 1 Δzk− 1
2

2

2

2

2Δzk

−

Δzk+ 1 Δzk− 12

2

4

Δzk− 1 − Δzk+ 1
2
2
−
c2 ,
Δzk
2
⎧
 T −T
T 1 −T 3
k− 21
k+ 21
k+ 2
k+ 2
⎪
1
⎪
− Δz
⎨ Δzk +Δzk+1
Δzk+1
 T −Tk=
T 1 −T 1
3
1
k−
k−
k−
k+ 2
⎪
1
2
2
2
⎪
−
⎩ Δz +Δz
Δz
Δz

c2 ,

Tk− 1 − Tk+ 1
2

2

k−1

k

k−1

k

(13.7)(13.8)(w > 0),(13.9)(w < 0).

Next, equation (13.1) is integrated over one time step and one grid cell.
 t n+1

dt

tn

 zk



zk+1

dz

The r.h.s. of (13.10) can be written as
−

t
∂T
=−
∂t
tn

 t n+1
tn

n+1

dt

 zk
zk+1

dz

∂
(wT ).
∂zdt(wu Tu − wl Tl ),(13.10)(13.11)

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

dtTl =

 0−wl Δt[cn0 + cn1 ξ + cn2 ξ 2 + O(Δz3 )]

dξ
.
wl

(13.12)Thus expression (13.11) becomes−Δt(wu Tun − wl Tln ) + O(Δz3 wΔt),(13.13)

where
Tln

=
=

1
wl Δt

 0

(cn0 + cn1 ξ + cn2 ξ 2 )d ξ
−wl Δtcncn
cn0 − 1 wl Δt + 2 w2l Δt 2 .
2

3(13.14)Using up to the second order terms of a Taylor expansion, the l.h.s. of (13.10) can be written as follows:⎡⎤
 t n+1  zk
Δz2k+ 1
∂T
n+1
n
n+1
n
3
2
dt
dz
(Tzz 1 − Tzz k+ 1 ) + O(Δz )⎦ ,(13.15)= Δzk ⎣Tk+ 1 − Tk+ 1 +
k+ 2
∂t
24
2
2
2
tn
zk+1
– 178

–

13.2. UTOPIA for horizontal advection
where− Tzz nk+ 1
Tzz n+1
k+ 12
2

∂ Tzz n= Δt+ O(Δt 2 )∂ t k+ 1
2
,n
+
∂2 ∂
+ O(Δt 2 )= −Δt 2
(uT )
1
∂z ∂z
k+
2

∂
= −Δt (wTzz )nk+ 1 + O(Δt 2 )∂z2nwu Tzz u − wl Tzz nl+ O(wΔtΔx) + O(Δt 2 ).
= −ΔtΔzk+ 1(13.16)2

The expression for the r.h.s. of (13.15) becomes⎡⎤Δz2k+ 1 w T n − u T n
u zz ul zz ln
2
ΔtΔzk ⎣T n+11 − Tk++ O(Δz3 )⎦ + O(wΔtΔx3 ) + O(Δz3 Δt 2 ).
1 −
k+ 2
24
Δzk+ 1
2(13.17)

2

Based on (13.13) and (13.17), the discretized forecasting equation is expressed as follows:⎡⎤
Δz2k+ 1Δt
2
⎣wu Tun − wl T n −Tkn+1 = Tkn −(wu Tzz nu − wl Tzz nl )⎦ + O(α Δz3 ) + O(Δz2 Δt 2 ),lΔzk+ 1
24(13.18)

2

where

α

≡

Tzz nl

=

wΔt
< 1,
Δz
2c2 + O(Δz).

(13.19)(13.20)

The accuracy of equation (13.18) is max(O(Δz3 ), O(Δz2 Δt 2 )).

13.2

UTOPIA for horizontal advection

The Uniformly Third Order Polynomial Interpolation Algorithm (UTOPIA; Leonard et al., 1993) is an advection
scheme that can be regarded as a multi-dimensional version of QUICKEST. In MRI.COM, horizontally twodimensional 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 boarderlines of thevelocity cells into four boxes with (almost) identical area.
Consider an equation of advection:

∂T
1 ∂
1 ∂
(hψ uT ) +(hμ vT ) = 0.
+∂thμ hψ ∂ μhμ hψ ∂ ψ

(13.21)

Integrated over a tracer cell and for one time step, ψL +ΔψL /2=dψ

 μL +ΔμL /2

d μ (χ n+1 − χ n )ψL −ΔψL /2
μL −ΔμL /2n n−Δt(ur Tr Δyr − unl Tln Δyl + vnu Tun Δxu − vnd Tdn Δxd ),(13.22)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 (13.22), the secondorder interpolation of χ is used to integrate the terms. The Taylor expansion of χ about L is given as follows (seeFigure 13.1 for the label of the point):

χ = χL + a10 (μ − μL ) + a20 (μ − μL )2 + a01 (ψ − ψL ) + a02 (ψ − ψL )2 + a11 (μ − μL )(ψ − ψL ).
– 179

–

(13.23)

Chapter 13

Tracer advection schemes

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

NW

N

W

L

E

SW

S

SE

WW

u

r

SS
l

d

Figure 13.1. Labels of tracer grid points (upper case characters) and faces (lower case characters).

χE=

χL + a10 Δμr + a20 Δμr2 ,(13.24)χW=

χL − a10 Δμl + a20 Δμl2 ,(13.25)

χN=

χL + a01 Δψu + a02 Δψu2 ,(13.26)=

χL − a01 Δψd + a02 Δψd2 ,(13.27)

χS
whereΔψL + ΔψN,
2
ΔψL + ΔψS,Δψd ≡2
Δ μL + Δ μE
Δ μr ≡,
2
ΔμL + ΔμW
Δ μl ≡.
2
Using these known values, the following parameters are obtained,χ E − χL
χL − χW
Δ μl
+ Δμr
Δ μr
Δ μl
,
a10 =
Δμr + Δμl
χE − χL χL − χW
−
Δ μr
Δ μl
a20 =
,
Δ μr + Δ μl
χ N − χL
χL − χ S
Δψd
+ Δψu
Δψ u
Δψd
a01 =
,
Δψu + Δψd
χN − χL χ L − χ S
−
Δψu
Δψd
a02 =
.
Δψu + Δψd
Δ ψu≡Substituting (13.23) into the l.h.s. of (13.22) yields


2
2
Δ
Δ
μ
ψ
L n+1
− an20 ) +− an02 ) .
ΔμL ΔψL χLn+1 − χLn + L (an+1
(a
12 20
12 02
– 180

–

(13.28)(13.29)(13.30)(13.31)(13.32)(13.33)(13.34)(13.35)(13.36)

13.2. UTOPIA for horizontal advection
Using equation (13.21), the following approximation is allowed:,+hψ r unr Tμnμ r − hψ l unl Tμnμ l hμ u vnu Tμnμ u − hμ d vnd Tμnμ d
n+1
n
,
+
a20 − a20 = −Δt
Δ μL
ΔψL
,
+
n
n n
n
n n
hμ u vnu Tψψhψ r unr Tψψ
r − hψ l ul Tψψ lu − hμ d vd Tψψ d
n
,
+
−
a
=
−Δt
an+1
02
02
Δ μL
ΔψL

(13.37)(13.38)where Tμnμ r is the value of the second order derivative at the right face r, whose expression is similar to that of c20
described later. Therefore, under a suitable approximation,
TLn+1 = TLn −Δt n ˜n(u T Δyr − unl T˜ln Δyl + vnu T˜un Δxu − vnd T˜dn Δxd ),ΔSL r r(13.39)

where
T˜ln
T˜dn

ΔμL2 nΔψL2 nTμ μ l −T ,2424 ψψ l
2
2
Δμ
ΔψL n
= Tdn − L Tμnμ d −.
T2424 ψψ d= Tln −

(13.40)(13.41)

Next, the expressions for Tln and Tdn are required. The term Tln is the average over the hatched area of Figure
13.2, and the values of T n are given as the second order interpolation about l of Figure 13.1. Similar operations
will be used to obtain the expression for Tdn .

W

L

vlΔt

ulΔt

Figure 13.2. 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 − ψL ) + c02 (ψ − ψL )2 + c11 (μ − μl )(ψ − ψL ),(13.42)

T |d =d00 + d10 (μ − μl ) + d20 (μ − μl ) + d01 (ψ − ψL ) + d02 (ψ − ψL ) + d11 (μ − μl )(ψ − ψL ).

(13.43)

n

2

2

–

181

–

Chapter 13

Tracer advection schemes

The T n values at eight points around l are,
Δ μL
Δμ 2
+ c20 L ,
2
4
ΔμW
ΔμW2
c00 − c10
+ c20
,
2
4




Δ μE
Δ μE 2+ c20 ΔμL +c00 + c10 ΔμL +
,
2
2




ΔμWW
ΔμWW 2+ c20 ΔμW +c00 − c10 ΔμW +
,
2
2
Δ μL
TLn + c01 Δψu + c02 Δψu2 + c11
Δψu ,
2
Δ μL
TLn − c01 Δψd + c02 Δψd2 − c11
Δ ψd ,2ΔμW
TWn + c01 Δψu + c02 Δψu2 − c11
Δψu ,
2
ΔμW
TWn − c01 Δψd + c02 Δψd2 + c11
Δψ d .
2TLn= c00 + c10(13.44)TWn=(13.45)

TEn

=

n
TWW

=

TNn

=

TSn

=

n
TNW

=

n
TSW

=

(13.46)(13.47)(13.48)(13.49)(13.50)(13.51)

To obtain all six coefficients, six of these equations (points) are used. The equations are chosen according to thefollowing flow direction.
unl > 0,vnl > 0⇒ L, W, WW, S, NW, SW(13.52)

unl < 0,vnl > 0⇒ L, W, E, N, S, SW(13.53)unl > 0,vnl < 0⇒ L, W, WW, N, NW, SW(13.54)

unl < 0,vnl < 0⇒ L, W, E, N, S, NW(13.55)

From equations (13.44) and (13.45),
c00

=

c10

=

ΔμL ΔμWΔμW TLn + ΔμL TWn
− c20
,
2Δμl
4
ΔμL − ΔμW
TLn − TWn
− c20
.
Δμl
2

(13.56)(13.57)

When unl > 0, from (13.44) and (13.47),

c20

=

n
TLn − TWn TWn − TWW
−
Δ μl
Δμll
,
Δμl + Δμll
ΔμW + ΔμWW
where Δμll ≡.
2(13.58)Using equations (13.50) and (13.51),nn− TWn TWn − TSW
TNW
−
Δψu
Δψd
c02 =
.
Δψu + Δψd(13.59)When unl < 0, from (13.45) and (13.46),
TEn − TLn TLn − TWn
−
Δ μr
Δ μl
c20 =
.
Δμr + Δμl
–

182

–

(13.60)13.2. UTOPIA for horizontal advection
Using equations (13.48) and (13.49),

TNn − TLn TLn − TSn
−
Δψu
Δψd
c02 =
.
Δψu + Δψd(13.61)

When vnl > 0, from (13.49) and (13.51),
c01

=

c11

=

n
)
ΔμW (TLn − TSn ) + ΔμL (TWn − TSW+ c02 Δψd ,2Δμl Δψdn− TWn − TSn + TLn
TSW.
Δμl Δψd(13.62)(13.63)

When vnl < 0, from (13.48) and (13.50),
c01

=

c11

=

n
ΔμW (TNn − TLn ) + ΔμL (TNW
− TWn )
− c02 Δψu ,2Δμl Δψun+ TWn
TNn − TLn − TNW.
Δ μl Δ ψ u(13.64)(13.65)

Next, using equation (13.43), the T n values at eight points around d are
TLn

=

TSn

=

TNn

=

n
TSS

=

TEn

=

TWn

=

n
TSE

=

n
TSW

=

ΔψL
Δψ 2
+ d02 L ,
2
4
ΔψS
Δψ 2
d00 − d01
+ d02 S ,
2
4




ΔψN
ΔψN 2
+ d02 ΔψL +d00 + d01 ΔψL +
,
2
2




ΔψSS
ΔψSS 2+ d02 ΔψS +d00 − d01 ΔψS +
,
2
2
Δψ L
TLn + d10 Δμr + d20 Δμr2 + d11
Δ μr ,2ΔψL
TLn − d10 Δμl + d20 Δμl2 − d11
Δμl ,
2
Δψ S
TSn + d10 Δμr + d20 Δμr2 − d11
Δ μr ,2ΔψSTSn − d10 Δμl + d20 Δμl2 + d11
Δ μl .
2d00 + d01

(13.66)(13.67)(13.68)(13.69)(13.70)(13.71)(13.72)(13.73)

From equations (13.66) and (13.67),
d00

=

d01

=

ΔψL ΔψSΔψS TLn + ΔψL TSn
,
− d02
2Δψd
4
ΔψL − ΔψS
TLn − TSn
.
− d02
Δψd
2

(13.74)(13.75)

When vnd > 0, from (13.66) and (13.69),

d02

From (13.72) and (13.73),=

nTLn − TSn TSn − TSS
−
Δψd
Δψdd
,
Δψd + Δψdd
ΔψS + ΔψSS
where Δψdd ≡.
2nn− TSn TSn − TSW
TSE
−
Δμr
Δ μl
.
d20 =
Δ μr + Δ μl

–

183

–

(13.76)(13.77)

Chapter 13

Tracer advection schemes

When vnd < 0, from (13.67) and (13.68),
TNn − TL TLn − TSn
−
Δψu
Δψd
d02 =
.
Δψu + Δψd
From (13.70) and (13.71),(13.78)TEn − TLn TLn − TWn
−
Δ μr
Δ μl
.
d20 =
Δμr + Δμl(13.79)When und > 0, from (13.71) and (13.73),
d10

=

d11

=

n
ΔψS (TLn − TWn ) + ΔψL (TSn − TSW)+ d20 Δμl ,2Δψd Δμln
TLn − TSn − TWn + TSW.
Δψd Δμl(13.80)(13.81)

When und < 0, from (13.70) and (13.72),
d10

=

d11

=

n
− TSn )ΔψS (TEn − TLn ) + ΔψL (TSE
− d20 Δμr ,2Δψd Δμrn− TLn + TSn
TEn − TSE.
Δψd Δμr(13.82)(13.83)The value of Tln is the average of T n over the hatched area of Figure 13.2. Defining

ξln =we haveTln=+aξln ΔtΔψL

unl
,
hμ l

+ ψ +Δψ /2  μ ψL −ΔψL /2

LLψL −ΔψL /2lμl −ξln Δt

ηln =

vnl
,
hψ l

(13.84)T nd μ dψ⎤n μl + ξln (ψ −ψL +ΔψL /2)ηlψL −ΔψL /2−ηln Δt μl −ξln Δt{T n (ψ ) − T n (ψ + ΔψL )}d μ d ψ ⎦

,
+
1 n
1
1∗ n 22= c00 − ηl Δt c01 +Δψ + (ηl Δt) c02
2
12 L 31
1
1− ξln Δt c10 + (ξln Δt)2 c20 + † ξln ηln Δt 2 c11 .
23
3(13.85)This is the result for unl > 0 and vnl > 0. The result is the same independent of the sign of unl and vnl .
Similarly,
Tdn

=

1
1
d00 − ηdn Δt d01 + (ηdn Δt)2 d02
2
,
+ 3
1 n
1
1
1
− ξd Δt d10 +ΔμL2 + ‡ (ξdn Δt)2 d20 + § ξdn ηdn Δt 2 d11 ,
2
12
3
3

where

ξdn =

und
,
hμ d

ηdn =

∗ 1 in CCSR model2† 1 in CCSR model4‡ 1 in CCSR model2§ 1 in CCSR model
4

– 184

–

vnd
.
hψ d

(13.86)(13.87)

13.3. Second Order Moment (SOM) scheme
Therefore,
T˜ln



1 n 2 ΔμL2
1 n
c20
= c00 − ξl Δt c10 + (ξl Δt) −
2
3
12

T˜dn

1
1
1
− ηln Δt c01 + (ηln Δt)2 c02 + ξln ηln Δt 2 c11 ,
2
3 
3

1 n 2 ΔψL2
1 nd02
= d00 − ηd Δt d01 + (ηd Δt) −
2
3
12
1
1
1
− ξdn Δt d10 + (ξdn Δt)2 d20 + ξdn ηdn Δt 2 d11 .
2
3
3(13.88)(13.89)Finally, we describe how to derive the boundary conditions. Since the face values of the tracers are calculated
through the second order interpolation, the value of a tracer at a point over land is sometimes necessary. For thatcase, 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,and either NW or SW may be land. When N or S is land, the land-sea boundary runs at the center of L in thezonal 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 by second order interpolation using the values at grids N, L, and S. Thus, we set(TNn − TLn )Δψd2 = (TSn − TLn )Δψu2 .

(13.90)

When NW or SW is a land grid, the following should be assumed.
nn− TWn )Δψu2(TNW− TWn )Δψd2 = (TSW

(13.91)n(TWW− TWn )Δμl2 = (TLn − TWn )Δμll2 .

(13.92)(TEn − TLn )Δμl2 = (TWn − TLn )Δμr2 .

(13.93)

When WW is a land grid,

When E is a land grid,

Similar boundary conditions are specified for face d.

13.3 Second Order Moment (SOM) scheme
13.3.1 Outline
The Second Order Moment (SOM) advection scheme by Prather (1986) seeks to improve the accuracy bytreating the tracer distribution within a grid cell, unlike the scheme that aims to calculate the tracer flux atthe boundary of grid cells with 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 + ax x + axx x2 + ay y + ayy y2 + az z + azz z2 + axy xy + ayz yz + azx zx.

(13.94)

Prather (1986) expressed the above as a sum of orthogonal functions Ki (x, y, z);f (x, y, z) = m0 K0 + mx Kx + mxx Kxx + my Ky + myy Kyy + mz Kz + mzz Kzz + mxy Kxy + myz Kyz + mzx Kzx ,
– 185

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(13.95)

Chapter 13

Tracer advection schemes

where the orthogonal functions are given as follows:
= 1,

K0

Kx (x) = x − X/2,
Kxx (x) = x2 − Xx + X 2 /6,
Ky (y) = y −Y /2,
Kyy (y) = y2 −Y y +Y 2 /6,
Kz (z) = z − Z/2,

(13.96)Kzz (z) = z − Zz + Z /6,
2

2

Kxy (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),
and

Ki K j dV = 0 (i = j).

The constants for normalization are decided using






Kx2 dV = V X 2 /12,
Ky2 dV = VY 2 /12,
Kz2 dV = V Z 2 /12,2
Kxy
dV = V X 2Y 2 /144,







(13.97)2Kxx
dV = V X 4 /180,
2
Kyy
dV = VY 4 /180,

Kzz2 dV = V Z 4 /180,

2Kyz
dV = VY 2 Z 2 /144,



2
Kzx
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)

Sxx= (30/X 2 )Sy= (6/Y )Syy= (30/Y 2 )Sz= (6/Z)

Szz= (30/Z 2 )Sxy= (36/XY )Syz= (36/Y Z)

Szx= (36/ZX)f (x, y, z)Kx (x)dV = mxV X/2,

f (x, y, z)Kxx (x)dV = mxxV X 2 /6,

f (x, y, z)Ky (y)dV = myVY /2,

f (x, y, z)Kyy (y)dV = myyVY 2 /6,

f (x, y, z)Kz (z)dV = mzV Z/2,







(13.98)f (x, y, z)Kzz (x)dV = mzzV Z 2 /6,
f (x, y, z)Kxy (x, y)dV = mxyV XY /4,
f (x, y, z)Kyz (y, z)dV = myzVY Z/4,
f (x, y, z)Kzx (z, x)dV = mzxV ZX/4.

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
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13.3. Second Order Moment (SOM) scheme
describe the change of each moment caused by an advection procedure in one direction (x) in a two dimensional
plane (xy) 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,

(13.99)is removed from the cell and added to the adjacent cell on the right during time interval t. This part is expressedusing superscript R. The remaining part,0 ≤ x ≤ X − ct, 0 ≤ y ≤ Y, 0 ≤ z ≤ Z,

(13.100)

is expressed by superscript L. New orthogonal functions Ki R (Ki L ) 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:
K0 L = K0 R = 1,
Kx L = x − (X − ct)/2, Kx R = x − (2X − ct)/2,
Kxx L = x2 − (X − ct)x + (X − ct)2 /6,
Kxx R = x2 − (2X − ct)x + (X − ct)X + (ct)2 /6,
Ky L = Ky R = y −Y /2,(13.101)

Kyy L = Kyy R = y2 −Y y +Y 2 /6,
Kxy L = [x − (X − ct)/2](y −Y /2),
Kxy R = [x − (2X − ct)/2](y −Y /2).
The basic quantities for calculating the moments are
m0 R

=

R
m0 + K̄xR mx + K̄xx
mxx ,

mx R

=

mx + 2K̄xR mxx ,

mxx R

=

mxx ,

my R

=

my + 2K̄xR mxy ,

myy R

=

myy ,

mxy R

=

mxy ,

(13.102)where K̄ is the average of the new orthogonal function:
K̄xL = −ct/2, K̄xR = (X − ct)/2,
LR= ct(2ct − X)/6, K̄xx
= (X − ct)(X − 2ct)/6.
K̄xx

The moments in the right part to be removed are expressed as follows:
S0 R= α [S0 + (1 − α )Sx + (1 − α )(1 − 2α )Sxx ],Sx R= α 2 [Sx + 3(1 − α )Sxx ],Sxx R= α 3 Sxx ,Sy R= α [Sy + (1 − α )Sxy ],R= α Syy ,Sxy R= α 2 Sxy ,Syy(13.103)– 187–

Chapter 13

Tracer advection schemes

where α = α R = ct/X = V R /V . The moments in the remaining part are expressed as follows:
S0 L
Sx

L

Sxx L
Sy

L

= (1 − α )[S0 − α Sx − α (1 − 2α )Sxx ],= (1 − α )2 (Sx − 3α Sxx ),= (1 − α )3 Sxx ,= (1 − α )(Sy − α Sxy ),Syy L= (1 − α )Syy ,L= (1 − α )2 Sxy .

Sxy(13.104)

As the final step of the procedure, the orthogonal functions and moments transported from the adjacent celland 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 = S0 R + S0 L ,Sx = α Sx R + (1 − α )Sx L + 3[(1 − α )S0 R − α S0 L ],Sxx = α 2 Sxx R + (1 − α )2 Sxx L + 5{α (1 − α )(Sx R − Sx L ) + (1 − 2α )[(1 − α )S0 R − α S0 L ]},Sy = Sy R + Sy L ,Syy = Syy R + Syy L ,Sxy = α Sxy R + (1 − α )Sxy L + 3[(1 − α )Sy R − α Sy L ],where

α = α R = V R /(V R +V L ).

(13.105)

The original moments are conserved through these operations.

Flux limiter
Some limiters are necessary to guarantee that the tracer is positive (negative) definite. 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 )],





Sxx

Sxy 

(13.106)(13.107)= min[2S0 − |Sx |/3, max(|Sx | − S0 , Sxx )],(13.108)= min[+S0 , max(−S0 , Sxy )].

(13.109)

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

13.3.2 Calculating SOM advection in MRI.COM
It should be noted that the coordinate system is not Cartesian in ocean models. Since the coordinate systemcovers a spherical surface, the x direction in a grid cell is not identical to that in the adjacent cell, for instance.
–

188

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13.3. Second Order Moment (SOM) scheme
Thus, the exact conservation of moments cannot be realized. In addition, a tracer-cell including solid earth (seafloor 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 seawaterin the non-cuboid grid cell, and the zeroth 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 thesubroutine cont). Following Prather (1986), the procedures in three directions are executed in order, not simultaneously. 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 procedure in the SOM scheme does not calculate the flux of the
tracer across the boundary of grid cells, unlike other advection schemes. The change in the tracer value caused by
SOM advection is estimated in the subroutine tracer and added to the variable trcal directly.

NAMELIST
When the SOM advection scheme is used, the following namelist (njobsom) is required (see Table 13.1) on
execution.

name

Table 13.1. Variables defined in namelist njobsom
explanation
string or value

file som infile som out
limiter(numtrc)

base name of the restart file to be input
base name of the restart file to be output
flag to set limits of the moments{file som in} i n *{file som out} i n *.true. / .false.

lrstinsom
lrstoutsom

flag to read the restart file for the initial stateflag to write the restart file for the final state.true. / .false.
.true. / .false.

flag to monitor the conservation of the moments.true. / .false.
lsommonitor* i={x, xx, y, yy, z, zz, xy, yz, zx}, n={01, 02, ..., numtrc}; tracer number

References
Leonard, B. P., 1979: A stable and accurate convective modeling procedure based upon quadratic upstream
interpolation, J. Comput. Methods Appl. Mech. Eng., 19, 59-98.
Leonard, B. P., M. K. MacVean, and A. P. Lock, 1993: Positivity-Preserving Numerical Schemes for Multidimensional Advection, NASA Tech. Memo., 106055, ICOMP-93-05, 62pp.
Leonard, B. P., M. K. MacVean, and A. P. Lock, 1994: The flux integral method for multidimensional convection
and diffusion, NASA Tech. Memo., 106679, ICOMP-94-13, 27pp.
Prather, M. J., 1986: Numerical advection by conservation of second-order moments, J. Geophys. Res., 91, 66716681.

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Chapter 14

Generalized orthogonal curvilinear
coordinate grids

This chapter introduces general orthogonal curvilinear coordinates and presents related calculus.

14.1

OutlineAn ocean model does not have any problem concerning the South Pole because it does not calculate aroundthe 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 in a 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 thezonal grid intervals in low latitudes and the Arctic region are extremely different, the arguments about accuracies ofnumerical schemes and the parameters 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 islandby changing the model’s horizontal grid system. The MRI.COM scheme adopts the latter approach, which isoutlined in this section.
Because the MRI.COM code is written based on the generalized orthogonal coordinate system, the geographic
latitude (φ ) and longitude (λ ) are not of a great concern for the calculus in the model. However, it is necessary toknow the land and sea distribution, sea depth, scale factor, and the Coriolis parameter given as a function of λ andφ at every grid point of the model prior to the calculation. We describe the method of generating the orthogonal
coordinate grid system of the model in Section 14.2. Using a conformal transformation in the general sense, thefunctions 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 the geographic coordinates,
it is necessary to prepare tables for converting the surface atmospheric temperature, the wind stress, and so on. To
convert a vector 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 14.3. 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 theatmosphere should be equal to the total flux that the atmosphere gives to the ocean. The method for conserving
the total flux is explored in Section 14.4. The vector operation in a generalized orthogonal coordinate system isconcisely described in Section 14.5.

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Chapter 14

Generalized orthogonal curvilinear coordinate grids

14.2 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
conformal transformation 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 14.1). Moreover, if SN is assumed to be a complex plane, various conformal transformations can be
defined on it. Therefore, applying (i) the polar stereographic projection, (ii) a conformal transformation on thecomplex 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 sphere can be obtained (Bentzen et al., 1999).

Figure 14.1. Schematic illustration of a Polar stereographic projection (a conformal transformation in the general
sense between the sphere and SN ).

The functions μ (λ , φ ) and ψ (λ , φ ) are obtained by the following procedure:
1. From a point (λ , φ ) on the sphere to a point z on SN (polar stereographic projection). Defining colatitude

φ  = π /2 − φ ,z = tan

φ
2



eiλ ,

(14.1)

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 mapping MC on SN :

ζ = MC (z).

–

192

–

(14.2)

14.2. Generation of orthogonal coordinate system using conformal mapping
3. From a point ζ on SN to a point (μ , ψ ) on the sphere (reverse polar stereographic projection).

μ= arg(ζ ),(14.3)

= 2 arctan |ζ |,(14.4)

ψ= π /2 − ψ  .

(14.5)

ψ

Functions λ (μ , ψ ) and φ (μ , ψ ) are obtained by reversing the above procedure.
Defining ψ  = π /2 − ψ , ψei μ ,
ζ = tan2(14.6)

z = MC−1 (ζ ),(14.7)

λ= arg(z),(14.8)

= 2 arctan |z|,

and

φ

φ

(14.9)

= π /2 − φ .

(14.10)Thus, when a model coordinate grid point, (μ0 + Δμ × i, ψ0 + Δψ × j) 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) =(z − a)(b − c),(c − a)(b − z)where the three complex numbers a, b, and c expressed by
 
 
φ
φ
a = tan a eiλa , b = tan b eiλb ,22(14.11)c = tanφc
2eiλc ,(14.12)

correspond to the three geographic coordinate grid points (λa , φa ), (λb , φb ), and (λc , φc ), which are mapped tothe 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 (λa , φa ) at 75◦ N and 40◦ W. If
the option TRIPOLAR or JOT 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 (λa , φa ) and (λb , φb ), which are model
parameters (NPLON, NPLAT, SPLON, and SPLAT in degree) that should be specified in configure.in.
When option TRIPOLAR is specified, the parameters are set to φa = φb = 64◦ N, λa = 80◦ E, and λb = 100◦ W.
The transformed grids are used for the region north of 64◦ N, and the geographic coordinates are used for theregion south of 64◦ N. This tripolar coordinate system can express the Arctic Sea with a higher resolution than theSouthern Ocean. The adoption of geographical coordinates south of 64◦ N enables us to do the assimilation andanalysis with relative ease (Figure 14.2).
When option JOT is specified, the Joukowski conversion is used as a conformal transformation on SN . That is,ψ 2(14.13)z = MC−1 (ζ ) = ζ + 0 eiμ0 .
ζThis Joukowski conversion maps the area outside the circle with a radius of ψ0 centered at the origin of the ζ -planeto the whole domain of the z-plane, and rotates it by μ0 . The left panel of Figure 14.2 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
– 193

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Chapter 14

Generalized orthogonal curvilinear coordinate grids

Figure 14.2. Model coordinate grid arrangement in the Arctic sea. Left: Grid system made through the Joukowski
conversion (JOT). Right: Combination of the coordinates made through the linear fraction conversion and conventional geographic coordinate (TRIPOLAR).
spacing in this coordinate system, the singular points can be put at various positions. For instance, the singularpoint on the North American side can be put on the Labrador peninsula or in Greenland.
Functions λ (μ , ψ ) and φ (μ , ψ ) are defined as subroutine mp2lp, and functions μ (λ , φ ) and ψ (λ , φ ) are definedas 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 made before starting the main integration of model.

14.3 Rotation of vector
A vector expressed in geographic coordinates (λ , φ ) should be rotated when observed from model coordinates(μ , ψ ).
First, we set
z = f (ζ ),z = x + iy,ζ = u + iv,(14.14)

∂x
∂y
∂y
∂x
+i
=
−i .
(14.15)∂u∂u∂v∂v
At a certain point z0 = f (ζ0 ), 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 v0f  (ζ )=Then (see Figure 14.3)

θ = arg( f  (ζ0 )).

(14.16)

Assuming λ = arg(z) and μ = arg(ζ ), at point ζ0 the straight line (v = v0 ) meets the straight line (μ = μ0 ) atangle −μ0 and at point z0 the meridian (λ = λ0 ) meets the curve (v = v0 ) at angle λ0 − θ . The meridian (λ = λ0 )

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14.4. Mapping a quantity from geographic coordinates to transformed coordinates

Figure 14.3. A meridian in geographical coordinate (λ , φ ) (left) and a meridian in model coordinate (μ , ψ ) (right).
in λ -φ coordinates meets the line (μ = μ0 ) in μ -ψ coordinates at angle α given as follows::α= − μ 0 + λ0 − θ= λ0 − μ0 − arg( f  (ζ0 )).

(14.17)

Subroutine rot mp2lp defined in trnsfrm. ∗ .F90 returns (cos α , sin α ) at a specified grid point of the model.
A wind stress vector (τx , τy ) in geographical coordinates should appear in the model ocean described in the μ -ψcoordinate system as (τx cos α − τy sin α , τx sin α + τy cos α ).

14.4 Mapping a quantity from geographic coordinates to transformed coordinates
We consider a method to receive a quantity GI,J given at the geographic coordinate grids (I, J) as the quantity
HM,N at the model coordinate grids (M, N) (Figure 14.4). 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.
Grids (I, J) and (M, N) are suitably subdivided into finer grids (i, j) and (m, n). We call these filter grids. A
quantity Gi, j is assumed to be homogeneously distributed in the geographic filter grids (i, j) covered by grid (I, J),Gi, j = GI,J .

Assume the quantity at a model filter grid Hm,nis equal to that at the nearest geographic filter grid,Hm,n= Gi(m,n), j(m,n) .

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Chapter 14

Generalized orthogonal curvilinear coordinate grids

Figure 14.4. Grids (I, J) and (M, N) subdivided into finer grids (i, j) and (m, n).

The quantity at model grid (M, N) is obtained as the area-weighted average:
HM,N =

1
,∑ AH  m,n Hm,nAH M,N m,n(14.18)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,
HM,N = ∑ w(M, N, I, J)GI,J .

(14.19)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 (14.19) 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 theocean through evaporation and sublimation. Therefore, the global sea surface height rises or descends during
years of integration. It is not very important to pursue complete conservation of fresh-water under this condition.
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14.5. Vector operation and differentiation in a general orthogonal coordinate system
In such a case, the flux at a model grid point can be prepared beforehand using equation (14.19), to avoid thetime-consuming flux conversions in the model calculation.

14.5

Vector operation and differentiation in a general orthogonal coordinate system

To formulate the model equations, we have to know the vector operation and differentiation in general orthogonal
coordinates. Some basic formulae used in formulating primitive equations are presented here.
The line element vector δ x at a certain point (μ , ψ , r) in an arbitrary general orthogonal coordinate system isexpressed asδ x = hμ δ μ eμ + hψ δ ψ eψ + hr δ rer ,(14.20)where basis vectors eμ , eψ , and er are mutually orthogonal unit vectors, and hμ , hψ , and hr are scale factors.
Defining
eψ ∂
eμ ∂
er ∂
+
+
∇=
,
(14.21)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(14.22)and the divergence of vector A = Aμ eμ + Aψ eψ + Ar er is,+∂ (hψ hr Aμ ) ∂ (hr hμ Aψ ) ∂ (hμ hψ Ar )
1
+
+
∇·A =
.
hμ hψ hr∂μ∂ψ∂r
The r component of curlA is(14.23)+,∂ (hψ Aψ ) ∂ (hμ Aμ )
1
.
−
hμ hψ
∂μ
∂ψ

(14.24)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
a · ∇Aμ +

Aψ
hμ hψ


aμ

∂ hμ
∂ hψ
− aψ
∂ψ
∂μ


+

Ar
hr hμ


aμ

∂ hμ
∂ hr
− ar
∂r
∂μ


.

(14.25)The second and third terms are so-called ‘metric’ terms in the equation of motion in the spherical coordinates.
These expressions in spherical coordinates (λ , φ , r) are shown next. Defining longitude λ , latitude φ , and radiusof the earth r, scale factors are hλ = r cos φ , hφ = r, and hr = 1.
Velocity vector v is
v = ueλ + veφ + wer ,(14.26)

where eλ , eφ , and er are the eastward, northward, and upward unit vectors, respectively, and (u, v, w) = (r cos φ λ̇ , rφ̇ , ṙ).
The gradient of scalar function A(λ , φ , r) is,∇A =∂Aeλ ∂ A eφ ∂ A++ er,r cos φ ∂ λr ∂φ∂r(14.27)

∇=

eφ ∂
∂
∂
eλ
+
+ er .
r cos φ ∂ λr ∂φ∂r(14.28)

where

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Chapter 14

Generalized orthogonal curvilinear coordinate grids

For vector A = Aλ eλ + Aφ eφ + Ar er ,divergence is
∇·A =

+
,
∂ Aλ ∂ (cos φ Aφ )∂ (r2 Ar )
1
+ 2
+
.
r cos φ ∂ λ∂φr ∂r(14.29)and the r component of curlA is+,∂ Aφ ∂ (cos φ Aλ )1.
−[curlA]r =r cos φ ∂ λ∂φ(14.30)Aφ aλ tan φ Ar aλ+.
rr(14.31)The λ component of (a · ∇)A is[(a · ∇)A]λ = a · ∇Aλ −The Coriolis force in a generalized orthogonal coordinates (μ , ψ , r) system is given as2Ω × v = (2Ωψ w − 2Ωr v)eμ + (2Ωr u − 2Ωμ w)eψ + (2Ωμ v − 2Ωψ u)er ,(14.32)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 , fψ = 2Ωψ , and f = fr = 2Ωr in Chapter 2. The rotation vector of the Earth is(Ωλ , Ωφ , Ωr ) = (0, Ω cos φ , Ω sin φ ) in the geographic coordinate (λ , φ , r) system.

References
Bentzen, M., G. Evensen, H. Drange, and A. D. Jenkins, 1999: Coordinate transformation on a sphere using
conformal mapping, Mon. Weather Rev., 127, 2733-2740.

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Chapter 15

Nesting

In MRI.COM, a high-resolution regional model could be embedded in a low-resolution model using a nesting
method. Only one-way nesting from a low- to a high-resolution model is implemented. This chapter briefly
describes the nesting method employed by MRI.COM, its program structure, and how to construct a set of lowand high-resolution models in nesting.

15.1

FeatureIn a set of nested grid models, a high-resolution model is embedded in a low-resolution model, and values at theside boundary of the high-resolution model are given by the low-resolution model. The side boundary data could
be exchanged both off-line and on-line. In off-line mode, the data needed to calculate side boundary values are
output to files by first running the low-resolution model, and the high-resolution model is executed reading thesedata and calculating the side boundary values. In on-line mode, the pre-communicator of Scup (simple coupler) by
Yoshimura and Yukimoto (2008) is used to exchange data, and the low-resolution and the high-resolution models
are run at the same time.
The main region of the high-resolution model (hereinafter called core region, the hatched region of Figure 15.1)
is constructed by connecting tracer points (T-points) of the low-resolution model to form a rectangular region.
Two rows or columns of velocity grid cells are added outside the core region at each side boundary (hereinafter
called margin; the unhatched region just outside the core region in Figure 15.1), and the side boundary condition
is imposed by the low-resolution model. Tracers at the boundary of the core region and one grid outside of itand velocities at two grid points outside the core region are replaced every time step using values from the lowresolution model and linear interpolation. These are referred to as the input interface (Figure 15.1).
A problem arises for velocity points of the high-resolution model between an ocean point and a land point of the
low-resolution model, since only data at a single point is available for interpolation. In this case, the high-resolution
model velocity perpendicular to the coast line is assumed to be zero at the coast and linearly interpolated from thenearest ocean grid point of the low-resolution model. Furthermore, the high-resolution model velocity tangent tothe coast line is set to the same value at the nearest ocean grid point of the low-resolution model. By doing so, theinterpolated velocity field satisfies the continuity equation.
The boundary values supplied from the low-resolution model should be enough to compute values at the marginof the high-resolution model using linear interpolation. In practice, three columns or rows are output; they areenough to compute the margin as well as the region one grid equivalent of the low-resolution model inside theboundary of the core region. For the northern boundary, T1-T3 and U1-U3 in Figure 15.1 are sent to the highresolution model.
In this way, prognostic variables are directly replaced by those from the low-resolution model, which is usuallycalled the clamped method (Cailleau et al., 2008). This method does not guarantee conservation of tracers incontrast to the method where fluxes are given at the boundary, but the integration is quite stable.

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Chapter 15

Nesting
T3
U3
Input interface for U

T2

Input interface for T

U2

T1

U1

:Temperature & Salinity points: Velocity points

Figure 15.1. Relation between the low-resolution (large symbols) and the high-resolution model (small symbols)
model. The hatched region is the main region of the high-resolution model. The tracer points at the boundaryof this region and at one grid point outside this boundary and the velocity points at two grid points outside thisboundary are replaced by values from the low-resolution model. At the northern boundary, three rows (T1 - T3
and U1 - U3) are sent to the high-resolution model from the low-resolution model.

15.2 Low-resolution model
There is no particular problem in constructing the low-resolution model. Since the grid size ratio between thehigh- and low-resolution models must be odd, the grid size of the low-resolution model should be determinedaccording to that of the high-resolution model. A grid size ratio of 1:3 or 1:5 is recommended.
In off-line mode, the following files are output from the low-resolution model to be read by the high-resolution
model.
• Three rows or columns of baroclinic mode data for each side boundary.
• Three rows or columns of barotropic mode data for each side boundary.
• Land-sea index of the low-resolution model.
• Latitude and longitude of the low-resolution model.
The filenames are set in parinit.F90 using namelist outflpar. In on-line mode, this namelist is not required.
The daily output of side boundary data will work in running the high-resolution model. In on-line mode, sideboundary data are sent every time step.

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15.3. High-resolution model

15.3

High-resolution model

15.3.1Required data

To run the high-resolution model, prepare two data files that contain• The contribution ratio between low- and high-resolution models around the boundary,• Two dimensional distribution of horizontal diffusivity and viscosity.
In addition, prepare the following data from the low-resolution model.
• Three rows or columns of baroclinic mode data for each side boundary.
• Three rows or columns of barotropic mode data for each side boundary.
• Land-sea index of the low-resolution model.
• Latitude and longitude of the low-resolution model.
Issues to be carefully considered in creating variable horizontal grid size information and bottom topography aredetailed in the next subsection.
Note the following when preparing barotropic boundary data. Since time filtering is used to feedback the result of
the barotropic equations to baroclinic modes, the barotropic equations are integrated past the baroclinic time. Thus,
the high-resolution model needs future barotropic data in addition to the original output from the low-resolution
model. To fulfill this need, prepare barotropic data in one file so that the second data in time sequence can be usedas data after the last one for repeating year cycle run. For a historical run, append the second data of the following
year of the low-resolution model to the present year data.

15.3.2Creating data

We briefly describe how to create data that will be read from the high-resolution model.

a. Horizontal grid size
The same grid size as in the low-resolution model should be used in the vertical direction. The high-resolution
model is nested in the horizontal directions. The distance between the velocity points of the low-resolution model
is divided equally in the high-resolution model. There should be two marginal velocity cells outside the core region
at the western and the southern boundary even if the western or the southern boundary is filled by land. However,if the eastern or the northern boundary is filled by land and does not receive boundary data, one marginal velocity
cell will be enough.

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Chapter 15

Nesting

How to divide low-resolution cells

‘‘dxt’’ of the low-resolution model (U-point distance: variable name is dxtdeg)
should be divided equally in the high-resolution model.
There should be two marginal U-points in the high-resolution model(one cell will do if the eastern or northern boundary is filled by land)***** low-resolution model *****
ifst is the western end grid point number of the low-resolution 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)|***** high-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)How to create a grid size information file (for variable grid size only)! the number of grid points of the high-resolution model
integer(4), parameter :: imut = 535, jmut = 431
integer(4), parameter :: ivgrid = 57 ! arbitrary
real(8) :: dxt(imut), dyt(jmut)! dxtdeg, dytdeg in the model
write(unit=ivgrid) (dxt(i),i=1,imut), (dyt(j),j=1,jmut)

b. Topography of the high-resolution model
To avoid serious discontinuities in mass transport, the high-resolution model should have the same topography
as the low-resolution model around the side boundary of the high-resolution model.
It is recommended that the topography of the high-resolution model should have the same topography as the
low-resolution model in the two inner low-resolution velocity cells and the one outer low-resolution velocity cellfrom the boundary of the core region.

c. Weighting ratio between the low- and high-resolution model
For smoothness, values of the high-resolution model around the side boundary could be given as a weighted
average of the low- and high-resolution models. However, we recommend that the weight for the low-resolution
model to be unity at the input interface and zero elsewhere.

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15.4. Usage

weighting ratio between the low and high-resolution model! the number of grid points of high-resolution model
integer(4), parameter :: imut = 535, jmut = 431
integer(4), parameter :: ibuffer = 57
! arbitrary
real(8) :: wbuft(imut,jmut), wbufu(imut,jmut)
write(unit=ibuffer) wbuft, wbufu



d. 2-D distribution of diffusion and viscosity coefficient
Diffusion and viscosity coefficients of a high-resolution model should be the same as those of a low-resolution
model around the side boundary. The coefficients are made small in the interior.

2-D distribution of diffusion and viscosity coefficient! the number of grid points of the high-resolution model
integer(4), parameter :: imut = 535, jmut = 431
integer(4), parameter :: ihdtsuv = 57
! arbitrary
real(8) :: hdts(imut,jmut), hduv(imut,jmut)
write(unit=ihdtsuv) hdts, hduv





15.4

Usage

15.4.1

Compilation

The model option for the low-resolution model is PARENT, and that for the high-resolution model is SUB. For
on-line mode, the option NESTONLINE should be specified along with the name of the model such as NAME MODEL= modelname. For a calculation using parallel processors (option PARALLEL), only one-dimensional partitioning(in the meridional direction) of model region is supported. The number of zonally partitioned regions (NPARTX)
must be one. These model options and the model name should be specified in configure.in.
The following describes parameters that a user should define before compilation.

a. Low resolution model
The following model parameters should be defined in parent/parapar.F90 prior to compilation.

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Chapter 15

Nestingifsto = 35
ifedo = 157
jfsto = 61
jfedo = 159

:
:
:
:

the western end point to be used to interpolate data (T-point)
the eastern end point to be used to interpolate data (T-point)
the southern end point to be used to interpolate data (T-point)
the northern end point to be used to interpolate data (T-point)

ipareo = ifedo - ifsto + 1
jpareo = jfedo - jfsto + 1... the total grid number to be sent to the high-resolution model
ioutrts = 5, ioutrtn = 0, ioutrtw = 5, ioutrte = 3... nesting ratio at each side boundary
south(s), north(n), west(w), east(e)b. High resolution model
Model parameters that should be defined in sub2/parasub.F90 are listed on Table 15.1.
Table 15.1. Parameters to be defined in sub2/parasub.F90
variable nameipmut, jpmut
ifst, ifed, jfst, jfed

description
the number of grid points of the low-resolution model
define the boundary of the low-resolution model used
for interpolation in the high-resolution model (considered in terms of T-points ... should be the same as

ipare, jpare

iinrts, iinrtn, iinrtw, iinrte
intbcl, intbtr
parapar.F90)
the total grid number of the low-resolution model tobe used for interpolation in the high-resolution model(... should be the same as parapar.F90)
nesting ratio at each side boundary
the time interval in seconds of the boundary data
(used only for off-line mode)e.g.
iinrts=5, iintrn=3, iinrtw=5, iinrte=3
5 for nesting ratio 5:1
3 for nesting ratio 3:1






e.g.,
intbcl = 21600, intbtr = 21600



15.4.2 Running the models
In off-line mode, specify the names of data files and how boundary data are handled in namelist. Users shouldsee parent/parinit.F90 for the low-resolution model and sub2/subinit.F90 for the high-resolution model.
The filename for the weighting ratio and the two-dimensional distribution of horizontal diffusion and viscosity
for the high-resolution model (namelist inflsub) should be specified even in on-line mode.

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15.4. Usagea. Low resolution model
The following namelists are read from parent/parinit.F90 on execution.
Table 15.2. Namelist for the nested low-resolution model
variable namegroupdescription

usage

nwrtpcnjobpar

The interval of time steps by which baroclinic mode is written to file

off line only

nwrtptnjobpar

The interval of time steps by which barotropic mode is writ-off line only

ircfstonjobparten to file
The record number in file at the beginning of the integration(baroclinic mode)

off line only

irtfstonjobpar

The record number in file at the beginning of the integration(barotropic mode)

off line only

ibfirstfile btrs out
file btrn out

njobparoutflparoutflparOutput (1) or no-output(0) the initial state as boundary data
the filename of the barotropic mode (southern end)
the filename of the barotropic mode (northern end)

off line only
off line only
off line only

file btrw out
file btre out
file bcls out
file bcln out

outflparoutflparoutflparthe filename of the barotropic mode (western end)
the filename of the barotropic mode (eastern end)
the filename of the baroclinic mode (southern end)

off line only
off line only
off line only

outflparoutflparthe filename of the baroclinic mode (northern end)
the filename of the baroclinic mode (western end)

off line only
off line only

outflparoutflparoutflparthe filename of the baroclinic mode (eastern end)
land-sea index of the low-resolution model
latitude-longitude of the low resolution model

off line only
off line only
off line only

file bclw out
file bcle out
file lidx out
file pgrd out

b. High resolution model
The following namelists are read from sub2/subinit.F90 on execution.
Table 15.3. namelist for the nested high-resolution model
variable namegroupdescription

usage

ircfstnjobsub

The record number in file at the beginning of the integration

off line only

(baroclinic mode)
irtfstnjobsub

The record number in file at the beginning of the integration(barotropic mode)

off line only

ircendnjobsub

The number of records written in the boundary file (baro-

off line only

irtendnjobsub

clinic mode).
The number of records written in the boundary file

off line only

irtcyclenjobsub(barotropic mode).
Return to the second record (1) or repeat the last data (0) if

off line only

inflp2s
inflp2s

the last barotropic data is reached.
the filename of the barotropic mode (southern end)
the filename of the barotropic mode (northern end)

off line only
off line only

file btrs
file btrn

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Chapter 15

Nestingvariable namegroupdescription

usage

file btrw
file btre
file bcls

inflp2s
inflp2s

the filename of the barotropic mode (western end)
the filename of the barotropic mode (eastern end)

off line only
off line only

inflp2s
inflp2s
inflp2s

the filename of the baroclinic mode (southern end)
the filename of the baroclinic mode (northern end)
the filename of the baroclinic mode (western end)

off line only
off line only
off line only

file lidx

inflp2s
inflp2s

the filename of the baroclinic mode (eastern end)
land-sea index of the low-resolution model

off line only
off line only

file pgrd
file bwgt

inflp2s
inflsub

latitude-longitude of the low-resolution model
weighting ratio between the low and high-resolution modelaround the boundary

off line only

file bhdf

inflsub

2-D distribution of diffusion and viscosity of the highresolution model

file bcln
file bclw
file bclec. On-line mode
In on-line mode, the pre-communicator of Scup (simple coupler) by Yoshimura and Yukimoto (2008) is used to
exchange data and the low and high-resolution models are run at the same time. User should tell the coupler how
data are exchanged between the low and high-resolution models via a namelist file NAMELIST SCUP. An exampleof NAMELIST SCUP is listed below. In this example, the model name of the low-resolution model is GLOBAL andthat of the high-resolution model is WNP01.

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15.4. Usagean example of NAMELIST SCUP

&nam_scup_pre model_put=’GLOBAL’, model_get=’WNP01’, 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’ /&nam_scup_pre var_put=’AEXLP’, var_get=’AEXLP’, dst_get=’ALL’ /&nam_scup_pre var_put=’ATXLP’, var_get=’ATXLP’, dst_get=’ALL’ /&nam_scup_pre var_put=’ULS’, var_get=’ULS’, dst_get=’FIRST’ /&nam_scup_pre var_put=’VLS’, var_get=’VLS’, dst_get=’FIRST’ /&nam_scup_pre var_put=’TLS’, var_get=’TLS’, dst_get=’FIRST’ /&nam_scup_pre var_put=’SLS’, var_get=’SLS’, dst_get=’FIRST’ /&nam_scup_pre var_put=’HTS’, var_get=’HTS’, dst_get=’FIRST’ /&nam_scup_pre var_put=’UMS’, var_get=’UMS’, dst_get=’FIRST’ /&nam_scup_pre var_put=’VMS’, var_get=’VMS’, dst_get=’FIRST’ /&nam_scup_pre var_put=’ULN’, var_get=’ULN’, dst_get=’LAST’ /&nam_scup_pre var_put=’VLN’, var_get=’VLN’, dst_get=’LAST’ /&nam_scup_pre var_put=’TLN’, var_get=’TLN’, dst_get=’LAST’ /&nam_scup_pre var_put=’SLN’, var_get=’SLN’, dst_get=’LAST’ /&nam_scup_pre var_put=’HTN’, var_get=’HTN’, dst_get=’LAST’ /&nam_scup_pre var_put=’UMN’, var_get=’UMN’, dst_get=’LAST’ /&nam_scup_pre var_put=’VMN’, var_get=’VMN’, dst_get=’LAST’ /&nam_scup_pre var_put=’ULW’, var_get=’ULW’, dst_get=’ALL’ /&nam_scup_pre var_put=’VLW’, var_get=’VLW’, dst_get=’ALL’ /&nam_scup_pre var_put=’TLW’, var_get=’TLW’, dst_get=’ALL’ /&nam_scup_pre var_put=’SLW’, var_get=’SLW’, dst_get=’ALL’ /&nam_scup_pre var_put=’HTW’, var_get=’HTW’, dst_get=’ALL’ /&nam_scup_pre var_put=’UMW’, var_get=’UMW’, dst_get=’ALL’ /&nam_scup_pre var_put=’VMW’, var_get=’VMW’, dst_get=’ALL’ /&nam_scup_pre var_put=’ULE’, var_get=’ULE’, dst_get=’ALL’ /&nam_scup_pre var_put=’VLE’, var_get=’VLE’, dst_get=’ALL’ /&nam_scup_pre var_put=’TLE’, var_get=’TLE’, dst_get=’ALL’ /&nam_scup_pre var_put=’SLE’, var_get=’SLE’, dst_get=’ALL’ /&nam_scup_pre var_put=’HTE’, var_get=’HTE’, dst_get=’ALL’ /&nam_scup_pre var_put=’UME’, var_get=’UME’, dst_get=’ALL’ /&nam_scup_pre var_put=’VME’, var_get=’VME’, dst_get=’ALL’ /&nam_scup_pre var_put=’ICECATS’, var_get=’ICECATS’, dst_get=’FIRST’ /&nam_scup_pre var_put=’ICECATN’, var_get=’ICECATN’, dst_get=’LAST’ /&nam_scup_pre var_put=’ICECATW’, var_get=’ICECATW’, dst_get=’ALL’ /&nam_scup_pre var_put=’ICECATE’, var_get=’ICECATE’, dst_get=’ALL’ /&nam_scup_pre var_put=’ICEDYNS’, var_get=’ICEDYNS’, dst_get=’FIRST’ /&nam_scup_pre var_put=’ICEDYNN’, var_get=’ICEDYNN’, dst_get=’LAST’ /&nam_scup_pre var_put=’ICEDYNW’, var_get=’ICEDYNW’, dst_get=’ALL’ /&nam_scup_pre var_put=’ICEDYNE’, var_get=’ICEDYNE’, dst_get=’ALL’ /&nam_scup_pre model_put=’WNP01’, model_get=’GLOBAL’, type=’REAL8’ /&nam_scup_pre var_put=’ATEXSUB’, var_get=’ATEXSUB’, dst_get=’ALL’ /&nam_scup_pre var_put=’TSUB’, var_get=’TSUB’, dst_get=’ALL’ /&nam_scup_pre var_put=’SSUB’, var_get=’SSUB’, dst_get=’ALL’ /





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Nesting

15.5 Program structure
The program structure of a low-resolution model is as follows.
ogcm__ini|+- parinit (parinit.F90)|+- off-line mode|+- if (ibfirst == 1)|+- out[snwe]cli|+- out[snwc]tro(initialize low-resolution model)(output boundary data of the initial state)ogcm__run|+-- part_1 (on-line mode)|||+-- surfce|||+-surfce_integ|||+-DO LOOP_N
|
|
|
|
|
+- outmrgnt(send barotropic boundary data|||
to high-resolution model)||+-- out[snwe]tro|||+- outmrgnt(send time-filtered barotropic boundary||
data to high-resolution model)|+-- out[snwe]tro|+-- part_2 (on-line mode)|||+-- outmrgn(send baroclinic boundary data|||
to high-resolution model)||+-- out[snwe]cli|||+-- outmrgnvd(send viscosity/diffusivity||
to high-resolution model)|+-- out[snwe]vd|+-- outmrgn (off-line mode at the end of each time step)|||+-- if (mod(nnmats,nwrtpc)== 0)|||+-- out[snwe]cli|+-- outmrgnvd (off-line mode at the end of each time step)|||+-- if (mod(nnmats,nwrtpc)== 0)|||+-- out[snwe]vd|+-- outmrgn (off-line mode at the end of each time step)|+-- if (mod(nnmats,nwrtpt)== 0)

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15.5. Program structure|+-- out[snwe]tro

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Chapter 15

Nesting

The program structure of a high-resolution model is as follows.
ogcm__ini|+- subinit (subinit.F90)(initialize high-resolution model)|+- mknlvect(make list vectors for interpolation|of boundary data)+- mknlvecv|+- off-line mode|+- rd[snwe]cli(read first two baroclinic boundary data)|||+ inplsubuf (interpolation of boundary data||
to high-resolution model grid points)|+ inplsubvf|||+ inplsubtf|+- rd[snwc]tro(read first two baroclinic boundary data)|+ inplsubus|+ inplsubvs|+ inplsubts
ogcm__run|+- part_1|||+- surfce|||+-surfce_integ|||+- DO LOOP_N
|
|
|
|
|
+- inpfacet (read/receive barotropic boundary data)
|
|
|
|
|
+- rd[snwe]tro
|
|
|
|
|
|
|
+ inplsubus
|
|
|
|
|
|
|
+ inplsubvs
|
|
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|
|
|
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+ inplsubts
|
|
|
|
|
+- flmrgn[snwe]t (replace barotropic prognistic variables||
to those from the low-resolution model)|+- inpfacet(read/receive time-filtered barotropic boundary data)|||+- rd[snwe]tro
|
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+ inplsubus
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+ inplsubvs
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+ inplsubts

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15.5. Program structure|||+- flmrgn[snwe]t|+- part_2|+- inpface(read/receive baroclinic boundary data||
after ‘‘trcimp’’ and before ‘‘stable’’)|+- rd[snwe]cli
|
|
|
|
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+ inplsubuf
|
|
|
|
|
+ inplsubvf
|
|
|
|
|
+ inplsubtf|||+- flmrgn[snwe]c(replace baroclinic prognistic variables|
to those from the low-resolution model)|+- inpfacevd(read/receive viscosity/diffusivity data|
after mixed layer model)+- rd[snwe]vd|||+ inplsubtf|+- flmrgn[snwe]vd (replace baroclinic prognistic variables
to those from the low-resolution model)

References
Cailleau, S., V. Fedorenko, B. Barnier, E. Blayo, and L. Debreu, 2008: Comparison of different numerical methods used to handle the open boundary of a regional ocean circulation model of the Bay of Biscay, Ocean
Modell., 25, 1-16, doi:10.1016/j.ocemod.2008.05.009.
Yoshimura, H., and S. Yukimoto, 2008: Development of simple coupler (Scup) for earth sysyem modeling.,
Pap. Meteor. Geophys., 59, 19-29, doi:10.2467/mripapers.59.19.

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Chapter 16

User’s Guide

This chapter explains the procedure to run MRI.COM using a long term integration of a global model as a standard
case. The description in this chapter is based on MRI.COM version 3.0 (MRICOM-3 0-20091130) and may notcorrespond to the latest version of MRI.COM, since the run procedure (user interface) is continuously updated. It
is recommended that users refer to README.First in src directory when they start setting up the 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 16.1 and Tables 16.1 and 16.2).
• Input data: Topography, surface forcing, and climatology to be read at run time (Section 16.2 and Table
16.3).
• Execution: An example shell script and namelist (Section 16.3).
• Post process: A description of output files (Section 16.4 and Table 16.4).
The input and main output files are listed in Tables 16.1 through 16.4. Note that cgs units are employed to express
physical values in the model.

16.1

Model setup

This section describes the procedure necessary to set up the model and compile its programs. First, prepare
configure.in that contains the information about the model options and grid sizes. In addition, edit several
Fortran 90 files to specify parameters corresponding to the selected model options.
Table 16.1. Files used for compilation and their related program files (see Section 16.1)
parameter to be specified
file nameincluded frommodel size, model option, compile op-

configure.inparam.F90 via configuretionvertical resolution

dz.F90param.F90.invertical diffusion coefficient

vdbg.F90

rdjobp.F90

grid of surface forcing data

intpolpar.F90

force.F90

vertical grid of climatological data (formodel option TSINTPOL)

depclim.F90

tsclim.F90

3-D

tsintpol.F90

tsclim.direct.F90(for model option FLXINTPOL)

grid

of

climatological

data(for model options TSINTPOL and
TSCLDIRECT)

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16.1.1 Files needed for compilationa. configure.in
The model options and configuration parameters should be specified in configure.in as follows.

An example configure.in for Global tripolar 1◦ × 0.5◦ grid model
DEFAULT_OPTIONS="ICE ICECAT SIDYN CALALBSI SFLUXW SFLUXR ISOPYCNALSMAGOR VIS9P DIFAJS NOHKIM VVDIMP UTOPIA ULTIMATE ZQUICKEST ZULTIMATEBBL HIST HISTFLUX HFLUX TAUBULK WFLUX RUNOFF Y365D CLMFRC LWDOWNBULKNCAR BULKITER INILEV CYCLIC VARIABLE TRIPOLAR PARALLEL FREESURFACE"#
IMUT=364
JMUT=368
KM=51
KSGM=5
KBBL=1
SLAT0=-78.D0
SLON0=0.D0
NPARTX=8
NPARTY=4
DXTDGC=1.0D0
DYTDGC=0.5D0
NPLAT=64.D0
NPLON=80.D0
SPLAT=64.D0
SPLON=260.D0

The parameters required for each major option are listed in Table 16.2 (see also README.Options).
Table 16.2. model parameters to be set in configure.in
option namevariable namedescription

default

IMUT, JMUT, KM

zonal/meridional/vertical grid number

SLON0, SLAT0DXTDGC, DYTDGC

longitude/latitude of the first tracer grid point excluding theboundary land or ghost cells
zonal/meridional grid sizes in degree for the case of uniform in-

TRIPOLAR, JOT

NPLAT, NPLON

crement; unless VARIABLE
geographical latitude and longitude of the displaced North Pole

FREESURFACESPLAT, SPLONKSGM
NSFMRGNgeographical latitude and longitude of the displaced South Pole
the number layers in the sea surface sigma-layer; see Chapter 4
the number of side-boundary ghost cells to reduce the communication cost in parallel computation (see Ishizaki and Ishikawa,

BBL
PARALLELTSINTPOL

KBBL
NPARTX, NPARTY2006)
the number layers of bottom boundary layer model; must be 1
the number of zonally/meridionally partitioned region for a calculation using parallel processors: the number of parallel pro-

IMT, JMT, KK

cesses should be NPARTX × NPARTY
3-D size of climatological data

SLATC, SLONCDLATC, DLONC

latitude/longitude of the first grid point data of climatology
grid increment for latitude/longitude of climatology

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

16.1. Model setup
option namevariable namedescription

FLXINTPOL(RUNOFF)

IMF, JMF
IMROF, JMROF

2-D size of sea surface data
2-D size of river discharge data

INTPWIND
passive tracer

NUMTRC P

horizontal interpolation type for the wind data; 1:linear, 2:thirdorder spline
only when any passive tracer is calculated (NUMTRC P ≥ 1)

b. dz.F90
Describe the discretization in the vertical direction in dz.F90. The widths of the discrete cells that fill the verticalcolumn should be written from top to bottom. Usually the depth of the bottom level is determined first, and thecolumn is separated into discrete cells. In general, the levels are finer from the sea surface to about 1000 m andbecome coarser towards the bottom. Tracer and velocity levels are placed at the middle level of the cell (Chapter
3). The number of vertical cells is described as KM in configure.in. The file dz.F90 is included by param.F90.

Example for dz.F90
real(8), parameter :: dz(km) = (/&& 3.0d2, 4.0d2, 6.0d2, 8.0d2, 1.0d3, &............
& 2.5d4, 2.5d4, 3.75d4, 7.50d4 /)c. depclim.F90
For option TSINTPOL, define the information about the depth of climatological temperature and salinity data indepclim.F90. It is used to interpolate the data to model grid points. Enter the number of grid points (IMT, JMT,and , KK) and horizontal grid-point information (DLATC, DLONC, SLATC, and SLONC) in configure.in (Table 16.2).
A sample can be found in directory src/intpolpar.

d. tsintpol.F90
When option TSCLDIRECT is selected in addition to TSINTPOL, define the information about the 3-D grid ofclimatological temperature and salinity data in tsintpol.F90. With this choice, the data should be prepared indirect access files. As in the default case, enter the number of grid points (IMT, JMT, and KK) and horizontal
grid-point information (DLATC, DLONC, SLATC, and SLONC) in configure.in (Table 16.2). But the horizontal gridof data may be defined in tsintpol.F90, overriding the above definition. A sample can be found in directory
src/intpolpar.

e. intpolpar.F90
For option FLXINTPOL, define the information about the latitude and longitude of the surface data intpolpar.F90.
It is used to interpolate the data to the model grids points for sea surface flux calculation. Enter the number of zonal
and meridional grid points (IMF and JMF) in configure.in (Table 16.2). Samples for the datasets of NCEP-NCAR
and JRA25 reanalyses and CORE can be found in the directory src/intpolpar.

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User’s Guidef. vdbg.F90
The file vdbg.F90 is necessary if the vertical diffusion equation is solved using the implicit method (option
VVDIMP). While the vertical diffusion coefficient is replaced by the large one from the mixed layer model andthe convection scheme (option DIFAJS), the lower limit (vdbg(1 : km)) should be set in vdbg.F90. The sample of
vdbg.F90 and the sample program to prepare it (vdmricom.F90) are in the directory src/vd. Here, the vertical
diffusion coefficient used in the GFDL-MOM is shown.

Example for vdbg.F90

do k = 1, km
vdbg(k) = .8d0 + 1.05d0 * datan(4.5d0*(dep(k+1)*1.d-5-2.5d0)) / pi
end do16.1.2 Compilation of the model
The standard compiling script is prepared as compile.sh in the src directory. The part depending on the system(OS, Fortran compiler, and compiler option) in compile.sh and configure have to be edited by users for their
computer (hardware) environment.
To compile the programs, execute compile.sh by issuing a shell prompt from the front-end machine. The function of compile.sh is to create param.F90 and Makefile from configure.in, param.F90.in, and Makefile.inby 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.inis newer than param.F90, the parameter values defined in configure.in replace those in param.F90.in to createparam.F90. The Makefile is created from Makefile.in. Finally, make is carried out, and the executable file ogcm
is obtained. 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.

16.2 Preparation of input data files for execution
The topographic data (file topo), temperature and salinity climatological data (file tscl), and wind stressdata (file wind) are always necessary.
According to user’s the specification of model options and job parameters, the following additional data files
should be prepared:• zonal and meridional grid spacing (units in degrees, file vgrid; for VARIABLE)• information about the area and the length for each grid (file scale; except for SPHERICAL)• information about grid-wise nudging “on” or “off” (file frc; for namelist parameter iforcev = 1)
• sea surface radiation data (file hflx; for HFLUX)• sea surface meteorological data (file bulk; for HFLUX + selected bulk formula, see Chapter 8)• precipitation data (file prcp; for WFLUX)• river discharge data (file rnof; for RUNOFF)
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16.2. Preparation of input data files for execution

Table 16.3. Main input data files and their related program files (see Section 16.2)
file name specified in (namelist)
read fromsubject

rdjobp.F90 etc.

run-time job parameters (as namelist parameters)

standard input (or redirection from file)variable horizontal grid spacing (for themodel option VARIABLE)

file vgrid (inflg)

stmdlp.F90

topography

file topo (infltopo)

rdbndt.F90

area and length scale for each grid

file scale (inflscl)

rdbndt.F90

climatological data of temperature andsalinity

file tscl (infltscl)

tsclim.F90

on/off of nudging for tracer points

file frc (njobbdy)

rstcoef.F90

wind stressfile wind (inflw)

force.F90

surface radiation data

file hflx (inflh)

force.F90

surface meteorological data

file bulk (inflh)

force.F90

precipitation data

file prcp (inflp)

force.F90

river discharge data

file rnof (inflo)

force.F90

sea-ice fractional area data

file icec (inflic)

force.F90

initial values for restart

file restart in (infla)(for ice averaged over all categories)

file ice restart in (infli)

mod seaice.F90(for ice for each thickness category)

file icecat restart in (inflic)

ice restart.F90

rdinit.F90• sea-ice fractional area data (file icec; for ICECLIM)

16.2.1

Topographic and grid spacing dataa. Topography
The topographic data consist of the single precision integer array HO4(IMUT, JMUT) that contains the sea floor
depths of the velocity grid points (in cm) and the single precision 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)
write(unit=inidt) ho4, exnn

For an experiment with a realistic topography, the model topography is usually prepared by averaging the depth
data of ETOPO2 over each grid cell. ETOPO2 is 2-minute-grid topography data. An example of the topography
for global 1◦ × 0.5◦ model created by this method is shown in Figure 16.1.
On creating a model topography, especially for a low-resolution model, the user should be careful that theimportant gateways for the ocean circulation be kept open and that the land blocking the ocean circulation be kept
closed. The topography data is read from the file file topo (namelist infltopo) in rdbndt.F90.

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Figure 16.1. Example of ocean model topography (global 1◦ × 0.5◦ grid model).
b. Grid spacing
The grid spacing should be prepared when variable grid spacing is used for either the zonal or meridional
direction (the model option VARIABLE). The units are in degrees. It is read from the file file vgrid (namelist
inflg) in stmdlp.F90.
Format of grid spacing data (file vgrid; VARIABLE)

real(8) :: dxtdeg(imut), dytdeg(jmut) ! grid increment for T-point
write(ivgrid) dxtdeg, dytdegc. Grid cell area and distance
When the model grid points are defined based on the general orthogonal coordinates, the quarter cell area anddistance should be prepared. The units are in cgs. It is read from the file file scale (namelist inflscl) in
rdbndt.F90.
When spherical coordinates are used (option SPHERICAL), e.g., the grids are defined on geographical latitude andlongitude, the grid information is analytically calculated in stmdlp.F90, and the file file scale is not necessary.

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16.2. Preparation of input data files for executionFormat of grid cell area and distance (file scale; if not SPHERICAL)

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)
write(unit=N_T) a_bl ! U-box area of bottom-left 1/4 grid
write(unit=N_T) a_br ! U-box area of bottom-right 1/4 grid
write(unit=N_T) a_tl ! U-box area of top-left 1/4 grid
write(unit=N_T) a_tr ! U-box area of top-right 1/4 grid
write(unit=N_T) dx_bl ! U-box length of bottom-left 1/4 grid
write(unit=N_T) dx_br ! U-box length of bottom-right 1/4 grid
write(unit=N_T) dx_tl ! U-box length of top-left 1/4 grid
write(unit=N_T) dx_tr ! U-box length of top-right 1/4 grid
write(unit=N_T) dy_bl ! U-box length of bottom-left 1/4 grid
write(unit=N_T) dy_br ! U-box length of bottom-right 1/4 grid
write(unit=N_T) dy_tl ! U-box length of top-left 1/4 grid
write(unit=N_T) dy_tr ! U-box length of top-right 1/4 grid

16.2.2

Climatological dataa. Default
By default, define the climatological temperature and salinity data at model grid points (i.e., IMT = IMUT,
JMT = JMUT, KK = KM). However, when the option TSINTPOL is selected, data with uniform grid spacing are readand interpolated in the model. In this case, the grid point information (DLATC, DLONC, SLATC, and SLONC) shouldbe specified in configure.in (Table 16.2), and the data grid points (alonc, alatc) are calculated in tsclim.F90.
The climatological data are read from the file file tscl (namelist infltscl) in tsclim.F90.
Format of climatological data (file tscl) for default caseinteger(4), parameter :: imn = 12real(4) :: ttlev(imt,jmt,kk,imn),tslev(imt,jmt,kk,imn)

do m = 1, imn
write(unit=itscl) (((ttlev(i,j,k),i=1,imt),j=1,jmt),k=1,kk), &&(((tslev(i,j,k),i=1,imt),j=1,jmt),k=1,kk)
enddo

b. Option TSCLDIRECT
When option TSCLDIRECT is selected, data should be prepared in direct access files. Monthly data should be
prepared in separate files as follows.
At run time, another namelist njobpts is required to handle data. The variable itsrepeat specifies whether
the parepared data are repeatedly used (=1) or interannual variations are assumed (=0). When itsrepeat = 0, themonth 00 (13) means the December (January) of the previous (next) year. The variable itsmonfst specifies thefirst data for “this” run, the start time is between mid-(itsmonfst) and mid-(itsmonfst − 1).

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Chapter 16User’s Guide
Format of climatological data (file tscl) for option TSCLDIRECT



integer(4), parameter :: imn = 12
character(128) :: file_tscl_monthreal(4) :: ttlev(imt,jmt,kk,imn), tslev(imt,jmt,kk,imn)! separate file for each month
do m = 1, imn
write(file_tscl_month,’(1a,i2.2)’) trim(file_tscl), m
open (unit=itscl,file=file_tscl_month,access=direct,recl=4*imt*jmt*kk)
write(unit=itscl,rec=1) (((ttlev(i,j,k),i=1,imt),j=1,jmt),k=1,kk)
write(unit=itscl,rec=2) (((tslev(i,j,k),i=1,imt),j=1,jmt),k=1,kk)
close(itscl)
end do

16.2.3 Nudging (body forcing) data
The time integration could be conducted restoring temperature and salinity of the specified grid points to theclimatological data read from the file file tscl (often referred to as nudging or body forcing). This is done bysetting the namelist parameter iforcev (namelist njobp) to one. In this case, the data that contain information
about the grid-wise “on (=1)” or “off (=0)” of nudging should be prepared in a default case. In a case of TSINTPOL,
data that contain the grid number of points where nudging is done should be prepared. It is read from the file
file frc (namelist njobbdy) in rstcoef.F90. The restoring time for nudging should be specified as rtmscb(namelist njobbdy) in units of days.
Format for nudging grid points (file frc) for TSINTPOL case#ifdef OGCM_TSINTPOL! when using interpolated climatological data,! for example a large size model! numbf : the number of grid points for nudging!
numbf is a namelist parameter in the model (rstcoef.F90)
integer(4), save :: numbf = 1000
namelist /nblarge/ numbf
read(unit=5,nml=nblarge,iostat=ios)! iposbf : the grid number of grids where nudging is done
integer(4):: iposbf(numbf)
write(ifrcdt) (iposbf(n),n=1,numbf)! serial form

! in the subroutine rstcoef, iposbf(numbf) is treated as follows and! the nudging grid points at (i,j) are defined.
do n = 1, numbf
j = iposbf(n)/imut + 1
i = iposbf(n) - (j-1)*imut
chfb(i,j,1:km) = chfbc ! coefficient for nudging! (chfbc = 1.0d0 / rtmscb / 86400)
enddo#endif /* OGCM_TSINTPOL */



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16.2. Preparation of input data files for executionFormat for nudging grid points (file frc) for default case#ifndef OGCM_TSINTPOL#ifdef OGCM_BF2D

! for a small size model! for two dimensional distribution! chf2d : 2-D on(=1) off(=0) information for nudging! imut, jmut are the numbers of grid points in the model domainreal(4) :: chf2d(imut,jmut)
write(ifrcdt) chf2d#else /* OGCM_BF2D */! single precision! serial form
! distributed to each node
! and stored in chfb(imx,jmx,km)! multiplied by chfbc! (chfbc = 1.0d0 / rtmscb / 86400)! for three dimensional distribution! chf3d : 3-D on(=1) off(=0) information for nudging! imut, jmut, km are the numbers of grids in the model domainreal(8):: chf3d(imut,jmut,km)

write(ifrcdt) chf3d! serial form
! read using the subroutine restart_read! distributed to each node
! and stored in chfb(imx,jmx,km)! multiplied by chfbc! (chfbc = 1.0d0 / rtmscb / 86400)#endif /* OGCM_BF2D */#endif /* OGCM_TSINTPOL */

16.2.4

Atmospheric forcing data

By default, the surface forcing data are read at a uniform time interval (isrstb; in seconds), which should bespecified in namelist ifrcd (force.F90) along with a parameter ifna that specifies the record number of surface
forcing data corresponding to the nearest future from the start time of ”this” run. A leap year is set according to
the calendar subroutine. When climatological data is repeatedly used, specify the option CLMFRC. When monthly
data are used, specify the option MONFRC.
The following data files should be prepared according to the chosen model options. Each file is opened only
once at the beginning of the run-time and thus should contain all data needed for that run.

a. Wind stress data (mandatory): file wind (namelist inflw)
zonal component: wsx4(imf,jmf) [dyn · cm−2 ]meridional component: wsy4(imf,jmf) [dyn · cm−2 ]

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real(4) :: wsx4(imf,jmf),wsy4(imf,jmf)



do irec = 1, nrec ! ‘‘nrec’’ is the number of data
! data formation
write(unit=iwind,rec=irec) wsx4, wsy4 ! direct form
enddo



b. Radiation data (HFLUX): file hflx (namelist inflh)
short wave: qsh4(imf,jmf) [erg · s−1 · cm−2 = 10−3 W · m−2 ]long wave: qlo4(imf,jmf) [erg · s−1 · cm−2 = 10−3 W · m−2 ]sea surface temperature: sst4(imf,jmf) [◦ C]; in the case of LWDOWN (only downward radiation) dummy
data are used
real(4) :: qsh4(imf,jmf), qlo4(imf,jmf)
real(4) :: sst4(imf,jmf)



do irec = 1, nrec ! ‘‘nrec’’ is the number of data
! data formation
write(unit=ihflx,rec=irec) qsh4, qlo4, sst4 ! direct form
enddo



c. Data needed to calculate the latent and sensible heat fluxes using bulk formula (HFLUX): file bulk
(namelist inflh)
air temperature: sat4(imf,jmf) [◦ C]
specific humidity: qar4(imf,jmf)
wind speed: wdv4(imf,jmf) [cm · s−1 ]sea surface pressure: slp4(imf,jmf) [hPa]
real(4) :: sat4(imf,jmf), qar4(imf,jmf)
real(4) :: wdv4(imf,jmf), slp4(imf,jmf)



do irec = 1, nrec ! ‘‘nrec’’ is the number of data
! data formation
write(unit=ibulk,rec=irec) sat4,qar4,wdv4,slp4
enddo

d. Precipitation data (WFLUX): file prcp (namelist inflp)
precipitation: pcp4(imf,jmf) [cm · s−1 ]Note: fresh water flux data should be scaled by pure water density (1.036 g cm−3 )

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16.2. Preparation of input data files for execution
real(4) :: pcp4(imf,jmf)



do irec = 1, nrec ! ‘‘nrec’’ is the number of data
! data formation
write(unit=ipcpr,rec=irec) pcp4
enddo



e. River discharge data (RUNOFF): file rnof (namelist inflo)
river discharge: rof4(imrof,jmrof) [cm · s−1 ]Note: fresh water flux data should be scaled by pure water density (1.036 g cm−3 )real(4) :: rof4(imrof,jmrof) ! imrof, jmrof could be different than imf, jmf

do irec = 1, nrec ! ‘‘nrec’’ is the number of data
! data formation
write(unit=irnof,rec=irec) rof4
enddo



f. Sea-ice fractional area data (ICECLIM): file icec (namelist inflic)
sea-ice fractional area: aic4(imf,jmf)
real(4) :: aic4(imf,jmf)



do irec = 1, nrec ! ‘‘nrec’’ is the number of data
! data formation
write(unit=iicec,rec=irec) aic4
enddo

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16.3 Execution
To run a model, a shell script that handles input/output files, executes the compiled binary ogcm, and postprocesses is usually prepared. The namelist parameters that control the job have to be given to the standard input
or redirection from a file.
This section presents an example shell script for the global tri-polar grid model. The following three files areused in this example.
• run.sh :: A shell script that submits the job to a batch request controlling system NQS.
• runogcm.sh :: The shell script that includes the command ogcm, the binary that executes the model.
• NAMELIST :: The namelist file that is sent to ogcm via the standard input.
For example, use the following command to submit a job to the batch request controlling system.
% qsub run.sh

Example of shell script that should be submitted (run.sh)#!/bin/sh#PBS -l cpunum_job=8#PBS -l memsz_job=10gb#PBS -l cputim_job=120000#PBS -A K0001#PBS -N coreI#
RUNDIR=${HOME}/MRICOM/coreI/run # The directory of this shell script
cd ${RUNDIR}
echo ------------------------------------------------------------------echo START
dateecho ------------------------------------------------------------------pwd/usr/bin/mpirun -np 8 ./runogcm.sh # this example is for the 8MPI job
echo ------------------------------------------------------------------echo END
dateecho -------------------------------------------------------------------



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16.3. Execution
An example shell script runogcm.sh that is called from run.sh#!/bin/sh## RUNDIR ... the directory of this shell script#
set to reconfirm the path of the environmental variables#
for the sub processes
RUNDIR=$HOME/MRICOM/coreI/run
cd $RUNDIR
if [ ! -s logs ]; thenmkdir ./logs# output directory for execution logs (standard output)fi
if [ ! -s logs/ftrace ]; thenmkdir ./logs/ftrace # output directory for job diagnoses by the systemfi
F_FTRACE=FMT1; export F_FTRACE
F_SETBUF06=0; export F_SETBUF06
F_RECLUNIT=BYTE; export F_RECLUNITFTRACEDIR=logs/ftrace; export FTRACEDIR./ogcm> logs/out.txt 2> logs/stderr.txt < NAMELIST=======================


Example for NAMELIST==========================(see README.Namelist for details)# basic parameters for model time integration&njobpnfirst=-1,
nstep=48
neng=-1,nneng=-1,
nwrit=48
nwrt2=-1,
mampai=1,
mmpai2=1,
nkeisu=48
adtuv=30,
adtts=30,
adtsf=30,
hdts=1.0d7,
hduv=0.0d0,
hduv2=0.0d0,
vduv=0.1d0,
alpha=1.0d0,
gamma=51*1.0d0,
hupp=0.5d0,
vupp=0.7d0,
ifrcsf=1,
iforcv=0,
ispin=0,
matsuno_int=12,
nocbgt=0,
flg_timemonitor=.false.
/

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User’s Guide# (ibyr,ibmn,ibdy) model integration starting date (year, month, day)&njobpt
ibyr=1001,
ibmn=1,
ibdy=1,/# file name for variable grid spacing (option VARIABLE)&inflgfile_vgrid=’../data/vgrid.d’/# Smagorinsky parameterization for horizontal viscosity (option SMAGOR)# (cscl) scaling factor of Smagorinsky viscosity coefficient&njobsmg
cscl = 3.5d0/# parameters for isopycnal diffusion, diapycnal diffusion, layer thickness diffusion&njobpiai=1.0d7
ad=0.1d0
aitd=5.0d6/# the number of restart and history files which is output during the job&numflnum_restart=1,
num_hist=12/# the names of the mandatory input data files# snapshot data to start integration&inflafile_restart_in=’result/restart.1000’/# topography&infltopo
file_topo=’../data/topo.d’/#scale factor&inflscl
file_scale=’../data/scale_factor.d’/# temperature and salinity climatology&infltscl
file_tscl=’../data/tsclim.d’/# nudging# if iforcv (/njobp/) = 0, dummy file name for file_frc should be specified# (rtmsc) restoring time for surface forcing [day]# (rtmscb) restoring time for body forcing [day]# (kmb)the vertical level below which body forcing (nudging) is applied for all area
# (kmb_cnst) the vertical level below which body forcing (nudging) is applied

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16.3. Execution#
for the area specified in (file_frc)&njobbdy
file_frc=’../data/bforce.d’,rtmsc=8.0d0,
rtmscb=30.0d0,
kmb=52,
kmb_cnst=2,/# the name of the final state (restart) file
&outfrfile_restart_out_temp=’result/restart.1001’,/# The time step (in minutes) for the free surface equation and the way of time-filtering&njobpf
adttr=1,
ntflt=-1,/# surface forcing data
# (isrstb) the time interval of the surface forcing data
# (ifna) the first record number for the sea surface data used for ‘‘this’’ run#
the start time is between (ifna) and (ifna - 1)
&ifrcdisrstb=21600
ifna=1,/# wind stress&inflw
file_wind=’../data/file_wind_core.grd’,/# the names of the input data files for the sea surface flux
&inflhfile_hflx=’../data/file_hflux_core.grd’,
file_bulk=’../data/file_bulk_core.grd’,/# the name of the input data file for precipitation data
&inflpfile_prcp=’../data/file_prcp_core.grd’,/# the name of the input data file for river discharge data
&inflofile_rnof=’../data/file_rnof_core.grd’,/# These parameters set up the diffusion around the river mouth&nrivermouth
CFLlim=0.2d0,
nspreadnum=0,
flg_enhance_vm_rivmouth=.false.,
avdrmax=1.0d4,
dep_rivmix=30.0d2,
afc1=10.0d0,

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afc2=7.0d0,
sal_lim=5.d0,
hdsal=1.d8,/# the name of the output data file for mean states&outfhfile_hist_temp=’result/hist.1001’,/# the name of the output data file for flux mean states&outfffile_hflux_temp=’result/hflux.1001’,/# These parameters set the sea surface albedo&njobalb
alb=0.066d0,
albedo_choice=1/# basic properties for sea ice&njobpsi
irstrt=1,
akh=1.0d2,
int_bgtice=0,
nstepi=1488,
ibyri=1001,
ibmni=1,
ibdyi=1,/# the time step interval for ice dynamics&njobidyn
adtdi=1.0d0/# These parameters set the sea-ice albedo&njobalbsi
alb_ice_visible_t0=0.8d0,
alb_ice_nearIR_t0=0.52d0,
alb_snw_visible_t0=0.98d0,
alb_snw_nearIR_t0=0.70d0,
alb_ice_visible_dec_ratio=0.075d0,
alb_ice_nearIR_dec_ratio=0.075d0,
alb_snw_visible_dec_ratio=0.10d0,
alb_snw_nearIR_dec_ratio=0.15d0,
hi_ref=0.5d0,
atan_ref=3.0d0,
tsfci_t0=-1.0d0,
tsfci_t1=0.0d0,
fsnow_patch=0.02d0,/# The name of the sea ice restart file (input)&inflifile_ice_restart_in=’result/ice_restart.1000’,
/

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16.3. Execution# The name of the sea ice restart file (output)&outfir
file_ice_restart_out_temp=’result/ice_restart.1001’,/# The name of the sea ice history file
&outfihfile_ice_hist_temp=’result/ice_hist.1001’,/# These parameters set the sea-ice thickness category&njobpscat
hbound=0.0d0,0.6d0,1.4d0,2.4d0,3.6d0,30.0d0,
lsicat_volchk=.false.
/# the number of intervals by which mean states of thickness-categorized ice are calculated&nhsticint
num_hint_ic=1/# These parameters set output of the mean states for thickness-categorized sea-ice&nhsticfile
maxnum_hist_ic=1,
nwrt_hist_ic=-1,
imin_hist_ic=1,
imax_hist_ic=364,
jmin_hist_ic=1,
jmax_hist_ic=368,
file_ice_hist_ic_temp=’result/sicathsta.1001’/# The name of the restart file (input) of the thickness-categorized sea ice&inflic
file_icecat_restart_in=’result/sicatrsta.1000’/# These parameters set the output of final state for the thickness-categorized sea ice&outflic
num_rst_ic=1,
maxnum_rst_ic=1,
nwrt_rst_ic=-1,
file_icecat_restart_out_temp=’result/sicatrsta.1001’/=====================================================================================

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subject

Table 16.4. Main output data files and their related program files (see Section 16.4)
file name specified in namelist as
output fromrestart data

file restart out temp

writdt.F90(for ice averaged over all categories)

file ice restart out temp

writdt.F90(for ice for each thickness category)

file icecat restart out temp

ice restart.F90

mean values

file hist temp

writdt.F90(for sea surface fluxes)

file hflux temp

writdt.F90(for ice averaged over all categories)

file ice hist temp

writdt.F90(for ice for each thickness category)

file ice hist ic temp

ice hist.F90

16.4 Structure of output files
This section summarizes the format of the final state (restart) data to resume the model integration and that of
the mean state (history) data used for monitoring and analyses (Table 16.4).
In addition to these, we are developing a simple module that outputs a mean state or a snap shot by only
specifying a namelist at run time. For example, to have monthly mean temperature, add a following namelist entry,
An example namelist for monthly mean temperature output&nmlhs_t
fname=’result/hs_tt’
wrtint=-1[undef_mask=-9.99e33][ymdhm=2]/where fname specifies the basename of file, wrtint specifies the output interval in terms of integration time
step (-1 for monthly output), and ymdhm specifies the depth of calendar date used in the file name. In the above
example, the temperature averaged for a month of mm of a year yyyy is output to file result/hs tt.yyyymm. See
README.Namelist for items available.

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16.4.1

Snapshot (restart)

Restart for the ocean model

integer(4) :: last, month, iday, ihour, iminreal(8) :: ahour(km), aday(km), ayear(km), pd(km), pm(km+1), dmn(km)real(8) :: ddmna, overreal(8) :: u(imut,jmut,km), v(imut,jmut,km)
real(8) :: t(imut,jmut,km), s(imut,jmut,km)
real(8) :: avd(imut,jmut,km), avm(imut,jmut,km)
real(8) :: avq(imut,jmut,km), eb(imut,jmut,km)
real(8) :: dbuf(imut,jmut,3)
integer(4) :: nu ! device number! The informations below are written to a single file in the following order! by serial form for the default case.
! last: the total number of the integrated time steps!
: at the time of writing the restart data
! month, iday, ihour, imin : month, day, hour and minute for restart data
! over: the parameter of relaxation for barotropic stream function!
: (dummy data for FREESURFACE)! ahour, aday, ayear : total integrated hour, day and year from the start!
: of the time integration (ahour is master)! pd, pm : averaged pressure at each vertical level!
: used for the calculation of the equation of state!
: pd: T-points!
: pm: middle point between adjacent T-points! ddmna : averaged density of the whole ocean! dmn: averaged density at each vertical level
write(nu) last, month, iday, ihour, imin, over, && ahour, aday, ayear, pd, pm, ddmna, dmn! In the case of SPLITREST the followings are written in the separate files! after the above informations about time.
write(nu) u
write(nu) v
write(nu) t
write(nu) s
write(nu) dbuf

write(nu) avd
write(nu) avm
write(nu) avq
write(nu) eb! zonal velocity! meridional velocity! temperature! salinity! 1: sea surface height! 2: vertically integrated zonal transport! 3: vertically integrated meridional transport! the vertical diffusion coeff. in the case of VVDIMP! the vertical viscosity coeff. in the case of VVDIMP! the vertical diffusion coeff. of turbulent kinetic energy! in the case of NOHKIM! in the case of NOHKIM turbulent kinetic energy

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For sea ice, the grid-averaged state and the thickness-categorized states are written to separate files.

Restart for grid-averaged sea-iceinteger(4) :: last, month, iday, ihour, imin ! from the ocean main partreal(8) :: a0iceo(imut,jmut) ! sea-ice concentration
real(8) :: hiceo(imut,jmut) ! sea-ice thickness
real(8) :: hi(imut,jmut) ! averaged thickness = a0iceo * hiceo
real(8) :: hsnwo(imut,jmut) ! snow depthreal(8) :: hsnw (imut,jmut) ! averaged snow depth = a0iceo * hsnwo
real(8) :: t0iceo(imut,jmut) ! skin temperature beneath the sea ice
real(8) :: t0icel(imut,jmut) ! skin temperature in the open leadsreal(8) :: tsfci (imut,jmut) ! sea-ice surface temperaturereal(8) :: uice(imut,jmut), vice(imut,jmut) ! drift vector! stress tensor
real(8) :: sigma1(imut,jmut), sigma2(imut,jmut), sigma3(imut,jmut)
real(8) :: dbuf(imut,jmut,10)
integer(4) :: nui ! device number
write (nui) last, month, iday, ihour, imin
dbuf(1:imut,1:jmut,1) = a0iceo(1:imut,1:jmut)
dbuf(1:imut,1:jmut,2) = hi(1:imut,1:jmut)
dbuf(1:imut,1:jmut,3) = hiceo(1:imut,1:jmut)
dbuf(1:imut,1:jmut,4) = hsnw(1:imut,1:jmut)
dbuf(1:imut,1:jmut,5) = hsnwo(1:imut,1:jmut)
dbuf(1:imut,1:jmut,6) = t0iceo(1:imut,1:jmut)
dbuf(1:imut,1:jmut,7) = t0icel(1:imut,1:jmut)
dbuf(1:imut,1:jmut,8) = tsfci(1:imut,1:jmut)
dbuf(1:imut,1:jmut,9) = uice(1:imut,1:jmut)
dbuf(1:imut,1:jmut,10) = vice(1:imut,1:jmut)
write(nui) dbuf(1:imut,1:jmut,1:10)
dbuf(1:imut,1:jmut,1) = sigma1(1:imut,1:jmut)
dbuf(1:imut,1:jmut,2) = sigma2(1:imut,1:jmut)
dbuf(1:imut,1:jmut,3) = sigma3(1:imut,1:jmut)
write(nui) dbuf(1:imut,1:jmut,1:3)





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

Restart data for thickness-categorized sea-iceinteger(4) :: nstep_job, month, iday, ihour, iminreal(8):: aihour ! total integrated time
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) :: nu_icecat_rst ! device number
write (nu_icecat_rst) nstep_job, month, iday, ihour, imin, aihour
do m = 0, ncat
write(nu_icecat_rst) aicen(1:imut,1:jmut,m)write(nu_icecat_rst) hin (1:imut,1:jmut,m)write(nu_icecat_rst) hsn (1:imut,1:jmut,m)write(nu_icecat_rst) hicen(1:imut,1:jmut,m)write(nu_icecat_rst) hsnwn(1:imut,1:jmut,m)write(nu_icecat_rst) tsfcin(1:imut,1:jmut,m)write(nu_icecat_rst) t1icen(1:imut,1:jmut,m)write(nu_icecat_rst) t0icen(1:imut,1:jmut,m)write(nu_icecat_rst) s0n(1:imut,1:jmut,m)end dowrite(nu_icecat_rst) a0iceo(1:imut,1:jmut)
write(nu_icecat_rst) hi(1:imut,1:jmut)
write(nu_icecat_rst) hiceo(1:imut,1:jmut)
write(nu_icecat_rst) hsnw(1:imut,1:jmut)
write(nu_icecat_rst) hsnwo(1:imut,1:jmut)
write(nu_icecat_rst) t0iceo(1:imut,1:jmut)
write(nu_icecat_rst) t0icel(1:imut,1:jmut)
write(nu_icecat_rst) tsfci(1:imut,1:jmut)
write(nu_icecat_rst) uice(1:imut,1:jmut)
write(nu_icecat_rst) vice(1:imut,1:jmut)
write(nu_icecat_rst) sigma1(1:imut,1:jmut)
write(nu_icecat_rst) sigma2(1:imut,1:jmut)
write(nu_icecat_rst) sigma3(1:imut,1:jmut)

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16.4.2 Averaged value (history)

Ocean model history data (HIST)

integer(4) :: nkai, month, iday, ihour, imin, mdaysreal(4) :: um(imut,jmut,km), vm(imut,jmut,km)
real(4) :: tm(imut,jmut,km), sm(imut,jmut,km)
real(4) :: hm(imut,jmut)
integer(4) :: nu_hst ! device number! The informations below are written to a single file in the following order! by serial form for the default case.
! nkai: the total number of the integrated time steps!
: at the time of writing the data
! month, iday, ihour, imin : month, day, hour and minute!
: at the time of writing the data
! mdays : the data are averaged over mdays
write(nu_hst) nkai, month, iday, ihour, imin, mdays! In the case of SPLITHIST the data below are written in a different file! after the above information about time
write(nu_hst) um
write(nu_hst) vm
write(nu_hst) tm
write(nu_hst) sm
write(nu_hst) hm! zonal velocity! meridional velocity! temperature! salinity! sea surface heightsea surface flux averaged value history file (HISTFLUX)

integer(4) :: nkai, month, iday, ihour, imin, mdaysreal(4) :: sqlw (imut,jmut) ! longwave radiation (downward minus upward)
real(4) :: sqsn (imut,jmut) ! sensible heat flux
real(4) :: sqla (imut,jmut) ! latent heat flux
real(4) :: shflux(imut,jmut) ! total heat flux
real(4) :: swflux(imut,jmut) ! total fresh water flux
real(4) :: sstrx(imut,jmut), sstry(imut,jmut) ! sea surface stressinteger(4) :: nuf_hst ! device number
write(nuf_hst) nkai, month, iday, ihour, imin, mdays
write(nuf_hst) sqlw (1:imut,1:jmut)
write(nuf_hst) sqsn (1:imut,1:jmut)
write(nuf_hst) sqla (1:imut,1:jmut)
write(nuf_hst) shflux(1:imut,1:jmut)
write(nuf_hst) swflux(1:imut,1:jmut)
write(nuf_hst) sstrx (1:imut,1:jmut)
write(nuf_hst) sstry (1:imut,1:jmut)





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For sea ice, the grid-averaged state and the thickness-categorized states are written to separate files.

Grid-averaged sea-ice history file (ICE)integer(4) :: nkai, month, iday, ihour, imin, mdaysreal(4) :: sar(imut,jmut) ! sea-ice concentration
real(4) :: shi(imut,jmut) ! averaged sea-ice thickness
real(4) :: ssn(imut,jmut) ! averaged snow depthreal(4) :: sti(imut,jmut) ! sea-ice surface temperaturereal(4) :: uice(imut,jmut), vice(imut,jmut) ! drift vector
real(4) :: sbuf(imut,jmut,6)
integer(4) :: nui_hst ! device number
write (nui_hst) nkai, month, iday, ihour, imin, mdays
sbuf(1:imut,1:jmut,1) = shi(1:imut,1:jmut)
sbuf(1:imut,1:jmut,2) = ssn(1:imut,1:jmut)
sbuf(1:imut,1:jmut,3) = sar(1:imut,1:jmut)
sbuf(1:imut,1:jmut,4) = sti(1:imut,1:jmut)
sbuf(1:imut,1:jmut,5) = sui(1:imut,1:jmut)
sbuf(1:imut,1:jmut,6) = svi(1:imut,1:jmut)
write(nui_hst) sbuf(1:imut,1:jmut,1:6)

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Thickness-categorized sea-ice history file (ICECAT)

integer(4) :: nstep_job, month, iday, ihour, imin, mdays
integer(4), parameter :: ncat = 5 ! number of thickness category
real(4) :: shi0(1:imut,1:jmut) ! sum of averaged sea-ice thickness! for all categories
real(4) :: shs0(1:imut,1:jmut) ! sum of snow depth for all category
real(4) :: sai0(1:imut,1:jmut) ! sum of concentration for all category
real(4) :: sui0(1:imut,1:jmut), svi0(1:imut,1:jmut) ! sea-ice drift velocity
real(4) :: sain(1:imut,1:jmut,0:ncat) ! sea-ice concentration
real(4) :: st0n(1:imut,1:jmut,0:ncat) ! sea surface skin temperaturereal(4) :: ss0n(1:imut,1:jmut,0:ncat) ! sea surface skin salinity
real(4) :: shin(1:imut,1:jmut,1:ncat) ! averaged sea-ice thickness
real(4) :: shsn(1:imut,1:jmut,1:ncat) ! averaged snow depthreal(4) :: stsn(1:imut,1:jmut,1:ncat) ! sea-ice surface temperaturereal(4) :: st1n(1:imut,1:jmut,1:ncat) ! sea-ice temperature
integer(4) :: nu_icecat_hst ! device number
write(nu_icecat_hst) nstep_job, month, iday, ihour, imin, mdays
write(nu_icecat_hst) = shi0(1:imut,1:jmut)
write(nu_icecat_hst) = shs0(1:imut,1:jmut)
write(nu_icecat_hst) = sai0(1:imut,1:jmut)
write(nu_icecat_hst) = sui0(1:imut,1:jmut)
write(nu_icecat_hst) = svi0(1:imut,1:jmut)
write(nu_icecat_hst) = sain(1:imut,1:jmut,0)
write(nu_icecat_hst) = st0n(1:imut,1:jmut,0)
write(nu_icecat_hst) = ss0n(1:imut,1:jmut,0)
do m = 1, ncat
write(nu_icecat_hst) = shin(1:imut,1:jmut,m)write(nu_icecat_hst) = shsn(1:imut,1:jmut,m)write(nu_icecat_hst) = sain(1:imut,1:jmut,m)write(nu_icecat_hst) = stsn(1:imut,1:jmut,m)write(nu_icecat_hst) = st1n(1:imut,1:jmut,m)write(nu_icecat_hst) = st0n(1:imut,1:jmut,m)write(nu_icecat_hst) = ss0n(1:imut,1:jmut,m)end do





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16.5. Appendix

16.5

Appendix

16.5.1Model options

The model options are as follows. Only major options are listed here. Description about those related to biogeochemical models can be found in chapter 11. The description of all options for the latest version can be foundin src/README.Options. In the source program an expression like OGCM PARALLEL is used but here OGCM isomitted for option expressions.
Table 16.5. Description of Model Options
Model option

Description

BBL
BF2D

uses the bottom boundary layer model

BIHARMONICBULKKARABULKKONDO2BULKNCAR

reads 2-D distribution of ”on”/”off” for nudging
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.
Kara (2000) is used for the surface flux bulk formula.
Kondo(1975) is used for the surface flux bulk formula.
Large and Yeager (2004) is used for the surface flux bulk formula. This option corresponds
to the COREs.

BULKITER(*) BULKKARA, BULKKONDO2, BULKNCAR is available only for HFLUX case.
Bulk transfer coefficient is calculated using iterative method

CALALBSI

if the observed wind speed is not at 10m.
(*) use with BULKKONDO2 and BULKNCAR
Sea-ice albedo is calculated using sea-ice conditions according to Los-Alamos model instead

CALPP
CARBONof using a constant value
considers the time variation of pressure for the equation of state

CBNHSTRUN

bio-geochemical process is included(*) NUMTRC P=4 for Obata-Kitamura model; NUMTRC P=8 for NPZD model
atmospheric pCO2 is given from file

CHFDIST(*) use with CARBON
employs horizontal distribution of surface restoring time for SST and SSS

CHLMA94
CLMFRCCYCLICDIAGTRANSP

shortwave penetration scheme with chlorophyll concentration by Morel and Antoine (1994)(*) use with NPZD and SOLARANGLE
uses climatological wind stress and surface heat flux (no leap year, repeating one year cycle)
uses zonally cyclic condition
outputs time averaged vertically integrated transport UM, VM

DIFAJS(*) available only for FREESURFACE
sets large vertical diffusion coefficient (1.0 m2 · s−1 ) between unstable points instead of convective adjustment

FLXINTPOL

interpolates the surface forcing data to model grid points

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Chapter 16

User’s Guide

Model option

description

FREESURFACE
FSEB
FSMOM

free surface (if undefined, rigid-lid)
free surface equations are advanced using the Eular-backward time-integration

FSVISC

uses MOM free surface scheme
calculates viscosity explicitly in the barotropic momentum equation(*) FSEB, FSMOM, FSVISC are available only in the case of FREESURFACEGMANISOTROP

Anisotropic horizontal variation of thickness diffusion is used(*) use with ISOPYCNAL

GMVAR
HFLUX
HIST

Horizontal thickness diffusion is allowed to vary in horizontal
calculates sea surface heat flux using bulk formulaoutputs averaged state of temperature, salinity and velocity

HISTFLUXHISTVARoutputs averaged state of surface fluxes
outputs variance of temperature, salinity and velocity

ICE
ICECAT(*) available only for HIST
sea ice is included
sea ice is categorized according to its thickness

ICECLIMINILEVreading climatological sea-ice fractional area from filesets Levitus climatological three-dimensional data for the initial value

ISOPYCNAL

uses isopycnal diffusion and Gent-McWilliams’ parameterization for eddy
induced tracer transport velocity (thickness diffusion)uses KPP for mixed layer model

KPP
LWDOWNMELYAMLong wave radiation data include only downward component instead of default net radiation
uses Mellor and Yamada Level 2.5 for mixed layer model

MON30DMONFRCNESTONLINE

sets one month as 30 days and one year as 360 days
uses monthly forcing for surface data
low resolution (PARENT) and high resolution (SUB) models exchange data on-line

NOHKIM(*) use with SCUP
uses Noh’s mixed layer model

NOMATSUNONPZD

uses forward finite difference instead of Matsuno scheme
NPZD process is included(*) use with CARBONPARALLEL

parallel calculation using MPI. The number of zonally and meridionally partitioned regions
should be specified as NPARTX and NPARTY, respectively.

PARENT
PRAJS

executed as low resolution model of the nesting calculation
adjusts the freshwater flux to suppress the increase/decrease of total ocean volume(*) available only for WFLUXRUNOFFuses river runoff data(*) available only for WFLUXSALCNSVRSSalinity restoring flux is corrected so that globally integrated salinity flux become zeroat each time step

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16.5. Appendix
Model option
SFLUXRSFLUXW

description(*) use with ”SFLUXR”
SSS is restored to the climatological sea surface salinity as the salinity flux

SIDYN

calculates salinity flux converting from the surface freshwater flux(*) available only for WFLUX
sea-ice dynamics model (EVP)

SLIMIT(*) available only for ICE
Tapers thickness diffusion near the sea surface

SMAGHDuses the Smagorinsky viscosity coefficient multiplied by a constant ratio as the horizontal
diffusion coefficient(*) available only for SMAGORSMAGORSOLARANGLE

uses the Smagorinsky parameterization for horizontal viscosity
solar insolation angle is considered in calculating short wave penetration

SOMADVECSPHERICALSPLITHIST

uses second order moment advection by Prather (1986)
calculates scale factor semi-analytically for the spherical coordinates
output averaged temperature, salinity, velocity and sea surface height to separate files

SPLITREST(*) available only for HIST
input(output) initial(final) state data from(to) separate files for each propertiesSUB
TAUBULK
TDEW

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

TRCBIHARM(*) available only for HFLUX
uses biharmonic operator for horizontal diffusion

TSCLDIRECT

TSINTPOL(*) Should not be used with ISOPYCNAL
Temperature and salinity climatology is read from direct access files
naming convention is (file tscl)mm (mm = month)see README.Namelistinterpolates climatological temperature and salinity data to model grids

ULTIMATE

applies ultimate limiter for the calculation of the horizontal advection of temperature andsalinity(*) available only for for UTOPIAUTOPIA
VARHIDuses UTOPIA scheme for horizontal advection of temperature and salinity
sets the horizontal (isopycnal) diffusion coefficient and viscosity as functions of grid size

VARIABLEVISANISOvariable horizontal grid spacing
Anisotropic viscosity coefficients are used(*) use with VIS9P

VIS9P
VISBIHARM
VMBG3D

calculates the viscosity using adjacent 9 grid points
uses biharmonic operator for both horizontal viscosity

VVDIMP

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 option ISOPYCNAL is selected

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Chapter 16

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Model option

description

WADJ

adjusts sea surface freshwater flux every time step to keep its global sum to be zero(*) available only for WFLUX

WFLUXuses the sea surface freshwater flux to force the modelno leap year
uses QUICKEST for the vertical advection of temperature and salinity

Y365
ZQUICKESTZULTIMATE

applies ultimate limiter for the vertical advection of temperature and salinity(*) available only for ZQUICKEST

CGCM
MOVE
SCUP

used as an ocean module for a coupled model
used as ocean module for data assimilation (MOVE) systemuse simple coupler (SCUP) library

SCUPCGCM

used as an ocean module for a coupled model using scup for communication

References
Ishizaki, H., and I. Ishikawa, 2006: High parallelization efficiency in barotropic-mode computation of ocean
models based on multi-grid boundary ghost area, Ocean Modell., 13, 238-254.

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Acknowledgments
The development of MRI.COM was first proposed by Yoshiteru Kitamura. During the course of its development,
significant contributions were made by Seiji Yukimoto and Atsushi Obata.
The authors owe special thanks to Hiroyasu Hasumi of the University of Tokyo for his valuable suggestions.
Application of the EVP dynamics to the sea ice model, of σ -coordinates to the free surface formulation, and ofhigh-accuracy advection schemes to the tracer equations would not have been successful without his support.
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. Amongthem, various comments made by Yosuke Fujii during his development of an adjoint code of this model wereindispensable for refining the model codes.
Thanks are extended to Shunji Konaga, Tatsushi Tokioka, Masahiro Endoh, Yoshihiro Kimura, and Noriya
Yoshioka for their numerous contributions to the original models that MRI.COM is based on.
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 Departmentof MRI contributed to the development and refinement of this model.
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 andworkshops. This fact strongly motivated us to write a new version in English. Continuous encouragement andnumerous suggestions by Stephen Griffies at Geophysical Fluid Dynamics Lab., NOAA, USA and Gurvan Madecat LODYC, Institute Pierre Simon Laplace, France, during panel meetings of the working group for ocean model
development (WGOMD) of Climate Variability and Predictability (CLIVAR) are gratefully acknowledged.

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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)
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(Takayuki Tokieda and Hisayuki Y. Inoue，2004)
第 44 号 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 CO2 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(SF6)の測定手法の高度化と SF6 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 (SF6) and evaluation of SF6 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 pHT 測定（斉藤 秀, 石井雅男, 緑
川 貴, 井上（吉川）久幸, 2008）
Precise Spectrophotometric Measurement of Seawater pHT 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.
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）