International Hydrological Programme

Numerical Prediction of High-Impact
Weather Systems-The Textbook for Seventeenth IHP Training Course in 2007-

Kazuhisa Tsuboki and Atsushi Sakakibara

Hydorospheric Atmospheric Research Center, Nagoya University

United Nations Educational Scientific and Cultural Organization

Prepared for the Sventeenth IHP Training Course on Numerical Prediction of High-Impact Weather
Systems, 2 - 15 December, 2007, Nagoya, Japan
Working Group for IHP Training Course,
Sub-Committee for IHP,
Japan National Commission for UNESCO
Chairperson: Prof. H. Uyeda, Nagoya University
Secretariat:
Mr. K. Akiyama, Ministry of Education, Culture, Sports, Science and Technology
Ms. E. Himata, Ministry of Education, Culture, Sports, Science and Technology
Ms. C. Akibayashi, Ministry of Education, Culture, Sports, Science and Technology
Mr. A. Yamamoto, Nagoya University
Members:
Prof. K. Takeuchi, Yamanashi University
Prof. Y. Fukshima, Research Institute for Humanity and Nature
Mr. T. Adachi, Ministry of Land Infrastructure and Transport
Dr. M. Sugi, Japan Meteorological Agency
Dr. K. Nakane, National Research Institute for Earth Science and Disaster Prevention
Dr. N. Horikawa, National Institute for Rural Engineering
Prof. T. Yanagi, Kyushu University
Prof. T. Hiyama, Nagoya University
Prof. A. Morimoto, Nagoya University
Prof. K. Tsuboki, Nagoya University
Prof. K. Nakamura, Nagoya University
Published in 2007 by the Hydrospheric Atmospheric Research Center (HyARC), Nagoya University,and United Nations Educational Scientific and Cultural Organization (UNESCO)
Printed by Nagoya University COOP
ISBN: 978-4-9980619-8-4
Cover: Beautiful clouds with precipitation over the Pacific Ocean near Palau. (Photo by K. Tsuboki)

iPreface
The general aim of the 17th IHP Training Course is to help participants develop their basic
knowledge of numerical prediction of high-impact weather systems, which is necessary for prediction and prevention/reduction of meteorological disasters. The course is part of UNESCO
International Hydrological Programme (IHP) and is held every Japanese ﬁscal year that starts
in April. The participants mainly from Asia-Paciﬁc regions are given a series of lectures andpractical trainings for about two weeks. Until 1996, the course was mainly focused on generalhydrology and later concentrated on speciﬁed themes such as: snow hydrology (1998), remotesensing (1999), limnology (1999), hydrology related to head water management (2000), hydrogenand oxygen isotopes in hydrology (2002), precipitation and water resources (2003), eﬀect of airpollutants on the atmospheric environments and climate changes (2004), water and carbon cycles
in terrestrial ecosystems (2005), and oceanography basics (2006).
This year, the Seventeenth IHP Training Course with a theme of ”Numerical Prediction ofHigh-Impact Weather Systems” will be held from 2 November to 15 December, 2007 at Nagoya
and Tokyo with many lectures and practices. This IHP short course will focus on understanding
basic knowledge of numerical prediction of high-impact weather systems. Participants will havepractice to perform some simulation experiments of weather systems to understand how to use
the cloud-resolving model, CReSS (cloud resolving storm simulator) developed in HyARC. Highresolution numerical simulation using the CReSS model will clarify a detailed structure of highimpact weather systems and make a quantitative prediction of the associated precipitation.
The Seventeenth course will be conducted by contributions from professionals/experts in universities and research institutes led by Prof. K. Tsuboki, Dr. T. Shinoda and Prof. T. Nagai,
Nagoya University, Dr. T. Nasuno, Frontier Research Center for Global Change, Dr. T. Maesaka, National Research Institute for Earth Science and Disaster Prevention, Dr. M. Yoshizaki,
Institute of Observational Research for Global Change, Dr. K. Saito, Meteorological ResearchInstitute, Prof. T. Aoki, Tokyo Institute of Technology, Prof. K. Tanaka, Kyoto University, Dr.
W. Ofuchi, The Earth Simulator Center, and Mr. A. Sakakibara, Chuden CTI Co.,Ltd.
The course would not have been possible to conduct without management supports. Dr.
Giuseppe Arduino and his staﬀ in UNESCO Jakarta Oﬃce worked for the arrangement forthe participants. The IHP Training Course is under continuous support from the Ministry of
Education, Culture, Sports, Science, and Technology.

November 28, 2007

Hiroshi Uyeda
Chairperson ofWorking Group for IHP Training Course
Director,
Hydrospheric Atmospheric Research Center,
Nagoya University

ii

Acknowledgments
I would like to express my sincere thanks to Prof. Hiroshi Uyeda, the chairperson of the working
group of the IHP Training Course and Director of the Hydrospheric Atmospheric Research Center(HyARC), Nagoya University, Japan.
The support from the Japan National Commission for UNESCO, sub-committee of IHP led by
Prof. Kuniyoshi Takeuchi from Yamanashi University, and the working group of IHP Training
Course is highly acknowledged. Thanks are also extended to Mr. Kazuo Akiyama and Ms.
Eriko Himata from the Ministry of Education, Culture, Sports, Science and Technology of Japan,
Mr. Akihiro Yamamoto from Nagoya University, Japan for sharing the role of secretariat, andthe staﬀs (Dr. Taro Shinoda, Ms. Machiko Toida, Ms. Saori Haga and Ms. Mariko Kayaba)of HyARC, Nagoya University, Japan for their local arrangements and technical editing of thistextbook.

November 22, 2007

Kazuhisa Tsuboki
Hydrospheric AtmosphericResearch Center (HyARC),
Nagoya University

Editor

Contents
I

Simulations of High-Impact Weather Systems

1 Introduction

5
7

1.1

Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

1.2

Brief description of CReSS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

1.3

History of CReSS development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 Simulations and Numerical Experiments

13

2.1

Mountain waves: dry experiment in 2-dimensional geometry . . . . . . . . . . . . . 14

2.2

Kelvin-Helmholtz billows: dry experiment in two-dimensional geometry . . . . . . 16

2.3

Simulation of Tornado within a Supercell: moist experiment in three-dimensional
geometory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.4

Simulation of a squall line: moist experiment in three-dimensional geometory . . . 19

2.5

Typhoons and the associated heavy rainfall: T0418 and T0423 . . . . . . . . . . . 20

2.6

Typhoon RUSA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.7

Localized heavy rainfall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.8

Idealized experiment of snow cloud bands . . . . . . . . . . . . . . . . . . . . . . . 28

2.9

Snow storms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.10 Heavy rainfall in Baiu season . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.11 Typhoon 0613 and associated tornadoes . . . . . . . . . . . . . . . . . . . . . . . . 38

II

Technical Description

3 Start to Use CReSS

41
43

3.1

Download source ﬁles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.2

Uncompress archived ﬁle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.3

Compilation and samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.4

To Display the result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4 Conﬁguration and Execution
4.1

49

User conﬁguration ﬁle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.1.1

Composition of user.conf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

iv

CONTENTS

4.2

4.3

4.4

4.1.2

Common region of the user conﬁguration ﬁle . . . . . . . . . . . . . . . . . 50

4.1.3

Region for the pre-processor, gridata.exe. . . . . . . . . . . . . . . . . . . . 75

4.1.4

Region for the pre-processor, radata.exe. . . . . . . . . . . . . . . . . . . . 78

4.1.5

Region for the pre-processor, terrain.exe. . . . . . . . . . . . . . . . . . . . 80

4.1.6

Region for the pre-processor, surf ace.exe. . . . . . . . . . . . . . . . . . . . 82

4.1.7

Region for the post-processor, unite.exe. . . . . . . . . . . . . . . . . . . . . 88

4.1.8

Region for the post-processor, rstruct.exe.

Data ﬂow in CReSS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.2.1

Outline of data ﬂow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

4.2.2

Data ﬂows of each conﬁguration . . . . . . . . . . . . . . . . . . . . . . . . 92

Formats of I/O data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
4.3.1

Format of the sounding data . . . . . . . . . . . . . . . . . . . . . . . . . . 94

4.3.2

Formats of the history dumped ﬁle and the geographic information ﬁle . . . 96

Execution of CReSS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
4.4.1

Execution of solver.exe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

4.4.2

Execution of unite.exe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

4.4.3

Execution of terrain.exe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

4.4.4

Execution of gridata.exe. . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

5 Program Lists
5.1

III

. . . . . . . . . . . . . . . . . . 88

107

Program list . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
5.1.1

Main programs of CReSS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

5.1.2

List of subroutines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

5.1.3

MPI subroutines list . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

Fundamental Theory and Formulations

6 The Basic Equations of CReSS

125
127

6.1

The basic equations without terrain . . . . . . . . . . . . . . . . . . . . . . . . . . 128

6.2

The basic equations system in terrain-following coordinates . . . . . . . . . . . . . 129

6.3

6.2.1

General curvilinear coordinates . . . . . . . . . . . . . . . . . . . . . . . . . 129

6.2.2

Terrain-following coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . 132

6.2.3

The basic equations in terrain-following coordinates . . . . . . . . . . . . . 136

Basic Equations — Map Projection . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
6.3.1

Basic equations with map projection — no topography . . . . . . . . . . . . 139

6.3.2

Basic equations with map projection and terrain . . . . . . . . . . . . . . . 143

6.3.3

Conformal map projections . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

7 Subgrid Scale Diﬀusion
7.1

147

Parameteriaation of Turbulence Transport . . . . . . . . . . . . . . . . . . . . . . . 148

CONTENTS

7.2

Eddy Viscosity Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
7.2.1

Formulation of Diﬀusion Term . . . . . . . . . . . . . . . . . . . . . . . . . 149

7.2.2

One order closure of Smagorinsky . . . . . . . . . . . . . . . . . . . . . . . . 151

7.2.3

One and a half order closure with turbulence kinetic energy . . . . . . . . . 154

8 Physical Processes of Cloud and Precipitation
8.1

8.2

9.2

157

Parametarization of “Warm Rain” with a Bulk Method . . . . . . . . . . . . . . . 158
8.1.1

Equations for “Warm Rain” Process of Cloud and Precipitation . . . . . . . 158

8.1.2

Microphysical Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

Parameterization of the bulk cold rain . . . . . . . . . . . . . . . . . . . . . . . . . 160
8.2.1

Parameterization of the cold rain . . . . . . . . . . . . . . . . . . . . . . . . 160

8.2.2

The Equation System for Cloud and Precipitation Processes . . . . . . . . . 162

8.2.3

Expression of Particles of Cloud and Precipitation . . . . . . . . . . . . . . 166

8.2.4

Formulizations of Physical Processes of Source terms . . . . . . . . . . . . . 171

8.2.5

Moist Saturation Adjustment . . . . . . . . . . . . . . . . . . . . . . . . . . 197

8.2.6

Formulization of Variational Terms on Mixing Ratio and Number Concentration with Precipitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

8.2.7

The other constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

9 Boundery Layer and Surface Process
9.1

v

201

Basic Theories of Atmospheric Boundary Layer . . . . . . . . . . . . . . . . . . . . 202
9.1.1

Structure of Atmospheric Boundary Layer . . . . . . . . . . . . . . . . . . . 202

9.1.2

Parameterization of Turbulence Transportation . . . . . . . . . . . . . . . . 202

9.1.3

Mixing Length Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

9.1.4

Surface Boundary Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

9.1.5

Vertical Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

Boundary Layer, Surface and Soil Temperature . . . . . . . . . . . . . . . . . . . . 209
9.2.1

Cloudiness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

9.2.2

Solar Radiation and Downward Long Wave Radiation . . . . . . . . . . . . 213

9.2.3

Flux at the Ground Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

9.2.4

Atmospheric Boundery Layer Process . . . . . . . . . . . . . . . . . . . . . 225

9.2.5

Ground Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

10 Numerical Computation Method

235

10.1 The discretization of basic equation system . . . . . . . . . . . . . . . . . . . . . . 236
10.1.1 The outline of the numerical solution . . . . . . . . . . . . . . . . . . . . . . 236
10.1.2 Grid in the model and setting of variables . . . . . . . . . . . . . . . . . . . 236
10.1.3 Discretization by ﬁnite diﬀerence method . . . . . . . . . . . . . . . . . . . 237
10.2 Artiﬁcial numerical viscosity term

. . . . . . . . . . . . . . . . . . . . . . . . . . . 252

10.2.1 Necessity of numerical viscosity term . . . . . . . . . . . . . . . . . . . . . . 252
10.2.2 Numerical viscosity with the second-order or fourth-order diﬀusion term . . 253

vi

CONTENTS

10.3 Boundary condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256
10.3.1 Lateral boundary condition . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
10.3.2 Boundary conditions at the top and bottom . . . . . . . . . . . . . . . . . . 262
10.3.3 Sponge layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
11 Application of Parallel Processing

265

11.1 Technique for parallelization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266
11.1.1 Two-dimensional domain decomposition . . . . . . . . . . . . . . . . . . . . 266
11.1.2 Example of parallel computing . . . . . . . . . . . . . . . . . . . . . . . . . 267
11.2 Test of the parallel program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
11.2.1 Veriﬁcation of coincidence of a calculation result . . . . . . . . . . . . . . . 269
11.2.2 Eﬃciency of parallel processing . . . . . . . . . . . . . . . . . . . . . . . . . 270

CONTENTS

Acknowledgments
The development of the CReSS model was supported by the Special Coordination Funds for
Promoting Science and Technology, ”Development of parallel softwares for large-scale simulation to predict the global climate change with high resolution” led by Professor A. Sumi at the
University of Tokyo, and by the Research Organization for Information Science and Technology(RIST) for the period from 1998 to 2003. The authors would like to thank Professor A. Sumi,
Professor N. Kamiya at Aichi-Gakusen University and Mr. H. Nakamura of the RIST for theirindispensable contributions to the model development.
The Hydrospheric Atmospheric Research Center (HyARC) at Nagoya University has supported
the development of the CReSS model as a research program since 2003. We used the mainframe
computers and supercomputers at the University of Tokyo and Nagoya University for the development and calculations of the CReSS model. We also used the Earth Simulator at the Earth
Simulator Center for simulations and calculations of the CReSS model. Part of the textbook
was written during K. Tsuboki’s visit to the International Paciﬁc Research Center (IPRC) at the
University of Hawaii. The IPRC is partly funded by the JAMSTEC in Japan. Tsuboki would
like to thank Dr. H. Kubota at the JAMSTEC and Professor Shang-Ping Xie at the Universityof Hawaii for this opportunity of visit.

1

2

CONTENTS

Notiﬁcations
The Cloud Resolving Storm Simulator (CReSS) version 2.2 was developed by Kazuhisa Tsuboki
and Atsushi Sakakibara, who reserve all the rights to the CReSS vers. 2.2 and its users’ guide.
The CReSS vers. 2.2 are free for scientiﬁc research and studies. If one wishes to use the CReSS
vers. 2.2 for operational or commercial purposes, he or she is requested to notify the developers.
This is to ensure that the CReSS vers. 2.2 are being used correctly and updated timely.
Users of the CReSS model should assume full responsibility for using the model. The developers
and administrators of any versions of CReSS will not be responsible for any damages resulted
from using the CReSS model, including those caused by errors in the code.
The developers kindly request that users acknowledge the CReSS model in their publications.
It would be appreciated very much if users would notify the use of the CReSS model and send a
copy of their publications to Tsuboki (tsuboki@rain.hyarc.nagoya-u.ac.jp).

CONTENTS

Preface
At the end of the twentieth century, massive computers became reality, with theoretical performance reaching Tera-FLOPS, main memory of Gigabytes, and storage of Terabytes. This led us tobelieve that we could perform super-massive calculations at the beginning of the twenty-ﬁrst century. What should we do with such a computing capability that is available now? Meteorologists
have longed for faster and more massive computers for their numerical models. A cloud-resolving
model is one of such models, which includes complicated interactions among water vapor, cloudparticles, and precipitation.
Research on cloud and precipitation using a cloud-resolving model is still at its developing stage.
There remain many unsolved problems, such as parameterizations of various physical processes,
techniques for eﬃcient computations, minimization of computational errors, etc. There are also a
lot of atmospheric phenomena to be discovered. Cloud-resolving models will help us to understandvarious phenomena related to cloud and precipitation systems.
What is needed for the development of cloud-resolving models? Researches with diﬀerent
viewpoints are needed for the development of cloud-resolving models. A large number of numericalexperiments using these cloud-resolving models are also needed. We can advance our knowledge
through trials and errors using these models. Only then, cloud-resolving models will be reﬁned.
As numerical models become more sophisticated, it is ineﬃcient, even impossible, for individual researcher to develop his own cloud-resolving model from a scratch. Most scientists are
more interested in using a numerical model as a tool for their own research, which may involve
computational methods, physical process studies, simulations of observed phenomena, weather
forecasts for disaster prevention, and so on. Out of such a consideration, we have been developing a cloud-resolving model, which was supported by Special Coordination Funds for Promoting
Science and Technology (SCF), ”Study on Advanced Simulations of Downburst Phenomenon”(Principal Investigator: Professor Nobuhiko Kamiya). This cloud-resolving model could serve asa basic platform for these scientists.
Recently, some models have included non-hydrostatic and compressible equations for their cloudphysical processes, such as the Advanced Regional Prediction System (ARPS) developed by the
University of Oklahoma and the Non-Hydrostatic Model (NHM) developed by the MeteorologicalResearch Institute of the Japan Meteorological Agency. We would like to acknowledge the useful
information provided by these models. The development of our own cloud-resolving model isheavily based on many previous studies. We would like to express our appreciations for theseauthors.
Computers are been improved rapidly, especially their hard-wears. A few years ago, only asingle-vector processor was used for massive calculations. Now, parallel computers with multiple
processors are the main player for the computations; they are necessary for the amount of calculations involved in a cloud-resolving model. Our cloud-resolving model is designed to performon parallel computers with high eﬃciency. It includes many physical processes with a high resolution, while its computation domain is smaller than those of global climate models. Parallel
computations are necessary for the huge calculations using the cloud-resolving model.
We name our cloud-resolving model the Cloud Resolving Storm Simulator (CReSS), which cansimulate clouds and precipitation systems. CReSS has high resolutions in both horizontal andvertical directions, and includes detailed physical processes of clouds and precipitation. Processesof clouds and precipitation, however, still need improvement. Both observational studies andmodel development are needed for further understanding of clouds and precipitation systems.

3

4

CONTENTS

The goal of the development of CReSS is to predict heavy rainfalls based on this understanding.
CReSS is now open to the public. This textbook describes the usage of CReSS, the basics ofmodel, and the program codes. We hope that this document will be useful for all CReSS users.
You can help us to improve the document by contacting us at tsuboki@rain.hyarc.nagoya-u.ac.jp.
Your comments will be appreciated.

Part I

Simulations of High-Impact Weather
Systems

Chapter 1

Introduction

In this chapter, we present the objectives, history, and summary of theCReSS (Cloud Resolving Storm Simulator) model. CReSS is a non-hydrostaticand compressible model to simulate cloud and precipitation systems. Thesesystems are highly complex, and their development and evolution processes are
determined by non-linear interactions between the atmosphere and hydrometeors. CReSS is designed for a parallel computer that enables to performlarge computations, such as this involving cloud and precipitation systems.
Advantages of CReSS are as follows.
• Two versions of CReSS, the parallel version and the single version areavailable for users. The former is designed for parallel computations,and the latter for a single CPU. The computation output will not be
dependent on the versions or on domain decompositions.
• Detailed cloud micro-physical processes are included for moist simulations. On the other hand, CReSS can perform dry simulations without
condensation processes.
• CReSS can perform a simulation of a real weather system with a 3dimensional inhomogeneous initial and boundary conditions, as well as
a numerical experiment with idealized initial and boundary conditions.
• All codes of CReSS are written using the standard FORTRAN 90. They
are easy to read and easy to modify, after a brief initial instruction.
CReSS can be performed on most computers ranging from mainframe
computers to PC Unix machines without any modiﬁcations of the codes.

8

1 Introduction

1.1

Objectives

One of the most important objectives of regional models is a high-resolution simulation of highimpact weather systems, which can be used for detailed studies and accurate predictions. Highimpact weather systems are most signiﬁcant phenomena in the atmosphere and sometimes cause
huge disasters to human society. Understanding their mechanisms and structures is necessary forprediction and prevention/reduction of disasters. Most high-impact weather systems that cause
heavy rainfalls and/or violent winds consist of cumulonimbus clouds and their organized systems.
They are usually embedded within a larger weather system and occasionally have a multi-scale
structure, ranging from cloud-scale to synoptic-scale systems. These weather systems in East Asia
include the Baiu front, typhoons, and winter snowstorms associated with a cold-air outbreak.
In order to perform simulations and numerical experiments of high-impact weather systems, wehave been developing the cloud-resolving numerical model named CReSS. Since the multi-scale
structure of the weather systems has a wide range of horizontal scales, a large computational
domain and a very high-resolution grid to resolve individual classes of the multi-scale structure arenecessary to simulate the evolution of the weather systems. In particular, an explicit calculationof cumulonimbus clouds is important for a quantitative simulation of precipitation associatedwith the high-impact weather. It is also required to formulate accurately cloud physical processes
as well as the ﬂuid dynamic and thermodynamic processes. For this type of computation, a large
parallel computer with a huge memory is necessary.
The purpose of the present-day research is to explicitly simulate clouds and their organizedsystems in a large domain (larger than 1000 x 1000 km) with resolved individual clouds usinga very ﬁne grid system (less than 1 km horizontally). This will clarify a detailed structure of
the high-impact weather systems and permits a more quantitative prediction of the associatedprecipitation. It will contribute to the accurate prediction of precipitation and the reduction ofdisasters caused by the high-impact weather systems.

1.2

Brief description of CReSS

The formulation of CReSS is based on the non-hydrostatic and compressible equation using terrain-following coordinates. Prognostic variables are 3-dimensional velocity components,
perturbations of pressure and potential temperature, water vapor mixing ratio, sub-grid scale
turbulent kinetic energy (TKE), and cloud physical variables. A ﬁnite diﬀerence method is used
for the spatial discretization. The horizontal domain is rectangular, and variables are set on a
staggered grid: the Arakawa-C grid in the horizontal and the Lorenz grid in the vertical. For time
integration, the mode-splitting technique (Klemp and Wilhelmson 1978) is used. Terms related
to sound waves of the basic equation are integrated with a small time step, and other terms witha large time step.
Cloud physical processes are formulated by a bulk method of cold rain, which is based on Lin
et al. (1983), Cotton et al. (1986), Murakami (1990), Ikawa and Saito (1991), and Murakami etal. (1994). The bulk parameterization of cold rain considers water vapor, rain, cloud, ice, snow,
and graupel. The microphysical processes implemented in the model are described in Fig.1.1.

Parameterizations of the sub-grid scale eddy motions in CReSS are one-order closure of Smagorin-

1.2 Brief description of CReSS-VD vr9

VD vg

water vapor (q v)
NUA vi
VD vc

VD vi

NUF ci ,NUC ci ,NUH ci
ML ic
CL is ,CN is

VD vscloud water (q c)

CL cs

SP si ,SP N si

cloud ice (q i,N i)
AG N iCL crsnow (q s,N s)

CL rs
AG N sCL sr ,CL sg ,CN sg ,CN Nsg

CL ir ,CL ig

ML sr ,SH sr

SP gi ,SP N gi

CN cr

CL cg

Fall. q s , Fall. (N s /ρ )ML gr ,SH gr

rain water (q r)Fall. q r

CL ri ,CL N ri ,CL rs ,CL N rs ,CL rg ,FR rg ,FR N rg

graupel (q g ,N g )Fall. q g , Fall.( N g/ρ )Figure 1.1. Diagram describing water substances and cloud microphysical processes in the bulk schemeof CReSS.

sky (1963) or the 1.5-order closure of turbulent kinetic energy (TKE). In the latter parameterization, the prognostic equation of TKE is used. All numerical experiments in this textbook use
three-dimensional 1.5-order closure scheme. The surface process of CReSS is formulated by abulk method, whose bulk coeﬃcients are taken from Louis et al. (1981).
Several types of initial and boundary conditions are available. For a numerical experiment,a horizontally uniform initial ﬁeld provided by a sounding proﬁle will be used with an initial
disturbance of a thermal bubble or random temperature perturbation. The boundary conditions
are one of the following types; rigid wall, periodic, zero normal-gradient, and wave-radiation
types.
CReSS can be nested within a coarse-grid model for a prediction experiment. In such an experiment, the initial ﬁeld is provided by interpolating grid-point values and the boundary condition
is provided by the coarse-grid model. For a computation within a large domain, conformal map
projections are available, which include the Lambert conformal projection, the polar stereographic
projection, and the Mercator projection.
For parallel computing of a large computation, CReSS provides two-dimensional domain decomposition in the horizontal direction (Fig.1.2). Parallel processing is performed using the Massage Passing Interface (MPI). Communications between individual processing elements (PEs) are
performed by exchanging data of the two outermost grids. The OpenMP is also available.

10

1 Introduction

Figure 1.2. Schematic representation of a two-dimensional domain decomposition and the communication strategy for parallel computations using MPI.

1.3

History of CReSS development

CReSS was developed by Kazuhisa Tsuboki at the Hydrospheric Atmospheric Research Center(HyARC) of the University of Nagoya, and by Atsushi Sakakibara at Research Organization for
Information Science and Technology, whose current aﬃliation is Chuden CTI Co.,Ltd.. The
development of CReSS was performed as a part of the project named ”Study for Advanced
Simulation of Downburst Phenomena” (Principal Investigator: Professor N. Kamiya at AichiGakusen University). The project was supported by Special Coordination Funds for Promoting
Science and Technology and was one of a program to develop parallel software for large-scale
simulations to be used for predictions of the global climate change led by Professor A. Sumi of
University of Tokyo for the period from 1998 to 2003. At the end of the project, the version 1.4of the CReSS model was accomplished.
Ever since 2003,HyARC at Nagoya University has been supporting the development of theCReSS model. It was optimized for the Earth Simulator in 2003. All codes were rewritten in
FORTRAN 90 instead of FORTRAN 77 and the version 2.0 was accomplished in 2003. We
evaluated the performance of CReSS on the Earth Simulator in 2003. The result is summarizedin Table 1. The parallel operation ratio was measured using 128 nodes (1024 CPUs) and 64 nodes(512 CPUs) of the Earth Simulator. The vector and parallel operation ratios are suﬃciently highto perform a large computation on the Earth Simulator.
Using the CReSS model, HyARC carried out a research program entitled ”High Resolution
Modeling of Atmospheric Water Circulation Systems Using the Cloud-Resolving Model” for theperiod from 2003 to 2006. The purpose of this research program was to explicitly simulate clouds
and organized precipitation systems in a large domain and very ﬁne grid (less than 1 km). Using
the Earth Simulator, we performed simulation experiments of important cloud and precipitation
systems.
One of the experiments is the simulation of the Niigata-Fukushima heavy rainfall event, which
occurred in July 2004. It shows that an intense rainband forms and is maintained within thesubsynoptic-scale low along the Baiu front. Other experiments include simulations of snowstorms
over the Sea of Japan as well as those over the Great Lakes region of Canada. Snowstorm

1.3 History of CReSS development

experiments show various types of precipitation systems composed of intense convective clouds
in cold air streams. One of the important objectives of this research program is to simulate
typhoons, whose developments are attributed to close interaction between a large-scale motion
and cumulonimbus convections. Simulations of Typhoons 0418, 0423, 0613 were performed with ahorizontal resolution of 500-1,000 m. They show very realistic features of typhoon and individualclouds. All of these experiments show very detailed structures of individual precipitation clouds
as well as overall structures of the large-scale systems.
The simulation experiments using the CReSS model range from typhoon scale to tatsumaki
scale, and resolve every class of the multi-scale structure. Such super-high resolution calculations
provide detailed information of weather systems. Using the output of these experiments, we will
be able to clarify not only structures of weather systems but also interactions between diﬀerent
classes of scales. High-resolution simulations will contribute to an accurate and quantitative
prediction, as well as to prevention/reduction of disaster caused by high-impact weather systems.
As a result of the research program at the HyARC, CReSS was updated to version 2.2 in 2006.
This version is now ﬁxed and open to the public. The development of version 3.0 is on-going,with substantial improvements on the way.

11

Chapter 2

Simulations and Numerical Experiments

The CReSS model has been used for various purposes. Calculations areroughly categorized into simulations and numerical experiments. The simulations are referred to those calculations that simulate a real weather system orreal clouds. In these cases, initial and boundary conditions are provide by acoarse-mesh model or 3-dimensional grid-point data; artiﬁcial initial perturbations are generally not necessary. A real topography and surface conditions arealso used. The numerical experiments are referred to those using an idealized
condition of the atmosphere. The initial conditions for these experiments areprovided by a single sounding or by a theoretical function. The initial state isusually horizontally uniform, and artiﬁcial perturbations are necessary if theland surface is ﬂat and unvarying. The numerical experiments are further categorized into dry and moist types in 2- or 3-dimensions. The dry experimentdoes not consider condensation, while the moist one includes condensation inboth cloud and precipitation processes.
In this chapter, we give some examples of simulations and numerical experiments under diﬀerent conditions. We recommend users to start with a2-dimensional dry experiment, such as cats-eye and mountain waves. They
are relatively simple calculations and can be easily performed even on a laptop
computer. Three-dimensional experiments are computationally heavier, anda large computer is necessary. Some simulations of real weather systems suchas typhoons and storms are also presented in this chapter. Readers will ﬁnd
objectives and performance of the CReSS model at the end of the chapter.

14

2 Simulations and Numerical Experiments

2.1

Mountain waves: dry experiment in 2-dimensional geometry

When air is forced to ﬂow over a mountain, individual air parcels are displaced vertically.
If the atmosphere is stably stratiﬁed, the parcels descend and may oscillate about their equilibrium levels. The gravity waves that result, referred as mountain waves, have been observed in
mountainous regions all over the world.
The presence of mountain waves is frequently revealed by distinctive orographic clouds. Various
types of mountain waves produce diﬀerent types of wave clouds. An identiﬁcation of a particular
type of wave cloud can be used to make qualitative deductions about the vertical variation inwind and stability over the mountains.
Large-amplitude mountain waves can produce several weather phenomena that signiﬁcantly
aﬀect human activity and therefore require the attention of weather forecasters. The strong
downslope winds observed along the lee of mountain barriers are usually associated with large
amplitude waves. Dangerous regions of clear-air turbulence (CAT) are also produced by thesewaves.
Numerical experiments of mountain waves are performed using CReSS. Examples are shownas follows. The basic conditions of the experiments are• Calculation is in 2-dimensional.
• A bell-shaped mountain is located at the center of the calculation domain.
• No moist processes are included.
• Horizontal wind and buoyancy frequency of the basic ﬁeld are uniform in both horizontal
and vertical directions.
• The vertical grid spacing is uniform.
• The lateral boundary condition is cyclic.
• The bottom boundary condition is slip.
Scorer parameter, which is a ratio of the buoyancy frequency to horizontal wind speed in thiscase and has the dimension of wave number, determines characteristics of the mountain waves
theoretically. According to the Scorer parameter and the horizontal scale of bell-shaped mountain,
diﬀerent types of mountain waves are excited.
One example is mountain waves propagating to the downstream. The experimental design issummarized in Table 2.1.
Grid number
Grid spacing

2003 × 4 × 253
50 × 50 × 50 m

Mountain
Initial wind
Buoyancy frequency
Initial disturbanceBell-shaped and isolated, height: 500 m, half-width: 500 m
u=10 m s−1
0.01 s−1 and constant with heightnone

Table 2.1: Experimental design of mountain waves propagating to the downstream.

2.1 Mountain waves: dry experiment in 2-dimensional geometry

In the experiment, the averaged gradient of the mountain is 26 degrees and the maximum is 34
degrees. Although the mountain slope is rather steep, mountain waves are successfully simulated(Fig. 2.1). The mountain waves propagate upward in the downstream direction, just as the linear
theory of mountain waves predicts.

Figure 2.1. Numerical experiment of mountain waves. Velocity (u, w)[m s−1 ], pressure perturbation[Pa]at 6000 seconds from the initial time.

The other example is large amplitude and non-steady waves. The experimental design is shownin Table 2.2.

Initial wind2003 × 4 × 253
50 × 50 × 50 m
Bell-shaped and isolated, height: 1000 m, half-width:
500 m
u=10 m s−1

Buoyancy frequency
Initial disturbance

0.02 s−1 and constant with heightnone

Grid number
Grid spacing
Mountain

Table 2.2: Experimental design of large amplitude andnon-steady waves.

15

16

2 Simulations and Numerical Experiments

The result shows that the non-steady waves are propagating on the downstream side of themountain (Fig. 2.2). A very strong wind is present near the surface on the lee side of the mountainand a negative perturbation velocity occurs above the strong surface wind. This is a sort of thedown-slope wind.

Figure 2.2. Numerical experiment of non-steady mountain waves. Velocity (u, w)[m s−1 ], pressure perturbation[Pa] at 4200 seconds from the initial time.

2.2

Kelvin-Helmholtz billows: dry experiment in two-dimensional
geometry

The numerical experiment of Kelvin-Helmholtz billows is performed in a two-dimensional
geometry with a grid size of 20 m. The proﬁle of the basic ﬂow is the hyperbolic tangent type.
The setting of the experiment is summarized in Table 2.3.
Grid number

51 × 4 × 71

Grid spacing
Time step
Topography

20 × 20 × 10 m
HE-VE, large: 0.8 s, small: 0.04 s
none

Basic wind

A shear wind with tanh type

2.3 Simulation of Tornado within a Supercell: moist experiment in three dimensional
geometory
17

Basic temperaturetanh type

Initial perturbation

A sine-shaped disturbance with maximum amplitude of
0.4 K

Table 2.3: Experimental design of Kelvin-Helmholtz billows.

Streamlines of u and w components show a clear cats eye structure of the Kelvin-Helmholtz
billows (Fig.2.3). This result is very similar to that of Klaassen and Peltier (1985). The model
also simulates the overturning of potential temperature associated with the billows (Fig. 2.4).
This example shows that the model works well with a grid size of a few tens of meters in the dry
experiment.

2.3

Simulation of Tornado within a Supercell: moist experimentin three-dimensional geometory

For a simulation experiment of a moist atmosphere, we choose a tornado-producing supercell
observed on 24 September 1999 in the Tokai District of Japan. The experiment is aimed at
resolving the vortex of the tornado within the supercell.
The supercell is highly three-dimensional and its horizontal scale is several tens of kilometers.
A large domain in the order of 100 km is necessary to simulate a supercell using a cloud model.
A tornado, on the other hand, has a horizontal scale of a few hundred meters. The simulationof the tornado requires a ﬁne horizontal grid spacing in the order of 100 m or less. In order tosimulate a supercell and its associated tornado by a cloud model, a huge memory and high-speed
CPU are crucial.
Numerical experiments of a supercell thunderstorm using a cloud model have been performed
during the past 20 years (Wilhelmson and Klemp, 1978; Weisman and Klemp, 1982, 1984).
Recently, Klemp and Rotunno (1983) attempted to increase the horizontal resolution to simulate
the ﬁne structure of a meso-scale cyclone within the supercell. An intense tornado occasionallyoccurs within a supercell thunderstorm. It is, however, more diﬃcult to resolve a tornado. To
overcome this diﬃculty, Wicker and Wilhelmson (1995) used an adaptive grid method, a two-way
nesting technique, to simulate tornado genesis. Their grid spacing of the ﬁne mesh was 120 m.
Grasso and Cotton (1995) also used a two-way nesting procedure of a cloud model to simulatea tornadic vorticity. Nesting methods introduce complications of communication between thecoarse-grid model and the ﬁne-mesh model at the boundary. A cloud-resolving model does not
require any nesting methods. We simulate both the supercell and the tornado using a uniform grid,
therefore avoid any complication of the boundary communication. The computational domain of
the present simulation is about 50 × 50 km and the grid spacing is 100 m. The integration time
is about 2 hours.
The basic ﬁeld is give by a sounding at Shionomisaki, Japan at 00 UTC, 24 September 1999
(Fig.2.5). The initial perturbation is a warm thermal bubble placed near the surface, which can
induce an initial convective cloud.
One hour from the initial time, a quasi-stationary supercell is simulated by CReSS (Fig.2.6).
The hook-shaped precipitation area and the bounded weak echo region (BWER), which arecharacteristic features of a supercell, are formed in the simulation. An intense updraft occurredalong the surface ﬂanking line. At the central part of BWER or of the updraft, a tornadic vortex

18

2 Simulations and Numerical Experiments

Figure 2.3. Stream lines of the Kelvin-Helmholtz billow at 240 seconds fromthe initial simulated in the two-dimensional geometry.

was formed 90 minutes from the initial time.
A close view shows closed contours in the central part of the vorticity (Fig.2.7). The diameterof the vortex is about 500 m and the maximum vorticity is about 0.1 s−1 , similar to the observed
tornado. The pressure perturbation also shows closed contours that correspond to those of thevorticity (Fig.2.8). This indicates that the ﬂow in the vortex is in cyclostrophic balance. The
vertical cross section of the vortex shows that the axis of the vorticity and the associated pressure
perturbation is inclined to the left hand side and extends to a height of 2 km (Fig.2.9). At thecenter of the vortex, the downward extension of cloud is simulated.
Though this is a preliminary result of a supercell and tornado, some characteristic features ofthe observation are successfully simulated. The important point of this simulation is that boththe supercell and the tornado are simulated with the same grid size. The tornado is produced
purely by the physical processes in the model. A detailed analysis of the simulation will provide
us important information on tornado genesis within a supercell.

2.4 Simulation of a squall line: moist experiment in three-dimensional geometory 19

Figure 2.4. Same as Fig.2.3, but for potential temperature.

2.4

Simulation of a squall line: moist experiment in three-dimensional
geometory

A squall line is a signiﬁcant mesoscale convective system. It is usually composed of an intense
convective leading edge and a trailing stratiform region. An intense squall line was observed bythree Doppler radars on 16 July 1998 during an intensive ﬁeld observation of GAME/HUBEX(the GEWEX Asian Monsoon Experiment/Huaihe River Basin Experiment) in China. The squallline extended from the northwest to the southeast with a width of a few tens of kilometers andmoved northeastward at a speed of 11 m s−1 . Radar observation showed that the squall line
consisted of intense convective cells along its leading edge. Some of the cells reached a height of
17 km. The rear-inﬂow was present at a height of 4 km, which descended to cause the intenselower-level convergence at the leading edge. After the squall line passed over the radar sites, astratiform precipitation was extending behind the convective leading edge.
The design of the simulation experiment using CReSS is as follows. Both the horizontal andvertical grid sizes are 300 m within a domain of 170 km × 120 km. Cloud microphysics is the coldrain type. The boundary condition is the wave-radiating type. The initial condition is provided
by a dual Doppler analysis and sounding data. The inhomogeneous velocity ﬁeld within the

20

2 Simulations and Numerical Experiments

Figure 2.5. Vertical proﬁles of zonal component (thick line) and meridional
component (thin line) observed at Shionomisaki at 00 UTC, 24
September 1999.

storm is determined by the dual Doppler radar analysis directly, while that outside the storm andthermodynamic ﬁeld are provided by the sounding observation. Mixing ratios of rain, snow andgraupel are estimated from the radar reﬂectivity, while mixing ratios of cloud and ice are set tobe zero at the initial time. A horizontal cross section of the initial ﬁeld is shown in Fig.2.10.
The simulated squall line extending from the northwest to the southeast moved northeastward(Fig.2.11). The convective leading edge of the simulated squall line is maintained by the replacement of new convective cells and moved to the northeast, similar to the behavior of the observed
squall line. Convective cells reached a height of about 14 km with large production of graupel
above the melting layer. The rear inﬂow is as signiﬁcant as that in the observation. A stratiform
region extends with time behind the leading edge. Cloud extends to the southwest to form a cloudcluster. These results show that CReSS can successfully simulate the development and movementof a squall line.

2.5

Typhoons and the associated heavy rainfall: T0418 and T0423

Typhoons develop by close interaction between a large-scale disturbance and embedded intense cumulonimbus clouds. The horizontal scale of typhoons ranges from several 100 km to afew 1000 km while that of the cumulonimbus clouds is an order of 10 km. Typhoons often bring aheavy rain and a strong wind. The heavy rain is usually localized in the eyewall and spiral rainbands which develop within typhoons. Since cumulonimbus clouds are essentially important for
typhoon development, the cloud-resolving model is necessary for a detailed numerical simulationof typhoons.
Some typhoons usually attain Japan and its surroundings and cause severe disasters. In particular, ten typhoons landed over the main lands of Japan in 2004. In the present section, weshow two simulation experiments of the typhoons. One is the typhoon T0418 which brought avery intense wind and caused huge disasters due to the strong wind. The other is the typhoon
T0423 which brought heavy rainfalls and caused severe ﬂoods.

2.5 Typhoons and the associated heavy rainfall: T0418 and T0423

Figure 2.6. Horizontal display at 600 m of the simulated supercell at 5400 seconds from the initial. Mixing ratio of rain (gray scales, g kg−1 ),vertical velocity (thick lines, m s−1 ), the surface potential temperature at 15 m (thin lines, K) and horizontal velocity vectors.

Typhoon T0418 moved northwestward over the northwest Paciﬁc Ocean and passed Okinawa
Island on 5 September 2004. Its center passed Nago City around 0930 UTC, 5 September with theminimum sea level presser of 924.4 hPa. When T0418 pass over Okinawa Island, double eyewalls
were observed. This is a distinctive feature of the typhoon. T0418 was characterized by strong
winds and caused huge disasters due to the strong winds over Japan.
The main objectives of the simulation experiment of T0418 are to study the eyewall as well as
spiral rainbands, and to examine structure of the strong wind associated with the typhoon around
Okinawa Island. The simulation experiment of T0418 started from 0000 UTC, 5 September 2004.
The experimental design of T0418 is summarized in Table 2.4.
The simulation experiment shows very detailed structure of the eye and the spiral rainbands(Fig.2.12). Individual cumulus clouds are resolved. They are simulated within the eyewall andalong the spiral rainband. A weak precipitation forms around the central part of the eye. The
maximum tangential velocity is present along the eyewall and at a height of 1 km. It is larger than
70 m s−1 . The high-resolution experiment shows detailed structure of the cloud and precipitation

21

22

2 Simulations and Numerical Experiments

Figure 2.7. Close view of the simulated tornado within the supercell. The contour lines are vorticity (s−1 ) and the arrows are horizontal velocity.
The arrow scale is shown at the bottom of the ﬁgure.

Figure 2.8. Same as Fig.2.7, but for pressure perturbation.

systems associated with the typhoon, and simulates the overall structure of the typhoon and its
movement.

Typhoon T0423 moved along the Okinawa Islands on 19 October 2004 and landed over Shikoku
Island on 20 October. In contrast to T0418, T0423 is characterized by heavy rainfall over Japan.

2.5 Typhoons and the associated heavy rainfall: T0418 and T0423

Figure 2.9. Vertical cross section of the simulated tornado. Thick lines arevorticity (s−1 ), dashed lines are pressure perturbation and arrows
are horizontal velocity.
Table 2.4. Experimental design of Typhoon T0418

domaingrid number
grid size
integration time

x 1536 km, y 1280 km, z 18 km
x 1539, y 1283, z 63
H 1000m, V 150 ∼300m
18 hrsES node numbers

128 nodes (1024 CPUs)

Heavy rainfalls associated with T0423 occurred in the eastern part of Kyushu, Shikoku, the east
coast of the Kii Peninsula, and the Kinki District. They caused severe ﬂoods and disasters inthese regions.
The purpose of the simulation experiment of T0423 is to study process of the heavy rainfall.
Experimental design of T0423 is summarized in Table 2.5. At the initial time of 1200 UTC, 19
October 2004, T0423 was located to the NNE of Okinawa.

The movement of T0423 and the rainfall were successfully simulated. In the simulation, anorthward moisture ﬂux is large in the east side of the typhoon center. When the large moisture
ﬂux reaches to the Japanese Islands, heavy rainfalls occur along the Paciﬁc Ocean side. The heavy
rainfall moves eastward with the movement of the typhoon from Kyushu to Shikoku. When thetyphoon reaches to the south of Shikoku, heavy rainfall begins in the Kinki District (the rectangle
in Fig.6) and intensiﬁes at 0630 UTC, 20 October (Fig.2.13). The distribution and intensity of
precipitation well correspond to those of the radar observation.

23

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2 Simulations and Numerical Experiments

Figure 2.10. Horizontal cross section of the initial ﬁeld at a height of 2.5km
at 1033 UTC, 16 July 1998. The color levels mixing ratio of rain
(g kg−1 ). Arrows show the horizontal velocity obtained by thedual Doppler analysis and sounding.

Figure 2.11. Time series of horizontal displays (upper row) and vertical crosssections (lower row) of the simulated squall line. Color levels
indicate total mixing ratio (g kg−1 ) of rain, snow and graupel.
Contour lines indicate total mixing ratio (0.1, 0.5, 1, 2 g kg−1 ) of
cloud ice and cloud water. Arrows are horizontal velocity.

The close view of northern Kinki shows that a large amount of solid precipitation is present
around a height of 6 km (Fig.2.14). The heavy rainfall in the region forms below the large mixing
ratio of the solid precipitation. The heavy rainfall along the Paciﬁc Ocean side moved eastward,
while that in the Kinki District lasted until 12 UTC, 20 October. After the typhoon moved
to the east of the Kinki District, the northeasterly was intensiﬁed signiﬁcantly. Consequently,
orographic rainfall formed in the northern Kinki District. As a result, the accumulated rainfall

2.6 Typhoon RUSA

Figure 2.12. Surface pressure (contour lines; hPa) and rainfall intensity (color levels; mm hr−1 ) of thesimulated Typhoon T0418 at 0830 UTC, 5 September 2004.

Table 2.5. Experimental design of Typhoon T0423

domaingrid number
grid sizex 1536 km, y 1408 km, z 18 km
x 1539, y 1411, z 63
H 1000m, V 200 ∼300m

integration time
ES node numbers

30 hrs128 nodes (1024 CPUs)

became a large amount and the severe ﬂood occurred.

2.6

Typhoon RUSA

A typhoon named RUSA attacked Korea on 31 August 2002 and caused a heavy rain andﬂood. In particular, a large amount of rainfall was brought to the southern part of Korea andthe east coast. We performed a simulation experiment of Typhoon RUSA with a very highresolution using CReSS. The horizontal resolution is 1 km and the domain is 1320 times 1440 km
in horizontal and 18 km in vertical (Table 2.6). The initial and boundary conditions are provided
by the Regional Spectral Model of JMA.
Typhoon RUSA moved northward over the East China Sea and landed on the southern part of
the Korean Peninsula. The experiment well simulated the track of the typhoon. The distributionof total amount of rainfall well corresponds to observation. The overall characteristics of Typhoon

25

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2 Simulations and Numerical Experiments

Figure 2.13. Same as Fig.2.12 but for the Typhoon T0423 at 0630 UTC, 20 September 2004. Arrows
are horizontal wind velocity at a height of 974 m and warmer colored arrows means moister
air. The rectangle indicates the region of Fig.2.14.

RUSA are successfully simulated.
The result shows that an intense spiral band forms on the northeast side of the typhoon center(Fig.2.15). When the spiral band approaches to the southernmost part of the Korean Peninsula,
the rain intensiﬁes by the topographic forcing. This causes the heavy rainfall in the southern
Korean Peninsula.
Another intense rainfall occurs along the east coast of the Korean Peninsula. It was caused byconvergence of a northeasterly wind and the cyclonic ﬂow around the typhoon. The topography
along the east coast also intensiﬁes the rainfall. As a result, the intense rainfall occurs along theeast coast of the peninsula.
The experiment with the high resolution and the large domain successfully simulated the whole
typhoon and individual spiral bands as well as the intensiﬁcation of rain due to topography. This

Table 2.6. Experimental design of Typhoon RUSA

domainx 1320, y 1440, z 18 km

grid number
grid size
integration time

x 1323, y 1443, z 63
H 1000 m, V 300 m
24 hrsnode number

120 (960 CPUs)

2.7 Localized heavy rainfall

Figure 2.14. Mixing ratio of precipitation (color levels; g kg−1 ) and horizontal velocity (arrows) at aheight of 6142 m at 0630 UTC, 20 September 2004.

experiment enables to study the formation and development processes of the intense rain within
the typhoon.

2.7

Localized heavy rainfall

Precipitation systems associated with the Baiu front occasionally show a multi-scale structure.
The Baiu front extends zonally for several thousand kilometers while a localized heavy rainfall has
a horizontal scale of a few hundred kilometers. To clarify the water circulation process and the roleof each class of the multi-scale systems, we performed a simulation experiment of the heavy rainfall
in a large domain and with a high resolution. The explicit representation of cumulonimbus clouds
in the model is essentially important for accurate and quantitative simulation of the localized
heavy rainfall.
The localized heavy rainfall occurred in Niigata and Fukushima prefectures on 13 July 2004.
Radar observation of the Japan Meteorological Agency (JMA) showed that an intense rainband
extended zonally and maintained for more than 6 hours. The Baiu front was present to the northof Niigata and a sub-synoptic scale low (SSL) moved eastward along the Baiu front.
The experimental setting of the simulation is summarized in Table 2.7. The initial and boundary
condition were provided by the JMA Regional Spectral Model (RSM). Initial time is 1200 UTC, 12
July 2004. The simulation showed that the SSL moved eastward along the Baiu front. Figure 2.16
shows that the SSL reaches Japan at 0020 UTC, 13 July 2004. Moist westerly wind is intense to
the south of the SSL. Large precipitation extends to the east of the SSL. On the other hand, avery intense rainband forms to the south of the SSL. Enlarged display (Fig.2.17) of the rainband

27

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2 Simulations and Numerical Experiments

Figure 2.15. Pressure (hPa), mixing ratio of rain (gray scale; g kg−1 ) and horizontal wind ata height of 2.15km at 03 UTC, 31 August 2002 obtained from the simulation
experiment.

shows that it extends from the northern part of the Noto Peninsula and reaches Niigata withintensiﬁcation. The rainband forms between the southwesterly and westerly winds at the lowlevel. The rainband is composed of intense convective cells. It maintains until the SSL moves to
the Paciﬁc Ocean. The long time maintenance of the intense rainband results in the severe ﬂoodin Niigata Prefecture.
Table 2.7. Experimental design of the Niigata-Fukushima heavy rainfall event.

2.8

domaingrid number

x 1792 km, y 1536 km, z 18 km
x 1795, y 1539, z 63

grid size
integration time
ES node number

H 1000 m, V 100 ∼ 300 m
24 hrs128 nodes (1024 CPUs)

Idealized experiment of snow cloud bands

When an outbreak of a cold and dry polar airmass occurs over the sea, many cloud streetsor cloud bands form in the polar air stream. Large amounts of sensible heat and latent heat
are supplied from the sea to the atmosphere. Intense modiﬁcation of the airmass results indevelopment of the mixing layer and convective clouds develop to form the cloud bands along
the mean wind direction. Their length reaches an order of 1000 km while individual convective

2.8 Idealized experiment of snow cloud bands

Figure 2.16. Surface pressure (contour lines; hPa) and rainfall intensity (color levels;
mm hr−1 ) and horizontal velocity (arrows) at a height of 1610 m at 0020 UTC, 13
July 2004. Warmer colored arrows means moister air. The rectangle indicatesthe region of Fig.2.17.

Figure 2.17. Same as Fig.2.16 but for the region of the rectangle in Fig.2.16 and at a heightof 436 m.

29

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2 Simulations and Numerical Experiments

cells have a horizontal scale ranging from a few kilometer to a few tens kilometers. In order toperform a 3-dimensional simulation of cloud bands, a large computation is necessary. In order tostudy the detailed structure of convective cells and the formation process of the organized cloudbands, we performed 3-dimensional simulation using the CReSS model on the Earth Simulator.
The experimental setting is summarized in Table 2.8. The initial condition was provided by asounding observation on the east coast of Canada at 06 UTC, 8 February 1997.
Table 2.8. Experimental design of the snow cloud bands in the cold air stream.

domaingrid number
grid size

x
457
km,
y
153
z 11 km
x 1527, y 515, z 73
H 300 m, V 50 ∼150 m

integration time
ES node number

20 hrs32 nodes (256 CPUs)km,

The calculation domain in this simulation is 457 km and 153 km in x- and y-directions, respectively with a horizontal grid spacing of 300 m. Sea ice is placed on the upstream side. Eachmodel grid of the surface is occupied by ice or open sea according to the probability of sea iceor sea ice density. In the experiment, density of sea ice is 100 0 outbreak is used for the initialcondition.
The atmosphere over the packed ice is stably stratiﬁed and the vertical shear is large. Mixing
layer develops with the distance from the edge of the packed ice. The cloud bands develop within
the mixing layer. Figure 2.18 shows formation and development of cloud bands over the sea.
They begin to form in the region of sea ice density of 50-70 cloud bands form in the cold air
stream. Some cloud bands merge each other and selectively develop. Consequently, the numberof lines decreases with the distance along the basic ﬂow.

Close view of the upstream region (Fig.2.19) shows upward and downward motions are almost
uniform in the x-direction. As a result, cloud ice and precipitation extend in the x-direction
uniformly. This indicates that the convections are the roll convection type in the upstream
region.
In the downstream region, the roll convections change to alignment of cellular convections(Fig.2.20). While the upward motions are centered and downward motions are located on theirboth sides, cloud and precipitation show cellular pattern. In this region, the mixing layer fully
developed and the vertical shear almost vanishes in the mixing layer.
In the region of far downstream, convections change to randomly distributed cells (Fig.2.21).
Band shape of cloud almost disappears and convections become closed cell type. The morphological transformation from alignment of cells to random cells is often observed by satellite. The
experiment successfully simulates the formation process of cloud bands, their extension and merging processes, and the morphological changes of convection from the roll to cellular types.

2.9 Snow storms

Figure 2.18. Horizontal cross sections of vertical velocity (upper panel), mixing ratio of precipitation(snow, graupel and rain) (middle panel) at 1000 m in height and mixing ratio of cloud ice
at 1300 m in height (lower panel) for x=100-450 km at 18 hours from the initial time.

2.9

Snow storms

One of the major precipitation systems in East Asia is snow cloud in a cold polar air stream.
In particular, various types of precipitation systems develop over the Sea of Japan: longitudinal
and transversal cloud bands, convergence zone and vortexes. Their horizontal scale ranges from a
few hundred kilometers to 1000 km while they are composed of convective clouds whose horizontal
scale is a few kilometers.
To study development process and detailed structure of longitudinal and transversal cloudbands, we performed simulation experiment of the cold air outbreak over the Sea of Japan on
14 January 2001. The initial ﬁeld at 0600 UTC, 13 January 2001 and boundary condition were
provided by the JMA RSM. The Domain of the simulation covered most part of the Sea of Japanand horizontal resolution was 1 km to resolve convective clouds (Table 2.9).
Table 2.9. Experimental design of the snow storm over the sea of Japan.

domainx 1350, y 1350, z 16 km

grid number
grid size
integration time

x 1353, y 1353, z 43
H 1000 m, V 200 ∼400m
18 hrsES node number

36 (288 CPUs)

Snow cloud bands over the Sea of Japan is realistically simulated (Fig. 2.22). An intense
and thick cloud band composed of cumulonimbus clouds extends from the root of the Korean
Peninsula to the Japan islands. Plenty of thin cloud bands develop over the sea. Longitudinal

31

32

2 Simulations and Numerical Experiments

Figure 2.19. Horizontal cross sections of vertical velocity (upper panel), mixing ratio of precipitation(middle panel) at 900 m in height and mixing ratio of cloud ice at 1100 m in height (lower
panel) for x=160-220 km.

and transversal cloud bands form to the west and east of the intense cloud band, respectively. The
enlarged display of the transversal cloud bands shows that the cloud bands extend the SW-NE
direction, which is almost parallel to the vertical wind shear between levels of the top of clouds
and the surface (Fig. 2.23). This is consistent with the dynamic theory shown by Asai (1972)[4].
Cold air outbreak and associated precipitation systems occurs over the Great Lakes region andthe Labrador Sea to the east of Canada. The snow cloud bands over the Great Lakes are known
as the ’lake eﬀect storm (LES)’ and have been studied observationally and numerically. Liu etal. (2004) studied snow cloud bands using CReSS. The LES is also composed of cumulonimbus
clouds and brings severe snowfall. We performed a simulation experimet (Table 2.10) to study theformation of the LES associated with a cold air outbreak. Figure 2.24 shows vertically integrated
cloud and precipitation obtained from the simulation experiment. Cloud bands begin to develop
from the lake on the upwind side (Lake Superior). They are composed of convective clouds
over the lake while they become stratiform over the land. When they arrive at the lakes on
the downwind side (Lakes Michigan and Huron), convection in the cloud bands is restored. The
pattern and structure of cloud are very similar to those observed by meteorological satellites.
Table 2.10. Experimental design of the snow storm over the Great Lakes.

domaingrid number

x 1300, y 660, z 18 km
x 2603, y 1323, z 43

grid size
integration time
ES node number

H 500m, V 40 ∼250m
12 hrs120 (960 CPUs)

Cold air outbreak over the Labrador Sea is similar to that over the Sea of Japan. Sea ice
is present along the east coast of Canada. Observation shows that plenty of snow cloud bands

2.10 Heavy rainfall in Baiu season

Figure 2.20. Horizontal cross sections of vertical velocity (upper panel), mixing ratio of precipitation(middle panel) at 1000 m in height and mixing ratio of cloud ice at 1300 m in height (lower
panel) for x=240-300 km.

develop over the sea. Some of them begin to form over the sea ice. Liu et al. (2005) studied theeﬀect of the sea-ice zone on the development of the band clouds using CReSS. In order to study
more detailed structure and formation process of cloud bands and a vortex over the Labrador Sea,
we performed a simulation experiment using a large calculaton domain and very ﬁne horizontal
grid size (Table 2.11). In the simulation experiment, sea ice is placed along the coast according
to the density of sea ice.
Table 2.11. Experimental design of the snow storm over the Labrador Sea.

domainx 1000, y 840, z 12 km

grid number
grid size
integration time

x 2003, y 1683, z 48
H 500m, V 60 ∼250m
10 hrsES node number

80 (640 CPUs)

Figure 2.25 shows that cloud bands begin to develop from the sea ice region and convection
composing the cloud bands intensiﬁes over the open sea. In this experiment, a vortex is also
simulated over the Labrador sea as observed. This is a sort of polar lows. The simulation
experiment shows the mesoscale vortex, cloud bands as well as convective clouds composing theseprecipitation systems with the very high resolution.

2.10

Heavy rainfall in Baiu season

The heavy rain event occurred on 19–20 July 2003 in Kyushu which is the western part ofJapan. During this period, the Baiu front was located to the north of Kyushu. The most intenserain occurred during the period of 16–22 UTC, 20 July and the area of the heavy rain was roughly

33

34

2 Simulations and Numerical Experiments

Figure 2.21. Same in Fig.2.20 , but for x=390-450 km.

100×100 km. The total amount of rain of this period reached about 210 mm at Minamata which
is located in the western Kyushu. We consider that this is a localized heavy rain. The heavy raincaused a ﬂood and 21 people were killed.
Satellite images show that a convective cloud cluster developed on the west coast of Kyushu.
The AMeDAS-radar composite data provided by JMA (the Japan Meteorological Agency) showsthat intense echo systems developed within the cloud cluster and moved into the west coast of
Kyushu (Fig.2.26). Figure 2.26 shows that an intense rainband extends in east-west direction andsome orographic rainfall echo developed in the western Kyushu.
We performed a prediction experiment of the localized heavy rain using CReSS. The experimental design is summarized in Table 2.12. The initial time is 00 UTC, 19 July 2003 and 24-hour
experiment was performed. The horizontal domain is centered at the west coast of Kyushu.
Table 2.12. Experimental design of the localized heavy rain in Kyushu using 2 km horizontal
resolution.

domainx 600, y 600, z 18 km

grid number
grid size
integration timenode number

x 603, y 603, z 63
H 2km, V 100 ∼150m
24 hrs10 (80 CPUs)

The experiment successfully simulated the localized heavy rain. Figure 2.27 shows that an
intense rainband developed to the west of Kyushu and the intense rain occurred along the west
coast of Kyushu. The rainfall pattern in Fig.2.27 is quite similar to the observation (Fig.2.26)
with regard to the pattern and intensity of rain. The intense rain in the western Kyushu issimulated during the period from 16–22 UTC, 19 July 2003.
We also performed prediction experiment with a ﬁne gird size of 500 m (Table 2.13). The

2.10 Heavy rainfall in Baiu season

Figure 2.22. Mixing ratio of precipitation (color levels; g kg−1 ) and horizontal velocity (arrows) at a height of 1112 m at 0000 UTC, 15 January 2001.

Figure 2.23. Mixing ratio of precipitation (gray scale; g kg−1 ). Black and blue arrows arehorizontal wind velocity at height of 315 and 2766 m, respectively. Read arrows
are wind shear between these levels.

35

2 Simulations and Numerical Experiments

Figure 2.24. Vertically integrated mixing ratio of precipitation (gray levels; g m−2 ) over theGreat lakes at 12 hours from the initial time.

VERTICALLY INTEGRATED HYDROMETEOR (g m-2), t = 10.00 hours
1000

800

700

600

DISTANCE Y (km)

36

500

400

500

300

200
200
100
100
50
0
0

100

200

300

400

500

600

700

800

900

DISTANCE X (km)Figure 2.25. Verticall integrated mixing ratio of prcipitation (gray levelsg m−2 ) over the
Labrador Sea at 10 hours from the initial time. Dark green and light green
mean land and sea ice, respectively.

2.10 Heavy rainfall in Baiu season

Figure 2.26. Rainfall rate of AMeDAS-radar composite (mm hr−1 ) at 16 UTC, 19 July 2003.

Figure 2.27. Rainfall amount during 15-16 UTC and horizontal wind at 500 m ASL at
16 UTC, 19 July 2003 obtained from the prediction experiment of 2 km resolution.

37

38

2 Simulations and Numerical Experiments

increase of horizontal resolution shows the detailed structure of the convective system (Fig.2.28).
The rainband extends from SW to NE direction. It is composed of intense convective cells. The
simulation resolves individual convective cells. The experiment with the high resolution also
shows orographic precipitation. Figure 2.28 shows that another rainband is present to the southof the main rainband. It forms on the lee side of the small islands over the sea, which is named
the Koshiki islands. We consider that both the two rainbands is related to the localized heavy
rainfall.
Table 2.13. Experimental design of the localized heavy rain in Kyushu using 500 m horizontal
resolution.

domaingrid number
grid size
integration time

x 600, y 600, z 18 km
x 1203, y 1203, z 63
H 500m, V 100 ∼150m
21 hrsnode number

120 (960 CPUs)

Figure 2.28. Mixing ratio of rain (gray scale; g/kg) and horizontal velocity (arrows) at aheight of 2080m at 1715 UTC, 19 July 2003 obtained from the experiment of
500 m resolution.

2.11

Typhoon 0613 and associated tornadoes

Typhoons occasionally accompany tornadoes and cause disasters due to strong winds. Whenthe typhoon 0613 (T0613) approached the western Japan on 17 September 2006, it accompanieda severe tornado (tatsumaki in Japanese). It killed three people and caused the train accident inKyushu. We performed a simulation experiment of T0613 and the associated tornado using theCReSS model to reveal structure of clouds along the typhoon rainband and the mechanism of thetornado. The horizontal grid size is 500m in the simulation of the typhoon and the calculation

2.11 Typhoon 0613 and associated tornadoes

domain is suﬃciently large to simulate overall structure of the typhoon. The initial condition(00 UTC, 17 September 2006) and the boundary conditions were provided by the regional model
output of the Japan Meteorological Agency (JMA). We used 128 nodes of the Earth Simulator forthe simulation. Intense spiral rainbands were formed in the eastern part of T0613 in the simulation
at about 5hours from the initial time. One of them passed over Nobeoka City at around 05 UTC,
17 September. This corresponds to the observation of JMA. The enlarged display of the rainbands
showed that some clouds composing the rainband are intense supercells.
A high-resolution experiment was performed to study the relationship between the supercells
and the tornado with a grid spacing of 75m. The result shows that hook-shaped structure forms
in the southernmost part of the supercell and that the intense tornado is simulated in the hookshaped part of the supercell (Fig.2.29). The horizontal diameter of the tornado is about 300m andits maximum vorticity is larger than 0.9 s−1 (Fig.2.30). Pressure perturbation is about −24 hPa
at the center of the tornado and maximum horizontal velocity reaches to 70 m s−1 . The verticalcross-section (Fig.2.31) shows that the tornado extends up to a height of 2.5 km in strong upward
motion. The simulation showed the detailed structures of the tornado and the supercell in thetyphoon rainband.

Figure 2.29. Supercell cloud simulated in the experiment with a horizontal grid spacing of 75m at 0500
UTC, 17 September 2006. Color levels indicate rain mixing ratio (g kg−1 ) at 200 m inheight. Arrows indicate horizontal velocity. The circle indicates that the tornado occurs
in the southernmost part of the supercell.

39

40

2 Simulations and Numerical Experiments

Figure 2.30. Enlarged view of the southernmost part of the supercell. Contours are vorticity drawn
every 0.1 s−1 from 0.1 s−1 and color levels are cloud mixing ratio (g kg−1 ).

Figure 2.31. Vertical cross-section of the tornado in the north-south direction. Contours are verticalvorticity drawn every 0.1 s−1 from 0.1 s−1 and color levels are cloud mixing ratio (g kg−1 ).

Part II

Technical Description

Chapter 3

Start to Use CReSS

This chapter is devoted to those who start to use the CReSS model forthe ﬁrst time. Most documents of this chapter are included in the ﬁle ofreadme. If you are already familiar with CReSS, you can skip this chapter.
Uncompressing archived ﬁles, compilation and some example to use CReSS
will be shown in the preset chapter.
CReSS has two versions: one is single version and the other is parallelversion. The former is indicated by ”s” in the ﬁle name and the latter by”m”. These two versions are totally identical. The result of calculations by
these two versions will be completely same. The only diﬀerence of the two
versions is necessity of the MPI library for parallel computation. The single
version can be performed without the MPI library on a single processor.

44

3 Start to Use CReSS

3.1

Download source ﬁles

At ﬁrst, please download the following ﬁles from our download page at http://www.rain.hyarc.nagoyau.ac.jp/ cress/download/ with your ID and password. If you have not have your ID and password,
please send an e-mail to tsuboki@rain.hyarc.nagoya-u.ac.jp, to request for getting ID and password. From the URL, you can download the following ﬁles.

Table 3.1. CReSS archived ﬁlescress2.2m.tar.Z

all ﬁle of CReSS parallel version.

cress2.2s.tar.Z

all ﬁle of CReSS serial version.

Since the users’ guide is not always updated, ﬁnd the readme ﬁles in the tar ﬁles and referthem. They are always updated and give a proper description.
In order to start to use the CReSS model, take the following processes. We assume that you
use a UNIX system such as LINUX or other type of UNIX systems. If you use parallel system,
use the cress2.2m.tar.Z. In the following part, we will explain the single version. The basic usage
is totally the same except to specify the number of processing elements.

3.2

Uncompress archived ﬁleTo uncompress the archived ﬁle of CReSS, use the following command of UNIX.

% uncompress -c cress2.2s.tar.Z |tar -xvf -Then, the following ﬁles will be found in the directory.

% cd CReSS2.2s/
% ls
./
Doc/
../
Form/

Src/
User_Mod/

compile.conf
compile.csh*

The document ﬁles will be found in the directory.

% cd Doc
% ls
/

../

readme.first

readme.user.conf

3.3 Compilation and samples

45

Read the ”readme.ﬁrst”, then you can ﬁnd how to compile the source. Then, read ”readme.user.conf”.
This readme is important to modify the namelist input.

3.3

Compilation and samples

To compile the solver of CReSS which is main processor of the model, enter the following
commands. If necessary, you should edit the conﬁguration ﬁle ”compile.conf”.

% vi compile.conf% compile.csh solver

After you compile the executable of solver.exe, please try a sample.

%cd Form%ls./../sounding.txt.cats.eye.form
sounding.txt.dry.downburst.form
sounding.txt.mountain.wave.form

sounding.txt.toyohashi.form
user.conf.cats.eye.form
user.conf.dry.downburst.form
user.conf.mountain.wave.form
user.conf.toyohashi.form% cp user.conf.mountain.wave.form ../user.conf% cp sounding.txt.mountain.wave.form ../test.sounding.txt

The ﬁrst part of the name of the sounding ﬁle ”test” is important. All I/O ﬁles of the CReSS
ﬁles are controlled by the name which is ﬁnd in the user.conf:&runameexprim = ’test’/

The preﬁx of all ﬁles is should be the same. In this case, it is ”test”.
To run the model, send the following commands.

% ./solver.exe < user.conf > & log.solver &46

3 Start to Use CReSSTo check the log ﬁle,% tail -f

log.solver

The output history dump ﬁles will be produced such as,

% ls

-l

-rw-r--r--rw-r--r--rw-r--r--rw-r--r--rw-r--r--rw-r--r--rw-r--r--rw-r--r--rw-r--r--rw-r--r--rw-r--r--rw-r--r--rw-r--r--

3.4

*.bin
1 tsuboki
1 tsuboki
1 tsuboki
1 tsuboki
1 tsuboki
1 tsuboki
1 tsuboki
1 tsuboki
1 tsuboki
1 tsuboki
1 tsuboki
1 tsuboki
1 tsuboki

kisho
kisho
kisho
kisho
kisho
kisho
kisho
kisho
kisho
kisho
kisho
kisho
kisho

1866240 038 12:13 test.dmp000000.bin
1866240 038 12:14 test.dmp000600.bin
1866240 038 12:16 test.dmp001200.bin
1866240 038 12:18 test.dmp001800.bin
1866240 038 12:20 test.dmp002400.bin
1866240 038 12:21 test.dmp003000.bin
1866240 038 12:23 test.dmp003600.bin
1866240 038 12:25 test.dmp004200.bin
1866240 038 12:27 test.dmp004800.bin
1866240 038 12:29 test.dmp005400.bin
1866240 038 12:30 test.dmp006000.bin
1866240 038 12:32 test.dmp006600.bin
4320 038 12:13 test.geography.bin

To Display the result

The output history dump ﬁles are displayed by the application soft wear ”grads”. You canﬁnd it in the URL at http://www.iges.org/grads/grads.html
The Grads need control ﬁle. In the above case, the ctl ﬁle will be3.4 To Display the result%cat dmp.ctl
DSET ^test.dmp%y400.bin
OPTIONS template
UNDEF -1.E35
XDEF 360 LINEAR 0.200
0.400
YDEF
1 LINEAR 0.200
0.400
ZDEF 144 LINEAR 0.062.5 0.125
TDEF 100 LINEAR 00Z01jan0000 6yr
VARS 9ub 144 99 base state x components of velocity [m/s].
u
144 99 x components of velocity [m/s].
vb 144 99 base state y components of velocity [m/s].
v
144 99 y components of velocity [m/s].
w
144 99 z components of velocity [m/s].
pb 144 99 base state pressure [Pa].
pp 144 99 pressure perturbation [Pa].
ptb 144 99 base state potential temperature [K].
ptp 144 99 potential temperature perturbation [K].
ENDVARSUse the dmp.ctl ﬁle to display the result by Grads. Then, you will ﬁnd the beautiful mountainwaves.

47

Chapter 4

Conﬁguration and Execution

In this chapter, conﬁguration, ﬂowchart of data, and execution of CReSS
will be described. Two types of CReSS are present: one is the single version
for a single processor and the other is the parallel version for multiple processors. This chapter describes mostly about the parallel version of CReSS. The
conﬁguration and execution of the serial version are essentially the same as
those of the parallel version and the diﬀerences between the parallel and serial
versions will be unspeciﬁed.
CReSS has two conﬁguration ﬁles: one is compile.conf which conﬁguresthe compile commands and their options, and the other is user.conf which
controls all the executables of pre- and post processors and main solver. The
former depends on the machine to be used while the latter is independent of
mostly except &sysdep. In this chapter, the latter conﬁguration ﬁle will be
explained in detail.
The conﬁguration ﬁle ”user.conf” is written as the format of the Fortran
namelist. The user.conf is used for not only run but also the compile. All thedimensions of array arguments are speciﬁed by the user.conf. Users, therefore,
will never change the source codes of CReSS.
The ﬂowchart of data will also described as well as data formats. There
are diﬀerent data ﬂows which depend on the conﬁguration of calculation.
At the last part of this chapter, an example run of experiment of the
Kelvin-Helmholtz waves will be shown: compilation of solver, run the solver
and output the result. Compilations and executions of the pre-processors willbe also presented.

50

4 Conﬁguration and Execution

4.1

User conﬁguration ﬁleAt present, CReSS version 2.2 and 2.3 are open to the public. The version 2.3 is almost
the same as the version 2.2 but grads ctl ﬁle is automatically produced in the version 2.3. The
conﬁguration ﬁles are almost same. Most parts of the descriptions are based on the version 2.2.
The parameters used only in the version 2.2 is indicated by the double dagger (‡). Those used
only in the version 2.3 is indicated by the dagger (†).

4.1.1

Composition of user.conf

The user conﬁguration ﬁle of CReSS, ”user.conf” is composed of the following 7 regions.
common region: All pre-processors, solver, and post-processors refer to the region.
gridata region: The pre-processor ”gridata.exe” refers to create initial and boundary ﬁles.
radata.region: The pre-processor ”radata.exe” refers the region to create model grid values ofradar data.
terrain region: The pre-processor ”terrain.exe” refers the region to create model topographydata.
surface region: The pre-processor ”surface.exe” refers the region to set surface parameters on
the model grid.
unite region: The post-processor ”unite.exe” refers the region to unite output data in parallel
computations.
rstruct region: The post-processor ”rstruct.exe” refers the region to rearrange restart ﬁles of
CReSS. Ordinary users will never use this region. Description of this region will be, therefore, omitted.
The user conﬁguration ﬁle of CReSS is a namelist of Fortran. The followings are important
notices to use the ﬁle.
• Since the conﬁguration ﬁle is used by a shell script in the compilation of CReSS, a space isnecessary to separate between a variable name and “=” and between a value and “=”.
• The solver and the pre-processors have dependency of some namelist variables. When adiscrepancy of variables between pre-processors and the solver is found, the program willstop execution and output an error.

4.1.2

Common region of the user conﬁguration ﬁle& sysdep
The namelist variable is machine-dependent. All programs use the variable, while no dependence is present between the programs.

savmeminteger(4)4.1 User conﬁguration ﬁleTo avoid memory limitation of small computer, the 3-dimensional array isarranged into 2-dimensional array.
0:1:2-dim. arrays are used.
3-dim. arrays are used. [recommended]& runame
The namelist speciﬁes the name of the experiment, which is used by solver, all pre-processor,
and all post-processor. Dependency is present between these programs. All I/O data set iscontrolled by the namelist.

exprimcharacter(80)Name of experiment. The number of character used in the variable must be
less than or equal to 80 characters. The variable is used by all input andoutput data ﬁles as a part of name. For example, the sounding data ﬁle will
be exprim.sounding.txt. No special characters are allowed to be used.

& dmisetThis namelist variables set numbers of grid points of calculation domain and numbers ofdecomposition in parallel computing. All programs use the variables and have dependency.

xpedim,ypedim

integer(4)Numbers of decomposition of the calculation domain in x and y directions.
The total numbers of processing elements (CPUs) are x × y.

xdim,ydim

integer(4)Numbers of grid points of the domain in the x and y directions, respectively.
Since the horizontal grid system is the Arakawa C type, the numbers of thescalar grid points are xdim − 1 and ydim − 1 in the x and y directions,
respectively. These numbers must satisfy the following relations in order toeach processing element is loaded with the same number of grid points.
(xdim − 3)/ xpedim = integer(ydim − 3)/ ypedim = integer
These relations means that numbers of grid points of the physical domain(xdim − 3, ydim − 3) must be divided by each number of decomposition.

zdim

integer(4)Number of grid point in the z direction. Since the vertical grid system isthe Lorenz type, the number of scalar grid points is zdim − 1. Using theparameters of cphopt = 2 or cphopt = 3, zdim must be larger or equal 12
because of use of temporary array.

51

52

4 Conﬁguration and Execution& project
The namelists deﬁne the horizontal coordinate systems of the calculation domain. They areused solver, pre-processors, and post- processors. Dependency is present between these programs.

mpopt

nspol

integer(4)Option for map projection of the domain. The following four projections
are available.
1: polar stereo graphic projection
2:
3:
4:

Lambert conformal projection [recommended]
Mercator projection
No map projection (latitude and longitude)

13:

Mercator projection of zonal circuit of the globe

integer(4)Switch to deﬁne origin of the coordinate in the northern or southern hemisphere.
1:

the origin of coordinate in the northern hemisphere. [recommended]-1:

the origin of coordinate in the southern hemisphere.

tlat1,tlat2

real(4)
The true latitudes of map projections. No deformation due to projection
occurs at the latitudes. The unit is degree [◦ ] and they are negative in thesouthern hemisphere. If the Lambert conformal projection is used, two true
latitude (tlat1 and tlat2) are necessary. If no map projection is used, they
are not necessary to be speciﬁed.

tlon

real(4)
The true longitude of coordinate. The unit is degree [◦ ] and degree in theWest Longitude is negative. The y-axis of the model domain corresponds tothe longitude. Even in the case of mpopt = 4, tlon must be speciﬁed.

& gridset
The namelists determine grid intervals and reference longitude and latitude. They are usedin solver, pre-processors, and post- processors. Dependency is present between these programs.

dx,dy,dz

real(4)
Grid intervals in the x, y and z directions, respectively. The unit is meter. If
non-uniform vertical grid is used (sthopt = 1,2) , dz is the averaged gridinterval.

4.1 User conﬁguration ﬁleulat,ulonreal(4)
Reference longitude and latitude, respectively, which corresponds to a single
point within the domain. The unit is degree. Degrees in the southernhemisphere and in the West Longitude are negative. CReSS speciﬁes theregion of the domain using the reference longitude and latitude with thefollowing real number of riu and verb”rju”.

riu,rjureal(4)
The real number of the reference point which corresponds to the point determined by the reference longitude and latitude (ulat,ulon).

& gridsth
The namelists determine vertical grid spacings in non-uniform coordinate. They are used insolver, pre-processors, and post- processors. Dependency is present between these programs.

zsfc

real(4)
Altitude of the sea surface. The unit is meter. In most cases, zsfc = 0.0 m.

zflat

real(4)The lowest height of the ﬂat level. Unit is meter.

sthoptinteger(4)Option for vertical stretching.
0: No stretching (uniform grid interval).
1: Stretching using a cubic function.
2:

Stretching using a function of tanh.

dzminreal(4)
Minimum grid interval at the lowest level. Unit is meter.

layer1,layer2

real(4)In stretching, the grid interval of dzmin is used up to a height of layer1, theintervals determined by functions between heights of layer1 and layer2,
and the grid interval at a height of layer2 is used above the height oflayer2. Unit is meter.

& terrain
The namelists specify terrain topography of the model. They are used in the program ofsolver, pre-processors, and post- processors. Dependency is present between these programs.

53

54

4 Conﬁguration and Execution

trnoptinteger(4)Option for model terrain.
0: the ﬂat ground.
the bell-shaped mountain.
user speciﬁed terrain from the external data ﬁle.

1:2:mnthghreal(4), dimension(1 : 2)
For trnopt = 1, ﬁrst number is height (m) of the summit of the bell-shaped
mountain, and the second number is height (m) of the base of the bellshaped mountain. The second number is usually 0.0 (m) and is possible tobe omitted. The altitude of the summit is mnthgh(1)-mnthgh(2) .

mntwx,mntwy

real(4)
For trnopt = 1, half width (distance from the center of a half height) of thebell-shaped mountain in the x and y directions, respectively. Unit is meter.

mntcx,mntcy

real(4)
For trnopt = 1, The coordinates x, y of the center of the bell-shaped mountain. Unit is meter. The relationship between the coordinates and theirarray number i,j is as follows.
x = (i − 2) × dx
y = (j − 2) × dy

& ﬂength
The namelists are with respect to time of calculation. They are used in solver, pre-processors,and post- processors. The parameter with ∗ has dependency between these programs.
sfcast ∗character ∗ 16
Date and time of the initial time of calculation in the Universal Time
Coordinate (UTC). The format is sfcast = ’yyyy/mm/dd hh:mm’.
For example, when the initial time is 13:00 UTC, 22 September 2007,
sfcast = ’2007/09/22 13:00’.

stime

real(4)
Initial time of calculation. Unit is second. The time is measured from thetime of sfcast. In the ﬁrst calculation, stime = 0.0, and In a restart,
stime is must be the restart time.

etime

real(4)

4.1 User conﬁguration ﬁleTime of the termination of calculation. Unit is second. The time is alsomeasured from the time of sfcast. For example, etime must be 5400.0 to
calculate 1800.0 seconds from 3600.0 seconds.

& boundry
The namelists determine the lateral boundary condition. They are used in solver, preprocessors, and post- processors. The parameters with ∗ have dependency between these programs.

wbc∗

integer(4)Western lateral boundary condition. Option is shown below.

ebc∗

integer(4)Eastern lateral boundary condition. Option is shown below.
integer(4)

sbc∗
nbc

∗

Southern lateral boundary condition. Option is shown below.
integer(4)Northern lateral boundary condition. Option is shown below.
1: periodic boundary condition.
2:
3:
4:

rigid wall boundary condition.
zero-gradient boundary condition.
wave-radiating boundary condition.

5:wave-radiating boundary condition using vertically averaged normal velocity component as phase velocity.

6:

wave-radiating boundary condition using constant phase velocity
and advection speed.

7:wave-radiating boundary condition using constant phase velocity.
[recommended]bbc

integer(4)Bottom boundary condition. Option is shown below.

tbc

integer(4)Top boundary condition. Option is shown below.
2: rigid wall boundary condition.
3: zero-gradient boundary condition.
4:wave-radiating boundary condition at the top boundary condition.
If impopt = 1, then these parameters are forced to be 3.

lbcvarcharacter(80); lbcvar = ’xxxxxxxxxxxxx’

55

56

4 Conﬁguration and Execution

If exbopt=0, the lateral boundary can be dumped to the base state. The
parameter speciﬁes variables along the lateral boundary to be dumped. If
the base state is given, the variable is dumped to the base state. If no basestate is given, then, it is dumped to zero. The ﬁrst 13 characters of 80
are used. Each character from the ﬁrst corresponds to u and v-componentsof velocity, vertical velocity component, pressure, potential temperature (ortemperature), water vapor mixing ratio (or relative humidity), mixing ratiosof cloud water, rain, cloud ice, snow and graupel, the turbulent kinetic energy and mixing ratio of tracer (u, v, w, p, pt, qv, qc, qr, qi, qs, qg, tke, qtrace ).
If exbopt=0, this option is ignored.
o:x:lbnewsthe variable is dumped to the base state or zero along the lateralboundary.
the variable is not dumped.

real(4)
Dumping coeﬃcient of all variables except the normal velocity components
along the lateral boundary. Unit is [s−1 ].

lbnormreal(4)
Dumping coeﬃcient of the normal velocity components along the lateralboundary. Unit is [s−1 ].

† gwavereal(4)
Fastest gravity wave speed for constant phase speed for radiation boundary
condition. Unit is [m s−1 ].

& gpvpram
The namelists control GPV external data and specify lateral external forcing and upper sponge
dumping. They are used in solver, pre-processors, and post-processors. The parameters with ∗have dependency between these programs.

gpvvar ∗character(80); gpvvar = ’xoxxxxx’
The parameter speciﬁes variables to be inputted to the model from an external data. The ﬁrst 7 characters of 80 are used. Each character from the ﬁrst
corresponds to the vertical velocity component (w), water vapor mixing ratio(qv)(or relative humidity), mixing ratios of cloud water, rain, cloud ice, snow
and graupel (qc, qr, qi, qs, qg). On the other hand, z-coordinate, horizontal
velocity components, pressure, potential temperature (or temperature) arenecessary and omitted form this parameter. For example, the vertical velocity component, water vapor mixing ratio (or relative humidity) and mixing
ratio of rain are used (when they are available), then extvar = ’ooxoxxx’

4.1 User conﬁguration ﬁlegpvitv ∗o:the variable is available in the dataset.

x:

the variable is not available in the dataset.

real(4)
Time interval of the external data. Unit is second.

gsmoptinteger(4)Option for smoothing GPV data.
0: No smoothing is performed.
1: Perform smoothing for boundary data.
2: Perform smoothing for boundary and initial data.

gsmcntinteger(4)Iteration count for GPV data smoothing.

gsmcoereal(4)
Smoothing coeﬃcient using in the iteration. Unit is [s−1 ].

nggoptinteger(4)Option for nudging of analysis data.
0: no nudging.
1:

nggvarperform the nudging.

character(80); nggvar = ’xxxxxxxxxxx’
The parameter speciﬁes variables to be used in nudging. The ﬁrst 11 characters of 80 are used. Each character from the ﬁrst corresponds to u andv-components of velocity, vertical velocity component, pressure, potential
temperature (or temperature), water vapor mixing ratio (or relative humidity), mixing ratios of cloud water, rain, cloud ice, snow and graupel(u, v, w, p, pt, qv, qc, qr, qi, qs, qg). For example, when all the three components of velocity, pressure and potential temperature are used for nudging, nggvar = ’oooooxxxxxx’. All variables besides the necessary variables
must be speciﬁed by extvar.
o: the variable is used in nudging.
x: the variable is not used in nudging.

nggcoereal(4)
Coeﬃcient of dumping (e-folding time of dumping) of the nudging of theanalysis data. Unit is [s−1 ].

nggdltreal(4)
Time interval of nudging. Unit is second. The parameter must be divided
by dtb and larger than or equal to dtb.

57

58

4 Conﬁguration and Execution

nggendreal(4)
Time to terminate the nudging (measured from the initial). Unit is second.

exbopt ∗integer(4)Option for forcing to external data at the lateral boundary. The forcing is
performed with respect to the necessary variables and variables speciﬁed by
exbvar.
0:
1:
2:

11:
12:

exbvar

No forcing at the lateral boundary.
Forcing to the external data at the lateral boundary is performed.
forcing to the external data at the lateral boundary is performed
and the terrain is also smoothed to be the external terrain in thelateral sponge layer.
Forcing to the external data at the lateral boundary is performed
and horizontal velocity is adjusted.
Forcing to the external data at the lateral boundary is performed.
The terrain is also smoothed to be the external terrain in thelateral sponge layer. Horizontal velocity is adjusted.

character(80); exbvar = ’–x—xxxxx’
The parameter speciﬁes variables to be externally forced. The ﬁrst 11 characters of 80 are used. Each character from the ﬁrst corresponds to u andv-components of velocity, vertical velocity component, pressure, potential
temperature (or temperature), water vapor mixing ratio (or relative humidity), mixing ratios of cloud water, rain, cloud ice, snow and graupel(u, v, w, p, pt, qv, qc, qr, qi, qs, qg).
o:

-:

+:

x:

Both the vector and scalar variables at the outer most grids are
replaced by the external data. In this option, the coeﬃcients
exnews and exnorm are not used.
Both the vector and scalar variables along the boundary are
dumped to the external data and wave-radiating lateral boundary
condition is applied. The dumping coeﬃcient of normal velocity
to the boundary is exnorm. The dumping coeﬃcient of all othervariables is exnews.
The normal velocity component of (u, v) along the outermost girds
are replaced by the external data and the parallel velocity components and scalar variables are dumped to the external data withthe coeﬃcient of exnews.
No external forcing is performed the boundary condition is thewave-radiating type.

exnewsreal(4)

exnormCoeﬃcient of the external forcing at the outermost lateral boundary. Unitis [s−1 ].
real(4)

4.1 User conﬁguration ﬁleCoeﬃcient of the lateral forcing for normal velocity component at the outermost lateral boundary. Unit is [s−1 ].
exbwidinteger(4)For exbopt = 2 or exbopt = 12, the model terrain is modiﬁed to connect
the coarse data terrain smoothly along the boundary within the width speciﬁed by this parameter. Unit is grid number.

lspoptinteger(4)Option for lateral sponge.

lspvar0:
1:
2:

No lateral sponge dumping.
Lateral sponge damping to external data.
lateral sponge damping to base state or initial values.

11:

Lateral sponge damping to external data with smoothing within
the lateral sponge layer.

12:lateral sponge damping to base state or initial values with smoothing within the lateral sponge layer.

character(80); lspvar = ’++xxooxxxxxxx’
The parameter speciﬁes variables to be dumped in the lateral sponge layer.
The ﬁrst 11 characters of 80 are used. Each character from the ﬁrst corresponds to u and v-components of velocity, vertical velocity component,pressure, potential temperature (or temperature), water vapor mixing ratio(or relative humidity), mixing ratios of cloud water, rain, cloud ice, snow
and graupel (u, v, w, p, pt, qv, qc, qr, qi, qs, qg).
o: The lateral sponge dumping is applied with the coeﬃcient lsnews
to the speciﬁed variables.
+: This option is available for only (u, v). Both the normal and parallel components in the sponge layer dumped with the coeﬃcientlsnews and the normal velocity component additionally dumped
with the coeﬃcient lsnorm.
-: This option is available for only (u, v). Only the normal velocity
component of (u, v) is dumped with the coeﬃcient of ls norm,
but the parallel component is not dumped.
x:

lspsmt ∗No dumping is performed.

real(4)For lspopt = 11 or lspopt = 12, the parameter is coeﬃcient of smoothing
within the lateral sponge layer. Unit is [s−1 ].

lsnewsreal(4)
Coeﬃcient of the dumping in the lateral sponge layer. Unit is [s−1 ]. The
forcing decrease with the distance from the lateral boundary.

59

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4 Conﬁguration and Execution

lsnormreal(4)
Coeﬃcient of the dumping of the normal velocity components in the lateralsponge layer. Unit is [s−1 ]. The forcing decrease with the distance from thelateral boundary.

wdnewsinteger(4)Grid number of the lateral sponge layer.

wdnorminteger(4)Grid number of the lateral sponge layer for the normal velocity components.

vspoptinteger(4)Option for dumping in the top sponge layer.
0: no dumping in the top sponge layer.
1: dumping in the top sponge layer is performed. The variables areforced to the external grid data.
2:

dumping in the top sponge layer is performed. The variables areforced to the initial data (base state).

vspgpvreal(4)For vspopt = 1, dumping coeﬃcient in top sponge layer of variables forwhich external data is given. The forcing decreases with the distance fromthe top of the model. Unit is [s−1 ]. Variables which external data is not
given are dumped to zero.

vspgpvreal(4)For vspopt = 1 or vspopt = 2, dumping coeﬃcient in top sponge layer.
The coeﬃcient is used for all variables in vspopt = 1, and for variables forwhich external data is not given in vspopt = 2. The forcing decreases withthe distance from the top of the model. Unit is [s−1 ].

botgpvreal(4)For vspopt = 1, the lowest height of the top sponge layer for variables forwhich external data is given. Unit is meter.

botgpvreal(4)For vspopt = 1 or vspopt = 2, the lowest height of the top sponge layer
for all variables in vspopt = 1, and for variables for which external data is
not given in vspopt = 2.

4.1 User conﬁguration ﬁle& radpram
The namelists control radar data nudging. They are used in solver and the pre-processor
radata.exe.
radvar ∗character(80); radvar = ’xxxx’
The parameter speciﬁes variables to be inputted to the model from an external data. The ﬁrst 4 characters of 80 are used. Each character from theﬁrst corresponds to u- and v-components of horizontal velocity, the verticalvelocity component, radar echo intensity (dBZ) or mixing ratio of precipitation which is usually converted from the radar echo intensity (u, v, w, qp).
o: the variable is available in the dataset.
x: the variable is not available in the dataset.

raditv ∗real(4)
Time interval of the radar data. Unit is second.

ngroptinteger(4)Option for nudging of the radar data.
0: no nudging of radar data.
1:

ngrvarperform the nudging of radar data.

character(80); ngrvar = ’xxxxxxxxxxx’
The parameter speciﬁes variables to be used in radar nudging. The ﬁrst 6
characters of 80 are used. Each character from the ﬁrst corresponds to u
and v-components of velocity, vertical velocity component, mixing ratios ofrain, snow and graupel (u, v, w, qr, qs, qg).
o: the variable is used in nudging.
x: the variable is not used in nudging.

ngrcoereal(4)
Coeﬃcient of nudging the radar data. Unit is [s−1 ].

ngrdltreal(4)
Time interval of nudging. Unit is second. The parameter must be divided
by dtb and larger than or equal to dtb.

ngrstrreal(4)
Time to start the nudging (measured from the initial). Unit is second.

ngrendreal(4)
Time to terminate the nudging (measured from the initial). Unit is second.

61

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4 Conﬁguration and Execution

ngraffreal(4)
Time width of the nudging. The nudging is performed within twice of ngraffwith centering at each time when data is given. The parameter must be
smaller than half of raditv. Unit is second.

& sfcphys
The namelists determine the surface conditions and the ground model.

† sfcdat ∗character(80)The parameter speciﬁes the external data used for calculation of the surface
process. The ﬁrst 3 characters of 80 are used. Each character from theﬁrst corresponds to land use, sea surface temperature, and sea ice distribution. For example, when the land use and the land surface temperature areavailable, then, sfcdat = ’oox’.
o: The variable is available and will be used.
x:

sfcopt

The variable is not used.

integer(4)Option for the surface process.
0:
No surface process is performed.
1:
Surface process is performed. Ground temperature and sea surface
temperature are calculated and vary with time.
2:
Surface process is performed. Ground temperature is calculated
and varies with time while sea surface temperature is constantwith time.
11: Surface process is performed. Calculations of ground temperatureand sea surface temperature are started from the previous calculation. The re-start ﬁle of the previous calculation is deﬁned by
prvres12:

levpblSurface process is performed. Calculation of ground temperature isstarted from the previous calculation while sea surface temperature
is constant with time. The re-start ﬁle of the previous calculation
is deﬁned by prvres

integer(4)For tubopt = 0, Vertical 1-dimensional diﬀusion of the Meller-Yamada closure 2 is performed. The parameter speciﬁes grid number of the planetary
boundary layer. The thickness of each layer is dz and 1 < levpbl <zdim - 2. If For tubopt = 0, then, this optional process is ignored.

levundinteger(4)Number of layers of the ground and sea.

4.1 User conﬁguration ﬁledzgrd

real(4)Thickness of the layers of the ground model. Unit is meter. The total depthof the ground model is desirable to be larger than a few meters.

dzsea

real(4)Thickness of the layers of the sea model. Unit is meter. The total depth of
the sea model is desirable to be larger than a few meters.

tgdeepreal(4)
Constant temperature at the bottom of the ground model. Unit is [K].

prevrescharacter(80); prevres=’test.res000000
Restart ﬁle name without extension of previous calculation. This ﬁle is used
for sfcopt=11 or sfcopt=12.

‡ sfcdat ∗character(80)The parameter speciﬁes the external data used for calculation of the surface
process. The ﬁrst 3 characters of 80 are used. Each character from theﬁrst corresponds to land use, sea surface temperature, and sea ice distribution. For example, when the land use and the land surface temperature areavailable, then, sfcdat = ’oox’.
o: The variable is available and will be used.
x: The variable is not used.

lnduseinteger(4)For sfcdat (1)= ’x’, ground surface condition is speciﬁed by this parameter. For every options, the area lower than 0 m in altitude is considered asa sea.
1: The ground higher than 0 m in altitude is not covered with snow
nor ice. The surface parameters are speciﬁed by user with thefollowing parameters.
2: The ground higher than 0 m in altitude is covered with snow. The
surface parameters are speciﬁed inside of solver.
3: The ground higher than 0 m in altitude is covered with ice. The
surface parameters are speciﬁed inside of solver.

gralbereal(4); gralbe=0.2
For sfcdat (1)= ’x’ and lnduse=1, a constant albedo of the ground.

grbetareal(4); grbeta=0.5
For sfcdat (1)= ’x’ and lnduse=1, a constant evapotranspiration coeﬃcient (moisture availability form the surface).

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4 Conﬁguration and Execution

grz0m

real(4); grz0m=0.4
For sfcdat (1)= ’x’ and lnduse=1, a constant roughness length for velocity of the ground surface [m].

grz0hreal(4); grz0h=0.1
For sfcdat (1)= ’x’ and lnduse=1, a constant roughness length for scalarof the ground surface [m].

grcap

real(4); grcap=2.3e6
For sfcdat (1)= ’x’ and lnduse=1, a constant thermal capacity of ground[J m−3 ].

grnuu

real(4); grnuu=7.0e-7
For sfcdat (1)= ’x’ and lnduse=1, a constant thermal diﬀusivity ofground [m−2 ].

sstcstreal(4); sstcst=300.15e0
For sfcdat (2)= ’x’, a constant sea surface temperature [K].

dstoptreal(4); dstopt=1
For sfcdat (3)= ’o’, Option to produce of model sea ice distibution.
1: Ice density weighted average distribution.
2:

All-or-nothing distribution.

& initype
The namelists determine the initialization of model. Used in only solver.

inioptinteger(4)Option for the model initialization.
1:
Initial data is from the sounding data.
2:
From restart ﬁles.
3:
12:

snddimFrom 3-dim. non-uniform grid data (GPV data).
From restart ﬁles without checking conﬁguration of the previous
run. variables.

integer(4)For iniopt = 1, number of levels of the sounding data.

‡ sndhed

integer(4)4.1 User conﬁguration ﬁle65

For iniopt = 1, number of comment lines in the header of the sounding
data. The comment lines must be at the beginning of the sounding ﬁle.
sndtypcharacter(80)For iniopt = 1, specify data type of the sounding. The following combinations are available except the horizontal velocity.
option

1st column

2nd column

5th column

ppm:
ppm:
zpm:

pressure [Pa],pressure [Pa],height [m],θ [K] ,
θ [K] ,
θ [K] ,

qv [kg kg−1 ]qv [kg kg−1 ]qv [kg kg−1 ]

ptm:
ztm:
ppr:
zpr:

pressure [Pa],height [m],pressure [Pa] ,height [m],temperature [K] ,temperature [K] ,θ [K] ,
θ [K] ,

qv [kg kg−1 ]qv [kg kg−1 ]Rh [%]
Rh [%]

ptr:
ztr:

pressure [Pa],height [m],temperature [K] ,temperature [K] ,Rh [%]
Rh [%]

zsnd0,psnd0

real(4)Height [m] and pressure [Pa] of the lowest level of the sounding
data, respectively. If sndtyp(1:1) = ’p’, then zsnd0 is used and if
sndtyp(1:1) = ’z’, then psnd0 is used.

masoptinteger(4)Option of mass-conservation calculation for the initial ﬁeld.
0:
1:

Mass-conservation model is not performed.
Mass-conservation model is performed.

masepsreal(4)
Value of convergence in iteration of the mass-conservation model calculation.

alpha1real(4)For masopt = 1, horizontal weighting coeﬃcient of the mass-conservation
calculation.

alpha2real(4)For masopt = 1, vertical weighting coeﬃcient of the mass-conservation calculation.

& gridmove
The namelists conﬁgure the horizontal displacement of the calculation domain and are used
only in solver.

66

4 Conﬁguration and Execution

movoptinteger(4)Option for the displacement of the domain. This option is valid whentrnopt = 0, iniopt = 1 and exbopt = 0.
0: No displacement of the domain.
1: Displacement of the domain is switched on.

umove,vmove

real(4)
Speeds of the displacement of the domain in the x and y directions, respectively. Unit is [m s−1 ].

& ptinicon
The namelists set perturbations of the potential temperature at the initial ﬁeld and are used
only in solver.

pt0optinteger(4)Option for perturbations of potential temperature at the initial time. Thisparameter is valid when iniopt = 1.

pt0num

0:
1:
2:
3:

No initial perturbation of potential temperature.
One or more elliptic perturbations are placed in the x direction.
One or more elliptic perturbations are placed in the y direction.
Perturbation of trigonometric function in the x direction is set.

4:
5:Perturbation of trigonometric function in the y direction is set.
Random perturbations are generated between the speciﬁed twolevels.

integer(4)Number of the elliptic perturbation of potential temperature when
pt0opt = 1 or 2.

ptp0

real(4)
Maximum potential perturbation at the initial time . Unit is Kelvin.

pt0rx,pt0ry,
pt0rz

real(4)
Radii of the elliptic perturbations, or half wave length of the trigonometric function or half width of the random perturbation layer in z direction(pt0opt = 5). Unit is meter.

pt0cx,pt0cy,
pt0cz

real(4)Center of the elliptic perturbations, or the origin of the trigonometricfunction or middle height of the random perturbation layer in z direction(pt0opt = 5). Unit is meter.

4.1 User conﬁguration ﬁlept0ds

real(4)
Distance between bubbles of potential temperature in the case of
pt0opt = 1 or 2. Unit is meter.

& integrat
The namelists deﬁne the parameters of time integration such as time intervals and coeﬃcientsof the Asselin’s time ﬁlter. Only in solver.

dtbig

real(4)
Time intervals of the large time step. The terms which are not related
acoustic waves are integrated in the large time step. Unit is second. Thesetime intervals must satisfy the CFL condition.

dtsml

real(4)
Time intervals of the small time step. The terms related with the acoustic
waves are integrated in the small time step. Unit is second. These timeintervals must satisfy the CFL condition.

gwmoptinteger(4)Option for the time integration of the terms related with the gravity waves.
[gwmopt = 0 is recommended.]
0: They are integrated in the large time step.
1: They are integrated in the small time step.

impoptinteger(4)Option for time integration scheme of the acoustic wave mode in the verticaldirection. [impopt = 1 is recommended.]
0: Explicit scheme.
1:
2:
3:

† advopt

Implicit scheme using Gaussian elimination.
Implicit scheme using Gaussian elimination with pivot option.
Implicit scheme using Gauss-Zaidel method.

integer(4)Option for the advection scheme. The center diﬀerence is used in the advection scheme. The numerical smoothing is, therefore, necessary to suppress
the computational instability.
0: the second order center diﬀerence.
1: the fourth order center diﬀerence in horizontal and the second
order in vertical directions.
2: the fourth order center diﬀerence in both horizontal and verticaldirections.

67

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4 Conﬁguration and Execution

gsdepsreal(4)For impopt = 3, threshold value for iteration convergence of the GaussSeidel method.

weicoereal(4)Weighting coeﬃcient of the implicit scheme.
[weicoe = 0.6e0 is recommended.]filcoereal(4)Weighting coeﬃcient of the Asselin’s time ﬁlter.
[filcoe = 0.1e0 is recommended.]& advction
The namelist is the option of the advection scheme. Only in solver. This namelist is used
only in the version 2.2. This namelist is included in the namelist ”& integrat” in the version 2.3.

‡ ad4opt

integer(4)Option for the advection scheme. The center diﬀerence is used in the advection scheme. The numerical smoothing is, therefore, necessary to suppress
the computational instability.
0: the second order center diﬀerence.
1:
2:

the fourth order center diﬀerence in horizontal and the second
order in vertical directions.
the fourth order center diﬀerence in both horizontal and verticaldirections.

& smootherConﬁgurations of the numerical smoothing are determined in the namelists. Only in solver.

smtoptinteger(4)Option for the numerical smoothing which works as an artiﬁcial viscosity
to suppress the numerical noise of the advection scheme. [smtopt = 3 isrecommended.]4.1 User conﬁguration ﬁle0:

No numerical smoothing.

1:
2:
3:

The second order numerical smoothing.
The fourth order numerical smoothing in horizontal direction.
The fourth order numerical smoothing both in horizontal and vertical directions.
The fourth order numerical smoothing in horizontal direction andnonlinear smoothing.
The fourth order numerical smoothing both in horizontal and vertical directions and nonlinear smoothing.

12:
13:

smhcoereal(4); smhcoe=0.0008333e0
Coeﬃcient of smoothing in the horizontal direction. Unit is [s−1 ].

smvcoereal(4); smvcoe=0.0008333e0
Coeﬃcient of smoothing in the vertical direction. Unit is [s−1 ].

nlhcoereal(4)
Coeﬃcent of nonlinear smoothing in the horizontal direction. Unit is [s−1 ].

nlvcoereal(4)
Coeﬃcient of nonlinear smoothing in the vertical direction. Unit is [s−1 ].

& mapfcter
The namelist is an option for map factors. Only in solver.

mfcoptinteger(4)Option for map factor.
0: No map factor is used.
1: Map factor is used.

& coriolis
Conﬁguration of the Coriolis force terms. Only in solver.

coroptinteger(4)Option for the Coriolis force terms.

69

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4 Conﬁguration and Execution

0:

No Coriolis force terms.

1:
2:

Horizontal components of the Coriolis force are calculated.
Both horizontal and vertical components of the Coriolis force arecalculated.

& earthcrv
The namelist is an option for map factors. Only in solver.

crvoptinteger(4)Option for calculation of the earth curvature term.
0:1:

No earth curvature term is calculated.
The earth curvature term is switched on.

& buoyancy
Switching the buoyancy term of pressure perturbation in the vertical component of the momentum equation. This term is related to the acoustic waves. Only in solver.

buyoptinteger(4)Option for the buoyancy term.
0: The buoyancy term is omitted.
1: The buoyancy term is included.

& diabatic
Switching the diabatic term in the pressure equation. Only in solver.

diaoptinteger(4)Option for the diabatic term.
0: The diabatic term is omitted.
1: The diabatic term is included.

& ddamping
Conﬁguration of the divergence dumping of the pressure gradient force in the momentum
equations. Only in solver.

4.1 User conﬁguration ﬁledivoptinteger(4)Option for the divergence dumping of the pressure gradient force, which isan artiﬁcial term to suppress the numerical instability.
0: No divergence dumping is performed.
1: Divergence dumping is performed.

divndcreal(4)Non-dimensional dumping coeﬃcient. The recommended value is 0.05.

& turbulent
The subgrid-scale turbulence process is conﬁgured in the namelist. Only in solver.

tuboptinteger(4)Option for the sub-grid turbulence process. [tubopt = 2 is recommended.]
0: No turbulence process is performed.
1: The Smagorinsky scheme (1 order closure).
2:
3:

isooptThe 1.5 order closure with TKE.
The 1.5 order closure with TKE and set initial mixing coeﬃcient.

integer(4)Option for directionality of the grid.
1: Grid is directional in horizontal and vertical directions.
2: Grid is non-directional in horizontal and vertical directions.

prnum

real(4)
Turbulent Prandtl number (P r = ντ /νH ). It is used only for stability checkwhen tubopt = 2.

csnum

real(4)The Smagorinsky constant when tubopt = 1.

& cloudphy
Conﬁguration for the cloud microphysics. Only in solver.

cphoptinteger(4)Option for the cloud microphysics.

71

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4 Conﬁguration and Execution

nclcst0:

No cloud microphysics (the dry model).

1:
2:
3:

The bulk warm rain parameterization.
The bulk cold rain parameterization.
The bulk cold rain parameterization with solving the tendency
equation of number densities of ice phase hydrometeors (ice, snow
and graupel).

real(4); nclcst=1.e8
The constant concentration of cloud water. Unit is [m−3 ].

thresqreal(4); thresq=1.e-12
The Minimum threshold value of mixing ratios. Unit is [kg kg−1 ].

& mixtrace
The tracer option is available using a tracer mixing ratio which has no source terms nor falling
velocity. The time change of the mixing ratio is due to only the advection. The namelists set
conﬁguration of tracer mixing ratio in the initial ﬁeld. Used only in solver.

trkoptqt0optinteger(4)Option for tracer capability.
0: No tracer is used.
1:

Calculation of tracer mixing ratio is performed. The source oftracer is set only in the initial ﬁeld.

2:

Calculation of tracer mixing ratio is performed. The source oftracer is set in the initial ﬁeld and constant source of the tracer ispresent at all time in the initial locations.

integer(4)Options for conﬁguration of tracer.
1: One or more elliptic source of tracer are placed in the x direction.
2:
3:
4:
5:

One or more elliptic source of tracer are placed in the y direction.
Source of tracer of trigonometric function in the x direction is set.
Source of tracer of trigonometric function in the y direction is set.
Random source of tracer are generated between the speciﬁed twolevels.

qt0numinteger(4)Number of the elliptic source of tracer when qt0opt = 1 or 2.

qt0

real(4)

4.1 User conﬁguration ﬁleMagnitude or amplitude of source of tracer at the initial time . Unit is [kgkg−1 ].
qt0rx,qt0ry,
qt0rz

real(4)
Radii of the elliptic source of tracer, or half wave length of the trigonometric function or half width of the random source of tracer in z direction(qt0opt = 5). Unit is meter.

qt0cx,qt0cy,
qt0cz

real(4)Center of the elliptic source of tracer, or the origin of the trigonometricfunction or middle height of the random source of tracer in z direction(qt0opt = 5). Unit is meter.

qt0ds

real(4)
Distance between bubbles of source of tracer in the case of qt0opt = 1 or
2. Unit is meter.

qtdt

real(4)
Intensity of tracer source in the case of trkopt = 2. Unit is [s−1 ].

& outfomat
The namelists determine the format and time intervals of outputs. Only in solver.

dmpfmtinteger(4)Option for the output format of the history and geographic ﬁles.
1: ASCII text format.
2:

dmpcmpBinary data of the unformatted direct access.

integer(4)Option of the vertical coordinate of the history ﬁle. When dmpcmp = 2 or
1, An undeﬁned value of −1.0 × 1035 are outputted below the surface andabove the top of the domain.

73

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4 Conﬁguration and Execution

1

The vertical coordinate of history ﬁles is the terrain following type(z∗ (ζ)). Horizontal velocity components (u, v) are x and y components of the coordinate. In this option, all grids are not theundeﬁned value. For nesting calculation, this option should be
used for coarse grid calculation.

2

The output variables are interpolated at horizontal levels with aninterval of dz. Horizontal velocity components (u, v) are x and ycomponents of the coordinate.

3

The output variables are interporlated at horizontal levels deﬁned
by the stretching functions. Horizontal velocity components (u, v)
are x and y components of the coordinate.
Same as dmpcmp=1, but horizontal velocity components (u, v) arezonal and meridional components.
Same as dmpcmp=2, but horizontal velocity components (u, v) arezonal and meridional components.
Same as dmpcmp=3, but horizontal velocity components (u, v) arezonal and meridional components.

11
12
13

dmpvarcharacter(80); dmpvar = ’–o—oooxoox’
The parameter speciﬁes variables to be outputted to the history ﬁles. The
ﬁrst 13 characters of 80 are used. Each character from the ﬁrst corresponds
todmpvar(1:1)=’-’/’o’/’x’: x- or zonal component of horizontal velcity.
dmpvar(2:2)=’-’/’o’/’x’: y- or meridioanl component of horizontal velcity.
dmpvar(3:3)=’o’/’x’ :vertical component of velocity.
dmpvar(4:4)=’-’/’o’/’x’:
dmpvar(5:5)=’-’/’o’/’x’:
dmpvar(6:6)=’-’/’o’/’x’:

pressure.
potential temperature.
mixing ratio of water vapor.

dmpvar(7:7)=’o’/’x’ :dmpvar(8:8)=’o’/’x’ :dmpvar(9:9)=’o’/’x’ :dmpvar(10:10)=’o’/’x’:

mixing ratios of hydrometeors.
concentrations of ice, snow and graupel.
TKE (turbulent kinetic energy).
mixing ratio of tracer.

dmpvar(11:11)=’o’/’x’:
dmpvar(12:12)=’o’/’x’:
dmpvar(13:13)=’o’/’x’:

surface monitor variables.
precipitation at the surface.
z physical coordinates.

The options areo: the variable is outputted to history ﬁles.
-:x:

For (u, v, qv), the variable is outputted to history ﬁles while basestate (initial state) of these variables are not outputted. For pressure and potential temperature, sum of base state and perturbation is outputted.
the variable is not outputted.

4.1 User conﬁguration ﬁledmpitvreal(4)
Time interval to be outputted the history-ﬁle outputs. Unit is second. Theinterval is counted from the initial time even the calculation is started froma restart ﬁle. For example, if dmpitv=300.0 seconds and restarted from450.0 sec, the ﬁrst output time will be 600.0 second from the initial time.

resoptinteger(4)Option for restart ﬁles.
0: no restart ﬁles are issued.
1: restart ﬁles are outputted at the speciﬁed time interval.

resitvreal(4)
Time interval of the restart-ﬁle outputs. Unit is second. The time interval
is also counted from the initial time even if started from a restart ﬁle.

mxnoptinteger(4)Option for log ﬁle.
0: no log ﬁle is issued.
1:

log ﬁle is outputted at the speciﬁed time interval.

mxnvarcharacter(80); mxnvar = ’oooooooooo’
The parameter speciﬁes variables to be logged to the log ﬁle (the standard
output). The ﬁrst 10 characters of 80 are used. Each character from the ﬁrst
corresponds to u, v, w, p, θ, qv, mixing ratios of hydrometeors, concentrationsof ice, snow and graupel, TKE and mixing ratio of tracer.
o: maximum and minimum values are logged.
x: no log of the variable is issued.

mxnitvreal(4)
Time interval to output maximum and minimum of prognostic variables tothe standard output. Unit is second. The time interval is also counted fromthe initial time even if started from a restart ﬁle.

4.1.3

Region for the pre-processor, gridata.exe

The pre-processor ”gridata.exe” converts grid point value (GPV) data of the data grid toinitial and boundary data of the model grid in 3-dimension. This region speciﬁes the grid systemof the GPV data and options of interpolation.
& project grd
The namelists describe the coordinate system of a data grid. The parameters are used in only
gridata.exe.

75

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4 Conﬁguration and Execution

mpopt_grd

integer(4)Option for the coordinate of the external data grid.
0:
latitude and longitude coordinates
1:
2:
3:

The polar stereo graphic projection
The Lambert conformal projection
The Mercator projection

4:
10:
13:

No map projection
The global latitude and longitude coordinates
The cyclic Mercator projection around the globe

nspol_grd

integer(4)Switch to deﬁne origin of the coordinate in the northern or southern hemisphere.
1: the origin of coordinate in the northern hemisphere.
-1: the origin of coordinate in the southern hemisphere.

tlat1_grd,
tlat2_grd

real(4)
The true latitudes of map projections. No deformation due to projection
occurs at the latitudes. The unit is degree [◦ ] and they are negative in thesouthern hemisphere. If the Lambert conformal projection is used, two true
latitudes (tlat1_grd and tlat2_grd) are necessary. If no map projection
is used, they are not necessary to be speciﬁed.

tlon_grdreal(4)
The true longitude of coordinate. The unit is degree [◦ ] and degree in theWest Longitude is negative.

& gridset grd
The namelists describe grid intervals and the reference points.
gridata.exe.

xdim_grd,
ydim_grd
zdim_grd

integer(4)dx_grd,dy_grdreal(4)

They are used only in

Dimensions of external data in the x, y and z directions, respectively. The
vertical dimension includes the surface.

4.1 User conﬁguration ﬁleGrid intervals of the external data in the x and y directions, respectively.
Unit is degree if mpopt_grd = 0, 10 and otherwise meter. The verticallevels should be included variables of the external data.
ulat_grd,
ulon_grdreal(4)
Longitude and latitude of a reference point of the external data. Unit isdegree and they are negative in the southern hemisphere and/or the West
Longitude. The point is corresponded to the real number of the external
data (riu_grd,rju_grd) to determine the region in gridata.exe.

riu_grd,rju_grdreal(4)
The real number of the reference point which corresponds to the point determined by the reference longitude and latitude (ulat_grd, ulon_grd).

& datconf grd
The namelists determine the scheme to interpolate the external GPV data to the model gridand indicate properties of the GPV data.

intopt_grd

integer(4)Option for scheme of horizontal interpolation of the external GPV data to
the model grid. The vertical interpolation is always the linear interpolation.
1:
2:

The linear interpolation.
The interpolation using the parabolic function.

rotopt_grd

integer(4)This name-list indicates property of horizontal wind of the GPV data.
0: The u and v components of the GPV horizontal velocity are zonal
and meridional components, respectively.
1: The u and v components of the GPV horizontal velocity are x
and y components of the projection speciﬁed by the namelist
mpopt_grd, respectively.

datype_grd

character(80)The name-list indicates data type of variables in the external GPV data.
The following combinations are available.
tm:
pm:
tr:

temperature [K] and water vapor mixing ratio [kg kg−1 ]potential temperature [K] and water vapor mixing ratio [kg kg−1 ]temperature [K] and relative humidity [%]pr:potential temperature [K] and relative humidity [%]

77

78

4 Conﬁguration and Execution

refsfc_grd

integer(4)The name-list indicates data property of the ﬁrst level of the GPV data.
This option is related to the vertical interpolation. If refsfc_grd =1, theﬁrst level of zdim_grd is considered as the surface. No extrapolation is
performed vertically. If refsfc_grd =0, the model grids lower than theﬁrst level of the data grid are extrapolated.
0: The ﬁrst level of the data grid is not a surface data and the model
grids lower than the ﬁrst data level is extrapolated.
1: The ﬁrst level of the data grid is considered as the surface and allmodel grids are determined by interpolation.

extvar_grd

character(80); extvar_grd = ’xxxxxxx’
For refsfc_grd=0, the parameter speciﬁes extrapolation method. The ﬁrst
7 characters of 80 are used. Each character from the ﬁrst corresponds tou, v, w, p, θ, qv, mixing ratios of hydrometeors.
o: Extrapolation is performed using the ﬁrst and second levels of the
GPV data grid.
x:

4.1.4

No extrapolation is performed but the data of the ﬁrst GPV levelis copied to all model grids lower the ﬁrst GPV level.

Region for the pre-processor, radata.exe

The pre-processor ”radata.exe” converts radar data from the data grid to the model grid in
3-dimension. This region speciﬁes the grid system of radar data and options of interpolation.
& project rad
The namelists describe the coordinate system of a data grid. The parameters are used in only
radata.exe.

mpopt_radinteger(4)Option for the coordinate of radar data grid.
0:
latitude and longitude coordinates
1:
The polar stereo graphic projection
2:
The Lambert conformal projection

nspol_rad

3:
4:
10:

The Mercator projection
No map projection
The global latitude and longitude coordinates

13:

The cyclic Mercator projection around the globe

integer(4)Switch to deﬁne origin of the coordinate in the northern or southern hemisphere.

4.1 User conﬁguration ﬁletlat1_rad,tlat2_rad1:

the origin of coordinate in the northern hemisphere.

-1:

the origin of coordinate in the southern hemisphere.

real(4)
The true latitudes of map projections. No deformation due to projection
occurs at the latitudes. The unit is degree [◦ ] and they are negative in thesouthern hemisphere. If the Lambert conformal projection is used, two true
latitudes (tlat1_rad and tlat2_rad) are necessary. If no map projection
is used, they are not necessary to be speciﬁed.

tlon_radreal(4)
The true longitude of coordinate. The unit is degree [◦ ] and degree in theWest Longitude is negative.

& gridset rad
The namelists describe grid intervals and the reference points.

xdim_rad,ydim_radzdim_radinteger(4)dx_rad,dy_radreal(4)

Dimensions of radar data in the x, y and z directions, respectively.

Grid intervals of radar data in the x and y directions, respectively. Unit isdegree if mpopt_rad = 0, 10 and otherwise meter.
ulat_rad,ulon_radreal(4)
Longitude and latitude of a reference point of radar data. Unit is degree
and they are negative in the southern hemisphere and/or the West Longitude. The point is corresponded to the real number of the external data
(riu_rad,rju_rad) to determine the region in radata.exe.

riu_rad,rju_radreal(4)
The real number of the reference point which corresponds to the point determined by the reference longitude and latitude (ulat_rad, ulon_rad).

79

80

4 Conﬁguration and Execution& datconf rad
The name-lists determine the scheme to interpolate radar data to the model grid and indicate
properties of radar data.

rotopt_rad

integer(4)This name-list indicates property of horizontal wind of radar data.
0:

The u and v components of the horizontal velocity are zonal andmeridional components, respectively.

1:

The u and v components of the horizontal velocity are x and ycomponents of the projection speciﬁed by the namelist mpopt_rad,respectively.

datype_radcharacter(80)The name-list indicates data type of variables in radar data. The following
combinations are available.
m: mixing ratio [kg kg−1 ]r: radar echo intensity [dBZ]

radcoe_radreal(4)
Coeﬃcient of the Z-M relationship to convert the radar echo intensity to
mixing ratio of rain: Z = BM β . In this equation, radcoe_rad is the coeﬃcient B.

radpwr_radreal(4)The exponent of the Z-M relationship to convert the radar echo intensity
to mixing ratio of rain: Z = BM β . In this equation, radpwr_rad is the
exponent β.

4.1.5

Region for the pre-processor, terrain.exe

The pre-processor ”terrain.exe” converts terrain data of the data grid to the model grid in
2-dimension. This region speciﬁes the grid system of terrain data and options of interpolation.
& project trn
The name-lists describe the coordinate system of a data grid. The parameters are used inonly terrain.exe.

mpopt_trn

integer(4)Option for the coordinate of terrain data grid.

4.1 User conﬁguration ﬁle81

0:latitude and longitude coordinates

1:
2:
3:

The polar stereo graphic projection
The Lambert conformal projection
The Mercator projection

4:
10:
13:

No map projection
The global latitude and longitude coordinates
The cyclic Mercator projection around the globe

nspol_trn

integer(4)Switch to deﬁne origin of the coordinate in the northern or southern hemisphere.
1: the origin of coordinate in the northern hemisphere.
-1: the origin of coordinate in the southern hemisphere.

tlat1_trn,
tlat2_trn

real(4)
The true latitudes of map projections. No deformation due to projection
occurs at the latitudes. The unit is degree [◦ ] and they are negative in thesouthern hemisphere. If the Lambert conformal projection is used, two true
latitudes (tlat1_trn and tlat2_trn) are necessary. If no map projection
is used, they are not necessary to be speciﬁed.

tlon_trnreal(4)
The true longitude of coordinate. The unit is degree [◦ ] and degree in theWest Longitude is negative.

& gridset trn
The namelists describe grid intervals and the reference points.
terrain.exe.

xdim_trn,
ydim_trn

They are used only in

integer(4)Dimensions of external data in the x, y directions, respectively.

dx_trn,dy_trnreal(4)
Grid intervals of the external data in the x and y directions, respectively.
Unit is degree if mpopt_trn = 0, 10, otherwise, meter.

ulat_trn,
ulon_trnreal(4)82

4 Conﬁguration and Execution

Longitude and latitude of a reference point of terrain data. Unit is degree and they are negative in the southern hemisphere and/or the West
Longitude. The point is corresponded to the real number of terrain data
(riu_trn,rju_trn) to determine the region in terrain.exe.
riu_trn,
rju_trnreal(4)
The real number of the reference point which corresponds to the point determined by the reference longitude and latitude (ulat_trn, ulon_trn).

& datconf trn
The namelists determine the scheme to interpolate terrain data to the model grid.

intopt_trn

integer(4)Option for scheme of horizontal interpolation of terrain data to the model
grid.
1:
2:

4.1.6

The linear interpolation.
The interpolation using the parabolic function.

Region for the pre-processor, surface.exe

The pre-processor ”surf ace.exe” converts surface data from the data-grid system to themodel-grid system in 2-dimension. CReSS enable to use 3 types of surface data: land use data,
sea surface temperature data, and snow and ice distribution data. The region of the conﬁgurationﬁle for surf ace.exe includes name-list parameters for these data.
& project lnd
The name-lists describe the coordinate system of a data grid. The parameters are used onlyin surf ace.exe.

mpopt_lnd

integer(4)Option for the coordinate of land-use data grid.
0:
latitude and longitude coordinates
1:
2:
3:
4:

The polar stereo graphic projection
The Lambert conformal projection
The Mercator projection
No map projection

10:
13:

The global latitude and longitude coordinates
The cyclic Mercator projection around the globe

4.1 User conﬁguration ﬁle83

nspol_lnd

integer(4)Switch to deﬁne origin of the coordinate in the northern or southern hemisphere.
1: the origin of coordinate in the northern hemisphere.
-1: the origin of coordinate in the southern hemisphere.

tlat1_lnd,
tlat2_lnd

real(4)
The true latitudes of map projections. No deformation due to projection
occurs at the latitudes. The unit is degree [◦ ] and they are negative in thesouthern hemisphere. If the Lambert conformal projection is used, two true
latitudes (tlat1_lnd and tlat2_lnd) are necessary. If no map projection
is used, they are not necessary to be speciﬁed.

tlon_lndreal(4)
The true longitude of coordinate. The unit is degree [◦ ] and degree in theWest Longitude is negative.

& gridset lnd
The namelists describe grid intervals and the reference points.
surf ace.exe.

xdim_lnd,
ydim_lnd

They are used only in

integer(4)Dimensions of external data in the x, y directions, respectively.

dx_lnd,dy_lndreal(4)
Grid intervals of the external data in the x and y directions, respectively.
Unit is degree if mpopt_lnd = 0, 10, otherwise, meter.

ulat_lnd,
ulon_lndreal(4)
Longitude and latitude of a reference point of surface data. Unit is degree and they are negative in the southern hemisphere and/or the West
Longitude. The point is corresponded to the real number of surface data
(riu_lnd,rju_lnd) to determine the region in surf ace.exe.

riu_lnd,rju_lndreal(4)84

4 Conﬁguration and Execution

The real number of the reference point which corresponds to the point determined by the reference longitude and latitude (ulat_lnd, ulon_lnd).

& datconf lnd
The namelists determine the scheme to interpolate surface data from data-grid system to themodel grid system, and deﬁne parameters of surface condition.

intopt_lnd

integer(4)Option for scheme of horizontal interpolation of land use data to the model
grid.
0:
1:

numctg_lnd

A model grid point gets values of nearest data points. No interpolation is performed. [recommended]
The linear interpolation of land use data is performed.

integer(4)Number of land-use categories.

lnduse_lnd

integer(4), dimension(1 : 100)lnduse_lnd = -1, 11, 12, 13, 14 . . .
User speciﬁed values of land use. The number of values is the number ofcategories numctg_lnd.

albe_lndreal(4), dimension(1 : 100)
albe_lnd = 0.e0, 0.e0, 0.e0, 0.e0, 0.e0 . . .
Values of abledo corresponding to the parameter lnduse_lnd. The numberof values is the number of categories numctg_lnd.

beta_lndreal(4), dimension(1 : 100)
beta_lnd = 0.e0, 0.e0, 0.e0, 0.e0, 0.e0 . . .
Values of evapotranspiration eﬃciency corresponding to the parameter
lnduse_lnd. The number of values is the number of categories numctg_lnd.

z0m_lndreal(4), dimension(1 : 100)z0m_lnd = 0.e0, 0.e0, 0.e0, 0.e0, 0.e0 . . .
Values of roughness length ([m]) for velocity corresponding to the parameter
lnduse_lnd. The number of values is the number of categories numctg_lnd.

z0h_lndreal(4), dimension(1 : 100)z0h_lnd = 0.e0, 0.e0, 0.e0, 0.e0, 0.e0 . . .

4.1 User conﬁguration ﬁleValues of roughness length ([m]) for scalar variables (heat and moisture)corresponding to the parameter lnduse_lnd. The number of values is thenumber of categories numctg_lnd.
cap_lndreal(4), dimension(1 : 100)
cap_lnd = 0.e0, 0.e0, 0.e0, 0.e0, 0.e0 . . .
Values of thermal capacity of ground ([K J m−3 ]) corresponding to theparameter lnduse_lnd. The number of values is the number of categories
numctg_lnd.

nuu_lndreal(4), dimension(1 : 100)
nuu_lnd = 0.e0, 0.e0, 0.e0, 0.e0, 0.e0 . . .
Values of thermal diﬀusivity of ground ([m2 s−1 ]) corresponding to the parameter lnduse_lnd. The number of values is the number of categories
numctg_lnd.

& project sst
The name-lists describe the coordinate system of a data grid. The parameters are used onlyin surf ace.exe.

mpopt_sst

nspol_sst

integer(4)Option for the coordinate of SST data grid.
0:
latitude and longitude coordinates
1:
The polar stereo graphic projection
2:
3:
4:

The Lambert conformal projection
The Mercator projection
No map projection

10:
13:

The global latitude and longitude coordinates
The cyclic Mercator projection around the globe

integer(4)Switch to deﬁne origin of the coordinate in the northern or southern hemisphere.
1: the origin of coordinate in the northern hemisphere.
-1: the origin of coordinate in the southern hemisphere.

tlat1_sst,
tlat2_sst

real(4)85

86

4 Conﬁguration and Execution

The true latitudes of map projections. No deformation due to projection
occurs at the latitudes. The unit is degree [◦ ] and they are negative in thesouthern hemisphere. If the Lambert conformal projection is used, two true
latitudes (tlat1_sst and tlat2_sst) are necessary. If no map projection
is used, they are not necessary to be speciﬁed.
tlon_sstreal(4)
The true longitude of coordinate. The unit is degree [◦ ] and degree in theWest Longitude is negative.

& gridset sst
The namelists describe grid intervals and the reference points.
surf ace.exe.

xdim_sst,
ydim_sst

They are used only in

integer(4)Dimensions of external data in the x, y directions, respectively.

dx_sst,dy_sstreal(4)
Grid intervals of the external data in the x and y directions, respectively.
Unit is degree if mpopt_sst = 0, 10, otherwise, meter.

ulat_sst,
ulon_sstreal(4)
Longitude and latitude of a reference point of surface data. Unit is degree and they are negative in the southern hemisphere and/or the West
Longitude. The point is corresponded to the real number of surface data
(riu_sst,rju_sst) to determine the region in surf ace.exe.

riu_sst,rju_sstreal(4)
The real number of the reference point which corresponds to the point determined by the reference longitude and latitude (ulat_sst, ulon_sst).

& project ice
The name-lists describe the coordinate system of a data grid. The parameters are used onlyin surf ace.exe.

mpopt_ice

integer(4)4.1 User conﬁguration ﬁle87

Option for the coordinate of ice and snow distribution data grid.

nspol_ice

0:
1:
2:

latitude and longitude coordinates
The polar stereo graphic projection
The Lambert conformal projection

3:
4:
10:

The Mercator projection
No map projection
The global latitude and longitude coordinates

13:

The cyclic Mercator projection around the globe

integer(4)Switch to deﬁne origin of the coordinate in the northern or southern hemisphere.
1: the origin of coordinate in the northern hemisphere.
-1:

tlat1_ice,
tlat2_ice

the origin of coordinate in the southern hemisphere.

real(4)
The true latitudes of map projections. No deformation due to projection
occurs at the latitudes. The unit is degree [◦ ] and they are negative in thesouthern hemisphere. If the Lambert conformal projection is used, two true
latitudes (tlat1_ice and tlat2_ice) are necessary. If no map projection
is used, they are not necessary to be speciﬁed.

tlon_ice

real(4)
The true longitude of coordinate. The unit is degree [◦ ] and degree in theWest Longitude is negative.

& gridset ice
The namelists describe grid intervals and the reference points.
surf ace.exe.

xdim_ice,
ydim_ice

They are used only in

integer(4)Dimensions of external data in the x, y directions, respectively.

dx_ice,dy_icereal(4)
Grid intervals of the external data in the x and y directions, respectively.
Unit is degree if mpopt_ice = 0, 10, otherwise, meter.

ulat_ice,
ulon_ice

real(4)

88

4 Conﬁguration and Execution

Longitude and latitude of a reference point of surface data. Unit is degree and they are negative in the southern hemisphere and/or the West
Longitude. The point is corresponded to the real number of surface data
(riu_ice,rju_ice) to determine the region in surf ace.exe.
riu_ice,
rju_icereal(4)
The real number of the reference point which corresponds to the point determined by the reference longitude and latitude (ulat_ice, ulon_ice).

4.1.7

Region for the post-processor, unite.exe

The post-processor ”unite.exe” is used for parallel version. The output ﬁles of the parallelversion are separately created. The post-processor ”unite.exe” unites all the output ﬁles.
& uniconf uni
The name-lists control the ﬁles to be united. The parameters are used only in unite.exe.

fltyp_unicharacter(80)Control ﬂag of ﬁles to be processed.
dmp:
History-dump ﬁles are united.
geo:† all:

Geography ﬁles are united
Both History-dump ﬁles and Geography ﬁles are united

flitv_uni

real(4)
Time interval to unite history-dump ﬁles. Unit is second.

rmopt_uni

integer(4)Control ﬂag whether separately outputted ﬁles are deleted or remain.
0: Separately outputted ﬁles remain.
1:

4.1.8

Separately outputted ﬁles are deleted after they are united. [recommended for saving storage]

Region for the post-processor, rstruct.exe

The post-processor ”rstruct.exe” rearranges restart ﬁles from the original processing element
number to another processing element number.

4.1 User conﬁguration ﬁle& rstconf rst
The name-lists control dimension and restart ﬁles to be rearranged. The parameters are used
only in rstruct.exe.

xpedim_rst,
ypedim_rst

integer(4)Numbers of decomposition of rearranged restart ﬁles in x and y directions,
respectively. They are numbers of processing elements which is used forrestart computation.

flitv_rst

real(4)
Time interval of the original restart ﬁles. Unit is second.

rmopt_rst

integer(4)Control ﬂag whether original restart ﬁles are deleted or remain.
0:1:

Original restart ﬁles remain.
Original restart are deleted after they are rearrenged. [recommended for saving storage]

89

90

4 Conﬁguration and Execution

4.2

Data ﬂow in CReSS

4.2.1

Outline of data ﬂow

The outline of the data ﬂow in CReSS is shown in Fig.4.1. The ﬁgure includes all types ofdata ﬂows. According to the conﬁguration, some of programs and data depicted in the ﬁgure are
not necessary. Each case of conﬁguration will be described in the following subsections.
The italic character part of ﬁle names in Fig.4.1 are summarized in the following table.
exprimxxxxxxThe experimental name which is deﬁned by the parameter exprim. The beginningpart of all input and output ﬁles is the name.
Elapsed time from the initial (second). For example, if the elapsed time is 1200.0
seconds, then xxxxxx = 001200.

These ﬁle names are used in the serial version of CReSS. In the parallel version, the number ofprocessing element (CPU or nodes) yyyy is appended to the ﬁle names as .peyyyy.
Details of the data format will be described in Subsection 4.3.
configuration fileuser.conf

gridata
GPV data

model GPV files
exprim .grd xxxxxx .bin
exprim .grd.check.txt

restart files
exprim .res xxxxxx .bin
exprim .res.check.txt

geography file
exprim .geography .txt
exprim .geography .bin

history-dumped files
exprim .dmp xxxxxx .txt
exprim .dmp xxxxxx .binterrainterrain data

model terrain files
exprim .terrain.bin
exprim .terrain.check.txt

solverconfiguration fileuser.conf

configuration fileuser.conf

radar dataland-use data

surfacemodel surface files
exprim .surface.bin
exprim .surface.check.txt

model radar files
exprim .rad xxxxxx .bin
exprim .rad.check.txt

configuration fileuser.conf
SST data

Figure 4.1. The outline of data ﬂow in CReSS.

radataconfiguration fileuser.conf

4.2 Data ﬂow in CReSS

The data ﬁles in Fig.4.1 are summarized in the following table.
ﬁle name

description

exprim.terrain.bin

Altitude of the model topography. (unformatted directaccess binary ﬁle).

exprim.terrain.check.txt

Parameter ﬁle to check the conﬁguration of the above
exprim.terrain.bin. (formatted text ﬁle).

exprim.grdxxxxxx.bin

GPV data coordinated at the model grid. Time seriesof the ﬁle is necessary when the nudging is performed. If
nudging is not performed, the data ﬁle is used at the initialtime. (unformatted direct access binary ﬁle).

exprim.grd.check.txt

Parameter ﬁle to check the conﬁguration of the above
exprim.grdxxxxxx.bin (formatted text ﬁle).

exprim.sounding.txt

One-dimensional sounding data to provide the horizontally
uniform initial ﬁeld. (formatted text ﬁle).

exprim.dmpxxxxxx.txt or .bin History dumped ﬁle. The format is optional: formatted
text ﬁle or unformatted direct access binary ﬁle.
exprim.geography.txt or .bin

Geographic information of the domain. The format is optional: formatted text ﬁle or unformatted direct accessbinary ﬁle.

exprim.resxxxxxx.bin

File to restart. (unformatted direct access binary ﬁle).

exprim.res.check.txt

Parameter ﬁle to check the conﬁguration of the above
exprim.resxxxxxx.bin. (formatted text ﬁle).

91

92

4 Conﬁguration and Execution

4.2.2

Data ﬂows of each conﬁguration

Using sounding data
(1) In the case of no terrain data
The conﬁguration in user.conf is summarized in the following table and the data ﬂow isshown in the ﬁgure.
iniopt = 1
trnopt = 0

Initial data is provided by the sounding data.
No external terrain data is used.

exbopt = 0

No external boundary forcing is performed.

sounding profile file
exprim .sounding.txt

solverrestart files
exprim .res xxxxxx .bin
exprim .res.check.txt

configuration fileuser.conf

history-dumped files
exprim .dmp xxxxxx .txt
exprim .dmp xxxxxx .bin

geography file
exprim .geography .txt
exprim .geography .bin

Figure 4.2. The outline of data ﬂow in CReSS using a sounding data.

(2) Using the external terrain data
The Conﬁguration in user.conf and the data ﬂow are as follows.
iniopt = 1

Initial data is provided by the sounding data.

trnopt = 2
exbopt = 0

External terrain data is used for model topography.
No external boundary forcing is performed.

Using the 3-dimensional GPV data
(1) No external lateral boundary data and no terrain data
The Conﬁguration in user.conf and the data ﬂow are as follows. (This conﬁguration will
not generally used.)
iniopt = 3
trnopt = 0
exbopt = 0

The initial data is a 3-dimensional GPV. Time series of the data isnecessary if the nudging is performed.
No external terrain data is used.
No external boundary forcing is performed.

4.2 Data ﬂow in CReSS(2) A lateral boundary data is used while no external terrain data
The Conﬁguration in user.conf and the data ﬂow are as follows. (This conﬁguration will
not generally used.)
iniopt = 3
trnopt = 0
exbopt = 1

The initial data is a 3-dimensional GPV. Time series of the data isnecessary if the nudging is performed.
No external terrain data is used.
External boundary forcing is performed.

(3) An external terrain data is used while no external lateral boundary data
The Conﬁguration in user.conf and the data ﬂow are as follows.
iniopt = 3
trnopt = 2
exbopt = 0

The initial data is a 3-dimensional GPV. Time series of the data isnecessary if the nudging is performed.
External terrain data is used.
No external boundary forcing is performed.

(4) Using both external lateral boundary data and terrain data
The Conﬁguration in user.conf and the data ﬂow are as follows.
iniopt = 3
trnopt = 2
exbopt = 1

The initial data is a 3-dimensional GPV. Time series of the data isnecessary if the nudging is performed.
External terrain data is used.(Boundary smoothing is performed optionally.)
External lateral boundary forcing is performed.

Restarting from restart ﬁle
The Conﬁguration in user.conf and the data ﬂow are as follows.
iniopt = 2
trnopt = anything
exbopt = anything

Begin from the restart ﬁle. Time series of GPV is necessary if
nudging is performed.
Terrain condition is already determined in the previous run. (Nodata is necessary.)
External lateral boundary data is necessary if the external lateral
forcing is performed.

93

94

4 Conﬁguration and Execution

configuration fileuser.confmodel GPV files
exprim .grd xxxxxx .bin
exprim .grd.check.txt
geography file
exprim .geography .txt
exprim .geography .bin

gridata
GPV data

model GPV-dumped-terrain files
exprim .terrain.damp.bin
exprim .terrain.damp.check.txt

restart files
exprim .res xxxxxx .bin
exprim .res.check.txt

history-dumped files
exprim .dmp xxxxxx .txt
exprim .dmp xxxxxx .binterrainterrain data

model terrain files
exprim .terrain.bin
exprim .terrain.check.txt

configuration fileuser.conf

solverconfiguration fileuser.conf

sounding profile file
exprim .sounding.txt

radar dataland-use data

surfacemodel surface files
exprim .surface.bin
exprim .surface.check.txt

model radar files
exprim .rad xxxxxx .bin
exprim .rad.check.txt

configuration fileuser.conf

radataSST data

configuration fileuser.conf

Figure 4.3. The outline of data ﬂow in the single version of CReSS.

4.3

Formats of I/O data

In this section, formats of I/O data which controlled by users will be described: sounding data
(exprim.sounding.txt), history dumped data (exprim.dmpxxxxxx.txt or exprim.dmpxxxxxx.bin)and geography data (exprim.geography.txt or exprim.geography.bin).

4.3.1

Format of the sounding data

The sounding data is a text data composed of ﬁve columns of variables. The data types of
the ﬁrst, second and ﬁfth columns are deﬁned by users. All types of combination are allowed.
Therefore, there are eight combinations of variables.

4.3 Formats of I/O data

95

configuration fileuser.conf
united geography file
exprim .geography .united.bin

configuration fileuser.conf

unite
model GPV files
exprim .grd xxxxxx .pe yyyy .bin
exprim .grd.check.txt

united history-dumped files
exprim .dmp xxxxxx .united.bin

gridata
GPV data

model GPV-damped-terrain files
exprim .terrain.damp.pe yyyy .bin
exprim .terrain.damp.check.txt

geography files
exprim .geography .pe yyyy .txt
exprim .geography .pe yyyy .bin
restart files
exprim .res xxxxxx .pe yyyy .bin
exprim .res.check.txt

history-dumped files
exprim .dmp xxxxxx .pe yyyy .txt
exprim .dmp xxxxxx .pe yyyy .binterrainterrain data

model terrain files
exprim .terrain.pe yyyy .bin
exprim .terrain.check.txt

configuration fileuser.conf

solverconfiguration fileuser.conf

sounding profile data
exprim .sounding.txt

radar dataland-use data

surfacemodel surface files
exprim .surface.pe yyyy .bin
exprim .surface.check.txt

model radar files
exprim .rad xxxxxx .pe yyyy .bin
exprim .rad.check.txt

configuration fileuser.conf

radataSST data

configuration fileuser.conf

Figure 4.4. The outline of data ﬂow in the parallel version of CReSS.

First column
Second column

height [m] or pressure [Pa]temperature [K] or potential temperature [K]Third column
fourth column
ﬁfth column

x-component of horizontal velocity [m s−1 ]y-component of horizontal velocity [m s−1 ]water vapor mixing ratio [kg kg−1 ] or relative
humidity [%]96

4 Conﬁguration and Execution

This is an example of the sounding ﬁle. It is easy to make a sounding ﬁle following the example.
Some samples are found in Form directory of CReSS.

#############################################################################
One dimensional sounding input file, sounding.txt.cats.eye.form## This is the cats eye simulation data.
##
Author : SAKAKIBARA Atsushi#
Date: 1999/07/23# Modification : 1999/07/28#: 1999/11/19
#
#
#
#
#
#
#

First column: height [m]Second column: tempereture [K]##Third column: x components of velocity [m/s]Fourth column: y components of velocity [m/s]Fifth column: water vapor relative humidity [%]

#
#
#
#

###########################################################################
780.e0 300.200e0 8.0000e0 0.e0 0.e0
720.e0 300.200e0 7.9999e0 0.e0 0.e0
660.e0 300.200e07.9993e0 0.e0 0.e0::
120.e0 299.800e0 -7.9946e0 0.e0 0.e0
60.e0 299.800e0 -7.9993e0 0.e0 0.e0
0.e0 299.800e0 -7.9999e0 0.e0 0.e0

4.3.2

Formats of the history dumped ﬁle and the geographic information ﬁleFormatted text data or unformatted binary data is optionally chosen for the history dumped
data and the geographic information data.
In the case of the formatted text data (dmpfmt = 1), a variable will be outputted as follows.

do xxx k=2,nk-2
write(ionum,*,err=errnum) (variable(i,j,k),i=2,nx-2,j=2,ny-2)
xxx continue

4.3 Formats of I/O data

97

If the unformatted binary data is selected (dmpfmt = 2), a variable will be outputted as follows.

do xxx k=2,nk-2
recnum=recnum+1
write(ionum,rec=recnum,err=errnum).
(variable(i,j,k),i=2,nx-2,j=2,ny-2)
xxx continue(refer to outdmp3d.f and outdmp2d.f). where nx,ny,nz are numbers of the grid points in thex, y and z directions, respectively. The number of nz is unity in the geographic information ﬁle.
The region of the output data is indicated by the thick lines in Fig.4.5. The data points areindicated by the crosses which is the scalar points. The vector variables deﬁned at the closed
circles are interpolated to the scalar points. The dimensions of the x, y and z directions arerespectively smaller than those of the model grid by three.

physical domain
ny

nz

ny-1

nz-1
nz-1

ny-1
ny-2

nz-2

ny-2

nz-2
nz-3

physical domain

physical domain

ny-3yorigin(x,y ) = (0.0, 0.0)

ζ

3

3
3

3

2

2
2

2
1

1
1

1
1
1

2
2

nx-3

3
3

x

nx -2
nx-2

nx-1
nx-1

nx

Figure 4.5. Setting of variables in the calculation domain. The closed circles are points of vector variables
and the crosses are of scalar variables.

98

4 Conﬁguration and Execution

Options for vertical grid of the output data are shown in Fig.4.6.

dmpcmp = 2

dmpcmp = 1

dmpcmp = 3

Figure 4.6. Vertical coordinates of the history dumped data.

Output variables in the geographic information data and the history dumped data are summarized in the following tables. The types of output variables of the history dumped data aredepend on the conﬁgurations.

4.3 Formats of I/O data

99

History dumped ﬁlealtitude of terrain
latitudelongitude
Coriolis parametermflu

map factor
land use

x-component of the horizontal velocity

m s−1

v

y-component of the horizontal velocity
vertical component of velocity
basic state of pressure
perturbation pressure
basic state of potential
temperature
perturbation
potential
temperature

m s−1

w

Geographic information ﬁle
ht
lat
lon
fc

u

m
◦
◦

pbar
pp
ptbar
ptp

m s−1
Pa
Pa
K
K

qv

mixing ratio of water vapor

kg kg−1qc

mixing ratio of cloud waterkg kg−1qr
qiqs

mixing ratio of rain water
mixing ratio of cloud ice
mixing ratio of snow

kg kg−1kg kg−1kg kg−1

qg
ncikg kg−1
m−3

zph
prr

mixing ratio of graupel
number density of cloudice
number density of snow
number density of graupelheight of grid points
rainfall intensity

par
prs
pas

total rainfall
snow intensity
total snowfall

m s−1 , m
m s−1
m s−1 , mprg
paggraupel intensity
total graupel fall

m s−1
m s−1 , m

ncs
ncg

m−3
m−3
m
m s−1

100

4 Conﬁguration and Execution

4.4

Execution of CReSS

4.4.1

Execution of solver.exe

Using an example experiment of the Kelvin-Helmholtz waves, the procedures to execute thesolver program solver will be explained.
First, users uncompress and expand the compressed archive ﬁle of CReSS. (This example is
CReSS ver.2.3 while the essence is the same in other version.)% lscress2.3m.tar.Z% uncompress -c cress2.3m.tar.Z | tar xvf messages:% ls
CReSS2.3m%

cress2.3m.tar.Z

The directory structure will be as follows.
% cd CReSS2.3m
% ls
Doc
Src
Form
Tmp

compile.conf
compile.csh%
Directory begin with a capital letter while ﬁles with a small letter. These are summarized in
the following table.
Doc
Form
Src

Documents and Readme are included.
Examples of conﬁguration and the related sounding data are included.
All source codes are archived in the directory.

Tmpcompile.conf
compile.csh

A temporary directory used in compilation of CReSS.
Conﬁguration ﬁle of compilation.
Shell script to control the compilation.

Some conﬁguration ﬁles and the related sounding ﬁles are archived in the directory of Form.
Copy the conﬁguration ﬁle and the sounding ﬁle of the Kelvin-Helmholtz waves to the directoryof CReSS2.3m as the following names. We will use test as the name of experiment. All I/O data
must begin with test..
% cp Form/user.conf.cats.eye.form.2 user.conf% cp Form/sounding.txt.cats.eye.form test.sounding.txt
% ls
Doc

Src

compile.conf

test.sounding.txt

4.4 Execution of CReSS

Form

Tmp

compile.csh

101user.conf%
Second, the conﬁguration ﬁle of compilation compile.conf will be edited if necessary. If thecompiler is diﬀerent from f90 or mpif90 or special options of compilation are used, the ﬁle shouldbe edited. Otherwise, the ﬁle is not changed.
Third, compile the solver as follows. The dimension is deﬁned in the user conﬁguration ﬁleuser.conf and include ﬁles are produced automatically. It is, therefore, not necessary to change
the source codes of the solver.
% compile.csh solver [compile.conf]
cd Src; messages:%
When the compilation is completed, the executable ﬁle solver.exe is generated. (Actually,solver.exe is a symbolic link to that in the directory of Src.)
% ls
Doc
Form
Src

Tmp
compile.conf
compile.csh

solver.exe
test.sounding.txt
user.conf%
Forth, execute the solver with reading the user conﬁguration ﬁle user.conf from the standard
input and outputting test.out to the standard output. In order to use main frame computer,job script and NQS could be necessary. In that case, ask the system manager how to use NQS.
% solver.exe < user.conf > test.log &%If the calculation is terminated normally, CReSS leaves the message “This program stopped normally.”at the end of the standard output test.out and the history dumped ﬁles (the names include dmp),
the geographic information ﬁle (the names include geography) and restart ﬁles (the names include res) are produced. The standard output includes history of calculation and maximum andminimum of prognostic variables.
% ls
Doc

test.dmp000120.pe0003.bin

Form
Src
Tmp

test.dmp000160.pe0000.bin
test.dmp000160.pe0001.bin
test.dmp000160.pe0002.bincompile.conf
compile.csh
solver.exe

test.dmp000160.pe0003.bin
test.dmp000200.pe0000.bin
test.dmp000200.pe0001.bin102

4 Conﬁguration and Execution

test.dmp000000.pe0000.bin

test.dmp000200.pe0002.bin

test.dmp000000.pe0001.bin
test.dmp000000.pe0002.bin
test.dmp000000.pe0003.bin

test.dmp000200.pe0003.bin
test.dmp000240.pe0000.bin
test.dmp000240.pe0001.bin

test.dmp000040.pe0000.bin
test.dmp000040.pe0001.bin
test.dmp000040.pe0002.bin

test.dmp000240.pe0002.bin
test.dmp000240.pe0003.bin
test.geography.bin

test.dmp000040.pe0003.bin
test.dmp000080.pe0000.bin
test.dmp000080.pe0001.bin
test.dmp000080.pe0002.bin

test.out
test.res000240.check.txt
test.res000240.pe0000.bin
user.res000240.pe0001.bin

test.dmp000080.pe0003.bin
test.dmp000120.pe0000.bin
test.dmp000120.pe0001.bin

test.res000240.pe0002.bin
test.res000240.pe0003.bin
test.sounding.txt

test.dmp000120.pe0002.bin%

user.conf

4.4.2

Execution of unite.exe

In parallel calculation, the history dumped ﬁles are outputted separately by each processing
element. To combine the ﬁles, the post-processor unite is used.
The compilation of unite is as follows.
% compile.csh unite [compile.conf]
cd Src; messages:%
Then the post-processor unite.exe is generated. unite.exe is an interactive executable. The
following example combines the history dumped ﬁle at 240 seconds. The name of experiment is
test.
% unite.exe
Now the program, unite start.
messages:
This program stopped normally.
%

The combined ﬁle is test.dmp000240.united.bin . The format of the ﬁle is described in
Subsection 4.3.2. Types and order of output are also found in the standard output test.log.
The ﬁle is used in a graphic application such as Grads, then the result will be displayed as in4.4 Execution of CReSS

Fig.4.7 and Fig.4.8.

Figure 4.7. Stream function of (u − w) at 240 sec- Figure 4.8. Potential temperature [K] at 240 seconds from the initial.
onds from the initial.

4.4.3

Execution of terrain.exe

The pre-processor terrain is a program to interpolate a terrain data into the model grids. Thisis necessary in a prediction experiment as well as a numerical experiment using a non-analytic
topography.
While an external terrain data is necessary to produce the model terrain using terrain, theformat of the external terrain data is not speciﬁed in CReSS. In order to read the data, it isnecessary to modify the subroutine rdtrn.f which will be found in the directory Src. User will
modify the parts of the open statement and read statement in rdtrn.f which is indicated by #to read the external terrain data.

* ##### You need to modify the following lines. #####
siz=nid*njd*wlngth
write(trnfl(1:16),’(a16)’) ’data.terrain.bin’open(iotrn,iostat=stat,file=trnfl(1:16),status=’old’,.
access=’direct’,form=’unformatted’,recl=siz)* ####################

103

104

4 Conﬁguration and Execution* ##### You need to modify the following lines. #####
read(iotrn,rec=1,err=100) ((htdat(id,jd),id=1,nid),jd=1,njd)* ####################The unit of the terrain data is meter. If the unit is not meter, it must be corrected here.
After the namelists of the user conﬁguration ﬁle are speciﬁed (refer to Section 4.1), compile asfollows.
% compile.csh terrain [compile.conf]
cd Src; messages:%
When the compilation is completed, the executable program terrain.exe is generated. Actually, terrain.exe is a symbolic link to that in the directory of Src.
The execution of terrain.exe is as follows. The input data ﬁle must be in the current directory.
% terrain.exe < user.conf > test.log.terrain &messages:
This program stopped normally.
%
If the execution succeeded, the terrain data ﬁle with the experimental name at the beginning(in the case, it is test) will be generated. (For example, test.terrain.bin). The program
terrain.exe of the serial and parallel versions is performed in a single processing element. A
single ﬁle will be produced by the serial version while multiple ﬁles by the parallel version withan extention of .peyyyy. Where yyyy is the unit number of the processing element. This isnecessary to execute the parallel version of solver.exe.

4.4.4

Execution of gridata.exe

The pre-processor gridata produces 3-dimensional grid point data ﬁles and the lateral boundary data ﬁles.
The format of the external GPV data such as an objective analysis data or a coarse model
output is not speciﬁed in CReSS. In order to read the data, it is necessary to modify the subroutine
rdgpv.f which will be found in Src. User will change the open statement and read statement of
the subroutine rdgpv.f, which are indicated by #. The ﬁle names of the input data must include
date and time of the Gregorian calendar yyyymmddhhmm (UTC).

4.4 Execution of CReSS

105* ##### You need to modify the following lines. #####
write(gpvfl(1:24),’(a8,a12,a4)’) ’data.gpv’,cdate(1:12),’.bin’open(igpv,iostat=stat,file=gpvfl(1:24),status=’old’,.
access=’sequential’,form=’unformatted’)* ####################* ##### You need to modify the following lines. #####
read(igpv,end=100,err=100) zdat
read(igpv,end=100,err=100) udat
read(igpv,end=100,err=100) vdat
read(igpv,end=100,err=100) pdat:
* ####################The necessary variables are summarized in the following table.
zdat
udat
vdat

height of the each grid point
x-component of horizontal velocity
y-component of horizontal velocity

m
m s−1
m s−1

pdat
ptdat

pressurepotential temperature or temperaturePa
K

The following variables are used optionally.
wdat

vertical velocity

m s−1

qvdat
qcdat
qrdat
qidat

water vapor mixing ratio or relative humidity
mixing ratio of cloud water
mixing ratio of rain water
mixing ratio of cloud icekg kg−1 or %kg kg−1kg kg−1kg kg−1

qsdat
qgdat

mixing ratio of snow
mixing ratio of graupel

kg kg−1kg kg−1

If the units of the variable are diﬀerent from the above, correction is necessary in the subroutine.

106

4 Conﬁguration and Execution

If data at the surface is present (refsfc = 1), the data is allocated at the ﬁrst level (kd = 1).
If the altitude of the surface is allocated at the ﬁrst level of zdat and the corresponded data isallocated at the ﬁrst level of each data, the level which is lower than the altitude will be ignoredin the vertical interpolation when refsfc = 1. If refsfc = 0, the interpolation or extrapolation
will be performed in the model grid even if the data is below the ground.
Compilation and execution of gridata.exe are essentially the same as terrain.exe. Afterthe namelists are conﬁgured (refer to Section 4.1), compile the program as follows.
% compile.csh gridata [compile.conf]
cd Src; messages:
This program stopped normally.
%%
If the compilation is succeeded, the executable ﬁle gridata.exe is generated. Actually gridata.exe
is also a symbolic link to that in the directory Src.
The execution of terrain.exe is as follows. The input data ﬁle must be in the current directory.
% gridata.exe < user.conf > test.log.gridata &cd Src; messages:%
If the execution succeeded, 3-dimensional grid data ﬁles (in this case test.grdxxxxxx.bin).
If the GPV data include abnormal data, gridata.exe will be stopped with messages.
The program gridata.exe of the serial and parallel versions is performed in a single processing
element. A single ﬁle will be produced by the serial version while multiple ﬁles by the parallelversion with an extension of .peyyyy. Where yyyy is the unit number of the processing element.
This is necessary to execute the parallel version of solver.exe.

Chapter 5

Program Lists

In this chapter, structures of solver and other processors of CReSS version
2.3m are summarized as a reference. The tree diagrams of the programs will
be presented. Most descriptions are about the parallel version of CReSS. Since
the serial version of CReSS is essentially the same as the parallel version, thereaders will have no trouble with the descriptions even if they use the serial
version.
Each source code of CReSS includes a single subroutine and each ﬁle has
the same name as subroutine. This is helpful to read the source code as wellas the tree diagram.

108

5 Program Lists

5.1

Program list

In this section, the programs of CReSS version 2.3, the composing subroutines, include ﬁlesand the library subroutine of MPI will be described.

5.1.1

Main programs of CReSS

programs

description

solversolver.exe

The solver program is the main program to perform the time
integration from an initial value with boundary conditions.

check

check.exe

A pre-processor to check the consistency of parameters in the
conﬁgulation ﬁle ”user.conf”.

gridata

gridata.exe

A pre-processor to produce an initial and boundary values for thesolver by interpolating grid point values to the grid points of thesolver.

terrainterrain.exe

A pre-processor to produce model terrain data by interpolatingof topography data.

surf ace

surface.exe

A pre-processor to produce model surface data by interpolatingof surface data.

radataradata.exe

A pre-processor to produce radar data on the model grid by interpolating of a topography data.

unite

unite.exe

A post-processor to combine multiple-output history ﬁles and/or
the multiple terrain ﬁle in to a single ﬁle after parallel calculation.

rstruct

5.1.2

rstruct.exe

A post-processor to rearrange the re-start ﬁles for diﬀerent number of processor.

List of subroutines

CReSS is composed of many subroutines. All subroutines are listed below and their functions
are described. Each ﬁle includes its corresponding subroutine. The names in the list will also
refer to the ﬁle names.

solvercheck
terrainthis is the main program for the solver processer solver.
this is the main program for the pre processor ckeck.
this is the main program for the pre processor terrain.

5.1 Program list

surfacethis is the main program for the pre processor surface.

gridataradataunite

this is the main program for the pre processor gridata.
this is the main program for the pre processor radata.
this is the main program for the post processor unite.

rstruct
abortslv
addqt

this is the main program for the post processor rstruct.
check the ﬁle to abort the solver.
add the source intensity to the current tracer.

adjstbw
adjstnc

adjust the mean water mass to be between their boundaries.
adjust the concentrations of the cloud ice, snow and graupel for the given
mixing ratio.
adjust the concentrations of the cloud ice.
adjust the concentrations of the snow and graupel for the given mixing ratio.
force the hydrometeor mixing ratio more than user speciﬁed value.

adjstniadjstnsg
adjstqadjstuv
advbspeadvbspt
advp

adjust the x and y components of velocity.
calculate the base state pressure advection for the horizontally explicit andvertically explicit method.
calculate the base state pressure advection for the horizontally explicit andvertically implicit method.
calculate the base state potential temperature advection.
calculate the pressure advection.

advs
advuvwaggregatcalculate optional scalar advection.
calculate the velocity advection.
calculate the aggregation rate for the cloud ice and snow.

allocbuf
allocgrd
allocgrp
allocmph

allocate the communication buﬀer array.
allocate the array for gridata.
allocate the grouped domain arrangement array.
allocate the array for cloud physics.

allocnio
allocradallocrst

allocate the unit numbers table.
allocate the array for radata.
allocate the array for rstruct.

allocsfc
allocslv
alloctrn

allocate the array for surface.
allocate the array for solver.
allocate the array for terrain.

allocuni
baserhobbcs
bbcw

allocate the array for unite.
the base state density is multiplyed by the Jacobian.
set the vertical boundary condition for optional scalar variable.
set the bottom boundary conditions for the z components of velocity.

bc2d
bc4news
bc8u

set the boundary conditions for two dimensional variables.
set the boundary conditions at the four corners.
set the boundary conditions for optional variable at the u points.

bc8v
bc8w

set the boundary conditions for optional variable at the v points.
set the boundary conditions for optional variable at the w points.

advbspi109

110

5 Program Lists

bcbaseset the bottom and the top boundary conditions for the base state variables.

bcgrd
bcgsmsbcini

set the boundary conditions for GPV data at current forecast time.
set the boundary conditions for optional scalar variable in GPV datasmoothing.
set the boundary conditions for the x components of velocity in GPV datasmoothing.
set the boundary conditions for the y components of velocity in GPV datasmoothing.
set the boundary conditions for the z components of velocity in GPV datasmoothing.
set the boundary conditions for GPV data at forecast start time.

bcphi
bcrad
bcradqp
bcten

set the boundary conditions for the parabolic pertial diﬀerential equation.
set the boundary conditions for radar data.
set the boundary conditions for radar data.
set the boundary conditions for optional tensor.

bcyclebcyclexbcycley

set the periodic boundary conditions.
set the periodic boundary conditions in x direction.
set the periodic boundary conditions in y direction.

bruntvbulksfc
buoytkebuoywbcalculate the Brunt-Vaisala frequency squared.
calculate the bulk coeﬃcients of surface ﬂux.
calculate the buoyancy production in the T.K.E. equation.
calculate the buoyancy in the large time steps integration.

buoywsecastgrp.single
castname.multicastname.single

calculate the buoyancy in the small time steps integration for the horizontally explicit and vertically explicit method.
calculate the buoyancy in the small time steps integration for the horizontally explicit and vertically implicit method.
broadcast the grouped domain arrangemnet array from the root processor
element to the others.
do nothing because this subroutine is dummy.
broadcast the namelist array from the root processor element to the others.
do nothing because this subroutine is dummy.

castvar
cg2mkbw
charging

broadcast optional array from the root processor element to the others.
change measurement from [c] [g] to [m] [k] for warm bin cloud physics.
perfom useless calculations, because this routine is imitation.

chkerr.multichkerr.single
chkfall
chkﬁlecheck the error descriptor for calling subroutine, destroy.
check the error descriptor for calling subroutine, destroy.
check the time interval for upwind diﬀerential of the sedimentation to holdthe common values.
check the namelist variables which are read out from the input ﬁles.

chkgpvchkitrchkkindchkmoistcheck the GPV data variables.
check the convergence of the iteration.
check the kind of the Fortran veriable.
check the air moisture.

bcgsmubcgsmv
bcgsmw

buoywsicastgrp.multi

5.1 Program list

chkmxnread in the maximum and minimum data variable to the standard i/o andcheck the maximum and minimum data variable.

chknamechknlgrd
chknlradchknlrst

read out and set the namelist variables.
check the namelist variables for gridata.
check the namelist variables for radata.
check the namelist variables for rstruct.

chknlsfc
chknlslv
chknltrn

check the namelist variables for surface.
check the namelist variables for solver.
check the namelist variables for terrain.

chknlunichkopen.multichkopen.single
chkradcheck the namelist variables for unite.
check the common ﬁles open between all processor elements.
check the common ﬁles open between all processor elements.
check the radar data variables.

chkrainchkstd.multi
chkstd.single

check the precipitation on the surface.
check the control ﬂag of message type for standard i/o.
do nothing because this subroutine is dummy.

closedmp
cloudamt
cloudphyclose the opening history ﬁle.
calculate the cloud amount.
control the inferior subroutines for the cloud micro physics.

cnt2phy
coldblk
collbw
collc2r

calculate the z components of velocity.
calculate the source amounts of the bulk cold cloud micro physics.
perform the collection processes between water bins.
calculate the collection rate between the cloud water and the rain water.

collect
combincomblkcalculate the collection rate.
declare the array for warm bin cloud physics.
declare the array for bulk cold rain model.

combufcomcapcomdaycomdmpdeclare the common used buﬀer array.
declare the caption for dumped variables.
declare and deﬁne common used and referenced calendar.
declare the control variables of history ﬁle.

comerrcomgrdcomgrpdeclare and deﬁne common used error list array.
declare the array for gridata.
declare the grouped domain arrangement array in entire domain.

comindx
comkindcommathdeclare the unique index for each namelist variable in the namelist array.
declare and deﬁne the kind of veriable for the long integer.
declare and deﬁne the mathmatical constants.

commpicomnamecomniocomphydeclare the parameters of parallelizing.
declare and deﬁne common used namelist array.
declare the unit numbers table.
declare and deﬁne the physical constants.

comraddeclare the array for radata.

111

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5 Program Lists

comrstdeclare the array for rstruct.

comsave
comsfccomslvdeclare the external referenced variables to avoid save attribute statement.
declare the array for surface.
declare the array and variable for solver.

comstdcomtable
comtrndeclare the control ﬂag of standard i/o.
declare and deﬁne common used and referenced table array.
declare the array for terrain.

comuniconvc2r
convers
convlandcopy1ddeclare the array for unite.
calculate the auto conversion rate from the cloud water to the rain water.
calculate the conversion rate from the cloud water to the rain water, fromthe cloud ice to the snow and from the snow to the graupel.
convert veriable type of the land use categories.
copy 1-dimensional invar to outvar.

copy2dcopy3dcopy4dcopy 2-dimensional invar to outvar.
copy 3-dimensional invar to outvar.
copy invar to outvar.

coriuv

calculate the Coriolis force in the x and the y components of velocity equation.

coriuvw

calculate the Coriolis force in the x, the y and the z components of velocity
equation.

cpondht
cpondpe.multicpondpe.single

correspond and damp the model height to the external data height.
wait for the other processor elements’ signals.
do nothing because this subroutine is dummy.

cpondsfc
culintgcorrespond the surface data to the land use data.
control the inferior subroutines for the time steps integration of the Cubic
Lagrange advection.
set the parameters of parallelizing from user arrangement.
calculate the curvature of earth.

currpecurveuvw

depsitbw

convert the hydrometeor mesuremement from [dBZe] to [kg m−3 ]declare the dimensions of all array.
declare the namelist variables and blocks.
calculate the evaporation rate from the rain water to the water vapor andthe deposition rate from the water vapor to the ice hydrometeor.
perform deposition for water bin.

destroy
dfomssq
dfomtenread in the error messages to the standard i/o and stop the program.
calculate the deformation squared.
calculate the deformation tensor.

diabatdiagncdiagnicalculate the diabatic in the pressure equation.
get the diagnostic concentrations of the cloud ice, snow and graupel.
get the diagnostic concentrations of the cloud ice.

diagnsg
diﬀequa

get the diagnostic concentrations of the snow and graupel.
solve the parabolic pertial diﬀerential equation.

dbz2kgdefdimdefnamedepsit5.1 Program list

disptkecalculate the dissipation in the T.K.E. equation.

distrpg

calculate the distribution ratio at which the collisions between rain waterand snow and reset the collection rate.
distribute the observed precipitation to the rain, snow and graupel mixing
ratio.
calculate the negative divergence horizontally.
calculate the three dimensional negative divergence.
calculate the divergence in the pressure equation with the horizontally explicit and vertically explicit method.
calculate the divergence horizontally in the pressure equation with the horizontally explicit and vertically implicit method.

distrqp
diver2ddiver3ddiverpediverpihdiverpiv

calculate the divergence vertically in the pressure equation with the horizontally explicit and vertically implicit method.

eddydifeddypbleddyviseddyvisj

calculate the eddy diﬀusivity.
calculate the eddy viscosity and diﬀusivity in planetaty boundary layer.
calculate the eddy viscosity and the turbulent length scale.
the eddy viscosity is devided by Jacobian.

endmpi.multi

read in the ﬁnal message to the standard i/o and call the function to ﬁnalize
the MPI processes.

endmpi.single
estimsfc
evapr2v

read in the ﬁnal message to the standard i/o.
re-estimete the surface value on the water, ice and snow surface.
calculate the evaporation rate from the rain water to the water vapor.

exbcptforce the lateral boundary values to the external boundary values for potential temperature.

exbcq

force the lateral boundary values to the external boundary values for optional mixing ratio.

exbcssforce the lateral boundary values to the external boundary values for scalarvariables.

exbcu

force the lateral boundary values to the external boundary values for the xcomponents of velocity.
force the lateral boundary values to the external boundary values for the ycomponents of velocity.
force the lateral boundary values to the external boundary values for the z
components of velocity.
perform the fall out.
perform the fall out.
perform the fall out of the rain water.

exbcv
exbcw
fallblk
fallbw
fallqrfdmdrvﬁtuvwcforcedrv
forcepforcept

control the inferior subroutines for the ﬁnite diﬀerence time steps integration.
ﬁt the x and the y components of velocity and the zeta components ofcontravariant velocity to the mass consistent equation by the lamb.
control the inferior subroutines for the forcing terms calculation.
calculate the forcing term in the pressure equation.
calculate the forcing term in the potential temperature equation.

113

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5 Program Lists

forceqcalculate the forcing terms in the water hydrometeor and the ice hydrometeor and concentrations equations.

forceqcg
forceqt
forceqv
forcesforcesfc

calculate the forcing terms in the charging distribution equation.
calculate the forcing term in the tracer equation.
calculate the forcing term in the water vapor mixing ratio equation.
control the inferior subroutines for the scalar forcing terms calculation andbrige the complicated array, rkh, rkh8u and rkh8v.
get the surface ﬂux to bottom boundary.

forcetke
forceuvw
freezing

calculate the forcing terms in the T.K.E. equations.
calculate the forcing terms in the velocity equations.
calculate the freezing rate from the rain water to the graupel.

gausselget1dgetareagetbasesolve the tridiagonal equation with the Gauss elimination.
get the horizontally averaged variables.
get the area of each boundary plane.
extract the base state variables for the model grid from the three dimensional
data variables.

getbufgx
getbufgy
getbufsx
getbufsy

get the receiving buﬀer in x direction for grouped domain.
get the receiving buﬀer in y direction for grouped domain.
get the receiving buﬀer in x direction for sub domain.
get the receiving buﬀer in y direction for sub domain.

getcname
getdategetexner

get the character namelist variable.
calculate the current date from the time with ssssss format.
calculate the Exner function.

getinamegetindx

get the integer namelist variable.
get the maximum and minimum indexes of do loops for optional variable ordata grid.
get the index of the base state reference pressure.
calculate the latitude and the longitude at south-west corner.

getkref
getllsw
getmxnget the maximum and minimum prognostic variables in the grouped grid
points.

getnews
getpt0getqt0getrichget the mininum and maximum data index to create the base state variables.
set the initial potential temperature perturbation.
set the initial tracer.
calculate the bulk Richardson number on the surface.

getrijgetrname
getta2d

calculate the real indexes at the model grid points in the data region.
get the real namelist variable.
calculate the air temperature.

getta3dgettimegettrncalculate the air temperature.
calculate the model time in the large time steps integration.
get the terrain height.

getunitgetvdens

get the unit number to open the ﬁle.
calculate the inverse of base state density.

5.1 Program list

getvprescalculate the inverse of the pressure.

getxy
getz
getz11

calculate the x and the y coordinates.
calculate the zeta coordinates.
calculate the zeta coordinates.

getzlow
getzph
grddrv

calculate the z physical coordinates at lowest plane.
calculate the z physical coordinates.
control the inferior subroutines to interpolate the grid data.

grdstep
gseidelgsmoosgsmooucalculate the number of steps of the main do loop in grddrv.
solve the tridiagonal equation with the Gauss-Seidel method.
smooth the interpolated scalar GPV data.
smooth the x components of velocity of interpolated GPV data.

gsmoov
gsmoowhculs

smooth the y components of velocity of interpolated GPV data.
smooth the z components of velocity of interpolated GPV data.
calculate the horizontal scalar advection by the Cubic Lagrange scheme.

hculuvw
heatsfc
heve

hint2dcalculate the horizontal velocity advection by the Cubic Lagrange scheme.
calculate the sensible and latent heat on the surface.
control the inferior subroutines for the horizontally explicit and vertically
explicit method.
control the inferior subroutines for the horizontally explicit and vertically
implicit method.
interpolate the data variables to the model grid horizontally.

hint3dhintlndhintradini0nameinterpolate the data variables to the model grid horizontally.
interpolate the land use data to the model grid.
interpolate the data variables to the model grid horizontally.
initialize the namelist variables.

inibinbw
inicharinicomcontrol the inferior subroutines for the initial bin distribution of total water.
initialize the character variable with space.
initialize the common used module variables.

inidefinidisbw
inierrinitialize the dimension and namelist variables.
set the initial bin distribution of total water.
initialize the common used error list array.

inimpi.multicall the function to initialize the MPI processes and set the parameters ofparallelizing.

inimpi.single

set the parameters of parallelizing in the case the program run one after
another.

ininameinisfcinitkeinitundinitialize the common used namelist array.
initialize the surface physical parameters.
initialize the T.K.E.
initialize the temperature under the ground and sea.

inivar
instvel
intgdrv

set the initial conditions.
calculate the maximum instantaneous wind velocity.
control the inferior subroutines for the time steps integration.

hevi

115

116

5 Program Lists

intrpgpv

control the interior subroutines for interpolating the GPV data to the model
grid points.

intrpradcontrol the interior subroutines for interpolating the radar data to the model
grid points.

intrpsnd
jacobian
kh8uv
lbcs

extract the base state variables from interpolated sounding data.
calculate the transformation Jacobian.
set the horizontal eddy diﬀusivity at the u and v points.
set the lateral boundary conditions for optional scalar variable.

lbcu
lbculslbculuset the lateral boundary conditions for the x components of velocity.
set the lateral boundary conditions for optional scalar variable.
set the lateral boundary conditions for x components of velocity.

lbculvlbculw
lbcv
lbcw

set the lateral boundary conditions for y components of velocity.
set the lateral boundary conditions for w components of velocity.
set the lateral boundary conditions for the y components of velocity.
set the lateral boundary conditions for the z components of velocity.

lbcwc
ll2xy
llnewslspdmpset the lateral boundary conditions for the zeta components of contravariant
velocity.
calculate the x and the y coordinates from the latitude and the longitude.
calculate the latitude and the longitude at model corner.
calculate the relaxed lateral sponge damping coeﬃcients.

lspp
lspqv
lsps

calculate the lateral sponge damping for pressure.
calculate the lateral sponge damping for the water vapor mixing ratio.
calculate the lateral sponge damping for optional scalar variable.

lsps0
lspuvw
mapfctmassconcalculate the lateral sponge damping for optional scalar variable to initial.
calculate the lateral sponge damping for the velocity.
calculate the map scale factors.
ﬁt the x, y and z components of velocity to the mass consistent equation.

melting

calculate the melting rate from the cloud ice to the cloud water and fromthe snow and graupel to the rain water.
change measurement from [m] [k] to [c] [g] for warm bin cloud physics.
force the mixing ratio to not being less than zero.
set the grid moving velocity.

mk2cgbw
more0qmove2dmvrst
ndgstep
newblknewsindx

rename restructed restart ﬁles.
get the control ﬂag of analysis nudging.
solve the new potential temperature perturbation, the mixing ratio and concentrations.
get the mininum and maximum data index to create the base state variables.

nlsmqv
nlsms
nlsmuvw

perform the non linear smoothing for the water vapor mixing ratio.
perform the non linear smoothing for optional scalar variable.
perform the non linear smoothing for the velocity.

nuc1stc
nuc1stv

calculate the nucleation rate of the condensation, contact and homogeneous.
calculate the nucleation rate of the deposition or sorption.

5.1 Program list

nuc2ndcalculate the secondary nucleation rate of the ice crystals.

numcharopendmpoutcheckcount the number of the character variable.
open the history ﬁle when the current time reaches marked time.
open and read in the data to the history data checking ﬁle.

outctloutdmpoutdmp2d

create the GrADS control ﬁle.
control the inferior subroutines to read in the variables to the history ﬁle.
read in the two dimensional variables to the history ﬁle.

outdmp3d
outgeooutgrdoutllsw

read in the three dimensional variables to the history ﬁle.
open and read in data to the geographical ﬁle.
open and read in the data to the three dimensional grided data ﬁle.
read in the latitude and the longitude at south-west corner to standard i/o.

outmxnread in the maximum and minimum prognostic variables to the standard
i/o when the current time reaches marked time.

outpbloutradoutresoutrstread in the surface variables to the history ﬁle.
open and read in the data to the radar data ﬁle.
open and read in the data to the restart ﬁle when the current time reaches
marked time.
open and read in the data to the restructed restart ﬁle.

outsfcoutstd1
outstd10

open and read in the data to the two dimensional grided surface ﬁle.
read in the ﬁrst message to standard i/o.
read in the messages to standard i/o.

outstd11
outstd12
outstd13
outstd14

read in the message to standard i/o.
read in the messages to standard i/o.
read in the messages to standard i/o.
read in message to standard i/o.

outstd15
outstd16
outstd2

read in message to standard i/o.
read in message to standard i/o.
read in the ﬁnal message to standard i/o.

outstd3
outstd4
outstd5

read in the message to standard i/o.
read in message to standard i/o.
read in message to standard i/o.

outstd6
outstd7
outstd8
outstd9

read in the message to standard i/o.
read in the messages to standard i/o.
read in the messages to standard i/o.
read in the messages to standard i/o.

outtrnp2gpv
paractl

open and read in the data to the two dimensional grided terrain ﬁle.
perform the analysis nudging to GPV data of pressure.
set the parameters of GrADS control ﬁle.

pbldrv

control the inferior subroutines for the virtical diﬀusion in the planetary
boundary layer.

pblptv
pblqv

calculate the virtical diﬀusion for the virtual potential temperature.
calculate the virtical diﬀusion for the water vapor mixing ratio.

117

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5 Program Lists

pblu

calculate the virtical diﬀusion for the x components of velocity.

pblv
pc2kg
pgrad

calculate the virtical diﬀusion for the y components of velocity.
convert the relative humidity to the water vapor mixing ratio.
calculate the pressure gradient force in the velocity equations.

pgradiv

calculate the pressure gradient force vertically with the horizontally explicit
and vertically implicit method.

phasevphasevbc

phvs
phvuvw

calculate the phase speed for the open boundary conditions.
calculate the diﬀrential phase speed term between the external boundaryand model grid variables.
calculate the diﬀrential phase speed term between the external boundaryand model grid for scalar variables.
calculate the diﬀrential phase speed term between the external boundaryand model grid for velocity variables.
calculate the scalar phase speed for the open boundary conditions.
calculate the velocity phase speed for the open boundary conditions.

phy2cnt
phycoodprodctwg

calculate the zeta components of contravariant velocity.
calculate the three dimensional z physical coordinates.
calculate the graupel production rate.

putbufgx
putbufgy
putbufsx
putbufsy

put the sending buﬀer in x direction for grouped domain.
put the sending buﬀer in y direction for grouped domain.
put the sending buﬀer in x direction for sub domain.
put the sending buﬀer in y direction for sub domain.

putunitqp2radraddrvradiatreturn the unit number to close the ﬁle.
perform the analysis nudging to radar data of optional precipitation mixing
ratio.
control the inferior subroutines to interpolate the radar data.
calculate the short and long wave radiation.

radstep
rbcpt
rbcq

calculate the number of steps of the main do loop in raddrv.
set the radiative lateral boundary conditions for potential temperature.
set the radiative lateral boundary conditions for optional mixing ratio.

rbcqv

set the radiative lateral boundary conditions for the water vapor mixing
ratio.

rbcs
rbcs0
rbcss
rbcu

set the radiative lateral boundary conditions for optional scalar variable.
set the radiative lateral boundary conditions for optional scalar variable.
set the radiative lateral boundary conditions for optional scalar variable.
set the radiative lateral boundary conditions for the x components of velocity.

rbcv

set the radiative lateral boundary conditions for the y components of velocity.

rbcw
rdcheck
rdconfrdgrd

set the radiative lateral boundary conditions for the z components of velocity.
open and read out the data from the history data checking ﬁle.
read out the namelist variables.
read out the data from the three dimensional grided data ﬁle.

phvbcs
phvbcuvw

5.1 Program list

rdgrp

open and read out the data from the grouped domain arrangement ﬁle andcheck them.

rdini

open and read out the data from the three dimensional grided data ﬁle atforcast start time.

rdnamerdradmrk
rdradnxt
rdres

read out and set the namelist variables.
read out the data from the radar data ﬁle.
read out the data from the radar data ﬁle.
open and read out the data from the restart ﬁle.

rdsfc
rdsnd
rdtrn

open and read out the data from the two dimensional grided surface ﬁle.
open and read out the data from the sounding data ﬁle.
open and read out the data from the two dimensional grided terrain ﬁle.

rdtundopen and read out the temperature under the ground and sea data from therestart ﬁle.

reducelb
reducevb
remapbw

reduce the integrated value on lateral boundary.
reduce the integrated value on bottom and top boundary.
remap the shifted water mass and concentrations for origianl distributed
water bins.
control the inferior subroutines for the repositon.

repdrv
repsit2d

reposition the restructed boundary array form original restart boundary
array.

repsit3d
resetagrmres

reposition the restructed array form original restart array.
reset the message tag.
remove the original restart ﬁles.

rotuvm2s

rotate the x and the y components of velocity from the projected grid to thelatitude and longitude grid.

rotuvs2m

rotate the x and the y components of velocity from the latitude and longitude
grid to the projected grid.

roughitr

calculate the roughness length on the sea surface by iteration at forcast starttime.

roughnxt
rstdrvrstindx

reset the roughness parameter on the sea surface to the next time step.
restruct the restart ﬁles.
calculate the minimum and maximum and diﬀerential do loops indexes toreposition.
calculate the number of steps of the main do loop in rstdrv.

rststep
rstuvwc
s2gpv
sadjstbw
satadjst
set1d

the base state density x the Jacobian is multiplyed by u, v and w.
perform the analysis nudging to GPV data of optional scalar variable.
perform the saturation adjustment for water in the case the generated total
water is more than critical value.
control the inferior subroutines for the saturation adjustment.
set the horizontally averaged values from the read out sounding data.

setbasesetbinset the base state variables.
set the bin parameters.

119

120

5 Program Lists

setblksetcapcalculate the air temperature, saturation mixing ratio, latent heat, viscosityof air, molecular diﬀusivity of water vapor, mean mass of cloud ice, mean
diameters and ventilation factors.
set the common used and referenced caption for dumped variables.

setcomsetcst1d
setcst2d
setcst3d

set the common used and referenced array.
ﬁll in the 1-dimensional array with the constant value.
ﬁll in the 2-dimensional array with the constant value.
ﬁll in the 3-dimensional array with the constant value.

setcst4d
setdaysetdimﬁll in the 4-dimensional array with the constant value.
set the common used and referenced calendar.
set the array dimension.

setgpv
setgridsethalosetnameset the three dimensional grided point variables.
set the model grid.
set the parameters of sendding and receiving.
set the namelist variables.

setproj
setradsetradqp

calculate the map projection parameters.
control the inferior subroutines to interpolate the radar data.
set the three dimensional grided point radar variables.

setsfccalculate the magnitude of velocity, virtual potential temperature and water
vapor mixing ratio on the surface.

setvar1d
setvar3d
sfcdrv

set the initial velocity and the scalar perturbations.
set the initial and the boundary conditions.
control the inferior subroutines to interpolate the surface data.

sfcﬂx
sfcphysheartke
shedding

calculate the bulk and the exchange coeﬃcients of surface ﬂux.
call the surface process driver routine to brige the complicated array.
calculate the shear production in the T.K.E. equation.
calculate the shedding rate from the snow and graupel to the rain water.

shift2nd
shiftgx
shiftgyshiftsx

exchange the values horizontally in the case the fourth order calculation is
performed.
exchange the values in x direction between grouped domain.
exchange the values in y direction between grouped domain.
exchange the values in x direction between sub domain.

shiftsy
siadjstslvdrvexchange the values in y direction between sub domain.
perform the saturation adjustment for ice.
control the inferior subroutines for the time steps integration.

slvstep
smalldrv
smoo2psmoo2qv

calculate the number of steps in slvdrv.
control the inferior subroutines for the small time steps integration.
perform the second order smoothing for the pressure.
perform the second order smoothing for the water vapor mixing ratio.

smoo2ssmoo2uvw
smoo4pperform the second order smoothing for optional scalar variable.
perform the second order smoothing for the velocity.
perform the fourth order smoothing for the pressure.

5.1 Program list

smoo4qv

perform the fourth order smoothing for the water vapor mixing ratio.

smoo4ssmoo4uvw
smoogpv

perform the fourth order smoothing for optional scalar variable.
perform the fourth order smoothing for the velocity.
smooth the interpolated GPV data.

sndwavesoildrv
sparprtcalculate the base state density x sound wave speed squared.
control the inferior subroutines for the surface physics.
separate the pressure and the potential temperature to the base state andthe perturbation values.
control the inferior subroutines for source term of the bulk cold rain cloudphycics.
solve the variables to the next time step by the Cubic Lagrange scheme.

srcblkstepcul
steppesolve the pressure to the next time step with the horizontally explicit andvertically explicit method.

steppisolve the pressure to the next time step with the horizontally explicit andvertically implicit method.

steppts
steps
steptundsolve the potential temperature perturbation to the next time step.
solve the scalar variables to the next time step.
solve the temperature under the ground and sea to the next time step.

stepuv
stepwestretchstrstensolve the x components of velocity to the next time step.
solve the z components of velocity to the next time step with the horizontally
explicit and vertically explicit method.
solve the z components of velocity to the next time step with the horizontally
explicit and vertically implicit method.
calculate the one dimensional physical coordinates.
calculate the stress tensor.

sumbinswadjstswp2nxt
t2pt

get the total mixing ratio.
perform the saturation adjustment for water.
swap the prognostic variables to the next time step.
convert the temperature to the potential temperature.

termblk
termbw2d
termbw3d

calculate the terminal velocity of the rain water, the snow and the graupel.
calculate the terminal velocity for water bin.
calculate the terminal velocity for water bin.

termqrtimeﬂttkeﬂx
totalbincalculate the terminal velocity of the rain water.
perform the Asselin time ﬁlter for the prognostic variables.
calculate the turbulent ﬂuxes for the T.K.E..
get the total water and ice mixing ratio and concentrations for bin categories.

totalqtrilattrndamp

get the total water and ice mixing ratio.
calculate the Coriolis parameters x 0.25.
calculate the damped terrain height.

trndrv
turbdrv
turbﬂxcontrol the inferior subroutines to interpolate the terrain data.
control the interior subroutines to solve the turbulent mixing.
calculate the turbulent ﬂuxes for optional scalar variable.

stepwi121

122

5 Program Lists

turbs

calculate optional scalar turbulent mixing.

turbtketurbuvw
undeﬁcecalculate the T.K.E. turbulent mixing.
calculate the velocity turbulent mixing.
convert the undeﬁned values to the interpolated values in the sea ice distribution data.
convert the undeﬁned values to the interpolated values in the sea surface
temperature data.
generate the united history ﬁle from the dumped ﬁles in entire domain.

undefsst
unidmpen
unidmpgr
unidmprdunidrv
unigeoen
unigeogr
unigeord
unistep
upwmbingenerate the united history ﬁle from the dumped ﬁles in grouped domain.
generate the united history ﬁle from the dumped ﬁles in reductional entire
domain.
control the inferior subroutines for uniting.
generate the united geography ﬁle from the dumped ﬁles in entire domain.
generate the united geography ﬁle from the dumped ﬁles in grouped domain.
generate the united geography ﬁle from the dumped ﬁles in reductional entire
domain.
calculate the number of steps of the main do loop in unidrv.
calculate the sedimentation and precipitation for optional bin mass.

upwnbinupwnsgupwqcgupwqp

calculate the sedimentation for optional bin concentrations.
calculate the sedimentation for optional precipitation concentrations.
calculate the sedimentation for optional charging distribution.
calculate the sedimentation and precipitation for optional precipitation mixing ratio.

uvw2gpv
uvw2radvar8uvw
var8w8s

perform the analysis nudging to GPV data of the velocity.
perform the analysis nudging to radar data of the velocity.
optional variable be averaged to u, v and w points.
optional variable at w points be averaged to scalar points.

var8w8u
var8w8v
vbcp

optional variable at w points be averaged to u points.
optional variable at w points be averaged to v points.
set the vertical boundary conditions for the pressure.

vbcqcg
vbcs
vbcu
vbcv

set the vertical boundary condition for the charging distribution.
set the vertical boundary condition for optional scalar variable.
set the vertical boundary conditions for the x components of velocity.
set the vertical boundary conditions for the y components of velocity.

vbcw
vbcwc
vculs
vculs0

set the vertical boundary conditions for the z components of velocity.
set the vertical boundary conditions for the zeta components of contravariant
velocity.
calculate the vertical scalar advection by the Cubic Lagrange scheme.
calculate the vertical scalar advection by the Cubic Lagrange scheme.

vculuvw
vextlow
vextundcalculate the horizontal velocity advection by the Cubic Lagrange scheme.
extrapolate the base state variables in lowest layer.
extrapolate the base state variables in undeﬁned layer.

5.1 Program list

123

vint11interpolate the one dimensional data to the ﬁne horizontally averaged levels
vertically.

vint113
vint133g
vint133r
vint31g

interpolate the variable to the model or data grid vertically.
interpolate the variable to the model grid vertically.
interpolate the variable to the model grid vertically.
interpolate the three dimensional input variable to the one dimensional ﬂat
plane.
interpolate the three dimensional input variable to the one dimensional ﬂat
plane.
interpolate the three dimensional input variable to the one dimensional ﬂat
plane.

vint31r
vint31s
vintbase

control the interior subroutines for vertical interpolating the GPV data to
the ﬂat plane to create the base state variables.

vintgpv

control the interior subroutines for vertical interpolating the GPV data to
the ﬂat plane.

vintradcontrol the interior subroutines for vertical interpolating the radar data to
the ﬂat plane.

vintsndvspdmpvspp

interpolate the sounding data to ﬁne interval levels vartically.
calculate the relaxed vertical sponge damping coeﬃcients.
calculate the vertical sponge damping for pressure.

vspqv
vsps
vsps0
vspuvw

calculate the vertical sponge damping for the water vapor mixing ratio.
calculate the vertical sponge damping for optional scalar veriables.
calculate the vertical sponge damping for optional scalar variable to initial.
calculate the vertical sponge damping for the velocity.

warmbinwarmblkxy2ij

control the inferior subroutines for the warm bin cloud physics.
calculate the source amounts of the bulk warm cloud micro physics.
calculate the real indexes from the two dimensional x and the y coordinates
at the model grid points.
calculate the latitude and the longitude from the x and the y coordinates.

xy2ll

5.1.3

MPI subroutines list

The program codes of CReSS have MPI subroutines as well as the above subroutines. MPI
subroutines used in CReSS are listed in this subsection. The detailed description of MPI routine
is found in the manuals of MPI.
subroutine

description

mpi_abort
mpi_allreduce
mpi_barrier

Halt MPI processes.
Perform the same process in all nodes.
Adjust timing between nodes.

mpi_bcast

Broadcast from a node to all nodes.
continued to the next page124

5 Program Lists

continued

subroutine

description

mpi_comm_rank

Get the number of node.

mpi_comm_size
mpi_comm_split
mpi_finalize

Get a number of all nodes.
Split communicator.
Terminate MPI.

mpi_gather
mpi_init
mpi_sendrecv

Gather data to a node from other nodes.
Initialize MPI.
Send and receive data simultaneously.

Part III

Fundamental Theory and
Formulations

Chapter 6

The Basic Equations of CReSS

The basic equations of CReSS are composed of the Navier-Stokes equation
including the eﬀect of the earth rotation, the equation of thermodynamics, thecontinuity equation, the conservation equations of water vapor mixing ratio,
equations of mixing ratios of hydrometeors, and equations of number densityof solid hydrometeors. The CReSS model includes the formulation of various
physical processes and the formulation of boundary value. In this chapter, we
describe the formulation of the basic equations in the constitution.
CReSS can consider the eﬀect of terrain using the terrain-following coordinate. For understanding the basic dynamics of the model, ﬁrstly, we describe
the basic equations without terrain-following coordinates. Secondly, we describe those in the terrain-following coordinates.

128

6 The Basic Equations of CReSS

6.1

The basic equations without terrain

The independent variables of the model are the distance coordinates, x, y, z and time, t. The
dependent variables are horizontal velocity, u, v, vertical velocity, w, perturbation of potential
temperature, θ , perturbation pressure, p , mixing ratio of water vapor, qv , mixing ratios ofhydrometeors, qx , and number density of solid hydrometrors Nx . Here, The basic state of thepotential temperature, pressure and density considering water contents are constant with timeand satisfy the hydrostatic equilibrium,∂ p̄= −ρ̄g∂z(6.1)

The relation between the basic states and the deviation from them are given as,θ = θ̄ + θ 

(6.2)



(6.3)



(6.4)

p = p̄ + p

ρ = ρ̄ + ρ

The density is given by the equation of state diagnostically,
p
ρ=
Rd T



qv
1−
 + qv

1 + qv +





qx

(6.5)

Here, g is the gravity acceleration,  is the ratio of molecule number between water vapor anddry air and Rd is gas constant of dry air.
All of the dependent variables except density are described in time-developing equations. In
the coordinates without terrain, the equations of the dependent variables are given as follows.
The basic equations with terrain will be shown in Section 6.2.
Equation of motion


∂u
∂u
∂u
∂ ρ̄u
= −ρ̄ u
+v
+w
∂t
∂x
∂y
∂z


∂ ρ̄v
∂v
∂v
∂v
= −ρ̄ u
+v
+w
∂t
∂x
∂y
∂z




−

∂p
+ ρ̄ (fs v − fc w) + Turb.u
∂x

(6.6)

−

∂p
− fs ρ̄u + Turb.v
∂y

(6.7)

∂p
− ρ̄Buoy.w + fc u + Turb.w
∂z

(6.8)



∂w
∂w
∂ ρ̄w
∂w
= −ρ̄ u
+v
+w
∂t
∂x
∂y
∂z



−

Here, fs, fc is Coriolis-parameter and Buoy.w is the term of buoyancy.
Equation of pressure6.2 The basic equations system in terrain-following coordinates


129



∂p
∂p
∂p
∂p
=− u
+v
+w
+ ρ̄gw
∂t
∂x
∂y
∂z




1 dQ
∂u ∂v ∂w
1 dθ++−+ ρ̄c2s−ρ̄c2s
∂x ∂y
∂z
θ dt
Q dt

Here, cs is the speed of sound in atmosphere and Q = 1 + 0.61qv +



(6.9)

qx .

Equation of potential temperature∂ ρ̄θ 
∂θ 
∂θ 
∂θ 
= −ρ̄ u
+v
+w
∂t
∂x
∂y
∂z



− ρ̄w

∂ θ̄
+ Turb.θ + ρ̄Src.θ
∂z(6.10)

Equations of water vapor and mixing ratio of wear contents∂ ρ̄qv∂qv∂qv∂qv= −ρ̄ u
+v
+w
∂t
∂x
∂y
∂z


∂ ρ̄qx∂qx∂qx∂qx= −ρ̄ u
+v
+w
∂t
∂x
∂y
∂z



+ Turb.qv + ρ̄Src.qv(6.11)+ Turb.qx + ρ̄Src.qx + ρ̄Fall.qx(6.12)



Equation of number density of water contents










∂ Nx
∂ Nx
∂
∂Nx
= −ρ̄ u
+v
+w
∂t
∂x ρ̄
∂y ρ̄
∂z
Nx
Nx
Nx
+ ρ̄Src.
+ ρ̄Fall.
+Turb.
ρ̄
ρ̄
ρ̄



Nx
ρ̄



(6.13)The detail of other dependent variables are described in Section 6.2. The term of diﬀusion in
sub-grid scale Turb.φ which appears except in equation of pressure is described in Chapter 7, andthe source terms of cloud and precipitation Src.φ and falling of precipitation Fall.φ which appear
in the equation of potential temperature and water contents are described in Chapter 8.

6.2

The basic equations system in terrain-following coordinates

6.2.1

General curvilinear coordinates

CReSS adopts the terrain-following coordinate to consider terrain eﬀect. The basis of thiscoordinate system doesn’t necessarily become to orthonormal, while the vector in Cartesian coordinate is expressed by orthogonal basis. This is categorized in curvilinear coordinate of liner
algebra. Here, the basic background is summarized.
Contravariant components and covariant components
Introducing general linear independent bases (e1 , e2 , e3 ), which are neither regular nor orthogonal,
facultative vector A is expressed as the linear combination,

130

6 The Basic Equations of CReSS

A = A1 e 1 + A2 e 2 + A3 e 3(6.14)On the other hand, since the inverse bases f 1 , f 2 , f 3 , is deﬁned as

f i · ej = δji(6.15)

f i · A = Ai

(6.16)

then, the components are,Here, Kronecker’s delta is deﬁned as follows,

δji =

1,

i=j

0,

i = j(6.17)In the case of orthonormal basis, coordinate components are given by scalar product of basis
vector and facultative vector. In the case of general basis, however, the components must be
given by scalar product of inverse basis vector and facultative vector. Thus components of coordinates for original basis are named contravariant components and expressed in a superscript.
On the other hand, components of coordinates for inverse basis are named covariant components,
expressed in a subscript.
The facultative vector B is expressed by inverse basic vector as
B = B1 f 1 + B2 f 2 + B3 f 3(6.18)The solution of scalar product of them becomes

A · B = Ai Bi(6.19)

Thus, scalar product is expressed easily by the sum of product of components in the relative coordinate when both contravariant and covariant components are used. In the case of orthonormal
basis, as inverse basis becomes same to original basis, there are no diﬀerence between contravariant
and covariant components.

A · B = Ai B i(6.20)

6.2 The basic equations system in terrain-following coordinates

131

Scalar product of facultative vector A and bases (e1 , e2 , e3 ) gives covariant components forbases of vector A, (e1 , e2 , e3 ),Ai = A · e i(6.21)

Next, for bases (e1 , e2 , e3 ), 9 scalar product can be made,
Gij = ei · ej

(6.22)

This matrix is named metric matrix. Moreover, this is quadratic tensor which is named metrictensor. Using this, the relationship between contravariant and covariant components is expressed
as follows.
Ai = Gji Aj(6.23)

curvilinear coordinates
As a function of Cartesian coordinate, three function F i (x, y, z) which deﬁned at a certain regionof a space is considered. The diﬀerentiation of them for (x, y, z) is possible for any number oftimes.
ξ = F 1 (x, y, z)(6.24)

2

η = F (x, y, z)(6.25)

3(6.26)

ζ = F (x, y, z)(ξ, η, ζ) correlates to the each point in a region P (x, y, z). When this correlation is 1 to 1, it
can be said that curvilinear coordinates is made in a region. Furthermore, the following conditionis assumed.

∂ (x, y, z)=∂ (ξ, η, ζ)

∂x
∂ξ

∂x
∂η

∂x
∂ζ

∂y
∂ξ

∂y
∂η

∂y
∂ζ

∂z
∂ξ

∂z
∂η

∂z
∂ζ

= 0

(6.27)

132

6 The Basic Equations of CReSSFor Cartesian coordinate, vector (e1 , e2 , e3 ) is
⎛

⎞

⎛

⎞

⎛

⎞

⎜
⎜ ∂y
⎜
e1 = ⎜
⎜ ∂ξ
⎜
⎜
⎝ ∂z

⎟
⎟
⎟
⎟,
⎟
⎟
⎟
⎠

⎜
⎜ ∂y
⎜
e2 = ⎜
⎜ ∂η
⎜
⎜
⎝ ∂z

⎟
⎟
⎟
⎟,
⎟
⎟
⎟
⎠

⎜
⎜ ∂y
⎜
e3 = ⎜
⎜ ∂ζ
⎜
⎜
⎝ ∂z

⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠

∂x
⎜ ∂ξ ⎟⎜⎟∂x⎜ ∂η ⎟
⎜
⎟

∂ξ

∂η

∂x
⎜ ∂ζ ⎟⎜⎟(6.28)∂ζ

Here, vector (e1 , e2 , e3 ) is named ’basis’ or ’fundamental vector’ of curvilinear coordinates (ξ, η, ζ).
Using above, facultative vector ﬁeld A is expressed as the linear combinatio of them.
A = A1 e 1 + A2 e 2 + A3 e 3(6.29)

Then, Ai is named ’contravariant components’ of curvilinear coordinates (ξ, η, ζ) of vector ﬁeld
A. The covariant component is mentioned in (6.21). Samely, for the bases (e1 , e2 , e3 ), 9 scalar
product can be made.

Gij = ei · ej

(6.30)

Matrix Gij is named metric matrix of curvilinear coordinates (ξ, η, ζ). The relationship between
contravariant and covariant components is given by (6.23).

6.2.2

Terrain-following coordinates

Many cloud models adopt terrain-following coordinate system to consider the eﬀect of terrain.
Such coordinate system are oriented on the special relationship of curvilinear coordinates.
CReSS adopts the terrain-following coordinate as well as NHM and ARPS.

ξ=x(6.31)

η=y(6.32)

ζ = ζ (x, y, z)(6.33)In this case, velocity vector of Cartesian coordinate can be expressed as well as (6.29) bycomponents of velocity vector (contravariant components) (U, V, W ) in the terrain-following coordinate.(Ordinally, (U, V, W ) are contravariant components and they should be expressed in the
superscript u1 , u2 , u3 . However, since it is easy, it is written in this way.)6.2 The basic equations system in terrain-following coordinates

133

u=U

∂x
∂x
∂x
+V
+W
∂ξ
∂η
∂ζ

(6.34)

v=U

∂y
∂y
∂y
+V
+W
∂ξ
∂η
∂ζ

(6.35)

w=U

∂z
∂z
∂z
+V
+W
∂ξ
∂η
∂ζ

(6.36)

Since it is assumed that condition (6.27) is realized, the inverse of the velocity vector can be
asked by solving (6.34)∼(6.36) about (U, V, W ).

1

yz
xy
zx
U G 2 = uJηζ
+ vJηζ
+ wJηζ

1yzxyzx
V G 2 = uJζξ
+ vJζξ
+ wJζξ

1yzxyzx
W G 2 = uJξη
+ vJξη
+ wJξη

(6.37)(6.38)(6.39)

Here, J is Jacobian. For example, it is deﬁned as

yz
≡
Jηζ

∂ (y, z)=∂ (η, ζ)

∂y
∂η

∂y
∂ζ

∂z
∂η

∂z
∂ζ

(6.40)

1G 2 is deﬁned as Jacobian of the coordinate conversion between (ξ, η, ζ) and (x, y, z) and expressed
as

1

G2 ≡

∂ (x, y, z)=∂ (ξ, η, ζ)

∂x
∂ξ

∂x
∂η

∂x
∂ζ

∂y
∂ξ

∂y
∂η

∂y
∂ζ

∂z
∂ξ

∂z
∂η

∂z
∂ζ

(6.41)In the case of terrain-following coordinate deﬁned by (6.31)∼(6.33), Jacobians which appear in(6.37)∼(6.39) are as follows.

134

6 The Basic Equations of CReSS

yz
Jηζ
=

∂z
,
∂ζ

yz
Jζξ
= 0,
yz
=−
Jξη

xy
Jηζ
= 0,

zx = 0,
Jηζ

∂z
,
∂ζ

zx
Jζξ
=

∂z
,
∂ξ

xy
= 0,
Jζξ

∂z
,
∂η

zx
=−
Jξη

(6.42)xy=1JξηIn the case of three-dimension, Jacobian of the coordinate conversion between (ξ, η, ζ) and(x, y, z) is

1

G2 =

∂z
∂ζ

(6.43)

Here, variable components of Jacobian’s components are deﬁned asyz
J31 ≡ Jξη
=−

∂z
∂ξ

(6.44)zx
J32 ≡ Jξη
=−

∂z
∂η

(6.45)

yz
zx
= Jζξ
=
Jd ≡ Jηζ∂z∂ζ(6.46)and ζ is deﬁned by using the altitude of surface zsf c (x, y) and the height of the region of model
ztop ,
ζ (x, y, z) =ztop [z − zsf c (x, y)]ztop − zsf c (x, y)(6.47)or

zsf c (ξ, η)z (ξ, η, ζ) = zsf c (ξ, η) + ζ 1 −ztop

In this case, Jacobian’s various components are(6.48)

6.2 The basic equations system in terrain-following coordinates

135

z top
w
W

U
u

z sfc

Figure 6.1. Terrain-following coordinates and direction of the vector.

∂z
J31 = −=
∂ξ∂z=J32 = −
∂η
Jd =



ζ
ztop



ζ
ztop



−1


−1

∂zsf c (ξ, η)∂ξ

(6.49)∂zsf c (ξ, η)∂η

(6.50)

zsf c (ξ, η)
∂z
= 1−
∂ζ
ztop

(6.51)

Like in this case, when ζ is a monotonically increasing function about z, it is expressed that1

G 2 = |Jd | = Jd

(6.52)

Velocity of the terrain-following coordinate (contravariant velocity) which is given in (6.37)∼(6.39)
is transformed as follows,

U =u(6.53)V =v



W = (uJ31 + vJ32 + w) G

(6.54)
1
2(6.55)

When the transformation from Cartesian coordinate to the coordinate along terrain is performed using above relation, the space diﬀerential of a various value φ is transformed as follows,
1
∂φ
= 1
∂x
G2
∂φ
1
= 1
∂y
G2




∂
∂
(Jd φ) +(J31 φ)∂ξ∂ζ(6.56)∂∂(Jd φ) +(J32 φ)∂η∂ζ(6.57)

136

6 The Basic Equations of CReSS

∂φ
1 ∂φ
= 1
∂z
G 2 ∂ζ

6.2.3(6.58)The basic equations in terrain-following coordinates

In the coordinates of terrain-following, three dependent variables are separated into the valuesof base state and deviations from the base state: potential temperature, pressure and density
which considered eﬀects of water contents and water vapor. The values of base state are deﬁned
as those are in hydrostatic balance taken account of eﬀects of terrain as1∂ ρ̄= −G 2 ρ̄g∂ζ(6.59)

To simplify those representation, we use1ρ∗ = G 2 ρ̄

(6.60)and transform each predicted variable as follows:u∗ = ρ∗ u(6.61)∗∗(6.62)∗∗(6.63)∗∗(6.64)∗∗ θ =ρ θ

(6.65)

qv∗ = ρ∗ qv
qx∗ = ρ∗ qx(6.66)

v =ρ vw =ρ w
W =ρ W

(6.67)

Using the above, the basic equations system in terrain-excluding coordinates shown in Section
6.1 of this chapter are transformed in terrain-following coordinates as follows.
Equation of motion








∂u
∂u
∂u
∂u∗
= − u∗
+ v∗
+ W∗
∂t
∂ξ
∂η
∂ζ



1
∂ 
∂ 
−
Jd p − αDiv ∗ +J31 p − αDiv ∗+ (fs v∗ − fc w∗ ) + G 2 Turb.u(6.68)
∂ξ
∂ζ
∂v∗
∂v
∂v
∂v
= − u∗
+ v∗
+ W∗
∂t
∂ξ
∂η
∂ζ



1
∂ 
∂ 
−
Jd p − αDiv ∗ +J32 p − αDiv ∗− fs u∗ + G 2 Turb.v∂η∂ζ

(6.69)

6.2 The basic equations system in terrain-following coordinates



137



∂w∗
∂w
∂w
∂w
= − u∗
+ v∗
+ W∗
∂t
∂ξ
∂η
∂ζ
1
∂
−
p − αDiv ∗ − ρ∗ Buoy.w + fc u∗ + G 2 Turb.w∂ζ(6.70)

where the buoyancy term Buoy.w is expressed as




θ
ρ
qv
q  + qxp
Buoy.w = −g = g
− v
− 2+
ρ̄
ρ̄cs
 + q̄v
1 + q̄v
θ̄

(6.71)

Here qv is not the deviation from the value of base state but the deviation from the initial value, is the ratio of molecular weight to water vapor and dry air. cs is the speed of sound in air given
by


cs =

γRd T̄ ,γ ≡ Cp /Cv(6.72)

where g is the gravity acceleration、T̄ is the temperature of base state and Rd is the gas constantfor dry air. Cp , Cv are the speciﬁc heat at constant pressure and the speciﬁc heat at constantvolume for dry air, respectively. fs , fc are the Coriolis coeﬃcients:

fs = 2ω sin ϕ(6.73)

fc = 2ω cos ϕ(6.74)

where ω is the angular velocity of the earth and ϕ is the latitude. Furthermore, αDiv ∗ shown in
the pressure term is the divergence damping to suppress soundwaves, which is given byDiv∗ =

1



1

G2

∂u∗ ∂v∗
∂W ∗
+
+
∂ξ
∂η
∂ζ



(6.75)

Equation of pressure

1




1
1 ∂p
1
∂G 2 p∂p∂p
= − G2 u+ G2 v+ G2 W
∂t
∂ξ
∂η
∂ζ



−ρ̄c2s1

1

1∂G 2 u ∂G 2 v ∂G 2 W
+
+
∂ξ
∂η
∂ζ



1

+ G 2 ρ̄gw

1

+ G 2 ρ̄c2s



1 dθ
1 dQ
−
θ dt
Q dt



(6.76)

138

6 The Basic Equations of CReSS

where Q = 1 + 0.61qv +qx was used.

Equation of potential temperature


∂θ 
∂θ 
∂θ 
∂θ ∗
= − u∗
+ v∗
+ W∗
∂t
∂ξ
∂η
∂ζ



− ρ̄w

1
∂ θ̄
+ G 2 Turb.θ + ρ∗ Src.θ
∂ζ

(6.77)

Equations of mixing ratio of water vapor and water contents


∂qv∗
∂qv
∂qv
∂qv
= − u∗
+ v∗
+ W∗
∂t
∂ξ
∂η
∂ζ


∂qx
∂qx
∂qx
∂qx∗
= − u∗
+ v∗
+ W∗
∂t
∂ξ
∂η
∂ζ




1

+ G 2 Turb.qv + ρ∗ Src.qv(6.78)

1+ G 2 Turb.qx + ρ∗ Src.qx + ρ∗ Fall.qx(6.79)

Equations of number concentration per unit volume1∂G 2 Nx∂= − u∗
∂t
∂ξ









Nx
∂ Nx
∂
+ v∗
+ W∗
ρ̄
∂η ρ̄
∂ζ
1
Nx
N
N
x
x
+G 2 Turb.
+ ρ∗ Src.
+ ρ∗ Fall.
ρ̄
ρ̄
ρ̄



Nx
ρ̄



(6.80)

Similar to the no terrain, there is the prognostic equation of turbulence kinetic energy E in
addition to these equations. In these equations, Turb.φ is the diﬀusion term of sub-grid scale.
Src.φ is the production or loss term and Fall.φ is the sedimentation term in the equations ofpotential temperature and water contents.

6.3

Basic Equations — Map Projection

The computational domain of a cloud-resolving model has been small enough to ignore theearth curvature. Map projection, therefore, was not necessary for a simulation using the cloudresolving model. Recently, parallel computers rapidly developed and their performance and memory have increased. Computational dimensions of the cloud-resolving model became large and theearth curvature is not small enough to ignore. For instance, a computational domain of a typhoon
simulation is large and the eﬀect of the earth curvature should be taken into the computation.
To consider the earth curvature, map projections are necessary. In this section, we introduce themap factor into the basic equations and map projections will enable to use in the CReSS model.
Longitude-latitude coordinates and conformal map projections are orthogonal curvilinear coordinates, if no topography is considered. When topography is considered, the vertical coordinate is
no longer orthogonal to the horizontal coordinates. They are general curvilinear coordinates andtensor transformation is necessary. However, we take the following two steps to introduce map
projection and terrain-following coordinates for simplicity. First, the equation system including
map projection will be derived without topography. Then, the terrain-following coordinates will
be introduced.

6.3 Basic Equations — Map Projection

6.3.1

139

Basic equations with map projection — no topography

The longitude-latitude coordinates and conformal map projections with a vertical coordinate
are 3-dimensional orthogonal curvilinear coordinates if no topography is considered. Assuming
the horizontal coordinates are (ξ, η) and the vertical coordinate is z, the basic equations on amap projection will be derived.
Metric factors (stretching factors) are introduced as

h1 =

1
m

(6.81)

h2 =1n(6.82)

h3 = 1(6.83)The metric factors m, n will be used as map factors. In general, they are called as the metricfactors. For example, metric factors in the spherical coordinate arem=1
a cos φ(6.84)

1(6.85)awhere a is the radius of the earth and φ is latitude. If the metric factors are these forms, thebasic equations which will be derived later are the equations on the spherical coordinate.
n=Now, we take a point P and a neighbor point. Their coordinates are (ξ, η, z) and (ξ + dξ, η +dη, z + dz). Then, the distances between the two points along the each coordinate ds1 , ds2 , ds3
are
dξ
ds1 =
(6.86)mdη
n(6.87)

ds3 = dz

(6.88)

ds2 =Derivatives of tangential unit vector along the each coordinate are
∂e1
∂
= −n
∂ξ
∂η
∂e2
∂
=n
∂ξ
∂η



1
∂e3
=
e1 ,
∂ξ
am





1
1
e3 ,
e2 −
m
am


1
e1 ,
m

∂
∂e1
=m
∂η
∂ξ

 

1
e2 ,
n

∂e2
∂
1
= − e3 − m
∂η
an
∂ξ
1
∂e3
=
e2 ,
∂η
an

∂e1
= 0,
∂ζ
 

1
e1 ,
n

∂e2
= 0,
∂ζ
∂e3
=0
∂ζ

(6.89)

140

6 The Basic Equations of CReSS

Using these formulas, the equations of momentum, the equation of thermodynamics, the equation of continuity of compressible ﬂuid, the equations of mixing ratios of water vapor and hydrometeors, the equations of number density of solid hydrometeors are rewritten in the following
part of this section.
The momentum equations with no topography (6.6)〜(6.8) will be derived using the metricfactors of the orthogonal coordinates(ξ, η, z).
The velocity vectors u on the orthogonal coordinate (ξ, η, z) are represented asu = ue1 + ve2 + we3(6.90)

Considering the relationships between distance and coordinate (6.86)〜(6.88), the each component of velocity will be

u=

ds1
1 dξ
=
,
ddt
m dt

(6.91)

v=

1 dη
ds2
=
,
ddt
n dt

(6.92)

w=

ds3
dz
=
.
ddt
dt

(6.93)The total derivative on the map projection is
d
∂
dξ ∂
dη ∂
dz ∂
=
+
+
+
.
dt
∂t dt ∂ξ
dt ∂η
dt ∂z

(6.94)

Using equations (6.91)〜(6.93), then,
∂
∂
∂
∂
d
=
+ mu
+ nv
+w .
dt
∂t
∂ξ
∂η
∂z

(6.95)

Consequently, time derivative of velocity vector (6.90) on the curvilinear coordinates is
du
du
de2
de3
dv
dw
de1
=
e1 + e2 +e3 + u
+v
+w
.
dt
dt
dt
dt
dt
dt
dt

(6.96)

Using (6.89) and (6.95), the time derivatives of the tangential unit vectors in (6.96) are modiﬁedas6.3 Basic Equations — Map Projection



 



141



de2
de3
1
uw
de1
∂ 1
∂
+v
+w
= −e1 mnv v
u
−u
+ e1
dt
dt
dt
∂ξ n
∂η m
a

 
 
1
vw
u2 + v2
∂ 1
∂
−u
+ e1
+e2 mnu v− e3. (6.97)
∂ξ n
∂η m
a
a

These terms appear since the unit vectors changes with the curvilinear coordinates. They arecalled as curvature terms or metric terms.
On the coordinates (ξ, η, z), each component (fξ , fη , fz ) of the Coriolis force 2Ω are represented
as

e1

e2

e3

2Ω × u = 2Ωξ 2Ωη 2Ωz
u v w= e1 (fη w − fz v) + e2 (fz u − fξ w) + e3 (fξ v − fη u).

(6.98)

Using the relationship of the gradient of the scalar variable in the orthogonal coordinates, thepressure gradient force is∇p = m

∂p
∂p
∂p
e1 + n
e2 +
e3 .
∂ξ
∂η
∂z

(6.99)

Using the above formulations, the equations of momentum (6.6)〜(6.8) will be rewritten as
The equations of momentum








∂u
∂u
∂u
∂u
= −ρ̄ mu
+ nv
+w
ρ̄
∂t
∂ξ
∂η
∂z

 
 
1
∂p
∂ 1
∂
uw
−m
−u
− ρ̄
+ ρ̄(fη w − fz v) + ρ̄mnv v+ Turb.u, .(6.100)
∂ξ
∂ξ n
∂η m
a
ρ̄

∂v
∂v
∂v
∂v
= −ρ̄ mu
+ nv
+w
∂t
∂ξ
∂η
∂z

 
 

1
∂p
∂ 1
∂
vw
+ ρ̄(fz u − fξ w) − ρ̄mnu v+ Turb.v, (6.101)
−n
−u
− ρ̄
∂η
∂ξ n
∂η m
a


ρ̄



∂w
∂w
∂w
∂w
= −ρ̄ mu
+ nv
+w
∂t
∂ξ
∂η
∂z

∂p
u2 + v2−− ρ̄Buoy.w + ρ̄(fξ v − fη u) + ρ̄+ Turb.w.
∂za(6.102)

142

6 The Basic Equations of CReSS

where ρ̄ = ρ̄(z) is density of the basic ﬁeld and it is a function of only the vertical coordinate.
The term Buoy.w is buoyancy term.
In the same manner, the equation of pressure perturbation (6.9), the equation of potential temperature perturbation (6.10), and the equations of water substances (6.11)〜(6.13) are rewritten
as follows.
The equation of pressure perturbation




∂p
∂p
∂p
∂p
+ ρ̄gw= − mu
+ nv
+w
∂t
∂ξ
∂η
∂z





∂ u
1 dθ
∂w
∂ v
1 dQ
+
+ ρ̄c2s−ρ̄c2s mn+−∂ξ m ∂η n
∂z
θ dt
Q dt

(6.103)The equation of potential temperature perturbation


ρ̄

∂θ 
∂θ 
∂θ 
∂θ 
= −ρ̄ mu
+ nv
+w
∂t
∂ξ
∂η
∂z



− ρ̄w

∂ θ̄
+ ρ̄Src.θ + Turb.θ
∂z(6.104)The equations of mixing ratios of water vapor and hydrometeors


∂qv
∂qv
∂qv
∂qv
ρ̄
= −ρ̄ mu
+ nv
+w
∂t
∂ξ
∂η
∂z


∂qx
∂qx
∂qx
∂qx
ρ̄
= −ρ̄ mu
+ nv
+w
∂t
∂ξ
∂η
∂z



+ ρ̄Src.qv + Turb.qv(6.105)+ ρ̄Fall.qx + ρ̄Src.qx + Turb.qx(6.106)6.3 Basic Equations — Map Projection

143

The equations of number density of hydrometeors










∂Nx
∂ Nx
∂ Nx
∂
+ nv
+w
= −ρ̄ mu
∂t
∂ξ ρ̄
∂η ρ̄
∂z
Nx
Nx
Nx
+ρ̄Src.
+ ρ̄Fall.
+ Turb.
ρ̄
ρ̄
ρ̄



Nx
ρ̄



(6.107)The equation of the turbulence kinetic energy is necessary to evaluate the turbulence term
Turb.φ. The equation on the map projection will be shown in Section of 7.

6.3.2

Basic equations with map projection and terrain

In this section, the basic equations using terrain-following coordinate with map projections aredescribed. Introducing the terrain-following coordinate, the coordinate system is not orthogonal
coordinate but a general curvilinear coordinate. Because the map factor is not a function of thevertical coordinate, we take the same way to introduce the terrain-following vertical coordinate
as shown in Section 6.2.
In the previous section, the metric factors m, n are a stretched factor. In this section, they are
the map factors. Most map projections in meteorology are a conformal projection such as thepolar stereographic projection, the Lambert conformal projection and Mercator projection. In
the conformal projection, direction is correct and two metric factors in the horizontal directions
are the same at any point on the map. That is


1
h1 =
m





1
= h2 =
n(6.108)In general, the map factors are a function of longitude and latitude.
Using (6.91)〜(6.93), the total derivative on the conformal map projections is
d
∂
dξ ∂
dη ∂
dζ ∂
=
+
+
+
dt
∂t dt ∂ξ
dt ∂η
dt ∂z
∂
∂
∂
∂
=
+ mu
+ mv
+W
∂t
∂ξ
∂η
∂ζ

(6.109)Then, the vertical velocity W is

W =

∂ζ
∂ζ
∂ζ
dζ
= mu
+ mv
+w
dt
∂ξ
∂η
∂z





∂z
∂z
∂ζ ∂ζ
= mu −+ mv −
+w
∂ξ
∂η
∂z ∂z
1
= 1 (muJ31 + mvJ32 + w)
G2

(6.110)144

6 The Basic Equations of CReSS

Derivatives of a dependent variable φ with distances (x, y, z) are

m

∂φ
1
→m 1
∂ξ
G2

1
∂φ
→m 1
m
∂η
G2




∂
∂
(Jd φ) +(J31 φ) ,∂ξ∂ζ(6.111)∂∂(Jd φ) +(J32 φ) ,∂η∂ζ(6.112)

∂φ
1 ∂φ
→ 1
.
∂z
G 2 ∂ζ

(6.113)Using these derivatives, the basic equations are rewritten as follows.
Equations of motion






∂
∂u
1
∂
∂u
∂u
∂u
−m 1
Jd p +
J31 pρ̄= −ρ̄ mu
+ mv
+W
∂t
∂ξ
∂η
∂ζ
∂ξ
∂ζ
2
G

 
 
1
1
∂
∂
uw
2
+ρ̄(fη w − fz v) + ρ̄m v v−u− ρ̄+ Turb.u,
∂ξ m
∂η m
a






∂v
∂v
∂
∂v
1
∂
∂v
= −ρ̄ mu
+ mv
+W
−m 1
Jd p +
J32 p
ρ̄
∂t
∂ξ
∂η
∂ζ
∂η
∂ζ
2
G

 
 
1
1
∂
∂
vw
2
+ Turb.v,+ρ̄(fz u − fξ w) − ρ̄m u v
−u
− ρ̄
∂ξ m
∂η m
a


(6.114)(6.115)



∂w
∂w
∂w
∂w
ρ̄
= −ρ̄ mu
+ mv
+W
∂t
∂ξ
∂η
∂ζ

1 ∂p
u2 + v2− 1− ρ̄Buoy.w + ρ̄(fξ v − fη u) + ρ̄+ Turb.w.
aG 2 ∂ζ

(6.116)Equation of the pressure perturbation


∂p
∂p
∂p
∂p
= − mu
+ mv
+W
∂t
∂ξ
∂η
∂ζ
−ρ̄c2s

1
1

G2



m



2

1



+ ρ̄gw
1∂ G2 u∂ G2 v
+
∂ξ m
∂η m



1





1 dθ
∂G 2 w
1 dQ++ ρ̄c2s
−
∂ζ
θ dt
Q dt



(6.117)Equation of the potential temperature perturbation


∂θ 
∂θ 
∂θ 
∂θ 
= −ρ̄ mu
+ mv
+W
ρ̄
∂t
∂ξ
∂η
∂ζ



− ρ̄w

1 ∂ θ̄
+ ρ̄Src.θ + Turb.θ.
1
G 2 ∂ζ

(6.118)

6.3 Basic Equations — Map Projection

145

Equations of mixing ratios of water substances




∂qv
∂qv
∂qv
∂qv
ρ̄
= −ρ̄ mu
+ mv
+W
∂t
∂ξ
∂η
∂ζ


+ ρ̄Src.qv + Turb.qv ,(6.119)+ ρ̄Fall.qx + ρ̄Src.qx + Turb.qx .

(6.120)



∂qx
∂qx
∂qx
∂qx
= −ρ̄ mu
+ mv
+W
ρ̄
∂t
∂ξ
∂η
∂ζ

Equations of number density of water substances










∂Nx
∂ Nx
∂ Nx
∂
= −ρ̄ mu
+ mv
+W
∂t
∂ξ ρ̄
∂η ρ̄
∂ζ
Nx
Nx
Nx
+ ρ̄Fall.
+ Turb.
.
+ρ̄Src.
ρ̄
ρ̄
ρ̄



Nx
ρ̄



(6.121)

The above equations will be rewritten more simply using variables with the asterisk (6.61)〜(6.67) as well as using the following expression.


m2 v

∂
∂ξ





1
∂
−u
m
∂η



1
m



=u

∂m
∂m
−v
∂η
∂ξ

(6.122)

Then, the basic equations are rewritten as follows.
Equations of motion






∂u∗
∂u
∂u
∂u
∂
∂
= − mu∗+ mv ∗+ W∗−m
Jd p +
J31 p
∂t
∂ξ
∂η
∂ζ
∂ξ
∂ζ

1
w
∂m
∂m
+(fη w∗ − fz v∗ ) + v∗ u
− u∗ + G 2 Turb.u,
−v
∂η
∂ξ
a




(6.123)



∂v
∂v
∂v
∂
∂v∗
∂
−m
Jd p +
J32 p
= − mu∗+ mv ∗
+ W∗
∂t
∂ξ
∂η
∂ζ
∂η
∂ζ

1
w
∂m
∂m
+(fz u∗ − fξ w∗ ) − u∗ u− v∗ + G 2 Turb.v,
−v
∂η
∂ξ
a


(6.124)



∂w∗
∂w
∂w
∂w
= − mu∗+ mv ∗
+ W∗
∂t
∂ξ
∂η
∂ζ

1
∂p
u∗ u + v∗ v
− ρ∗ Buoy.w + (fξ v∗ − fη u∗ ) ++ G 2 Turb.w.
−∂ζa(6.125)Equation of the pressure perturbation
1

1∂G 2 p= −G 2∂t−ρ̄c2s



mu


m

∂p
∂p
∂p
+ mv
+W
∂ξ
∂η
∂ζ


2

1

1

∂ G2 u∂ G2 v
+
∂ξ m
∂η m





+ gw∗
1





1
1 dθ
1 dQ
∂G 2 w−++ G 2 ρ̄c2s
∂ζ
θ dt
Q dt



(6.126)

146

6 The Basic Equations of CReSS

Equation of the potential temperature perturbation


∂θ ∗
∂θ 
∂θ 
∂θ 
= − mu∗+ mv ∗
+ W∗
∂t
∂ξ
∂η
∂ζ



− ρ̄w

1
∂ θ̄
+ ρ∗ Src.θ + G 2 Turb.θ
∂ζ(6.127)Equations of mixing ratios of water vapor and water substances


∂qv∗
∂qv
∂qv
∂qv
= − mu∗+ mv ∗
+ W∗
∂t
∂ξ
∂η
∂ζ


∂qx∗
∂qx
∂qx
∂qx
= − mu∗+ mv ∗
+ W∗
∂t
∂ξ
∂η
∂ζ




1

+ ρ∗ Src.qv + G 2 Turb.qv ,(6.128)1+ ρ∗ Fall.qx + ρ∗ Src.qx + G 2 Turb.qx . (6.129)

Equations of number density of solid water substances
1











∂G 2 Nx∂ Nx∂ Nx∂= − mu∗+ mv ∗
+ W∗
∂t
∂ξ
ρ̄
∂η ρ̄
∂ζ
1
N
N
N
x
x
x
+ ρ∗ Fall.
+ G 2 Turb.
+ρ∗ Src.
ρ̄
ρ̄
ρ̄



Nx
ρ̄



(6.130)To suppress computational instability due to the acoustic waves, an artiﬁcial term with respect
to pressure αDiv ∗ is used in calculation. The pressur term p − αDiv ∗ will be rewritten in map
projection as

∗

Div =

1
G

1
2



m

2



∂ u∗
∂ v∗
+
∂ξ m
∂η m



+

∂W ∗
.
∂ζ

(6.131)The equation of the turbulent kinetic energy, which is related with the turbulent term Turb.φwill be shown in Section 7.

6.3.3

Conformal map projections

Most map projections in meteorology are a conformal map projection. The stretching coeﬃcients m, n are independent of direction in the conformal map projection. This means thatm = n. The basic equations will be simpler.
CReSS can use the following three conformal map projections.
• The stereographic projection for middle and high latitudes.
• The Lambert conformal projection for middle latitude.
• The Mercator projection for lower latitudes.

Chapter 7

Subgrid Scale Diﬀusion
The atmosphere is represented by grid point value in a numerical simulation.
However, air motions of smaller than the grid spacing always exist in theactual atmosphere. No matter how small grid spacing, such motions can exist.
They are called the subgrid scale motions, which act as diﬀusion in general.
In addition, they correspond to turbulence and are often called ‘turbulence
diﬀusion’.
Since the subgrid-scale motion exist even if we use very small grid sipaceing,
prognostic equations of the subgrid-scale motion is necessary. For example, we
can separate velocity into the grid-scale motion and deviations. In the equations of averaged variables, the second-degree correlation of deviation appears
as unknown variables which is called Reynolds stress. Then, their prognostic equations are necessary. However, the third-degree correlation appears in
the prognostic equations. As a result, the system of equations is not closed.
Such problem is due to the non-linearity of turbulent ﬂow. Kellar and Friedmann (1924) got ﬁrst recognition on this problem, which is called ‘closure
problem’.
To solve the closure problem, the unknown variables should be expressed by
known variables. In this chapter, we discuss the formulation of subgrid-scale
motion using the following two manner.
• Modeling of unknown variables in the prognostic equations by dealing
with two-degree correlation• Modeling of the prognostic equations on the two-degree correlation represented by scalar, which can indicate averaged velocity, turbulence kinetic energy and turbulent ﬂow of dissipation ratio through a concept
of eddy viscosity

148

7 Subgrid Scale Diﬀusion

7.1

Parameteriaation of Turbulence Transport

There are various scale motion in atomosphere. The motion, which can be expressed withgrid of numerical simulation, is called grid-scale motion, mean motion, subgrid scale motion or
eddy motion.
To separate these motions, we assume that the ﬁeld variables A, velocity, temperature, mixing
ratio and so on, can be separated into mean ﬁeld and deviation components.
A = Ā + A(7.1)

where the corresponding means are indicated by¯and the deviation components by  .
There are various ways when we average variables, but we do not show them in detail. For
your information, average of deviations and average of products of two variabeles are given byA = 0

(7.2)

AB = ĀB̄ + A B 

(7.3)

In other words, if you average products, you do not always get the variable which correspond withproducts of each averaged, and the second term of above equation appears. We apply them to xcomponents in equations of motion. Here, to simplify we discuss incompressible ﬂuid ρ = const,
and we can obtain a variable consisting of mean and deviation components.
∂ ū
∂ ū
∂ ū
∂ ū
1 ∂ p̄
+ ū
+ v̄
+ w̄
− f v̄ = −
∂t
∂x
∂y
∂z
ρ ∂x


∂
∂
1 ∂
ρu u +ρu v +ρu w + ν∇2 ū
−
ρ ∂x
∂y
∂z

(7.4)

The terms −ρu u , −ρu v , −ρu w in this equation represent stress by turbulent ﬂow, which iscalled eddy stress or Reynolds stress. We can regard them as transport of momentum, so stressare made by trasport of momentum by eddy.
Similarly, the equations on potential temperature and mixing ratio are given by
∂ θ̄
∂ θ̄
∂ θ̄
∂ θ̄
∂
∂  
∂  
+ ū
+ v̄
+ w̄
= − u θ  −
v θ −
w θ
∂t
∂x
∂y
∂z
∂x
∂y
∂z

(7.5)

Even if you use grid, the variables indicated by prime cannot be expressed. If you use them innumerical simulation, you have to represent them with the variables indicated by overbar (mean7.2 Eddy Viscosity Model

149

components). If you not, you cannot get eﬀects of transport by eddy motion which has smallerscale than grid distance. The problem of turbulent parametarization is how we have to express
such deviation by eddy, which is indicated by prime, by using limited grid. In addition, you needto pay attention to a diﬀerence between the deﬁnition of these variables indicated by ¯ and thedeﬁnition of variables indicated by¯shown at the second chapter.

7.2

Eddy Viscosity Model

7.2.1

Formulation of Diﬀusion Term
1

In this section, we formulate diﬀusion term (the term of turbulence mixing) G 2 Turb.φ. The
diﬀusion term appears in Equations of motion (6.123)〜(6.125), Equation of potential temperature(6.127), Equations of mixing ratio of water vapor and hydrometeor (6.128) and Equations of
number concentration per unit volume (6.130), which shown in the basic dynamical equations interrain-following coordinates in the section 6.3.2.
The diﬀusion term in the equation of motions is expressed using stress tensor τij as follows:

1





∂τ11 ∂τ12 ∂τ13
+
+
∂x
∂y
∂z
∂
∂
∂
(Jd τ11 ) +(Jd τ12 ) +(τ13 + J31 τ11 + J32 τ12 )
=
∂ξ
∂η
∂ζ
1

G 2 Turb.u = G 2

1





∂τ21 ∂τ22 ∂τ23
+
+
∂x
∂y
∂z
∂
∂
∂
=
(Jd τ21 ) +(Jd τ22 ) +(τ23 + J31 τ21 + J32 τ22 )
∂ξ
∂η
∂ζ
1

(7.6)

G 2 Turb.v = G 2

1





∂τ31 ∂τ32 ∂τ33
+
+
∂x
∂y
∂z
∂
∂
∂
=
(Jd τ31 ) +(Jd τ32 ) +(τ33 + J31 τ31 + J32 τ32 )
∂ξ
∂η
∂ζ
1

(7.7)

G 2 Turb.w = G 2(7.8)

where stress tensor τij consists of shear stress and Reynolds stress. Reynolds stress consists of
ﬂuctuation from averaged variables, so we need modeling in any way to the form with averaged
variables. Reynolds stress can express in the form of gradient diﬀusion using viscosity coeﬃcientfrom an analogy of shear stress.
τ11 = ρ̄ντ h2
S11 − Div
3



(7.9)

τ12 = ρ̄ντ hS12

(7.10)

τ13 = ρ̄ντ v S13(7.11)

τ21 = ρ̄ντ hS12

(7.12)

150

7 Subgrid Scale Diﬀusion2
S22 − Div3

τ22 = ρ̄ντ h

(7.13)

τ23 = ρ̄ντ v S23(7.14)

τ31 = ρ̄ντ h S13(7.15)

τ32 = ρ̄ντ h S232τ33 = ρ̄ντ v S33 − Div3

(7.16)(7.17)

where ντ handντ v are horizontal and vertical eddy viscosity coeﬃcients regarding kinetic momentum, respectively. The molecular viscosity coeﬃcient of shear stress is so small to eddy viscosity
coeﬃcient that it can be neglected. Sij is deformation rate tensor. In curvilinear coordinate
system, it is given by∂∂(Jd u) +(J31 u)∂ξ∂ζ(7.18)∂∂(Jd v) +(J32 v)∂η∂ζ

(7.19)

S11 = 2

∂u
2
= 1
∂x
G2

S22 = 2

∂v
2
= 1
∂y
G2

S33 = 2

2 ∂w
∂w
= 1
∂z
G 2 ∂ζ



∂u ∂v
1
S12 =
+
= 1
∂y
∂x
G2

(7.20)
∂∂∂(Jd u) +(Jd v) +(J32 u + J31 v)
∂η
∂ξ
∂ζ



(7.21)∂∂(Jd w) +(u + J31 w)∂ξ∂ζ

(7.22)∂∂(Jd w) +(v + J32 w)∂η∂ζ

(7.23)∂  1 ∂  1 ∂  1 G2 u +
G2 v +
G2W
∂ξ
∂η
∂ζ

(7.24)

1
∂u ∂w
+
= 1
S13 =
∂z
∂x
G2
∂v ∂w
1
S23 =
+
= 1
∂z
∂y
G2



and Div is divergence.

Div =

1
G

1
2



The diﬀusion terms of potential temperature, mixing ration of hydrometeor and water vapor
and the number concentration per unit volume are formalized by using φ as follows:∂Hφ1 ∂Hφ2 ∂Hφ3
G Turb.φ = G
+
+
∂x
∂y
∂z
∂
∂
∂
(Jd Hφ1 ) +(Jd Hφ2 ) +(Hφ3 + J31 Hφ1 + J32 Hφ2 )
=
∂ξ
∂η
∂ζ
1
2

1
2

(7.25)

7.2 Eddy Viscosity Model

151

where Hφ1 , Hφ2 and Hφ3 are the molecular diﬀusions of the corresponding scalar φ and turbulent
ﬂuxes in the x, y and z directions, respectively. They have forms of the gradient diﬀusion similar
to velocity


Hφ1 = ρ̄νHh∂φ1= ρ̄νHh 1∂xG2

Hφ2 = ρ̄νHh∂φ1= ρ̄νHh 1∂yG2

Hφ3 = ρ̄νHv∂φ1 ∂φ= ρ̄νHv 1∂z
G 2 ∂ζ

∂∂(Jd φ) +(J31 φ)∂ξ∂ζ(7.26)∂∂(Jd φ) +(J32 φ)∂η∂ζ(7.27)(7.28)

where νHh andνHv are horizontal and vertical eddy viscosity coeﬃcients about scalar, respectively.
The molecular diﬀusion coeﬃcient is so small that it is neglected.
We have the modeling of Reynolds to the form with eddy viscosity coeﬃcient. The manner is called eddy viscosity model, with which we can appreciate the eddy viscosity coeﬃcientsντ handντ v andtheessydif f usioncoef f icientsνHhandνHv appeard in avobe equations.
In following section, two eddy viscosity model are explained, which are actually used in CReSS.
• One order closure of Smagorinsky• One and a half order closure with turbulence kinetic energy

7.2.2

One order closure of Smagorinsky

Smagorinsky (1963) and Lilly (1962) give the eddy viscosity coeﬃcient in the case it is isotropic
vertically and horizontally, using ντ h = ντ v = ντ
⎧


2
⎪
N
⎪
2
⎪
,
⎨ (CS Δ) Def 2 −

ντ =

⎪
⎪
⎪
⎩ 0,

Pr

ντ > 0(7.29)ντ ≤ 0where CS is smagorinsky constant, CS = 0.21 by Deardorﬀ (1972a). Δ is averaged grid intervalof numerical simulation.
1

Δ = (ΔxΔyΔz) 3(7.30)

Def , which is measurement of transformation, can be obtained
Def 2 =


1 2
2
2
2
2
2
2
S11 + S22
+ S33
+ S12
+ S13
+ S23
− Div22
3(7.31)

152

7 Subgrid Scale Diﬀusion

and

N2 =

⎧
g ∂ ln θ
⎪
⎪
,
⎪
1
⎪
⎪
∂ζ
2
⎪
G
⎪
⎪
⎪
⎪
⎨
⎡

⎤
Lv qvsw 

⎪
1+
⎪
⎪
g ⎢
∂ ln θ
∂qw ⎥Lv ∂qvsw
Rd T
⎢
⎥
⎪
⎪
−
+
⎢
⎥,
⎪
1
2
⎪
⎪
∂ζ
Cp T ∂ζ∂ζ ⎦G 2 ⎣ 1 + Lv qvsw⎪⎪⎩

Cp Rv T 2

qv < qvsw(7.32)
qv ≥ qvsw

7.2 Eddy Viscosity Model

153

where N is a constant Brunt-Väisälä frequency, and

Pr =

ντ
,
νH

νHh = νHv = νH

(7.33)

where P r is Turbulent Prandtl number. So we can obtain the eddy viscosity coeﬃcient regarding
scalar φ. g is gravity acceleration, T is temperature, Rd andRv are gas constants for dry air andwet air, respectively. Cp is speciﬁc heat at constant pressure and qw is a sum of mixing ratio of
molecular weight for water vapor, cloud liquid water and rainwater. With the equation of Tetens,
mixing ratio of water saturation qvsw is given by

qvsw = 

610.78T − 273.16
exp 17.269
p
T − 35.86

(7.34)and latent heat for water evaporation Lv is given by6

Lv = 2.50078 × 10


−4
273.16 (0.167+3.67×10 T )T(7.35)

where  is the ratio of molecular weight of water vapor and of dry air.
Next, in the case it is anisotropic vertically and horizontally, the eddy viscosity coeﬃcients ofeach direction are represented as
⎧


2
⎪
N
⎪
2
⎪
,
⎨ (CS Δh ) Def 2 −ντ h =

Pr

⎪
⎪
⎪
⎩ 0,

(7.36)
ντ h ≤ 0

⎧


2
⎪
N
⎪
2
⎪
,
⎨ (CS Δv ) Def 2 −ντ v =

ντ h > 0

Pr

⎪
⎪
⎪
⎩ 0,

ντ v > 0(7.37)
ντ v ≤ 0

where Δh and Δv are given by1

Δh = (ΔxΔy) 2(7.38)

Δv = Δz(7.39)

154

7 Subgrid Scale Diﬀusion

Furthermore,

Pr =

ντ v
ντ h
=
νHh
νHv

(7.40)

Thus, we obtain the eddy viscosity coeﬃcients νHh , νHv of each direction regarding scalar φ, using
the Turbulent Prandtl number P r.

7.2.3

One and a half order closure with turbulence kinetic energy

With regard to one and a half order closure, we use prognostic equations on turbulence kinetic
energy. We mark deviation from averaged ﬂow each velosity component with  , so this turbulencekinetic energy is represented asE=
1  2u + v2 + w2
2

(7.41)

and, the prognostic equations are given by








∂ρ∗ E
∂E
∂E
∂E
1
Ce 3
2
+ Buoy.E + ρ∗
= − u∗+ v∗+ W∗
νE Def 2 − EDiv − ρ∗ E 2
∂t
∂ξ
∂η
∂ζ
2
3
lh

∂
∂
∂
+
(7.42)(Jd HE1 ) +(Jd HE2 ) +(HE3 + J31 HE1 + J32 HE2 )
∂ξ
∂η
∂ζ

Buoy.E which appeard in equations (7.42), is given as follows with the converting term ofpotential energy and kinetic energy.
⎧∂θe ∂qall⎪∗⎪⎪−−gρ νHv A
,
⎪
⎪
∂ζ
∂ζ
⎨

Buoy.E =



⎪
⎪
⎪
∂qv
1 ∂θ
⎪
∗
⎪
+ 0.61,⎩ −gρ νHv

θ ∂ζ

∂ζ

qv ≥ qvsw or qc + qi > 0(7.43)
qv < qvsw or qc + qi = 0

where, A is expressed as
1.61Lv qv
1
Rd T
A=
θ
L2v qv
1+
Cp Rd T 2
1+

(7.44)

where, g is gravity acceleration, T is temperature,  is the ratio of molecular weight for water
vapor and dry air, qall is a sum of mixing ratio of molecular weight for water vapor, cloud liquid7.2 Eddy Viscosity Model

155

water and cloud ice, and θe is equivalent potential temperature. Cp andRd are gas constants fordry air and wet air, respectively. Lv is latent heat for water evaporation.
Next, the third term of right hand Def, Div is shown at the section 7.2.2. The coeﬃcient Ceof the forth term, the dissipating term is represented as

Ce =

3.9,
0.93,

thelowestlayer
theotherlayer(7.45)In addition, the last term of right hand, ﬂux of turbulence kinetic energy is given by


∂E
1
∂
∂
= ρ̄νE 1(Jd E) +(J31 E)HE1 = ρ̄νE∂x∂ζG 2  ∂ξ∂E
1∂∂HE2 = ρ̄νE
= ρ̄νE 1(Jd E) +(J32 E)∂y∂ζ
G 2 ∂η
HE3 = ρ̄νE∂E
1 ∂E= ρ̄νE 1∂z
G 2 ∂ζ

(7.46)(7.47)(7.48)

where νE is the eddy viscosity coeﬃcient for turbulence kinetic energy.
Thus, the eddy viscosity coeﬃcients ντ h, ντ v are represented as a function of turbulent kinetic
energy E,
1ντ h = 0.1E 2 lh

(7.49)

12

ντ v = 0.1E lv(7.50)

where lh andlv are horisontal and vertical mixing length scales. In the case where the diﬀerenceis small between horisontal and vertical grid interval,l = lh = lv =Δs,min (Δs, ls ) ,unstableorneutral
stable(7.51)

where $Deltas and ls are expressed as
1

Δs = Δsh = Δsv = (ΔxΔyΔz) 3
1

ls = 0.76E 2

g ∂ θ̄
θ̄ ∂z(7.52)− 12(7.53)

156

7 Subgrid Scale Diﬀusion

On the other hand, in the case where the diﬀerence is large between horisontal and vertical gridinterval,
lh = Δshlv =(7.54)Δsv ,min (Δsv , ls ) ,unstableorneutral
stable(7.55)

where Δsh and Δsv are expressed as
1

Δsh = (ΔxΔy) 2(7.56)

Δsv = Δz

(7.57)

Finally, the eddy viscosity coeﬃcients ντ handντ v are represented as a function of turbulencekinetic energy E,


1



1



ντ h = max 0.1E 2 lh , αΔs2hντ v = max 0.1E 2 lv , αΔs2v(7.58)(7.59)

where α is smaller number like α = 10−6 . The eddy viscosity coeﬃcient νHv , νHh for scalar φ andνE for the turbulence kinetic energy E can be obtained as follows:
In the case of the small diﬀerence between horisontal and vertical grid interval,
ντ v
1
ντ h
( = Pr )
=
=
νHh
νHv
1 + 2l/ Δs(7.60)

νE = 2ντ h = 2ντ v(7.61)In the case of the large diﬀerence between horisontal and vertical grid interval,
ντ h
1
1
=
=
νHh
1 + 2lh / Δsh3(7.62)

ντ v
1=( = Pr )νHv
1 + 2lv / Δsv(7.63)

νE = 2ντ h(7.64)

Chapter 8

Physical Processes of Cloud and Precipitation
Cloud physics is largely divided into cloud microphysics and cloud dynamics.
Because they are closely related each other, knowledge of cloud physics isnecessary for understanding cloud dynamics.
Ascending a humid air parcel in atmosphere, clouds generate in association
with a conversion from water vapor to cloud and precipitation particles andprecipitation is induced. We generally classify the formation processes ofprecipitation into “warm rain” and “cold rain (ice-phase rain).” The “warmrain” is rainfall which has no ice-phase process, and clouds form below thealtitude of 0◦ C. Such clouds is called “warm clouds.” The “cold rain” israinfall of which ice-phase processes considerably contribute for the growth of
precipitation particles. Such clouds is called “cold clouds.” The clouds form
not only below the altitude of 0◦ C but also above the altitude. The clouds
have both a liquid phase and an ice-phase. There are the following categories
in modeling of these clouds.
• Parametarization of “warm rain” with a bulk method• Parametarization containing an ice-phase with a bulk method• A method predicting time development of size distribution with dividing
the sizes of liquid-phase particles into several bins (a bin method)• A bin method as same as the above but containing an ice-phase• A bin method containing categorized aerosols• A method of which a bin method is used for a liquid phase and a bulk
method for an ice-phase (A hybrid type)
Bulk method parametarizations of “warm rain” and “cold rain” are used in
the CReSS. Hereafter, modelings of these two methods are described.

158

8 Physical Processes of Cloud and Precipitation

8.1

Parametarization of “Warm Rain” with a Bulk Method

8.1.1

Equations for “Warm Rain” Process of Cloud and Precipitation

The parametarization of “warm rain” with a bulk method has the following three categories ofwater substance1 .
Sign

Meaning

Contentsqv

Mixing ratio of watervapor
Mixing ratio of cloudwater
Mixing ratio of rain
water

Water which exists in the atmosphere in the state of gas

qc

qr

Its fall velocity can be regarded as zero. It corresponds to cloud drops
in the actual atmosphere. The cloud drop is ordinary a small particle ofwhich its diameter is less than 100 μm.
It corresponds to raindrops, which have a meaningful fall velocity, in theactual atmosphere.

Modeling variables involved in the “warm rain” process are three variables for water substance[kg kg−1 ] and potential temperature (temperature) [K]. Using equations shown in section 6.1(wecan explain by using the equations in which orography is not considered), the equations for the“warm rain” process are simply written as follows.

∂ ρ̄θ
∂ θ̄
ρ̄Lv
= Adv.θ + Turb.θ − ρ̄w+(CNvc − EVcv − EVrv )∂t∂z Cp Π(8.1)

∂ ρ̄qv= Adv.qv + Turb.qv − ρ̄ (CNvc − EVcv − EVrv )
∂t

(8.2)

∂ ρ̄qc= Adv.qc + Turb.qc + ρ̄ (CNvc − EVcv − CNcr − CLcr )
∂t

(8.3)

∂ ρ̄qr∂= Adv.qr + Turb.qr + ρ̄ (CNcr + CLcr − EVrv ) +(ρ̄Ur qr )
∂t
∂z

(8.4)

where Adv.φ and Turb.φ represent the terms of advection and subgrid-scale turbulance. Lv , Cpand Π are latent heat of evaporation of water [J kg−1 ], speciﬁc heat at constant pressure in a dry
air [J K kg−1 ] and the Exner function, respectively. The last term of the right side of Eq. (8.4)
represents a ﬂux divegence of qr in association with falling of rainwater. Microphisical processes
considered here are as follows.
Sign

Contents

CNvc

Conversion from water vapor to cloud water by condensation (condensation)

1

Rain water which has a diameter of 0.1-0.5 mm is sometimes called drizzle. Here, drizzle is contained in a
category of rain.

8.1 Parametarization of “Warm Rain” with a Bulk Method

159

EVcv

Conversion from cloud water to water vapor by evaporation (evaporation)

EVrv
CNcr

Conversion from rain water to water vapor by evaporation (evaporation)
Conversion from cloud water to rain water by coalescence growth. It corresponds togrowing from a cloud droplet to the size of a raindrop by coalescence or diﬀusion ofwater vapor (autoconversion).
Conversion from cloud water to rain water by collision. A process of which a large
waterdrop captures a small waterdrop (collection).

CLcr

All of these amounts are deﬁned as a positive value. A process of condensation from water
vapor to rain water is neglected. Calculations of those processes are shown in the next section.

8.1.2

Microphysical Processes

Conversion between Water Vapor and Cloud Water: −CNvc + EVcv
As the same as Klemp and Wilhelmson (1978), we use the method of the moist saturation adjustment by Soong and Ogura (1973). This method will be explained in Section 8.2.5.
Saturation Mixing Ratio (Tetens Equation): qvsw
Saturation mixing ratio over water qvsw is represented by using the Tetens equation.


qvsw = 

610.78Πθ − 273.16
exp 17.269
pΠθ − 35.86



(8.5)

Here,  is the ratio with the molecular weight of water vapor and the molecular weight of dry air.
Conversion from Cloud Water to the Rain Water: CNcr , CLcr
The conversion from cloud water to rain water with coalescence growth (CNcr ) and conversion
from cloud water to rain water (CLcr ) is calculated by using the Kessler (1969)’s parameterization.
CNcr = k1 (qc − a)(8.6)

CLcr = k2 qc qr0.875(8.7)

k1 = 0.001 [s−1 ]

(8.8)

where

a = 0.001 [kg kg
k2 = 2.2

[s

−1

].

−1

]

(8.9)
(8.10)

160

8 Physical Processes of Cloud and Precipitation

Evaporation of Rain Water: EVrv
In the same way as Ogura and Takahashi (1971) and Klemp and Wilhelmson (1978), evaporationof rain water is represented as a following.

EVrv =

1 (1 − qv /qvsw ) C (ρ̄qr )0.525
ρ̄ 5.4 × 105 + 2.55 × 106 /(pqvsw )(8.11)

where C is a ventilation factor given by
C = 1.6 + 124.9 (ρ̄qr )0.2046 .

(8.12)

Fall Velocity of Rain: Ur
Adding change of density to the equation of Soong and Ogura (1973), fall velocity Ur of the lastterm in the right side of the equation (8.4) is given by

Ur = 36.34 ( ρ̄qr )0.1346



ρ0
ρ̄



.

(8.13)

where ρ0 is density [kg m−3 ] at the surface in the basic state and the unit of Ur is [m s−1 ].
Precipitation at the surface is calculated by using this fall velocity. Note that a diﬀerence in z∗
cordinate system (ζ cordinate system) is multiplied by a matrix, as shown in equations (6.58).

8.2

Parameterization of the bulk cold rain

8.2.1

Parameterization of the cold rain

We formulate the parameterization of the cloud and precipitation processes containing ice-phasewith a bulk method which are used in a cloud model. Here, conversions of water substanceand changes of temperature and mixing ratio of water vapor by the conversions are considered.
When we use a bulk method, the water in the atmosphere is divided into some categories (e.g.
rain, snow and graupel) The categories are formulized with the typical variables (usually, mixing
ratio or mixing ratio and number concentration), and their time developments are calculated.
Therefore, when we use a bulk method, the deﬁnition of each variable must be clear. The way of
categorization and variables are diﬀerent in each model.
The following physical processes are considered in formulization of parameterization containing
an ice-phase. In the formulization, the type of a particle must be considered.
• Primary and secondary nucleation of ice crystals• Growth and decline of the particle with water vapor diﬀusion.
• Growth with collision between particles

8.2 Parameterization of the bulk cold rain

• Breakup of a particle (breakup of a raindrop)• Conversion to another category (Conversion from the cloud water to rain water, and fromcloud water to snow, and from snow to graupel. etc.)• Freezing and melting• Shedding of un-freezing water• Gravity falling
Although the deﬁnitions of the cloud physical valiables and its treatment are diﬀerent in eachmodel, we show two types of models containing an ice-phase in the following.
• A model which calculates only the time development equation of the mixing ratio of eachcategory• A model which calculates the graupel, cloud ice, snow, and their number concentrations
adding the above equation
Hereafter, we summarize the formulizations by Murakami (1999), Murakami et al. (1994) and
Murakami (1990).
We consider the six categories for the physical processes of cloud and precipitation: water
vapor, cloud water, rain water, cloud ice, snow and graupel. In addition to the six categories,
there is a model which makes fog water and hail another categories.
Sign

Meaning

ContentsθPotential
tempertureθ = θ̄ + θ 

qv

Mixing ratio of water
vapor

Water which exists in the atmosphere in the state of gas.

qc

Mixing ratio of cloud
water

Liquid water of which a particle diameter is small and fall velocity is
negligible. It can move with atmospheric motion.

qr

Mixing ratio of rain
water

Usually, the particle of a liquid water with a diameter more than 100 μm is
called rain. It is expressed as ”rain water” in a model. Fall velocity
depending on the size of a particle is meaningful. Although it moves inassociation with the atmospheric motion in horizontal, it falls and drops
out of an air mass in vertical.

qi

Mixing ratio of cloudice

A minute crystal of ice which is called ”ice crystal” in cloud physics.
Usually, a diameter is less than 100 μm . Fall velocity is so small that itis negligible.

161

162

8 Physical Processes of Cloud and Precipitation

qs

Mixing ratio of snow

In a model, it means a solid precipitation particle with density of about
0.1 g cm−3 and fall velocity of about 1 m s−1 . It corresponds to a snow
crystal or snow ﬂakes, etc.

qg

Mixing ratio of graupel

In a model, it means a solid precipitation particle with density of about
0.4 g cm−3 and fall velocity of about 1〜4 m s−1 . It corresponds to asnow crystal with cloud droplets, snow ﬂakes with cloud droplets, graupel,
etc.

qh

Mixing ratio of hail

In a model, it means a solid precipitation particle with density of about
0.9 g cm−3 and fall velocity of about 10 m s−1 fall speed.In an actual
cloud particle, frozen rain, hail, etc. correspond. hail is contained ingraupel in CReSS.

Ni

numberconcentration ofcloud ice

Expressing cloud ice in a model, number concentration is sometimes regarded as a variable.

Nsnumberconcentration ofsnow

The same in case of snow.

Ngnumberconcentration ofgraupelThe same in case of graupel.

Here, the units of potential temperature, mixing ratio and number concentration are [K],[kg kg−1 ] and [m−3 ], respectively. We often use [g kg−1 ] as the unit of mixing ratio.

8.2.2

The Equation System for Cloud and Precipitation Processes

The time development equations of potential temperature, water vapor and particles of cloud andprecipitation are used here. Considering number concentrations of particles, the time development
equations of number concentration of cloud ice, snow, and graupel are needed. These describe
brieﬂy the equation system shown in Section 6.1.
The time development equation of a potential temperature and a mixing ratio of water isexpressed as

∂ ρ̄θ
∂ θ̄
= Adv.θ + Turb.θ − ρ̄w+ ρ̄ (Src.θV + Src.θS + Src.θF )∂t∂z(8.14)

8.2 Parameterization of the bulk cold rain163

∂ ρ̄qv= Adv.qv + Turb.qv + ρ̄Src.qv∂t(8.15)∂ ρ̄qc= Adv.qc + Turb.qc + ρ̄Src.qc + ρ̄Fall.qc∂t(8.16)∂ ρ̄qr= Adv.qr + Turb.qr + ρ̄Src.qr + ρ̄Fall.qr∂t(8.17)∂ ρ̄qi= Adv.qi + Turb.qi + ρ̄Src.qi + ρ̄Fall.qi∂t(8.18)∂ ρ̄qs= Adv.qs + Turb.qs + ρ̄Src.qs + ρ̄Fall.qs∂t(8.19)∂ ρ̄qg= Adv.qg + Turb.qg + ρ̄Src.qg + ρ̄Fall.qg∂t(8.20)

where characters of v, c, r, i, s, and g, which are attached at the bottom, express cloud water,rain water, cloud ice, snow, and graupel, respectively. Hereafter, we sometimes use x or y as
representative characters.
The meaning of each term is shown below.
Adv.φTurb.φSrc.θV
Src.θS
Src.θF
Src.qxFall.qx

Advection term of potential temperature or mixing ratio of water substanceDiﬀusion term of potential temperature or mixing ratio of water substance by a subgrid-scale turbulance
Source term of potential temperature in association with condensationand evaporation
Source term of potential temperature in association with sublimation
Source term of potential temperature in association with freezeing andmelting
Source term of the mixing ratio of water
Term of falling water substance (precipitation)

The time development equations of number concentrations of cloud ice, snow, and graupel are
∂Ni
Ni
Ni
Ni
Ni
= Adv.
+ Turb.
+ ρ̄Src.
+ ρ̄Fall.
∂t
ρ̄
ρ̄
ρ̄
ρ̄

(8.21)

Ns
Ns
Ns
Ns
∂Ns
= Adv.
+ Turb.
+ ρ̄Src.
+ ρ̄Fall.
∂t
ρ̄
ρ̄
ρ̄
ρ̄

(8.22)

∂Ng
Ng
Ng
Ng
Ng
= Adv.
+ Turb.
+ ρ̄Src.
+ ρ̄Fall.
∂t
ρ̄
ρ̄
ρ̄
ρ̄

(8.23). where characters of i, s, g, which are attached at the bottom, express water vapor, cloud ice,
snow, and graupel, respectively. Hereafter, we sometimes use x or y as representative characters.

164

8 Physical Processes of Cloud and Precipitation

The meaning of each term is shown below.
Adv. Nx / ρ̄Turb. Nx / ρ̄Src. Nx / ρ̄Fall. Nx / ρ̄

Advection term of number concentration of water in the solid state
Diﬀusion term of number concentration of water in the solid state by a
subgrid-scale turbulence
Source term of the number concentration of water in the solid state
Term of change of number concentration of water in the solid state inassociation with precipitation

Source terms of those equations are as follows.
Source term of the equation (8.14) of potential temperature θ: Src.θV +Src.θS +Src.θF
Src.θV =

Lv
V Dvr
Cp Π

(8.24)

Src.θS =Ls(N U Avi + V Dvi + V Dvs + V Dvg )Cp Π(8.25)

Src.θF =Lf(N U Fci + N U Cci + N U Hci + CLcs + CLcg + CLri + CLrs + CLrg
Cp Π
−M Lic − M Lsr − M Lgr + F Rrg − SHsr − SHgr )(8.26)

Source term of the equation for mixing ratio of water vapor qv (8.15):
Src.qv = −N U Avi − V Dvr − V Dvi − V Dvs − V Dvg(8.27)

Source term of the equation for mixing ratio of cloud water qc (8.16):
Src.qc = −N U Fci − N U Cci − N U Hci − CLcr − CLcs − CLcg − CNcr + M Lic (8.28)
Source term of the equation for mixing ratio of rain water qr (8.17):
Src.qr = V Dvr + CLcr − CLri − CLrs − CLrg + CNcr
+M Lsr + M Lgr − F Rrg + SHsr + SHgr(8.29)

Source term of the equation for mixing ratio of cloud ice qi (8.18):
Src.qi = N U Avi + N U Fci + N U Cci + N U Hci+V Dvi − CLir − CLis − CLig − CNis − M Lic + SPsi + SPgi

(8.30)

Source term of the equation for mixing ratio of snow qs (8.19):
Src.qs = −SPsi + V Dvs + CLcs + CLrs αrs + CLis − CLsr (1 − αrs ) − CLsg
+CNis − CNsg − M Lsr − SHsr

(8.31)

Source term of the equation for mixing ratio of graupel qg (8.20):
Src.qg = −SPgi + V Dvg + P Gg + CLri + CLir + (CLrs + CLsr ) (1 − αrs )+CNsg − M Lgr + F Rrg − SHgr(8.32)

8.2 Parameterization of the bulk cold rain

165

Source term of the equation for number concentration of cloud ice
Src.

Nc
1
Ni
N
N U Avi +(N U Fci + N U Cci + N U Hci ) + SPsiN + SPgi
=
ρ̄
mi0
ρ̄qc
Ni
1
+
(V Dvi − CLir − CLis − CLig − M Lic ) − AGN
CNis
i −
ρ̄qi
ms0

Source term of the equation for number concentration of snow
Src.

Ni
(8.21):
ρ̄

(8.33)

Ns(8.22):
ρ̄

Ns
Ns
1
N
N
N
=
(V Dvs − M Lsr ) − CLN
CNis − CNsg
(8.34)sr (1 − αrs ) − CLsg − AGs +
ρ̄
ρ̄qs
ms0

Source term of the equation for number concentration of graupel
Src.

Ng(8.23):
ρ̄

Ng
Ng
N
N
N
(V Dvg − M Lgr ) + CLN=
ri + CLrs (1 − αrs ) + CNsg + F Rrg
ρ̄ρ̄qg(8.35)

were Lv , Ls, and Lf are latent heat with evaporation, sublimation, and melting [J kg−1 ],respectively. Cp is speciﬁc heat at the constant pressure [J K kg−1 ] in dry air. Π is the Exner
function. mi0 and ms0 are the minimum masses of cloud ice and snow [kg], respectively. The nexttable shows the meaning of each term in the above equations. Figure 8.1 shows the correlation
between categories. In Section 8.2.4 conversion terms, which consist of these source terms, areformulated.
Sign

Contents

N U Avi
N U FciN U Cci

Deposition or sorption nucleation
Condensation-freezing nucleation
Contact-freezing nucleation

N U HciSP
VD
CL

Homogeneous-freezing nucleation
Secondary nucleation of ice crystals
Vapor deposition, evaporation and sublimation
Collision collection

PG
AG
CN

Growth of graupel with collisional collection （graupel produciton）
Aggregation
Conversion from a certain category to the other categories （conversion）

ML
FR
SH

MeltingFreezing
Shedding of liquid water

SP N
CLN
AGN

Secondary nucleation of ice crystals for number concentration
Collisional collection for number concentration （collection）
Aggregation for number concentration （aggregation）

166

8 Physical Processes of Cloud and Precipitation-VD vrVD vg

water vapor (q v)
NUA vi
VD vc

VD vi

NUF ci ,NUC ci ,NUH ci
ML ic
CL is ,CN is

VD vscloud water (q c)

CL cs

SP si ,SP N si

cloud ice (q i,N i)
AG N iCL crsnow (q s,N s)

CL rs
AG N sCL sr ,CL sg ,CN sg ,CN Nsg

CL ir ,CL ig

ML sr ,SH sr

SP gi ,SP N gi

CN cr

CL cg

Fall. q s , Fall. (N s /ρ )ML gr ,SH gr

rain water (q r)

CL ri ,CL N ri ,CL rs ,CL N rs ,CL rg ,FR rg ,FR N rg

Fall. q r

graupel (q g ,N g )Fall. q g , Fall.( N g/ρ )Figure 8.1. Correlation of cloud microphysical processes in a bulk method

CN N

Conversion from a certain category to the other categories for number concentration （conversion）

F RN
αrs

Freezing for number concentration （freezing）
1 − αrs : Production rate of graupel with the collision of raindrops and snow

There is no exchange term V Dvc between water vapor and an cloud water in (8.24), (8.27), and(8.28). The exchange term is calculated with the moist saturation adjustment method. Section
8.2.5 describes about the calculation.

8.2.3

Expression of Particles of Cloud and Precipitation

Particle-size Distribution
When we use a bulk method, we need to give particle-size distribution with a certain suitablefunction because only mixing ratio, or mixing ratio and number concentration can be directly
calculated. A particle-size distribution is needed for the calculations of averaged mass and averaged fall velocity.
The particle-size distribution in the bulk method is often given with the exponential function.
This is given bynx (Dx ) = nx0 exp(−λx Dx )& '( )[m−4 ]&'()[m−4 ]& '( )[m−1 ](8.36)

8.2 Parameterization of the bulk cold rain

167

using the parameters of an inclination in an exponential function for particle size distribution
λx and y-section density nx0 . The parameters are based on Marshall and Palmer (1948). This
distribution is sometimes called the Marshall-Palmer distribution. The distribution is sometimes
expressed by using a gamma function2
A gamma function distribution3 expressing particle size distribution of a category is

1
fx (Dx ) =& '( )
Γ(νx )

[m−1 ]Dx
Dnx

νx −1



1
Dx
exp −
Dnx
Dnx



(8.43)

where Dx is the diameter of particle [m] and Γ (νx ) is a standardization (Integral from 0 to
∞ is made to be set to 1.) constant, νx is the form parameter of a gamma function and Dnx
is a characteristic diameter. Number concentrations of particles of cloud and precipitation are
expressed as
2

A gamma function is one of the special functions which extended Factorial n! even to the complex number.
Using this function, we can mechanically solve a certain kind of deﬁnite integral. Especially, we often integratedusing the exponential function distribution (8.36) in cloud physics. In such a case, deﬁnite integral can be solved
mechanically. Although there are some deﬁnitions for the gamma function Γ (x), it is deﬁned with integration as
follows.

* ∞Γ (x) =

exp (−t) tx−1 dt(8.37)0Γ (x + 1) = xΓ (x)(8.38)

Γ (1) = 1

(8.39)

where especially x is the positive integer NNN, it is set toΓ (n + 1) = n (n − 1) (n − 2) · · · 2 · 1 · Γ (1) = n!

(8.40)and is

 
Γ

1
2

=

√
π.

(8.41)For example, if this is used instead of carrying out integration by parts 3 times, in the deﬁnite integral of cloudphysics, the following is mechanically calculable like

* ∞
0

3

Murakami (1999)

Dx3 exp (−λx Dx ) dDx =16Γ (4) = 4λ4xλx(8.42)

168

8 Physical Processes of Cloud and Precipitation

nx (Dx ) = nxt fx (Dx ) .
& '( )[m−4 ]&'() & '( )[m−3 ](8.44)[m−1 ]

The averaged diameter of a particle D̄x in this distribution is
* ∞

D̄x =

Dx fx (Dx ) dDx =0Γ (νx + 1)
Dnx = νx Dnx .
Γ (νx )

(8.45)

Here, the relation of (8.38) is used for the last conversion. Generally, the P’th moment of a
gamma function distribution is expressed as follows (a complex number is suﬃcient as P).
* ∞0

DxP fx (Dx ) dDx =Γ (νx + P ) P
Dnx .
Γ (νx )

(8.46)The exponential function distribution (8.36) is one of the special form of a gamma function
distribution (8.43). The distribution (8.43) is the exponential function distribution (8.36) when
we set as follows.

νx = 1(8.47)

1
λx(8.48)Dnx =In this case, (8.46) can be expressed as* ∞0DxP λx exp (−λx Dx ) dDx =1
Γ (P + 1) .
λPx

(8.49)

It is an availability of a gamma function that deﬁnite integral is made simply. The average radiusof a particle D̄x is

D̄x =

1
.
λx

(8.50)A single distribution is used for an cloud water and cloud ice, and the exponential function
distribution4 is used for precipitation particles of rain water, snow, and graupel. Here, theaveraged diameters of cloud water and cloud ice are4

The exponential function distribution is used by Murakami (1999), Ikawa and Saito (1991), Murakami etal. (1994), Ikawa et al. (1991), Lin et al. (1983). Ferrier (1994) uses the exponential function distribution for cloudwater and the gamma function distribution for the other precipitation particles.

8.2 Parameterization of the bulk cold rain



D̄c =


D̄i =

6ρ̄qc
πρw Nc6ρ̄qiπρi Ni169

1
3(8.51)

1
3.

(8.52)

where ρw is density of cloud water [kg m−3 ] and ρi is density of cloud-ice [kg m−3 ]. The constantvalue 1 × 108 m−3 is used for the number concentration of cloud water Nc .
Particle size distributions of rain water, snow, and graupel are given as

nr (Dr ) = nr0 exp (−λr Dr )(8.53)

ns (Ds ) = ns0 exp (−λs Ds )(8.54)

ng (Dg ) = ng0 exp (−λg Dg ) .

(8.55)

Note that the unit of nx is [m−4 ].
Averaged Mass and Averaged Fall Velocity
The mass of a particle of cloud or precipitation, which depends on the form of the particle, isgiven by the empirical equation of an exponentiation of a particle diameter.

mx (Dx ) = αux Dxβux(8.56)

βux = 3 in the case of a spherical particle. Using a gamma function distribution, the averaged
mass isβuxm̄x = αux Dnx

Γ (νx + βux ).
Γ(νx )(8.57)

The fall velocity of a particle is also given by the empirical equation of the exponentiation of adiameter.
Ux (Dx ) = αux Dxβux

ρ0
ρ̄

γux

(8.58)In the same way, using a gamma function distribution, the averaged fall velocity of which theweight with the particle number concentration per unit diameter [m−4 ] is multiplied is170

8 Physical Processes of Cloud and Precipitation

βux Γ (νx + βux )ŪxN = αux Dnx
Γ(νx )



ρ0
ρ̄

γux

.

(8.59)

The averaged fall velocity of which the weight of mass is multiplied is

βux
Ūxq = αux Dnx

Γ (νx + 3 + βux )Γ (νx + 3)



ρ0
ρ̄

γux

.

(8.60)

These relations include an exponential function distribution as a special case. Using the relational
expression with a gamma function in the cases of (8.47), (8.60) and (8.59) are

ŪxN = αux
Ūxq = αuxΓ (1 + βux )



λβxux
Γ (4 + βux )

6λβxux

ρ0
ρ̄
ρ0
ρ̄

γux

(8.61)
γux.

(8.62)

where ρ0 is the air density at the surface in the basic state [kg m−3 ].
In the exponential function distribution5 (8.53)∼(8.55), the total particle number for categories
x = r, s, g: Nx is
* ∞

Nx =

0

nx0 exp (−λx Dx ) dDx =

nx0
.
λx

(8.63)The total mass of water substance x per unit volume isρ̄qx =* ∞0π
πρx nx0.
ρx Dx3 nx0 exp (−λx Dx ) dDx =6
λ4x(8.64)

From these two equations, an inclination parameter in an exponential function paritlce size
distribution λx andthe y-section density nx0 areλx =πρx Nxρ̄qx

nx0 = Nx
5

Ikawa and Saito (1991)

1
3

πρx Nxρ̄qx(8.65)
1
3.

(8.66)

8.2 Parameterization of the bulk cold rain

171

Then, the terminal fall velocity of particle in category x which is weighted by number concentration isŪxN =

* ∞

1
Nx

0

= αux

Ux (Dx ) nx0 exp (−λx Dx ) dDx

Γ (1 + βux )
λβxux



ρ0
ρ̄

γux

(8.67)and this is consistent with (8.61). In the same way, the mass-weighted terminal fall speed ofcategory x isŪxq =

1
ρ̄qx

= αux

* ∞
0

π
Ux (Dx ) Dx3 ρx nx0 exp (−λx Dx ) dDx
6Γ (4 + βux )6λβxux



ρ0
ρ̄

γux

.

(8.68)and this is consistent with (8.62). These averaged fall velocities are used for calculations ofchange of mixing ratio and number concentration in association with a falling, as described in
Section 8.2.6.
Finally, the form parameter of each category is summarized below.
variable

8.2.4

the density of ysection[m−4 ]the form parameter of fall speed

density[kg m−3 ]（αuc = 2.98×107 , βuc = 2.0, γuc = 1.0） ρw = 1.0 × 103

qc

———

qr

nr0 = 8.0 × 106

αur = 842, βur = 0.8, γur = 0.5

ρw = 1.0 × 103

qi
qs
qg

———
（ns0 = 1.8 × 106 ）（ng0 = 1.1 × 106 ）（αui = 700, βui = 1.0, γui = 0.33）

ρi = 5.0 × 102
ρs = 8.4 × 101
ρg = 3.0 × 102

αus = 17, βus = 0.5, γus = 0.5
αug = 124, βug = 0.64, γug = 0.5

Formulizations of Physical Processes of Source terms

Hereafter, we will explain physical processes of source terms. Signs are summarized into table ineach section (Sections 8.2.5 and 8.2.7 are the same as this section).
Nuclear Formation of Primary Ice¦ N U Avi , N U Fci , N U Cci , N U Hci
The primary ice nucleation are shown as below.

172

8 Physical Processes of Cloud and Precipitation

homogeneous nucleation

heterogeneous
nucleationwater vapor →ice crystals
supercooled water drops → ice
crystals

×:not occur

sublimation nuclei
sorption nuclei

water vapor →ice crystals:N U Avi

contact-freezing
nucleisupercooled water drops → ice
crystals
supercooled water drops → ice
crystals

sublimation nucleation
freezing
nucleation
sublimation nucleation
freezing
nucleation

immersionfreezing nuclei:N U Hci

×:unknown:N U Cci:N U Fci

Our model adopts N U Avi , N U Fci , N U Cci , N U Hci .
(1)Sublimation Nucleation: N U Avi(a)Number Concentration of Sublimation Nuclei as a Function of Supercooling Temperature6
Number concentration of sublimation nuclei as a function of supercooled temperature Ts is
written bySi − 1
mi0
N U Avi =β2 Ni0 exp (β2 Ts )ρ̄Swi − 1N U AN
vi =

B

∂Ts
w
∂z

N U Avi.
mi0(8.69)(8.70)

Note that a diﬀerence in z ∗ coordinate system (ζ coordinate system) is multiplied by a matrix,
as shown in equations (6.58) since the vertical diﬀerence is a diﬀerence in a real space.
(b)Number Concentration of Sublimation Nuclei as a Function of Supersaturation7
Number concentration of sublimation nuclei as a function of supersaturation is written by

6

N U Avi =mi0∂SSi
15.25 exp (5.17 + 15.25SSi )wρ̄∂z(8.71)N U ANvi =N U Avi
mi0(8.72)

Ikawa and Saito (1991), Cotton et al. (1986), Murakami (1990), Ikawa et al. (1991), Murakami et al. (1994),
村上 (1999)
7
Meyers et al. (1992), 村上 (1999)

8.2 Parameterization of the bulk cold rain

173

Note that a diﬀerence in z∗ coordinate system (ζ cordinate system) is multiplied by a matrix, as
well as the same way in (a).
(c)Considering both Supercooling Temperature and Supersaturation8
Ferrier (1994) adopted a method that uses two equations owing to temperature for heterogeneous sublimation nucleation. In the method, the equation of Murakami (1990), Cotton etal. (1986) is used when temperature is more than −5 ◦ C , and that of Meyers et al. (1992) is used
when temperature is less than −5 ◦ C .

N U Avi =mi0 ∂Niwρ̄∂z(8.73)N U ANvi =N U Avi
mi0(8.74)

where Ni is divided by −5 ◦ C as follows.

Ni =

⎧


Si − 1 B⎪⎪⎨ Ni01 exp (β2 Ts ),⎪⎪⎩Swi − 1

Ni02 exp (a1 SSi − b1 ) ,T ≥ −5 [◦ C](8.75)

T < −5 [◦ C]

Note that a diﬀerence in z ∗ coordinate system (ζ coordinate system) is multiplied by a matrix,
as well as the same way in (a) and (b).
Meanings of the signs used in (a)∼(c) are as follows.
a1b1
B

Coeﬃcient of the Ferrier equation in below −5 ◦ C
Coeﬃcient of the Ferrier equation in below −5 ◦ C
Coeﬃcient of the Huﬀmann and Vail equation

12.96
0.639
4.5

mi0
Ni0
Ni01

10−12
10−2
103

kg
m−3
m−3

50

m−3

qvsi

Minimum mass of cloud ice
Coeﬃcient of Fletcher equation
Number of particles of the Ferrier equation in morethan −5 ◦ C
Number of particles of the Ferrier equation in below−5 ◦ C
Saturation mixing ratio over ice

qvsw
T
T0
Ts

Saturation mixing ratio over water
Temperature
Melting point of ice
Supercooling temperature（T0 − T ）

Ni02

8

Ferrier (1994)

kg kg−1

273.16kg kg−1
K
K
K

174

8 Physical Processes of Cloud and Precipitation

Si

Ratio of mixing ratio of water vapor and saturation
mixing ratio of ice in an air parcel

Swi
SSi
w
β2

Ratio of the saturation mixing ratios of water and ice
Ice supersaturation（Si − 1）
Vertical motion in z coordinate system
Coeﬃcient of the Fletcher equationρ̄

Air density in the basic state

0.6

m s−1
K−1
kg m−3(2)Immesion Freezing Nucleation: N U Fci
Heterogeneous nucleation of cloud droplets depends on a size, physical and chemical featuresof freezing nucleus and temperature and size of a cloud drop. We use the nucleation that is
extrapolated outside to the size of a cloud droplet with the Bigg (1953) empirical equation9 .
+, ρ̄qc2N U Fci = B  exp A Ts − 1+(8.76)

ρw Nc

, qc

N U FciN = B  exp A Ts − 1(8.77)

ρw

Meanings of the signs used here are as follows.
A

Coeﬃcient of the Bigg empirical equation

0.66

K−1

B
Nc
T

Coeﬃcient of the Bigg empirical equation
Number concentration of cloud droplets
Temperature

100.0
1×108

m−3 s−1
m−3
K

T0
Ts
ρ̄
ρw

Melting point of ice
Supercooled temperature（T0 − T ）Air density in the basic state
Density of water

273.16K
K
kg m−3kg m−3

1×103(3)Contact Freezing Nucleation¦ N U Cci
The contact freezing nucleation is occurred with a collision of an ice particle core (freezing core)
and a supercooled cloud droplet. The following three processes are considered to it.
• Brownian diﬀusion• diﬀusiophoresis• thermophoresis
The generation rate of ice particles by these processes is given by9

村上 (1999), Ikawa and Saito (1991)

8.2 Parameterization of the bulk cold rain




175

dNc
= F1 Dardt b(8.78)dNc
Rv T= F1 F2
dt v
Lv(8.79)

dNc= F1 F2 ft
dt t(8.80)

where

F1 = 2πDc Nc Nar(8.81)F2 =κa(T − Tcl )p(8.82)ft =0.4 [1 + 1.45Kn + 0.4 exp (−1 /Kn )] (κ + 2.5Kn κa ).
(1 + 3Kn ) (2κ + 5κa Kn + κa )(8.83)

Kn in the equations is the Knudsen number which is deﬁned as

Kn = λa0

p00 T.
T00 Ra p(8.84)The diﬀusion coeﬃcient of an aerosol particle is

Dar =

kTcl
(1 + Kn ) .
6πRa μ(8.85)The number concentration Nar of the contact freezing nuclei activated by temperature Tcl 10 isNar = Na0 (270.16 − Tcl )1.3 .

(8.86)

The formation of an ice particle with the contact freezing nuclei is ﬁnally represented by the sumof the three processes, as shown in follows.

10

Cotton et al. (1986)







dNc
dNc
dNc
+
+
dt b
dt v
dt t

N
=
N U Cci1ρ̄N U Cci =ρ̄qcN
N U CciNc(8.87)(8.88)

176

8 Physical Processes of Cloud and Precipitation

Meanings of the signs used here are as follows.
Dc
k
Lv
Na0

Diameter of a cloud dropletmNc

The Boltzmann number
Latent heat of evaporation of water
Coeﬃcient in the number concentration ofcontact freezing nuclei
Number concentration of cloud droplets

p
p00
Ra

PressureStandard pressure
Radius of an aerosol particle

Rv
T
T00

The gas constant of water vapor
Temperature
Temperature in the basic state

Tcl
κ
κa
λa0

Temperature of a cloud droplet
Thermal conductivity of air
Thermal conductivity of an aerosol particle
Mean free path at p00 , T00μρ̄

Coeﬃcient of viscosity of air
Air density in the basic state

−23

2×105

J K−1
J kg−1
m−3

1×108

m−3

101325
3×10−7Pa
Pam

1.380658×10

J K−1 kg−1
K
K

461.0
293.15−2

2.4 × 10

6.6×10−8K
J m−1 s−1 K−1
J m−1 s−1 K−1m
kg m−1 s−1
kg m−3(4)Homogeneous Freezing Nuclei¦ N U Hci
We assume that cloud water freezes momentarily when temperature is below −40 ◦ C 11 . Rateof the homogeneous freezing nuclei is written byN=
N U Hci

1 Nc
ρ̄ 2Δt(8.89)N U Hci =qc
2Δt(8.90)

where Δt is the time step in the leap frog method. Note that generation of cloud water must be
given by the moist saturation adjustment before this calculation (Ferrier, 1994).
Secondary Nucleation of Ice Crystals: SP
Although there are still many points which are not known about the nucleation of secondary ice
crystals, the following processes are mainly known.
11

Ikawa and Saito (1991), Ferrier (1994)

8.2 Parameterization of the bulk cold rain

177

• While snow and graupel grow up with collection of cloud droplets, minute particles of ice
are generated (Hallett and Mossop, 1974).
• Snow and graupel collide each other while they are falling, and the fragment of small ice isgenerated (Vardiman, 1978).
• When the large supercooled water droplet, which is generated near the cloud top, freezes,
ice crystals of high-concentration are generated (Hobbs and Rangno, 1985).
We consider only the Hallett-Mossop rime splintering mechanism12 . Generation rates of thesecondary nucleation of ice crystals are
SPsiN =

1
× 3.5 × 108 f (Ts ) CLcs
ρ̄

(8.91)

SPsi = mi0 SPsiN
1
N
SPgi= × 3.5 × 108 f (Tg ) CLcg
ρ̄

(8.92)N
SPgi = mi0 SPgi.

(8.94)(8.93)

Note that the secondary nucleation of ice crystals has never occurred in the wet growth of graupel.
Function of the temperature of a particle f (Tx ) (Tx is Ts or Tg ) is deﬁned as follows.

f (Tx ) =

⎧
⎪
⎪ 0,
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪ Tx − 268.16 ,⎪⎨2⎪
268.16 − Tx
⎪
⎪
⎪
,
⎪
⎪
3
⎪
⎪
⎪
⎪
⎪
⎩ 0,

Tx > 270.16 [K]268.16 ≤ Tx ≤ 270.16 [K](8.95)
265.16 ≤ Tx ≤ 268.16 [K]Tx < 265.16 [K]Although f (Tx ) will become a negative value in (72) of Cotton et al. (1986), if an absolutevalue is taken, it will become the same as the equation (8.95). Although (8.95) is a function thatit may be f (Tx ) = 0 at Tx = 268.16［K］, Ikawa et al. (1991), Ikawa and Saito (1991) use afunction that it may be f (Tx ) = 1 at Tx = 268.16[K] as follows.
⎧
⎪
⎪ 0,
⎪
⎪
⎪
⎪
⎪
⎪
270.16 − Tx
⎪
⎪
,
⎪
⎪
⎪
2
⎪
⎪
⎨

f (Tx ) =

1,

⎪
⎪
⎪
⎪
⎪
⎪
Tx − 265.16
⎪
⎪
,
⎪
⎪
⎪
3
⎪
⎪
⎪
⎪
⎩

0,

12

Tx ≥ 270.16 [K]268.16 < Tx < 270.16 [K]Tx = 268.16 [K]265.16 ≤ Tx < 268.16 [K]Tx < 265.16 [K]村上 (1999), Ikawa and Saito (1991), Cotton et al. (1986)(8.96)

178

8 Physical Processes of Cloud and Precipitation

This function has the maximum value at −5 ◦ C and is considered to be more reasonable.
Meanings of the signs used here are as follows.
CLcg

mi0
Ts
Tg

Growth rate of graupel in the collision with cloud
droplets
Growth rate of snow in the collision with cloud
droplets
The mass of the minimum cloud ice
Temperature of snow
Temperature of graupel

Txρ̄

Temperature of snow or graupel
Air density in the basic state

CLcs

s−1
s−1
10−12

kg
K
K
K
kg m−3

Water Vapor Diﬀusion Growth¦ V D
Diﬀusion growth is the generation, the growth, disappearance, and consumption by exchange of
the water molecule between moisture and a particle. There are the following processes and someof them are considered in our model.
gaseous phase −liquid phasecondensationwater vapor → cloud waterrain water → water vapor

calculated by the moist
saturation adjustment
neglected because ofsmall amountcalculated by the moist
saturation adjustment
V Dvr < 0water vapor → cloud ice

V Dvi > 0water vapor → snow

V Dvs > 0water vapor → graupel
water vapor → hailcloud ice → water vapor

V Dvg > 0
V Dvh > 0
V Dvi < 0snow → water vapor

V Dvs < 0graupel → water vapor
hail → water vapor

V Dvg < 0
V Dvh < 0water vapor → rain water
evaporation

gaseous phase −solid phasesublimation

sublimation
evaporation

cloud water → water vapor(1)Evaporation of Rain Water¦ V Drv

8.2 Parameterization of the bulk cold rain

179

Variations of mixing ratio and number concentration of rain water 13 are represented by

V Dvr = −V Drv =2π(Sw − 1) Gw (T, p) V EN Tr
ρ̄(8.97)where

Gw (T, p) =

L2v
1
+
2
κRv T
ρ̄qvsw Dv

−1.

(8.98)The term of the ventilation coeﬃcient of rain water is represented by


1

1

1
3 −2
2
αur
Γ
V EN Tr = nr0 0.78λ−2
r + 0.31Sc ν

5 + βur2−λr(5+βur )
2



ρ0
ρ̄

1 
4

.

(8.99)(2)Sublimation Condensation of Snow and Graupel (Sublimation Evaporation)¦V Dvs , V Dvg
14

Sublimation condensation (sublimation evaporation) rates of snow and graupel are formulized
like rain water. However, they are divided in the cases more and less than the melting point ofice. Here, snow and graupel are simultaneously formulized as x = s, g. That is,considering the latent heat of freezing in a collecting supercooled cloud droplets at T < T0 ,V Dvx =Ls Lf
2π
(Si − 1) Gi (T, p) V EN Tx −Gi (T, p) CLcx .
ρ̄κRv T 2(8.100)At T > T0 ,when a melting process occurs,
V Dvx = 2πDv (qv − qvs (T0 )) V EN Tx(8.101)

when a melting process does not occur,

V Dvx =
13
14

2π
(Sw − 1) Gw (T, p) V EN Tx .
ρ̄

村上 (1999)
村上 (1999), Ikawa and Saito (1991), Lin et al (1983)(8.102)

180

8 Physical Processes of Cloud and Precipitation

where


Gi (T, p) =

L2s
1
+
2
κRv T
ρ̄qvsi Dv

−1(8.103)where Gw (T, p) is given by the equation (8.98). A ventilation coeﬃcient at x = s, g is as follows.


V EN Tx = nx0

1

1

1
3 −2
2
0.78λ−2αux
Γx + 0.31Sc ν(5+βux )5 + βux
−
λx 2
2



ρ0
ρ̄

1 
4

(8.104)(3)Growth of Cloud Ice in Gaseous Phase¦ V Dvi 15
Variation of growth of cloud ice in gaseous phase is represented as follows.

V Dvi =

qv − qvsi
Ni
a1 (m̄i )a2
qvsw − qvsi
ρ̄

(8.105)where a1 , a2 is a parameter depending on temperature (Köenig (1971)) and m̄i is mean mass ofcloud ice given as following equation.

m̄i =

qi ρ̄
Ni

(8.106)

Meanings of the signs used in (1)-(3) are as follows.
s−1

Dv
Lf

Growth rate of graupel in coalescence with cloud
droplets
Growth rate of snow in coalescence with cloud
droplets
Diﬀusion coeﬃcient of water vapor
Latent heat of melting of water

Ls
Lv
ng0

Latent heat of sublimation of water
Latent heat of evaporation of water
Density of graupel in the y section

J kg−1J kg−1m−4

nr0
ns0
qvs (T0 )Density of rain water in the y section
Density of snow in the y section
Saturation mixing ratio of the melting point of
water

CLcg
CLcs

15

Ikawa and Saito (1991), Ikawa et al. (1991)s−1
m2 s−1J kg−1

8.0 × 106m−4
m−4
kg kg−1

8.2 Parameterization of the bulk cold rain

181

qvsi

Ice saturation mixing ratio

kg kg−1

qvsw
Rv
Sc

Water saturation mixing ratio
The gas constant of water vapor
Schmidt number

kg kg−1
J K−1 kg−1

Si − 1Sw − 1T

Ice supersaturation of an air parcel
Water supersaturation of an air parcel
Temperature

T0
αug

κ
ν

Melting point of ice
Coeﬃcient in the relational expression of fall velocity and diameter of graupel
Coeﬃcient in the relational expression of fall velocity and diameter of rain water
Coeﬃcient in the relational expression of fall velocity and diameter of snow
Coeﬃcient in the relational expression of fall velocity and diameter of graupel
Coeﬃcient in the relational expression of fall velocity and diameter of rain water
Coeﬃcient in the relational expression of fall velocity and diameter of snow
Inclination in a reversed exponential function expressing particle size distribution of graupel
Inclination in a reversed exponential function expressing particle size distribution of rain water
Inclination in a reversed exponential function expressing particle size distribution of snow
Thermal conductivity of air
Coeﬃcient of kinematic viscosity of air

ρ̄ρ0

Air density in the basic state
Density at the surface in the basic state

αur
αus
βug
βur
βus
λg
λr
λs

461.0
0.6

K
273.16124Km1−βug s−1842

m1−βur s−117m1−βus s−1

0.64
0.8
0.5
m−1
m−1
m−1
2.4×10−2

J m−1 s−1 K−1
m2 s−1kg m−3kg m−3

Particle Collection: CL
In this section, we formulize the growth process of the particle which captures a certain particle
by the collision between various particles. For the particle collision, the following processes canbe respectively considered to the categories of the particles which are deﬁned before.
processsink

sourcesign

meaning

Collectionof cloud
water

qc

qi

CLci

Growth of cloud ice by collection with cloud water

qc
qc
qc

qs
qg
qh

CLcs
CLcg
CLch

Growth of snow by collection with cloud water
Growth of graupel by collection with cloud water
Growth of hail by collection with cloud water

model
×

182

8 Physical Processes of Cloud and Precipitation

Collection
of
rain
water

Collectionof cloudiceCollectionof snow

qc

qr

CLcr

Growth of rain water by collection with cloud water

qr

qi

CLri

Growth of cloud ice by collection with rain water

qr
qr
qr

qs
qg
qh

CLrs
CLrg
CLrh

Growth of snow by collection with rain water
Growth of graupel by collection with rain water
Growth of hail by collection with rain water

qi

qr

CLir

Growth of rain water by collection with cloud ice

qi
qi

qs
qg

CLis
CLig

Growth of snow by collection with cloud ice
Growth of graupel by collection with cloud ice

qi

qh

CLih

Growth of hail by collection with cloud ice

qs

qr

CLsr

Growth of rain water by collection with snow

qs
qs

qg
qh

CLsg
CLsh

Growth of graupel by collection with snow
Growth of hail by collection with snow

In this table, ” ” and ”×” denote the signs of which it is included or is not included in ourmodel, respectively, and ” ” is the sign of suspending for including in our model at present.
The number concentration will change for the collection of snow particles (aggregation) andthat of rain water (accretion) but the mixing ratio do not change. These are described in theother section.
In the following section, we formulize the coalescences between large particles (rain water, snow,
and graupel), which have a suﬃcient value of fall velocity, a large particle and a small particle(cloud water and cloud ice) of which its fall velocity can be neglected, and the growth of graupelby collection of supercooled rain water and ice crystals16 .
(1)Coalescence between Rain Water, Snow and Graupel¦ CLxy (x, y = r, s, g; x = y)
Variations of mixing ratio and number concentration by the collection between precipitation
particles (rain water, snow, and hail) are formulized as follows.

CLxy = π
CLN
xy =

2 ρx

ρ̄



Exy

π
Exy
2ρ̄





Ūx − Ūy2

+ αŪx Ūy nx0 ny0Ūx − Ūy2

+ αŪx Ūy nx0 ny052
0.5+ 5 2+ 4 3
6
λx λy
λx λy
λx λy

1
1
1
+ 2 2 +
3
λx λy
λx λy
λx λ3ywhere x, y = r, s, g; x = y. Meanings of the signs used are as follows.
16

Murakami (1999), Lin et al. (1983), Murakami (1990), Ikawa and Saito (1991)



(8.107)(8.108)8.2 Parameterization of the bulk cold rain

Exy

Collection eﬃciency of a particle

nx0
ny0
Ūx
Ūy

Density of a category x in the y section
Density of a category y in the y section
Mean fall velocity weighted by a mass of a category x
Mean fall velocity weighted by mass of a category yαλx

Coeﬃcient of an adjustment term
Inclination in a reversed exponential function expressing particle size distribution of a category x
Inclination in a reversed exponential function expressing particle size distribution of a category y
Air density in the basic state
Density of a particle of a category x

λy
ρ̄
ρx

183

m−4
m−4
m s−1
m s−1
0.04
m−1
m−1
kg m−3kg m−3

184

8 Physical Processes of Cloud and Precipitation(2)Collection of Rain water, Snow, and Graupel with Cloud Water and Cloud Ice¦
CLcy , CLiy (y = r, s, g)
Because fall velocities of cloud water and cloud ice are relatively small to those of rain water,snow and graupel, the collection is represented as follows.

πuy )CLxy = Ēxy ny0 qx αuy Γ (3 + βuy ) λ−(3+β
y
4



ρ0
ρ̄

1
2

(8.109)The mean collection eﬃciency for coalescence with cloud ice Ēiy is constant, and that of coalescence with cloud water Ēcy is given byĒcy =

Stk2(Stk + 0.5)2(8.110)where Stk is the Stokes number which is given with the average radius of cloud water, cloud ice,and precipitation particle. Ikawa and Saito (1991) calculates as follows.

Stk = D̄c2 ρw

Ūy
9μD̄y

(8.111)

Meanings of the signs used here are as follows.
Dc
Dy

Diameter of a particle of cloud water
Diameter of a particle of a category y

m
m

ny0
Ūy

m−4
m s−1

μ
ρ̄

Density of a category y in the y section
Mean fall velocity multiplied by an weight coeﬃcientof mass of a category y
Coeﬃcient in the relational expression of fall velocity
and diameter of a category y
Coeﬃcient in the relational expression of fall velocity
and diameter of a category y
Inclination in a reversed exponential function expressing particle size distribution of a category y
Coeﬃcient of viscosity of air
Air density in the basic state

ρ0ρw

Density at the surface in the basic state
Density of water

kg m−3kg m−3αuy
βuy
λym1−βuy s−1

m−1
kg m−1 s−1
kg m−3
1×103(3)Formation Process of Graupel by Collection of a Supercooled Raindrop and Ice8.2 Parameterization of the bulk cold rain

185

Crystals¦ CLriWhen it is assumed that a raindrop freezes instantaneously, variations of mixing ratio andnumber concentration of graupel with its formation by the collision of a supercooled raindrop andice crystals are given as follows17 .
π2
ur )CLri =
Eir Ni nr0 αur Γ (6 + βur ) λ−(6+β
r
24
CLN
ri =

π
ur )
Eir Ni nr0 αur Γ (3 + βur ) λ−(3+β
r
4ρ̄





ρ0
ρ̄
ρ0
ρ̄

1
2

(8.112)

1
2(8.113)Meanings of the signs used here are as follows.
Eir

Collection eﬃciency of a particle

1.0

nr0
αur

Density of rain water in the y section
Coeﬃcient in the relational expression of fall velocity and diameter of rain water
Coeﬃcient in the relational expression of fall velocity and diameter of rain water
Inclination in a reversed exponential function expressing particle size distribution of rain water
Air density in the basic state
Density at the surface in the basic state

8.0 × 106
842

βur
λr
ρ̄
ρ0

m−4
m1−βur s−1

0.8
m−1
kg m−3kg m−3(4)Summary of Collection Eﬃciencies for Coalescences¦ Exy 18
Collection eﬃciencies Exy for coalescences between particles shown in (1)-(3) are summarized
in the following table.


EcrCollection eﬃciency for the coalescence of rain waterwith cloud water

Stk2 (Stk + 0.5)2

Ecs

Collection eﬃciency for the coalescence of snow withcloud water

Stk2 (Stk + 0.5)2

Ecg

Collection eﬃciency for the coalescence of graupel withcloud water
Collection eﬃciency for the coalescence of snow withrain water
Collection eﬃciency for the coalescence of graupel withrain water
Collection eﬃciency for the coalescence of rain waterwith cloud ice

Stk2 (Stk + 0.5)2

Ers
Erg
Eir
17
18

村上 (1999)
Ikawa and Saito (1991), Ikawa et al. (1991)




1.0
1.0
1.0

186

8 Physical Processes of Cloud and Precipitation

Eis

Collection eﬃciency for the coalescence of snow withcloud ice

1.0

Eig

Collection eﬃciency for the coalescence of graupel withcloud ice

0.1

Esr

Collection eﬃciency for the coalescence of rain waterwith snow

1.0

Esg

Collection eﬃciency for the coalescence of graupel with
snow

0.001

(5)Category Allocation Ratios of Snow and Raindrop after the Coalescence¦ αrs
In the coalescence of snow and raindrop, it is diﬃcult that we decide an adequate category aparticle formed by the coalescence in the layer less than 0 ◦ C . Here, the allocation ratios of thesnow and graupel, which appear in the equations (8.31) and (8.32), is given by using the average
masses of rain water m̄r and snow m̄s as follows.

αrs =

m̄2s
m̄2s + m̄2r

(8.114)

The average masses of rain water m̄r and snow m̄s are given as follows.
m̄r = ρrm̄s = ρs

4
λr
4
λs

3

(8.115)3(8.116)Using these equations, we can ﬁnd that the generation ratio of graupel by the coalescence of araindrop and snow is (1 − αrs ). Note that the ratios in (8.114) are tentative values.
Production of Graupel¦ P G
Production of graupel is an important and complicated problem, since graupel is producted by
the collision between other particles. In this section, production process of graupel is explained indetail especially, although it should be explained in the section of ”Collision between particles”.
Production of graupel has two processes: dry growth and wet growth. By the former process,all supercooled cloud droplets freeze in an instant on contact with graupel, and surface of graupel
remains dry. By the latter process, not all supercooled cloud droplets freeze by the latent heat
which supercooled cloud droplets emit when they collided with graupel. Surface of graupel getswet.
In dry growth process, although all supercooled cloud droplets contribute to growth, cloud
ice and snow hardly contribute to growth because the collection eﬃciency of cloud ice or snow
colliding with graupel is small. On the contrary, in wet growth process, a freezing amount of

8.2 Parameterization of the bulk cold rain

187

supercooled water is decided by the budget of sensible and latent heat. Cloud ice or snow are
eﬃciently captured by graupel.
Dry growth process is deﬁned as following19 .
P Gdry = CLcg + CLrg + CLig + CLsg

(8.117)On the contrary, wet growth process is deﬁned as following.






2π [κTs + Lv Dv ρ̄ (qvs (T0 ) − qv )]Ci T sV ENTg + CLig + CLsg
1+
P Gwet =ρ̄ (Lf − Cw Ts )Lf − Cw Ts (8.118)

Where V EN Tg is ventilation eﬃciency (8.104). Whether graupel grows by dry growth process
or wet growth process is decided by following relation of P Gdry and P Gwet .

P Gg = P Gdry ,P Gdry ≤ P Gwet

(8.119)P Gg = P Gwet ,P Gdry > P Gwet

(8.120)Meanings of signs used in this section are shown as follows.

188

8 Physical Processes of Cloud and Precipitation

Aggregation¦ AG
Aggregation, as well as production of graupel, should be categorized in ”Collision between particles,” however, we explain it especially in this section.
We consider the following two phenomenons for aggregation. First, the number of cloud ice
per unit volume decreases by the collision between cloud ices (ice crystals). Second, the numberof snow per unit volume decreases by the collision between snows (snowﬂakes).
By aggregation process, only the number concentration changes but mixing ratio does notchange.
20(1)Aggregation between Cloud Ice Particles¦ AGN
i
Decrease of the number concentration of cloud ices (ice crystals) by aggregation process isdeﬁned as following.


AGN
i =

d
dt



Ni
ρ̄



=−
aggr

c1
Ni
2ρ̄

(8.121)

where c1 is deﬁned as following.

c1 =ρ̄qi αui Eii X
ρi



ρ0
ρ̄

1
3

(8.122)Meanings of signs used in this section are shown as follows.
Eii
X
αui

ρ̄

Collection eﬃciency between ice crystals
Spectrum dispersion of fall velocity of ice crystal
The coeﬃcient which appears in the equation offall velocity and diameter of ice crystal
The coeﬃcient which appears in the equation offall velocity and diameter of ice crystal
Air density in the basic stateρ0ρi

Air density at the surface in the basic state
Density of cloud ice

βui

0.1
0.25
700

m1−βui s−1

1.0
kg m−3
5.0×102kg m−3kg m−321(2)Aggregation between Snow Particles¦ AGNs

A decrease of the number concentration of snow (snowﬂakes) by aggregation process is deﬁned
as following. By this process, only the number concentration of snow (Ns ) changes but mixing
ratio of snow (qs ) does not change.
2021

村上 (1999), Ikawa and Saito (1991)
村上 (1999), Ikawa and Saito (1991)

8.2 Parameterization of the bulk cold rain



AGN
s =

d
dt



Ns
ρ̄



=−
aggr

189

2+βus
4−βus
us
1 αus Ess I (βus ) 1−βus 2+βus −2−β
qs 3 Ns 3(8.123)π 3 ρ̄ 3 ρs 3
ρ̄
4 × 720

where I (βus ) is deﬁned as following.

I (βus ) =* ∞* ∞
00βx3 y 3 (x + y)2 xβus − yus
exp [− (x + y)] dxdy

(8.124)This equation is solved with Gauss’s hypergeometric function as following. 22

I (βus ) = Γ (βus ) 21−d

3




Ci

i=1

F (1, d; 8 − i; 0.5) F (1, d; 4 + βus ; 0.5)−7−i
3 + βus + i(8.129)where

d = 10 + βus
C1 = 1C2 = 3C3 = 1.

We adopted the following values as typical values of I (βus ) (Ikawa and Saito, 1991; Mizuno, 1990).
22

The hypergeometric function is one of solutions of hypergeometric equation which is two order linear ordinary
diﬀerential equation, and has regular sigularity at x = 0, 1, ∞.

F (x, a; b; c) = 1 +a (a + 1) b (b + 1) x2
a·b x
+
+ ···
c 1!
c (c + 1)
2!

(8.125)where c = 0, −1, −2, · · ·. With Pockhimer’s sign,(a)n = a (a + 1) (a + 2) · · · (a + n − 1) =(a + n − 1)!
(a − 1)!

(a)0 = 1(8.126)(8.127)F (x, a; b; c) =∞(a)n (b)n xn
n=0

(c)n

n!

As special limit of this hypergeometric function, there is conﬂuent hypergeometric function(8.128)190

8 Physical Processes of Cloud and Precipitation

βus

0.4

0.5

0.6

I (βus )

1108

1610

2566

Meanings of signs used in these equations are shown as follows.
Ess
αus
βus
ρ̄
ρs

Collection eﬃciency between snows
The coeﬃcient which appears in the equation offall velocity and diameter of snow
The coeﬃcient which appears in the equation offall velocity and diameter of snow
Air density in the basic state
Density of snow

0.117m1−βus s−1

0.5

8.4×101kg m−3kg m−3

Conversion¦ CN
Particle conversion from a certain category ’x’ into other category ’y’ is expressed by CNxy .
For example, Kessler (1969)’s expression of conversion from cloud water into rain water is oneof the earliest formulation of category conversion.
We consider the following types of category conversions.
process

consumed

growingsign

meaning

conversion
from
cloud
water to rainwater
conversion
from cloud ice
to snow

qc

qr

CNcr

by aggregation

qi

qs

CNis

by aggregation and sublimation

conversion
from snow to
graupelconversion
from snow tohailconversion
from graupel
to hailconversion
from snow to
graupelconversion
from snow to
hail

qs

qg

CNsg

by growth with capturing cloud particle

qs

qh

CNsh

by growth with capturing cloud droplets

qg

qh

CNgh

by growth with capturing cloud particle

qs

qg

CNsg

by freezing of rain water

qs

qh

CNsh

by freezing of rain water

qg

qh

CNgh

by freezing of rain water

conversion
from graupel
to hail

8.2 Parameterization of the bulk cold rain

191

Hereafter, we consider about conversion processes of CNcr , CNis , CNsg .
(1)Conversion from Cloud to Rain¦ CNcr 23
The conversion from cloud water to rain water is based on coalescence growth of cloud droplets.
Kessler (1969) parameterized this process ﬁrst. This process is investigated in detail by Berry (1968),andBerry and Reinhardt (1974). Recently, the method for the conversion based on this process isused in many cases.
(a)Conversion Based on Berry(1968), Berry and Reinhardt(1974) 24
The method for the conversion based on Berry (1968), Berry and Reinhardt (1974) is shownas following.

CNcr =⎧
0.104gEcc 4 7 13⎪⎪
ρ̄ qc ,⎪⎪⎨ μ (N ρ ) 13

qc ≥ qcm

c w(8.130)

⎪
⎪
⎪
⎪
⎩ 0,

qc < qcm

where the collection eﬃciency between cloud water is Ecc = 0.55. The critical value of mixing
ratio of cloud water to convert from cloud water to rain water (qcm ) is shown as following.

qcm =

ρw
3
Nc
πDcm
6ρ̄

(8.131)

where the critical mean value of diameter of cloud is Dcm = 20［μm］. We adopt a typical value(Nc = 108［m−3 ］) of the number concentration of cloud (Nc ). However, it is better for this to be
calculated by time integration as a forecast variable of a time development equation essentially.
(b)KesslerÉ1969Ê’s Deﬁnition of Coalescence Growth 25
Kessler’s parametarization is the most classic parameterization, and is also used in many models
now.
CNcr = a (qc − qcm ) H (qc − qcm )(8.132)where H is set function. Usually a is selected as 10−3 [s−1 ],qcm is selected as 10−3 [kg kg−1 ]. In
Cotton and Anthes (1989), qcm and a are given by the function of qc as following.

qcm =

3
4πρw Nc Dcm= 4 × 10−12 Nc ,
3ρ̄

4
−1
a = πEcc Udc Nc Dc2 = 1.3 × qc3 Nc 3
23

Dcm = 10−5 [m]


ρ0
ρ̄



村上 (1999), Lin et al. (1983), Ferrier (1994), Ikawa and Saito (1991)
村上 (1991)
25
Ikawa and Saito (1991)
24(8.133)(8.134)

192

8 Physical Processes of Cloud and Precipitation(c)Lin et al.(1983)’s way
Lin et al. (1983) improved Berry (1968)’s method as following.
−12

CNcr = ρ̄ (qc − qcm )2 1.2 × 10−4 + 1.569 × 10

Nc
σ2 (qc −qcm )(8.135)where σ 2 indicates dispersion of the number concentration of cloud (= 0.15). qcm is selected as
2 × 10−3 [kg kg−1 ].
Meanings of signs used in (a)∼(c) are shown as following.
gNcμ

Acceleration of gravity
Number concentration of cloud particle
Viscosity Coeﬃcient of air

ρ̄
ρ0
ρw

Air density in the basic state
Air density at the surface in the basic state
Density of water

9.8
1×108

1×103

m s−2
m−3
kg m−1 s−1
kg m−3kg m−3kg m−3(2)Conversion from Cloud Ice to Snow¦ CNis
The conversion from cloud ice to snow is formulized with an assumption of which the conversion
occurs by two processes of sublimation growth of cloud ice and condensation26 .
By sublimation condensation growth, it takes Δtis1 for ice crystal with R̄i radius to grow into
snow with Rs0 radius. Δtis1 isΔtis1 =

R2s0 − R̄2i
ρi
2a1

(8.136)

where a1 , as well as (8.100) and (8.97), is expressed as following.


L2s
1
+
a1 = (Si − 1)2
κRv Tρ̄qvsi Dv

−1(8.137)

Variation of mixing ratio from ice cloud to snow per unit time by sublimation condensation
dep
growth (conversion rate) CNis
is showed as following.
dep
CNis
=

26

村上 (1999), Murakami (1990), Ikawa and Saito (1991)

qi
Δtis1

(8.138)8.2 Parameterization of the bulk cold rain

193

By aggregation growth, it takes Δtis1 for cloud ice with R̄i radius to grow into snow with Rs0
radius. We suppose that the Δtis1 is the same time as it takes for the number concentration ofcloud ice to decrease from Ni to Ni (Ri /Rs0 )3 on condition ρi = const.. Therefore, the Δtis1 isexpressed as following.


Rs0
2
Δtis2 =
log
c1
R̄i

3

(8.139)where c1 is given by the equation (8.122). Variation of mixing ratio from ice cloud to snow perunit time by aggregation growth (conversion rate) is showed as following.

agg
CNis
=

qi
Δtis2

(8.140)

Finaly, the conversion rate from cloud ice to snow (CNis ) is given as following.
dep
agg
CNis = CNis
+ CNis(8.141)

Meaning of signs used here are shown as following.
Dv
Ls
qvsi
Rv

Diﬀusion coeﬃcient of water vapor
Latent heat of sublimation of water
Saturation mixing ratio over ice
Gas constant of water vapor

461.0

m2 s−1J kg−1kg kg−1
J K−1 kg−1

Si − 1TκSupersaturation of air to ice
Temperature
Conduction eﬃciency of heat of air

2.4×10−2

K
J m−1 s−1 K−1ρ̄ρi

Air density in the basic state
Density of cloud ice

5.0×102kg m−3kg m−3(3)Conversion from Snow to Graupel: CNsg
We suppose that the conversion from snow to graupel takes place when CLcs (growth rate bycaputuring cloud droplets), which is given in (8.109), exceeds V Dvs (growth rate by sublimation
condensation), which is given in (8.100)27 。When the conversion takes place, the rate of changeof mixing ratio per unit time by converting from snow to graupel is shown as following.
27

村上 (1999), Murakami (1990)

194

8 Physical Processes of Cloud and Precipitation

When CLcs > V Dvs ,CNsg = CLcs

ρs.
ρg − ρs(8.142)The variation of the number concentration is shown here,
123αus Ecs qc (ρ0 ρ̄) Γ (βus ) 1−βus
N
λs
=
Ns .
CNsg
2ρ̄ (ρg − ρs )(8.143)Meanings of signs used here are shown as following.


Ecs

Collection eﬃciency of snow in the coalescence with cloud ice

Stk2 (Stk + 0.5)2

αus

17

ρg

The coeﬃcient which appears in theequation of fall velocity and a diameterof snow.
The coeﬃcient which appears in theequation of fall velocity and a diameterof snow
Inclination in the reverse exponentialfunction expressing particle size distribution of snow
Air density in the basic state
Air density at the surface in the basic
stateDensity of graupel

3.0×102kg m−3ρs

Density of snow

8.4×101kg m−3

βus

λs

ρ̄
ρ0

m1−βus s−1

0.5

m−1

kg m−3kg m−3

Melting of Solid Particle: M L(1)Melting of Cloud Ice: M Lic
We suppose that cloud ice melts and is converted into cloud water in an instant when T > T0 ,because the radius of cloud ice (ice crystal) is very small.
When Tc > T0 ,M Lic =qi.
2Δt(2)Melting of Snow and Graupel: M Lsr , M Lgr(8.144)

8.2 Parameterization of the bulk cold rain

195

With regard to snow and graupel, M Lxr （x = s, g ）, which indicates the conversion rate fromsolid particles to rain, is controled by heat budget28 , that is,When T > T0 ,M Lxr =2π
Cw Tc
[κTc + Lv Dv ρ̄ (qv − qvs (T0 ))] V EN Tx +(CLcx + CLrx ) .
ρ̄LfLf(8.145)

If M Lxr > 0, snow or graupel melts and is converted to rain, however if M Lxr < 0, even if
T > T0 , melting does not take place, that is, M Lxr = 0. V EN Tg is ventiration eﬃciency(8.104).
Meanings of signs used in (1),(2) are shown as following.
CLcx

Cw
Dv

Growth rate of a particle in x catogory by collision with cloud water
Growth rate of a particle in x catogory by collision with rain waterSpeciﬁc heat at constant pressure of waterDiﬀusion coeﬃcient of water vapor

LfLv
qvs (T0 )T

Latent heat of fusion of water
Latent heat of evaporation of water
Saturated mixing ratio at melting point of water
Temperature

T0
Tc
κ

Melting point of ice
Temperature in Celsius scaleEﬃciency of conduction of heat of air

ρ̄

Air density in the basic state

CLrx

s−1
s−1
4.17×103

J K−1 kg−1
m2 s−1J kg−1J kg−1kg kg−1
K

273.16KC
J m−1 s−1 K−1◦

2.4×10−2

kg m−3Freezing of Rain: F R
We suppose that freezing of rain takes place in an instant. All frozen rain is categorized into
graupel because we do not consider hail. If there is a category of hail, frozen rain is categorized
into the category of hail.
The rate of variation of mixing ratio by freezing rain (F Rrg ) is based on Bigg (1953)’s empiricalequation29 , and is given as following.
F Rrg = 20π2 B  nr0,
ρw +exp A Ts − 1 λ−7rρ̄(8.146)

The rate of variation of the number concentration is given as following.
F RN
rg =
28
29

Ikawa and Saito (1991)
Lin et al. (1983), 村上 (1999)+,
π 
B nr0 exp A Ts − 1 λ−4r
6ρ̄(8.147)

196

8 Physical Processes of Cloud and Precipitation

Meanings of signs used here are shown as following.
A
B
nr0
T

A coeﬃcient in Bigg’s empirical equation
A coeﬃcient in Bigg’s empirical equation
Density of rain in y section
Temperature

0.66
100.0
8.0 × 106K−1
m−3 s−1
m−4
K

T0
Ts
λr

273.16

K
K
m−1

ρ̄

Melting point of ice
Supercooled temperature （T0 − T ）Inclination in the reverse exponential function expressing particle size distribution of rain
Air density in the basic stateρw

Density of water

1×103

kg m−3kg m−3

Shedding of Water from Snow and Graupel: SH
In our model, we do not consider that snow and graupel partly contain liquid water in them.
Therefore, melted water sheds from snow or ice, and becomes rain water. The process is expressed
as following.
When T > T0 ,SHsr = CLcs + CLrs

(8.148)SHgr = CLcg + CLrg

(8.149)If graupel grows by the wet growth process (even when T ≤ T0 ),SHgr = CLcg + CLrg + CLig + CLsg − P Gwet .

(8.150)Meanings of signs used here are shown as following.
CLcg
CLcs

Growth rate of graupel by collision with cloud water
Growth rate of snow by collision with cloud water

s−1
s−1

CLrg
CLrs

CLig

s−1
s−1
s−1

P GwetGrowth rate of graupel by collision with rain water
Growth rate of snow by collision with rain water
Growth rate of graupel by collision with cloud water in
wet growth process
Growth rate of graupel by collision with snow in wet
growth process
Growth rate of graupel in wet growth process

T
T0

Temperature
Melting point of ice

K
K



CLsg

s−1
s−1
273.16

8.2 Parameterization of the bulk cold rain

197

Ferrier (1994) formulizes the wet growth process of snow, graupel and hail (frozen ice) with atime development equation, and takes the wet snow and wet graupel and wet hail into consideration.
Such formulization will be required in the future.
Breakup of Waterdrop
It is known that if it becomes large to some extent (it is 8mm at the diameter of a sphere), rainwater becomes unstable and breaks up.
From this reason, the waterdrop more than the size of 8 mm does not fall down in an actual
nature. Although the breakup of waterdrop does not change mixing ratio of rain water, it changes
number concentration of rain water. However, since the number concentration of rain water is
not predicted in this model, the process is not considered.

8.2.5

Moist Saturation Adjustment

The exchange between water vapor and cloud water is expressed by the moist saturation adjustment30 .
We add the mark ∗ to the variable which is calculated by forecast equation. The supersaturated
mixing ratio which is deﬁned as∗
Δqc = qv∗ − qvsw.

(8.151)

It Δqc > 0, the moist saturation adjustment is applied. The the moist saturation adjustment
caluculates θ, qv , qc with following way.
-∗
θ t+Δt = θ ∗ + γ (qv∗ − qvsw)



qvt+Δt = qv∗ + θ ∗ − θ t+Δt



4093Π
∗
1 + γqvsw(Πθ ∗ − 36)2γ

qct+Δt = qv∗ + qc∗ − qvt+Δt



(8.152)(8.153)(8.154)First, provisional θ, qv , qc are calculated with these equations. If qct+Δt > 0, the calculation withthese equations (8.152)∼(8.154) is repeatedly done until when the value of θ, qv , qc converges.
Therefore, ﬁnal θ, qv , qc is decided by the moist saturation adjustment. Here, we deﬁned γ asγ ≡ Lv / (Cp Π).
If Δqc ≤ 0, we apply following equations.

30

Soong and Ogura (1973)

θ t+Δt = θ ∗ − γqc∗

(8.155)

qvt+Δt = qv∗ + qc∗
qct+Δt = 0(8.156)(8.157)

198

8 Physical Processes of Cloud and Precipitation

Meanings of signs used here are shown as following.
Cp
Lv
qvsw
Π

8.2.6

Speciﬁc heat at constant pressure of dry air
Latent heat of evaporation of water
Saturation mixing ratio over water
Exner function

1004

J K kg−1
J kg−1kg kg−1

Formulization of Variational Terms on Mixing Ratio and Number Concentration with Precipitation

The rate of variation of mixing ratio by fall of particles of cloud and precipitation are shown as
following.

Fall.qx =1 ∂ ρ̄Ūxq qxρ̄ ∂z(8.158)The mass-weighted terminal fall velocity of particle in category x (Ūxq ) is given by the equation(8.68) in the section 8.2.3.
The rate of variation of the number concentration is shown as following.

Fall.

Nx
1 ∂Nx ŪxN
=
ρ̄
ρ̄
∂z

(8.159)ŪxN indicates mean fall velocity of particle in category x. Similarly, it is given by the equation(8.67) in the section 8.2.3.
When the grid interval in the vertical direction is small, Δt must be enough small so that CFL
condition is ﬁlled, and so that a falling precipitation particle should not pass next grid for 1 time
step.
Since the diﬀerentiation in the vertical direction is the diﬀerentiation in real space, the diﬀerentiation in z∗ coordinate （ζ coordinate） must be multiplied by a metric, as shown in equation(6.58).

8.2.7

The other constants

Here, we show variables and constants, which are used in equations but are not explained.
Water Saturation Mixing Ratio and Ice Saturation Mixing Ratio¦ qvsw , qvsi 31
610.78T − T0qvsw = 
exp 17.269
p
T − 35.86


610.78T − T0qvsi = exp 21.875
p
T − 7.86
31

Orville and Kopp (1977), Murray (1966)



(8.160)(8.161)

8.2 Parameterization of the bulk cold rain

199

Latent Heat with Evaporation, Sublimation and Melting of Water¦ Lv , Ls , Lf［J kg−1 ］




−4 T

T0 (0.167+3.67×10
Lv = 2.50078 × 10
T
Ls = 2.834 × 106 + 100 (T − T0 )6)
(8.162)(8.163)

5

Lf = 3.34 × 10 + 2500 (T − T0 )(8.164)

Coeﬃcient of Kinematic Viscosity of Air¦ ν ［m2 s−1 ］−5 p0

ν = 1.328 × 10

p



T
T0

1.754(8.165)

Coeﬃcient of Viscosity of Air: μ［kg m−1 s−1 ］
μ = ρν

(8.166)

Coeﬃcient of Water Vapor Diﬀusion: Dv ［m2 s−1 ］
Dv = 2.23 × 10−5

p0
p



T
T0

1.81

(8.167)

Meanings of signs used here are shown as following.
p
p0
T
T0

PressureStandard pressure
Temperature
Melting point of iceRatio of molecular weight of water vapor to molecularweight of dry air

ρAir density

101325273.16

Pa
Pa
K
K

0.622
kg m−3Chapter 9

Boundery Layer and Surface Process
The integration time of a cloud model is short ranging from few to ten
hours. Therefore, in the model, most of phenomena provided by the atmospheric processes. When energy ﬂux from the ground surface (land or sea)is large in a cloud / precipitation system as in the case of a thunderstorm insummer or a snow cloud over Japan Sea in winter, however, ground surface
process is substantially important for the development of the system.
Following processes are considered as the ground surface processes and they
are expressed by simple bulk methods.
• vertical one-dimensional diﬀusion in the atmospheric boundary layer• ﬂuxes of sensible heat and latent heat from the ground to atmosphere• momentum ﬂux (friction of the ground surface)• soil temperature variation
In CReSS, these processes are expressed with vertical one-dimensional equations. We adopt the ground surface ﬂux with a bulk method and soil temperature forecasting which are used in JSM (Japan Spectral Model; Segami etal., 1989). Complicated soil and vegetation models are incorporated in somecloud models (e.g. RAMS, ARPS). However, they are important only when
integration for a long time (month order) in a wide area (e.g. the region covering all the Eurasian Continent) is conducted. Therefore, the complicatedsoil and vegitation models will be added to CReSS in the future.

202

9 Boundery Layer and Surface Process

9.1

Basic Theories of Atmospheric Boundary Layer

Here, we summarize brieﬂy basic theories for the atmospheric boundary layer processes1 .

9.1.1

Structure of Atmospheric Boundary Layer

Atmosphere is bordered on the lower boundary by the ground surface. Generally, eﬀects ofthe ground surface which includes various conditions like land and sea become stronger as thealtitude becomes lower. The eﬀects exist from the surface to about 1-2 km high though the depth
is aﬀected with conditions of the ground surface and atmosphere. Such atmospheric layer whichis thermodynamically and dynamically aﬀected with the ground surface is called atmosphericboundary layer or planetary boundary layer; PBL.
Atmospheric boundary layer is devided into two layers; surface boundary layer which ranges
from the surface to 20-50 m high and is aﬀected by the ground surface, and Ekman layer or outerboundary layer above the surface boundary layer. Atmosphere above the atmospheric boundary
layer is called free atmosphere.
The surface boundary layer has a property that vertical ﬂuxes of sensible heat, latent heat,
momentum and so on are uniform vertically and equal to their values on the ground. In otherwords, the atmospheric layer where these vertical ﬂuxes are equal to the values on the ground canbe called the surface boundary layer. The surface boundary layer is sometimes called constant ﬂux
layer. Because momentum ﬂux is vertically constant, stress of turbulence is vertically constantin the surface boundary layer, that is, wind direction is also constant vertically.

9.1.2

Parameterization of Turbulence Transportation

Motions of various scale exist in the atmosphere. Among them, motion which can be expressedon the grid points of a numerical model is called grid-scale motion or mean motion. Motion which
has a scale smaller than a grid size is called subgrid-scale motion or eddy motion.
To classify these motions, physical quantity A (e.g. wind speed, temperature, mixing ratio ofwater vapor) is devided into two quantity; averaged quantity which can be expressed on the grid
points and deviation from that, as follows.

A = Ā + A(9.1)

There are some ways to average physical quantities, we will not mentione about them here.
Average of deviation and that of product of two physical quantities are expressed as follows;A = 0(9.2)

AB = ĀB̄ + A B 

(9.3)

The average of the pruducts of some quantities is not always equal to the product of averages ofeach quantity, and the second term of the equation above appears. This equation is applied to1

Introduction to dynamic meteorology (Ogura, 1978) and Atmospheric Science Course 1 - Atmosphere near the
ground surface (Takeuchi and Kondo) are mainly reﬀered here.

9.1 Basic Theories of Atmospheric Boundary Layer203

the x component of equations of motion. Considering an uncompressed ﬂuid (ρ = const.), eachterm of the equation can be expressed with the average and deviation from that as follows;

∂ ū
∂ ū
∂ ū
∂ ū
1 ∂ p̄
+ ū
+ v̄
+ w̄
− f v̄ = −
∂t
∂x
∂y
∂z
ρ ∂x


∂
∂
1 ∂
ρu u +ρu v +ρu w + ν∇2 ū.
−
ρ ∂x
∂y
∂z

(9.4)

The terms of −ρu u , −ρu v and − ρu w express the stress of turbulence and are called eddy
stress or Reynolds stress. These stresses can be regarded as a momentum transportation, and themomentum transportation by eddy is the stress.
Similarly, equations of potential temperature and mixing ratio of water vapor are as follows.

∂ θ̄
∂ θ̄
∂ θ̄
∂ θ̄
∂
∂  
∂  
+ ū
+ v̄
+ w̄
= − u θ  −
v θ −
w θ
∂t
∂x
∂y
∂z
∂x
∂y
∂z

(9.5)

The quantities with a prime cannot be expressed with the grid points. It is neccessary to
express them with quantities on the grid points (averaged quantities) for including eﬀects ofmomentum transportation by eddy which is smaller than the grid size. A problem of turbulent
parameterization is to decide the way to express deviation of quantity by eddy with prime on
the grid points. Note that deﬁnitions of quantities with − used in this chapter and in the otherchapter (e.g. Chapter 2) are diﬀerent.

9.1.3

Mixing Length Theory

As mentioned in the chapter of turbulence, atmospheric motions should have a motion whichis smaller than the grid size. Therefore, the equation system is not closed due to correlation
quantities of deviation as in (9.4) and (9.5). This is called closure problem and the terms oftransportation by turbulence must be expressed as functions of quantities on grid points to closethe equation system. One of the turbulent parameterization methods is mixing length theory byPrandtl (1925). This is explained by an analogy of kinetic theory of gases as follows.
We assume that the mean vertical speed of air mass at the altitude of z − l is ū(z − l ), andgives momentum to the environment when the air mass moves upward for l reaching to thealtitude of z. The variation of the speed is∂ ūu = ū(z − l ) − ū(z) ∼= −l .
∂z(9.6)

Raynolds tension isτzx = −ρu w = ρl w

∂ ū
∂ ū
= ρKmz
∂z
∂z

(9.7)

204

9 Boundery Layer and Surface Process

where

Kmz = l w

(9.8)

is eddy viscosity coeﬃcient.
Considering ﬂuid with neutral stratiﬁcation, the eﬀect of buoyancy can be neglected, and it
can be assumed |w | ∼ |u |. We can assume the following relation.

w ∼ u ∼= −l

∂ ū
∂z

(9.9)

∂ ū ∂ ū.
∂z ∂z(9.10)Then,τzx = −ρu w = ρl2

and,Kmz = l2

∂ ū
.
∂z

(9.11)

Using these equations,

−



1 ∂
1 ∂
ρu w = −
ρ ∂z
ρ ∂z
∂
∂
− w θ  = −
∂z
∂z

ρKmz



∂ θ̄
Khz
∂z

∂ ū
∂z



(9.12)

(9.13)are gained.
As mentioned above, the transportation of physical quantities with turbulence could be explained by using mean ﬁeld based on an analogy of kinetic theory of gases. That is called
Prandtl’s mixing length theory. It is diﬃcult to apply generally this theory for turbulent transportation. For the surface boundary layer, we can apply the theory as a basic theory to calculate
ground surface ﬂux. The other theories are needed to solve a closure problem for the whole of
the atmospheric boundary layer including Ekman layer.

9.1 Basic Theories of Atmospheric Boundary Layer

9.1.4

205

Surface Boundary Layer

The surface boundary layer is called constant ﬂux layer where a value of vertical momentum ﬂux
at all altitudes is equal to that of the ground surface, and stress is constant. Therefore, winddirection is constant in the surface boundary layer. We set that the direction of an averaged windis the x direction. Considering (9.10),
∂ ū
1
=
∂z
l

.

τzx
ρ

(9.14)/

where l = l2 . The size of turbulence which transports momentum within the surface boundary
layer is restricted by the ground surface. The mixing length l is assumed as follows;

l=kz(9.15)

where z is the distance from the surface and k is nondimensional quantity called Karman constant.
k is decided as k = 0.4 from wind tunnel experiments and observations2 .
We deﬁne u∗ as follows;
u2∗ ≡

τzx
= −u w .
ρ

(9.16)

This is called friction velocity and has a dimension of velocity, corresponding to the scale ofturbulent strength.
Observations showed that the amount of vertical transportation of momentum (or stress) isconstant to altitudes within the surface boundary layer. Using these equations,
.

u∗ =

∂ ū
τzx
= kz
= const.
ρ∂z(9.17)

is gained. Integrating this equation for z,

ū =

u∗
ln z + C.
k(9.18)

Since the actual ground surface has various irregurality, it can not say that ū = 0 at z = 0.
Then, the constant of integration C is decided as the value which makes ū = 0 at z = Z0m .
ū =
2

u∗
z
ln
k
Z0m

(9.19)

Karman constant is decided as k = 0.4 ± 0.01 by simultaneous observations of Raynolds stress τ = −ρuw
and vertical proﬁle of wind speed (Meteorology of water environment, p99, Kondo).

206

9 Boundery Layer and Surface ProcessThis is the logarithmic proﬁle of wind speed. Z0m means the extent of roughness on the groundand is called aerodynamical roughness parameter (or length) or ground surface roughness forwind proﬁle. The logarithmic proﬁle expressed with this equation corresponds well with theactual wind proﬁle when the atmosphere has a neutral stratiﬁcation.
Eddy viscosity coeﬃcient in the surface boundary layer Kmz can be expressed by (9.11) and(9.17) as follows;
Kmz = (kz)2

9.1.5

u∗
= kzu∗ .
kz(9.20)

Vertical Flux

Momentum ﬂux and Bulk coeﬃcient
In the surface boundary layer, vertical ﬂux is constant for altitude and it is enough to estimate ﬂux
at an altitude in the surface boundary layer for the estimation of ﬂux on the ground. Momentum
ﬂux, i.e., wind stress on the ground is as follows;
⎡

⎤2

⎢

k ⎥
2
z ⎦ = ρCm ū
ln
Z0m

τ0 = −ρu w = ρu2∗ = ρū2 ⎣(9.21)ū can be gained by measuring average wind speed at an altitude (usually 10 m) in the surface
boudnary layer. Cm deﬁned as
⎡

⎤2

⎢

k ⎥
z ⎦
ln
Z0m

Cm = ⎣(9.22)

is nondimensional quantity and called drag coeﬃcient or bulk transfer coeﬃcient.
Potential Temperature Flux
In the surface boundary layer, it can be considered that the amount of transportations (ﬂuxes)of sensible heat and latent heat Hs , HL are constant at all altitudes. If turbulence is decided byu∗ and z in the surface boundary layer, equation which appears in (9.13);

Kmz

∂ θ̄
= w θ ∂z

(9.23)

9.1 Basic Theories of Atmospheric Boundary Layer

207

is expressed as follows using eddy viscosity coeﬃcient (9.20) as in the case of momentum;w θ 
∂ θ̄
=
∂z
kzu∗

(9.24)

Sensible heat ﬂux from the ground Hs ［J m−2 s−1 ］is oﬀered as follows;

Hs = −ρCp w θ 

(9.25)

where potential temperature scale θ∗ which is deﬁned byθ∗ =w θ u∗(9.26)and potential temperature ﬂux is expressed as follows;Hs∂ θ̄= −w θ  = −kz u∗ = −θ∗ u∗
ρCp∂z(9.27)

θ∗ deﬁned here is constant in the surface boundary layer. Therefore,
∂ θ̄
θ∗
=
∂z
kz

(9.28)

is obtained. By integrating this vertically and deﬁning Z0h as the altitude of θ̄ = θG as in thecase of momentum,θ̄ − θG =

θ∗
z
ln
k
Z0h

(9.29)

is gained. Potential temperature ﬂux is expressed with this as follows;Hskk= −w θ  = −θ∗ u∗ = −zz (θ̄ − θG )ū
ρCp
ln
ln
Z0h Z0m

where(9.30)

208

9 Boundery Layer and Surface Process

Hs
k
k
=−
z
z (θ̄ − θG )ū = −Ch (θ̄ − θG )ū.
ρCp
ln
ln
Z0h Z0m(9.31)

Bulk coeﬃcients of sensible heat and potential temperature can be deﬁned as follows;

Ch =

k
k
z
z .
ln
ln
Z0h Z0m(9.32)

Mixing Ratio of Water Vapor Flux
Latent heat ﬂux HL can be regarded to be constant in the surface boundary layer,
HL = l E = −Lv w qv(9.33)∂ q̄v
w qv=∂zkzu∗(9.34)

andwhere L is latent heat of evaporation. Flux of mixing ratio of water vapor is expressed as follows.

E = −w qv = −kzu∗

∂ q̄v
∂z

(9.35)In the same way, we deﬁne the scale of mixing ratio of water vapor (friction mixing ratio) qv∗ asqv∗ =w qvu∗(9.36)and ﬂux of mixing ratio of water vapor is expressed as

E = −w qv = −kz∂ q̄vu∗ = −qv∗ u∗∂z(9.37)

9.2 Boundary Layer, Surface and Soil Temperature

209

where qv∗ deﬁned here is also constant in the surface boundary layer. By integrating this verticallyand deﬁning an altitude Z0h of which q̄v = qvsf c as same as the case of momentum and potential
temperature,q̄v − qvsf c =

qv∗
z
ln
k
Z0h

(9.38)

is given. This equation is expressed with Mixing ratio of water vapor as follows;−w qv = −qv∗ u∗ = −kkzz (q̄v − qvsf c )ūln
ln
Z0h Z0m(9.39)

This is expressed by using the bulk coeﬃcient of potential temperature deﬁned by (9.32) as
follows;−kzz (q̄v − qvsf c )ū = −Ch (q̄v − qvsf c )ūln
ln
Z0h Z0mk(9.40)

Vertical ﬂux of momentum, potential temperature and water vapor mixing ratio in the atmospheric boundary layer have been described with bulk coeﬃcient. In the ground surface processof CReSS, the exchange of sensible and latent heat between land and atmosphere is describedby calculating these bulk coeﬃcients. The eﬀect of ground surface friction is included in a caluculation of momentum ﬂux.

9.2

Boundary Layer, Surface Process and Soil Temperature

The ways of calculation used in JSM (Japan Spectral Model) by Segami et al. (1989) are basicallyadopted in the calculations of atmospheric boundary layer, ground surface process and soil temperature in CReSSTheir calculations are one-dimensional vertically. However, following main
diﬀerences exist between the calculations in CReSSand those in JSM.
• On the calculation of soil temperature, a four layer model is used in JSM but we can selectan arbitrary number (m) of layers in CReSS.
• On the calculation of soil temperature, it starts at n − 1 step and uses a complicated calculation method adopting diﬀerential of ﬂux on temperature in JSM. On the other hand in
CReSS, the calculation method is a simple implicit one, just solving simultaneous equations.
• Cloudiness is estimated from relative humidity by the method of Ohno and Isa (1983) in
JSM, but it is estimated from mixing ratio of cloud water and cloud ice in CReSS.

210

9 Boundery Layer and Surface Process• In JSM, quantities of atmosphere are calculated at ﬁrst and after that, sensible heat ﬂux
and latent heat ﬂux on the ground are calculated from temperature and relative humidity of
the atmosphere which are modiﬁed by the ﬁrst calculation of the atmosphere. In CReSSon
the other hand, they are calculated from temperature and mixing ratio of water vapor ofn − 1 step and the result is applied for the atmosphere and the ground.
• The calculation of vertical diﬀusion is conducted for all layers of the atmosphere at every
step in JSM and ﬂux entering the atmosphere from the ground aﬀects instantly (within thetime step of Δt) to top of the atmosphere. In CReSS, the layers aﬀected by the ﬂux canbe designed arbitrary.
• On the calculation of vertical diﬀusion, an implicit method modifying dependent variables
is used in JSM but an explicit method is used in CReSS. Rate of change for time can becalculated and it is added as a forcing term for a time integration.
The calculations of the atmospheric boundary layer, ground surface process and soil temperature are conducted mainly as follows. They are independently conducted for the vertical
one-dimension on each horizontal grid.
• Cloudiness is calculated. Usually cloudinesses are calculated for upper, middle and lower
layers.
• Solar radiation and downward long wave radiation are calculated.
• Coeﬃcients of ﬂuxes of momentum, sensible heat and latent heat on the ground surface arecalculated.
• Horizontal wind speed, potential temperature and mixing ratio of water vapor are modiﬁedby calculation of the atmospheric boundary layer process.
• Soil temperature is calculated.
It is notable that, on the ground surface process, ﬂux is not calculated but its coeﬃcient iscalculated. The reason is that sensible and latent heat ﬂuxes,

HS = −ρa Cp Ch |Va|(Ta − TG )(9.41)

l E = −ρa Lv Ch |Va |β [qva − qvs (TG )](9.42)are given as external forces on the ground surface, and potential temperature ﬂux 3 and ﬂux of
mixing ratio of water vapor,3Fθ = −ρa Ch |Va |(θva − θvG )(9.43)

Fqv = −ρa Ch |Va |β [qva − qvs (TG )](9.44)

Virtural potential temperature is used here instead of potential temperature. It makes no diﬀerence because
ﬂux is proportional to the diﬀerence of quantities. However, it is notable that it makes big diﬀerence when they
are used to calculate CAPE.

9.2 Boundary Layer, Surface and Soil Temperature211

are given as external forces from the ground surface in the atmosphere. Coeﬃcients common to
the two above ρa Ch |Va | (actually it is diﬀerent between for potential temperature and for mixing
ratio of water vapor) are calculated in the ground surface process and are given to the processesof the atmospheric boundary layer and soil temperature. Then sensible and latent heat ﬂuxes areincluded in the process of soil temperature, while the eﬀects of ﬂuxes of potential temperatureand mixing ratio of water vapor are included in the process of the atmospheric boundary layer.
Mixing ratio of water vapor corresponding to the ground surface temperature is calculated in the
ground surface process and given to each process simultaneously.
Processes for the calculation of the atmospheric boundary layer, the ground surface process
and soil temperature are explained in the next section.

9.2.1

Cloudiness

The amount of solar radiation reaching to the ground decreases by the eﬀect of cloud. Cloudiness
is used to estimate this eﬀect. The way of Ohno and Isa (1984) to estimate it by an empiricalformula of cloudiness and relative humidity is explained in this section.
Cloudiness is nondimensional quantity from 0 to 1. Cloudiness is given as a function of relative
humidity for upper, middle and lower layers, respectively. The height of each layer is not exact
and 400hPa is for the upper layer, 500 and 700hPa for the middle layer and 850hPa for the lower
layer in Ohno and Isa (1984). Upper layer corresponds to 7-7.5km, middle corresponds to 5-6
and 3km, and lower corresponds to 1.5km.
Cloudiness is given by 21 data every 5% ranging from 0 to 100% from the result of Ohno andIsa (1984). When average relative humidity in each layer is deﬁned as Rh[%], an integer kn isdeﬁned as follows;

kn = 1 + 20 ×Rh
100

(9.45)and cloudiness of lower layer CDL 、middle layer CDM and upper layer CDH is deﬁned as follows(written in the format of data description of Fortran);data cdldef / 0.e0, 0.e0, 0.e0, 0.e0, 0.e0, 0.e0, 0.e0, 0.e0,.
0.e0, 0.e0, 0.e0, 0.e0,.07e0,.11e0,.19e0,.40e0,.
.85e0, 1.e0, 1.e0, 1.e0, 1.e0 /data cdmdef / 0.e0, 0.e0, 0.e0, 0.e0, 0.e0, 0.e0, 0.e0, 0.e0,.
0.e0, 0.e0, 0.e0,.05e0,.12e0,.30e0,.40e0,.50e0,.
.70e0,.95e0, 1.e0, 1.e0, 1.e0 /data cdhdef / 0.e0, 0.e0, 0.e0, 0.e0, 0.e0, 0.e0, 0.e0, 0.e0,.
0.e0,.05e0,.15e0,.30e0,.55e0,.75e0,.87e0,.95e0,.
1.e0, 1.e0, 1.e0, 1.e0, 1.e0 /212

9 Boundery Layer and Surface Process

To calculate relative humidity from mixing ratio of water vapor qv and air temperature Ta ,partial pressure of water vapor estimated from qv and saturated water vapor pressure at the airtemperature is calculated by using Tetens (1930)’s equation. Relative humidity is given from theratio of the partial pressure of water vapor and saturated water vapor pressure.
Then, air pressure, air temperature and mixing ratio of water vapor are deﬁned as p (= p̄ + p ),Ta and qv , respectively, partial pressure of water vapor is given as follows;

ea =

qv p
 + (1 − ) qv(9.46)

where  is the ratio of molecular mass of water vapor to that of dry air, which is 0.622 whenmixing ratio of water vapor is provided with the unit of [kg kg−1 ], and is 622.0 when it is provided
with the unit of [g kg−1 ]. The unit of air pressure p and partial pressure of water vapor ea is [Pa].
We assume that air temperature T [◦ C] (note that the unit is degrees Celsius) is given, saturated
water vapor pressure of the air temperature (Tetens (1930)’s equation) becomesa·T

eas = es · 10 b+T(9.47)

or




a·T
eas = es exp
ln 10
b+T

(9.48)

where es (= 610.78 Pa) is saturated water vapor pressure at 0◦ C. The constant values a and b
are deﬁned as follows;◦（T ≥ 0 C）on the water（T < 0◦ C）on the ice

a

b

7.5
9.5

237.3
265.3

By using these, relative humidity Rh is gained as follows.

Rh =

ea
× 100
eas

(9.49)

Cloudiness is derived with this.
By using the method above, cloudiness is decided irrespective of the actual cloud distribution. Because mixing ratios of water substances are calculated in CReSS, cloudiness should be
estimated from these values. The total amount of cloud water [m−2 ] is calculated as follows.

9.2 Boundary Layer, Surface and Soil Temperature

213

* zL

qL =

ρ (qc + qr + qi + qs + qg ) dz

(9.50)

ρ (qc + qr + qi + qs + qg ) dz

(9.51)

ρ (qc + qr + qi + qs + qg ) dz

(9.52)

zs

* zM

qM =
zL

* zt

qH =
zM

where, zL , zM are the heights of interfaces between upper, middle and lower layer, respectively.
Then, cloudiness is estimated from the relation between these total amount of cloud water andthe solar radiation4 .

9.2.2

Solar Radiation and Downward Long Wave Radiation

Solar radiation（short wave radiation） and downward long wave radiation from atmosphere arecalculated as a function of cloudiness, zenith angle of the sun, near–surface temperature, water
vapor pressure and albedo at the ground surface.
The estimation of solar radiation and atmospheric long wave radiation are summarized asfollows.
• Partial pressure of water vapor ea is obtained by the temperature and mixing ratio of water
vapor qv at the near surface layer (around the ﬁrst or second layer).
• The zenith angle of the sun for each grid at the time is calculated.
• Downwarad short wave radiation is estimated by the partial pressure of water vapor ea ,zenith angle and cloud amount with the consideration for albedo.
• Downward long wave radiation is represented as a function of atmospheric temperature Ta ,partial pressure of water vapor ea and cloud amount.

Following subsections explain about net (downward) short wave radiation and downward longwave radiation, respectively.

Partial Pressure of Water Vapor at the Ground Surface
Partial pressure of water vapor at lower layer of atmosphere is estimated utilizing temperatureand mixing ratio of water vapor at the lowest atmospheric layer or averaged among some lower
atmospheric layers5 .
4
5

Kondo (2000)
In JSM, averaged value of ﬁrst and second atmospheric layers is used214

9 Boundery Layer and Surface Process

Zenith Angle of the Sun
Zenith angle of the sun is calculated as a function of the number of days from the ﬁrst day ofJanualy jday, local time TLC and latitude at the point lat.
Local time is a function of longuitude based on the universal standard time U T C. It is expressedbyTLC = U T C + lon/15(9.53)

φs = 23.44 cos(172 − jday)(9.54)

Angle of the sun φs is expressed as

then, zenith angle of the sun is expressed as follows (see ﬁg.9.1):

ζ = sin(lat) sin(φs ) + cos(lat) cos(φs ) cos[0.2618(TLC − 12)].

(9.55)zζFigure 9.1. Zenith angle of the sunζ .

Net (Downward) Short Wave Radiation
The solar radiation which reaches at the ground surface is called downward short wave radiation.
Considering it’s reﬂection from the ground surface with a rate of albedo, downward short wave
radiation which is absorbed into the ground is estimated. It is called net downward short wave
solar radiation.
In order to estimate total solar radiation which reaches at the ground surface, it is needed toestimate solar radiation which arrive at the top of atmosphere S∞ 6 .

S∞ = S0 cos ζ
6(S0 = 1, 367[W m−2 ] : solar constant)in actual, S∞ = S0 (r̄E /rE )2 cos ζ. rE is distance between earth and sun, r̄E is its average(9.56)

9.2 Boundary Layer, Surface and Soil Temperature

215

where ζ is zenith angle of the sun. If we now let

a = 1.12 − b − 0.06 log 10 ea(1 ≤ ea ≤ 3000[Pa] : water vapor pressure near surface) (9.57)

b = 0.43 + 0.00016ea(9.58)

then solar radiation of the clear weather isS↓ =S∞ (a + b × 10−0.13secζ )0cos ζ > 0cos ζ ≤ 0

(9.59)

Adding the eﬀect of the cloud and its albedo, net downward short wave radiation which isabsorbed into the ground is
RS ↓ = (1 − Al )S ↓ (1 − cdL · CDL )(1 − cdM · CDM )(1 − cdH · CDH )(9.60)

where
cdL
cdM
cdH

absorbing and refrecting eﬀects with the lower cloudabsorbing and refrecting eﬀects with the middle cloudabsorbing and refrecting eﬀects with the upper cloud

0.7
0.6
0.3

As mentioned before, a cloud amount is calculated by empirical formula using relative humidity(Ohno and Isa, 1984). Albedo Al is shown in the following table.
land surface

given from datasea ice or snow surface

constant to 0.6

At the sea surface, there is no need to consider solar radiation because sea temperature isconstant during a calculation, so that there is no need to give the albedo.
Downward Long Wave Radiation from the Atmospere at the Ground Surface
Using empirical formula of Kondo (1976), downward long wave radiation from atmosphere including eﬀects of cloud and water vapor is√L↓ = εG σTa4 [1 + (−0.49 + 0.0066 ea )(1 − CD · C · Ac )](9.61)

216

9 Boundery Layer and Surface Process

where Ta is estimated by a temperature at the lowest atmospheric layer or averaged for somelower atmospheric layers7 and

C = 0.75 − 0.005ea
Ac =(9.62)Nr
CDL + 0.85CDM + 0.5CDH
+ 0.1
CD
N

(9.63)

CD = CDL + CDM + CDH

8(9.64)

Then, utilized constants are expressed in the following table.

εG
σ

9.2.3

emissivity for infrared region at the ground surface
Stefan-Boltzman constant

0.95
5.67×10−8

W m−2 K−4

Flux at the Ground Surface

Momentum ﬂux at the ground surface τx , τy [kg m s−1 m−2 s−1 = N m−2 ] is expressed as follows
with bulk coeﬃcient and the absolute value of wind speed.

τx = ρa Cm |Va |ua(9.65)

τy = ρa Cm |Va |va(9.66)

Sendible and latent heat ﬂux HS , l E ［J m−2 s−1 = W m−2 ］ is given as

HS = −ρa Cp Ch |Va |(Ta − TG )(9.67)

l E = −ρa Lv Ch |Va |β [qva − qvs (TG )](9.68)(9.69)

where a and G mean the ﬁrst layer of the atmosphere and the ground surface (ﬁrst layer of theground temperature), Cm and Ch are the bulk coeﬃcient (with no dimension) of the momentum,
and heat and water vapor, qvs (TG ) is saturated mixed ratio of ground temperature TG , β isevapotranspiration coeﬃcient and Lv is latent heat of evaporation of water. They are multiplied
atmospheric density and rewrited to equivarent temperature ﬂux Fθ [kg K m−2 s−1 ] and water
vapor mixed ratio ﬂux Fqv [kg m−2 s−1 ] as7

In JSM, averaged value of ﬁrst and second atmospheric layers is used
Nr /N means appearance rate of precipitating time. In temporal integration, it is set to 0 in case of no
precipitation and to 1 in case of precipitation through the time. In JMS, on the other hand, it is set to 0 constantly
though there is no conﬁdence to be ﬁnest method.
89.2 Boundary Layer, Surface and Soil Temperature

HS
Fθ =
Cp



p0
p

R

Cp

= −ρa Ch |Va |(θva − θvG )= −ρa u∗ θ∗

Fqv =

E

217

(9.70)= −ρa Ch |Va |β [qva − qvs (TG )]= −ρa u∗ qv∗(9.71)

where θ∗ is scale of potential temperature (frictional temperature) and qv∗ is scale of mixing ratio(frictional mixing ratio). Friction velocity u∗ is deﬁned asu2∗ = Cm |Va |2(9.72)

where |Va | is absolute value of ground surface wind speed which expressed as1|Va | = (u2a + va2 ) 2(9.73)

1 + iv qv
θ
1 + qv(9.74)

Virtual potential temperature θv isθv =where iv is ratio of molecular weight of water vapor to that of dry air.
Non-dimensional bulk coeﬃcients Cm , Ch are expressed by the scheme of Louis et al. (1980)
considering momentum, heat and roughness parameter of water vapor as follows:⎡⎢

Cm = ⎢
⎣

Ch =

⎤2
⎥
k
⎥
f (Ri, za , Z0m )za ⎦ m
ln
Z0m


k
k
 
 fh (Ri, za , Z0m , Z0h )
za
za
ln
ln
Z0m
Z0h


(9.75)(9.76)

where k (=0.4), za , Z0m , Z0h and Ri are Kalman coeﬃcient, height of the ﬁrst atmosphericlayer, roughness of momentum, roughness of heat and water vapor, and Richardson number,respectively.
In unstable case (Ri < 0),218

9 Boundery Layer and Surface Process

fm = 1 −⎢1 + 3bc ⎢⎣fh = 1 −2b · Ri

⎤2

⎡

⎥
k
⎥
za ⎦
ln
Z0m

(9.77)

3b · Ri−

Ri · za
Z0m

1
2



k
k
Ri · za  −1 + 3bc 
za
za
Z0h
ln
ln
Z0m
Z0h

(9.78)

1
2

b=c=5

(9.79)In stable case (Ri > 0),

fm =

1
√
1 + 2b · Ri · 1 + d · Ri

(9.80)fh =1√
1 + 3b · Ri · 1 + d · Ri

(9.81)

b=d=5(9.82)

Roughness parameters of momentum Z0m and of heat and water vapor Z0h are shown in thefollowing table. Roughness of the sea surface is reﬁned at the every calculating step.
roughness
Z0m
Z0h

land surface

sea surface

sea ice or snow surface

given from data set
constant value (0.1 m)

calculating as a function of u∗same with Z0m

optional constant value
same with Z0m

Standing on the above, actual calculation of land surface ﬂux is shown as follows.
(1) Calculation of Absolute Value of Horizontal Wind Speed
Its absolute value |Va | is calculated by using horizontal wind speed at the ﬁrst atmosphericlayer (ua , va ) (9.73).
(2) Calculataion of Saturated Mixing Ratio of Water Vapor Corresponding to Ground
Surface Temperature
Saturated mixing ratio of water vapor qvs (TG ) corresponding to ground temperature at the ﬁrst
ground layer or sea surface temperature is obtained by the equation of Tetens (1930) or (9.48).
From partial pressure of saturated water vapor by the equation of Tetens (1930), saturated
mixing ratio of water vapor [kg/kg] is expressed as9.2 Boundary Layer, Surface and Soil Temperature

qvs (TG ) = 

eas
p

219

(9.83)

Originally this is the equation to deﬁne speciﬁc humidity, but we can regard it as mixing ratiowith a small error (diﬀerence). p is expressed as p = p̄ + p , and  is ratio of molecular weight ofdry air to water vapor.
(3) Deﬁnition of Mixing Ratio of Water Vapor at the Ground Surface
Mixing ratio of water vapor at the ground surface is deﬁned for land, sea, sea ice and snow
surfaces, respectively. Considering evapotranspirate eﬃciency W ET which is given to land surface
only, it is deﬁned as follows:
for land surface,qvsf c = β [qvs (TG ) − qva ] + qva(9.84)

for sea, sea ice and snow surfaces,

qvsf c = qvs (TG )(9.85)(4) Calculation of Richardson Number
Richardson number of the ground surface is calculated in order to decide stability at the groundsurface and to calculate fm, fh in the scheme of Louis et al. (1980).
Richardson number Ri can be expressed asRi =gza Δθv
θva |Va |2(9.86)

where iv indicate ratio of the molecular weight between dry air and water vapor, then1 + iv qvsf c
θvG = TG
1 + qvsf c
Δθv = θva − θvG



p0
p

R

Cp

(9.87)(9.88)and za is the height of ﬁrst atmospheric layer and g is the acceleration of gravity.
(5) Calculation of Z0m and u∗ at the Sea Surface at the First Step by Iteration
Though both roughnesses at the land surface Z0m and Z0h are constant value during the calculation, those at the sea surface are estimated by the iteration with frictional velocity considering

220

9 Boundery Layer and Surface Process

its dependence on wind speed. The iteration method is used at the only ﬁrst step and Z0m is
on and after the second step.
estimated by frictional velocity of the one step before; un−1∗
Z0m is expressed as a function of u∗ based on the formula of Kondo (1975) as follows:

Z0m = −34.7 × 10−6 + 8.28 × 10−4 u∗−2Z0m = −0.277 × 10

−3

+ 3.39 × 10

u∗

for

u∗ ≤ 1.08[m s−1 ](9.89)

for−1(9.90)u∗ > 1.08[m s]where maximum of Z0m is 1.5 × 10−5 m. From (9.72), u∗ would beu∗ = |Va |/Cm(9.91)

Although bulk coeﬃcient Cm is estimated by (9.75), its coeﬃcient fm is depended on Richardson
number which given by (9.86); in unstable case, it is estimated by (9.77) while in stable case, itby (9.80).
Z0m is calculated with 9.89 or 9.90 using the friction velocity u∗ . The calculation using theroughness is iterated untill the value of Z0m , u∗ converges. Then, we can decide Z0m , u∗ at thesea surface.
(6) Calculation of fm, fh for the Whole of a Region
In the subsection (5), the roughness parameter of momentum at the sea surface Z0m is estimated. Roughness parameters of heat and water vapor Z0h are deﬁned as Z0h = Z0m . At theland surface, the roughness of momentum Z0m is given as a data and the roughnesses of heat
and water vapor Z0h are deﬁned as constant value 0.1 m. In case of sea ice and snow cover,roughnesses of momentum, heat and water vapor are deﬁned as constant values.
Roughness parameters of momentum, heat and water vapor for the whole of a region are
decided, so coeﬃcients fm , fh which are used in the scheme of Louis et al. (1980) and multiply
by bulk coeﬃcient are calculated by using (9.77), (9.78), (9.80), and (9.81).
(7) Calculation of Bulk Coeﬃcients Cm , Ch for the Whole of a Region
Using fm , fh and roughness parameters Z0m andZ0h which are estimated from (5) and (6), bulk
coeﬃcients Cm, Ch are calculated with (9.75) and (9.76).
(8) Calculation of Friction Velocity u∗ for the Whole of a Region
According to (9.91), friction velocity u∗ is estimated for the whole of a region using bulk
coeﬃcients of momentum Cm and |Va |.
(9) Calculation of Correction Term for Potential Temperature and Flux of Mixing
Ratio Used in Kondo(1975).
The correction term by Kondo (1975) is estimated to calculate ﬂuxes of water vapor and heatat the sea surface.
Kondo (1975) suggest the following equations of correction factor Bh−1 andBe−1 by the observation.
At the sea surface,9.2 Boundary Layer, Surface and Soil Temperature1
Bh−1 = lnkBe−1 =

ν + ku∗ Z0m
Daν + ku∗ Z0m
1
ln
k
Dv

221



(9.92)
= Bh−1 +



Da
1
ln
k
Dv



(9.93)or,Bh−1 =

1 
ln 0.71 + 4.64 × 104 ku∗ Z0m
k(9.94)

1k(9.95)

Bh−1 = 0(9.96)

Be−1 = Bh−1 − 0.168

At the land surface, sea ice and snow cover,Be−1 = 0(9.97)

where ν is kinetic viscosity coeﬃcient of air [m2 s−1 ], Da is diﬀusion coeﬃcient of air [m2 s−1 ]and Dv is diﬀusion coeﬃcient of water vapor [m2 s−1 ]9 . 10
9Kinetic viscosity coeﬃcient of air ν[m2 s−1 ] is expressed by,ν = ν0101325p

T
273.16

1.754(9.98)

ν0 = 1.328 × 10−5

(9.99)

Diﬀusion coeﬃcient of air Da [m2 s−1 ] is expressed by,
Da = Da0101325p

T
273.16

1.78(9.100)Da0 = 1.87 × 10−5

(9.101)Diﬀusion coeﬃcient of water vapor Dv [m2 s−1 ] is expressed by,
Dv = Dv0101325p

Dv0 = 2.23 × 10−5T
273.16

1.81(9.102)(9.103)

However, it is enough to use apploximation in the text.
10

Bh is Stanton number at the bottom layer and Be is Dalton number at the bottom layer (大気科学講座 1 P95).

222

9 Boundery Layer and Surface Process(10) Calculation of Flux Coeﬃcients of Potential Temperature and Mixing Ratio of
Water Vapor
In order to simplify the estimation of ﬂuxes of potential temperature and mixing ratio of water
vapor, the coeﬃcients Eθ andEqv are calculated.


Eθ =

−1
u∗
θ∗
+ Bh−1=
θva − θvG
Ch |Va |(9.104)



−1
qv∗
u∗
+ Be−1=Eqv =qva − qvsf c
Ch |Va |(9.105)Though same bulk coeﬃcient Ch has been used in case of water vapor and heat ﬂuxes, the use ofabove coeﬃcients adopting Bh−1 andBe−1 make diﬀerent coeﬃcient value in case of the sea surface.
In cases of the land surface, sea ice and snow cover, however, these values become equally 0.
(11) Calculation of Flux
Standing on the above, ﬂuxes of momentum, sensible heat, latent heat, potential temperatureand mixing ratio of water vapor at the ground surface are given as follows.
At ﬁrst, absolute value of momentum ﬂux is expressed byτ = ρa u2∗(9.106)

τx = τ

ua|Va |(9.107)τy = τ

va|Va |(9.108)and its x and y components are

Secondly, sensible heat and latent heat ﬂuxes are expressed by

HS = −ρa Cp u∗ Eθ (Ta − TG )(9.109)l E = −ρa Lv u∗ Eqv (qva − qvsf c )(9.110)and, ﬂuxes of potential temperature Fθ [kg K m−2 s−1 ] and mixing ratio of water vapor Fqv[kg m−2 s−1 ] which are multiplied by atmospheric density are expressed as follows:Fθ = −ρa u∗ Eθ (θva − θvG )(9.111)

Fqv = −ρa u∗ Eqv (qva − qvsf c )(9.112)

9.2 Boundary Layer, Surface and Soil Temperature223

At land surface, qva − qvsf c = β [qva − qvs (TG )] is obtained using (9.84). Then, (9.110) and (9.112)
take account of evapolation eﬃciency β through qvsf c .
The above is the calculation of ground surface ﬂux, however, (9.109) and (9.110) are usedin the calculating process of ground temperature actually. (9.111) and (9.112) are used in thecalculating process of atmosperic boundary layer. Therefore, this process returns following threecoeﬃcients which are common to these equations,

1
|Va |

(9.113)Cθ = ρa u∗ Eθ(9.114)

Cqv = ρa u∗ Eqv(9.115)

Cw = ρa u2∗

and mixing ratio at the ground surface qvsf c which include β in case of the land surface. Utilisingthese coeﬃcients, x, y components of momentum ﬂux and ﬂux of mixing ratio are expressed asfollows:

τx = Cw ua(9.116)τy = Cw va(9.117)Fθ = Cθ (θva − θvG )(9.118)

Fqv = Cqv (qva − qvsf c )(9.119)When we don’t use an implicit method in the calculation of atmospheric boundary layer process,we can calculate ﬂuxes of momentum, sensible heat, latent heat, potential temperatre and mixing
ratio, instead.
(12) Calculation of Monitor Data at the Ground Surface
Temperature and wind speed at the ground surface are important to be compared with observed
physical parameter at the ground surface. Surface boundary layer touching the ground surface iscalled as constant ﬂux layer where vertical ﬂux is constant regardless of height. According to thenature, temperature and mixing ratio of water vapor at a height of 1.5 m and wind speed at 10
m are calculated as a monitor value.
The method of calculation is simple. u∗ is estimated previously and wind speed at a height of
10 m is calculated by (9.72) or (9.91) asu∗|Va10 | = √Cm10(9.120)

Cm10 is estimated by (9.75)–(9.81) in wnich fm andCm are calculated with application of za = 10.
Since wind direction is constant in vertical within the surface ground layer, x and y components
are

224

9 Boundery Layer and Surface Process

ux10 = |Va10 |

ua
|Va |

(9.121)vy10 = |Va10 |

va
|Va |

(9.122)

In case of temperature and mixing ratio of water vapor, za = 1.5 is used to calculate fmandCmby (9.75)–(9.81). Using the obtained bulk coeﬃcient Cm1.5 and friction velocity u∗ at a height of
1.5 m, wind speed at a height of 1.5 m is calculated as follows.
|Va1.5 | = √

u∗
Cm1.5

(9.123)

Simultaneously, bulk coeﬃcient Ch1.5 of potential temperature and mixing ratio at a height of
1.5m is calculated. Substituting these coeﬃcients into (9.104) and (9.105), we getEθ1.5 =


Eqv 1.5 =−1u∗+ Bh−1Ch1.5 |Va1.5 |(9.124)−1u∗+ Be−1Ch1.5 |Va1.5 |(9.125)

Virtual potential temperature at a height of 1.5m θv1.5 is related to that at the surface layer
θva andatthegroundθvG asθv1.5 − θvGEθ=θva − θvGEθ1.5

(9.126)

Then, vertual potential temperature at height of 1.5m is given by

θv1.5 = θvG + (θva − θvG )

Eθ
Eθ1.5

(9.127)

Similarly, mixing ratio of water vapor at height of 1.5m qva1.5 is also related to that at thesurface layer qva andatthegroundqvsf c as
qva1.5 − qvsf c
Eqv=
qva − qvsf c
Eqv 1.5

Then, mixing ratio of water vapor at a height of 1.5m qva1.5 is given by(9.128)9.2 Boundary Layer, Surface and Soil Temperature

qva1.5 = qvsf c + (qva − qvsf c )Eqv
Eqv 1.5

225(9.129)where qvsf c includes the eﬀect of evapotranspiration coeﬃcient.
(13) Calculation of the Roughness Parameter at the Sea Surface Z0m for the Next
Step
At the end of this calculation process, roughness Z0m which relate to the momentum at the sea
surface is calculated for the next step. Adopting the friction velocity u∗ into (9.89) and (9.90),
Z0m at the sea surface is calculated and conserved untill the next step.

9.2.4

Atmospheric Boundery Layer Process

Momentum, sensible heat and latent heat which are given to the atmosphere from the groundsurface are transported upward by the vertical diﬀusion. Although diﬀusion is caused by the
turbulent ﬂow, we should consider at some points for the diﬀusion in the boundary layer.
In CReSS, the vertical diﬀusion other than the usual turbulent ﬂow is taken into considerationin the atmospheric boundary layer process. The height of boundary layer is speciﬁed by the user.
Since vertical diﬀusion becomes so small that it is far from the ground surface, the height of which
diﬀusion attains to should just be given. Vertical diﬀusion of the atmospheric boundary layer
process is fundamentally calculated using level 2 of the turbulent closure model by Mellor and
Yamada (1974) which is used in JSM (Segami et al. (1989)), and that is also calculation about
only 1 dimension in vertical.
The equation of vertical diﬀusion is expressed with a z coordinate as follows.
∂u
1 ∂
=
∂t
ρ ∂z
∂v
1 ∂
=
∂t
ρ ∂z
1 ∂
∂θv
=
∂t
ρ ∂z
∂qv
1 ∂
=
∂t
ρ ∂z






∂u
ρKm
∂z
∂v
ρKm
∂z
∂θv
ρKh
∂z
∂qv
ρKh
∂z



(9.130)(9.131)(9.132)

where the turbulent mixing coeﬃcient added K0 = 1.0[m2 − s(9.133)−1] was used. That is,

Km = K0 + l2

∂V
Sm
∂z

(9.134)

Kh = K0 + l2

∂V
Sh
∂z

(9.135)

226

9 Boundery Layer and Surface Process

where Sm , Sh are the function of ﬂux Richardson number (Mellor and Yamada, 1974) andlength scale l is

l=

kz
kz
1+
l0

(9.136)k is Karman’s constant and l0 is expressed by making E into turbulent energy as follows.
* ∞l0 = 0.10 ×

ρEzdz

*zs∞

(9.137)ρEdz

zs

Based on the above, the calculation method of boundary layer process is as follows.
(1) Calculation of Air Density between Grids
Air density ρw
k is calculated between each vertical grid which is needed in the case of diﬀerence
specialization (see Fig. 9.2).
É2ÊCalculation of the square of vertical shear

Because of the use for following calculation of Richardson number, or of ﬂux, the absolute valueof vertical shear is calculated here.
∂Vk 2
=
∂z



∂uk
∂z

2



+

∂vk
∂z

2

(9.138)É3ÊGradient Richardson Number

In the level 2 by Mellor and Yamada (1974), ﬂux Richardson number Rf is given as a functionof gradient Richardson number Ri . It is deﬁned as follows.
∂θv
∂z
Ri =
∂Vk 2
θ̄v
∂z
g

(9.139)1
where θ̄v = (θvk+1 + θvk ).
2
É4ÊFlux Richardson Number

Following to Mellor and Yamada (1974), ﬂux Richardson number Rf is given as follows.

9.2 Boundary Layer, Surface and Soil Temperature

Rf = 0.725 Ri + 0.186 −

R2i − 0.316Ri + 0.0346

227

(9.140)É5ÊCalculation of S̃H andS̃M

The constants used by Mellor and Yamada (1974) are as follows.
A1

A2

B1

B2

C1

0.78

0.78

15.0

8.0

0.056

Using these values, the following constants are estimated.

γ1 ≡

1 2A1
= 0.2293333−
3B1(9.141)γ2 ≡B2 6A1+= 0.8453333
B1
B1(9.142)Rf
1 − Rf(9.143)and,Γ≡

Using these values, S̃H andS̃M are expressed as follows.

S̃H = 3A2 (γ1 − γ2 Γ)(9.144)Γγ1 − C1 − (6A1 + 3A2 )B1S̃M = 3A1 (γ1 − γ2 Γ)Γ
γ1 − γ2 Γ + 3A1
B1(9.145)

These can be simpliﬁed more in the actual calculation. Eliminating Γ and considering A1 = A2 ,they can expresses as
γ1 − (γ1 + γ2 )Rf
1 − Rf(9.146)

X1 − X2 Rf
X3 − X4 Rf(9.147)

S̃H = 3A2
S̃M = S̃H

The constants used here are as follows.

228

9 Boundery Layer and Surface Process

γ1 + γ2 = 1.074667(9.148)

3A2 = 2.34

(9.149)X1 = 0.173333(9.150)X2 = 0.641333(9.151)X3 = 0.229333(9.152)

X4 = 0.918667(9.153)(6)Calculation of SH andSM
Using S̃H andS̃M which are calculated previously, SH andSM are estimated. As the factor
common to both SH andSM ,
0

Se =

∂V 2
S̃M
∂z

B1 (1 − Rf )(9.154)is deﬁned. Then, SH andSM are expressed as follows.

SH = Se S̃H(9.155)

SM = Se S̃M(9.156)(7)Calculation of Standard of a Length Scale l0
In Mellor and Yamada (1974), standard of a length scale l0 is calculated as a function of
turbulent kinetic energy (9.137). A turbulent kinetic energy which is estimated in Chapter 7”diﬀusion in sub-grid scale” is used.
(8)Calculation of a Length Scale lLength scale l is given by (9.136) as follows.
l=z(9.157)

1
z
+
k l0

(9)Calculation of Km andKh
Although Km andKh are deﬁned as (9.134) and (9.135), the absolute value of vertical shear isalready applied to SM andSH in the above-mentioned calculation. Then, they are rewritten as
follows.
Km = K0 + l2 SM
2

Kh = K0 + l SH

(9.158)(9.159)9.2 Boundary Layer, Surface and Soil Temperature

229

Here, K0 = 1.0［m2 s−1 ］.
(10)The Finite Diﬀerence Method for Equation of Vertical Diﬀusion
The vertical distributions of Km andKh in boundary layer were calculated on the level between
grids. Then, the vertical diﬀusion equations (9.130)–(9.133) are solved and a time changing rate
and a correction value (integration value) are calculated in the cases of the explicit method andthe inplicit method, respectively.
φzm

ρm ,u m ,φm ,d m

m

z m-1
φ

z m-1

m-1

ρm-1 ,u m-1 ,φm-1 ,d m-1

m-1

z2
φ

z2

2

φ

z1

w

ρ2 ,Km 2 ,Kh 2 ,f 2
ρ2 ,u 2 ,φ2 ,d 22
z1

ρwm-1 ,Km m-1 ,Kh m-1 ,f m-1

11w

ρ1 ,Km 1 ,Kh 1 ,f 1
ρ1 ,u 1 ,φ1 ,d 1

Figure 9.2. vertival grid setting used for atmospheric boundary layer process

We conduct a discretization for these equations with the arrangement as shown in Fig.9.2.
unk − un−1
1
1
k
=
Δt
ρk zk − zk−1
φnk − φn−1
1
1
k
=
Δt
ρk zk − zk−1





N
N
uN
uN
k+1 − ukk − uk−1
w
ρw
K
−
ρ
K
m
m
k
k−1
k
k−1
φ
φ
zk+1
− zkφ
zkφ − zk−1

ρw
k Kmk

N
φN
k+1 − φk
φ
zk+1
− zkφ

− ρw
k−1 Kmk−1

NφN
k − φk−1φzkφ − zk−1

(9.160)(9.161)

where unk and φnk (subscript with bottom k represents a vertical grid number and that with top n
represents the time step) are used as a representative of the velocity component and of the scalar,respectively. N which appears in the right-hand side of diﬀerence equation becomes discretizationof the explicit method in the case of N = n − 1, and becomes discretization of the implicit methodin the case of N = n.
Coeﬃcients of surface ﬂuxes; Cw in (9.113), Cθ in (9.114) and Cqv in (9.115) are used at thebottom layer (k = 1).




N
uN
un1 − un−1
1 1
N
2 − u1
1
ρw
K− Cw (uN=m1 φ
1 − usf c )Δt
ρ1 z1 1z2 − z1φ(9.162)

230

9 Boundery Layer and Surface ProcessN
φn1 − φ1n−1
φN
1 1
N
2 − φ1
ρw− Cφ (φN=
1 Km1 φ
1 − φsf c )
φ
Δt
ρ1 z1
z2 − z1(9.163)uN
sf c = 0(9.164)

where

and representative of the scalar is φ. In these equations, z0 = 0 can deﬁne the forrowing signs,

dk =

1
1
ρk zk − zk−1

fk = ρw
k Kmk

1
φ
zk+1
− zkφ

(9.165)(9.166)

We can understand Khk using fk .
In the case of k = 1,12
un1 − un−1
NN1
= d1 f1 (uN
2 − u1 ) − Cw u1
Δt(9.167)12
φn1 − φn−1
NNN1
= d1 f1 (φN
−
φ
)
−
C
(φ
−
φ
)
φ
2
1
1
sf c
Δt

(9.168)1
2
unk − ukn−1
NNN= dk fk (uN
k+1 − uk ) − fk−1 (uk − uk−1 )Δt(9.169)1
2
φnk − φn−1NNNk= dk fk (φN
−
φ
)
−
f
(φ
−
φ
)
k−1
k+1
k
k
k−1
Δt

(9.170)12
unm−1 − un−1
m−1
N
N
N
= dm−1 fm−1(un−1m − um−1 ) − fm−2 (um−1 − um−2 )Δt(9.171)n−1
12
φnm−1 − φm−1
n−1
N
N
− φN
= dm−1 fm−1(φmm−1 ) − fm−2 (φm−1 − φm−2 )Δt(9.172)In the case of k ≥ 2,

About k = m − 1,

9.2 Boundary Layer, Surface and Soil Temperaturen−1 in the case of k = m − 1 are given as boundary condition, this process
where since un−1and φmm
is eternal.

In the case of N = n − 1, these equations are solved by the explicit method and time change
rate is acquired. In the case of N = n, they are solved by the implicit method and integration
value is acquired.

9.2.5

Ground Temperature

The ground temperature is calculated by the model with m layers.

G0
ΔZ 1

T 1=T G

k=1

ΔZ 2

T2

k=2

ΔZ m

T m-1

k = m-1

ΔZ m-1T m = const.

k=m

Figure 9.3. A underground setup used for calculation ofground temperature.

Figure 9.3 shows a ground grid setting used for calculation of ground temperature. In thecalculation, only the vertical heat conduction is taken into consideration, while the horizontal
heat diﬀusion does not take into consideration.
Here, time evolution equation system of the ground temperature in the model with m layer isexpressed as follows.

231

232

9 Boundery Layer and Surface Process

G0ρG CG ΔZ1

∂T1
∂t

=

∂T2
∂t

=−

2νG
(T2 − T1 )
ΔZ1 (ΔZ2 + ΔZ1 )

+

2νG
2νG
(T2 − T1 ) +(T3 − T2 )
ΔZ2 (ΔZ2 + ΔZ1 )
ΔZ2 (ΔZ3 + ΔZ2 )·······(9.173)∂Tk∂t

2νG (Tk − Tk−1 )=−ΔZk (ΔZk + ΔZk−1 )2νG (Tk+1 − Tk )+ΔZk (ΔZk+1 + ΔZk )·······∂Tm−1
2νG (Tm−1 − Tm−2 )2νG (Tm − Tm−1 )=−+∂tΔZm−1 (ΔZm−1 + ΔZm−2 )ΔZm−1 (ΔZm + ΔZm−1 )In the m layer, temperature Tm is constant during a calculation period. Heat capacitiy of theground per unit volume ρG CG and heat diﬀusion coeﬃcient νG are
ρG CG = 2.3 × 106[Jm−3 K−1 ](9.174)2 −1(9.175)−7

νG = 7.0 × 10

[m s

]

In（9.173）, G0 which is appeared in the ﬁrst ground layer（k = 1）is heat ﬂux which goes to
the ground. It is sum of net radiation Rnet , sensible heat HS and latent heat l E.
G0 = Rnet − HS − l E(9.176)

They are given as follows, respectively.
Rnet = RS ↓ + L↓ − L↑(9.177)HS = −Cp Cθ (Ta − TG )(9.178)l E = −Lv Cqv β [qva − qvs (TG )](9.179)where the last term of net radiation Rnet shows net upward longwave radiation T1 = TG and,L↑ = εG σTG4

(9.180)β in the equation of latent heat l E is evapotranspiration coeﬃcient which is supposed to takeconstant value during the integration period. Then, radiation process is used only for calculationof the heat balance at the ground surface.

9.2 Boundary Layer, Surface and Soil Temperature

233

Time integration of the ground temperature solved by implicit scheme. In order to disperse
the time evolution equation of ground temperature (9.173),
about k = 1 ∼ m − 1, the followings are deﬁned.

a1 = 0(9.181)

ak = −Δt2νGΔzk (Δzk + Δzk−1 )(9.182)

bk = −Δt2νGΔzk (Δzk+1 + Δzk )(9.183)

Then, (9.173) is dispersed as follows.

T1n= T1n−1+

ΔtG0
ρG CG ΔZ1

+ b1 (T1n − T2n )T2n= T2n−1+ a2 (T2n − T1n)

+ b2 (T2n − T3n )·······(9.184)Tknn+ ak (Tkn − Tk−1)

= Tkn−1n+ bk (Tkn − Tk+1)·······n−1
nTm−1= Tm−1nnn+ am−1 (Tm−1− Tm−2) + bm−1 (Tm−1− Tm )The simultaneous equations can be expressed using a procession as follows.

⎛

T1n

⎜
⎜ T2n
⎜
⎜ .
⎜ ..
⎜
⎜
⎜ Tn
⎜ k
⎜ .
⎜ ..
⎝
n
Tm−1

⎞

⎛

⎞

n−1
+
⎜ T1

⎟ ⎜
⎟ ⎜ n−1⎟ ⎜ T2⎟ ⎜⎟ ⎜ ..
⎟ ⎜.
⎟=⎜⎟ ⎜ n−1⎟ ⎜ Tk⎟ ⎜⎟ ⎜ ..
⎠ ⎜.
⎝ΔtG0
ρG CG ΔZ1 ⎟
⎟

n−1
Tm−1
− bm−1 Tm

⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠

(9.185)

234

9 Boundery Layer and Surface Process
⎛

b1

−b1

···

0

···

⎜
⎜ −a2 a2 + b2 −b2
0
···
⎜
⎜ .
.
.
.
..
⎜ ..
..
..
..
.
⎜
+⎜
⎜ 0
· · · −ak ak + bk −bk
⎜
⎜ .
..
..
..
..
⎜ ..
.
.
.
.
⎝

0

···

···

···

···

0

···
..
.

0
..
.

···
..
.

0
..
.

⎞⎛

T1n

⎟⎜
⎟ ⎜ T2n⎟⎜⎟⎜ .
⎟ ⎜ ..
⎟⎜
⎟⎜
⎟⎜ Tn
⎟⎜ k
⎟⎜ .
⎟ ⎜ ..
⎠⎝· · · −am−1 am−1 + bm−1

⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠

n
Tm−1

Furthermore, it can be expressed brieﬂy like

Tn = F + ATn(9.186)

If the term of Tn is shifted using unit procession I of (m − 1) × (m − 1),(I − A)Tn = F(9.187)

is obtained. While it can be solved easily by the elimination of a gauss (refer to the 10.1.3 forexplanation of a concrete solution method), ground temperature of all layers at the time of n arecalculated.

Chapter 10

Numerical Computation Method

Generally, the equation of motion, the equation of potential temperature
derived by the equation of thrmodynamics, the equation of pressure derived
by the continity eqution and the eqution of gaseous state, and the equationsof water vapor and hydrometeors are solved simultaneously and integrated
with respect to time under appropriate boundary and initial values in cloud
resolving models.
In CReSS, the equation of motion, the equations of the potential temperature perturbation and the pressure perturbation, and the equations ofhydrometeors and water vapor which are shown in Chapter 6 ’Formulationof the system of basic equations’ are also temporally and spatially integrated
by using ﬁnite diﬀerence method. There are many studies about the methods
which are discretizated and integrated these equation systems and the various
calculation techniques are established.
In this chapter, the structure of a grid point, the method of discretization,
the method of time integration and so on are explained about the numericalcomputation method used by CReSS.

236

10 Numerical Computation Method

10.1

The discretization of basic equation system

10.1.1The outline of the numerical solution

There are various techniques from the selection of dependent variables to the method of time
integration in the numerical solution by cloud resolving models. The outline of the numericalsolution used in CReSS is as follows.
• The basic equation system is the equation of motion, potential temperature perturbation,pressure perturbaton, water vapor and hydrometeors.
• The dependent variables are three-dimensionl velocity components u, v, w, potential temperature perturbation θ , pressure perturbation p , mixing ratio of watar vapor, and mixing
ratios and number concentrations of hydrometeors.
• The basic equations are deﬁned by a terrain-following curvilinear coordinates are used toinclude the eﬀect of orography.
• The ﬁnite diﬀerences are used for the derivation of the dependent variable with respect
to space. The types of Arakawa-C and Lorenz staggered grids are used for hrizontal andvertical grid arrangement, respectively.
• The sound wave is contained as a solution by using compressible equation system. The
terms related to the sound wave are integrated with small time step Δτ , the other terms
are integrated with large time step Δt.
• The leap-frog scheme with the Asselin time ﬁlter is used for time integration with respect
to the large time step.
• The explicited forward-backward scheme or the inplicited Crank-Nicolson scheme in only
vertical is used for time integration with respect to the small time step.
• The eﬀects of sound wave are weakened by adding the divergence terms explicit schemewith the small time step.
Furthermore, ξ, η are expressed as x, y in this chapter as it is ξ = x, η = y in a terrain-following
coordinates.

10.1.2

Grid in the model and setting of variables

As the space are represented by the method of grid points in CReSS, ﬁnite diﬀerence method isused for the spatial derivation. Here, the strucuture of grid points and the setting of variables in
CReSS are shown.
The strucuture of grid points and the setting of variables are set on the staggered grid 1 as
Fig10.1 in horizontal and vertical. Arakawa-C grid and Lorenz grid are used for the settings ofhorizontal and vertical grid points, respectively.
1

There are various setting in the staggered grid which variables are deﬁned alternately.

10.1 The discretization of basic equation system

In this case, the point of the all scalars, such as pressure perturbations, potential temperature
perturbation, mixing ratios of watar vapor and physical quantities of hydrometeors, are locatedat the center of the grid box2 .
The point of x-component velocity u is dislocated to a half interval from the center of the gridin the x-direction, the point of y-component velocity v is dislocated to a half interval from thecenter of the grid in the y-direction in horizontal. The point of u and v are deﬁned at the samelevel as the scalars points. The point of z-component velocity w is vertically dislocated at a halfinterval just above and below the scalar variables points.
J 32
w

J 31

J 31
J 32

v

u

u
φ, J d
v

J 32

J 31
J 32

w

J 31

Figure 10.1. Structure of the staggered grid and setting of dependent variables.

Furthermore, as a terrain-following coordinate is used in CReSS, the Jacobian with transform1
1
for coordinates G 2 and the metric tensor which is not zero or 1, that is, J31 , J32 , Jd (= G 2 ) are1
shown as follows. The G 2 (= Jd ) point is deﬁned at the same level as the scalars points. The
J31 point is dislocated to a half interval from the center of the grid in the x-direction and thez-direction. The J32 is dislocated to a half interval from the center of the grid in the y-direction
and the z-direction.
The coordinates x, y, ζ which are set on the staggered grid as shown in Fig 10.1 are diﬁned as
the vectors points. The domain in a thick line except the most outside grid points is the physicaldomain as shown in Fig10.2. So four grid points in the direction without a dimension need to be
calculated at the time of the 2-dimensional caluculation.

10.1.3Discretization by ﬁnite diﬀerence method

Generally, since elastic equations without any approximations are used in the cloud resolving
model as the basic equations, all waves in the atmosphere are represented in the model. Waves in
the atmosphere are divided into the sound wave, the gravity wave and the Rossby wave by phasevelocity or frequency, and the phase velocities become small in this order. The phase velocity
of sound wave is the fastest but we have no meteorological interest for it. On the other hand,
the gravity wave and the Rossby wave are important for a cloud and a mesocale meteorological2

The point of potential temperature (temperature) and the point of pressure are located at the same level in
Lorenz grid. On the other hand, the point of potential temperature (temperature) is dislocated at a half interval
below the pressure point, and deﬁned at the same level as the vertical velocity point in the Charney Phillips’ grid.

237

238

10 Numerical Computation Method

physical domain
ny

nz

ny-1

nz-1
nz-1

ny-1
ny-2

nz-2

ny-2

nz-2
nz-3

physical domain

physical domain

ny-3yorigin(x,y ) = (0.0, 0.0)

ζ

3

3
3

3
2

2
2

2
1

1
1

1
1
1

2
2

nx-3

3
3

x

nx -2
nx-2

nx-1
nx-1

nx

Figure 10.2. Setting of variables in a calculation domain (● represents a point for velocities. Þ represents a point for scalars).

phenomena3 However, the sound wave is contained in models with elastic equation system, andthe special technique is needed to be used for the calculation about the wave. So the time stepis very small to satisfy CFL condition for the sound wave which the phase velocity is fastest
if the time integration is calculated explicitly in meteorological models the generall numericalintegration for ﬂuid. However, the time steps for the gravity and the Rossby waves which aremeteorologically important are satisﬁed with the much larger time step than that of the soundwave, since the phase velocities of these waves are much slower than the sound wave.
There are various calculation techniques not to make the time step of the sound wave small4 .
The well-known calculation techniques are as follows.
HI-VI
methodThe terms of the sound wave are solved inplicitly, and the other terms are solved
explicitly. This technique is called semi-inplicit time integration method5 .

3
The Rossby wave is important for larger scale phenomena in mesocale meteorology, synoptic meteorologicalphenomena and planetary meteorological phenomena. Weather is inﬂuenced by the Rossby-scale atmosphericphenomena every day, and usual forecast models represent these phenomena.
4
If the anelastic equation system by Ogura and Phillips (1962) is used for the basic equation system in anumerical model, the time step can be deﬁned by the phase velocity of gravity wave because the sound wave is notcontained in this equation system. In this case, pressure is calculated by the equation of motion and the continity
eqution. There are the advantage that the sound wave is removed, but the demerits of which the equation ofpressure is complicated and a error is large at the place with large variations of density in this techniques.
5
The terms with large time step are explicitly calculated and the other terms are inplicitly solved in the semiinplicit time integration method. For example, the semi-inplicit time integration method of which the terms forthe gravity wave are explicitly solved, and the other terms are solved by using the leap-frog scheme is used in themodel of JMA.

10.1 The discretization of basic equation system

HE-VE
methodHE-VImethodThe terms of the sound wave and the other terms are solved explicitly using
diﬀerent time steps, respectively. This technique which is called the modesplitting technique (Klemp and Wilhelmson, 1978) is often used.
This technique is almost the same as the mode-splitting technique. But theterms of the sound wave are solved inplicitly in vertical.

The schematic representation of this mode-splitting technique is shown in Fig10.3.

t -Δt

t

t +Δ
τ

t +Δt

Figure 10.3. The schematic representation of the mode-splitting technique.

Many data are exchanged between some nodes in the case of using parallel computer sincelarge simultaneous equations are solved in the HI-VI method. The HE-VE method or HE-VI
method are used in CReSS since it is assumed to be mainly used on a parallel computer. These
techniques are easy to be applied to the large parallel computing since the phisycal space arecorrsponded to the setting of a node and only the data in halo region of a node communicates
between nodes. The detail description is shown in 11 ’Mounting of a parallel processing’. The
eﬃciency of calculation does not become so bad to integrate the terms with respect to sound wave
in small time step.

239

240

10 Numerical Computation Method

The method of discretization for physical quanities with the sound wave mode
The method of discretization for physical quantities with the sound wave mode is shown from here.
The gravity wave mode is calculated with the large time step. In this case, the discretizations forpotential temperature, mixing ratios of watar vapor, mixing ratios and number concentrations ofhydrometeors is not took into consideration as they are calculated in the large time step.
Physical quantities with the sound wave mode are shown in the basic equation system in
CReSS. In the following equations, [rm] is indicates terms which are related to the Rossby wave
mode (rotational mode), [gm] the gravity wave mode (divergence mode) and [am] the sound wave
mode, respectively. Terms of physical processes are indicated by [physics].
The equation of motion


∂u
∂u
∂u
∂u∗
= − u∗
+ v∗
+ W∗
∂t
∂x
∂y
∂ζ
'(

&



−

)

[rm]



1
∂ 
∂ 
Jd p − αDiv∗ +J31 p − αDiv ∗+ (fs v∗ − fc w∗ ) + &G 2 Turb.u
'( )
&
'(
)
∂x
∂ζ
'(

&

)

[am]



∂v∗
∂v
∂v
∂v
= − u∗
+ v∗
+ W∗
∂t
∂x
∂y
∂ζ
'(

&



−
&



[rm]

[physics]

[rm]


)



1
∂ 
∂ 
Jd p − αDiv ∗ +J32 p − αDiv ∗− fs u∗ + &G 2 Turb.v'( )& '( )
∂y
∂ζ
'(

)

[am]



∂w∗
∂w
∂w
∂w
= − u∗
+ v∗
+ W∗
∂t
∂x
∂y
∂ζ
'(

&

[rm]

(10.2)[physics]

[rm]


)

⎛

⎞


⎜

⎟

⎜ θ
1
∂
p
qv
qv + qx ⎟⎟ + fc u∗ + G 2 Turb.w−p − αDiv ∗ −ρ∗ g ⎜
−
+
−
⎜
⎟
2
&
'(
)
∂ζ
ρ̄cs  + q̄v
1 + q̄v ⎠ & '( )
θ̄
⎝&'()

&

'(

)

[am]

&'()

[gm]

&

'(

)

[physics]

[am]

[rm]

[physics]

The equation of pressure


1




1
1 ∂p
1
1
∂G 2 p∂p∂p
= − G2 u+ G2 v+ G2 W+ G 2 ρ̄gw
& '( )
∂t
∂x
∂y
∂ζ

'(

&



− ρ̄c2s
&

)

[rm]
1

1

1

∂G 2 u ∂G 2 v ∂G 2 W
+
+
∂x
∂y
∂ζ
'(

[am]

[am]



+G
)

&

1
2

ρ̄c2s



1 dQ
1 dθ
−
θ dt
Q dt
'(

[am]



(10.4))

242

10 Numerical Computation Method

1Here, the density ρ̄ and the Jacobian G 2 for coordinate transformation are deﬁned at the same
point as the scalars. The vertical velocity W in a terrain-following coordinates and the verticalvelocity w in the Cartesian coordinates are deﬁned at the same point.
xyuζ J31 + v ζ J32 + w

W =

G



(10.15)1ζ
2

In the case of explicit scheme, descretization of the equations of (10.1) ∼ (??) are given withrespect to u, v, w, p as follows.

ρ∗

τ +Δτ
− uτ
xu

ρ∗

ρ∗

G

ζw

1ζ
2



ζx

2τ

+ Fut

(10.16)

1

2
− vτ
ζy τ
+ Fvt
= − ∂y (Jd Pα ) + ∂ζ J32 PαΔτ

(10.17)Δτyv

1= − ∂x (Jd Pα ) + ∂ζ J31 Pα

τ +Δτ

τ +Δτ

⎡

ζ

⎤τ

−w
p
= − [∂ζ Pα ]τ − ⎣gG 2 ⎦ + Fwt
Δτ
cs
τ

1
2







1

2τ +Δτ

(10.18)





τ +Δτ


x
y
1x
1y
pτ +Δτ − pτ= −ρ̄c2s ∂x G 2 u + ∂ζ J31 uζ + ∂y G 2 v + ∂ζ J32 vζ + ∂ζ w
Δτ+gρ∗ w ζ+ Fpt(10.19)where
Pα = p − αDiv ∗
∗

Div =

1
1

G2



∂u∗
∂v∗ ∂W ∗
+
+
∂x
∂y
∂ζ

(10.20)(10.21)

where α is a factor of damping for divergence terms to weaken the eﬀects of the sound wave.
The terms concerning the sound wave, which are integrated with a small time step Δτ , areexplicitly written, the other terms are included in Fφt (φ = u, v, w, p ). The ﬁrst term on theright side in the equations of (10.16) and (10.17), and the second term on the right side in theequation of (10.18) are pressure gradient force term. Buoyancy force with pressure perturbations
and divergence damping are shown by the ﬁrst and third term on the right side in the equation(10.18), respectively. The ﬁrst term on the right side in the equation of (10.19) is a divergenceterm and the second term of that is a vertical advection term.

10.1 The discretization of basic equation system

243

When terms for sound wave are solved explicitly, u, v, and w are calculated by the equationsof (10.16) ∼ (10.18) with forward-diﬀerence for Δτ . Using the values of u, v, and w, pressure
perturbation p is calculated by the equation of (10.19) with backward-diﬀerence for Δτ . During
this integration with a small time step for 2Δτ (= nΔτ ), Fφt is constant. Terms of the sound wave
in most of non-hydrostatic models are not solved explicitly in vertical and horizontal. Althoughsome models (e.g. ARPS) include the HE-VE scheme as an option, the NHM of MeteorologicalResearch Institute adopts the HI-VI and HE-VI schemes. The technique to solve inplicitly in only
vertical is described later. However, it is not necessarily that terms for sound wave are calculated
inplicitly in vertical when the horizontal resolution is the same as the verical resolution as an
object of CReSS. There is a option to calculate the gravity mode in the small time step in
CReSS. This technique is also shown later. The gravity wave mode is integrated in the small
time step in ARPS and in the large time step in NHM.
In the case of using vertical implicit scheme, the equations of verical velocity w and pressure
perturbation p in the equations (10.18 and 10.19) are replaced by the following equations.

ρ∗

ζw

1
2
− wτ
τ +Δτ
τ
+ (1 − β) ∂ζ p= [∂ζ (αDiv∗ )]τ − β ∂ζ p
Δτ
⎡
⎤

τ +Δτ

⎛
⎞τ +Δτ
⎛
⎞τ
ζ
ζ
1 p
1 p
⎢ ⎝
⎥
− ⎣β gG 2 2 ⎠+ (1 − β) ⎝gG 2 2 ⎠ ⎦ + Fwt

cs

G

1ζ
2





cs





(10.22)xy  τ +Δτ
1x1ypτ +Δτ − pτ= −ρ̄c2s ∂x G 2 u + ∂ζ J31 uζ + ∂y G 2 v + ∂ζ J32 v ζΔτ1−ρ̄c2s β (∂ζ w)τ +Δτ + (1 − β) (∂ζ w)τ
∗

+gρ

 

β wζ

τ +Δτ

+ (1 − β) w ζ

τ

2

+ Fpt

(10.23)

These equations are gained by averaging for τ and τ + Δτ of the equations (10.18) and (10.19)
weighted by β. The equations of w (10.22) with β = 0 and p (10.23) with β = 1 result in
the equations of (10.18) and (10.19), respectively. When β = 0, (10.22) and (10.23) are teh
simultaneous equations for wτ +Δτ and pτ +Δτ . They are implicitly calculated in vertical wtih the
Crank-Nicolson scheme.
In the vertical implicit scheme, the forward diﬀerence for the equations of (10.16) and (10.17)
are calculated and the values of u and v at τ + Δτ are gained. In the vertical implicit scheme,
the forward diﬀerence for the equations of (10.16) and (10.17) are calculated and the values of uand v at τ + Δτ are gained. Concerning pτ +Δτ from (10.22) and (10.23), we get the following
equations for w and p.
w
∗ζ

ρ

τ +Δτ

− wτ
= −Δτβ 2 ∂ζ
Δτ



⎛

−Δτβ 2 g ⎝

gρ∗
1

G2
∗

wζ −
ζ

ρ̄c2s
1

G2

τ +Δτ

∂ζ w⎞τ +Δτgρ ζ
ζw − ρ̄∂ζ w ⎠
c2s

+ Fw

(10.24)

244

10 Numerical Computation Method

where Fw and Fp are
Fp =

Δτ 1
1
2



Fpt + (1 − β) gρ∗ w ζ − ρ̄c2s ∂ζ w

τ 2

G
 





x
y  τ +Δτ
1x1yΔτ
− 1 ρ̄c2s ∂x G 2 u + ∂ζ J31 uζ + ∂y G 2 v + ∂ζ J32 vζ
G2(10.25)

⎡

ζ
1 p

⎤τ

⎡
⎢ gG

Fw = − [∂ζ Pα ]τ − ⎣gG 2 2 ⎦ − β ⎣
cs

1
2

Fp

c2s

⎤

ζ

⎥

+ ∂ζ Fp ⎦ + Fwt .

(10.26)The second-order discretization in vertical (10.24) is ordered as follows for the unknown wτ +Δτ .
τ +Δτ
τ +Δτ
Ak wk−1+ Bk wkτ +Δτ + Ck wk+1
= Fk

(10.27)

Simultaneous equations whose factors are a triple diagonal matrix are obtained as this equation.
This equation is solved simply if the top and bottom boundary conditions are given as follows,for example.

w=0
w=u

∂zs
∂zs
+v
∂x
∂y

k = nk − 1(10.28)k=2(10.29)where Ak , Bk , Ck , Fk are known quantities at τ . These quantities are given under the boundary
conditions of (10.28) and (10.29) as follows.
⎧
⎪
0
⎪
⎪
⎨
Ak =


⎪
1
⎪
⎪
2
2
⎩ −Qk csk−1 + Pk Gk−1 (Rk−1 + Sk−1 ) ,3

k=3(10.30)4 ≤ k ≤ nk − 2

Bk = 1 + Qk (Rk + Sk ) c2s k − (Rk−1 − Sk−1 ) c2s k−1
5

1
2

1
2

4

6

+Pk (Rk + Sk ) Gk − (Rk−1 − Sk−1 ) Gk−1 ,3 ≤ k ≤ nk − 2(10.31)

10.1 The discretization of basic equation system

⎧ 
1
⎪
2
2
⎪ Qk cs k + Pk Gk (Rk + Sk ) ,
⎪
⎨

Ck =

⎪
⎪
⎪
⎩

3 ≤ k ≤ nk − 3(10.32)k = nk − 2

0,

⎧
ζ
ζ

τ
⎪
⎪
⎨ Fwk + wk + uk (∂x zs )k + v k (∂y zs )k ,

Fk =

k=3
(10.33)⎪⎪⎩ F

4 ≤ k ≤ nk − 2

τw k + wk ,

Pk =

245

Δτ 2 β 2 g
2ρ∗k

ζ

,

Qk =

Δτ 2 β 2Δζk ρ∗k

Rk =

,
ζ

gρ∗k

,

1

2Gk2 c2s k

Sk =

ρ∗k
Δζk Gk

(10.34)This simultaneous equation (10.27) is represented by a matrix by replacing wkτ +Δτ with wk asfollows.
⎛

B3 C3 0 · · · · · ·⎜⎜ A4 B4 C4 0 · · ·⎜⎜ . . . . .
⎜ .. .. .. .. ..
⎜⎜⎜ 0 · · · Ak Bk Ck⎜⎜ . . . . .
⎜ .. .. .. .. ..
⎝

···

0

···
..
.

0
..
.

···
..
.

0
..
.

⎞⎛

w3

⎟⎜
⎟ ⎜ w4⎟⎜⎟⎜ .
⎟ ⎜ ..
⎟⎜⎟⎜⎟ ⎜ wk⎟⎜⎟⎜ .
⎟ ⎜ ..
⎠⎝

0 · · · · · · · · · · · · Ank−2 Bnk−2

⎞

⎛

F3

⎟ ⎜
⎟ ⎜ F4⎟ ⎜⎟ ⎜ .
⎟ ⎜ ..
⎟ ⎜⎟=⎜⎟ ⎜ Fk⎟ ⎜⎟ ⎜ .
⎟ ⎜ ..
⎠ ⎝

wnk−2

⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠

(10.35)

Fnk−2

The forward elimination and the backward substitution are carried out to solve this matrix by
the Gaussian Elimination. The ﬁrst line in (10.35) are divided by B3 to carry out the forwardelimination.
⎛

1 c3 0 · · · · · ·⎜⎜ A4 B4 C4 0 · · ·⎜⎜ . . . . .
⎜ .. .. .. .. ..
⎜⎜⎜ 0 · · · Ak Bk Ck⎜⎜ . . . . .
⎜ .. .. .. .. ..
⎝

···

0

···
..
.

0
..
.

···
..
.

0
..
.

0 · · · · · · · · · · · · Ank−2 Bnk−2

⎞⎛

w3

⎟⎜
⎟ ⎜ w4⎟⎜⎟⎜ .
⎟ ⎜ ..
⎟⎜⎟⎜⎟ ⎜ wk⎟⎜⎟⎜ .
⎟ ⎜ ..
⎠⎝

wnk−2

⎞

⎛

f3

⎟ ⎜
⎟ ⎜ F4⎟ ⎜⎟ ⎜ .
⎟ ⎜ ..
⎟ ⎜⎟=⎜⎟ ⎜ Fk⎟ ⎜⎟ ⎜ .
⎟ ⎜ ..
⎠ ⎝

⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠

(10.36)Fnk−2

where c3 = C3 /B3 and f3 = F3 /B3 . The diagonal components of the values which subtracted
from the second line to the values which multiplied the ﬁrst line by A4 are standardized as follows.

246

10 Numerical Computation Method⎛

1 c3 0 · · · · · ·⎜⎜ 0 1 c4 0 · · ·⎜⎜. . . . .
⎜ .. .. .. .. ..
⎜⎜⎜ 0 · · · Ak Bk Ck⎜⎜. . . . .
⎜ .. .. .. .. ..
⎝

···
···
..
.

0
0
..
.

···
..
.

0
..
.

⎞⎛

w3
w4
..
.

⎟⎜
⎟⎜
⎟⎜
⎟⎜
⎟⎜
⎟⎜
⎟⎜
⎟ ⎜ wk⎟⎜⎟⎜ .
⎟ ⎜ ..
⎠⎝

0 · · · · · · · · · · · · Ank−2 Bnk−2

⎞

⎛

f3
f4
..
.

⎟ ⎜
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎟=⎜
⎟ ⎜ Fk⎟ ⎜⎟ ⎜ .
⎟ ⎜ ..
⎠ ⎝

wnk−2

⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠

(10.37)Fnk−2

where c4 = C4 /(B4 − A4 c3 ) and f4 = (F4 − A4 f3 ) /(B4 − A4 c3 ) . Similarly, using ck = Ck /(Bk − Ak ck−1 )and fk = (Fk − Ak fk−1 ) /(Bk − Ak ck−1 ) ,
⎛

⎞⎛

⎞

⎠⎝

⎟ ⎜
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎟=⎜
⎟ ⎜ fk⎟ ⎜⎟ ⎜ .
⎟ ⎜ ..
⎠ ⎝1 c3 0 · · · · · · · · · 0
w3⎜⎟⎜⎜ 0 1 c4 0 · · · · · · 0 ⎟ ⎜ w4⎜⎟⎜
⎜ . . . . . . . ⎟⎜ .
⎜ .. .. .. .. .. .. .. ⎟ ⎜ ..
⎜
⎟⎜
⎜
⎟⎜
⎜ 0 · · · 0 1 ck · · · 0 ⎟ ⎜ wk⎜⎟⎜
⎜ . . . . . . . ⎟⎜ .
⎜ .. .. .. .. .. .. .. ⎟ ⎜ ..
⎝0 ··· ··· ··· ··· 0 1

⎛

wnk−2

⎞

f3
f4
..
.

⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠

(10.38)fnk−2

Finally, solutions for all k are obtained to carry out the backward substitution in order of k =nk − 2, nk − 3 and · · · , 4.
wnk−2 = fnk−2
wnk−3 = fnk−3 − cnk−3 wnk−2..
.
wk= fk − ck wk+1
..
.

w4

= f4 − c4 w5

w3= f3 − c3 w4(10.39)The term (Fφt ) which is calculated with the large time step is represented as follows.
1x

Fut = −Adv.ut + ρ∗ fs v y − ρ∗ fc wζ
1

Fvt = −Adv.vt − ρ∗ fsux
1

2

y t

1

x 2t

1

1

1

+ G 2 Turb.v
ζ(10.40)2t−ΔtFwt = −Adv.wt + ρ∗ (Buoy.θ + Buoy.q) + ρ∗ fc ux
Fpt = −Adv.pt

2t−Δt+ G 2 Turb.u(10.41)
2
ζ t

1

1

2t−Δt

+ G 2 Turb.w(10.42)(10.43)

10.1 The discretization of basic equation system247

where Buoy.θ and Buoy.q are buoyancy terms for temperature perturbations, watar vapor andhydrometeors, and are deﬁned as follows.

Buoy.θ ≡ gθθ̄(10.44)






qv
q  + qx
Buoy.q ≡ g− v
 + q̄v
1 + q̄v(10.45)The advection term is calculated with the second or fourth-order accuracy by using the values
at the time (t). The second-order advection term is descretized as follows.

x

x

x

y

x

ζ

y

x

y

y

y

ζ

Adv.ut = u∗ ∂x u + v∗ ∂y u + W ∗ ∂ζ uAdv.vt = u∗ ∂x v + v∗ ∂y v + W ∗ ∂ζ v
ζ

x

y

ζ

ζ

Adv.wt = u∗ ∂x w + v∗ ∂y w + W ∗ ∂ζ v
1x

x

1y

y

(10.46)(10.47)ζ(10.48)ζ

1yAdv.pt = G 2 u∂x p + G 2 v∂y p + G 2 W ∂ζ p

(10.49)The fourth-order advection term is descretized as follows.


ζ
x
y
4 ∗x
x
x
Adv.u =u ∂x u + v ∗ ∂y u + W ∗ ∂ζ u
3


2ζ
2x
2y
1 ∗ 2xxζxy−
u ∂2x u + v∗ ∂2y u + W ∗ ∂2ζ u
3
t(10.50)



ζ
x
y
4 ∗y
y
y
Adv.v =u ∂x v + v ∗ ∂y v + W ∗ ∂ζ v
3


2ζ
2y
2x
1 ∗ xyyζ2y∗∗−u ∂2x v + v ∂2y v + W ∂2ζ v
3
t



ζ
x
y
4 ∗ζ
ζ
ζ
Adv.w =u ∂x w + v ∗ ∂y w + W ∗ ∂ζ v
3

(10.51)



t



2ζ
2x
2y
1 ∗ xζ2ζyζ−u ∂2x w + v∗ ∂2y w + W ∗ ∂2ζ v
3



x
y
ζ
1x
1y
1y
4
G 2 u∂x p + G 2 v∂y p + G 2 W ∂ζ p
Adv.p =3t

(10.52)

248

10 Numerical Computation Method⎡

1 ⎢ 1x x− ⎣G 2 u ∂2x p
3

2x

+G

1y
2

2y

y

v ∂y p

+G

1ζ
2

2ζ

ζ

⎤
⎥

W ∂2ζ p ⎦(10.53)The computing instability is caused in the calculation with discretization of central diﬀerence. So
an artiﬁcial numerical viscosity term which is shown in the following section 10.2 is added to theabove terms.
Variables except for pressure in a turbulent diﬀusion term are applied in the turbulent diﬀsion
terms. The tensor for deformation velocity which are indicated in (7.18) ∼ (7.23) are solved asfollows by using calculus of ﬁnite diﬀerences.

S11 =
S22 =
S33 =

2 1
G

2 1
G

S23 =

1
2

2

x





x 2



y





y 2

∂y Jd v + ∂ζ J32 v ζ

∂ζ w

1

1

1 xy

G2
1

1

1 xζ

G2
1
G



∂x Jd u + ∂ζ J31 uζ

1

G2

S12 =
S13 =

1
2

(10.54)(10.55)(10.56)



x







ζ





ζ



y





∂x Jd w + ∂ζ u + J31 wx

1





∂y Jd w + ∂ζ v + J32 wy

1 yζ
2

x

y

∂y Jd u + ∂x Jd v + ∂ζ J32 uζy + J31 vζx
ζ

ζ

2

(10.57)

2(10.58)2(10.59)The divergence term Div which is indicated in 7.24 are solved as follows by using calculus of ﬁnite
diﬀerences.

Div =

1
1

G2





∂x G

1x
2





u + ∂y G

1y
2





v + ∂ζ G

1ζ
2



W

(10.60)Using these, the stress tensor which are indicated in (7.9) ∼ (7.17) are written by ﬁnit diﬀerences
as follows.
2
τ11 = ρ̄ντ h S11 − Div3

(10.61)τ12 = ρ̄ντ h xy S12(10.62)

τ13 = ρ̄ντ v xζ S13(10.63)

10.1 The discretization of basic equation system249

τ21 = ρ̄ντ hyx S21τ22 = ρ̄ντ h(10.64)

2
S22 − Div3



(10.65)

τ23 = ρ̄ντ v yζ S23(10.66)

τ31 = ρ̄ντ hζx S31(10.67)

τ32 = ρ̄ντ hζy S32(10.68)2
τ33 = ρ̄ντ v S33 − Div3

(10.69)Finally, the turbulent diﬀusion terms are indicated in diﬀerences as follows.


1



xy



x

G 2 Turb.u = ∂x (Jd τ11 ) + ∂y Jd τ12 + ∂ζ τ13 + J31 τ11 xζ + J32 τ12 ζ
1



yx



1



ζx





x

y

y

G 2 Turb.v = ∂x Jd τ21 + ∂y (Jd τ22 ) + ∂ζ τ23 + J31 τ21 ζ + J32 τ22 yζ


ζy





ζ

x

ζ

(10.70)

G 2 Turb.w = ∂x Jd τ31 + ∂y Jd τ32 + ∂ζ τ33 + J31 τ31 ζ + J32 τ32 ζ(10.71)y(10.72)The method of discretization for physical quanities without the sound wave mode
When the gravity wave mode is calculated with the large time step, the equations of potential
temperature, watar vapor and hydrometeors are solved with only large time step. These equations
are shown by indicating watar vapor and hydrometeors as x as follows.
θ t+Δt − θ t−Δt
= Fθt
2Δt

(10.73)

q t+Δt − qxt−Δt
= Fqt
ρ∗ x
2Δt

(10.74)

Nxt+Δt − Nxt−Δt
= FNt
2Δt

(10.75)

ρ∗

1

G2

where

Fθt = −Adv.θ t +1

1
2

G Turb.θ

2t−Δt

∗

t



+ [ρ Src.θ] − ρ̄ζ w∂ζ θ̄ζ t(10.76)

250

10 Numerical Computation Method
11

Fqt = −Adv.qxt + G 2 Turb.qx

2t−Δt+ [ρ∗ Src.qx ]t + [ρ∗ Fall.qx ]tFNt = −Adv.

(10.77)1
Nx t
Nx t−Δt
Nx t
Nx t
∗
∗
2 Turb.
+
G
+
ρ
Src.
+
ρ
Fall.
.
ρ∗
ρ∗
ρ∗
ρ∗

(10.78)The fourth term on the right side of the equation of potential temperature perturbation (10.76)
is The term of gravity wave with vertical advection of potential temperature in the basic state.
The advection term is calculated as physical quanities with the sound wave mode by using thevalues at the time t. The advection term is solved by using the second-order scheme as follows.
xyAdv.θ t = u∗ ∂x θ + v∗ ∂y θ + W ∗ ∂ζ θx

ζ(10.79)yζ

Adv.qxt = u∗ ∂x qx + v∗ ∂y qx + W ∗ ∂ζ qx
x(10.80)

y

Nx
Nx
Nx
Nx t
Adv. ∗ = u∗ ∂x ∗ + v∗ ∂y ∗ + W ∗ ∂ζ ∗
ρ
ρ
ρ
ρ

ζ

(10.81)

The advection terms are solved by using the fourth-order scheme as follows.
2
41 ∗
x
y
ζ
u ∂x θ + v ∗ ∂y θ + W ∗ ∂ζ θ
3

2ζ
2x
2y
1 ∗x
ζ
y
−
u ∂2x θ  + v∗ ∂2y θ  + W ∗ ∂2ζ θ 
3Adv.θ t =(10.82)

2
41 ∗
x
y
ζ
u ∂x qx + v∗ ∂y qx + W ∗∂ζ qx
3

2ζ
2x
2y
1 ∗x
ζ
y
−
u ∂2x qx + v∗ ∂2y qx + W ∗ ∂2ζ qx
3Adv.qxt =



x

y

Nx
Nx
Nx t 4 ∗ Nx
u ∂x ∗ + v ∗ ∂y ∗ + W ∗ ∂ζ ∗Adv. ∗ =
ρ
3
ρ
ρ
ρ


Nx
1 ∗x
−
u ∂2x ∗3ρ2xy+ v∗ ∂

Nx
2y ∗
ρ

2y

(10.83)ζζ+ W∗ ∂

Nx
2ζ ∗
ρ

2ζ 

(10.84)

The turbulent ﬂux Hφ1 , Hφ2 , and Hφ3 for one scalar φ which is represented by (7.26) ∼ (7.28)
are solved by using diﬀerence as follows.


Hφ1 = ρ̄νHh

Hφ2 = ρ̄νHh

1
G



xζ

2

y 1



yζ

2

∂x (Jd φ) + ∂ζ J31 φ

1
2

1
G

x 1

1
2

∂y (Jd φ) + ∂ζ J32 φ(10.85)(10.86)

10.1 The discretization of basic equation systemHφ3 = ρ̄νHv

ζ

1
G

∂ζ φ

1
2

251

ζ

(10.87)

Finally, the turbulent diﬀsion term is indicated as follows.


1
2

x



x









y

G Turb.θ = ∂x Jd Hθ1 + ∂y Jd Hθ2 + ∂ζ Hθ3 + J31 Hθ1


1



x

+ J32 Hθ2





y

ζ

G 2 Turb.qx = ∂x Jd Hqx 1 + ∂y Jd Hqx 2 + ∂ζ Hqx 3 + J31 Hqx 1
1

ζ

ζ

y

x

+ J32 Hqx 2(10.88)ζy

G 2 Turb.

(10.89)





x
y
Nx
ζ
ζ
x
y
=
∂
J
H
+
∂
J
H
+ J32 HNx 2
x
Nx 1
y
Nx 2 + ∂ζ HNx 3 + J31 HNx 1
d
d
∗
ρ



The method of discretization in the case of calculating the gravity wave mode withsmall time step
When the gravity wave mode is calculated with small time step, the equation of potential temperature is indicated as follows.




1
∂θ 
∂θ 
∂θ 
∂ θ̄
∂θ ∗
∗
− ρ̄w + &G 2 Turb.θ
= − u∗+ v∗+ W∗'( ) + &ρ Src.θ
'( )
∂t
∂x
∂y
∂ζ
∂ζ

'(

&

)

[rm]

& '( )[gm][physics](10.91)[physics]

The second term on the right side of is calculated by using the values at the time step Δτ . The
time integration of this equation is shown as follows.

ρ∗



ζ τ
θ τ +Δτ − θ τ+ Fθt= − ρ̄ζ w∂ζ θ̄
Δτ

(10.92)The time integration of vertical velocity (w) is shown since the buoyancy terms are calculated byusing the values at the time(Δτ).
⎡

⎤
ζ τ

τ +Δτ
1
2
− wτ
θ
τ +Δτ
τ
ζw
= [∂ζ (αDiv ∗ )]τ + g ⎣ρ∗ ⎦ − β ∂ζ pρ∗+ (1 − β) ∂ζ p
Δτ
θ̄

⎡ ⎛
⎞
⎛
⎞ ⎤
ζ τ +Δτ
ζ τ


1 p
1 p
⎢
⎥
− ⎣β ⎝gG 2 2 ⎠+ (1 − β) ⎝gG 2 2 ⎠ ⎦ + Fwt

cs

cs(10.93)

252

10 Numerical Computation Method

Terms which are calculated with large time step are modiﬁed as follows by this.
1ζFwt = −Adv.wt + ρ∗ Buoy.q + ρ∗ fc ux
1

1

Fθt = −Adv.θ t + G 2 Turb.θ

2t−Δt

2
ζ t

1

1

2t−Δt

+ G 2 Turb.w+ [ρ∗ Src.θ]t(10.94)(10.95)time ﬁlter
The large time step integration is solved by using the leap-frog scheme. The separation of solutions
between the odd time step and the even time step is caused in the case of using this technique.
So the Asselin time ﬁlter (Asselin, 1972) is used to control the separation of solutions. This is
the technique to ﬁlter physical quanities at the time t after physical quanities are calculated at
the time t + Δt.
The Asselin time ﬁlter is indicated by using a representative variable ψ for forecasted variablesu, v, w, p , θ , qx .
ψ t = ψ t + μa ψ t−Δt − 2ψ t + ψ t+δt(10.96)

where μa is a factor of ﬁlter, and μa = 0.1 is a standard value.

10.2

Artiﬁcial numerical viscosity term

10.2.1

Necessity of numerical viscosity term

In previous section, it is shown that advection term is expressed by second-order or fourth-order
central diﬀerence, and calculation instability will be generated if the artiﬁcial numerical viscosity
term is not added. The reason for that is shown as follows.
One-dimensional linear wave equation :
∂u
∂u
+c
= 0,
∂t
∂x

(c > 0)(10.97)The equation is discretized as follows. Arrangement of shifted grid points is not considered here.
ut+Δt − uti
∂u
= i
,
∂t
Δt

The equation (10.97) is written by− uti−Δxut∂u= i+Δx∂x2Δx(10.98)

10.2 Artiﬁcial numerical viscosity termc
ut+Δt
= uti −
i



2



Δt  tui+Δx − uti−ΔxΔx253(10.99)From this equation, the value at time t + Δt is calculated by using the value at time t. However,when this calculation is performed, this calculation will break even if the relation of Δt and Δxsatisfys the CFL condition.
As the other discrete method, we consider ﬁrst-order windward diﬀerence because its calculation
can be stable. However, it does not mean higher accuracy for the calculation. If this discretization
is applied to an equation (10.97) and solved as follows.
= uti − c
ut+Δt
i



Δt  tui − uti−ΔxΔx(10.100)

However, if this is transformed into the form using the equation (10.99) which is the ﬁrst discretization,c= uti −
ut+Δt
i

2




 c  Δt  Δt  tt
ui+Δx − ui−Δx +uti+Δx − 2uti + uti−ΔxΔx
2 Δx(10.101)∂2u
The added term is diﬀerence approximation of space second-degree diﬀerentiation, and aﬀects∂x2
as the diﬀusion term. Calculation is stably preformed using the ﬁrst-order forward diﬀerence fromthis reason. In central diﬀerence, however, for making the diﬀusion term, the artiﬁcial numericalviscosity term is added to the advection term.

10.2.2

Numerical viscosity with the second-order or fourth-order diﬀusion
term

Since the eventh-degree diﬀerentiation generally has the eﬀect of diﬀusion, the deﬃsion term
explained in the previous subsection can change the degree of the diﬀerentiation. Here, numericalviscosities with the second-order and the fourth-order diﬀusion terms are explained.
Firstly, the second-order viscosity term is represented by the arbitrary physical quantities φ,1
2

G Diﬀ.φ = ν2h∂ 2 (ρ̄φ ) ∂ 2 (ρ̄φ)∂ 2 (ρ̄φ)
+
+
ν
.
2v
∂x2
∂y 2
∂ζ 2

(10.102)

When φ are u, v, w, qx , Nx /ρ∗ , φ is perturbation from an initial value. When φ is θ, φ is theperturbation from the basic state.
ν2h and ν2v are viscosity coeﬃcient, and are deﬁned as follows.

254

10 Numerical Computation Method

ν2h =α2h Δ2hΔt(10.103)ν2v =α2v Δ2v
Δt(10.104)

where,
1

Δh = (ΔxΔy) 2(10.105)

Δv = Δζ

(10.106)

In these equations, Δh and Δv are the horizontal and vertical averaged grid interval in the domain.
Moreover, α2h and α2v are the non-dimensional quantities. For a stable calculation, the following
requirement needs to be satisﬁed.
1
α2h ≤ ,
8

1
8

α2v ≤

(10.107)

In this way, this viscosity term is applied to forecast variables other than pressure perturbation.
We deﬁne the second-order diﬀerentiation operator as


∂x2 φ


i

=

1
(φi+Δx − 2φi + φi−Δx )Δx(10.108)

When u , v , w , qx , Nx /ρ∗ are made into perturbation from the initial value and θ  is made into
perturbation from the basic state, diﬀerence notation are shown as follows.
1

1

2

1

1

2

1

1

1

1

2

1

2

G 2 Diﬀ.u = ν2h ∂x2 ρ̄x u + ∂y2 ρ̄x u
G 2 Diﬀ.v = ν2h ∂x2 ρ̄y v + ∂y2 ρ̄y v

G 2 Diﬀ.w = ν2h ∂x2 ρ̄ζ w + ∂y2 ρ̄ζ w
G 2 Diﬀ.θ = ν2h ∂x2 ρ̄θ  + ∂y2 ρ̄θ 
1G 2 Diﬀ.qx = ν2h ∂x2 ρ̄qx + ∂y2 ρ̄qx




Nx
N
G Diﬀ. ∗ = ν2h ∂x2 ρ̄ ∗x
ρ
ρ
1
2



+ ∂y2



1

2

1

2

+ ν2v ∂ζ2 ρ̄x u+ ν2v ∂ζ2 ρ̄y v

2

1



1

1





2

(10.111)2+ ν2v ∂ζ2 ρ̄qx

N
ρ̄ ∗x
ρ

(10.110)+ ν2v ∂ζ2 ρ̄ζ w

+ ν2v ∂ζ2 ρ̄θ (10.109)+ ν2v ∂ζ2

(10.112)2(10.113)


N
ρ̄ ∗x
ρ



(10.114)10.2 Artiﬁcial numerical viscosity term

255

The fourth-order numerical viscosity term is considered as well as the second-order. When φis u, v, w, qx , Nx /ρ∗ , φ is perturbation from an initial value. When φ is θ, φ is the perturbationfrom the basic state.

1
2

G Diﬀ.φ = −ν4h

∂ 4 (ρ̄φ ) ∂ 4 (ρ̄φ)∂ 4 (ρ̄φ)
+
− ν4v
4
4
∂x
∂y
∂ζ 4

(10.115)

ν2h and ν2v are a viscosity coeﬃcient, and are deﬁned as follows.

ν4h =α4h Δ4hΔt(10.116)

ν4v =α4v Δ4v
Δt(10.117)

where α2h and α2v are the non-dimensional quantities, and the values which are often used are
as follows.

α4h = 0.001,

α4v = 0.001.

(10.118)

As well as the second-order, this viscosity term is applied to forecast variables other than pressure
perturbation.
When u , v , w , qx , Nx /ρ∗ are made into perturbation from initial value and θ  is made into
perturbation from the basic state, diﬀerence notation are shown as follows.
1

1

3

1

1

3

1

1

3

1

1

3

4

1

3

4



5

4

G 2 Diﬀ.u = −ν4h ∂x2 ∂x2 ρ̄x u

4

G 2 Diﬀ.v = −ν4h ∂x2 ∂x2 ρ̄y v


G 2 Diﬀ.w = −ν4h ∂x2 ∂x2 ρ̄ζ w
G 2 Diﬀ.θ = −ν4h ∂x2 ∂x2 ρ̄θ 
1G 2 Diﬀ.qx = −ν4h ∂x2 ∂x2 ρ̄qx3+ ∂y2 ∂y2 ρ̄y v

4

Nx
N
G Diﬀ. ∗ = −ν4h ∂x2 ∂x2 ρ̄ ∗x
ρ
ρ
1
2

3

42

+ ∂y2 ∂y2 ρ̄x u

3

42



+ ∂y2 ∂y2 ρ̄ζ w
42

3

42

+ ∂y2 ∂y2 ρ̄qx

6

+ ∂y2

5

∂y2



1

− ν4v ∂ζ2 ∂ζ2 ρ̄y v

423+ ∂y2 ∂y2 ρ̄θ 1

1(10.119)2− ν4v ∂ζ2 ∂ζ2 ρ̄ζ w
11− ν4v ∂ζ2 ∂ζ2 ρ̄qx
6− ν4v ∂ζ2

(10.120)
2(10.121)2− ν4v ∂ζ2 ∂ζ2 ρ̄θ 

N
ρ̄ ∗x
ρ

2

− ν4v ∂ζ2 ∂ζ2 ρ̄x u(10.122)2∂ζ2



(10.123)
N
ρ̄ ∗x
ρ



(10.124)

It is more desirable to use this numerical viscosity term which has fourth-order diﬀusion term,since it can distinguish from the diﬀusion term Turb.φ in Section 7 - “subgrid-scale diﬀusion”,
and can diﬀuse the ingredient of high wave number.

256

10 Numerical Computation Method

As a result, including these numerical viscosity term, the term Fφt calculated with large time
step t shown in (10.40) ∼ (10.43), (10.76) and (10.77) are able to be expressed as follows.
1

x

1

y t

Fut = −Adv.ut + ρ∗ fs vy − ρ∗ fc wζ
Fvt = −Adv.vt − ρ∗ fs ux

2

1

x 2t

1

1

1

1

+ G 2 Turb.v + G 2 Diﬀ.v

1

ζ

Fwt = −Adv.wt + ρ∗ (Buoy.θ + Buoy.q) + ρ∗ fc ux
1

1

1

1

1

Fθt = −Adv.θ t + G 2 Turb.θ + G 2 Diﬀ.θ
1

2t−Δt

Fqt = −Adv.qxt + G 2 Turb.qx + G 2 Diﬀ.qx(10.125)

2t−Δt

2
ζ t

1

(10.126)
1

1+ G 2 Turb.w + G 2 Diﬀ.w

+

,t



+ ρ∗ Src.θ − ρ̄ζ w∂ζ θ̄

2t−Δt



2t−Δt

(10.127)ζ t(10.128)+ [ρ∗ Src.qx ]t + [ρ∗ Fall.qx ]tFNt = −Adv.

2t−Δt

1

+ G 2 Turb.u + G 2 Diﬀ.u(10.129)



1
1
Nx t
Nx
Nx t−Δt
Nx t
Nx t
∗
∗
2 Turb.
2 Diﬀ.
+
G
+
G
+
ρ
Src.
+
ρ
Fall.
(10.130)
ρ∗
ρ∗
ρ∗
ρ∗
ρ∗

The terms Fwt and Fθt in the case of calculating gravity wave mode with small time step shownin (10.94) and (10.95) is able to be expressed as follows.
1ζFwt = −Adv.wt + ρ∗ Buoy.q + ρ∗ fc ux
1

1

1

2
ζ t

Fθt = −Adv.θ t + G 2 Turb.θ + G 2 Diﬀ.θ

10.3

1

1

1

+ G 2 Turb.w + G 2 Diﬀ.w

2t−Δt+ [ρ∗ Src.θ]t2t−Δt(10.131)(10.132)

Boundary condition

Finit diﬀerence of the basic equations shown in the previous section includes operators at i −1, i + 1, j − 1, .... For calculation of forecast variables at he grid points which locate along eachboundary:

u;
v;
w;

i = 1, nx,
j = 1, ny − 1, k = 1, nz − 1
i = 1, nx − 1, j = 1, ny,
k = 1, nz − 1
i = 1, nx − 1, j = 1, ny − 1, k = 1, nz

φ;i = 1, nx − 1, j = 1, ny − 1, k = 1, nz − 1,

(10.133)

where φ of the arbitrary scalar, and a boundary condition is needed to be given. We will explain
about some kinds of a boundary condition in CReSS in the followings. In the fourth-order
calculation, values at the grid points shifted one grid from the boundary are calculated with thesecond-order precision.

10.3 Boundary condition

10.3.1

257

Lateral boundary condition

Four boundary conditions of periodic, ﬁxed wall (mirror condition), non-gradient, and radiation,
are applicable to a lateral boundary.
Periodic boundary condition
The periodic boundary condition is the condition that the values along the eastern (northern)boundary are equal to those along the western (southern) boundary during an integration.
In the cast that we use the periodic boundary condition in the east-west direction,

u1,j,k = unx−2,j,k ,unx,j,k = u3,j,k

v1,j,k = vnx−2,j,k ,w1,j,k = wnx−2,j,k ,
φ1,j,k = φnx−2,j,k ,W1,j,k = Wnx−2,j,k ,vnx−1,j,k = v2,j,k
wnx−1,j,k = w2,j,kφnx−1,j,k = φ2,j,k
Wnx−1,j,k = W2,j,k ,(10.134)

or in the cast that we use the periodic boundary condition in the north-south direction,

ui,1,k = ui,ny−2,k ,vi,1,k = vi,ny−2,k ,wi,1,k = wi,ny−2,k ,ui,ny−1,k = ui,2,kvi,ny,k = vi,3,k
wi,ny−1,k = wi,2,k

φi,1,k = φi,ny−2,k ,Wi,1,k = Wi,ny−2,k ,

φi,ny−1,k = φi,2,kWi,ny−1,k = Wi,2,k ,(10.135)

where φ is the arbitrary scalar.
Fixed wall boundary condition
In order to explain ﬁxed wall boundary condition, for example, an east-and-west wall is shown
here. Along the east and west boundaries, x diﬀerentiation in uncompressible equation of conservation of mass and free-slip condition are


∂u ∂w
+
∂
∂x
∂ζ

7

∂x = 0,

∂w
= 0.
∂x

(10.136)

It is drawn that the velocity u of x direction satisfys the following relation.
∂ 2u
=0
∂x2

=⇒

u (x = −Δx, y, z) = u (x = Δx, y, z)(10.137)

258

10 Numerical Computation Method

Under the situation, the ﬁxed wall condition is determined as follows.
In the cast that we use the periodic boundary condition in the east-west direction,
u1,j,k = −u3,j,k ,v1,j,k = v2,j,k ,unx,j,k = −unx−2,j,k
vnx−1,j,k = vnx−2,j,k

w1,j,k = w2,j,k ,
φ1,j,k = φ2,j,k ,W1,j,k = W2,j,k ,wnx−1,j,k = wnx−2,j,kφnx−1,j,k = φnx−2,j,k
Wnx−1,j,k = Wnx−2,j,k ,(10.138)

or in the cast that we use the periodic boundary condition in the north-south direction,

ui,1,k = ui,2,k ,vi,1,k = −vi,3,k ,wi,1,k = wi,2,k ,ui,ny−1,k = ui,nx−2,kvi,ny,k = −vi,nx−2,kwi,ny−1,k = wi,nx−2,kφi,1,k = φi,2,k ,Wi,1,k = Wi,2,k ,

φi,ny−1,k = φi,nx−2,kWi,ny−1,k = Wi,nx−2,k ,(10.139)

where φ is the arbitrary scalar.
Non-gradient boundary condition
Non-gradient boundary condition is the condition that the gradient of forecast variables acrossboundary plane may be set to grad (φ) = 0.
In the cast that we use the periodic boundary condition in the east-west direction,

u1,j,k = u3,j,k ,v1,j,k = v2,j,k ,unx,j,k = unx−2,j,k
vnx−1,j,k = vnx−2,j,k

w1,j,k = w2,j,k ,
φ1,j,k = φ2,j,k ,W1,j,k = W2,j,k ,wnx−1,j,k = wnx−2,j,kφnx−1,j,k = φnx−2,j,k
Wnx−1,j,k = Wnx−2,j,k ,(10.140)or in the cast that we use the periodic boundary condition in the north-south direction,

ui,1,k = ui,2,k ,vi,1,k = vi,3,k ,wi,1,k = wi,2,k ,ui,ny−1,k = ui,nx−2,kvi,ny,k = vi,nx−2,kwi,ny−1,k = wi,nx−2,kφi,1,k = φi,2,k ,Wi,1,k = Wi,2,k ,

φi,ny−1,k = φi,nx−2,kWi,ny−1,k = Wi,nx−2,k ,(10.141)

10.3 Boundary condition259

where φ is the arbitrary scalar.
Radiational boundary condition
Radiational boundary condition is set up so that the radiational conditions with one-dimensional
linear wave equation
∂φ
∂φ
+c
=0
∂t
∂x

(10.142)is satisﬁed for stopping the action of the unnatural solution produced in a boundary plane. Adding
the attenuation term,
∂φ
∂φ
+c
= −γφ
∂t
∂x

(10.143)is used here. As nesting from a model output for a larger domain, such as objective analysis data,as an initial and a boundary values. The diﬀerence between CReSS and the model for a larger
domain for each physical amount is used for a boundary condition.
∂ (φf − φc )∂ (φf − φc )+c
= −γ (φf − φc )∂t∂x(10.144)

where φ is the arbitrary physical quantities, bottom characters f and c indicate the values of thelarger model and this model, respectively.
There are various methods in the way to solve phase velocity c in the above radiational condition. The three methods can be chosen in CReSS. φ and φf − φc is expressed with Φ below.
[ (1) Method to calculate phase velocity for each grid point](1) Method to calculate phase velocity
for each grid point
In this case, equations excluded the attenuation term on the right side from the equations(10.143), (10.144) is solved about phase velocity c. Non-dimensional phase velocity is directly
solved.

Cbt = ctb

Φt+Δt − Φt−Δt
Δtb,=− t bΔx2Φb−1 − Φt+Δt− Φt−Δtbb−1 ≤ Cbt ≤ 0, western edge or southern edge

Cbt = ctb

Φt+Δt − Φt−Δt
Δtb= − t+Δt b,
ΔxΦb+ Φt−Δt− 2Φtb−1b0 ≤ Cbt ≤ 1, eastern edge or northern edge

260

10 Numerical Computation Method

These phase velocities should be calculated for each forecast variables. In the actual calculation,
the following values are used.

Cbt = ctb

Φt − Φt−2Δt
Δt= − t−Δtb−1 t b−1 t−2Δt ,Δx2Φb−2 − Φb−1 − Φb−1

Δt
Cbt = ctb=−Δx

Φtb−1 − Φt−2Δtb−1
Φtb−1 + Φt−2Δt− 2Φt−Δtb−1b−2−1 ≤ Cbt ≤ 0, westernmostorsouthernmost
0 ≤ Cbt ≤ 1, easternmostornorthernmost,

where bottom character b is the number of grid point on the boundary given by the equation(10.133).
[ (2) Method to calculate constant phase velocity in vertical direction](2) Method to calculate
constant phase velocity in vertical direction
Wave equation which assumed constant phase velocity in vertical is considered as follows.
* ztop
zsf c*

ztop ∂Φ∂Φdz + cdz = −γΦ∂tzsf c ∂x(10.149)

Using the value calculated at the inner grid point on preceding time step,Δt
Cbt = ctb
=−
Δx

kmax

3

k=kmin





t−2Δt
t
Φtb−1 − Φt−2Δtsgn 2Φt−Δtb−1
b−2 − Φb−1 − Φb−1

kmax

k=kminkmax

3

k=kmin−1 ≤ Cbt ≤ 0, western edge or southern edge

Φtb−1 − Φt−2Δtsgn Φtb−1 + Φt−2Δt− 2Φt−Δt
b−1
b−1
b−2
kmax

k=kmin,t−2Δtt2Φt−Δtb−2 − Φb−1 − Φb−1

····
Δt
=−
Cbt = ctbΔx4

Φtb−1 + Φt−2Δt− 2Φt−Δt
b−1
b−2
····

4

,

0 ≤ Cbt ≤ 1, eastern edge or northern edge

where kmin and kmax areu, v, φ ; kmin = 2, kmax = nz − 2(10.152)

w ; kmin = 2, kmax = nz − 1.

(10.153)[ (3) Method to add advectional velocity to constant phase velocity](3) Method to add advectional

10.3 Boundary condition

261

velocity to constant phase velocity
Phase velocities should be calculated for each forecast variable. c∗ is given as the propagation
velocity of typical gravity wave, such as about 30 m s−1 . Using x-component velocity u forthe eastern and western boundaries and y-component velocity v for the northern and southernboundaries, a boundary condition is given as follows.

Cbt = ctb

ΔtΔt
= (u − c∗ ),Δx
Δx

−1 ≤ Cbt ≤ 0, westernmost(10.154)

Cbt = ctb

ΔtΔt
= (u + c∗ ),Δx
Δx

0 ≤ Cbt ≤ 1, easternmost(10.155)

Cbt = ctb

ΔtΔt= (v − c∗ ),ΔyΔy−1 ≤ Cbt ≤ 0, southernmost(10.156)

Cbt = ctb

ΔtΔt= (v + c∗ ),ΔyΔy0 ≤ Cbt ≤ 1, northernmost(10.157)[ (4) Method to use constant phase velocity](4) Method to use constant phase velocity
Only c∗ in (3) is used on assumption that a wave always goes away out of the calculation
domain. Therefore, phase velocities at all grid points on the lateral boundary plane are the samevalue. However, a sign becomes reverse between eastern and western sides, and southern andnorthern sides.
In the case of (3) and (4), the same phase velocity as (2) is used for a vertical velocity component,such as u on the eastern and western planes and v on the southern and northern planes.
By using the phase velocity solved in (1), (2), (3), and (4), the value at a grid point on theboundary at next time step is,for the variable integrated with small time step Δτ ,

Φτb +Δτ = Φτb −
Cbt  tt−Δt ΔτΦ−ΦΔτ,− γΦt−Δtb−1bb
1 − Cbt
Δtwestern edge or southern edgeΦτb +Δτ = Φτb +
Cbt  tt−Δt ΔτΦ−ΦΔτ,− γΦt−Δtb−1bb
1 + Cbt
Δteastern edge or northern edge

for the variable integrated with large time step Δt,= Φt−Δt
−
Φt+Δt
b
b


2Cbt  tΦb−1 − Φt−Δt− γΦt−ΔtΔt,bbt
1 − Cbwesternmostorsouthernmost

Φt+Δt= Φt−Δt+bb


2Cbt  tΦb−1 − Φt−Δt− γΦt−ΔtΔt,bbt
1 + Cbeasternmostornorthernmost

262

10 Numerical Computation Method

where attenuation constant γ has the dimension of time inverse number. It is thought that theinverse number 1/γ is suitable more than one hour and less than twelve hours.

10.3.2

Boundary conditions at the top and bottom

Two kinds of a boundary condition, ﬁxed wall (mirror condition) and non-gradient, are applicable
to boundary conditions at the top and bottom.
Fixed wall boundary condition
Basically, it can be considered as same as lateral ﬁxed wall boundary condition. However, when
we use the terrain-following coordinate system, the relation between the velocity on Cartesian
coordinate system and contravariant velocity should be considered as follows.


1

W = (uJ31 + vJ32 + w) G 2(10.162)

Note that vertical velocity W when velocity is 0 on the boundary plane.
On the boundary plane at the top (k = nz − 1),
W = 0,

w=0(10.163)

and on the boundary plane at the bottom (k = 2),

W = 0,

w = −uJ31 − vJ32.

(10.164)

Therefore, boundary condition is given as follows.
ui,j,1 = ui,j,2 ,vi,j,1 = vi,j,2 ,
φi,j,1 = φi,j,2 ,Wi,j,2 = 0, Wi,j,1 = −Wi,j,3 ,

ui,j,nz−1 = ui,j,nz−2vi,j,nz−1 = vi,j,nz−2
φi,j,nz−1 = φi,j,nz−2
Wi,j,nz−1 = 0, Wi,j,nz = −Wi,j,nz−2(10.165)

where φ is the arbitrary scalar. By replacing vertical velocity W at the top and bottom boundaries
with the equation (10.162),
on boundary plane at the top (k = nz − 1),
wi,j,nz−1 = 0(10.166)


x

y

1ζ

wi,j,nz = − uζ J31 + vζ J32 + W G 2

(10.167)
i,j,nz−2

10.3 Boundary condition

263

on the boundary plane at the bottom (k = 2),xwi,j,2 = − uζ J31 + vζ J32


x

y

(10.168)

i,j,2
y

1ζ

wi,j,1 = − uζ J31 + vζ J32 + W G 2

(10.169)
i,j,3

where pressure perturbation p’ on the boundary plane at the bottom is given by extrapolation as
follows.
pi,j,1 = 2pi,j,2 − pi,j,3(10.170)

Non-gradient boundary condition
Similarly to the lateral boundary,ui,j,1 = ui,j,2 ,vi,j,1 = vi,j,2 ,
wi,j,1 = wi,j,2 ,
φi,j,1 = φi,j,2 ,

ui,j,nz−1 = ui,j,nz−2vi,j,nz−1 = vi,j,nz−2
wi,j,nz = wi,j,nz−1
φi,j,nz−1 = φi,j,nz−2

Wi,j,1 = Wi,j,2 ,Wi,j,nz = Wi,j,nz−1(10.171)

where φ is the arbitrary scalar.

10.3.3Sponge layer

In CReSS, the sponge layer set as arbitrary thickness is able to be set up. This is a layer for
suppressing and absorbing wave reﬂections near the boundary plane. An additional term is added
to the basic equation as follows.
∂ (ρ∗ φ)= −Adv.φ + ... − γh (x, y) ρ∗ (φ − φe) − γv (z) ρ∗ (φ − φe)∂t(10.172)

where φ is an arbitrary forecast variable, φe is a variable of external data, such as an objective
analysis, γh (x, y) is an attenuation coeﬃcient which becomes small toward each horizontal boundary plane. γv (z) is attenuation coeﬃcient which becomes small toward each vertical boundary
plane.
For example, in the case of the west lateral boundary layer which made thickness of sponge
layer d in horizontal, γh (x, y) is given as follows.

264

10 Numerical Computation Method

⎧


x 3
⎪
⎪
α
1
−
,
⎪
⎨ h

γh =

x≤d

d

⎪
⎪
⎪
⎩

(10.173)0,x>d

In the case of the bottom boundary layer which made minimum height of sponge layer in vertical,
γv (z) is given as follows.
⎧
⎪
0,
⎪
⎪
⎪
⎨

γv =

⎪
⎪
⎪
⎪
⎩ αv

z < zlowz − zlow
1 − cos πztop − zlow

8(10.174),z ≥ zlow

where αh and αv are horizontal and vertical attenuation constants, and have dimension of theinverse number of time. These inverse numbers 1/αh and 1/αv is called e-folding time. It isappropriate that these constants are set up in the range about 1/100 ∼ 1/300 s−1 . In addition,
it is suitable for the thickness of sponge layer to set up in the number of several grid points inhorizontal and the thickness of vertical layer about the upper surface to 1/3 in vertical.
This term is calculated at t − Δt. About the forecast variable calculated at a small time step,
it is treated as a value of a big time step 2Δt like advection term and numerical viscosity term.
The adjusted value to adjust was external data’s value φe , such as objective analysis, in theequation (10.172). However, when a model is performed independently, and there is no objective
analysis data for a forecast variable, it can adjust to initial value φ0 or basic state value φ̄ asfollows.
∂ (ρ∗ φ)= −Adv.φ + ... − γh (x, y) ρ∗ (φ − φ0 ) − γv (z) ρ∗ (φ − φ0 )∂t

(10.175)∂ (ρ∗ φ)= −Adv.φ + ... − γh (x, y) ρ∗ φ − φ̄ − γv (z) ρ∗ φ − φ̄∂t(10.176)

Chapter 11

Application of Parallel Processing

In this section, we will describe the computation method of CReSS with aparallel computing.
The parallel program of CReSS is fundamentally designed for running ona multiple memory type parallel computer. The parallel program is writtenwith Message Passing Interface (MPI). CReSS is available to almost all parallel computers, because the MPI is a defactor standard library. Using someworkstations, CReSS can also be performed in a cluster environment.

266

11 Application of Parallel Processing

11.1

Technique for parallelization

11.1.1

Two-dimensional domain decomposition

CReSS is fundamentally designed for a multiple memory type parallel computer. A parallel
program is available for not only large multiple memory type parallel computers but also cluster
systems of workstations and PC-UNIX computers.
It is easy to use the parallel processing by adopting two-dimensional domain decomposition inhorizontal, because a spacial diﬀerence in horizontal is explicitly calculated in CReSS. Various
physical processes, such as cloud microphysics, can be fundamentally calculated only with thephysical quantities in the caluculating cell, there are some references between upper and lower
cells. A z ∗ coordinate system (ζ system) makes parallelization easy that a coordinate system.
Firstly, we show the advantage of parallel processing by adopting two-dimensional domain
decomposition in horizontal.
• It is easy to establish the boundary condition, because coordination system is z ∗ -system(ζ-system) has no loss of data near the surface.
• It is easy to understand the technique of parallelization intuitively.
• It is available to use the same function on nodes except the node included the boundary
condition.
• The load balance is high. (That is, a waiting time for other node’s computation would beshort.)• Parallelization eﬃciency for a calculation with a number of grid points would be high.
• As a calculation domain for each node is larger, data communication between each node isrelatively smaller.
We show a schematic illustration of two-dimensional domain devomposition as follows.
Figure 11.1 shows a schematic illustration of the parallel computing in the case of calculatingwith the second-order precision. In the case of adopting central diﬀerence approximation withthe second order precision, six grid points must be referred for a certain grid point. Considering
the horizontal two-dimensional domain decomposition, there is no problem of reference in thevartical direction. In horizontal ,however, when the approximate solution of the grid point on theboundary of decomposition is calculated,the reference between nodes is needed. Then showing
the enlarged ﬁgure on right side, we set the halo region in each node, and it is enable to obtainthe same result as by one node on the computational domain computed by decomposed nodes
had the halo region. For that purpose, communication between the nodes are performed only
for getting the essential data from neighboring the node whenever the value in the halo region isneeded (The detail is shown in subsection 11.1.2)
In addition, CReSS is available to compute with fourth-order precision. For that purpose, itis necessary to communicate much the value which is needed by doubling a halo region like thecase of second order precision. It is possible to change the precision of the diﬀerential equationby the same method, for example, sixth-order, eighth-order, ... and N-th-order.

11.1 Technique for parallelization

Figure 11.1. Schematic illustration of two-dimensional domain decomposition and exchange of the value
in the halo region

11.1.2

Example of parallel computing

In this subsection, we describe the communication between halo regions explained in subsection
11.1.1.
The source code is written with Message Passing Interface (MPI). The MPI is the interface
speciﬁcations for the call form of various communication routines required for a multiple memory
type parallel computer and the function and subroutines of C and Fortran. But it is interface
speciﬁcation to the last, and the concrete application method is not determined.
As for the details about MPI, it is good to refer to the following homepages.
http://www.mpi-forum.org
For communicating by MPI such as ﬁgure 11.1, it is essential to repeat a shift in all directions
except the boundary plane. Figure 11.2 is the example of a shift to the halo region of east sidefrom the halo region of west side except the boundary plane. In this ﬁgure, nipe is numbers of
node in the east-west direction and ipe is node number.

267

268

11 Application of Parallel Processing

ipe = 0ipe = 1

ipe = nipe - 2

ipe = nipe - 1

Figure 11.2. This is a example of a shift to the halo region of east side from the halo region of west side.

The source code of the MPI parts in CReSSis shown as following. The part of it is shown
here, so for understanding the detail it is necessary to refer exchansn.f and exchanwe.f in the
directory Src.

! A node at the westernmost end has no transmitting buﬀer
if(ipe.eq.0) thendst=mpi_proc_null
end if! A node at the easternmost end has no transmitting buﬀer
if(ipe.eq.nipe-1) thensrc=mpi_proc_null
end if! A setup of transmitting buﬀers other than a node at the westernmost end
if(ipe.ne.0) thendo 140 k=kbmin,kbmax
do 140 j=jbmin,jbmax
ib=(k-kbmin)*jbmxn1-jbmin1+jsbuf(ib)=var(iwsend,j,k)140

continueend if! The call of the MPI routine for transmission and reception
call mpi_sendrecv(sbuf,siz,mpi_real,dst,tag,rbuf,siz,mpi_real,src,

11.2 Test of the parallel program.

tag,mpi_comm_world,stat,ierr)! A setup of receiving buﬀers other than a node at the easternmost end
if(ipe.ne.nipe-1) thendo 170 k=kbmin,kbmax
do 170 j=jbmin,jbmax
ib=(k-kbmin)*jbmxn1-jbmin1+jvar(ierecv,j,k)=rbuf(ib)170continueend ifAlthough the shift is mounted by mpi_sendrecv here, it is better not to mount this by mpi_send
and mpi_recv. Although it is not impossible, various programming can be considered and may
carry out a deadlock depending on the case. Moreover, by the formal speciﬁcation of the MPI,
the guarantee of operation of a program will be impossible, since it seems that it has not deﬁned
whether a deadlock is carried out by this or it does not carry out.

11.2

Test of the parallel program

11.2.1

Inspection of coincidence of a calculation result

Since parallel programming adopted in CReSS should just do a shift fundamentally shown in
the previous section, the result performed using several nodes must be completely same as theresult performed using one node (in calculation for the whole average value, it is not necessarily to
get complete agreement. Since it does not apply operation to the value in shifting, disagreement
cannot happen.). We checked the agreement by CReSS as follows.
When several nodes are used for calculation, a result is outputted from every node.
ex.： exprim.dmpxxxxxx.pe0000.bin 〜 exprim.dmpxxxxxx.peyyyy.bin
For comparing to the result by one node and that by several nodes, we gather the above dataﬁles up to one ﬁle. (A post processor “unite” can gather the output ﬁles up to a binary unformatedﬁle with direct access.) The ﬁle was compared with the result of calculation with one node. If theﬁles of both one node and several nodes are text format, we can conﬁrm their complete agreement
as the following command.
ex.： % diff exprim.dmpxxxxxx.united.txt exprim.dmpxxxxxx.txt
As a result of executing this command, if nothing happens (0 byte of ﬁle will be made if a ﬁle

269

270

11 Application of Parallel Processing

is outputted to a standard input/output), the result must completely be in agreement. exprim
of ﬁle name is an experiment name (see section 4.1).
We perform this veriﬁcation to various experiments with CReSS, and there is no problem inthe parallelization.

11.2.2Eﬃciency of parallel processing

Finally, the eﬃciency of parallel processing is shown by a comparison of calculations with thevarious numbers of processing elements (PEs). The performance of parallel processing of CReSS
was tested by a simulation which had 67 × 67 × 35 grid points and which was integrated for 50
steps on HITACHI SR2201.
As increase of the numbers of PEs, the calculation time decreased almost linearly (Fig. 11.3).
The results of the test showed a suﬃciently high performance of the parallel computing of CReSS.
In ﬁgure 11.4, it can see that the eﬃciency of parallel processing is falling little by little. Whenthe number of PEs was 32, the eﬃciency decreased signiﬁcantly. Because the number of grid
points was too small to use the 32 PEs, the communication between PEs became relatively large
for the present veriﬁcation setting. This result of the veriﬁcation shows that the eﬃciency ofparallel processing of CReSS is higher when each PE shares larger number of grid points. The
calculation with a lot of PEs is more useful for larger calculation.

Linear Scaling ,Real-time

Efficiency
1.1
1

1000

Efficiency

Real-time

10000

100

0.9
0.8
0.7

10
1

2

0.6
4

8

16

32

Number of Processors

100

1

2

4

8

16 32 100

Number of Processors

Figure 11.3. Time for parallel processing from a Figure 11.4. Eﬃciency of parallel processing fromtest experiment [s]a test experiment

However, there is a problem. Especially, it is becoming more remarkable in the case of using
the option for cloud microphysics. Because the program for cloud microphysics has so many if
sentences. For example, when a node without clouds must wait for the processing of the nodewith clouds, eﬃciency would be small. In the present condition, CReSS does not have advanced
parallel processing changed from the ﬁrst setup to the more eﬃcient setup in the calculation.
The values of eﬃciency in the graph is somewhat wavy in ﬁgure 11.4, since it is based on the11.2 Test of the parallel program

diﬀerence between the ways of a decomposition on two-dimensional domain decomposition. (Forexample, when a calculation is performed with four nodes, there is the ways of dividing of 1 × 4,
2 × 2, or 4 × 1.) However, it is suﬃcient for seeing the tendency of the eﬃciency of parallel
processing.

271

272

11 Application of Parallel Processing

References
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University of Oklahoma, 380pp.

Outline of the IHP Training
Course in 2007

Outline
As a part of the Japanese contribution to the International Hydrological Programme (IHP), a shortcourse on numerical prediction of high-impact weather systems will be conducted for participantsfrom the Asia-Pacific regions. The course will be held for the period from 2 to 15 December 2007 at
Hydrospheric Atmospheric Research Center (HyARC) and Information Technology Center, Nagoya
University, Nagoya, Japan. The course includes a series of lectures in English, practical sessions andtechnical tours.

Objectives
High-impact weather systems are most significant phenomena in the atmosphere and occasionallycause huge disasters to human society. Understanding their mechanisms and structures is necessary forprediction and prevention/reduction of disasters. One of the most important objectives of numericalmodels is high-resolution simulation of high-impact weather systems for detailed studies and accurate
predictions. In the 17th IHP training course, the participants will study characteristics and performanceof the state-of-the-art numerical models such as regional and global cloud resolving models.
In order to perform simulations and numerical experiments of the high-impact weather systems,
HyARC is developing a cloud-resolving numerical model named “the Cloud Resolving Storm
Simulator” (CReSS). The participants will have practice to perform some simulation experiments ofweather systems to understand how to use the cloud-resolving model. High-resolution numericalsimulation using the CReSS model will clarify a detailed structure of the high-impact weather systems
and make a quantitative prediction of the associated precipitation.

Course Contents (convener: K. Tsuboki)
Lecturers
K. Tsuboki (Hydrospheric Atmospheric Research Center, Nagoya University)
T. Shinoda (Hydrospheric Atmospheric Research Center, Nagoya University)
T. Nasuno (Frontier Research Center for Global Change)
T. Maesaka (National Research Institute for Earth Science and Disaster Prevention)
M. Yoshizaki (Institute of Observational Research for Global Change)
M. Sato (Center for Climate System Research, University of Tokyo)
T. Nagai (Information Technology Center, Nagoya University)
K. Saito (Meteorological Research Institute, Japan Meteorological Agency)
T. Aoki (Tokyo Institute of Technology)
K. Tanaka (Disaster Prevention Research Institute, Kyoto University)
W. Ofuchi (The Earth Simulator Center)
A. Sakakibara (Chuden CTI Co.,Ltd.)

List of Participants

1) UNESCO-IHP Participants
Mr. Fierra Setyawan Sastropawiro (Indonesia)
Affiliation: Meteorological and Geophysical Agency, Research and Development Center
Address: Jl. Angkasa I/2, Kemayoran, Jakarta 10720, Indonesia
Ms. Elza Surmaini (Indonesia)
Affiliation: Indonesian Agroclimate & Hydrology Research Institute, Indonesian Center for
Agricultural Land Resources Research and Developmet, Ministry of Agriculture
Address: Jl. Tentara Pelajar no. 1A, Bogor 16111, West Java, Indonesia
Ms. Sinthaly Chanthana (Lao. P.D.R.)
Affiliation: Ministry of Agriculture and Forestry, Department of Meteorology and Hydrology (DMH)
Address: Bane Arkarth, Luangprabang Road, Sikhottabong District, Vientiane Capital, Lao PDR
Mr. Sazali Bin Osman (Malaysia)
Affiliation: Division of Hydrology and Water Resources, Department of Irrigation & Drainage
Address: KM 7, Jl. Ampang, 68000 Ampang, Kuala Lumpur, Malaysia
Mr. Tirtha Raj Adhikari (Nepal)
Affiliation: Central Department of Hydrology and Meteorology, Tribhuvan University
Address: Kirtipur, Kathmandu, Nepal
Ms. Thanh Thi Nguyen (Vietnam)
Affiliation: Vietnam Institute of Meteorology, Hydrology and Environment; Center for Marine andOcean - Atmosphere Interaction Research
Address: 5/62 Nguyen Chi Than, Hanoi, Viet Nam
Ms. Josefina Calapan Argete (Philippines)
Affiliation: Institute of Environmental Science & Meteorology College of Science, University of thePhilippines
Address: Diliman, Quezon City, Philippines
Ms. Sirithorn Noiduang (Thailand)
Affiliation: Dpartment of Water Resources, Bureau of Research Development and Hydrology
Address: 180/3 Rama VIRd., Samsanenai, Payathai District, Bangkok, 10400, Thailand

2) Participants supported by other funds
Mr. Liew Ju Neng (Malaysia)
Affiliation: School of Environmental and Natural Resource Sciences, Faculty of Science andTechnology, University Kevangsaan Malaysia
Address: 43600 Ukm Bangi, Selangor , Malaysia
Mr. Palikone Thalongsengchanh (Lao. P.D.R.)
Affiliation: Graduate School of Science Kyoto University
Address: Oiwake-Cho, Kitashirakawa, Sakyo-Ku, Kyoto 606-8502 Japan

3) Participants from Graduate School of Environmental Studies, Nagoya University
Mr. Uddin Md. Rafi (Bangladesh)
Affiliation: Graduate School of Environmental Studies, Nagoya University, Japan
Address: Furo-cho, Chikusa-ku, Nagoya, Aichi 464-8601, Japan
Ms. Singh Prasamsa (Nepal)
Affiliation: Graduate School of Environmental Studies, Nagoya University, Japan
Address: Furo-cho, Chikusa-ku, Nagoya, Aichi 464-8601, Japan
Ms. Nasreen Akter (Bangladesh)
Affiliation: Graduate School of Environmental Studies, Nagoya University, Japan
Address: Furo-cho, Chikusa-ku, Nagoya, Aichi 464-8601, Japan
Mr. Ningning Zhang(China)Affiliation: Graduate School of Environmental Studies, Nagoya University, Japan
Address: Furo-cho, Chikusa-ku, Nagoya, Aichi 464-8601, JapanLectures
L0 Introduction ······································································································· K. Tsuboki- Guidance and outline of the IHP short course
L1 Mesoscale meteorology and numerical models········································· M. Yoshizaki- Introductory lecture of meso-scale meteorology and overview of non-hydrostatic
numerical model
L2 Computational Fluid Dynamics··············································································T. Aoki
- Basics of computational fluid dynamics and environmental simulation- Visualization of computational fluid dynamics
L3 Main frame computer in Nagoya University ······················································T. Nagai- Computer system at the Information Technology Center, Nagoya University- Basics of UNIX and usage of the main frame computer
L4 Cloud resolving model ······················································································ K. Tsuboki- Dynamic process and physical process of a cloud resolving model- Introduction of the Cloud Resolving Storm Simulator (CReSS)
L5 Parallel processing of the cloud resolving model····································· A. Sakakibara- Parallel computation using the Massage Passing Interface- OpenMP
L6 The non-hydrostatic model of the Japan Meteorological Agency ···················· K. Saito- Introduction of the JMAMRI non-hydrostatic model- Experimental and operational utilization of the NHM
L7 Surface process ··································································································· K. Tanaka- Surface process model, SiBUC- Coupling the surface model and the cloud resolving mode
L8 Planetary boundary layer and convective clouds···········································T. Shinoda- Simulation and observation of the planetary boundary layer- Interaction of PBL and convective clouds
L9 AGCM for the Earth Simulator ········································································W. Ofuchi- Introduction of the AFES (the AGCM for the Earth Simulator)- Simulation experiments using the AFES model
L10 Global non-hydrostatic model ·············································································M. Sato- Introduction of the global non-hydrostatic model, NICAM- Simulation experiments using NICAM
L11 Typhoon simulation ···························································································T. Nasuno- Overview and recent researches of typhoon simulations

L 12 Radar observation and cloud resolving model··········································· T. Maesaka- Radar observation of severe storms- Data assimilation of radar to the cloud resolving model

Practices
P1 Basic usage of the CReSS model- Numerical experiments of cats-eye, mountain waves and supercells
P2 Simulation experiments of high impact weather systems- Numerical experiments of typhoons, heavy rainfall, snow storms and so on

Technical Tours
T1 The Earth Simulator Center in Yokohama
T2 The Japan Meteorological Agency in Tokyo

Schedule (2-15 December 2007)
Date

DescriptionCity

2 (Sunday)
December 2007
3 (Monday)

Arrival at Nagoya

Nagoya10:00-12:00: Guidance (Lecture 0) by K. Tsuboki and
Lecture 1 by M. Yoshizaki
14:00-16:00: Lecture 3 by T.Nagai
Reception at Nagoya University in the eveneing
10:00-12:00: Lecture 7 by K. Tanaka
14:00-16:00: Lecture 8 by T. Shinoda
10:00-12:00: Lecture 4 by K. Tsuboki
14:00-16:00: Lecture 5 by A. Sakakibara
10:00-12:00 Lecture 9 by W. Ofuchi
14:00 -: Practice 1
10:00-12:00: Lecture 6 by K. Saito
14:00-: Practice 2
Move to TokyoJapanese Culture Introduction and Free Time

NagoyaTechnical Tour 1: The Japan Meteorologycal Agency inTokyo (Move to Yokohama)Technical Tour 2: The Earth Simulation Center in
Yokohama (Move to Nagoya)
10:00-12:00: Lecture 2 by T. Aoki
14:00-: Practice 2 (continued)
10:00-12:00: Lecture 11 by T. Nasuno
14:00-: Practice 2 (continued)
10:00-12:00: Lecture 10 by M. Sato
14:00-: Practice 2 (continued)
10:00-12:00: Lecture 12 by T. Maesaka
14:00-: Closing Ceremony
Departure from Nagoya

Nagoya4 (Tuesday)
5 (Wednesday)
6 (Thursday)
7 (Friday)
8 (Saturday)
9 (Sunday)
10 (Monday)

11 (Tuesday)
12 (Wednesday)
13 (Thursday)
14 (Friday)
15 (Saturday)

NagoyaNagoyaNagoyaNagoyaTokyo
Tokyo

NagoyaNagoyaNagoyaNagoya