FVCOM使用手册(完整版)

An Unstructured Grid, Finite-Volume Coastal Ocean Model

（无规则网格的有限体积海岸海洋模型）

FVCOM User Manual

（FVCOM用户手册）

FVCOM软件用户许可协议 ........................................................................................................... 3 第一章 序言 ............................................................................................................................... 4 第二章：模型公式 ........................................................................................................................... 6

2.1 直角坐标系下的原始方程 ................................................................................................ 7 2.2 ?-坐标下的控制方程 .................................................................................................... 12 2.3 二维（垂直积分）方程 .................................................................................................. 13 2.4 湍流闭合模型 .................................................................................................................. 15

2.4.1 水平扩散系数 ....................................................................................................... 15 2.4.2 垂直旋转粘性和热扩散系数 ............................................................................... 16 2.5 球面坐标系下的原始方程 .............................................................................................. 24 第三章 有限体积离散法 ............................................................................................................... 27

3.1 不规则三角网格的设计 .................................................................................................. 27 3.2 笛卡尔坐标下的离散方法 .............................................................................................. 29

3.2.1 二维外部模式 ....................................................................................................... 29 3.2.2 三维内模式 ........................................................................................................... 37 3.3 外部与内部模式的输运一致性 ...................................................................................... 44 3.4 干/湿处理方法 ................................................................................................................. 46

3.4.1 标准 ....................................................................................................................... 48 3.4.2 Isplit的上限 .......................................................................................................... 52 3.5 球坐标系下的有限体积离散方法 .................................................................................. 57 3.6 岸边界条件的微元处理 .................................................................................................. 63 第四章：外部强迫 ......................................................................................................................... 66

4.1 风应力、热通量和降水/蒸发 ......................................................................................... 66 4.2 潮汐强迫 .......................................................................................................................... 67 4.3 增加海岸或江河流量的方法 .......................................................................................... 69

4.3.1 TCE方法 ............................................................................................................... 69 4.3.2 MCE方法 .............................................................................................................. 72 4.4 水平分辨率和时间步长的规范 ...................................................................................... 74 4.5 通过底部输入地下水 ...................................................................................................... 77

4.5.1 简单盐平衡地下水通量形式 ............................................................................... 77 4.5.2 地下水输入的完全格式 ....................................................................................... 78

第五章：开边界处理 ..................................................................................................................... 79

5.1 开边界处理的初始设定 .................................................................................................. 79 5.2 普遍辐射开边界条件 ...................................................................................................... 82 5.3 新的有限体积开边界条件模块 ...................................................................................... 87 第六章：数据同化方法 ................................................................................................................. 97

6.1 推导方法 ........................................................................................................................ 100 6.2 OI方法 .......................................................................................................................... 102

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6.3 Kalman筛选 ................................................................................................................... 104

6.3.1减小序列Kalman筛选(RRKF) .......................................................................... 106 6.3.2 集合Kalman筛选(EnKF) .................................................................................. 109 6.3.3 集合平方根Kalman过滤(EnSRF) .................................................................... 111 6.3.4. 集合变换 Kalman筛选 (ETKF) ...................................................................... 113 6.3.5确认实验 .............................................................................................................. 114

第七章：FVCOM沉积模块 ....................................................................................................... 120

7.1 控制方程 ........................................................................................................................ 121 7.2 简单测试情况 ................................................................................................................ 122 第八章：FVCOM生物模块 ....................................................................................................... 123

8.1灵活生物模块（FBM） ................................................................................................ 124

8.1.1 FBM流程图 ........................................................................................................ 124 8.1.2 FBM中的方程和函数 ........................................................................................ 126 8.2 提前选择生物模块 ........................................................................................................ 157

8.2.1 养分-浮游植物-浮游动物（NPZ模型） .......................................................... 158 8.2.2 磷限制低养分层食物网模型 ............................................................................. 160 8.2.3. The Multi-Species NPZD Model ........................................................................ 168 8.2.3 多物种NPZD模型 ............................................................................................ 168 8.2.4 水质量模型 ......................................................................................................... 171

第九章：示踪-追踪模型 ............................................................................................................. 174 第十章：三维拉格朗日粒子追踪 ............................................................................................... 175 第十二章：代码平行 ................................................................................................................... 193

12.1 区域分解 ..................................................................................................................... 194 12.2 区域设置 ..................................................................................................................... 195 12.3 数据交换 ..................................................................................................................... 196 12.4数据收集 ...................................................................................................................... 197 12.5 执行 ............................................................................................................................. 198 第十三章：模型代码描述和总说明 ........................................................................................... 199

13.1 在使用FVCOM前的用户应知 ..................................................................................... 199 13.3 数值稳定的标准 ......................................................................................................... 206 13.4子程序和函数描述 ...................................................................................................... 207 第14章 模式安装，编译和运行 ............................................................................................... 231

14.1 获得FVCOM .................................................................................................................. 232

14.2a 编译METIS库 .................................................................................................. 233 14.2b 编译FVCOM .................................................................................................... 233 14.3a 运行FVCOM（连续） ..................................................................................... 238 14.3b 运行FVCOM（平行） ..................................................................................... 239

第十五章：模型设置 ................................................................................................................... 240

15.1 FVCOM运行时间控制变量文件casename_run.dat .................................................. 240 15.2 FVCOM输入文件 .......................................................................................................... 253 15.3特殊设置的必需输入文件 .......................................................................................... 256 15.4 原始输入文件的输入文件格式 ................................................................................. 257 15.5 建立和使用FVCOM模块 ............................................................................................. 268 第十六章：FVCOM测试例子 ................................................................................................... 292

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第十七章：不规则三角形网格产生 ........................................................................................... 319

17.1数据准备 ...................................................................................................................... 320 17.2 网格产生 ..................................................................................................................... 324 感谢 .............................................................................................................................................. 347 参考文献....................................................................................................................................... 348

FVCOM Software Users’ License Agreement

FVCOM软件用户许可协议

All users should read this agreement carefully. A user, who receives any version of the source code of FVCOM, must accept all the terms and conditions of this

agreement and also agree that this agreement is like any written negotiated agreement signed by you. You may be required to have another written agreement directly with Dr. Changsheng Chen at SMAST/UMASS-D and Dr. Robert C. Beardsley at WHOI

所有用户须仔细阅读此协议。收到FVCOM源代码译本的用户必须接受本协议的所有条款并与陈常胜博士和罗伯特C.比尔兹利博士直接签署书面协议。

The Finite-Volume Coastal Ocean Model (―FVCOM‖) source code has been developed in the Marine Ecosystem Dynamics Modeling Laboratory led by Dr. C. Chen at the University of Massachusetts – Dartmouth (UMASS-D) in collaboration with Dr. R. Beardsley at the Woods Hole Oceanographic Institution. All copyrights to the FVCOM code are reserved. Unauthorized reproduction and redistribution of this code are expressly prohibited except as allowed in this License.

有限体积海岸海洋模型（FVCOM）源代码由陈常胜博士领导的马萨诸塞州达特默斯大学海洋生态动力学模型实验室与伍兹霍尔海洋学协会的罗伯特C.比尔兹利博士合作开发。我们保留FVCOM代码的所有版权，在未经许可的情况下禁止复制和重新分配本代码。

A. Permitted Use and Restrictions on Redistribution

The user agrees that he/she will use the FVCOM source code, and any

modifications to the FVCOM source code that the user may create, solely for internal, non-commercial purposes and shall not distribute or transfer the FVCOM source code or modifications to it to any person or third parties not participating in their primary research project without prior written permission from Dr. Chen. The term

\other scholarly research which (a) is not undertaken for profit, or (b) is not intended to produce work, services, or data for commercial use.

A． 对于重新分配的许可使用和限制

用户须同意使用FVCOM源代码，可以对源代码做出的个人的，非商业性的修改，不得传播或转让FVCOM源代码 ，不得将修改的源代码给没有预先与陈常胜博士签署书面协议任何人或没有参加主要研究工作的第三方。在最终用户许可协议中使用的“非商业性”的意思是（a）不获得利润或（b）不用以商业用途的生产、服务、数据的理论或学术研究。

B. Mandatory Participation in the FVCOM Community

The user agrees to openly participate in the FVCOM community through three primary mechanisms. These are (a) reporting code bugs and problems, (b) sharing

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major modifications made to the code, and (c) contributing to an open and ongoing discussion of model deficiencies, needed improvements and additions, and major

successes. (Contact Drs. C. Chen, G. Cowles, or R. Beardsley). These mechanisms are intended to benefit the entire FVCOM user community through quick notification of code problems, possible solutions, major code improvements, and, in general, the further development of the FVCOM source code and the associated software tools needed to process, visualize and interpret FVCOM model output.

B．参与FVCOM社区

用户可以通过三种主要途径公开参与FVCOM社区（a）报告代码缺陷和问题（b）分享对代码的修正（c）公开讨论模式缺陷、必须的改进和增加以及主要成就（与陈常胜博士或罗伯特C.比尔兹利博士接触）。通过这些途径可以使整个FVCOM社区快速通知用户代码问题、可能的解决方法、主要代码改进、FVCOM源代码和程序所需要的联合软件工具进一步改进、显示和解释FVCOM模式输出。

C. FVCOM Validation

The user agrees to inform Dr. Chen about any FVCOM model validation test case conducted by the user before formal publication of the test case results. This step is intended to minimize potential errors in gridding, model setup, boundary conditions and coding that could contribute to poor FVCOM performance in the validation test case. There is no intent here to exercise any prior restraint on publication.

C． FVCOM确认

用户在测试结果正式出版以前可以通知陈常胜博士任何FVCOM模式确认测试情况。这一步可以减少网格、模式设置、边界条件和译码的潜在错误以改善FVCOM模式确认测试的性能。这些不受出版的限制。

D. Publication of FVCOM Results

The user agrees to acknowledge FVCOM in any publications resulting from the use of the FVCOM source code. The user agrees to use the name ―FVCOM‖ to refer to the model.

D． FVCOM结果的发布

用户发表由FVCOM源程序得到的结果时必须注明并用“FVCOM”来指代模式。

Chapter 1: Introduction

第一章 序言

Throughout much of the world oceans, the inner continental shelves and estuaries are characterized by barrier island complexes, inlets, and extensive intertidal salt marshes. Such an irregularly-shaped ocean-land margin system presents a serious challenge for oceanographers involved in model development even though the governing equations of ocean circulation are well defined and numerically solvable in terms of discrete mathematics. Two numerical methods have been widely used in ocean circulation models: (1) the finite-difference method (Blumberg and Mellor, 1987; Blumberg, 1994; Haidvogel et al., 2000) and (2) the finite-element method (Lynch and Naimie, 1993; Naimie, 1996). The finite-difference method is the most basic discrete scheme

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and has the advantage of computational and coding efficiency. Introducing an

orthogonal or non-orthogonal curvilinear horizontal coordinate transformation into a finite-difference model can provide adequate boundary fitting in relatively simple coastal regions but these transformations are incapable of resolving the highly irregular inner shelf/estuarine geometries found in many coastal areas (Blumberg 1994; Chen et al. 2001; Chen et al. 2004a). The greatest advantage of the

finite-element method is its geometric flexibility. Triangular grid meshes of arbitrary spatially-dependent size are commonly used in this method, and can provide an

accurate fitting of the irregular coastal boundary. The P-type Finite-Element Method (Maday and Patera, 1988) or Discontinuous Galerkin Method (Reed and Hill, 1973; Cockburn et al., 1998) has recently been applied to ocean and have shown promise in improving both computational accuracy and efficiency.

全球海洋内部大陆架和河口有复杂的礁岛、水湾、广阔的高潮线与低潮线之间的盐碱湾等特征。即使定义并可用离散数学解得大洋环流控制方程，这种不规则的海岸边界系统对于研究模型发展的海洋学家是一个严峻的挑战。有两种常用的数学方法解决大洋环流模型：（1）有限差分方法（Blumberg and Mellor, 1987; Blumberg, 1994;Haidvogel et al., 2000）（2）有限元方法(Lynch and Naimie, 1993;Naimie, 1996)。有限差分法基于离散方法并具有计算和编码效率的优点。在有限差分法中引入正交或非正交水平曲线坐标转换可以为简单海岸区域提供适当的边界，但这种转换不能解决许多海岸的高度不规则内部陆架/河口几何学(Blumberg 1994; Chen et al. 2001; Chen et al.2004a)。有限元法最大的优点是几何学的灵活性。任意空间尺寸的三角网格通常用于这种方法，并可精确的适用于不规则海岸边界。P型有限元法(Maday and Patera, 1988)或不连续Galerkin方法(Reed and Hill, 1973; Cockburn et al., 1998)已应用于解决海洋问题并取得了较好的计算精确性和效率。

We have developed a 3-D unstructured-grid, free-surface, primitive equation, Finite-Volume Coastal Ocean circulation Model (called FVCOM) (Chen et al. 2003a; Chen et al.2004b). Unlike the differential form used in finite-difference and finite-element models, FVCOM discretizes the integral form of the governing equations. Since these integral equations can be solved numerically by flux

calculation (like those used in the finite-difference method) over an arbitrarily-sized triangular mesh (like those used in the finite-element method), the finite-volume

approach is better suited to guarantee mass conservation in both the individual control element and the entire computational domain. From a technical point of view,

FVCOM combines the best attributes of finite-difference methods for simple discrete coding and computational efficiency and finite-element methods for geometric flexibility. This model has been successfully applied to study several estuarine and shelf regions that feature complex irregular coastline and topographic geometry, including inter-tidal flooding and drying (see http://codfish.smast.umassd.edu or http://fvcom.smast.umassd.edu for descriptions of these initial applications).

我们开发了三维自由网格、自由表面、原始方程、有限体积海岸大洋环流模型（FVCOM）(Chen et al. 2003a; Chen et al.2004b).与有限差分法和有限元法使用的微分形式不同，FVCOM是对控制方程进行离散。在自由尺度三角网格中（与有限元方法相同）用通量计算（与有限差分法相同）可以从数学上解得这些积分方程，有限体积近似可以保证单独控制要素和整体计算范围的质量守恒。从技术角度来看，FVCOM结合了用于简单离散编码和计算功率的有限差分法以及用于几何灵活性的有限元法的优良特征。这种模型已成功应用于研究几种河口

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和陆架区域的特征复杂的不规则海岸线和地形几何学，包括内潮涨落（对这些早期应用的描述见http://codfish.smast.umassd.edu 或 http://fvcom.smast.umassd.edu）。

This manual is provided to help users to 1) understand the basic discrete structure and numerical methods used in FVCOM and 2) learn how to use the model for their own applications. Detailed instructions are given for all steps (e.g., grid generation, model input and output, compilation, parallel computation, etc.). Several experiments are included to provide new users with simple examples of model setup and execution

本手册将为用户提供以下帮助：（1）理解FVCOM使用的基本离散结构和数学方法；（2）学习怎样应用本模式。给出了所有步骤的详细说明（如网格生成、模式输入和输出、汇编、平行计算等）。包括几个实验结果为新用户提供模式建立和运行的几个简单例子。

The remaining chapters are organized as follows. Chapter 2: the model

formulation; Chapter 3: the finite-volume discrete method; Chapter 4: the external forcings; Chapter 5: the open boundary treatments; Chapter 6: the 4-D data

assimilation methods; Chapter 7: the sediment module; Chapter 8: the biological modules; Chapter 9: the tracer-tracking model; Chapter 10: the 3-D Lagrangian

particle tracking; Chapter 11: the sea ice module, Chapter 12: the code parallelization; Chapter 13: the model coding description and general information; Chapter 14: the model installation; Chapter 15: the model setup; Chapter 16: examples of model applications, and Chapter 17: an example of the unstructured grid generation.

剩余章节结构如下。第二章：模型公式；第三章：有限体积离散方法；第四章：外强迫；第五章：开边界处理；第六章：四维数据同化方法；第七章：沉积模块；第八章：生物模块；第九章：示踪物追踪模型；第十章：三维拉格朗日粒子追踪模型；第十一章：海冰模块；第十二章：代码平行计算；第十三章：模式编码和总说明；第十四章：模型安装；第十五章：

模型设置；第十六章： 模型应用的举例；第十七章：自由网格产生的一个例子。

Users should be aware that this manual is only useful for the current version of FVCOM. FVCOM is in continually testing and improvement by a

SMAST/UMASSDWHOI effort led by Changsheng Chen and Robert C. Beardsley. Some very recent modifications may not have been included in this manual. If users find any inconsistency between this manual and the FVCOM code, it is likely to be due to a typo in the manual. Please report any problems with this manual as well as suggestions for improvement, so that future versions can be enhanced.

用户应该知道本手册仅适用于FVCOM的当前版本。FVCOM由陈常胜博士和罗伯特C.比尔兹利博士领导的SMAST/UMASSDWHOI不断地测试和改进。本手册可能不包括一些近期的修正。如果用户发现任何这本手册和FVCOM代码之间的矛盾，可能是手册中的印刷问题。为了提高下一版本的质量，请提出本手册存在的问题以及改进建议。

Chapter 2: The Model Formulation

第二章：模型公式

2.1. The Primitive Equations in Cartesian Coordinates

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2.1 直角坐标系下的原始方程

The governing equations consist of the following momentum, continuity, temperature, salinity, and density equations:

控制方程包括下列动量方程、连续方程、温度方程、盐度方程和密度方程：

where x, y, and z are the east, north, and vertical axes in the Cartesian coordinate system; u, v, and w are the x, y, z velocity components; T is the temperature; S is the salinity; ? is the density; P is the pressure; f is the Coriolis parameter; g is the gravitational acceleration; Km is the vertical eddy viscosity coefficient; and Kh is the thermal vertical eddy diffusion coefficient. Fu,Fv,FT,andFS represent the horizontal momentum, thermal, and salt diffusion terms. The total water column depth is D ??H ??z , where H is the bottom depth (relative to z = 0) and z is the height of the free surface (relative to z = 0).

其中在直角坐标系中x，y，z分别表示东，北和竖直坐标轴；u，v，w是x，y，z方向的速度分量；T为温度；S为盐度；?为密度；P为压强；f为科氏参量； g为重力加速度；Km为垂直旋转粘性系数；Kh为热量垂直旋转扩散系数；Fu,Fv,FT,和FS代表水平动量，热量和盐度的扩散项。整体水柱深度为

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其中H为底部深度（z=0）；?为自由面高度（z=0）。

Fig. 2.1: Illustration of the orthogonal coordinate system: x: eastward; y: northward; z: upward.

图2.1 直角坐标系图解：x：东；y：北；z：竖直向上。

The surface and bottom boundary conditions for temperature are:

温度的表面和底边界条件为：

where Qn(x,y,t) is the surface net heat flux, which consists of four components: downward shortwave, longwave radiation, sensible, and latent fluxes, SW( x, y,0,t ) is the shortwave flux incident at the sea surface, and cp is the specific heat of seawater.

AH is the horizontal thermal diffusion coefficient, ?is the slope of the bottom bathymetry, and n is the horizontal coordinate shown in Figure 2.2 (Pedlosky, 1974; Chen et al., 2004b).

其中Qn(x,y,t)为表面净热量通量，包括四部分：向下的短波，长波辐射，显通量和潜通量；

SW(x,y,0,t)为海表面的短波通量；cp为海水比热；AH水平热量扩散系数;?为底面地形；

n为图2.2所示的水平坐标(Pedlosky, 1974; Chen et al.2004b)。

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Fig. 2.2: Schematic of the no-flux boundary condition on the bottom slope.

图2.2 底面倾斜无通量边界条件示意图

The longwave, sensible and latent heat fluxes are assumed here to occur at the ocean surface, while the downward shortwave flux SW( x, y, z, t ) is approximated by:

假设长波通量、显热通量和潜热通量发生在海表面，向下的短波通量SW(x,y,z,t)近似由下式给出：

where a and b are attenuation lengths for longer and shorter (blue-green) wavelength components of the shortwave irradiance, and R is the percent of the total flux associated with the longer wavelength irradiance. This absorption profile, first

suggested by Kraus (1972), has been used in numerical studies of upper ocean diurnal heating by Simpson and Dickey (1981a, b) and others. The absorption of downward irradiance is included in the temperature (heat) equation in the form of

其中a和b为组成短波辐照度的长波长和短波长（蓝-绿）的衰减长度；R为长波长辐照度占总通量的百分比。这种吸收面首先由Kraus（1972）提出，Simpson和 Dickey (1981a, b)以及其他学者将其用于上部海洋日热量的数学研究。向下辐照度的吸收包含于下式的温度（热量）方程：

This approach leads to a more accurate prediction of near-surface temperature than the flux formulation based on a single wavelength approximation (Chen et al., 2003b).

与基于一种单一波长近似的通量公式相比，这种近似可以得到近表面温度更精确的预报

(Chen et al., 2003b)。

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The surface and bottom boundary conditions for salinity are： 盐度的表面和底边界条件如下：

whereP andE are precipitation and evaporation rates, respectively.

.Note that a groundwater flux can be easily added into the model

by modifying the bottom boundary conditions for vertical velocity and salinity.

其中

^^P和E分别为降水率和蒸发率；

^^；可以通过改变底边界条件的垂

直速度和盐度将地下水流量加入模型。

The surface and bottom boundary conditions for u, v, and w are: u，v，w的表面和底边界条件如下：

where and are the x and y components of

surface wind and bottom stresses, Qbis the groundwater volume flux at the bottom and ??is the area of the groundwater source. The drag coefficient Cd is determined by matching a logarithmic bottom layer to the model at a heightzab above the bottom, i.e.,

其中

和

为表面风和底压力的x，y方向的成

分；Qb为底部地下水流量；?为地下水源的面积。牵引系数Cd为在底面高度zab出将对数底层引入模型，例如

where k = 0.4 is the von Karman constant and zo is the bottom roughness parameter.

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Specifying

al., 1992; Janssen, 2001) and

其中

(Charnock‘s relation), (Smith et

, (2.51) can be rewritten as

(Smith et al., 1992;

（Charnock关系）；

，（2.51）式可改写为

Janssen, 2001)；

According to field data,

根据野外资料

(Stacey, 1999).

(Stacey, 1999)。

2.4.2.2. The k ?e Turbulence Model 2.4.2.2 k ??湍流模型 In the boundary layer approximation (Rodi, 1980), the k ?? model can be simplified as

在边界层近似 (Rodi,1980)下，k ??模型可简化为

wherevt is the eddy viscosity (which is the same as K q in the MY level 2.5 model),

?k is the turbulent Prandtl number that is defined as the ratio of turbulent eddy viscosity to conductivity, P is the turbulent shear production, and G is the turbulent buoyancy production. These two variables have the same definitions as P s and P b in the MY level 2.5 model. c1，c2, and c3 are empirical constants. A detailed description of the standard and advanced k ?? models was given by Burchard and Baumert (1995) and is briefly summarized next.

其中vt为旋转粘性（与MT-2.5模型中的Kq相同）；?k为湍流普朗特数，等于湍流旋转粘性与传导率的比；P为湍流切分量；G为湍流浮力分量。这两个变量与MY-2.5模型中的Ps和Pb定义相同。c1，c2，c3为经验常数，k ??模型的详细描述和改进由 Burchard 和 Baumert

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^^(1995)提出，并将在下面简单介绍。

In the standard k ?? model,

在标准k ??模型中

where

其中

and R i is the gradient Richardson number defined as

Ri为理查森数梯度，定义如下

The eddy viscosityvtcan be estimated by

旋转粘度vt可由下式估计给出

wherec? is a constant. In this standard k ?? model, the empirical constants are specified as

其中c?为常数。在标准k ??模型中，经验常数为

In the advanced k ?? model, the turbulence model consists of the k and ? equations plus 6 transport equations for the Reynolds stresses (

) and the turbulent heat fluxes

.

In this model, the eddy viscosity (vt ) is still given by (2.59), but c? is a function of the vertical shear of the horizontal velocity and vertical stratification. This function corresponds to the stability function Sm in the MY- 2.5 model. vtand vT (thermal

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diffusion coefficient) are given as

在改进的k ??模型中，湍流模型由k和?方程，雷诺压力的6个输运方程

（）和湍流热通量

组成。在此模型中，

旋转粘性（vt）仍由式（2.59）给出，但c?由水平速度和垂直成层结构的垂直剪切函数给出。这个函数与MY-2.5模型中的稳定函数Sm相符。vt和vT（热扩散系数）由下式给出：

where ?P and ?Gare functions of dimensionless turbulent shear and turbulent buoyancy numbers in the forms of

其中无量纲湍流剪切函数和湍流浮力数?P和?G如下式

F is a near-wall correction factor

F为近壁修正因子。

The 8-component advanced turbulence model is mathematically closed with the

specification of 11 empirical constants (Burchard and Baumert, 1995).

8-要素改进湍流模型在数学上接近11经验常数的规格(Burchard and Baumert, 1995)。 The surface boundary conditions for k and ? in the k - ? turbulent closure model described above are specified as

在上述k ??湍流闭合模型中，k 和?的表面边界条件由下式给出：

The bottom boundary conditions for k and ?are given as

k 和?的底边界条件如下

where k is the von Karman constant.

其中k为卡曼常数。

The wave- induced turbulent kinetic energy flux at the surface was recently taken into account for the k ?? model. A detailed description of the modified surface

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boundary conditions for k and ? is given in Burchard (2001).

k ??模型中引入了表面的波感应湍流动能通量。Burchard (2001)给出了k 和?的改进表面边界条件的详细描述。

2.5. The Primitive Equations in Spherical Coordinates

2.5 球面坐标系下的原始方程

The FVCOM was originally coded for the local Cartesian coordinate system in which f may vary with latitude but the curvature terms due to the spherical shape of the earth were not included in the momentum equations. Therefore, it is suitable for regional applications but not for basin- or global-scale applications. To make FVCOM flexible for either regional or global application, we have built a spherical-coordinate version of FVCOM (Chen et al., 2006b).

FVCOM最初在局地笛卡尔坐标系下，其f随纬度变化但地球的球形曲率项没有包括在动力方程中。因此，它适合局部应用而不适合盆地或全球尺度应用。为了使FVCOM可在局部或全球灵活应用，我们建立了一个球坐标下的FVCOM版本(Chen et al., 2006b)。

Consider a spherical coordinate system in which the x (eastward) and y (northward) axes are expressed as

考虑球坐标系x轴（东）和y轴（北）表示如下

where r is the earth‘s radius; ??is longitude; ??is latitude, and ?0 and ?0 are the

reference longitude and latitude, respectively. The vertical coordinate z is normal to the earth‘s surface and positive in the upward direction. This coordinate system is shown in Fig. 2.3.

其中r为地球半径；?为经度；?为纬度；?0和?0分别为参考经度和参考纬度。竖直坐标z为地球表面法向并取向上为正。这种坐标系如图2.3所示。

图2.3 球坐标系图解Fig. 2.3: Illustration of the spherical coordinate system.

The three-dimensional (3-D) internal mode flux forms of the governing equations of motion in the spherical and ??coordinates are given as

球坐标和?坐标下，三维（3-D）运动控制方程的内部模式通量形式如下

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where u, v, and? are zonal, meridional and ?-coordinate vertical components of the velocity, T is the temperature; S is the salinity; ? is the total density that is equal to a sum of perturbation density???and reference density ?0 , P is the pressure; f is the Coriolis parameter; g is the gravitational acceleration; and Km is the vertical eddy viscosity and Kh the thermal vertical eddy diffusion coefficients that are calculated using one of the above turbulence closure models (Chen et al., 2004). H is the vertical gradient of the short-wave radiation. Fu , Fv , F T , and F S represent the

horizontal momentum, thermal, and salt diffusion terms and the horizontal diffusion is calculated using the Smagorinsky eddy parameterization method (Smagorinsky, 1963). The relationship between? and the true vertical velocity (w) is given as

其中u，v和?分别为速度的纬向分量，经向分量和?坐标垂直分量；T为温度；S为盐度；

^?为总密度，等于微扰密度??和参考密度?0之和；P为压强；f为科氏参数；g为重力加速

度；Km为垂直旋转粘性，Kh为热量的垂直旋转扩散率，可由上面给出的湍流闭合模型算出(Chen et al., 2004)；H为短波辐射垂直梯度；Fu，Fv，FT和FS由水平动量，热量和盐度扩散项以及水平扩散表达，并可由Smagorinsky旋转参数法(Smagorinsky, 1963)算出。?和

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^

垂直速度（w）的关系如下：

The 2-D (vertically integrated) momentum and continuity equations are written as 二维（垂直积分）动量方程和连续方程如下：

where Gu and Gv are defined as

其中Gu和Gv定义如下：

并且

where the definitions of variables are the same as those described in the Cartesian coordinates. The spherical-coordinate version of FVCOM was developed based on the

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Cartesian coordinate version, in which all the boundary cond itions and forcing used in the spherical-coordinate system are the same. The only difference is in the discrete approach, which is described later in chapter 3.

其中变量的定义与笛卡尔坐标系相同。FVCOM的球坐标版本基于笛卡尔坐标版本，用于球坐标系中的所有边界条件和强迫与笛卡尔坐标系下的相同。唯一的不同是离散逼近，浙江在第三章中详细给出。

Chapter 3: The Finite-Volume Discrete Method

第三章 有限体积离散法

3.1. Design of the Unstructured Triangular Grids

3.1 不规则三角网格的设计

Similar to a triangular finite element method, the horizontal numerical computational domain is subdivided into a set of non-overlapping unstructured

triangular cells. An unstructured triangle is comprised of three nodes, a centroid, and three sides (Fig. 3.1). Let N and M be the total number of centroids and nodes in the computational domain, respectively, then the locations of centroids can be expressed as:

与三角有限元方法相似，水平数学计算域可细分为许多不重复的无规则三角元。一个不规则三角形包含三个节点、一个质心和三条边（图3.1）。令N和M分别为计算域中的质心总数和节点总数，质心位置可表示为

图3.1 FVCOM不规则三角网格示意图。变量位置：节点?：H，?，?，D，s，?，q，

2q2l，Am，Kh；质心?：u，v。

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and the locations of nodes can be specified as:

节点位置可由下式给出：

Since none of the triangles in the grid overlap, N should also be the total number of triangles. On each triangular cell, the three nodes are identified using integral numbers defined as Ni(j)where j is counted clockwise from 1 to 3. The surrounding triangles that have a common side are counted using integral numbers defined as

NBEi(j) wherejis counted clockwise from 1 to 3. At open or coastal solid

^^^^boundaries, NBEi(j) is specified as zero. At each node, the total number of the surrounding triangles with a connection to this node is expressed as NT( j) , and they are counted using integral numbers NB i (m) where m is counted clockwise from 1 to NT( j) .

因为网格中三角形没有交叠，所以N也是三角形总数。在每个三角元中，三个节点由积分数Ni(j)确定其中j为顺时针方向从1到3积分。有一条共同边的相邻三角形可由积分数

^^^NBEi(j)计算，其中j为顺时针方向从1到3积分。在开边界或海岸固边界，NBEi(j)为

0。在每个节点处，相邻三角形的总数与这个节点有关，表示为NT（j），并可由积分数NBi(m)计算，其中m为顺时针方向从1到NT（j）积分。

^^^To provide a more accurate estimation of the sea-surface elevation, currents and salt and temperature fluxes, u and v are placed at centroids and all scalar variables,

such as ?, H, D, w, S, T, ?, K m ,K h ,A m and A h are placed at nodes. Scalar variables at each node are determined by a net flux through the sections linked to centroids and the mid-point of the adjacent sides in the surrounding triangles (called the ―tracer control element‖ or TCE), while u and v at the centroids are calculated based on a net flux through the three sides of that triangle (called the ―momentum control element‖ or MCE).

为了给出海面上升、水流通量、盐度通量和温度通量的精确估计，将u，v放置在质心处，所有的标量变量放置在节点处，如?，H，D，?，S，T，?，Km，Kh，Am和Ah。在节点处的标量变量由穿过连接质心和相邻三角形邻边中点截面的净通量决定（称为“追踪控制元”或TCE）；在质心处的u和v由穿过三角形三条边的净通量计算（称为“动量控制元”或MCE）。

Similar to other finite-difference models such as POM and ROM, all the model variables except ??(vertical velocity on the sigma- layer surface) and turbulence

variables (such as q2and q2l ) are placed at the mid-level of each ?? layer (Fig. 3.2). There are no restrictions on the thickness of the ?- layer, which allows users to use

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either uniform or non-uniform ?-layers.

与其他的有限差分法（如POM和ROM）类似，除了?（?层表面的垂直速度）和湍流变量（如q2和q2l）外，所有的模型变量位于每个?层的中间（图3.2）。对?层的厚度没有约束，用户可使用统一的或不统一的?层。

图3.2 垂直?坐标中模型变量的位置

3.2. The Discrete Procedure in the Cartesian Coordinates

3.2 笛卡尔坐标下的离散方法

3.2.1. The 2-D External Mode.

3.2.1 二维外部模式

Let us consider the continuity equation first. Integrating Eq. (2.30) over a given triangle area yields:

首先考虑连续方程。结合方程（2.30）给出三角形面积流量：

wherevn is the velocity component normal to the sides of the triangle and s??is the closed trajectory comprised of the three sides. Eq. (3.3) is integrated numerically

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using the modified fourth-order Runge-Kutta time-stepping scheme. This is a

multi-stage timestepping approach with second-order temporal accuracy. The detailed procedure for this method is described as follows:

其中vn为三角形边的垂直法线分量；s?为包含三条边的闭合曲线。方程（3.3）是改进的四阶Runge-Kutta时间步长方案的积分。这是多阶时间步长趋近于二阶时间精确性。这种方法的具体步骤如下：

where k =1,2,3,4 and (?1，?2，?3，?4 ) = (1/4, 1/3,1/2, 1). Superscript n represents the nth time step. ??j?is the area enclosed by the lines through centroids and mid-points of the sides of surrounding triangles connected to the node where ?j is

nnlocated. um and vm are defined as:

其中k=1,2,3,4；（?，?，?，?）=（1/4，1/3，1/2，1）；上标n代表第n个时间步长；

nn和vm定义如下： ??j为质心、相邻三角形边中点与?j所在节点围成的面积。um1234

?t is the time step for the external mode, and

?t为外部模式的时间步长，并有：

Similarly, integrating Eqs. (2.31) and (2.32) over a given triangular area gives:

相似的，在给定的三角区域积分方程（2.31）和（2.32）

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