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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^