2.1 笛卡尔坐标系下的三维控制方程
2.1.1 浅水方程
局地连续方程写为：
$$
\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial w}{\partial z}=S
\tag{2.1}
$$
\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial w}{\partial z}=S
\tag{2.1}
$$
分别对应 $x$ 与 $y$ 分量的两条水平动量方程为：
$$
\begin{align}
\frac{\partial u}{\partial t}
+\frac{\partial u^{2}}{\partial x}
+\frac{\partial (vu)}{\partial y}
+\frac{\partial (wu)}{\partial z}
&= fv-g\frac{\partial \eta}{\partial x}
-\frac{1}{\rho_{0}}\frac{\partial p_{a}}{\partial x}
-\frac{g}{\rho_{0}}\int_{z}^{\eta}\frac{\partial \rho}{\partial x}\,dz \notag\\
&\quad -\frac{1}{\rho_{0}h}\left(\frac{\partial s_{xx}}{\partial x}+\frac{\partial s_{xy}}{\partial y}\right)
+F_{u}\\
&\quad+\frac{\partial}{\partial z}\!\left(\nu_{t}\frac{\partial u}{\partial z}\right)
+u_{s}S
\tag{2.2}
\end{align}
$$
\begin{align}
\frac{\partial u}{\partial t}
+\frac{\partial u^{2}}{\partial x}
+\frac{\partial (vu)}{\partial y}
+\frac{\partial (wu)}{\partial z}
&= fv-g\frac{\partial \eta}{\partial x}
-\frac{1}{\rho_{0}}\frac{\partial p_{a}}{\partial x}
-\frac{g}{\rho_{0}}\int_{z}^{\eta}\frac{\partial \rho}{\partial x}\,dz \notag\\
&\quad -\frac{1}{\rho_{0}h}\left(\frac{\partial s_{xx}}{\partial x}+\frac{\partial s_{xy}}{\partial y}\right)
+F_{u}\\
&\quad+\frac{\partial}{\partial z}\!\left(\nu_{t}\frac{\partial u}{\partial z}\right)
+u_{s}S
\tag{2.2}
\end{align}
$$
$$
\begin{align}
\frac{\partial v}{\partial t}
+\frac{\partial v^{2}}{\partial y}
+\frac{\partial (uv)}{\partial x}
+\frac{\partial (wv)}{\partial z}
&= -fu-g\frac{\partial \eta}{\partial y}
-\frac{1}{\rho_{0}}\frac{\partial p_{a}}{\partial y}
-\frac{g}{\rho_{0}}\int_{z}^{\eta}\frac{\partial \rho}{\partial y}\,dz \notag\\
&\quad-\frac{1}{\rho_{0}h}\left(\frac{\partial s_{yx}}{\partial x}+\frac{\partial s_{yy}}{\partial y}\right)
+F_{v}\\
&\quad+\frac{\partial}{\partial z}\!\left(\nu_{t}\frac{\partial v}{\partial z}\right)
+v_{s}S
\tag{2.3}
\end{align}
$$
\begin{align}
\frac{\partial v}{\partial t}
+\frac{\partial v^{2}}{\partial y}
+\frac{\partial (uv)}{\partial x}
+\frac{\partial (wv)}{\partial z}
&= -fu-g\frac{\partial \eta}{\partial y}
-\frac{1}{\rho_{0}}\frac{\partial p_{a}}{\partial y}
-\frac{g}{\rho_{0}}\int_{z}^{\eta}\frac{\partial \rho}{\partial y}\,dz \notag\\
&\quad-\frac{1}{\rho_{0}h}\left(\frac{\partial s_{yx}}{\partial x}+\frac{\partial s_{yy}}{\partial y}\right)
+F_{v}\\
&\quad+\frac{\partial}{\partial z}\!\left(\nu_{t}\frac{\partial v}{\partial z}\right)
+v_{s}S
\tag{2.3}
\end{align}
$$
其中：
$t$ 为时间；$x,y,z$ 为笛卡尔坐标；$\eta$ 为自由表面高程；$d$为静水水深；总水深$h=\eta+d$。$u,v,w$分别为$x,y,z$方向速度分量；科氏参数 $f=2\Omega\sin\phi$$(\Omega$ 为地球自转角速度，$\phi$ 为地理纬度）；$g$ 为重力加速度。$\rho$ 为水体密度；$s_{xx},s_{xy},s_{yx},s_{yy}$为辐射应力张量分量；$\nu_t$为垂向湍动（涡）黏性系数；$p_a$为大气压；$\rho_0$ 为参考密度。$S$ 为点源（汇）排放强度；$(u_s,v_s)$为排入环境水体时的水平速度分量。
水平应力项采用梯度–应力关系表示，并简化为：
$$
F_u=\frac{\partial}{\partial x}\!\left(2A\frac{\partial u}{\partial x}\right)+\frac{\partial}{\partial y}\!\left(A\left(\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}\right)\right)
\tag{2.4}
$$
F_u=\frac{\partial}{\partial x}\!\left(2A\frac{\partial u}{\partial x}\right)+\frac{\partial}{\partial y}\!\left(A\left(\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}\right)\right)
\tag{2.4}
$$
$$
F_v=\frac{\partial}{\partial x}\!\left(A\left(\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}\right)\right)
+\frac{\partial}{\partial y}\!\left(2A\frac{\partial v}{\partial y}\right)
\tag{2.5}
$$
F_v=\frac{\partial}{\partial x}\!\left(A\left(\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}\right)\right)
+\frac{\partial}{\partial y}\!\left(2A\frac{\partial v}{\partial y}\right)
\tag{2.5}
$$
其中：$A$ 为水平涡黏系数。
$u,v,w$ 的自由表面与底边界条件为：
在 $z=\eta$处：
$$
\begin{align}
\frac{\partial \eta}{\partial t}
+u\frac{\partial \eta}{\partial x}
+v\frac{\partial \eta}{\partial y}
-w &= 0,\quad
\left(\frac{\partial u}{\partial z},\frac{\partial v}{\partial z}\right)
=\frac{1}{\rho_{0}\nu_{t}}\left(\tau_{sx},\tau_{sy}\right)
\tag{2.6}
\end{align}
$$
\begin{align}
\frac{\partial \eta}{\partial t}
+u\frac{\partial \eta}{\partial x}
+v\frac{\partial \eta}{\partial y}
-w &= 0,\quad
\left(\frac{\partial u}{\partial z},\frac{\partial v}{\partial z}\right)
=\frac{1}{\rho_{0}\nu_{t}}\left(\tau_{sx},\tau_{sy}\right)
\tag{2.6}
\end{align}
$$
在 $z=-d$ 处：
$$
\begin{align}
u\frac{\partial d}{\partial x}
+v\frac{\partial d}{\partial y}
+w &= 0, \quad
\left(\frac{\partial u}{\partial z},\frac{\partial v}{\partial z}\right)
=\frac{1}{\rho_{0}\nu_{t}}\left(\tau_{bx},\tau_{by}\right)
\tag{2.7}
\end{align}
$$
\begin{align}
u\frac{\partial d}{\partial x}
+v\frac{\partial d}{\partial y}
+w &= 0, \quad
\left(\frac{\partial u}{\partial z},\frac{\partial v}{\partial z}\right)
=\frac{1}{\rho_{0}\nu_{t}}\left(\tau_{bx},\tau_{by}\right)
\tag{2.7}
\end{align}
$$
其中$(\tau_{sx},\tau_{sy})$与$(\tau_{bx},\tau_{by})$分别为表面风应力与底应力在$x$、$y$ 方向的分量。
已知由动量方程与连续方程得到的速度场后，可通过自由表面运动学边界条件求得总水深$h$。但更稳健的形式是对局地连续方程做垂向积分，得到：
$$
\frac{\partial h}{\partial t}
+\frac{\partial (h\bar{u})}{\partial x}
+\frac{\partial (h\bar{v})}{\partial y}
=hS+\hat{P}-\hat{E}
\tag{2.8}
$$
\frac{\partial h}{\partial t}
+\frac{\partial (h\bar{u})}{\partial x}
+\frac{\partial (h\bar{v})}{\partial y}
=hS+\hat{P}-\hat{E}
\tag{2.8}
$$
其中 $\hat{P}$与$\hat{E}$分别为降水率与蒸发率；$\bar{u}$、$\bar{v}$为水深平均速度：
$$
h\bar{u}=\int_{-d}^{\eta}u\,dz,\qquad
h\bar{v}=\int_{-d}^{\eta}v\,dz
\tag{2.9}
$$
h\bar{u}=\int_{-d}^{\eta}u\,dz,\qquad
h\bar{v}=\int_{-d}^{\eta}v\,dz
\tag{2.9}
$$
$$
\tag{2.10}
$$