2.3 笛卡尔坐标下的二维控制方程

      对水平动量方程与连续方程在水深 $h=\eta+d$ 上做垂向积分，可得到二维浅水方程：

$$ \frac{\partial h}{\partial t}+\frac{\partial (h\overline{u})}{\partial x}+\frac{\partial (h\overline{v})}{\partial y}=hS\tag{2.65} $$

$$ \begin{aligned}\frac{\partial (h\overline{u})}{\partial t}+\frac{\partial (h\overline{u}^{2})}{\partial x}+\frac{\partial (h\overline{vu})}{\partial y}=&f\overline{v}h-gh\frac{\partial\eta}{\partial x}-\frac{h}{\rho_{0}}\frac{\partial p_{a}}{\partial x}-\frac{gh^{2}}{2\rho_{0}}\frac{\partial\rho}{\partial x}+\frac{\tau_{sx}}{\rho_{0}}-\frac{\tau_{bx}}{\rho_{0}}-\frac{1}{\rho_{0}}\Bigg(\frac{\partial s_{xx}}{\partial x}+\frac{\partial s_{xy}}{\partial y}\Bigg)+\frac{\partial}{\partial x}\big(hT_{xx}\big)+\frac{\partial}{\partial y}\big(hT_{xy}\big)+hu_{s}S\end{aligned}\tag{2.66} $$

$$ \begin{aligned}\frac{\partial (h\overline{v})}{\partial t}+\frac{\partial (h\overline{uv})}{\partial x}+\frac{\partial (h\overline{v}^{2})}{\partial y}=&-f\overline{u}h-gh\frac{\partial\eta}{\partial y}-\frac{h}{\rho_{0}}\frac{\partial p_{a}}{\partial y}-\frac{gh^{2}}{2\rho_{0}}\frac{\partial\rho}{\partial y}+\frac{\tau_{sy}}{\rho_{0}}-\frac{\tau_{by}}{\rho_{0}}-\frac{1}{\rho_{0}}\Bigg(\frac{\partial s_{yx}}{\partial x}+\frac{\partial s_{yy}}{\partial y}\Bigg)+\frac{\partial}{\partial x}\big(hT_{xy}\big)+\frac{\partial}{\partial y}\big(hT_{yy}\big)+hv_{s}S\end{aligned}\tag{2.67} $$

      上划线表示水深平均。例如，$\overline{u}$ 与 $\overline{v}$ 为水深平均速度：

$$ h\overline{u}=\int_{-d}^{\eta}u\,dz,\quad h\overline{v}=\int_{-d}^{\eta}v\,dz\tag{2.68} $$

      侧向应力 $T_{ij}$ 包含黏性摩擦、湍流摩擦与差异平流等效应。其采用基于水深平均速度梯度的涡黏表达式估算：

$$ T_{xx}=2A\frac{\partial\overline{u}}{\partial x},\quad T_{xy}=A\Bigg(\frac{\partial\overline{u}}{\partial y}+\frac{\partial\overline{v}}{\partial x}\Bigg),\quad T_{yy}=2A\frac{\partial\overline{v}}{\partial y}\tag{2.69} $$

      对盐度与温度输运方程沿水深积分，得到二维输运方程：

$$ \frac{\partial (h\overline{T})}{\partial t}+\frac{\partial (h\overline{u}\,\overline{T})}{\partial x}+\frac{\partial (h\overline{v}\,\overline{T})}{\partial y}=hF_{T}+h\widehat{H}+hT_{s}S\tag{2.70} $$

$$ \frac{\partial (h\overline{s})}{\partial t}+\frac{\partial (h\overline{u}\,\overline{s})}{\partial x}+\frac{\partial (h\overline{v}\,\overline{s})}{\partial y}=hF_{s}+hs_{s}S\tag{2.71} $$

其中：$\overline{T}$ 与 $\overline{s}$ 分别为水深平均温度与盐度。

      对标量输运方程沿水深积分，得到二维输运方程：

$$ \frac{\partial (h\overline{C})}{\partial t}+\frac{\partial (h\overline{u}\,\overline{C})}{\partial x}+\frac{\partial (h\overline{v}\,\overline{C})}{\partial y}=hF_{C}-hk_{p}\overline{C}+hC_{s}S\tag{2.72} $$

其中：$\overline{C}$ 为水深平均标量浓度。