2.1 笛卡尔坐标系下的三维控制方程
2.1.1 浅水方程
局部连续方程为：
$$
\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial w}{\partial z}=S
$$
$$
\frac{\partial u}{\partial t}
+\frac{\partial u^{2}}{\partial x}
+\frac{\partial (vu)}{\partial y}
+\frac{\partial (wu)}{\partial z}
=
fv
-g\frac{\partial \eta}{\partial x}\\
-\frac{1}{\rho_0}\frac{\partial p_a}{\partial x}
-\frac{g}{\rho_0}\int_{z}^{\eta}\frac{\partial \rho}{\partial x}\,dz\\dz
-\frac{1}{\rho_0 h}\left(\frac{\partial s_{xx}}{\partial x}+\frac{\partial s_{xy}}{\partial y}\right)
+F_u
+\frac{\partial}{\partial z}\left(\nu_t\frac{\partial u}{\partial z}\right)
+u_s S
\tag{2.2}
$$
\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}\,
-\frac{1}{\rho_0 h}\left(\frac{\partial s_{xx}}{\partial x}+\frac{\partial s_{xy}}{\partial y}\right)
+F_u
+\frac{\partial}{\partial z}\left(\nu_t\frac{\partial u}{\partial z}\right)
+u_s S
\tag{2.2}
$$