2.4 球坐标下的二维控制方程

      在球坐标中，自变量为经度 $\lambda$ 与纬度 $\phi$。水平速度场 $(\overline{u},\overline{v})$ 定义为：
$$ \overline{u}=R\cos\phi\frac{d\lambda}{dt}\quad\overline{v}=R\frac{d\phi}{dt}\tag{2.73} $$
其中：$R$ 为地球半径。  
      球坐标下的二维控制方程可写为：
$$ \frac{\partial h}{\partial t}+\frac{1}{R\cos\phi}\Bigg(\frac{\partial (h\overline{u})}{\partial\lambda}+\frac{\partial (h\overline{v}\cos\phi)}{\partial\phi}\Bigg)=0\tag{2.74} $$
$$ \begin{aligned}\frac{\partial (h\overline{u})}{\partial t}+\frac{1}{R\cos\phi}\Bigg(\frac{\partial (h\overline{u}^{2})}{\partial\lambda}+\frac{\partial (h\overline{v}\,\overline{u}\cos\phi)}{\partial\phi}\Bigg)=&\Bigg(f+\frac{\overline{u}}{R}\tan\phi\Bigg)\overline{v}h-\frac{1}{R\cos\phi}\Bigg(gh\frac{\partial\eta}{\partial\lambda}-\frac{h}{\rho_{0}}\frac{\partial p_{a}}{\partial\lambda}+\frac{gh^{2}}{2\rho_{0}}\frac{\partial\rho}{\partial\lambda}+\frac{1}{\rho_{0}}\Bigg(\frac{\partial s_{xx}}{\partial\lambda}+\cos\phi\frac{\partial s_{xy}}{\partial\phi}\Bigg)\Bigg)+\frac{\tau_{sx}}{\rho_{0}}-\frac{\tau_{bx}}{\rho_{0}}+\frac{\partial}{\partial x}\big(hT_{xx}\big)+\frac{\partial}{\partial y}\big(hT_{xy}\big)+hu_{s}S\end{aligned}\tag{2.75} $$
$$ \begin{aligned}\frac{\partial (h\overline{v})}{\partial t}+\frac{1}{R\cos\phi}\Bigg(\frac{\partial (h\overline{u}\,\overline{v})}{\partial\lambda}+\frac{\partial (h\overline{v}^{2}\cos\phi)}{\partial\phi}\Bigg)=&-\Bigg(f+\frac{\overline{u}}{R}\tan\phi\Bigg)\overline{u}h-\frac{1}{R}\Bigg(gh\frac{\partial\eta}{\partial\phi}-\frac{h}{\rho_{0}}\frac{\partial p_{a}}{\partial\phi}+\frac{gh^{2}}{2\rho_{0}}\frac{\partial\rho}{\partial\phi}+\frac{1}{\rho_{0}}\Bigg(\frac{1}{\cos\phi}\frac{\partial s_{yx}}{\partial\lambda}+\frac{\partial s_{yy}}{\partial\phi}\Bigg)\Bigg)+\frac{\tau_{sy}}{\rho_{0}}-\frac{\tau_{by}}{\rho_{0}}+\frac{\partial}{\partial x}\big(hT_{xy}\big)+\frac{\partial}{\partial y}\big(hT_{yy}\big)+hv_{s}S\end{aligned}\tag{2.76} $$
$$ \frac{\partial (h\overline{T})}{\partial t}+\frac{1}{R\cos\phi}\Bigg(\frac{\partial (h\overline{u}\,\overline{T})}{\partial\lambda}+\frac{\partial (h\overline{v}\,\overline{T}\cos\phi)}{\partial\phi}\Bigg)=hF_{T}+h\widehat{H}+hT_{s}S\tag{2.77} $$
$$ \frac{\partial (h\overline{s})}{\partial t}+\frac{1}{R\cos\phi}\Bigg(\frac{\partial (h\overline{u}\,\overline{s})}{\partial\lambda}+\frac{\partial (h\overline{v}\,\overline{s}\cos\phi)}{\partial\phi}\Bigg)=hF_{s}+hs_{s}S\tag{2.78} $$
$$ \frac{\partial (h\overline{C})}{\partial t}+\frac{1}{R\cos\phi}\Bigg(\frac{\partial (h\overline{u}\,\overline{C})}{\partial\lambda}+\frac{\partial (h\overline{v}\,\overline{C}\cos\phi)}{\partial\phi}\Bigg)=hF_{C}-hk_{p}\overline{C}+hC_{s}S\tag{2.79} $$