2.2 球坐标与 σ 坐标下的三维控制方程

      在球坐标系中，自变量为经度 $\lambda$ 与纬度 $\phi$。水平速度场 $(u,v)$ 定义为：

$$ u=R\cos\phi\frac{d\lambda}{dt}\quad v=R\frac{d\phi}{dt}\tag{2.54} $$

其中：$R$ 为地球半径。  

      在该坐标系下，控制方程可写为（为书写简便，下文省略表示新坐标系的水平坐标上标）：

$$ \frac{\partial h}{\partial t}+\frac{1}{R\cos\phi}\Bigg(\frac{\partial (hu)}{\partial\lambda}+\frac{\partial (hv\cos\phi)}{\partial\phi}\Bigg)+\frac{\partial (h\omega)}{\partial\sigma}=hS\tag{2.55} $$

$$ \begin{aligned}\frac{\partial (hu)}{\partial t}+\frac{1}{R\cos\phi}\Bigg(\frac{\partial (hu^{2})}{\partial\lambda}+\frac{\partial (hvu\cos\phi)}{\partial\phi}\Bigg)+\frac{\partial (h\omega u)}{\partial\sigma}=&\Bigg(f+\frac{u}{R}\tan\phi\Bigg)vh-\frac{1}{R\cos\phi}\Bigg(gh\frac{\partial\eta}{\partial\lambda}+\frac{h}{\rho_{0}}\frac{\partial p_{a}}{\partial\lambda}+\frac{gh}{\rho_{0}}\int_{z}^{\eta}\frac{\partial\rho}{\partial\lambda}\,dz+\frac{1}{\rho_{0}}\Bigg(\frac{\partial s_{xx}}{\partial\lambda}+\cos\phi\frac{\partial s_{xy}}{\partial\phi}\Bigg)\Bigg)+hF_{u}+\frac{\partial}{\partial\sigma}\Bigg(\frac{\nu_{t}}{h}\frac{\partial u}{\partial\sigma}\Bigg)+hu_{s}S\end{aligned}\tag{2.56} $$

$$ \begin{aligned}\frac{\partial (hv)}{\partial t}+\frac{1}{R\cos\phi}\Bigg(\frac{\partial (huv)}{\partial\lambda}+\frac{\partial (hv^{2}\cos\phi)}{\partial\phi}\Bigg)+\frac{\partial (h\omega v)}{\partial\sigma}=&-\Bigg(f+\frac{u}{R}\tan\phi\Bigg)uh-\frac{1}{R}\Bigg(gh\frac{\partial\eta}{\partial\phi}+\frac{h}{\rho_{0}}\frac{\partial p_{a}}{\partial\phi}+\frac{gh}{\rho_{0}}\int_{z}^{\eta}\frac{\partial\rho}{\partial\phi}\,dz+\frac{1}{\rho_{0}}\Bigg(\frac{1}{\cos\phi}\frac{\partial s_{yx}}{\partial\lambda}+\frac{\partial s_{yy}}{\partial\phi}\Bigg)\Bigg)+hF_{v}+\frac{\partial}{\partial\sigma}\Bigg(\frac{\nu_{t}}{h}\frac{\partial v}{\partial\sigma}\Bigg)+hv_{s}S\end{aligned}\tag{2.57} $$

$$ \begin{aligned}\frac{\partial (hT)}{\partial t}+\frac{1}{R\cos\phi}\Bigg(\frac{\partial (huT)}{\partial\lambda}+\frac{\partial (hvT\cos\phi)}{\partial\phi}\Bigg)+\frac{\partial (h\omega T)}{\partial\sigma}=hF_{T}+\frac{\partial}{\partial\sigma}\Bigg(\frac{D_{v}}{h}\frac{\partial T}{\partial\sigma}\Bigg)+h\hat{H}+hT_{s}S\end{aligned}\tag{2.58} $$

$$ \begin{aligned}\frac{\partial (hs)}{\partial t}+\frac{1}{R\cos\phi}\Bigg(\frac{\partial (hus)}{\partial\lambda}+\frac{\partial (hvs\cos\phi)}{\partial\phi}\Bigg)+\frac{\partial (h\omega s)}{\partial\sigma}=hF_{s}+\frac{\partial}{\partial\sigma}\Bigg(\frac{D_{v}}{h}\frac{\partial s}{\partial\sigma}\Bigg)+hs_{s}S\end{aligned}\tag{2.59} $$

$$ \begin{aligned}\frac{\partial (hk)}{\partial t}+\frac{1}{R\cos\phi}\Bigg(\frac{\partial (huk)}{\partial\lambda}+\frac{\partial (hvk\cos\phi)}{\partial\phi}\Bigg)+\frac{\partial (h\omega k)}{\partial\sigma}=hF_{k}+\frac{1}{h}\frac{\partial}{\partial\sigma}\Bigg(\frac{\nu_{t}}{\sigma_{k}}\frac{\partial k}{\partial\sigma}\Bigg)+h(P+B-\varepsilon)\end{aligned}\tag{2.60} $$

$$ \begin{aligned}\frac{\partial (h\varepsilon)}{\partial t}+\frac{1}{R\cos\phi}\Bigg(\frac{\partial (hu\varepsilon)}{\partial\lambda}+\frac{\partial (hv\varepsilon\cos\phi)}{\partial\phi}\Bigg)+\frac{\partial (h\omega\varepsilon)}{\partial\sigma}=hF_{\varepsilon}+\frac{1}{h}\frac{\partial}{\partial\sigma}\Bigg(\frac{\nu_{t}}{\sigma_{\varepsilon}}\frac{\partial\varepsilon}{\partial\sigma}\Bigg)+h\frac{\varepsilon}{k}\big(c_{1\varepsilon}P+c_{3\varepsilon}B-c_{2\varepsilon}\varepsilon\big)\end{aligned}\tag{2.61} $$

$$ \begin{aligned}\frac{\partial (hC)}{\partial t}+\frac{1}{R\cos\phi}\Bigg(\frac{\partial (huC)}{\partial\lambda}+\frac{\partial (hvC\cos\phi)}{\partial\phi}\Bigg)+\frac{\partial (h\omega C)}{\partial\sigma}=hF_{C}+\frac{\partial}{\partial\sigma}\Bigg(\frac{D_{v}}{h}\frac{\partial C}{\partial\sigma}\Bigg)-hk_{p}C+hC_{s}S\end{aligned}\tag{2.62} $$

      球坐标下的修正垂向速度（modified vertical velocity）定义为：

$$ \omega=\frac{1}{h}\Biggl[w+\frac{u}{R\cos\phi}\frac{\partial d}{\partial\lambda}+\frac{v}{R}\frac{\partial d}{\partial\phi}-\sigma\Biggl(\frac{\partial h}{\partial t}+\frac{u}{R\cos\phi}\frac{\partial h}{\partial\lambda}+\frac{v}{R}\frac{\partial h}{\partial\phi}\Biggr)\Biggr]\tag{2.63} $$

      球坐标下用于确定水深的方程为：

$$ \frac{\partial h}{\partial t}+\frac{1}{R\cos\phi}\Bigg(\frac{\partial (h\overline{u})}{\partial\lambda}+\frac{\partial (h\overline{v}\cos\phi)}{\partial\phi}\Bigg)=hS\tag{2.64} $$