前言

采用物质面作为垂直坐标面的最大优点是运动准二维，可以很大程度上避免垂直平流，且有利于保持位涡守恒。但是缺点是需要处理下垫面等位温面与地表的交叉。

等熵坐标下的方程组

状态方程

α=RdTp \alpha = \frac{R_d T}{p}

位温定义θ\theta为

T=θ(pp0)Rdcp T = \theta \left( \frac{p}{p_0} \right)^{\frac{R_d}{c_p}}

推导一些微分关系式，沿等位温面的水平梯度

∇θT=θ∇θ(pp0)Rdcp=T(p0p)RdcpRdcp(p0p)1−Rdcpp0−1∇θp \nabla_\theta T = \theta \nabla_\theta \left( \frac{p}{p_0} \right)^{\frac{R_d}{c_p}} = T \left( \frac{p_0}{p} \right)^{\frac{R_d}{c_p}} \frac{R_d}{c_p} \left( \frac{p_0}{p} \right)^{1 - \frac{R_d}{c_p}} p_0^{-1} \nabla_{\theta} p

经过整理，得

cp∇θT=α∇θp c_p \nabla_\theta T = \alpha \nabla_\theta p

类似时间导数

cp∂T∂t=α∂p∂t c_p \frac{\partial T}{\partial t} = \alpha \frac{\partial p}{\partial t}

再推导得垂直梯度

cp∂T∂θ=α∂p∂θ+Π c_p \frac{\partial T}{\partial \theta} = \alpha \frac{\partial p}{\partial \theta} + \Pi

其中Π\Pi为Exner函数

Π=cp(pp0)Rdcp \Pi = c_p \left( \frac{p}{p_0} \right)^{\frac{R_d}{c_p}}

热力学方程

此时的热力学方程变为了垂直坐标速度

θ˙=QΠ \dot{\theta} = \frac{Q}{\Pi}

其中QQ为单位质量的加热率。

静力平衡方程

∂ϕ∂θ=−α∂p∂θ \frac{\partial \phi}{\partial \theta} = - \alpha \frac{\partial p}{\partial \theta}

∂M∂θ=Π \frac{\partial M}{\partial \theta} = \Pi

其中MM为Montgomery势

M=cpT+ϕ M = c_p T + \phi

连续方程

∇p=∇θ−∇θp∂∂p \nabla_p = \nabla_\theta - \nabla_\theta p \frac{\partial}{\partial p}

ω=DpDt=(∂∂t+v⋅∇)θp+θ˙∂p∂θ \omega = \frac{D p}{D t} = \left( \frac{\partial}{\partial t} + \mathbf{v} \cdot \nabla \right)_\theta p + \dot{\theta} \frac{\partial p}{\partial \theta}

利用气压坐标的连续方程

∇p⋅v+∂ω∂p=0 \nabla_p \cdot \mathbf{v} + \frac{\partial \omega}{\partial p} = 0

∂m∂t+∇θ⋅(mv)+∂mθ˙∂θ=0 \frac{\partial m}{\partial t} + \nabla_\theta \cdot \left( m \mathbf{v} \right) + \frac{\partial m \dot{\theta}}{\partial \theta} = 0

其中m=−∂p∂θm = - \frac{\partial p}{\partial \theta}。

动量方程

气压梯度力在高度、气压、经典气压地形追随坐标系下的形式为

 −1ρ∇zp,−∇pϕ,−1ρ∇σp−∇σϕ \ - \frac{1}{\rho} \nabla_z p, \quad - \nabla_p \phi, \quad - \frac{1}{\rho} \nabla_\sigma p - \nabla_\sigma \phi

从高度坐标推等熵坐标的气压梯度力

 −1ρ∇zp=−1ρ∇θp+∂p∂z∇θz=−1ρ∇θp−∇θϕ \ - \frac{1}{\rho} \nabla_z p = - \frac{1}{\rho} \nabla_\theta p + \frac{\partial p}{\partial z} \nabla_\theta z = - {\color{red}\frac{1}{\rho} \nabla_\theta p} - \nabla_\theta \phi

其中红色项需要进一步推导（带入理想气体状态方程和位温定义式）

1ρ∇θp=RdTp∇θp=Rdpθ(pp0)Rdcp∇θp=θRdcpcp(pp0)−1(pp0)Rdcp∇θpp0=θ∇θcp(pp0)Rdcp=θ∇θΠ=∇θθΠ \begin{matrix} {\color{red}\frac{1}{\rho} \nabla_\theta p} & = \frac{R_d T}{p} \nabla_\theta p = \frac{R_d}{p} \theta \left( \frac{p}{p_0} \right)^{\frac{R_d}{c_p}} \nabla_\theta p \\ & = \theta \frac{R_d}{c_p} c_p \left( \frac{p}{\color{blue}p_0} \right)^{-1} \left( \frac{p}{p_0} \right)^{\frac{R_d}{c_p}} \nabla_\theta \frac{p}{\color{blue}p_0} \\ & = \theta \nabla_\theta c_p \left( \frac{p}{p_0} \right)^{\frac{R_d}{c_p}} = \theta \nabla_\theta \Pi = {\color{cyan} \nabla_\theta \theta \Pi} \end{matrix}

则等熵坐标系下的气压梯度力为

 −1ρ∇zp=−∇θ(θΠ+ϕ)=−∇θM \ - \frac{1}{\rho} \nabla_z p = - \nabla_\theta \left(\theta \Pi + \phi\right) = - \nabla_\theta M

其中MM是干静力能，或称为Montgomery势。从气压坐标推导也可以得到同样的结果

 −∇pϕ=−∇θM \ - \nabla_p \phi = - \nabla_\theta M

水平动量方程为

DvDt=−fk×v−∇θM+F \frac{D \mathbf{v}}{D t} = - f \mathbf{k} \times \mathbf{v} - \nabla_\theta M + \mathbf{F}

参考文献

  • Yueh-Jiuan G. Hsu and Akio Arakawa, 1990: Numerical Modeling of the Atmosphere with an Isentropic Vertical Coordinate. Monthly Weather Review, 1990, 118, 1933-1959.