我们关心的是球面正压浅水方程组的求解，首先给出它们的两种常用形式。

标准形式

∂u∂t=−uacos⁡φ∂u∂λ−va∂u∂φ+fv−∂ϕ∂λ∂v∂t=−uacos⁡φ∂v∂λ−va∂v∂φ−fu−∂ϕ∂φ∂h∂t=−1acos⁡φ(∂hu∂λ+∂hvcos⁡φ∂φ) \frac{\partial u}{\partial t} = - \frac{u}{a \cos{\varphi}} \frac{\partial u}{\partial \lambda} - \frac{v}{a} \frac{\partial u}{\partial \varphi} + f v - \frac{\partial \phi}{\partial \lambda} \\ \frac{\partial v}{\partial t} = - \frac{u}{a \cos{\varphi}} \frac{\partial v}{\partial \lambda} - \frac{v}{a} \frac{\partial v}{\partial \varphi} - f u - \frac{\partial \phi}{\partial \varphi} \\ \frac{\partial h}{\partial t} = - \frac{1}{a \cos{\varphi}} \left( \frac{\partial h u}{\partial \lambda} + \frac{\partial h v \cos{\varphi}}{\partial \varphi} \right)

其中科氏力参数f=2Ωsin⁡φf = 2 \Omega \sin{\varphi}。

涡度方程

矢量不变形式

矢量不变形式将非线性动量平流分为两项：

v⋅∇v=(∇×v)×v+∇∣v∣22 \mathbf{v} \cdot \nabla{\mathbf{v}} = \left({\color{red}\nabla \times \mathbf{v}}\right) \times \mathbf{v} + \nabla {\color{blue}\frac{\vert \mathbf{v} \vert^2}{2}}

其中红色项是相对涡度，蓝色项是动能。

∂u∂t=qhv−1acos⁡φ∂∂λ(K+ϕ)∂v∂t=−qhu−1a∂∂φ(K+ϕ)∂h∂t=−1acos⁡φ(∂hu∂λ+∂hvcos⁡φ∂φ) \frac{\partial u}{\partial t} = q h v - \frac{1}{a \cos{\varphi}} \frac{\partial}{\partial \lambda} \left( K + \phi \right) \\ \frac{\partial v}{\partial t} = - q h u - \frac{1}{a} \frac{\partial}{\partial \varphi} \left( K + \phi \right) \\ \frac{\partial h}{\partial t} = - \frac{1}{a \cos{\varphi}} \left( \frac{\partial h u}{\partial \lambda} + \frac{\partial h v \cos{\varphi}}{\partial \varphi} \right)

其中位涡为q=ζ+fhq = \frac{\zeta + f}{h}，动能为K=12(u2+v2)K = \frac{1}{2} \left( u^2 + v^2 \right)，相对涡度为

ζ=1a∂v∂λ−1acos⁡φ∂ucos⁡φ∂φ \zeta = \frac{1}{a} \frac{\partial v}{\partial \lambda} - \frac{1}{a \cos{\varphi}} \frac{\partial u \cos{\varphi}}{\partial \varphi}