# Earth Cubed

## Coriolis Forces in Hopkins and Simmons Vorticity Equation

In the thread vector operations in Hopkins and Simmons, I compute the components of the curl as:

$\left[ \nabla \times \vec A \right]_{\sigma}= \frac{1}{r\mu}\frac{\partial A_{\lambda}}{\partial \mu}+ \frac{RT}{\sigma m g}\frac{\partial A_{\mu}}{\partial \lambda}=\left[ \nabla \times \vec A \right]_{\sigma,\mu}+\left[ \nabla \times \vec A \right]_{\sigma,\lambda}$

$\left[ \nabla \times \vec A \right]_{\mu}= -\frac{RT}{\sigma m g}\frac{\partial A_{\sigma}}{\partial \lambda}- \frac{\sqrt{1-\mu^2}}{r} \frac{\partial A_{\lambda}}{\partial \sigma}=\left[ \nabla \times \vec A \right]_{\mu,\lambda}+\left[ \nabla \times \vec A \right]_{\mu,\sigma}$

$\left[ \nabla \times \vec A \right]_{\lambda}= \frac{\sqrt{1-\mu^2}RT}{r}\frac{\partial A_{\mu}}{\partial \sigma}-\frac{RT}{r\mu} \frac{\partial A_{\sigma}}{\partial \mu}=\left[ \nabla \times \vec A \right]_{\lambda,\sigma}+\left[ \nabla \times \vec A \right]_{\lambda,\mu}$

In my post Coriolis forces in Hopkins and Simmons I compute the coriolis force as:

$\boldsymbol{\Omega \times v} = \begin{pmatrix} \mu U \\ \pm \sqrt{1-\mu^2} U \\ \mp \sqrt{1-\mu^2} V - \mu W \end{pmatrix} \ = \begin{pmatrix} A_{\sigma} \\ A_{\mu} \\ A_{\lambda} \end{pmatrix} \$

And the partial derivatives are given by:

$\frac{\partial A_{\lambda}}{\partial \mu}=\frac{\partial}{\partial \mu}\left(\mp \sqrt{1-\mu^2} V - \mu W\right)=\pm \frac{2\mu}{\sqrt{1-\mu^2}}V \mp \sqrt{1-\mu^2} \frac{\partial V}{\partial \mu} - W-\frac{\partial W}{\partial \mu}$

$\frac{\partial A_{\mu}}{\partial \lambda}=\frac{\partial }{\partial \lambda}\left(\pm \sqrt{1-\mu^2} U\right)=\pm \sqrt{1-\mu^2} \frac{\partial U}{\partial \lambda}$

$\frac{\partial A_{\sigma}}{\partial \lambda}=\frac{\partial }{\partial \lambda}\left( \mu U \right)=\mu \frac{\partial U}{\partial \lambda}$

$\frac{\partial A_{\lambda}}{\partial \sigma}=\frac{\partial}{\partial \sigma}\left(\mp \sqrt{1-\mu^2} V - \mu W\right)=\mp \sqrt{1-\mu^2} \frac{\partial V}{\partial \sigma} - \mu \frac{\partial W}{\partial \sigma}$

$\frac{\partial A_{\mu}}{\partial \sigma}=\frac{\partial }{\partial \sigma}\left(\pm \sqrt{1-\mu^2} U\right)=\pm \sqrt{1-\mu^2} \frac{\partial U }{\partial \sigma}$

$\frac{\partial A_{\sigma}}{\partial \mu}=\frac{\partial }{\partial \mu}\left( \mu U \right)=\mu \frac{\partial U}{\partial \mu}$

I’ll derive the rest of this later but this doesn’t seem to be the form of the prognostic equation used by Hopkins and Simmons.

Divergence Free Flow

In the post Vector Operations in Hopkins and Simmons I derived the divergence operator as follows:

$\nabla \cdot = \frac{1}{h_{\mu}} \frac{\partial }{\partial \mu} + \frac{1}{h_{\lambda}}\frac{\partial}{\partial \lambda}+\frac{1}{h_{\sigma}}\frac{\partial}{\partial \sigma}=\frac{1-\mu^2}{r} \frac{\partial}{\partial \mu} + \frac{1}{r \mu}\frac{\partial}{\partial \lambda}-\frac{RT}{\sigma mg}\frac{\partial}{\partial \sigma}$

If the divergence of the velocity equals zero then:

$\nabla \cdot \begin{pmatrix} W \\ V \\ U \end{pmatrix}\ =\frac{1-\mu^2}{r} \frac{\partial V}{\partial \mu} + \frac{1}{r \mu}\frac{\partial U}{\partial \lambda}-\frac{RT}{\sigma mg}\frac{\partial W}{\partial \sigma}=0$

Which implies:

$\frac{\partial W}{\partial \sigma}=\frac{(1-\mu^2)\sigma mg}{r RT} \frac{\partial V}{\partial \mu} + \frac{\sigma mg}{r \mu \sigma mg}\frac{\partial U}{\partial \lambda}$