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Vector Operations in Hoskins and Simmons Coordinates

In my post Hoskins and Simmons (1974) Coordinate System, I derived the following scaling quantities which will be used to derive the vector operations of Grad, Div and Curl in Hoskins Coordinate system.

h_{\mu}= \left| \frac{r}{\sqrt{1-\mu^2}} \right|
h_{\lambda}=r \mu
h_{\sigma} =-\sigma \frac{mg}{RT}

The coordinates in Hoskins coordinate system are dimensionless . (see nondimentionalization of Navier Stokes).

The gradient is defined as (see lectures on coordinate transforms):

\nabla \phi = \frac{\vec e_{\mu}}{h_{\mu}} \frac{\partial \phi}{\partial \mu} + \frac{\vec e_{\lambda}}{h_{\lambda}}\frac{\partial \phi}{\partial \lambda}+\frac{\vec e_{\sigma}}{h_{\sigma}}\frac{\partial \phi}{\partial \sigma}=\left(\vec e_{\mu}\right)\frac{1-\mu^2}{r} \frac{\partial \phi}{\partial \mu} + \left(\vec e_{\lambda}\right)\frac{1}{r \mu}\frac{\partial \phi}{\partial \lambda}-\left(\vec e_{\sigma}\right)\frac{RT}{\sigma mg}\frac{\partial \phi}{\partial \sigma}

The divergence is defined as:

\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}

The curl is defined by:

\nabla \times \vec A = \frac{1}{h_{\sigma}h_{\mu}h_{\lambda}} \left| \begin{array}{ccc} \vec e_{\sigma} & \vec e_{\mu} & \vec e_{\lambda} \\ \frac{\partial}{\partial \sigma} & \frac{\partial}{\partial \mu} & \frac{\partial}{\partial \lambda} \\ A_{\sigma}h_{\sigma} & A_{\mu}h_{\mu} &  A_{\lambda}h_{\lambda} \end{array} \right|

(note, the direction of the longitudinal coordinate \lambda is defined to obey the right hand rule)

This gives for the components

\left[ \nabla \times \vec A \right]_{\sigma}=\frac{RT}{r \mu \sigma mg}\left(\sigma \frac{mg}{RT}\frac{\partial A_{\lambda}}{\partial \mu}+ r \mu\frac{\partial A_{\mu}}{\partial \lambda} \right)

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

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

Which Simplifies to:

\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]_{\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]_{\lambda}=  \frac{\sqrt{1-\mu^2}RT}{r}\frac{\partial A_{\mu}}{\partial \sigma}-\frac{RT}{r\mu} \frac{\partial A_{\sigma}}{\partial \mu}

August 29, 2009 - Posted by | Uncategorized

1 Comment »

  1. […] the post Vector Operations in Hopkins and Simmons I derived the divergence operator as […]

    Pingback by Coriolis Forces in Hopkins and Simmons Vorticity Equation « Earth Cubed | September 12, 2009 | Reply

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