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Next: 6.3.5 Tapering and stability Up: 6.3 Gent/McWiliams/Redi SGS Eddy Previous: 6.3.3 Griffies Skew Flux   Contents

6.3.4 Variable $ \kappa _{GM}$

Visbeck et al., 1996, suggest making the eddy coefficient, $ \kappa _{GM}$, a function of the Eady growth rate, $ \vert f\vert/\sqrt{Ri}$. The formula involves a non-dimensional constant, $ \alpha $, and a length-scale $ L$:

$\displaystyle \kappa_{GM} = \alpha L^2 \overline{ \frac{\vert f\vert}{\sqrt{Ri}} }^z $

where the Eady growth rate has been depth averaged (indicated by the over-line). A local Richardson number is defined $ Ri = N^2 / (\partial u/\partial z)^2$ which, when combined with thermal wind gives:

$\displaystyle \frac{1}{Ri} = \frac{(\frac{\partial u}{\partial z})^2}{N^2} = \f... ...\vert {\bf\nabla} \sigma \vert )^2 }{N^2} = \frac{ M^4 }{ \vert f\vert^2 N^2 } $

where $ M^2$ is defined $ M^2 = \frac{g}{\rho_o} \vert{\bf\nabla} \sigma\vert$. Substituting into the formula for $ \kappa _{GM}$ gives:

$\displaystyle \kappa_{GM} = \alpha L^2 \overline{ \frac{M^2}{N} }^z = \alpha L^2 \overline{ \frac{M^2}{N^2} N }^z = \alpha L^2 \overline{ \vert S\vert N }^z $


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