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2.21 Nonlinear Viscosities for Large Eddy Simulation

In Large Eddy Simulations (LES), a turbulent closure needs to be provided that accounts for the effects of subgridscale motions on the large scale. With sufficiently powerful computers, we could resolve the entire flow down to the molecular viscosity scales ( $ L_{\nu}\approx 1 \rm cm$ ). Current computation allows perhaps four decades to be resolved, so the largest problem computationally feasible would be about 10m. Most oceanographic problems are much larger in scale, so some form of LES is required, where only the largest scales of motion are resolved, and the subgridscale's effects on the large-scale are parameterized.

To formalize this process, we can introduce a filter over the subgridscale L: $ u_\alpha\rightarrow \overline{u}_\alpha$ and $ L:
b\rightarrow \overline{b}$ . This filter has some intrinsic length and time scales, and we assume that the flow at that scale can be characterized with a single velocity scale ($ V$ ) and vertical buoyancy gradient ($ N^2$ ). The filtered equations of motion in a local Mercator projection about the gridpoint in question (see Appendix for notation and details of approximation) are:

$\displaystyle \frac{\overline{D}{{\tilde {\overline{u}}}}}{\overline{Dt}} - \fr...
...heta_0}+\frac{{M_{Ro}}}{{\rm Ro}}{\frac{\partial{\overline{\pi}}}{\partial{x}}}$ $\displaystyle =$ $\displaystyle -\left({\overline{\frac{D{{\tilde u}}}{Dt} }}-{\frac{\overline{D}...
...}}}{\overline{Dt}} }\right)
+\frac{\nabla^2{{\tilde {\overline{u}}}}}{{\rm Re}}$ (2.206)
$\displaystyle \frac{\overline{D}{{\tilde {\overline{v}}}}}{\overline{Dt}} +\fra...
...eta_0}
+\frac{{M_{Ro}}}{{\rm Ro}}{\frac{\partial{\overline{\pi}}}{\partial{y}}}$ $\displaystyle =$ $\displaystyle -\left({\overline{\frac{D{{\tilde v}}}{Dt} }}-{\frac{\overline{D}...
...}}}{\overline{Dt}} }\right)
+\frac{\nabla^2{{\tilde {\overline{v}}}}}{{\rm Re}}$  
$\displaystyle \frac{\overline{D}{\overline{w}}}{\overline{Dt}} +\frac{{\frac{\partial{\overline{\pi}}}{\partial{z}}}-\overline{b}}{{\rm Fr}^2\lambda^2}$ $\displaystyle =$ $\displaystyle -\left(\overline{\frac{D{w}}{Dt}}-\frac{\overline{D}{\overline{w}}}{\overline{Dt}}\right)
+\frac{\nabla^2\overline{w}}{{\rm Re}}$  
$\displaystyle \frac{\overline{D}{\ \overline{b}}}{\overline{Dt}}+\overline{w}$ $\displaystyle =$ $\displaystyle -\left(\overline{\frac{D{b}}{Dt}}-\frac{\overline{D}{\ \overline{b}}}{\overline{Dt}} \right)
+\frac{\nabla^2 \overline{b}}{\Pr{\rm Re}}$  
$\displaystyle \mu^2\left({\frac{\partial{{\tilde {\overline{u}}}}}{\partial{x}}...
...rline{v}}}}}{\partial{y}}} \right)+{\frac{\partial{\overline{w}}}{\partial{z}}}$ $\displaystyle =$ 0 (2.207)

Tildes denote multiplication by $ \cos\theta/\cos\theta_0$ to account for converging meridians.

The ocean is usually turbulent, and an operational definition of turbulence is that the terms in parentheses (the 'eddy' terms) on the right of (2.206) are of comparable magnitude to the terms on the left-hand side. The terms proportional to the inverse of Re, instead, are many orders of magnitude smaller than all of the other terms in virtually every oceanic application.



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