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% $Name$ |
% $Name$ |
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\section{Nonlinear Viscosities for Large Eddy Simulation} |
\section{Nonlinear Viscosities for Large Eddy Simulation} |
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\label{sect:nonlin-visc} |
\label{sec:nonlin-visc} |
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In Large Eddy Simulations (LES), a turbulent closure needs to be |
In Large Eddy Simulations (LES), a turbulent closure needs to be |
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provided that accounts for the effects of subgridscale motions on the |
provided that accounts for the effects of subgridscale motions on the |
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\end{equation} |
\end{equation} |
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The coefficient {\sf viscC2Smag} is what an MITgcm user sets, and it |
The coefficient {\sf viscC2Smag} is what an MITgcm user sets, and it |
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replaces the proportionality in the Kolmogorov length with an |
replaces the proportionality in the Kolmogorov length with an |
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equality. Others \cite{grha00} suggest values of {\sf viscC2Smag} |
equality. Others \cite{griffies:00} suggest values of {\sf viscC2Smag} |
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from 2.2 to 4 for oceanic problems. Smagorinsky \cite{Smagorinsky93} |
from 2.2 to 4 for oceanic problems. Smagorinsky \cite{Smagorinsky93} |
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shows that values from 0.2 to 0.9 have been used in atmospheric |
shows that values from 0.2 to 0.9 have been used in atmospheric |
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modeling. |
modeling. |
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viscA4GridMin}, which if used, should be set to values $\ll 1$. $L$ |
viscA4GridMin}, which if used, should be set to values $\ll 1$. $L$ |
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is roughly the gridscale (see below). |
is roughly the gridscale (see below). |
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Following \cite{grha00}, we note that there is a factor of $\Delta |
Following \cite{griffies:00}, we note that there is a factor of $\Delta |
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x^2/8$ difference between the harmonic and biharmonic viscosities. |
x^2/8$ difference between the harmonic and biharmonic viscosities. |
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Thus, whenever a non-dimensional harmonic coefficient is used in the |
Thus, whenever a non-dimensional harmonic coefficient is used in the |
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MITgcm (\textit{eg.} {\sf viscAhGridMax}$<1$), the biharmonic equivalent is |
MITgcm (\textit{eg.} {\sf viscAhGridMax}$<1$), the biharmonic equivalent is |
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\frac{-\nabla^4_h \BFKav b}{\Pr\BFKRe_4} |
\frac{-\nabla^4_h \BFKav b}{\Pr\BFKRe_4} |
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+\frac{\BFKpds{z} {\BFKav b}}{\Pr\BFKRe_v}\nonumber |
+\frac{\BFKpds{z} {\BFKav b}}{\Pr\BFKRe_v}\nonumber |
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\end{eqnarray} |
\end{eqnarray} |
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\cite{grha00} propose that if one scales the biharmonic viscosity by |
\cite{griffies:00} propose that if one scales the biharmonic viscosity by |
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stability considerations, then the biharmonic viscous terms will be |
stability considerations, then the biharmonic viscous terms will be |
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similarly active to harmonic viscous terms at the gridscale of the |
similarly active to harmonic viscous terms at the gridscale of the |
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model, but much less active on larger scale motions. Similarly, a |
model, but much less active on larger scale motions. Similarly, a |
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+\left(\frac{{\sf viscC4LeithD}}{\pi}\right)^{12} |
+\left(\frac{{\sf viscC4LeithD}}{\pi}\right)^{12} |
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|\nabla^2 \nabla\cdot \BFKav {\bf \BFKtu}_h|^2} |
|\nabla^2 \nabla\cdot \BFKav {\bf \BFKtu}_h|^2} |
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\end{eqnarray} |
\end{eqnarray} |
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Thus, the biharmonic scaling suggested by \cite{grha00} implies: |
Thus, the biharmonic scaling suggested by \cite{griffies:00} implies: |
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\begin{eqnarray} |
\begin{eqnarray} |
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|D| & \propto & L|\nabla^2\BFKav {\bf \BFKtu}_h|\\ |
|D| & \propto & L|\nabla^2\BFKav {\bf \BFKtu}_h|\\ |
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|\nabla \BFKav \omega_3| & \propto & L|\nabla^2 \BFKav \omega_3| |
|\nabla \BFKav \omega_3| & \propto & L|\nabla^2 \BFKav \omega_3| |
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\end{eqnarray} |
\end{eqnarray} |
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It is not at all clear that these assumptions ought to hold. Only the |
It is not at all clear that these assumptions ought to hold. Only the |
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\cite{grha00} forms are currently implemented in MITgcm. |
\cite{griffies:00} forms are currently implemented in MITgcm. |
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\subsubsection{Selection of Length Scale} |
\subsubsection{Selection of Length Scale} |
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Above, the length scale of the grid has been denoted $L$. However, in |
Above, the length scale of the grid has been denoted $L$. However, in |
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