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% $Header$ |
% $Header$ |
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% $Name$ |
% $Name$ |
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\def\del{{\mathbf \nabla}} |
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\def\av#1{\overline{#1}} |
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\def\pd#1#2{{\frac{\partial{#2}}{\partial#1}}} |
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\def\pds#1#2{{\frac{\partial^2{#2}}{{\partial#1}^2}}} |
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\def\Dt#1{\frac{D{#1}}{Dt}} |
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\def\aDt#1{\frac{\av D{#1}}{\av{Dt}}} |
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\def\d#1{{\,\rm d#1}} |
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\def\Ro{{\rm Ro}} |
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\def\Re{{\rm Re}} |
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\def\Fr{{\rm Fr}} |
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\def\mr{{m_{Ro}}} |
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\def\Mr{{M_{Ro}}} |
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\def\eg{{\emph{e.g.,}\ }} |
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\def\ie{{\emph{i.e.,}\ }} |
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\def\tu{{\tilde u}} |
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\def\tv{{\tilde v}} |
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\def\atu{{\tilde {\av u}}} |
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\def\atv{{\tilde {\av v}}} |
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\def\lesssim{{<\atop\sim}} |
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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{sect:nonlin-visc} |
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on the large-scale are parameterized. |
on the large-scale are parameterized. |
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To formalize this process, we can introduce a filter over the |
To formalize this process, we can introduce a filter over the |
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subgridscale L: $u_\alpha\rightarrow \av u_\alpha$ and $L: |
subgridscale L: $u_\alpha\rightarrow \BFKav u_\alpha$ and $L: |
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b\rightarrow \av b$. This filter has some intrinsic length and time |
b\rightarrow \BFKav b$. This filter has some intrinsic length and time |
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scales, and we assume that the flow at that scale can be characterized |
scales, and we assume that the flow at that scale can be characterized |
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with a single velocity scale ($V$) and vertical buoyancy gradient |
with a single velocity scale ($V$) and vertical buoyancy gradient |
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($N^2$). The filtered equations of motion in a local Mercator |
($N^2$). The filtered equations of motion in a local Mercator |
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projection about the gridpoint in question (see Appendix for notation |
projection about the gridpoint in question (see Appendix for notation |
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and details of approximation) are: \newpage |
and details of approximation) are: |
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\begin{eqnarray} |
\begin{eqnarray} |
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\aDt \atu- \frac{\atv \sin\theta}{\Ro\sin\theta_0}+\frac{\Mr}{\Ro}\pd{x}{\av\pi}=-\left({\av{\Dt \tu}}-{\aDt \atu}\right)+\frac{\nabla^2{\atu}}{\Re}\label{eq:mercat}\\ |
\BFKaDt \BFKatu- \frac{\BFKatv |
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\aDt\atv+ \frac{\atu\sin\theta}{\Ro\sin\theta_0}+\frac{\Mr}{\Ro}\pd{y}{\av\pi}=-\left({\av{\Dt \tv}}-{\aDt \atv}\right)+\frac{\nabla^2{\atv}}{\Re}\nonumber\\ |
\sin\theta}{\BFKRo\sin\theta_0}+\frac{\BFKMr}{\BFKRo}\BFKpd{x}{\BFKav\pi} |
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\aDt {\av w} +\frac{\pd{z}{\av\pi}-\av b}{\Fr^2\lambda^2}=-\left(\av{\Dt w}-\aDt {\av{w}}\right)+\frac{\nabla^2\av w}{\Re}\nonumber\\ |
& = & -\left({\BFKav{\BFKDt \BFKtu}}-{\BFKaDt \BFKatu}\right) |
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\aDt{\ \av b}+\av w=-\left(\av{\Dt{b}}-\aDt{\ \av b} \right)+\frac{\nabla^2 \av b}{\Pr\Re}\nonumber \\ |
+\frac{\nabla^2{\BFKatu}}{\BFKRe}\label{eq:mercat}\\ |
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\mu^2\left(\pd x\atu + \pd y\atv \right)+\pd z {\av w} =0\label{eq:cont} |
\BFKaDt\BFKatv+\frac{\BFKatu\sin\theta}{\BFKRo\sin\theta_0} |
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+\frac{\BFKMr}{\BFKRo}\BFKpd{y}{\BFKav\pi} |
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& = & -\left({\BFKav{\BFKDt \BFKtv}}-{\BFKaDt \BFKatv}\right) |
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+\frac{\nabla^2{\BFKatv}}{\BFKRe}\nonumber\\ |
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\BFKaDt {\BFKav w} +\frac{\BFKpd{z}{\BFKav\pi}-\BFKav b}{\BFKFr^2\lambda^2} |
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& = & -\left(\BFKav{\BFKDt w}-\BFKaDt {\BFKav{w}}\right) |
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+\frac{\nabla^2\BFKav w}{\BFKRe}\nonumber\\ |
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\BFKaDt{\ \BFKav b}+\BFKav w & = & |
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-\left(\BFKav{\BFKDt{b}}-\BFKaDt{\ \BFKav b} \right) |
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+\frac{\nabla^2 \BFKav b}{\Pr\BFKRe}\nonumber \\ |
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\mu^2\left(\BFKpd x\BFKatu + \BFKpd y\BFKatv \right)+\BFKpd z {\BFKav w} |
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& = & 0\label{eq:cont} |
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\end{eqnarray} |
\end{eqnarray} |
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Tildes denote multiplication by $\cos\theta/\cos\theta_0$ to account |
Tildes denote multiplication by $\cos\theta/\cos\theta_0$ to account |
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for converging meridians. |
for converging meridians. |
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The ocean is usually turbulent, and an operational definition of |
The ocean is usually turbulent, and an operational definition of |
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turbulence is that the terms in parentheses (the 'eddy' terms) on the |
turbulence is that the terms in parentheses (the 'eddy' terms) on the |
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right of (\ref{eq:mercat}) are of comparable magnitude to the terms on |
right of (\ref{eq:mercat}) are of comparable magnitude to the terms on |
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the left-hand side. The terms proportional to the inverse of \Re, |
the left-hand side. The terms proportional to the inverse of \BFKRe, |
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instead, are many orders of magnitude smaller than all of the other |
instead, are many orders of magnitude smaller than all of the other |
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terms in virtually every oceanic application. |
terms in virtually every oceanic application. |
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just to increase the viscosity and diffusivity until the viscous and |
just to increase the viscosity and diffusivity until the viscous and |
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diffusive scales are resolved. That is, we approximate: |
diffusive scales are resolved. That is, we approximate: |
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\begin{eqnarray} |
\begin{eqnarray} |
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\left({\av{\Dt \tu}}-{\aDt \atu}\right)\approx\frac{\nabla^2_h{\atu}}{\Re_h}+\frac{\pds{z}{\atu}}{\Re_v}\label{eq:eddyvisc},\qquad |
\left({\BFKav{\BFKDt \BFKtu}}-{\BFKaDt \BFKatu}\right) |
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\left({\av{\Dt \tv}}-{\aDt \atv}\right)\approx\frac{\nabla^2_h{\atv}}{\Re_h}+\frac{\pds{z}{\atv}}{\Re_v}\nonumber\\ |
\approx\frac{\nabla^2_h{\BFKatu}}{\BFKRe_h} |
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\left(\av{\Dt w}-\aDt {\av{w}}\right)\approx\frac{\nabla^2_h\av w}{\Re_h}+\frac{\pds{z}{\av w}}{\Re_v}\nonumber,\qquad |
+\frac{\BFKpds{z}{\BFKatu}}{\BFKRe_v}\label{eq:eddyvisc}, & & |
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\left(\av{\Dt{b}}-\aDt{\ \av b} \right)\approx\frac{\nabla^2_h \av b}{\Pr\Re_h}+\frac{\pds{z} {\av b}}{\Pr\Re_v}\nonumber |
\left({\BFKav{\BFKDt \BFKtv}}-{\BFKaDt \BFKatv}\right) |
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\approx\frac{\nabla^2_h{\BFKatv}}{\BFKRe_h} |
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+\frac{\BFKpds{z}{\BFKatv}}{\BFKRe_v}\nonumber\\ |
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\left(\BFKav{\BFKDt w}-\BFKaDt {\BFKav{w}}\right) |
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\approx\frac{\nabla^2_h\BFKav w}{\BFKRe_h} |
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+\frac{\BFKpds{z}{\BFKav w}}{\BFKRe_v}\nonumber, & & |
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\left(\BFKav{\BFKDt{b}}-\BFKaDt{\ \BFKav b} \right) |
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\approx\frac{\nabla^2_h \BFKav b}{\Pr\BFKRe_h} |
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+\frac{\BFKpds{z} {\BFKav b}}{\Pr\BFKRe_v}\nonumber |
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\end{eqnarray} |
\end{eqnarray} |
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\subsubsection{Reynolds-Number Limited Eddy Viscosity} |
\subsubsection{Reynolds-Number Limited Eddy Viscosity} |
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One way of ensuring that the gridscale is sufficiently viscous (\ie |
One way of ensuring that the gridscale is sufficiently viscous |
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resolved) is to choose the eddy viscosity $A_h$ so that the gridscale |
(\textit{ie.} resolved) is to choose the eddy viscosity $A_h$ so that |
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horizontal Reynolds number based on this eddy viscosity, $\Re_h$, to |
the gridscale horizontal Reynolds number based on this eddy viscosity, |
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is O(1). That is, if the gridscale is to be viscous, then the |
$\BFKRe_h$, to is O(1). That is, if the gridscale is to be viscous, |
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viscosity should be chosen to make the viscous terms as large as the |
then the viscosity should be chosen to make the viscous terms as large |
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advective ones. \citet{Bryanetal75} note that a computational mode is |
as the advective ones. Bryan \textit{et al} \cite{Bryanetal75} notes |
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squelched by using $\Re_h<$2. |
that a computational mode is squelched by using $\BFKRe_h<$2. |
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The MITgcm user can select an horizontal eddy viscosity based on |
The MITgcm user can select an horizontal eddy viscosity based on |
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$\Re_h$ by two methods. 1) The user may estimate the velocity scale |
$\BFKRe_h$ by two methods. 1) The user may estimate the velocity |
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expected from the calculation and grid spacing and set the {\sf |
scale expected from the calculation and grid spacing and set the {\sf |
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viscAh} to satisfy $\Re_h<2$. 2) The user may use {\sf |
viscAh} to satisfy $\BFKRe_h<2$. 2) The user may use {\sf |
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viscAhReMax}, which ensures that the viscosity is always chosen so |
viscAhReMax}, which ensures that the viscosity is always chosen so |
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that $\Re_h<{\sf viscAhReMax}$. This last option should be used with |
that $\BFKRe_h<{\sf viscAhReMax}$. This last option should be used |
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caution, however, since it effectively implies that viscous terms are |
with caution, however, since it effectively implies that viscous terms |
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fixed in magnitude relative to advective terms. While it may be a |
are fixed in magnitude relative to advective terms. While it may be a |
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useful method for specifying a minimum viscosity with little effort, |
useful method for specifying a minimum viscosity with little effort, |
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tests have shown that setting {\sf viscAhReMax}=2 |
tests \cite{Bryanetal75} have shown that setting {\sf viscAhReMax}=2 |
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\citep[per][]{Bryanetal75} often tends to increase the viscosity |
often tends to increase the viscosity substantially over other more |
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substantially over other more 'physical' parameterizations below, |
'physical' parameterizations below, especially in regions where |
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especially in regions where gradients of velocity are small (and thus |
gradients of velocity are small (and thus turbulence may be weak), so |
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turbulence may be weak), so perhaps a more liberal value should be |
perhaps a more liberal value should be used, \textit{eg.} {\sf |
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used, \eg {\sf viscAhReMax}=10. |
viscAhReMax}=10. |
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While it is certainly necessary that viscosity be active at the |
While it is certainly necessary that viscosity be active at the |
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gridscale, the wavelength where dissipation of energy or enstrophy |
gridscale, the wavelength where dissipation of energy or enstrophy |
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Vertical eddy viscosities are often chosen in a more subjective way, |
Vertical eddy viscosities are often chosen in a more subjective way, |
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as model stability is not usually as sensitive to vertical viscosity. |
as model stability is not usually as sensitive to vertical viscosity. |
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Usually the 'observed' value from finescale measurements, etc., is |
Usually the 'observed' value from finescale measurements, etc., is |
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used (\eg {\sf viscAr}$\approx1\times10^{-4} m^2/s$). However, |
used (\textit{eg.} {\sf viscAr}$\approx1\times10^{-4} m^2/s$). However, |
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\citet{Smagorinsky93} notes that the Smagorinsky parameterization of |
Smagorinsky \cite{Smagorinsky93} notes that the Smagorinsky |
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isotropic turbulence implies a value of the vertical viscosity as well |
parameterization of isotropic turbulence implies a value of the |
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as the horizontal viscosity (see below). |
vertical viscosity as well as the horizontal viscosity (see below). |
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\subsubsection{Smagorinsky Viscosity} |
\subsubsection{Smagorinsky Viscosity} |
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\citet{sm63} and \citet{Smagorinsky93} suggest choosing a viscosity |
Some \cite{sm63,Smagorinsky93} suggest choosing a viscosity |
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that \emph{depends on the resolved motions}. Thus, the overall |
that \emph{depends on the resolved motions}. Thus, the overall |
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viscous operator has a nonlinear dependence on velocity. Smagorinsky |
viscous operator has a nonlinear dependence on velocity. Smagorinsky |
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chose his form of viscosity by considering Kolmogorov's ideas about |
chose his form of viscosity by considering Kolmogorov's ideas about |
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accordingly. |
accordingly. |
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Smagorinsky formed the energy equation from the momentum equations by |
Smagorinsky formed the energy equation from the momentum equations by |
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dotting them with velocity. \citep[There are some complications when |
dotting them with velocity. There are some complications when using |
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using the hydrostatic approximation, see][]{Smagorinsky93}. The |
the hydrostatic approximation as described by Smagorinsky |
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positive definite energy dissipation by horizontal viscosity in a |
\cite{Smagorinsky93}. The positive definite energy dissipation by |
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hydrostatic flow is $\nu D^2$, where D is the deformation rate at the |
horizontal viscosity in a hydrostatic flow is $\nu D^2$, where D is |
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viscous scale. According to Kolmogorov's theory, this should be a |
the deformation rate at the viscous scale. According to Kolmogorov's |
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good approximation to the energy flux at any wavenumber |
theory, this should be a good approximation to the energy flux at any |
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$\epsilon\approx\nu D^2$. Kolmogorov and Smagorinsky noted that using |
wavenumber $\epsilon\approx\nu D^2$. Kolmogorov and Smagorinsky noted |
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an eddy viscosity that exceeds the molecular value $\nu$ should ensure |
that using an eddy viscosity that exceeds the molecular value $\nu$ |
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that the energy flux through viscous scale set by the eddy viscosity |
should ensure that the energy flux through viscous scale set by the |
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is the same as it would have been had we resolved all the way to the |
eddy viscosity is the same as it would have been had we resolved all |
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true viscous scale. That is, $\epsilon\approx A_{hSmag} \av D^2$. If |
the way to the true viscous scale. That is, $\epsilon\approx |
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we use this approximation to estimate the Kolmogorov viscous length, |
A_{hSmag} \BFKav D^2$. If we use this approximation to estimate the |
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then |
Kolmogorov viscous length, then |
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\begin{eqnarray} |
\begin{equation} |
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L_\epsilon(A_{hSmag})\propto\pi\epsilon^{-1/4}A_{hSmag}^{3/4}\approx\pi(A_{hSmag} \av D^2)^{-1/4}A_{hSmag}^{3/4}=\pi A_{hSmag}^{1/2}\av D^{-1/2} |
L_\epsilon(A_{hSmag})\propto\pi\epsilon^{-1/4}A_{hSmag}^{3/4}\approx\pi(A_{hSmag} |
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\end{eqnarray} |
\BFKav D^2)^{-1/4}A_{hSmag}^{3/4} = \pi A_{hSmag}^{1/2}\BFKav D^{-1/2} |
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\end{equation} |
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To make $L_\epsilon(A_{hSmag})$ scale with the gridscale, then |
To make $L_\epsilon(A_{hSmag})$ scale with the gridscale, then |
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\begin{eqnarray} |
\begin{equation} |
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A_{hSmag}=\left(\frac{{\sf viscC2Smag}}{\pi}\right)^2L^2|\av D| |
A_{hSmag} = \left(\frac{{\sf viscC2Smag}}{\pi}\right)^2L^2|\BFKav D| |
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\end{eqnarray} |
\end{equation} |
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Where the deformation rate appropriate for hydrostatic flows with |
Where the deformation rate appropriate for hydrostatic flows with |
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shallow-water scaling is |
shallow-water scaling is |
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\begin{eqnarray} |
\begin{equation} |
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|\av D|=\sqrt{\left(\pd{x}{\av \tu}-\pd{y}{\av \tv}\right)^2+\left(\pd{y}{\av \tu}+\pd{x}{\av \tv}\right)^2} |
|\BFKav D|=\sqrt{\left(\BFKpd{x}{\BFKav \BFKtu}-\BFKpd{y}{\BFKav \BFKtv}\right)^2 |
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\end{eqnarray} |
+\left(\BFKpd{y}{\BFKav \BFKtu}+\BFKpd{x}{\BFKav \BFKtv}\right)^2} |
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\end{equation} |
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The coefficient {\sf viscC2Smag} is what the MITgcm user sets, and it |
The coefficient {\sf viscC2Smag} is what the 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. \citet{grha00} suggest values of {\sf viscC2Smag} from 2.2 |
equality. Others \cite{grha00} suggest values of {\sf viscC2Smag} |
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to 4 for oceanic problems. \citet{Smagorinsky93} shows that values |
from 2.2 to 4 for oceanic problems. Smagorinsky \cite{Smagorinsky93} |
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from 0.2 to 0.9 have been used in atmospheric modeling. |
shows that values from 0.2 to 0.9 have been used in atmospheric |
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modeling. |
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\citet{Smagorinsky93} shows that a corresponding vertical viscosity |
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should be used: |
Smagorinsky \cite{Smagorinsky93} shows that a corresponding vertical |
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\begin{eqnarray} |
viscosity should be used: |
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A_{vSmag}=\left(\frac{{\sf viscC2Smag}}{\pi}\right)^2H^2\sqrt{\left(\pd{z}{\av \tu}\right)^2+\left(\pd{z}{\av \tv}\right)^2}\nonumber |
\begin{equation} |
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\end{eqnarray} |
A_{vSmag}=\left(\frac{{\sf viscC2Smag}}{\pi}\right)^2H^2 |
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\sqrt{\left(\BFKpd{z}{\BFKav \BFKtu}\right)^2 |
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+\left(\BFKpd{z}{\BFKav \BFKtv}\right)^2} |
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\end{equation} |
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This vertical viscosity is currently not implemented in the MITgcm |
This vertical viscosity is currently not implemented in the MITgcm |
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(although it may be soon). |
(although it may be soon). |
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\subsubsection{Leith Viscosity} |
\subsubsection{Leith Viscosity} |
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\citet{Leith68,Leith96} notes that 2-d turbulence is quite different |
Leith \cite{Leith68,Leith96} notes that 2-d turbulence is quite |
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from 3-d. In two-dimensional turbulence, energy cascades to larger |
different from 3-d. In two-dimensional turbulence, energy cascades to |
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scales, so there is no concern about resolving the scales of energy |
larger scales, so there is no concern about resolving the scales of |
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dissipation. Instead, another quantity, enstrophy, (which is the |
energy dissipation. Instead, another quantity, enstrophy, (which is |
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vertical component of vorticity squared) is conserved in 2-d |
the vertical component of vorticity squared) is conserved in 2-d |
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turbulence, and it cascades to smaller scales where it is dissipated. |
turbulence, and it cascades to smaller scales where it is dissipated. |
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Following a similar argument to that above about energy flux, the |
Following a similar argument to that above about energy flux, the |
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enstrophy flux is estimated to be equal to the positive-definite |
enstrophy flux is estimated to be equal to the positive-definite |
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gridscale dissipation rate of enstrophy $\eta\approx A_{hLeith} |
gridscale dissipation rate of enstrophy $\eta\approx A_{hLeith} |
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|\nabla\av \omega_3|^2$. By dimensional analysis, the |
|\nabla\BFKav \omega_3|^2$. By dimensional analysis, the |
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enstrophy-dissipation scale is $L_\eta(A_{hLeith})\propto\pi |
enstrophy-dissipation scale is $L_\eta(A_{hLeith})\propto\pi |
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A_{hLeith}^{1/2}\eta^{-1/6}$. Thus, the Leith-estimated length scale |
A_{hLeith}^{1/2}\eta^{-1/6}$. Thus, the Leith-estimated length scale |
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of enstrophy-dissipation and the resulting eddy viscosity are |
of enstrophy-dissipation and the resulting eddy viscosity are |
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\begin{eqnarray} |
\begin{eqnarray} |
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L_\eta(A_{hLeith})\propto\pi A_{hLeith}^{1/2}\eta^{-1/6}=\pi A_{hLeith}^{1/3}|\nabla \av \omega_3|^{-1/3}\\ |
L_\eta(A_{hLeith})\propto\pi A_{hLeith}^{1/2}\eta^{-1/6} |
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A_{hLeith}=\left(\frac{{\sf viscC2Leith}}{\pi}\right)^3L^3|\nabla \av \omega_3|\\ |
& = & \pi A_{hLeith}^{1/3}|\nabla \BFKav \omega_3|^{-1/3} \\ |
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|\nabla\omega_3|\equiv\sqrt{\left[\pd{x}{\ }\left(\pd{x}{\av \tv}-\pd{y}{\av \tu}\right)\right]^2+\left[\pd{y}{\ }\left(\pd{x}{\av \tv}-\pd{y}{\av \tu}\right)\right]^2} |
A_{hLeith} & = & |
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|
\left(\frac{{\sf viscC2Leith}}{\pi}\right)^3L^3|\nabla \BFKav\omega_3| \\ |
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|
|\nabla\omega_3| & \equiv & |
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\sqrt{\left[\BFKpd{x}{\ } |
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\left(\BFKpd{x}{\BFKav \BFKtv}-\BFKpd{y}{\BFKav |
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\BFKtu}\right)\right]^2 |
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+\left[\BFKpd{y}{\ }\left(\BFKpd{x}{\BFKav \BFKtv} |
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-\BFKpd{y}{\BFKav \BFKtu}\right)\right]^2} |
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\end{eqnarray} |
\end{eqnarray} |
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|
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\subsubsection{Modified Leith Viscosity} |
\subsubsection{Modified Leith Viscosity} |
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instabilities near the gridscale. The combined viscosity has the |
instabilities near the gridscale. The combined viscosity has the |
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form: |
form: |
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\begin{eqnarray} |
\begin{eqnarray} |
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A_{hLeith}=L^3\sqrt{\left(\frac{{\sf viscC2Leith}}{\pi}\right)^6|\nabla \av \omega_3|^2+\left(\frac{{\sf viscC2LeithD}}{\pi}\right)^6|\nabla \nabla\cdot \av {\tilde u}_h|^2}\nonumber\\ |
A_{hLeith} & = & |
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|\nabla \nabla\cdot \av {\tilde u}_h|\equiv\sqrt{\left[\pd{x}{\ }\left(\pd{x}{\av \tu}+\pd{y}{\av \tv}\right)\right]^2+\left[\pd{y}{\ }\left(\pd{x}{\av \tu}+\pd{y}{\av \tv}\right)\right]^2} |
L^3\sqrt{\left(\frac{{\sf viscC2Leith}}{\pi}\right)^6 |
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|
|\nabla \BFKav \omega_3|^2 |
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|
+\left(\frac{{\sf viscC2LeithD}}{\pi}\right)^6 |
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|
|\nabla \nabla\cdot \BFKav {\tilde u}_h|^2} \\ |
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|\nabla \nabla\cdot \BFKav {\tilde u}_h| & \equiv & |
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\sqrt{\left[\BFKpd{x}{\ }\left(\BFKpd{x}{\BFKav \BFKtu} |
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+\BFKpd{y}{\BFKav \BFKtv}\right)\right]^2 |
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+\left[\BFKpd{y}{\ }\left(\BFKpd{x}{\BFKav \BFKtu} |
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+\BFKpd{y}{\BFKav \BFKtv}\right)\right]^2} |
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\end{eqnarray} |
\end{eqnarray} |
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Whether there is any physical rationale for this correction is unclear |
Whether there is any physical rationale for this correction is unclear |
| 253 |
at the moment, but the numerical consequences are good. The |
at the moment, but the numerical consequences are good. The |
| 268 |
by gridscale and timestep by the Courant--Freidrichs--Lewy (CFL) |
by gridscale and timestep by the Courant--Freidrichs--Lewy (CFL) |
| 269 |
constraint on stability: |
constraint on stability: |
| 270 |
\begin{eqnarray} |
\begin{eqnarray} |
| 271 |
A_h<\frac{L^2}{4\Delta t}\nonumber\\ |
A_h & < & \frac{L^2}{4\Delta t} \\ |
| 272 |
A_4 \le \frac{L^4}{32\Delta t}\nonumber |
A_4 & \le & \frac{L^4}{32\Delta t} |
|
%% A_4\lesssim\frac{L^4}{32\Delta t}\nonumber |
|
| 273 |
\end{eqnarray} |
\end{eqnarray} |
| 274 |
The viscosities may be automatically limited to be no greater than |
The viscosities may be automatically limited to be no greater than |
| 275 |
these values in the MITgcm by specifying {\sf viscAhGridMax}$<1$ and |
these values in the MITgcm by specifying {\sf viscAhGridMax}$<1$ and |
| 278 |
viscA4GridMin}, which if used, should be set to values $\ll 1$. $L$ |
viscA4GridMin}, which if used, should be set to values $\ll 1$. $L$ |
| 279 |
is roughly the gridscale (see below). |
is roughly the gridscale (see below). |
| 280 |
|
|
| 281 |
Following \citet{grha00}, we note that there is a factor of $\Delta |
Following \cite{grha00}, we note that there is a factor of $\Delta |
| 282 |
x^2/8$ difference between the harmonic and biharmonic viscosities. |
x^2/8$ difference between the harmonic and biharmonic viscosities. |
| 283 |
Thus, whenever a non-dimensional harmonic coefficient is used in the |
Thus, whenever a non-dimensional harmonic coefficient is used in the |
| 284 |
MITgcm (\eg {\sf viscAhGridMax}$<1$), the biharmonic equivalent is |
MITgcm (\textit{eg.} {\sf viscAhGridMax}$<1$), the biharmonic equivalent is |
| 285 |
scaled so that the same non-dimensional value can be used (\eg {\sf |
scaled so that the same non-dimensional value can be used (\textit{eg.} {\sf |
| 286 |
viscA4GridMax}$<1$). |
viscA4GridMax}$<1$). |
| 287 |
|
|
| 288 |
\subsubsection{Biharmonic Viscosity} |
\subsubsection{Biharmonic Viscosity} |
| 289 |
\citet{ho78} suggested that eddy viscosities ought to be focuses on |
\cite{ho78} suggested that eddy viscosities ought to be focuses on |
| 290 |
the dynamics at the grid scale, as larger motions would be 'resolved'. |
the dynamics at the grid scale, as larger motions would be 'resolved'. |
| 291 |
To enhance the scale selectivity of the viscous operator, he suggested |
To enhance the scale selectivity of the viscous operator, he suggested |
| 292 |
a biharmonic eddy viscosity instead of a harmonic (or Laplacian) |
a biharmonic eddy viscosity instead of a harmonic (or Laplacian) |
| 293 |
viscosity: |
viscosity: |
| 294 |
\begin{eqnarray} |
\begin{eqnarray} |
| 295 |
\left({\av{\Dt \tu}}-{\aDt \atu}\right)\approx\frac{-\nabla^4_h{\atu}}{\Re_4}+\frac{\pds{z}{\atu}}{\Re_v}\label{eq:bieddyvisc},\qquad |
\left({\BFKav{\BFKDt \BFKtu}}-{\BFKaDt \BFKatu}\right)\approx |
| 296 |
\left({\av{\Dt \tv}}-{\aDt \atv}\right)\approx\frac{-\nabla^4_h{\atv}}{\Re_4}+\frac{\pds{z}{\atv}}{\Re_v}\nonumber\\ |
\frac{-\nabla^4_h{\BFKatu}}{\BFKRe_4} |
| 297 |
\left(\av{\Dt w}-\aDt {\av{w}}\right)\approx\frac{-\nabla^4_h\av w}{\Re_4}+\frac{\pds{z}{\av w}}{\Re_v}\nonumber,\qquad |
+\frac{\BFKpds{z}{\BFKatu}}{\BFKRe_v}\label{eq:bieddyvisc}, & & |
| 298 |
\left(\av{\Dt{b}}-\aDt{\ \av b} \right)\approx\frac{-\nabla^4_h \av b}{\Pr\Re_4}+\frac{\pds{z} {\av b}}{\Pr\Re_v}\nonumber |
\left({\BFKav{\BFKDt \BFKtv}}-{\BFKaDt \BFKatv}\right)\approx |
| 299 |
|
\frac{-\nabla^4_h{\BFKatv}}{\BFKRe_4} |
| 300 |
|
+\frac{\BFKpds{z}{\BFKatv}}{\BFKRe_v}\nonumber\\ |
| 301 |
|
\left(\BFKav{\BFKDt w}-\BFKaDt |
| 302 |
|
{\BFKav{w}}\right)\approx\frac{-\nabla^4_h\BFKav |
| 303 |
|
w}{\BFKRe_4}+\frac{\BFKpds{z}{\BFKav w}}{\BFKRe_v}\nonumber, & & |
| 304 |
|
\left(\BFKav{\BFKDt{b}}-\BFKaDt{\ \BFKav b} \right)\approx |
| 305 |
|
\frac{-\nabla^4_h \BFKav b}{\Pr\BFKRe_4} |
| 306 |
|
+\frac{\BFKpds{z} {\BFKav b}}{\Pr\BFKRe_v}\nonumber |
| 307 |
\end{eqnarray} |
\end{eqnarray} |
| 308 |
\citet{grha00} propose that if one scales the biharmonic viscosity by |
\cite{grha00} propose that if one scales the biharmonic viscosity by |
| 309 |
stability considerations, then the biharmonic viscous terms will be |
stability considerations, then the biharmonic viscous terms will be |
| 310 |
similarly active to harmonic viscous terms at the gridscale of the |
similarly active to harmonic viscous terms at the gridscale of the |
| 311 |
model, but much less active on larger scale motions. Similarly, a |
model, but much less active on larger scale motions. Similarly, a |
| 330 |
substitution $h\rightarrow 4$. The MITgcm also supports a biharmonic |
substitution $h\rightarrow 4$. The MITgcm also supports a biharmonic |
| 331 |
Leith and Smagorinsky viscosities: |
Leith and Smagorinsky viscosities: |
| 332 |
\begin{eqnarray} |
\begin{eqnarray} |
| 333 |
A_{4Smag}=\left(\frac{{\sf viscC4Smag}}{\pi}\right)^2\frac{L^4}{8}|D|\nonumber\\ |
A_{4Smag} & = & |
| 334 |
A_{4Leith}=\frac{L^5}{8}\sqrt{\left(\frac{{\sf viscC4Leith}}{\pi}\right)^6|\nabla \av \omega_3|^2+\left(\frac{{\sf viscC4LeithD}}{\pi}\right)^6|\nabla \nabla\cdot \av {\bf \tu}_h|^2}\nonumber |
\left(\frac{{\sf viscC4Smag}}{\pi}\right)^2\frac{L^4}{8}|D| \\ |
| 335 |
|
A_{4Leith} & = & |
| 336 |
|
\frac{L^5}{8}\sqrt{\left(\frac{{\sf viscC4Leith}}{\pi}\right)^6 |
| 337 |
|
|\nabla \BFKav \omega_3|^2 |
| 338 |
|
+\left(\frac{{\sf viscC4LeithD}}{\pi}\right)^6 |
| 339 |
|
|\nabla \nabla\cdot \BFKav {\bf \BFKtu}_h|^2} |
| 340 |
\end{eqnarray} |
\end{eqnarray} |
| 341 |
However, it should be noted that unlike the harmonic forms, the |
However, it should be noted that unlike the harmonic forms, the |
| 342 |
biharmonic scaling does not easily relate to whether |
biharmonic scaling does not easily relate to whether |
| 344 |
similar arguments are used to estimate these scales and scale them to |
similar arguments are used to estimate these scales and scale them to |
| 345 |
the gridscale, the resulting biharmonic viscosities should be: |
the gridscale, the resulting biharmonic viscosities should be: |
| 346 |
\begin{eqnarray} |
\begin{eqnarray} |
| 347 |
A_{4Smag}=\left(\frac{{\sf viscC4Smag}}{\pi}\right)^5L^5|\nabla^2\av {\bf \tu}_h|\nonumber\\ |
A_{4Smag} & = & |
| 348 |
A_{4Leith}=L^6\sqrt{\left(\frac{{\sf viscC4Leith}}{\pi}\right)^{12}|\nabla^2 \av \omega_3|^2+\left(\frac{{\sf viscC4LeithD}}{\pi}\right)^{12}|\nabla^2 \nabla\cdot \av {\bf \tu}_h|^2}\nonumber |
\left(\frac{{\sf viscC4Smag}}{\pi}\right)^5L^5 |
| 349 |
|
|\nabla^2\BFKav {\bf \BFKtu}_h| \\ |
| 350 |
|
A_{4Leith} & = & |
| 351 |
|
L^6\sqrt{\left(\frac{{\sf viscC4Leith}}{\pi}\right)^{12} |
| 352 |
|
|\nabla^2 \BFKav \omega_3|^2 |
| 353 |
|
+\left(\frac{{\sf viscC4LeithD}}{\pi}\right)^{12} |
| 354 |
|
|\nabla^2 \nabla\cdot \BFKav {\bf \BFKtu}_h|^2} |
| 355 |
\end{eqnarray} |
\end{eqnarray} |
| 356 |
Thus, the biharmonic scaling suggested by \citet{grha00} implies: |
Thus, the biharmonic scaling suggested by \cite{grha00} implies: |
| 357 |
\begin{eqnarray} |
\begin{eqnarray} |
| 358 |
|D|\propto L|\nabla^2\av {\bf \tu}_h|\\ |
|D| & \propto & L|\nabla^2\BFKav {\bf \BFKtu}_h|\\ |
| 359 |
|\nabla \av \omega_3|\propto L|\nabla^2 \av \omega_3| |
|\nabla \BFKav \omega_3| & \propto & L|\nabla^2 \BFKav \omega_3| |
| 360 |
\end{eqnarray} |
\end{eqnarray} |
| 361 |
It is not at all clear that these assumptions ought to hold. Only the \citet{grha00} forms are currently implemented in the MITgcm. |
It is not at all clear that these assumptions ought to hold. Only the |
| 362 |
|
\cite{grha00} forms are currently implemented in the MITgcm. |
| 363 |
|
|
| 364 |
\subsubsection{Selection of Length Scale} |
\subsubsection{Selection of Length Scale} |
| 365 |
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 |
| 375 |
useAreaViscLength} is true, then the square root of the area of the |
useAreaViscLength} is true, then the square root of the area of the |
| 376 |
grid cell is used. |
grid cell is used. |
| 377 |
|
|
|
% The Appendices part is started with the command \appendix; |
|
|
% appendix sections are then done as normal sections |
|
|
% \appendix |
|
|
|
|
| 378 |
\subsection{Mercator, Nondimensional Equations} |
\subsection{Mercator, Nondimensional Equations} |
| 379 |
The rotating, incompressible, Boussinesq equations of motion |
The rotating, incompressible, Boussinesq equations of motion |
| 380 |
\citep{Gill1982} on a sphere can be written in Mercator projection |
\cite{Gill1982} on a sphere can be written in Mercator projection |
| 381 |
about a latitude $\theta_0$ and geopotential height $z=r-r_0$. The |
about a latitude $\theta_0$ and geopotential height $z=r-r_0$. The |
| 382 |
nondimensional form of these equations is: |
nondimensional form of these equations is: |
| 383 |
\begin{eqnarray} |
\begin{equation} |
| 384 |
\Ro\Dt\tu- \frac{\tv \sin\theta}{\sin\theta_0}+\Mr\pd{x}{\pi}+\frac{\lambda\Fr^2\Mr\cos \theta}{\mu\sin\theta_0} w=-\frac{\Fr^2\Mr \tu w}{r/H}+\frac{\Ro{\bf \hat x}\cdot\nabla^2{\bf u}}{\Re}\nonumber\\ |
\BFKRo\BFKDt\BFKtu- \frac{\BFKtv |
| 385 |
\Ro\Dt\tv+ \frac{\tu\sin\theta}{\sin\theta_0}+\Mr\pd{y}{\pi}=-\frac{\mu\Ro\tan\theta(\tu^2+\tv^2)}{r/L} -\frac{\Fr^2\Mr \tv w}{r/H}+\frac{\Ro{\bf \hat y}\cdot\nabla^2{\bf u}}{\Re}\nonumber\\ |
\sin\theta}{\sin\theta_0}+\BFKMr\BFKpd{x}{\pi} |
| 386 |
\Fr^2\lambda^2\Dt w -b+\pd{z}{\pi}-\frac{\lambda\cot \theta_0 \tu}{\Mr}=\frac{\lambda\mu^2(\tu^2+\tv^2)}{\Mr(r/L)}+\frac{\Fr^2\lambda^2{\bf \hat z}\cdot\nabla^2{\bf u}}{\Re}\nonumber\\ |
+\frac{\lambda\BFKFr^2\BFKMr\cos \theta}{\mu\sin\theta_0} w |
| 387 |
\Dt b+w=\frac{\nabla^2 b}{\Pr\Re}\nonumber, \qquad |
= -\frac{\BFKFr^2\BFKMr \BFKtu w}{r/H} |
| 388 |
\mu^2\left(\pd x\tu + \pd y\tv \right)+\pd z w =0\nonumber |
+\frac{\BFKRo{\bf \hat x}\cdot\nabla^2{\bf u}}{\BFKRe} |
| 389 |
|
\end{equation} |
| 390 |
|
\begin{equation} |
| 391 |
|
\BFKRo\BFKDt\BFKtv+ |
| 392 |
|
\frac{\BFKtu\sin\theta}{\sin\theta_0}+\BFKMr\BFKpd{y}{\pi} |
| 393 |
|
= -\frac{\mu\BFKRo\tan\theta(\BFKtu^2+\BFKtv^2)}{r/L} |
| 394 |
|
-\frac{\BFKFr^2\BFKMr \BFKtv w}{r/H} |
| 395 |
|
+\frac{\BFKRo{\bf \hat y}\cdot\nabla^2{\bf u}}{\BFKRe} |
| 396 |
|
\end{equation} |
| 397 |
|
\begin{eqnarray} |
| 398 |
|
\BFKFr^2\lambda^2\BFKDt w -b+\BFKpd{z}{\pi} |
| 399 |
|
-\frac{\lambda\cot \theta_0 \BFKtu}{\BFKMr} |
| 400 |
|
& = & \frac{\lambda\mu^2(\BFKtu^2+\BFKtv^2)}{\BFKMr(r/L)} |
| 401 |
|
+\frac{\BFKFr^2\lambda^2{\bf \hat z}\cdot\nabla^2{\bf u}}{\BFKRe} \\ |
| 402 |
|
\BFKDt b+w & = & \frac{\nabla^2 b}{\Pr\BFKRe}\nonumber \\ |
| 403 |
|
\mu^2\left(\BFKpd x\BFKtu + \BFKpd y\BFKtv \right)+\BFKpd z w |
| 404 |
|
& = & 0 |
| 405 |
\end{eqnarray} |
\end{eqnarray} |
| 406 |
Where |
Where |
| 407 |
\begin{eqnarray} |
\begin{equation} |
| 408 |
\mu\equiv\frac{\cos\theta_0}{\cos\theta},\qquad\tu=\frac{u^*}{V\mu},\qquad\tv=\frac{v^*}{V\mu} , \qquad \Dt\ \equiv \mu^2\left(\tu\pd x\ +\tv \pd y\ \right)+\frac{\Fr^2\Mr}{\Ro} w\pd z \nonumber \\ |
\mu\equiv\frac{\cos\theta_0}{\cos\theta},\ \ \ |
| 409 |
f_0\equiv2\Omega\sin\theta_0,\qquad x\equiv \frac{r}{L} \phi \cos \theta_0, \qquad y\equiv \frac{r}{L} \int_{\theta_0}^\theta\frac{\cos \theta_0 \d \theta'}{\cos\theta'}, \qquad z\equiv \lambda\frac{r-r_0}{L}\nonumber\\ |
\BFKtu=\frac{u^*}{V\mu},\ \ \ \BFKtv=\frac{v^*}{V\mu} |
| 410 |
t^*=t \frac{L}{V},\qquad b^*= b\frac{V f_0\Mr}{\lambda},\qquad \pi^*=\pi V f_0 L\Mr,\qquad w^*=w V \frac{\Fr^2\lambda\Mr}{\Ro}\nonumber\\ |
\end{equation} |
| 411 |
\Ro\equiv\frac{V}{f_0 L},\qquad \Mr\equiv \max[1,\Ro], \qquad \Fr\equiv\frac{V}{N \lambda L}, \qquad \Re\equiv\frac{VL}{\nu}, \qquad \Pr\equiv\frac{\nu}{\kappa}\nonumber |
%% EH3 :: This is the key bit thats messed up in the next equation |
| 412 |
\end{eqnarray} |
%% \Dt\ \equiv \mu^2\left(\tu\pd x\ +\tv \pd y\ \right)+\frac{\Fr^2\Mr}{\Ro} w\pd z |
| 413 |
|
\begin{equation} |
| 414 |
|
f_0\equiv2\Omega\sin\theta_0,\ \ \ |
| 415 |
|
%,\ \ \ \BFKDt\ \equiv \mu^2\left(\BFKtu\BFKpd x\ |
| 416 |
|
%+\BFKtv \BFKpd y\ \right)+\frac{\BFKFr^2\BFKMr}{\BFKRo} w\BFKpd z\ |
| 417 |
|
\frac{D}{Dt} \equiv \mu^2\left(\BFKtu\frac{\partial}{\partial x} |
| 418 |
|
+\BFKtv \frac{\partial}{\partial y} \right) |
| 419 |
|
+\frac{\BFKFr^2\BFKMr}{\BFKRo} w\frac{\partial}{\partial z} |
| 420 |
|
\end{equation} |
| 421 |
|
\begin{equation} |
| 422 |
|
x\equiv \frac{r}{L} \phi \cos \theta_0, \ \ \ |
| 423 |
|
y\equiv \frac{r}{L} \int_{\theta_0}^\theta |
| 424 |
|
\frac{\cos \theta_0 \BFKd \theta'}{\cos\theta'}, \ \ \ |
| 425 |
|
z\equiv \lambda\frac{r-r_0}{L} |
| 426 |
|
\end{equation} |
| 427 |
|
\begin{equation} |
| 428 |
|
t^*=t \frac{L}{V},\ \ \ b^*= b\frac{V f_0\BFKMr}{\lambda} |
| 429 |
|
\end{equation} |
| 430 |
|
\begin{equation} |
| 431 |
|
\pi^*=\pi V f_0 L\BFKMr,\ \ \ |
| 432 |
|
w^*=w V \frac{\BFKFr^2\lambda\BFKMr}{\BFKRo} |
| 433 |
|
\end{equation} |
| 434 |
|
\begin{equation} |
| 435 |
|
\BFKRo\equiv\frac{V}{f_0 L},\ \ \ \BFKMr\equiv \max[1,\BFKRo] |
| 436 |
|
\end{equation} |
| 437 |
|
\begin{equation} |
| 438 |
|
\BFKFr\equiv\frac{V}{N \lambda L}, \ \ \ |
| 439 |
|
\BFKRe\equiv\frac{VL}{\nu}, \ \ \ |
| 440 |
|
\BFKPr\equiv\frac{\nu}{\kappa} |
| 441 |
|
\end{equation} |
| 442 |
Dimensional variables are denoted by an asterisk where necessary. If |
Dimensional variables are denoted by an asterisk where necessary. If |
| 443 |
we filter over a grid scale typical for ocean models ($1m<L<100km$, |
we filter over a grid scale typical for ocean models ($1m<L<100km$, |
| 444 |
$0.0001<\lambda<1$, $0.001m/s <V<1 m/s$, $f_0<0.0001 s^{-1}$, $0.01 |
$0.0001<\lambda<1$, $0.001m/s <V<1 m/s$, $f_0<0.0001 s^{-1}$, $0.01 |
| 445 |
s^{-1}<N<0.0001 s^{-1}$), these equations are very well approximated |
s^{-1}<N<0.0001 s^{-1}$), these equations are very well approximated |
| 446 |
by |
by |
| 447 |
\begin{eqnarray} |
\begin{eqnarray} |
| 448 |
\Ro{\Dt\tu}- \frac{\tv \sin\theta}{\sin\theta_0}+\Mr\pd{x}{\pi}=-\frac{\lambda\Fr^2\Mr\cos \theta}{\mu\sin\theta_0} w+\frac{\Ro\nabla^2{\tu}}{\Re}\nonumber\\ |
\BFKRo{\BFKDt\BFKtu}- \frac{\BFKtv |
| 449 |
\Ro\Dt\tv+ \frac{\tu\sin\theta}{\sin\theta_0}+\Mr\pd{y}{\pi}=\frac{\Ro\nabla^2{\tv}}{\Re}\nonumber\\ |
\sin\theta}{\sin\theta_0}+\BFKMr\BFKpd{x}{\pi} |
| 450 |
\Fr^2\lambda^2\Dt w -b+\pd{z}{\pi}=\frac{\lambda\cot \theta_0 \tu}{\Mr}\nonumber+\frac{\Fr^2\lambda^2\nabla^2w}{\Re}\\ |
& =& -\frac{\lambda\BFKFr^2\BFKMr\cos \theta}{\mu\sin\theta_0} w |
| 451 |
\Dt b+w=\frac{\nabla^2 b}{\Pr\Re}\nonumber, \qquad |
+\frac{\BFKRo\nabla^2{\BFKtu}}{\BFKRe} \\ |
| 452 |
\mu^2\left(\pd x\tu + \pd y\tv \right)+\pd z w =0\nonumber\\ |
\BFKRo\BFKDt\BFKtv+ |
| 453 |
\nabla^2\approx\left(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\lambda^2\partial z^2}\right)\nonumber |
\frac{\BFKtu\sin\theta}{\sin\theta_0}+\BFKMr\BFKpd{y}{\pi} |
| 454 |
|
& = & \frac{\BFKRo\nabla^2{\BFKtv}}{\BFKRe} \\ |
| 455 |
|
\BFKFr^2\lambda^2\BFKDt w -b+\BFKpd{z}{\pi} |
| 456 |
|
& = & \frac{\lambda\cot \theta_0 \BFKtu}{\BFKMr} |
| 457 |
|
+\frac{\BFKFr^2\lambda^2\nabla^2w}{\BFKRe} \\ |
| 458 |
|
\BFKDt b+w & = & \frac{\nabla^2 b}{\Pr\BFKRe} \\ |
| 459 |
|
\mu^2\left(\BFKpd x\BFKtu + \BFKpd y\BFKtv \right)+\BFKpd z w |
| 460 |
|
& = & 0 \\ |
| 461 |
|
\nabla^2 & \approx & \left(\frac{\partial^2}{\partial x^2} |
| 462 |
|
+\frac{\partial^2}{\partial y^2} |
| 463 |
|
+\frac{\partial^2}{\lambda^2\partial z^2}\right) |
| 464 |
\end{eqnarray} |
\end{eqnarray} |
| 465 |
Neglecting the non-frictional terms on the right-hand side is usually |
Neglecting the non-frictional terms on the right-hand side is usually |
| 466 |
called the 'traditional' approximation. It is appropriate, with |
called the 'traditional' approximation. It is appropriate, with |
| 468 |
is used here, as it does not affect the form of the eddy stresses |
is used here, as it does not affect the form of the eddy stresses |
| 469 |
which is the main topic. The frictional terms are preserved in this |
which is the main topic. The frictional terms are preserved in this |
| 470 |
approximate form for later comparison with eddy stresses. |
approximate form for later comparison with eddy stresses. |
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