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revision 1.3 by edhill, Fri Dec 16 06:26:09 2005 UTC revision 1.4 by edhill, Wed Apr 5 02:27:33 2006 UTC
# Line 80  then the viscosity should be chosen to m Line 80  then the viscosity should be chosen to m
80  as the advective ones.  Bryan \textit{et al} \cite{Bryanetal75} notes  as the advective ones.  Bryan \textit{et al} \cite{Bryanetal75} notes
81  that a computational mode is squelched by using $\BFKRe_h<$2.  that a computational mode is squelched by using $\BFKRe_h<$2.
82    
83  The MITgcm user can select an horizontal eddy viscosity based on  MITgcm users can select horizontal eddy viscosities based on
84  $\BFKRe_h$ by two methods.  1) The user may estimate the velocity  $\BFKRe_h$ using two methods.  1) The user may estimate the velocity
85  scale expected from the calculation and grid spacing and set the {\sf  scale expected from the calculation and grid spacing and set the {\sf
86    viscAh} to satisfy $\BFKRe_h<2$.  2) The user may use {\sf    viscAh} to satisfy $\BFKRe_h<2$.  2) The user may use {\sf
87    viscAhReMax}, which ensures that the viscosity is always chosen so    viscAhReMax}, which ensures that the viscosity is always chosen so
# Line 137  gridscale, $L_\epsilon(A_h)\propto\pi\ep Line 137  gridscale, $L_\epsilon(A_h)\propto\pi\ep
137  Large Eddy Simulation or LES).  Large Eddy Simulation or LES).
138    
139  There are two methods of ensuring that the Kolmogorov length is  There are two methods of ensuring that the Kolmogorov length is
140  resolved in the MITgcm.  1) The user can estimate the flux of energy  resolved in MITgcm.  1) The user can estimate the flux of energy
141  through spectral space for a given simulation and adjust grid spacing  through spectral space for a given simulation and adjust grid spacing
142  or {\sf viscAh} to ensure that $L_\epsilon(A_h)>L$.  2) The user may  or {\sf viscAh} to ensure that $L_\epsilon(A_h)>L$.  2) The user may
143  use the approach of Smagorinsky with {\sf viscC2Smag}, which estimates  use the approach of Smagorinsky with {\sf viscC2Smag}, which estimates
# Line 172  shallow-water scaling is Line 172  shallow-water scaling is
172  |\BFKav D|=\sqrt{\left(\BFKpd{x}{\BFKav \BFKtu}-\BFKpd{y}{\BFKav \BFKtv}\right)^2  |\BFKav D|=\sqrt{\left(\BFKpd{x}{\BFKav \BFKtu}-\BFKpd{y}{\BFKav \BFKtv}\right)^2
173  +\left(\BFKpd{y}{\BFKav \BFKtu}+\BFKpd{x}{\BFKav \BFKtv}\right)^2}  +\left(\BFKpd{y}{\BFKav \BFKtu}+\BFKpd{x}{\BFKav \BFKtv}\right)^2}
174  \end{equation}  \end{equation}
175  The coefficient {\sf viscC2Smag} is what the MITgcm user sets, and it  The coefficient {\sf viscC2Smag} is what an MITgcm user sets, and it
176  replaces the proportionality in the Kolmogorov length with an  replaces the proportionality in the Kolmogorov length with an
177  equality.  Others \cite{grha00} suggest values of {\sf viscC2Smag}  equality.  Others \cite{grha00} suggest values of {\sf viscC2Smag}
178  from 2.2 to 4 for oceanic problems.  Smagorinsky \cite{Smagorinsky93}  from 2.2 to 4 for oceanic problems.  Smagorinsky \cite{Smagorinsky93}
# Line 186  A_{vSmag}=\left(\frac{{\sf viscC2Smag}}{ Line 186  A_{vSmag}=\left(\frac{{\sf viscC2Smag}}{
186  \sqrt{\left(\BFKpd{z}{\BFKav \BFKtu}\right)^2  \sqrt{\left(\BFKpd{z}{\BFKav \BFKtu}\right)^2
187  +\left(\BFKpd{z}{\BFKav \BFKtv}\right)^2}  +\left(\BFKpd{z}{\BFKav \BFKtv}\right)^2}
188  \end{equation}  \end{equation}
189  This vertical viscosity is currently not implemented in the MITgcm  This vertical viscosity is currently not implemented in MITgcm
190  (although it may be soon).  (although it may be soon).
191    
192  \subsubsection{Leith Viscosity}  \subsubsection{Leith Viscosity}
# Line 230  exist which are only horizontally diverg Line 230  exist which are only horizontally diverg
230  vorticity.  This causes a difficulty with the Leith viscosity, which  vorticity.  This causes a difficulty with the Leith viscosity, which
231  can only responds to buildup of vorticity at the grid scale.  can only responds to buildup of vorticity at the grid scale.
232    
233  The MITgcm offers two options for dealing with this problem.  1) The  MITgcm offers two options for dealing with this problem.  1) The
234  Smagorinsky viscosity can be used instead of Leith, or in conjunction  Smagorinsky viscosity can be used instead of Leith, or in conjunction
235  with Leith--a purely divergent flow does cause an increase in  with Leith--a purely divergent flow does cause an increase in
236  Smagorinsky viscosity.  2) The {\sf viscC2LeithD} parameter can be  Smagorinsky viscosity.  2) The {\sf viscC2LeithD} parameter can be
# Line 272  A_h & < & \frac{L^2}{4\Delta t} \\ Line 272  A_h & < & \frac{L^2}{4\Delta t} \\
272  A_4 & \le & \frac{L^4}{32\Delta t}  A_4 & \le & \frac{L^4}{32\Delta t}
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 MITgcm by specifying {\sf viscAhGridMax}$<1$ and
276  {\sf viscA4GridMax}$<1$.  Similarly-scaled minimum values of  {\sf viscA4GridMax}$<1$.  Similarly-scaled minimum values of
277  viscosities are provided by {\sf viscAhGridMin} and {\sf  viscosities are provided by {\sf viscAhGridMin} and {\sf
278    viscA4GridMin}, which if used, should be set to values $\ll 1$. $L$    viscA4GridMin}, which if used, should be set to values $\ll 1$. $L$
# Line 324  the choice of coefficients.  Using bihar Line 324  the choice of coefficients.  Using bihar
324  that one zeros the first non-vanishing term in the Taylor series,  that one zeros the first non-vanishing term in the Taylor series,
325  which is unsupported by any fluid theory or observation.  which is unsupported by any fluid theory or observation.
326    
327  Nonetheless, the MITgcm supports a plethora of biharmonic viscosities  Nonetheless, MITgcm supports a plethora of biharmonic viscosities
328  and diffusivities, which are controlled with parameters named  and diffusivities, which are controlled with parameters named
329  similarly to the harmonic viscosities and diffusivities with the  similarly to the harmonic viscosities and diffusivities with the
330  substitution $h\rightarrow 4$.  The MITgcm also supports a biharmonic  substitution $h\rightarrow 4$.  MITgcm also supports a biharmonic
331  Leith and Smagorinsky viscosities:  Leith and Smagorinsky viscosities:
332  \begin{eqnarray}  \begin{eqnarray}
333  A_{4Smag} & = &  A_{4Smag} & = &
# Line 359  Thus, the biharmonic scaling suggested b Line 359  Thus, the biharmonic scaling suggested b
359  |\nabla \BFKav \omega_3| & \propto & L|\nabla^2 \BFKav \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  It is not at all clear that these assumptions ought to hold.  Only the
362  \cite{grha00} forms are currently implemented in the MITgcm.  \cite{grha00} forms are currently implemented in 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
# Line 368  some locations.  In that case, the CFL c Line 368  some locations.  In that case, the CFL c
368  minimum of $L_x$ and $L_y$ be used.  On the other hand, other  minimum of $L_x$ and $L_y$ be used.  On the other hand, other
369  viscosities which involve whether a particular wavelength is  viscosities which involve whether a particular wavelength is
370  'resolved' might be better suited to use the maximum of $L_x$ and  'resolved' might be better suited to use the maximum of $L_x$ and
371  $L_y$.  Currently the MITgcm uses {\sf useAreaViscLength} to select  $L_y$.  Currently, MITgcm uses {\sf useAreaViscLength} to select
372  between two options.  If false, the geometric mean of $L^2_x$ and  between two options.  If false, the geometric mean of $L^2_x$ and
373  $L^2_y$ is used for all viscosities, which is closer to the minimum  $L^2_y$ is used for all viscosities, which is closer to the minimum
374  and occurs naturally in the CFL constraint.  If {\sf  and occurs naturally in the CFL constraint.  If {\sf
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