/[MITgcm]/manual/s_algorithm/text/nonlin_visc.tex

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-revision 1.3 by edhill,
Fri Dec 16 06:26:09 2005 UTC
+revision 1.6 by dfer,
Mon Jun 27 02:10:49 2011 UTC
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  %  $Name$
  \section{Nonlinear Viscosities for Large Eddy Simulation}
- \label{sect:nonlin-visc}
+ \label{sec:nonlin-visc}
  In Large Eddy Simulations (LES), a turbulent closure needs to be
  provided that accounts for the effects of subgridscale motions on the
 Line 80 
 then the viscosity should be chosen to m
  as the advective ones.  Bryan \textit{et al} \cite{Bryanetal75} notes
  that a computational mode is squelched by using $\BFKRe_h<$2.
- The MITgcm user can select an horizontal eddy viscosity based on
+ MITgcm users can select horizontal eddy viscosities based on
- $\BFKRe_h$ by two methods.  1) The user may estimate the velocity
+ $\BFKRe_h$ using two methods.  1) The user may estimate the velocity
  scale expected from the calculation and grid spacing and set the {\sf
    viscAh} to satisfy $\BFKRe_h<2$.  2) The user may use {\sf
    viscAhReMax}, which ensures that the viscosity is always chosen so
 Line 137 
 gridscale, $L_\epsilon(A_h)\propto\pi\ep
  Large Eddy Simulation or LES).
  There are two methods of ensuring that the Kolmogorov length is
- 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
  through spectral space for a given simulation and adjust grid spacing
  or {\sf viscAh} to ensure that $L_\epsilon(A_h)>L$.  2) The user may
  use the approach of Smagorinsky with {\sf viscC2Smag}, which estimates
 Line 172 
 shallow-water scaling is
  |\BFKav D|=\sqrt{\left(\BFKpd{x}{\BFKav \BFKtu}-\BFKpd{y}{\BFKav \BFKtv}\right)^2
  +\left(\BFKpd{y}{\BFKav \BFKtu}+\BFKpd{x}{\BFKav \BFKtv}\right)^2}
  \end{equation}
- The coefficient {\sf viscC2Smag} is what the MITgcm user sets, and it
+ The coefficient {\sf viscC2Smag} is what an MITgcm user sets, and it
  replaces the proportionality in the Kolmogorov length with an
- equality.  Others \cite{grha00} suggest values of {\sf viscC2Smag}
+ equality.  Others \cite{griffies:00} suggest values of {\sf viscC2Smag}
  from 2.2 to 4 for oceanic problems.  Smagorinsky \cite{Smagorinsky93}
  shows that values from 0.2 to 0.9 have been used in atmospheric
  modeling.
 Line 186 
 A_{vSmag}=\left(\frac{{\sf viscC2Smag}}{
  \sqrt{\left(\BFKpd{z}{\BFKav \BFKtu}\right)^2
  +\left(\BFKpd{z}{\BFKav \BFKtv}\right)^2}
  \end{equation}
- This vertical viscosity is currently not implemented in the MITgcm
+ This vertical viscosity is currently not implemented in MITgcm
  (although it may be soon).
  \subsubsection{Leith Viscosity}
 Line 230 
 exist which are only horizontally diverg
  vorticity.  This causes a difficulty with the Leith viscosity, which
  can only responds to buildup of vorticity at the grid scale.
- The MITgcm offers two options for dealing with this problem.  1) The
+ MITgcm offers two options for dealing with this problem.  1) The
  Smagorinsky viscosity can be used instead of Leith, or in conjunction
  with Leith--a purely divergent flow does cause an increase in
  Smagorinsky viscosity.  2) The {\sf viscC2LeithD} parameter can be
 Line 272 
 A_h & < & \frac{L^2}{4\Delta t} \\
  A_4 & \le & \frac{L^4}{32\Delta t}
  \end{eqnarray}
  The viscosities may be automatically limited to be no greater than
- these values in the MITgcm by specifying {\sf viscAhGridMax}$<1$ and
+ these values in MITgcm by specifying {\sf viscAhGridMax}$<1$ and
  {\sf viscA4GridMax}$<1$.  Similarly-scaled minimum values of
  viscosities are provided by {\sf viscAhGridMin} and {\sf
    viscA4GridMin}, which if used, should be set to values $\ll 1$. $L$
  is roughly the gridscale (see below).
- Following \cite{grha00}, we note that there is a factor of $\Delta
+ Following \cite{griffies:00}, we note that there is a factor of $\Delta
  x^2/8$ difference between the harmonic and biharmonic viscosities.
  Thus, whenever a non-dimensional harmonic coefficient is used in the
  MITgcm (\textit{eg.} {\sf viscAhGridMax}$<1$), the biharmonic equivalent is
 Line 305 
 viscosity:
  \frac{-\nabla^4_h \BFKav b}{\Pr\BFKRe_4}
  +\frac{\BFKpds{z} {\BFKav b}}{\Pr\BFKRe_v}\nonumber
  \end{eqnarray}
- \cite{grha00} propose that if one scales the biharmonic viscosity by
+ \cite{griffies:00} propose that if one scales the biharmonic viscosity by
  stability considerations, then the biharmonic viscous terms will be
  similarly active to harmonic viscous terms at the gridscale of the
  model, but much less active on larger scale motions.  Similarly, a
 Line 324 
 the choice of coefficients.  Using bihar
  that one zeros the first non-vanishing term in the Taylor series,
  which is unsupported by any fluid theory or observation.
- Nonetheless, the MITgcm supports a plethora of biharmonic viscosities
+ Nonetheless, MITgcm supports a plethora of biharmonic viscosities
  and diffusivities, which are controlled with parameters named
  similarly to the harmonic viscosities and diffusivities with the
- substitution $h\rightarrow 4$.  The MITgcm also supports a biharmonic
+ substitution $h\rightarrow 4$.  MITgcm also supports a biharmonic
  Leith and Smagorinsky viscosities:
  \begin{eqnarray}
  A_{4Smag} & = &
 Line 353 
 L^6\sqrt{\left(\frac{{\sf viscC4Leith}}{
    +\left(\frac{{\sf viscC4LeithD}}{\pi}\right)^{12}
    |\nabla^2 \nabla\cdot \BFKav {\bf \BFKtu}_h|^2}
  \end{eqnarray}
- Thus, the biharmonic scaling suggested by \cite{grha00} implies:
+ Thus, the biharmonic scaling suggested by \cite{griffies:00} implies:
  \begin{eqnarray}
  |D| & \propto &  L|\nabla^2\BFKav {\bf \BFKtu}_h|\\
  |\nabla \BFKav \omega_3| & \propto & L|\nabla^2 \BFKav \omega_3|
  \end{eqnarray}
  It is not at all clear that these assumptions ought to hold.  Only the
- \cite{grha00} forms are currently implemented in the MITgcm.
+ \cite{griffies:00} forms are currently implemented in MITgcm.
  \subsubsection{Selection of Length Scale}
  Above, the length scale of the grid has been denoted $L$.  However, in
 Line 368 
 some locations.  In that case, the CFL c
  minimum of $L_x$ and $L_y$ be used.  On the other hand, other
  viscosities which involve whether a particular wavelength is
  'resolved' might be better suited to use the maximum of $L_x$ and
- $L_y$.  Currently the MITgcm uses {\sf useAreaViscLength} to select
+ $L_y$.  Currently, MITgcm uses {\sf useAreaViscLength} to select
  between two options.  If false, the geometric mean of $L^2_x$ and
  $L^2_y$ is used for all viscosities, which is closer to the minimum
  and occurs naturally in the CFL constraint.  If {\sf

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