/[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 1.5 by jmc,
Mon Aug 30 23:09:18 2010 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}
  from 2.2 to 4 for oceanic problems.  Smagorinsky \cite{Smagorinsky93}
 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 $\ll 1$. $L$
 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 359 
 Thus, the biharmonic scaling suggested b
  |\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{grha00} 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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