/[MITgcm]/manual/s_examples/global_oce_latlon/climatalogical_ogcm.tex
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revision 1.3 by cnh, Thu Oct 25 18:36:55 2001 UTC revision 1.6 by adcroft, Tue Nov 13 19:01:42 2001 UTC
# Line 148  $ Line 148  $
148   \Delta z_{20}=815\,{\rm m}   \Delta z_{20}=815\,{\rm m}
149  $ (here the numeric subscript indicates the model level index number, ${\tt k}$).  $ (here the numeric subscript indicates the model level index number, ${\tt k}$).
150  The implicit free surface form of the pressure equation described in Marshall et. al  The implicit free surface form of the pressure equation described in Marshall et. al
151  \cite{Marshall97a} is employed. A Laplacian operator, $\nabla^2$, provides viscous  \cite{marshall:97a} is employed. A Laplacian operator, $\nabla^2$, provides viscous
152  dissipation. Thermal and haline diffusion is also represented by a Laplacian operator.  dissipation. Thermal and haline diffusion is also represented by a Laplacian operator.
153    
154  Wind-stress forcing is added to the momentum equations for both  Wind-stress forcing is added to the momentum equations for both
# Line 210  g\rho_{0} \eta + \int^{0}_{-z}\rho^{'} d Line 210  g\rho_{0} \eta + \int^{0}_{-z}\rho^{'} d
210  $v=\frac{Dy}{Dt}=r \frac{D \phi}{Dt}$  $v=\frac{Dy}{Dt}=r \frac{D \phi}{Dt}$
211  are the zonal and meridional components of the  are the zonal and meridional components of the
212  flow vector, $\vec{u}$, on the sphere. As described in  flow vector, $\vec{u}$, on the sphere. As described in
213  MITgcm Numerical Solution Procedure \cite{MITgcm_Numerical_Scheme}, the time  MITgcm Numerical Solution Procedure \ref{chap:discretization}, the time
214  evolution of potential temperature, $\theta$, equation is solved prognostically.  evolution of potential temperature, $\theta$, equation is solved prognostically.
215  The total pressure, $p$, is diagnosed by summing pressure due to surface  The total pressure, $p$, is diagnosed by summing pressure due to surface
216  elevation $\eta$ and the hydrostatic pressure.  elevation $\eta$ and the hydrostatic pressure.
# Line 219  elevation $\eta$ and the hydrostatic pre Line 219  elevation $\eta$ and the hydrostatic pre
219  \subsubsection{Numerical Stability Criteria}  \subsubsection{Numerical Stability Criteria}
220    
221  The Laplacian dissipation coefficient, $A_{h}$, is set to $5 \times 10^5 m s^{-1}$.  The Laplacian dissipation coefficient, $A_{h}$, is set to $5 \times 10^5 m s^{-1}$.
222  This value is chosen to yield a Munk layer width \cite{Adcroft_thesis},  This value is chosen to yield a Munk layer width \cite{adcroft:95},
223  \begin{eqnarray}  \begin{eqnarray}
224  \label{EQ:munk_layer}  \label{EQ:munk_layer}
225  M_{w} = \pi ( \frac { A_{h} }{ \beta } )^{\frac{1}{3}}  M_{w} = \pi ( \frac { A_{h} }{ \beta } )^{\frac{1}{3}}
# Line 233  boundary layer is adequately resolved. Line 233  boundary layer is adequately resolved.
233  \noindent The model is stepped forward with a  \noindent The model is stepped forward with a
234  time step $\delta t_{\theta}=30~{\rm hours}$ for thermodynamic variables and  time step $\delta t_{\theta}=30~{\rm hours}$ for thermodynamic variables and
235  $\delta t_{v}=40~{\rm minutes}$ for momentum terms. With this time step, the stability  $\delta t_{v}=40~{\rm minutes}$ for momentum terms. With this time step, the stability
236  parameter to the horizontal Laplacian friction \cite{Adcroft_thesis}  parameter to the horizontal Laplacian friction \cite{adcroft:95}
237  \begin{eqnarray}  \begin{eqnarray}
238  \label{EQ:laplacian_stability}  \label{EQ:laplacian_stability}
239  S_{l} = 4 \frac{A_{h} \delta t_{v}}{{\Delta x}^2}  S_{l} = 4 \frac{A_{h} \delta t_{v}}{{\Delta x}^2}
# Line 276  of $S_{l} \approx 0.5$. Line 276  of $S_{l} \approx 0.5$.
276  \\  \\
277    
278  \noindent The numerical stability for inertial oscillations  \noindent The numerical stability for inertial oscillations
279  \cite{Adcroft_thesis}  \cite{adcroft:95}
280    
281  \begin{eqnarray}  \begin{eqnarray}
282  \label{EQ:inertial_stability}  \label{EQ:inertial_stability}
# Line 287  S_{i} = f^{2} {\delta t_v}^2 Line 287  S_{i} = f^{2} {\delta t_v}^2
287  the $S_{i} < 1$ upper limit for stability.  the $S_{i} < 1$ upper limit for stability.
288  \\  \\
289    
290  \noindent The advective CFL \cite{Adcroft_thesis} for a extreme maximum  \noindent The advective CFL \cite{adcroft:95} for a extreme maximum
291  horizontal flow  horizontal flow
292  speed of $ | \vec{u} | = 2 ms^{-1}$  speed of $ | \vec{u} | = 2 ms^{-1}$
293    
# Line 302  limit of 0.5. Line 302  limit of 0.5.
302    
303  \noindent The stability parameter for internal gravity waves propagating  \noindent The stability parameter for internal gravity waves propagating
304  with a maximum speed of $c_{g}=10~{\rm ms}^{-1}$  with a maximum speed of $c_{g}=10~{\rm ms}^{-1}$
305  \cite{Adcroft_thesis}  \cite{adcroft:95}
306    
307  \begin{eqnarray}  \begin{eqnarray}
308  \label{EQ:cfl_stability}  \label{EQ:cfl_stability}
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