/[MITgcm]/manual/s_algorithm/text/mom_fluxform.tex
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revision 1.4 by cnh, Thu Oct 25 00:55:28 2001 UTC revision 1.6 by cnh, Thu Oct 25 18:36:53 2001 UTC
# Line 2  Line 2 
2  % $Name$  % $Name$
3    
4  \section{Flux-form momentum equations}  \section{Flux-form momentum equations}
5    \label{sec:flux-form_momentum_eqautions}
6    
7  The original finite volume model was based on the Eulerian flux form  The original finite volume model was based on the Eulerian flux form
8  momentum equations. This is the default though the vector invariant  momentum equations. This is the default though the vector invariant
# Line 193  $G_u^{metric}$, $G_v^{metric}$: {\bf mT} Line 194  $G_u^{metric}$, $G_v^{metric}$: {\bf mT}
194    
195  For the non-hydrostatic equations, dropping the thin-atmosphere  For the non-hydrostatic equations, dropping the thin-atmosphere
196  approximation re-introduces metric terms involving $w$ and are  approximation re-introduces metric terms involving $w$ and are
197  required to conserve anglular momentum:  required to conserve angular momentum:
198  \begin{eqnarray}  \begin{eqnarray}
199  {\cal A}_w \Delta r_f h_w G_u^{metric} & = &  {\cal A}_w \Delta r_f h_w G_u^{metric} & = &
200  - \overline{ \frac{ \overline{u}^i \overline{w}^k }{a} {\cal A}_c \Delta r_f h_c }^i \\  - \overline{ \frac{ \overline{u}^i \overline{w}^k }{a} {\cal A}_c \Delta r_f h_c }^i \\
# Line 257  represent the an-isotropic cosine scalin Line 258  represent the an-isotropic cosine scalin
258  ``lat-lon'' grid for Laplacian viscosity.  ``lat-lon'' grid for Laplacian viscosity.
259  \marginpar{Need to tidy up method for controlling this in code}  \marginpar{Need to tidy up method for controlling this in code}
260    
261  It should be noted that dispite the ad-hoc nature of the scaling, some  It should be noted that despite the ad-hoc nature of the scaling, some
262  scaling must be done since on a lat-lon grid the converging meridians  scaling must be done since on a lat-lon grid the converging meridians
263  make it very unlikely that a stable viscosity parameter exists across  make it very unlikely that a stable viscosity parameter exists across
264  the entire model domain.  the entire model domain.
# Line 307  G_v^{side-drag} & = & Line 308  G_v^{side-drag} & = &
308    
309  In fact, the above discretization is not quite complete because it  In fact, the above discretization is not quite complete because it
310  assumes that the bathymetry at velocity points is deeper than at  assumes that the bathymetry at velocity points is deeper than at
311  neighbouring vorticity points, e.g. $1-h_w < 1-h_\zeta$  neighboring vorticity points, e.g. $1-h_w < 1-h_\zeta$
312    
313  \fbox{ \begin{minipage}{4.75in}  \fbox{ \begin{minipage}{4.75in}
314  {\em S/R MOM\_U\_SIDEDRAG} ({\em mom\_u\_sidedrag.F})  {\em S/R MOM\_U\_SIDEDRAG} ({\em mom\_u\_sidedrag.F})
# Line 324  Vertical viscosity terms are discretized Line 325  Vertical viscosity terms are discretized
325  to the variable grid lengths introduced by the finite volume  to the variable grid lengths introduced by the finite volume
326  formulation. This reduces the formal accuracy of these terms to just  formulation. This reduces the formal accuracy of these terms to just
327  first order but only next to boundaries; exactly where other terms  first order but only next to boundaries; exactly where other terms
328  appear such as linar and quadratic bottom drag.  appear such as linear and quadratic bottom drag.
329  \begin{eqnarray}  \begin{eqnarray}
330  G_u^{v-diss} & = &  G_u^{v-diss} & = &
331  \frac{1}{\Delta r_f h_w} \delta_k \tau_{13} \\  \frac{1}{\Delta r_f h_w} \delta_k \tau_{13} \\
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