/[MITgcm]/manual/s_algorithm/text/mom_fluxform.tex
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revision 1.5 by cnh, Thu Oct 25 12:06:56 2001 UTC revision 1.7 by adcroft, Tue Nov 13 18:15:26 2001 UTC
# Line 194  $G_u^{metric}$, $G_v^{metric}$: {\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 258  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 291  handled using the lopped cells. Line 291  handled using the lopped cells.
291  The no-slip condition defines the normal gradient of a tangential flow  The no-slip condition defines the normal gradient of a tangential flow
292  such that the flow is zero on the boundary. Rather than modify the  such that the flow is zero on the boundary. Rather than modify the
293  stresses by using complicated functions of the masks and ``ghost''  stresses by using complicated functions of the masks and ``ghost''
294  points (see \cite{Adcroft+Marshall98}) we add the boundary stresses as  points (see \cite{adcroft:98}) we add the boundary stresses as
295  an additional source term in cells next to solid boundaries. This has  an additional source term in cells next to solid boundaries. This has
296  the advantage of being able to cope with ``thin walls'' and also makes  the advantage of being able to cope with ``thin walls'' and also makes
297  the interior stress calculation (code) independent of the boundary  the interior stress calculation (code) independent of the boundary
# Line 308  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 325  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} \\
# Line 343  In the interior the vertical stresses ar Line 343  In the interior the vertical stresses ar
343  \tau_{33} & = & A_v \frac{1}{\Delta r_f} \delta_k w  \tau_{33} & = & A_v \frac{1}{\Delta r_f} \delta_k w
344  \end{eqnarray}  \end{eqnarray}
345  It should be noted that in the non-hydrostatic form, the stress tensor  It should be noted that in the non-hydrostatic form, the stress tensor
346  is even less consistent than for the hydrostatic (see Wazjowicz  is even less consistent than for the hydrostatic (see
347  \cite{Waojz}). It is well known how to do this properly (see Griffies  \cite{wajsowicz:93}). It is well known how to do this properly (see
348  \cite{Griffies}) and is on the list of to-do's.  \cite{griffies:00}) and is on the list of to-do's.
349    
350  \fbox{ \begin{minipage}{4.75in}  \fbox{ \begin{minipage}{4.75in}
351  {\em S/R MOM\_U\_RVISCLFUX} ({\em mom\_u\_rviscflux.F})  {\em S/R MOM\_U\_RVISCLFUX} ({\em mom\_u\_rviscflux.F})
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