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revision 1.4 by cnh, Thu Oct 25 00:55:28 2001 UTC revision 1.9 by jmc, Tue Aug 2 20:54:44 2005 UTC
# Line 2  Line 2 
2  % $Name$  % $Name$
3    
4  \section{Flux-form momentum equations}  \section{Flux-form momentum equations}
5    \label{sect:flux-form_momentum_equations}
6    \begin{rawhtml}
7    <!-- CMIREDIR:flux-form_momentum_eqautions: -->
8    \end{rawhtml}
9    
10  The original finite volume model was based on the Eulerian flux form  The original finite volume model was based on the Eulerian flux form
11  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 197  $G_u^{metric}$, $G_v^{metric}$: {\bf mT}
197    
198  For the non-hydrostatic equations, dropping the thin-atmosphere  For the non-hydrostatic equations, dropping the thin-atmosphere
199  approximation re-introduces metric terms involving $w$ and are  approximation re-introduces metric terms involving $w$ and are
200  required to conserve anglular momentum:  required to conserve angular momentum:
201  \begin{eqnarray}  \begin{eqnarray}
202  {\cal A}_w \Delta r_f h_w G_u^{metric} & = &  {\cal A}_w \Delta r_f h_w G_u^{metric} & = &
203  - \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 261  represent the an-isotropic cosine scalin
261  ``lat-lon'' grid for Laplacian viscosity.  ``lat-lon'' grid for Laplacian viscosity.
262  \marginpar{Need to tidy up method for controlling this in code}  \marginpar{Need to tidy up method for controlling this in code}
263    
264  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
265  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
266  make it very unlikely that a stable viscosity parameter exists across  make it very unlikely that a stable viscosity parameter exists across
267  the entire model domain.  the entire model domain.
# Line 290  handled using the lopped cells. Line 294  handled using the lopped cells.
294  The no-slip condition defines the normal gradient of a tangential flow  The no-slip condition defines the normal gradient of a tangential flow
295  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
296  stresses by using complicated functions of the masks and ``ghost''  stresses by using complicated functions of the masks and ``ghost''
297  points (see \cite{Adcroft+Marshall98}) we add the boundary stresses as  points (see \cite{adcroft:98}) we add the boundary stresses as
298  an additional source term in cells next to solid boundaries. This has  an additional source term in cells next to solid boundaries. This has
299  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
300  the interior stress calculation (code) independent of the boundary  the interior stress calculation (code) independent of the boundary
# Line 307  G_v^{side-drag} & = & Line 311  G_v^{side-drag} & = &
311    
312  In fact, the above discretization is not quite complete because it  In fact, the above discretization is not quite complete because it
313  assumes that the bathymetry at velocity points is deeper than at  assumes that the bathymetry at velocity points is deeper than at
314  neighbouring vorticity points, e.g. $1-h_w < 1-h_\zeta$  neighboring vorticity points, e.g. $1-h_w < 1-h_\zeta$
315    
316  \fbox{ \begin{minipage}{4.75in}  \fbox{ \begin{minipage}{4.75in}
317  {\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 328  Vertical viscosity terms are discretized
328  to the variable grid lengths introduced by the finite volume  to the variable grid lengths introduced by the finite volume
329  formulation. This reduces the formal accuracy of these terms to just  formulation. This reduces the formal accuracy of these terms to just
330  first order but only next to boundaries; exactly where other terms  first order but only next to boundaries; exactly where other terms
331  appear such as linar and quadratic bottom drag.  appear such as linear and quadratic bottom drag.
332  \begin{eqnarray}  \begin{eqnarray}
333  G_u^{v-diss} & = &  G_u^{v-diss} & = &
334  \frac{1}{\Delta r_f h_w} \delta_k \tau_{13} \\  \frac{1}{\Delta r_f h_w} \delta_k \tau_{13} \\
# Line 342  In the interior the vertical stresses ar Line 346  In the interior the vertical stresses ar
346  \tau_{33} & = & A_v \frac{1}{\Delta r_f} \delta_k w  \tau_{33} & = & A_v \frac{1}{\Delta r_f} \delta_k w
347  \end{eqnarray}  \end{eqnarray}
348  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
349  is even less consistent than for the hydrostatic (see Wazjowicz  is even less consistent than for the hydrostatic (see
350  \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
351  \cite{Griffies}) and is on the list of to-do's.  \cite{griffies:00}) and is on the list of to-do's.
352    
353  \fbox{ \begin{minipage}{4.75in}  \fbox{ \begin{minipage}{4.75in}
354  {\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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