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revision 1.1 by adcroft, Thu Aug 9 19:48:39 2001 UTC revision 1.6 by jmc, Mon Jun 26 01:03:47 2006 UTC
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2  % $Name$  % $Name$
3    
4  \section{Vector invariant momentum equations}  \section{Vector invariant momentum equations}
5    \label{sect:vect-inv_momentum_equations}
6    \begin{rawhtml}
7    <!-- CMIREDIR:vector_invariant_momentum_eqautions: -->
8    \end{rawhtml}
9    
10  The finite volume method lends itself to describing the continuity and  The finite volume method lends itself to describing the continuity and
11  tracer equations in curvilinear coordinate systems but the appearance  tracer equations in curvilinear coordinate systems. However, in
12  of new metric terms in the flux-form momentum equations makes  curvilinear coordinates many new metric terms appear in the momentum
13  generalizing them far from elegant. The vector invariant form of the  equations (written in Lagrangian or flux-form) making generalization
14  momentum equations are exactly that; invariant under coordinate  far from elegant. Fortunately, an alternative form of the equations,
15  transformations.  the vector invariant equations are exactly that; invariant under
16    coordinate transformations so that they can be applied uniformly in
17    any orthogonal curvilinear coordinate system such as spherical
18    coordinates, boundary following or the conformal spherical cube
19    system.
20    
21  The non-hydrostatic vector invariant equations read:  The non-hydrostatic vector invariant equations read:
22  \begin{equation}  \begin{equation}
# Line 40  G_w & = & G_w^{fu} + G_w^{\zeta_1 v} + G Line 48  G_w & = & G_w^{fu} + G_w^{\zeta_1 v} + G
48  \end{eqnarray}  \end{eqnarray}
49    
50  \fbox{ \begin{minipage}{4.75in}  \fbox{ \begin{minipage}{4.75in}
51  {\em S/R CALC\_MOM\_RHS} ({\em pkg/mom\_vecinv/calc\_mom\_rhs.F})  {\em S/R MOM\_VECINV} ({\em pkg/mom\_vecinv/mom\_vecinv.F})
52    
53  $G_u$: {\bf Gu} ({\em DYNVARS.h})  $G_u$: {\bf Gu} ({\em DYNVARS.h})
54    
# Line 53  $G_w$: {\bf Gw} ({\em DYNVARS.h}) Line 61  $G_w$: {\bf Gw} ({\em DYNVARS.h})
61    
62  The vertical component of relative vorticity is explicitly calculated  The vertical component of relative vorticity is explicitly calculated
63  and use in the discretization. The particular form is crucial for  and use in the discretization. The particular form is crucial for
64  numerical stablility; alternative definitions break the conservation  numerical stability; alternative definitions break the conservation
65  properties of the discrete equations.  properties of the discrete equations.
66    
67  Relative vorticity is defined:  Relative vorticity is defined:
# Line 67  the vertical and $\Gamma$ is the circula Line 75  the vertical and $\Gamma$ is the circula
75  \fbox{ \begin{minipage}{4.75in}  \fbox{ \begin{minipage}{4.75in}
76  {\em S/R MOM\_VI\_CALC\_RELVORT3} ({\em mom\_vi\_calc\_relvort3.F})  {\em S/R MOM\_VI\_CALC\_RELVORT3} ({\em mom\_vi\_calc\_relvort3.F})
77    
78  $\zeta_3$: {\bf vort3} (local to {\em calc\_mom\_rhs.F})  $\zeta_3$: {\bf vort3} (local to {\em mom\_vecinv.F})
79  \end{minipage} }  \end{minipage} }
80    
81    
# Line 82  KE = \frac{1}{2} ( \overline{ u^2 }^i + Line 90  KE = \frac{1}{2} ( \overline{ u^2 }^i +
90  \fbox{ \begin{minipage}{4.75in}  \fbox{ \begin{minipage}{4.75in}
91  {\em S/R MOM\_VI\_CALC\_KE} ({\em mom\_vi\_calc\_ke.F})  {\em S/R MOM\_VI\_CALC\_KE} ({\em mom\_vi\_calc\_ke.F})
92    
93  $KE$: {\bf KE} (local to {\em calc\_mom\_rhs.F})  $KE$: {\bf KE} (local to {\em mom\_vecinv.F})
94  \end{minipage} }  \end{minipage} }
95    
96    
# Line 128  G_v^{fu} + G_v^{\zeta_3 u} & = & - Line 136  G_v^{fu} + G_v^{\zeta_3 u} & = & -
136  \end{eqnarray}  \end{eqnarray}
137    
138  \marginpar{Run-time control needs to be added for these options} The  \marginpar{Run-time control needs to be added for these options} The
139  disctinction between using absolute vorticity or relative vorticity is  distinction between using absolute vorticity or relative vorticity is
140  useful when constructing higher order advection schemes; monotone  useful when constructing higher order advection schemes; monotone
141  advection of relative vorticity behaves differently to monotone  advection of relative vorticity behaves differently to monotone
142  advection of absolute vorticity. Currently the choice of  advection of absolute vorticity. Currently the choice of
# Line 142  available only through commented subrout Line 150  available only through commented subrout
150    
151  {\em S/R MOM\_VI\_V\_CORIOLIS} ({\em mom\_vi\_v\_coriolis.F})  {\em S/R MOM\_VI\_V\_CORIOLIS} ({\em mom\_vi\_v\_coriolis.F})
152    
153  $G_u^{fv}$, $G_u^{\zeta_3 v}$: {\bf uCf} (local to {\em calc\_mom\_rhs.F})  $G_u^{fv}$, $G_u^{\zeta_3 v}$: {\bf uCf} (local to {\em mom\_vecinv.F})
154    
155  $G_v^{fu}$, $G_v^{\zeta_3 u}$: {\bf vCf} (local to {\em calc\_mom\_rhs.F})  $G_v^{fu}$, $G_v^{\zeta_3 u}$: {\bf vCf} (local to {\em mom\_vecinv.F})
156  \end{minipage} }  \end{minipage} }
157    
158    
# Line 171  G_v^{\zeta_1 w} & = & Line 179  G_v^{\zeta_1 w} & = &
179    
180  {\em S/R MOM\_VI\_V\_VERTSHEAR} ({\em mom\_vi\_v\_vertshear.F})  {\em S/R MOM\_VI\_V\_VERTSHEAR} ({\em mom\_vi\_v\_vertshear.F})
181    
182  $G_u^{\zeta_2 w}$: {\bf uCf} (local to {\em calc\_mom\_rhs.F})  $G_u^{\zeta_2 w}$: {\bf uCf} (local to {\em mom\_vecinv.F})
183    
184  $G_v^{\zeta_1 w}$: {\bf vCf} (local to {\em calc\_mom\_rhs.F})  $G_v^{\zeta_1 w}$: {\bf vCf} (local to {\em mom\_vecinv.F})
185  \end{minipage} }  \end{minipage} }
186    
187    
# Line 194  G_v^{\partial_y B} & = & Line 202  G_v^{\partial_y B} & = &
202    
203  {\em S/R MOM\_VI\_V\_GRAD\_KE} ({\em mom\_vi\_v\_grad\_ke.F})  {\em S/R MOM\_VI\_V\_GRAD\_KE} ({\em mom\_vi\_v\_grad\_ke.F})
204    
205  $G_u^{\partial_x KE}$: {\bf uCf} (local to {\em calc\_mom\_rhs.F})  $G_u^{\partial_x KE}$: {\bf uCf} (local to {\em mom\_vecinv.F})
206    
207  $G_v^{\partial_y KE}$: {\bf vCf} (local to {\em calc\_mom\_rhs.F})  $G_v^{\partial_y KE}$: {\bf vCf} (local to {\em mom\_vecinv.F})
208  \end{minipage} }  \end{minipage} }
209    
210    
211    
212  \subsection{Horizontal dissipation}  \subsection{Horizontal divergence}
213    
214  The horizontal divergence, a complimentary quantity to relative  The horizontal divergence, a complimentary quantity to relative
215  vorticity, is used in parameterizing the Reynolds stresses and is  vorticity, is used in parameterizing the Reynolds stresses and is
# Line 215  D = \frac{1}{{\cal A}_c h_c} ( Line 223  D = \frac{1}{{\cal A}_c h_c} (
223  \fbox{ \begin{minipage}{4.75in}  \fbox{ \begin{minipage}{4.75in}
224  {\em S/R MOM\_VI\_CALC\_HDIV} ({\em mom\_vi\_calc\_hdiv.F})  {\em S/R MOM\_VI\_CALC\_HDIV} ({\em mom\_vi\_calc\_hdiv.F})
225    
226  $D$: {\bf hDiv} (local to {\em calc\_mom\_rhs.F})  $D$: {\bf hDiv} (local to {\em mom\_vecinv.F})
227  \end{minipage} }  \end{minipage} }
228    
229    
# Line 246  D^* & = & \frac{1}{{\cal A}_c h_c} ( Line 254  D^* & = & \frac{1}{{\cal A}_c h_c} (
254  \fbox{ \begin{minipage}{4.75in}  \fbox{ \begin{minipage}{4.75in}
255  {\em S/R MOM\_VI\_HDISSIP} ({\em mom\_vi\_hdissip.F})  {\em S/R MOM\_VI\_HDISSIP} ({\em mom\_vi\_hdissip.F})
256    
257  $G_u^{h-dissip}$: {\bf uDiss} (local to {\em calc\_mom\_rhs.F})  $G_u^{h-dissip}$: {\bf uDiss} (local to {\em mom\_vecinv.F})
258    
259  $G_v^{h-dissip}$: {\bf vDiss} (local to {\em calc\_mom\_rhs.F})  $G_v^{h-dissip}$: {\bf vDiss} (local to {\em mom\_vecinv.F})
260  \end{minipage} }  \end{minipage} }
261    
262    
# Line 274  In the interior the vertical stresses ar Line 282  In the interior the vertical stresses ar
282    
283  {\em S/R MOM\_V\_RVISCLFUX} ({\em mom\_v\_rviscflux.F})  {\em S/R MOM\_V\_RVISCLFUX} ({\em mom\_v\_rviscflux.F})
284    
285  $\tau_{13}$: {\bf urf} (local to {\em calc\_mom\_rhs.F})  $\tau_{13}$: {\bf urf} (local to {\em mom\_vecinv.F})
286    
287  $\tau_{23}$: {\bf vrf} (local to {\em calc\_mom\_rhs.F})  $\tau_{23}$: {\bf vrf} (local to {\em mom\_vecinv.F})
288  \end{minipage} }  \end{minipage} }
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