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revision 1.1 by adcroft, Thu Aug 9 19:48:39 2001 UTC revision 1.4 by edhill, Sat Oct 16 03:40:12 2004 UTC
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2  % $Name$  % $Name$
3    
4  \section{Vector invariant momentum equations}  \section{Vector invariant momentum equations}
5    \begin{rawhtml}
6    <!-- CMIREDIR:vector_invariant_momentum_eqautions: -->
7    \end{rawhtml}
8    
9  The finite volume method lends itself to describing the continuity and  The finite volume method lends itself to describing the continuity and
10  tracer equations in curvilinear coordinate systems but the appearance  tracer equations in curvilinear coordinate systems. However, in
11  of new metric terms in the flux-form momentum equations makes  curvilinear coordinates many new metric terms appear in the momentum
12  generalizing them far from elegant. The vector invariant form of the  equations (written in Lagrangian or flux-form) making generalization
13  momentum equations are exactly that; invariant under coordinate  far from elegant. Fortunately, an alternative form of the equations,
14  transformations.  the vector invariant equations are exactly that; invariant under
15    coordinate transformations so that they can be applied uniformly in
16    any orthogonal curvilinear coordinate system such as spherical
17    coordinates, boundary following or the conformal spherical cube
18    system.
19    
20  The non-hydrostatic vector invariant equations read:  The non-hydrostatic vector invariant equations read:
21  \begin{equation}  \begin{equation}
# Line 53  $G_w$: {\bf Gw} ({\em DYNVARS.h}) Line 60  $G_w$: {\bf Gw} ({\em DYNVARS.h})
60    
61  The vertical component of relative vorticity is explicitly calculated  The vertical component of relative vorticity is explicitly calculated
62  and use in the discretization. The particular form is crucial for  and use in the discretization. The particular form is crucial for
63  numerical stablility; alternative definitions break the conservation  numerical stability; alternative definitions break the conservation
64  properties of the discrete equations.  properties of the discrete equations.
65    
66  Relative vorticity is defined:  Relative vorticity is defined:
# Line 128  G_v^{fu} + G_v^{\zeta_3 u} & = & - Line 135  G_v^{fu} + G_v^{\zeta_3 u} & = & -
135  \end{eqnarray}  \end{eqnarray}
136    
137  \marginpar{Run-time control needs to be added for these options} The  \marginpar{Run-time control needs to be added for these options} The
138  disctinction between using absolute vorticity or relative vorticity is  distinction between using absolute vorticity or relative vorticity is
139  useful when constructing higher order advection schemes; monotone  useful when constructing higher order advection schemes; monotone
140  advection of relative vorticity behaves differently to monotone  advection of relative vorticity behaves differently to monotone
141  advection of absolute vorticity. Currently the choice of  advection of absolute vorticity. Currently the choice of
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