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revision 1.1 by adcroft, Thu Aug 9 19:48:39 2001 UTC revision 1.3 by adcroft, Tue Nov 13 15:20:12 2001 UTC
# Line 4  Line 4 
4  \section{Vector invariant momentum equations}  \section{Vector invariant momentum equations}
5    
6  The finite volume method lends itself to describing the continuity and  The finite volume method lends itself to describing the continuity and
7  tracer equations in curvilinear coordinate systems but the appearance  tracer equations in curvilinear coordinate systems. However, in
8  of new metric terms in the flux-form momentum equations makes  curvilinear coordinates many new metric terms appear in the momentum
9  generalizing them far from elegant. The vector invariant form of the  equations (written in Lagrangian or flux-form) making generalization
10  momentum equations are exactly that; invariant under coordinate  far from elegant. Fortunately, an alternative form of the equations,
11  transformations.  the vector invariant equations are exactly that; invariant under
12    coordinate transformations so that they can be applied uniformly in
13    any orthogonal curvilinear coordinate system such as spherical
14    coordinates, boundary following or the conformal spherical cube
15    system.
16    
17  The non-hydrostatic vector invariant equations read:  The non-hydrostatic vector invariant equations read:
18  \begin{equation}  \begin{equation}
# Line 53  $G_w$: {\bf Gw} ({\em DYNVARS.h}) Line 57  $G_w$: {\bf Gw} ({\em DYNVARS.h})
57    
58  The vertical component of relative vorticity is explicitly calculated  The vertical component of relative vorticity is explicitly calculated
59  and use in the discretization. The particular form is crucial for  and use in the discretization. The particular form is crucial for
60  numerical stablility; alternative definitions break the conservation  numerical stability; alternative definitions break the conservation
61  properties of the discrete equations.  properties of the discrete equations.
62    
63  Relative vorticity is defined:  Relative vorticity is defined:
# Line 128  G_v^{fu} + G_v^{\zeta_3 u} & = & - Line 132  G_v^{fu} + G_v^{\zeta_3 u} & = & -
132  \end{eqnarray}  \end{eqnarray}
133    
134  \marginpar{Run-time control needs to be added for these options} The  \marginpar{Run-time control needs to be added for these options} The
135  disctinction between using absolute vorticity or relative vorticity is  distinction between using absolute vorticity or relative vorticity is
136  useful when constructing higher order advection schemes; monotone  useful when constructing higher order advection schemes; monotone
137  advection of relative vorticity behaves differently to monotone  advection of relative vorticity behaves differently to monotone
138  advection of absolute vorticity. Currently the choice of  advection of absolute vorticity. Currently the choice of
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