s_algorithm/text/mom_vecinv.tex

% $Header: $
% $Name: $

\section{Vector invariant momentum equations}

The finite volume method lends itself to describing the continuity and
tracer equations in curvilinear coordinate systems but the appearance
of new metric terms in the flux-form momentum equations makes
generalizing them far from elegant. The vector invariant form of the
momentum equations are exactly that; invariant under coordinate
transformations.

The non-hydrostatic vector invariant equations read:
\begin{equation}
\partial_t \vec{v} + ( 2\vec{\Omega} + \vec{\zeta}) \wedge \vec{v}
- b \hat{r}
+ \vec{\nabla} B = \vec{\nabla} \cdot \vec{\bf \tau}
\end{equation}
which describe motions in any orthogonal curvilinear coordinate
system. Here, $B$ is the Bernoulli function and $\vec{\zeta}=\nabla
\wedge \vec{v}$ is the vorticity vector. We can take advantage of the
elegance of these equations when discretizing them and use the
discrete definitions of the grad, curl and divergence operators to
satisfy constraints. We can also consider the analogy to forming
derived equations, such as the vorticity equation, and examine how the
discretization can be adjusted to give suitable vorticity advection
among other things.

The underlying algorithm is the same as for the flux form
equations. All that has changed is the contents of the ``G's''. For
the time-being, only the hydrostatic terms have been coded but we will
indicate the points where non-hydrostatic contributions will enter:
\begin{eqnarray}
G_u & = & G_u^{fv} + G_u^{\zeta_3 v} + G_u^{\zeta_2 w} + G_u^{\partial_x B}
+ G_u^{\partial_z \tau^x} + G_u^{h-dissip} + G_u^{v-dissip} \\
G_v & = & G_v^{fu} + G_v^{\zeta_3 u} + G_v^{\zeta_1 w} + G_v^{\partial_y B}
+ G_v^{\partial_z \tau^y} + G_v^{h-dissip} + G_v^{v-dissip} \\
G_w & = & G_w^{fu} + G_w^{\zeta_1 v} + G_w^{\zeta_2 u} + G_w^{\partial_z B}
+ G_w^{h-dissip} + G_w^{v-dissip}
\end{eqnarray}

\fbox{ \begin{minipage}{4.75in}
{\em S/R CALC\_MOM\_RHS} ({\em pkg/mom\_vecinv/calc\_mom\_rhs.F})

$G_u$: {\bf Gu} ({\em DYNVARS.h})

$G_v$: {\bf Gv} ({\em DYNVARS.h})

$G_w$: {\bf Gw} ({\em DYNVARS.h})
\end{minipage} }

\subsection{Relative vorticity}

The vertical component of relative vorticity is explicitly calculated
and use in the discretization. The particular form is crucial for
numerical stablility; alternative definitions break the conservation
properties of the discrete equations.

Relative vorticity is defined:
\begin{equation}
\zeta_3 = \frac{\Gamma}{A_\zeta}
= \frac{1}{{\cal A}_\zeta} ( \delta_i \Delta y_c v - \delta_j \Delta x_c u )
\end{equation}
where ${\cal A}_\zeta$ is the area of the vorticity cell presented in
the vertical and $\Gamma$ is the circulation about that cell.

\fbox{ \begin{minipage}{4.75in}
{\em S/R MOM\_VI\_CALC\_RELVORT3} ({\em mom\_vi\_calc\_relvort3.F})

$\zeta_3$: {\bf vort3} (local to {\em calc\_mom\_rhs.F})
\end{minipage} }


\subsection{Kinetic energy}

The kinetic energy, denoted $KE$, is defined:
\begin{equation}
KE = \frac{1}{2} ( \overline{ u^2 }^i + \overline{ v^2 }^j 
+ \epsilon_{nh} \overline{ w^2 }^k )
\end{equation}

\fbox{ \begin{minipage}{4.75in}
{\em S/R MOM\_VI\_CALC\_KE} ({\em mom\_vi\_calc\_ke.F})

$KE$: {\bf KE} (local to {\em calc\_mom\_rhs.F})
\end{minipage} }


\subsection{Coriolis terms}

The potential enstrophy conserving form of the linear Coriolis terms
are written:
\begin{eqnarray}
G_u^{fv} & = &
\frac{1}{\Delta x_c}
\overline{ \frac{f}{h_\zeta} }^j \overline{ \overline{ \Delta x_g h_s v }^j }^i \\
G_v^{fu} & = & -
\frac{1}{\Delta y_c}
\overline{ \frac{f}{h_\zeta} }^i \overline{ \overline{ \Delta y_g h_w u }^i }^j
\end{eqnarray}
Here, the Coriolis parameter $f$ is defined at vorticity (corner)
points.
\marginpar{$f$: {\bf fCoriG}}
\marginpar{$h_\zeta$: {\bf hFacZ}}

The potential enstrophy conserving form of the non-linear Coriolis
terms are written:
\begin{eqnarray}
G_u^{\zeta_3 v} & = &
\frac{1}{\Delta x_c}
\overline{ \frac{\zeta_3}{h_\zeta} }^j \overline{ \overline{ \Delta x_g h_s v }^j }^i \\
G_v^{\zeta_3 u} & = & -
\frac{1}{\Delta y_c}
\overline{ \frac{\zeta_3}{h_\zeta} }^i \overline{ \overline{ \Delta y_g h_w u }^i }^j
\end{eqnarray}
\marginpar{$\zeta_3$: {\bf vort3}}

The Coriolis terms can also be evaluated together and expressed in
terms of absolute vorticity $f+\zeta_3$. The potential enstrophy
conserving form using the absolute vorticity is written:
\begin{eqnarray}
G_u^{fv} + G_u^{\zeta_3 v} & = &
\frac{1}{\Delta x_c}
\overline{ \frac{f + \zeta_3}{h_\zeta} }^j \overline{ \overline{ \Delta x_g h_s v }^j }^i \\
G_v^{fu} + G_v^{\zeta_3 u} & = & -
\frac{1}{\Delta y_c}
\overline{ \frac{f + \zeta_3}{h_\zeta} }^i \overline{ \overline{ \Delta y_g h_w u }^i }^j
\end{eqnarray}

\marginpar{Run-time control needs to be added for these options} The
disctinction between using absolute vorticity or relative vorticity is
useful when constructing higher order advection schemes; monotone
advection of relative vorticity behaves differently to monotone
advection of absolute vorticity. Currently the choice of
relative/absolute vorticity, centered/upwind/high order advection is
available only through commented subroutine calls.

\fbox{ \begin{minipage}{4.75in}
{\em S/R MOM\_VI\_CORIOLIS} ({\em mom\_vi\_coriolis.F})

{\em S/R MOM\_VI\_U\_CORIOLIS} ({\em mom\_vi\_u\_coriolis.F})

{\em S/R MOM\_VI\_V\_CORIOLIS} ({\em mom\_vi\_v\_coriolis.F})

$G_u^{fv}$, $G_u^{\zeta_3 v}$: {\bf uCf} (local to {\em calc\_mom\_rhs.F})

$G_v^{fu}$, $G_v^{\zeta_3 u}$: {\bf vCf} (local to {\em calc\_mom\_rhs.F})
\end{minipage} }


\subsection{Shear terms}

The shear terms ($\zeta_2w$ and $\zeta_1w$) are are discretized to
guarantee that no spurious generation of kinetic energy is possible;
the horizontal gradient of Bernoulli function has to be consistent
with the vertical advection of shear:
\marginpar{N-H terms have not been tried!}
\begin{eqnarray}
G_u^{\zeta_2 w} & = &
\frac{1}{ {\cal A}_w \Delta r_f h_w } \overline{
\overline{ {\cal A}_c w }^i ( \delta_k u - \epsilon_{nh} \delta_j w )
}^k \\
G_v^{\zeta_1 w} & = &
\frac{1}{ {\cal A}_s \Delta r_f h_s } \overline{
\overline{ {\cal A}_c w }^i ( \delta_k u - \epsilon_{nh} \delta_j w )
}^k
\end{eqnarray}

\fbox{ \begin{minipage}{4.75in}
{\em S/R MOM\_VI\_U\_VERTSHEAR} ({\em mom\_vi\_u\_vertshear.F})

{\em S/R MOM\_VI\_V\_VERTSHEAR} ({\em mom\_vi\_v\_vertshear.F})

$G_u^{\zeta_2 w}$: {\bf uCf} (local to {\em calc\_mom\_rhs.F})

$G_v^{\zeta_1 w}$: {\bf vCf} (local to {\em calc\_mom\_rhs.F})
\end{minipage} }


\subsection{Gradient of Bernoulli function}

\begin{eqnarray}
G_u^{\partial_x B} & = &
\frac{1}{\Delta x_c} \delta_i ( \phi' + KE ) \\
G_v^{\partial_y B} & = &
\frac{1}{\Delta x_y} \delta_j ( \phi' + KE )
%G_w^{\partial_z B} & = &
%\frac{1}{\Delta r_c} h_c \delta_k ( \phi' + KE )
\end{eqnarray}

\fbox{ \begin{minipage}{4.75in}
{\em S/R MOM\_VI\_U\_GRAD\_KE} ({\em mom\_vi\_u\_grad\_ke.F})

{\em S/R MOM\_VI\_V\_GRAD\_KE} ({\em mom\_vi\_v\_grad\_ke.F})

$G_u^{\partial_x KE}$: {\bf uCf} (local to {\em calc\_mom\_rhs.F})

$G_v^{\partial_y KE}$: {\bf vCf} (local to {\em calc\_mom\_rhs.F})
\end{minipage} }


\subsection{Horizontal dissipation}

The horizontal divergence, a complimentary quantity to relative
vorticity, is used in parameterizing the Reynolds stresses and is
discretized:
\begin{equation}
D = \frac{1}{{\cal A}_c h_c} (
  \delta_i \Delta y_g h_w u
+ \delta_j \Delta x_g h_s v )
\end{equation}

\fbox{ \begin{minipage}{4.75in}
{\em S/R MOM\_VI\_CALC\_HDIV} ({\em mom\_vi\_calc\_hdiv.F})

$D$: {\bf hDiv} (local to {\em calc\_mom\_rhs.F})
\end{minipage} }


\subsection{Horizontal dissipation}

The following discretization of horizontal dissipation conserves
potential vorticity (thickness weighted relative vorticity) and
divergence and dissipates energy, enstrophy and divergence squared:
\begin{eqnarray}
G_u^{h-dissip} & = &
  \frac{1}{\Delta x_c} \delta_i ( A_D D - A_{D4} D^*)
- \frac{1}{\Delta y_u h_w} \delta_j h_\zeta ( A_\zeta \zeta - A_{\zeta4} \zeta^* )
\\
G_v^{h-dissip} & = &
  \frac{1}{\Delta x_v h_s} \delta_i h_\zeta ( A_\zeta \zeta - A_\zeta \zeta^* )
+ \frac{1}{\Delta y_c} \delta_j ( A_D D - A_{D4} D^* )
\end{eqnarray}
where
\begin{eqnarray}
D^* & = & \frac{1}{{\cal A}_c h_c} (
  \delta_i \Delta y_g h_w \nabla^2 u
+ \delta_j \Delta x_g h_s \nabla^2 v ) \\
\zeta^* & = & \frac{1}{{\cal A}_\zeta} (
  \delta_i \Delta y_c \nabla^2 v
- \delta_j \Delta x_c \nabla^2 u )
\end{eqnarray}

\fbox{ \begin{minipage}{4.75in}
{\em S/R MOM\_VI\_HDISSIP} ({\em mom\_vi\_hdissip.F})

$G_u^{h-dissip}$: {\bf uDiss} (local to {\em calc\_mom\_rhs.F})

$G_v^{h-dissip}$: {\bf vDiss} (local to {\em calc\_mom\_rhs.F})
\end{minipage} }


\subsection{Vertical dissipation}

Currently, this is exactly the same code as the flux form equations.
\begin{eqnarray}
G_u^{v-diss} & = &
\frac{1}{\Delta r_f h_w} \delta_k \tau_{13} \\
G_v^{v-diss} & = &
\frac{1}{\Delta r_f h_s} \delta_k \tau_{23}
\end{eqnarray}
represents the general discrete form of the vertical dissipation terms.

In the interior the vertical stresses are discretized:
\begin{eqnarray}
\tau_{13} & = & A_v \frac{1}{\Delta r_c} \delta_k u \\
\tau_{23} & = & A_v \frac{1}{\Delta r_c} \delta_k v
\end{eqnarray}

\fbox{ \begin{minipage}{4.75in}
{\em S/R MOM\_U\_RVISCLFUX} ({\em mom\_u\_rviscflux.F})

{\em S/R MOM\_V\_RVISCLFUX} ({\em mom\_v\_rviscflux.F})

$\tau_{13}$: {\bf urf} (local to {\em calc\_mom\_rhs.F})

$\tau_{23}$: {\bf vrf} (local to {\em calc\_mom\_rhs.F})
\end{minipage} }
1	adcroft	1.1	% $Header: $
2			% $Name: $
3
4			\section{Vector invariant momentum equations}
5
6			The finite volume method lends itself to describing the continuity and
7			tracer equations in curvilinear coordinate systems but the appearance
8			of new metric terms in the flux-form momentum equations makes
9			generalizing them far from elegant. The vector invariant form of the
10			momentum equations are exactly that; invariant under coordinate
11			transformations.
12
13			The non-hydrostatic vector invariant equations read:
14			\begin{equation}
15			\partial_t \vec{v} + ( 2\vec{\Omega} + \vec{\zeta}) \wedge \vec{v}
16			- b \hat{r}
17			+ \vec{\nabla} B = \vec{\nabla} \cdot \vec{\bf \tau}
18			\end{equation}
19			which describe motions in any orthogonal curvilinear coordinate
20			system. Here, $B$ is the Bernoulli function and $\vec{\zeta}=\nabla
21			\wedge \vec{v}$ is the vorticity vector. We can take advantage of the
22			elegance of these equations when discretizing them and use the
23			discrete definitions of the grad, curl and divergence operators to
24			satisfy constraints. We can also consider the analogy to forming
25			derived equations, such as the vorticity equation, and examine how the
26			discretization can be adjusted to give suitable vorticity advection
27			among other things.
28
29			The underlying algorithm is the same as for the flux form
30			equations. All that has changed is the contents of the ``G's''. For
31			the time-being, only the hydrostatic terms have been coded but we will
32			indicate the points where non-hydrostatic contributions will enter:
33			\begin{eqnarray}
34			G_u & = & G_u^{fv} + G_u^{\zeta_3 v} + G_u^{\zeta_2 w} + G_u^{\partial_x B}
35			+ G_u^{\partial_z \tau^x} + G_u^{h-dissip} + G_u^{v-dissip} \\
36			G_v & = & G_v^{fu} + G_v^{\zeta_3 u} + G_v^{\zeta_1 w} + G_v^{\partial_y B}
37			+ G_v^{\partial_z \tau^y} + G_v^{h-dissip} + G_v^{v-dissip} \\
38			G_w & = & G_w^{fu} + G_w^{\zeta_1 v} + G_w^{\zeta_2 u} + G_w^{\partial_z B}
39			+ G_w^{h-dissip} + G_w^{v-dissip}
40			\end{eqnarray}
41
42			\fbox{ \begin{minipage}{4.75in}
43			{\em S/R CALC\_MOM\_RHS} ({\em pkg/mom\_vecinv/calc\_mom\_rhs.F})
44
45			$G_u$: {\bf Gu} ({\em DYNVARS.h})
46
47			$G_v$: {\bf Gv} ({\em DYNVARS.h})
48
49			$G_w$: {\bf Gw} ({\em DYNVARS.h})
50			\end{minipage} }
51
52			\subsection{Relative vorticity}
53
54			The vertical component of relative vorticity is explicitly calculated
55			and use in the discretization. The particular form is crucial for
56			numerical stablility; alternative definitions break the conservation
57			properties of the discrete equations.
58
59			Relative vorticity is defined:
60			\begin{equation}
61			\zeta_3 = \frac{\Gamma}{A_\zeta}
62			= \frac{1}{{\cal A}_\zeta} ( \delta_i \Delta y_c v - \delta_j \Delta x_c u )
63			\end{equation}
64			where ${\cal A}_\zeta$ is the area of the vorticity cell presented in
65			the vertical and $\Gamma$ is the circulation about that cell.
66
67			\fbox{ \begin{minipage}{4.75in}
68			{\em S/R MOM\_VI\_CALC\_RELVORT3} ({\em mom\_vi\_calc\_relvort3.F})
69
70			$\zeta_3$: {\bf vort3} (local to {\em calc\_mom\_rhs.F})
71			\end{minipage} }
72
73
74			\subsection{Kinetic energy}
75
76			The kinetic energy, denoted $KE$, is defined:
77			\begin{equation}
78			KE = \frac{1}{2} ( \overline{ u^2 }^i + \overline{ v^2 }^j
79			+ \epsilon_{nh} \overline{ w^2 }^k )
80			\end{equation}
81
82			\fbox{ \begin{minipage}{4.75in}
83			{\em S/R MOM\_VI\_CALC\_KE} ({\em mom\_vi\_calc\_ke.F})
84
85			$KE$: {\bf KE} (local to {\em calc\_mom\_rhs.F})
86			\end{minipage} }
87
88
89			\subsection{Coriolis terms}
90
91			The potential enstrophy conserving form of the linear Coriolis terms
92			are written:
93			\begin{eqnarray}
94			G_u^{fv} & = &
95			\frac{1}{\Delta x_c}
96			\overline{ \frac{f}{h_\zeta} }^j \overline{ \overline{ \Delta x_g h_s v }^j }^i \\
97			G_v^{fu} & = & -
98			\frac{1}{\Delta y_c}
99			\overline{ \frac{f}{h_\zeta} }^i \overline{ \overline{ \Delta y_g h_w u }^i }^j
100			\end{eqnarray}
101			Here, the Coriolis parameter $f$ is defined at vorticity (corner)
102			points.
103			\marginpar{$f$: {\bf fCoriG}}
104			\marginpar{$h_\zeta$: {\bf hFacZ}}
105
106			The potential enstrophy conserving form of the non-linear Coriolis
107			terms are written:
108			\begin{eqnarray}
109			G_u^{\zeta_3 v} & = &
110			\frac{1}{\Delta x_c}
111			\overline{ \frac{\zeta_3}{h_\zeta} }^j \overline{ \overline{ \Delta x_g h_s v }^j }^i \\
112			G_v^{\zeta_3 u} & = & -
113			\frac{1}{\Delta y_c}
114			\overline{ \frac{\zeta_3}{h_\zeta} }^i \overline{ \overline{ \Delta y_g h_w u }^i }^j
115			\end{eqnarray}
116			\marginpar{$\zeta_3$: {\bf vort3}}
117
118			The Coriolis terms can also be evaluated together and expressed in
119			terms of absolute vorticity $f+\zeta_3$. The potential enstrophy
120			conserving form using the absolute vorticity is written:
121			\begin{eqnarray}
122			G_u^{fv} + G_u^{\zeta_3 v} & = &
123			\frac{1}{\Delta x_c}
124			\overline{ \frac{f + \zeta_3}{h_\zeta} }^j \overline{ \overline{ \Delta x_g h_s v }^j }^i \\
125			G_v^{fu} + G_v^{\zeta_3 u} & = & -
126			\frac{1}{\Delta y_c}
127			\overline{ \frac{f + \zeta_3}{h_\zeta} }^i \overline{ \overline{ \Delta y_g h_w u }^i }^j
128			\end{eqnarray}
129
130			\marginpar{Run-time control needs to be added for these options} The
131			disctinction between using absolute vorticity or relative vorticity is
132			useful when constructing higher order advection schemes; monotone
133			advection of relative vorticity behaves differently to monotone
134			advection of absolute vorticity. Currently the choice of
135			relative/absolute vorticity, centered/upwind/high order advection is
136			available only through commented subroutine calls.
137
138			\fbox{ \begin{minipage}{4.75in}
139			{\em S/R MOM\_VI\_CORIOLIS} ({\em mom\_vi\_coriolis.F})
140
141			{\em S/R MOM\_VI\_U\_CORIOLIS} ({\em mom\_vi\_u\_coriolis.F})
142
143			{\em S/R MOM\_VI\_V\_CORIOLIS} ({\em mom\_vi\_v\_coriolis.F})
144
145			$G_u^{fv}$, $G_u^{\zeta_3 v}$: {\bf uCf} (local to {\em calc\_mom\_rhs.F})
146
147			$G_v^{fu}$, $G_v^{\zeta_3 u}$: {\bf vCf} (local to {\em calc\_mom\_rhs.F})
148			\end{minipage} }
149
150
151			\subsection{Shear terms}
152
153			The shear terms ($\zeta_2w$ and $\zeta_1w$) are are discretized to
154			guarantee that no spurious generation of kinetic energy is possible;
155			the horizontal gradient of Bernoulli function has to be consistent
156			with the vertical advection of shear:
157			\marginpar{N-H terms have not been tried!}
158			\begin{eqnarray}
159			G_u^{\zeta_2 w} & = &
160			\frac{1}{ {\cal A}_w \Delta r_f h_w } \overline{
161			\overline{ {\cal A}_c w }^i ( \delta_k u - \epsilon_{nh} \delta_j w )
162			}^k \\
163			G_v^{\zeta_1 w} & = &
164			\frac{1}{ {\cal A}_s \Delta r_f h_s } \overline{
165			\overline{ {\cal A}_c w }^i ( \delta_k u - \epsilon_{nh} \delta_j w )
166			}^k
167			\end{eqnarray}
168
169			\fbox{ \begin{minipage}{4.75in}
170			{\em S/R MOM\_VI\_U\_VERTSHEAR} ({\em mom\_vi\_u\_vertshear.F})
171
172			{\em S/R MOM\_VI\_V\_VERTSHEAR} ({\em mom\_vi\_v\_vertshear.F})
173
174			$G_u^{\zeta_2 w}$: {\bf uCf} (local to {\em calc\_mom\_rhs.F})
175
176			$G_v^{\zeta_1 w}$: {\bf vCf} (local to {\em calc\_mom\_rhs.F})
177			\end{minipage} }
178
179
180
181			\subsection{Gradient of Bernoulli function}
182
183			\begin{eqnarray}
184			G_u^{\partial_x B} & = &
185			\frac{1}{\Delta x_c} \delta_i ( \phi' + KE ) \\
186			G_v^{\partial_y B} & = &
187			\frac{1}{\Delta x_y} \delta_j ( \phi' + KE )
188			%G_w^{\partial_z B} & = &
189			%\frac{1}{\Delta r_c} h_c \delta_k ( \phi' + KE )
190			\end{eqnarray}
191
192			\fbox{ \begin{minipage}{4.75in}
193			{\em S/R MOM\_VI\_U\_GRAD\_KE} ({\em mom\_vi\_u\_grad\_ke.F})
194
195			{\em S/R MOM\_VI\_V\_GRAD\_KE} ({\em mom\_vi\_v\_grad\_ke.F})
196
197			$G_u^{\partial_x KE}$: {\bf uCf} (local to {\em calc\_mom\_rhs.F})
198
199			$G_v^{\partial_y KE}$: {\bf vCf} (local to {\em calc\_mom\_rhs.F})
200			\end{minipage} }
201
202
203
204			\subsection{Horizontal dissipation}
205
206			The horizontal divergence, a complimentary quantity to relative
207			vorticity, is used in parameterizing the Reynolds stresses and is
208			discretized:
209			\begin{equation}
210			D = \frac{1}{{\cal A}_c h_c} (
211			\delta_i \Delta y_g h_w u
212			+ \delta_j \Delta x_g h_s v )
213			\end{equation}
214
215			\fbox{ \begin{minipage}{4.75in}
216			{\em S/R MOM\_VI\_CALC\_HDIV} ({\em mom\_vi\_calc\_hdiv.F})
217
218			$D$: {\bf hDiv} (local to {\em calc\_mom\_rhs.F})
219			\end{minipage} }
220
221
222			\subsection{Horizontal dissipation}
223
224			The following discretization of horizontal dissipation conserves
225			potential vorticity (thickness weighted relative vorticity) and
226			divergence and dissipates energy, enstrophy and divergence squared:
227			\begin{eqnarray}
228			G_u^{h-dissip} & = &
229			\frac{1}{\Delta x_c} \delta_i ( A_D D - A_{D4} D^*)
230			- \frac{1}{\Delta y_u h_w} \delta_j h_\zeta ( A_\zeta \zeta - A_{\zeta4} \zeta^* )
231			\\
232			G_v^{h-dissip} & = &
233			\frac{1}{\Delta x_v h_s} \delta_i h_\zeta ( A_\zeta \zeta - A_\zeta \zeta^* )
234			+ \frac{1}{\Delta y_c} \delta_j ( A_D D - A_{D4} D^* )
235			\end{eqnarray}
236			where
237			\begin{eqnarray}
238			D^* & = & \frac{1}{{\cal A}_c h_c} (
239			\delta_i \Delta y_g h_w \nabla^2 u
240			+ \delta_j \Delta x_g h_s \nabla^2 v ) \\
241			\zeta^* & = & \frac{1}{{\cal A}_\zeta} (
242			\delta_i \Delta y_c \nabla^2 v
243			- \delta_j \Delta x_c \nabla^2 u )
244			\end{eqnarray}
245
246			\fbox{ \begin{minipage}{4.75in}
247			{\em S/R MOM\_VI\_HDISSIP} ({\em mom\_vi\_hdissip.F})
248
249			$G_u^{h-dissip}$: {\bf uDiss} (local to {\em calc\_mom\_rhs.F})
250
251			$G_v^{h-dissip}$: {\bf vDiss} (local to {\em calc\_mom\_rhs.F})
252			\end{minipage} }
253
254
255			\subsection{Vertical dissipation}
256
257			Currently, this is exactly the same code as the flux form equations.
258			\begin{eqnarray}
259			G_u^{v-diss} & = &
260			\frac{1}{\Delta r_f h_w} \delta_k \tau_{13} \\
261			G_v^{v-diss} & = &
262			\frac{1}{\Delta r_f h_s} \delta_k \tau_{23}
263			\end{eqnarray}
264			represents the general discrete form of the vertical dissipation terms.
265
266			In the interior the vertical stresses are discretized:
267			\begin{eqnarray}
268			\tau_{13} & = & A_v \frac{1}{\Delta r_c} \delta_k u \\
269			\tau_{23} & = & A_v \frac{1}{\Delta r_c} \delta_k v
270			\end{eqnarray}
271
272			\fbox{ \begin{minipage}{4.75in}
273			{\em S/R MOM\_U\_RVISCLFUX} ({\em mom\_u\_rviscflux.F})
274
275			{\em S/R MOM\_V\_RVISCLFUX} ({\em mom\_v\_rviscflux.F})
276
277			$\tau_{13}$: {\bf urf} (local to {\em calc\_mom\_rhs.F})
278
279			$\tau_{23}$: {\bf vrf} (local to {\em calc\_mom\_rhs.F})
280			\end{minipage} }