s_algorithm/text/mom_fluxform.tex

% $Header: $
% $Name: $

\section{Flux-form momentum equations}

The original finite volume model was based on the Eulerian flux form
momentum equations. This is the default though the vector invariant
form is optionally available (and recommended in some cases).

The ``G's'' (our colloquial name for all terms on rhs!) are broken
into the various advective, Coriolis, horizontal dissipation, vertical
dissipation and metric forces:
\marginpar{$G_u$: {\bf Gu} }
\marginpar{$G_v$: {\bf Gv} }
\marginpar{$G_w$: {\bf Gw} }
\begin{eqnarray}
G_u & = & G_u^{adv} + G_u^{cor} + G_u^{h-diss} + G_u^{v-diss} +
G_u^{metric} + G_u^{nh-metric} \\
G_v & = & G_v^{adv} + G_v^{cor} + G_v^{h-diss} + G_v^{v-diss} +
G_v^{metric} + G_v^{nh-metric} \\
G_w & = & G_w^{adv} + G_w^{cor} + G_w^{h-diss} + G_w^{v-diss} +
G_w^{metric} + G_w^{nh-metric}
\end{eqnarray}
In the hydrostatic limit, $G_w=0$ and $\epsilon_{nh}=0$, reducing the
vertical momentum to hydrostatic balance.

These terms are calculated in routines called from subroutine {\em
CALC\_MOM\_RHS} a collected into the global arrays {\bf Gu}, {\bf Gv},
and {\bf Gw}.

\fbox{ \begin{minipage}{4.75in}
{\em S/R CALC\_MOM\_RHS} ({\em pkg/mom\_fluxform/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{Advection of momentum}

The advective operator is second order accurate in space:
\begin{eqnarray}
{\cal A}_w \Delta r_f h_w G_u^{adv} & = &
  \delta_i \overline{ U }^i \overline{ u }^i
+ \delta_j \overline{ V }^i \overline{ u }^j
+ \delta_k \overline{ W }^i \overline{ u }^k \\
{\cal A}_s \Delta r_f h_s G_v^{adv} & = &
  \delta_i \overline{ U }^j \overline{ v }^i
+ \delta_j \overline{ V }^j \overline{ v }^j
+ \delta_k \overline{ W }^j \overline{ v }^k \\
{\cal A}_c \Delta r_c G_w^{adv} & = &
  \delta_i \overline{ U }^k \overline{ w }^i
+ \delta_j \overline{ V }^k \overline{ w }^j
+ \delta_k \overline{ W }^k \overline{ w }^k \\
\end{eqnarray}
and because of the flux form does not contribute to the global budget
of linear momentum. The quantities $U$, $V$ and $W$ are volume fluxes
defined:
\marginpar{$U$: {\bf uTrans} }
\marginpar{$V$: {\bf vTrans} }
\marginpar{$W$: {\bf rTrans} }
\begin{eqnarray}
U & = & \Delta y_g \Delta r_f h_w u \\
V & = & \Delta x_g \Delta r_f h_s v \\
W & = & {\cal A}_c w
\end{eqnarray}
The advection of momentum takes the same form as the advection of
tracers but by a translated advective flow. Consequently, the
conservation of second moments, derived for tracers later, applies to
$u^2$ and $v^2$ and $w^2$ so that advection of momentum correctly
conserves kinetic energy.

\fbox{ \begin{minipage}{4.75in}
{\em S/R MOM\_U\_ADV\_UU} ({\em mom\_u\_adv\_uu.F})

{\em S/R MOM\_U\_ADV\_VU} ({\em mom\_u\_adv\_vu.F})

{\em S/R MOM\_U\_ADV\_WU} ({\em mom\_u\_adv\_wu.F})

{\em S/R MOM\_U\_ADV\_UV} ({\em mom\_u\_adv\_uv.F})

{\em S/R MOM\_U\_ADV\_VV} ({\em mom\_u\_adv\_vv.F})

{\em S/R MOM\_U\_ADV\_WV} ({\em mom\_u\_adv\_wv.F})

$uu$, $uv$, $vu$, $vv$: {\bf aF} (local to {\em calc\_mom\_rhs.F})
\end{minipage} }


\subsection{Coriolis terms}

The ``pure C grid'' Coriolis terms (i.e. in absence of C-D scheme) are
discretized:
\begin{eqnarray}
{\cal A}_w \Delta r_f h_w G_u^{Cor} & = &
  \overline{ f {\cal A}_c \Delta r_f h_c \overline{ v }^j }^i
- \epsilon_{nh} \overline{ f' {\cal A}_c \Delta r_f h_c \overline{ w }^k }^i \\
{\cal A}_s \Delta r_f h_s G_v^{Cor} & = &
- \overline{ f {\cal A}_c \Delta r_f h_c \overline{ u }^i }^j \\
{\cal A}_c \Delta r_c G_w^{Cor} & = &
 \epsilon_{nh} \overline{ f' {\cal A}_c \Delta r_f h_c \overline{ u }^i }^k
\end{eqnarray}
where the Coriolis parameters $f$ and $f'$ are defined:
\begin{eqnarray}
f & = & 2 \Omega \sin{\phi} \\
f' & = & 2 \Omega \cos{\phi}
\end{eqnarray}
when using spherical geometry, otherwise the $\beta$-plane definition is used:
\begin{eqnarray}
f & = & f_o + \beta y \\
f' & = & 0
\end{eqnarray}

This discretization globally conserves kinetic energy. It should be
noted that despite the use of this discretization in former
publications, all calculations to date have used the following
different discretization:
\begin{eqnarray}
G_u^{Cor} & = &
  f_u \overline{ v }^{ji}
- \epsilon_{nh} f_u' \overline{ w }^{ik} \\
G_v^{Cor} & = &
- f_v \overline{ u }^{ij} \\
G_w^{Cor} & = &
 \epsilon_{nh} f_w' \overline{ u }^{ik}
\end{eqnarray}
\marginpar{Need to change the default in code to match this}
where the subscripts on $f$ and $f'$ indicate evaluation of the
Coriolis parameters at the appropriate points in space. The above
discretization does {\em not} conserve anything, especially energy. An
option to recover this discretization has been retained for backward
compatibility testing (set run-time logical {\bf
useNonconservingCoriolis} to {\em true} which otherwise defaults to
{\em false}).

\fbox{ \begin{minipage}{4.75in}
{\em S/R MOM\_CDSCHEME} ({\em mom\_cdscheme.F})

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

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

$G_u^{Cor}$, $G_v^{Cor}$: {\bf cF} (local to {\em calc\_mom\_rhs.F})
\end{minipage} }


\subsection{Curvature metric terms}

The most commonly used coordinate system on the sphere is the
geographic system $(\lambda,\phi)$. The curvilinear nature of these
coordinates on the sphere lead to some ``metric'' terms in the
component momentum equations. Under the thin-atmosphere and
hydrostatic approximations these terms are discretized:
\begin{eqnarray}
{\cal A}_w \Delta r_f h_w G_u^{metric} & = &
  \overline{ \frac{ \overline{u}^i }{a} \tan{\phi} {\cal A}_c \Delta r_f h_c \overline{ v }^j }^i \\
{\cal A}_s \Delta r_f h_s G_v^{metric} & = &
- \overline{ \frac{ \overline{u}^i }{a} \tan{\phi} {\cal A}_c \Delta r_f h_c \overline{ u }^i }^j \\
G_w^{metric} & = & 0
\end{eqnarray}
where $a$ is the radius of the planet (sphericity is assumed) or the
radial distance of the particle (i.e. a function of height).  It is
easy to see that this discretization satisfies all the properties of
the discrete Coriolis terms since the metric factor $\frac{u}{a}
\tan{\phi}$ can be viewed as a modification of the vertical Coriolis
parameter: $f \rightarrow f+\frac{u}{a} \tan{\phi}$.

However, as for the Coriolis terms, a non-energy conserving form has
exclusively been used to date:
\begin{eqnarray}
G_u^{metric} & = & \frac{u \overline{v}^{ij} }{a} \tan{\phi} \\
G_v^{metric} & = & \frac{ \overline{u}^{ij} \overline{u}^{ij}}{a} \tan{\phi}
\end{eqnarray}
where $\tan{\phi}$ is evaluated at the $u$ and $v$ points
respectively.

\fbox{ \begin{minipage}{4.75in}
{\em S/R MOM\_U\_METRIC\_SPHERE} ({\em mom\_u\_metric\_sphere.F})

{\em S/R MOM\_V\_METRIC\_SPHERE} ({\em mom\_v\_metric\_sphere.F})

$G_u^{metric}$, $G_v^{metric}$: {\bf mT} (local to {\em calc\_mom\_rhs.F})
\end{minipage} }


\subsection{Non-hydrostatic metric terms}

For the non-hydrostatic equations, dropping the thin-atmosphere
approximation re-introduces metric terms involving $w$ and are
required to conserve anglular momentum:
\begin{eqnarray}
{\cal A}_w \Delta r_f h_w G_u^{metric} & = &
- \overline{ \frac{ \overline{u}^i \overline{w}^k }{a} {\cal A}_c \Delta r_f h_c }^i \\
{\cal A}_s \Delta r_f h_s G_v^{metric} & = &
- \overline{ \frac{ \overline{v}^j \overline{w}^k }{a} {\cal A}_c \Delta r_f h_c}^j \\
{\cal A}_c \Delta r_c G_w^{metric} & = &
  \overline{ \frac{ {\overline{u}^i}^2 + {\overline{v}^j}^2}{a} {\cal A}_c \Delta r_f h_c }^k
\end{eqnarray}

Because we are always consistent, even if consistently wrong, we have,
in the past, used a different discretization in the model which is:
\begin{eqnarray}
G_u^{metric} & = &
- \frac{u}{a} \overline{w}^{ik} \\
G_v^{metric} & = &
- \frac{v}{a} \overline{w}^{jk} \\
G_w^{metric} & = &
  \frac{1}{a} ( {\overline{u}^{ik}}^2 + {\overline{v}^{jk}}^2 )
\end{eqnarray}

\fbox{ \begin{minipage}{4.75in}
{\em S/R MOM\_U\_METRIC\_NH} ({\em mom\_u\_metric\_nh.F})

{\em S/R MOM\_V\_METRIC\_NH} ({\em mom\_v\_metric\_nh.F})

$G_u^{metric}$, $G_v^{metric}$: {\bf mT} (local to {\em calc\_mom\_rhs.F})
\end{minipage} }


\subsection{Lateral dissipation}

Historically, we have represented the SGS Reynolds stresses as simply
down gradient momentum fluxes, ignoring constraints on the stress
tensor such as symmetry.
\begin{eqnarray}
{\cal A}_w \Delta r_f h_w G_u^{h-diss} & = &
  \delta_i  \Delta y_f \Delta r_f h_c \tau_{11}
+ \delta_j  \Delta x_v \Delta r_f h_\zeta \tau_{12} \\
{\cal A}_s \Delta r_f h_s G_v^{h-diss} & = &
  \delta_i  \Delta y_u \Delta r_f h_\zeta \tau_{21}
+ \delta_j  \Delta x_f \Delta r_f h_c \tau_{22}
\end{eqnarray}
\marginpar{Check signs of stress definitions}

The lateral viscous stresses are discretized:
\begin{eqnarray}
\tau_{11} & = & A_h c_{11\Delta}(\phi) \frac{1}{\Delta x_f} \delta_i u
               -A_4 c_{11\Delta^2}(\phi) \frac{1}{\Delta x_f} \delta_i \nabla^2 u \\
\tau_{12} & = & A_h c_{12\Delta}(\phi) \frac{1}{\Delta y_u} \delta_j u
               -A_4 c_{12\Delta^2}(\phi)\frac{1}{\Delta y_u} \delta_j \nabla^2 u \\
\tau_{21} & = & A_h c_{21\Delta}(\phi) \frac{1}{\Delta x_v} \delta_i v
               -A_4 c_{21\Delta^2}(\phi) \frac{1}{\Delta x_v} \delta_i \nabla^2 v \\
\tau_{22} & = & A_h c_{22\Delta}(\phi) \frac{1}{\Delta y_f} \delta_j v
               -A_4 c_{22\Delta^2}(\phi) \frac{1}{\Delta y_f} \delta_j \nabla^2 v
\end{eqnarray}
where the non-dimensional factors $c_{lm\Delta^n}(\phi), \{l,m,n\} \in
\{1,2\}$ define the ``cosine'' scaling with latitude which can be
applied in various ad-hoc ways. For instance, $c_{11\Delta} =
c_{21\Delta} = (\cos{\phi})^{3/2}$, $c_{12\Delta}=c_{22\Delta}=0$ would
represent the an-isotropic cosine scaling typically used on the
``lat-lon'' grid for Laplacian viscosity.
\marginpar{Need to tidy up method for controlling this in code}

It should be noted that dispite the ad-hoc nature of the scaling, some
scaling must be done since on a lat-lon grid the converging meridians
make it very unlikely that a stable viscosity parameter exists across
the entire model domain.

The Laplacian viscosity coefficient, $A_h$ ({\bf viscAh}), has units
of $m^2 s^{-1}$. The bi-harmonic viscosity coefficient, $A_4$ ({\bf
viscA4}), has units of $m^4 s^{-1}$.

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

{\em S/R MOM\_U\_YVISCFLUX} ({\em mom\_u\_yviscflux.F})

{\em S/R MOM\_V\_XVISCFLUX} ({\em mom\_v\_xviscflux.F})

{\em S/R MOM\_V\_YVISCFLUX} ({\em mom\_v\_yviscflux.F})

$\tau_{11}$, $\tau_{12}$, $\tau_{22}$, $\tau_{22}$: {\bf vF}, {\bf
v4F} (local to {\em calc\_mom\_rhs.F})
\end{minipage} }

Two types of lateral boundary condition exist for the lateral viscous
terms, no-slip and free-slip.

The free-slip condition is most convenient to code since it is
equivalent to zero-stress on boundaries. Simple masking of the stress
components sets them to zero. The fractional open stress is properly
handled using the lopped cells.

The no-slip condition defines the normal gradient of a tangential flow
such that the flow is zero on the boundary. Rather than modify the
stresses by using complicated functions of the masks and ``ghost''
points (see \cite{Adcroft+Marshall98}) we add the boundary stresses as
an additional source term in cells next to solid boundaries. This has
the advantage of being able to cope with ``thin walls'' and also makes
the interior stress calculation (code) independent of the boundary
conditions. The ``body'' force takes the form:
\begin{eqnarray}
G_u^{side-drag} & = &
\frac{4}{\Delta z_f} \overline{ (1-h_\zeta) \frac{\Delta x_v}{\Delta y_u} }^j
\left( A_h c_{12\Delta}(\phi) u - A_4 c_{12\Delta^2}(\phi) \nabla^2 u \right)
\\
G_v^{side-drag} & = &
\frac{4}{\Delta z_f} \overline{ (1-h_\zeta) \frac{\Delta y_u}{\Delta x_v} }^i
\left( A_h c_{21\Delta}(\phi) v - A_4 c_{21\Delta^2}(\phi) \nabla^2 v \right)
\end{eqnarray}

In fact, the above discretization is not quite complete because it
assumes that the bathymetry at velocity points is deeper than at
neighbouring vorticity points, e.g. $1-h_w < 1-h_\zeta$

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

{\em S/R MOM\_V\_SIDEDRAG} ({\em mom\_v\_sidedrag.F})

$G_u^{side-drag}$, $G_v^{side-drag}$: {\bf vF} (local to {\em calc\_mom\_rhs.F})
\end{minipage} }


\subsection{Vertical dissipation}

Vertical viscosity terms are discretized with only partial adherence
to the variable grid lengths introduced by the finite volume
formulation. This reduces the formal accuracy of these terms to just
first order but only next to boundaries; exactly where other terms
appear such as linar and quadratic bottom drag.
\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} \\
G_w^{v-diss} & = & \epsilon_{nh}
\frac{1}{\Delta r_f h_d} \delta_k \tau_{33}
\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 \\
\tau_{33} & = & A_v \frac{1}{\Delta r_f} \delta_k w
\end{eqnarray}
It should be noted that in the non-hydrostatic form, the stress tensor
is even less consistent than for the hydrostatic (see Wazjowicz
\cite{Waojz}). It is well known how to do this properly (see Griffies
\cite{Griffies}) and is on the list of to-do's.

\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} }


As for the lateral viscous terms, the free-slip condition is
equivalent to simply setting the stress to zero on boundaries.  The
no-slip condition is implemented as an additional term acting on top
of the interior and free-slip stresses. Bottom drag represents
additional friction, in addition to that imposed by the no-slip
condition at the bottom. The drag is cast as a stress expressed as a
linear or quadratic function of the mean flow in the layer above the
topography:
\begin{eqnarray}
\tau_{13}^{bottom-drag} & = &
\left(
2 A_v \frac{1}{\Delta r_c}
+ r_b
+ C_d \sqrt{ \overline{2 KE}^i }
\right) u \\
\tau_{23}^{bottom-drag} & = &
\left(
2 A_v \frac{1}{\Delta r_c}
+ r_b
+ C_d \sqrt{ \overline{2 KE}^j }
\right) v
\end{eqnarray}
where these terms are only evaluated immediately above topography.
$r_b$ ({\bf bottomDragLinear}) has units of $m s^{-1}$ and a typical value
of the order 0.0002 $m s^{-1}$. $C_d$ ({\bf bottomDragQuadratic}) is
dimensionless with typical values in the range 0.001--0.003.

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

{\em S/R MOM\_V\_BOTTOMDRAG} ({\em mom\_v\_bottomdrag.F})

$\tau_{13}^{bottom-drag}$, $\tau_{23}^{bottom-drag}$: {\bf vf} (local to {\em calc\_mom\_rhs.F})
\end{minipage} }

\subsection{Derivation of discrete energy conservation}

These discrete equations conserve kinetic plus potential energy using the
following definitions:
\begin{equation}
KE = \frac{1}{2} \left( \overline{ u^2 }^i + \overline{ v^2 }^j +
\epsilon_{nh} \overline{ w^2 }^k \right)
\end{equation}

1	adcroft	1.1	% $Header: $
2			% $Name: $
3
4			\section{Flux-form momentum equations}
5
6			The original finite volume model was based on the Eulerian flux form
7			momentum equations. This is the default though the vector invariant
8			form is optionally available (and recommended in some cases).
9
10			The ``G's'' (our colloquial name for all terms on rhs!) are broken
11			into the various advective, Coriolis, horizontal dissipation, vertical
12			dissipation and metric forces:
13			\marginpar{$G_u$: {\bf Gu} }
14			\marginpar{$G_v$: {\bf Gv} }
15			\marginpar{$G_w$: {\bf Gw} }
16			\begin{eqnarray}
17			G_u & = & G_u^{adv} + G_u^{cor} + G_u^{h-diss} + G_u^{v-diss} +
18			G_u^{metric} + G_u^{nh-metric} \\
19			G_v & = & G_v^{adv} + G_v^{cor} + G_v^{h-diss} + G_v^{v-diss} +
20			G_v^{metric} + G_v^{nh-metric} \\
21			G_w & = & G_w^{adv} + G_w^{cor} + G_w^{h-diss} + G_w^{v-diss} +
22			G_w^{metric} + G_w^{nh-metric}
23			\end{eqnarray}
24			In the hydrostatic limit, $G_w=0$ and $\epsilon_{nh}=0$, reducing the
25			vertical momentum to hydrostatic balance.
26
27			These terms are calculated in routines called from subroutine {\em
28			CALC\_MOM\_RHS} a collected into the global arrays {\bf Gu}, {\bf Gv},
29			and {\bf Gw}.
30
31			\fbox{ \begin{minipage}{4.75in}
32			{\em S/R CALC\_MOM\_RHS} ({\em pkg/mom\_fluxform/calc\_mom\_rhs.F})
33
34			$G_u$: {\bf Gu} ({\em DYNVARS.h})
35
36			$G_v$: {\bf Gv} ({\em DYNVARS.h})
37
38			$G_w$: {\bf Gw} ({\em DYNVARS.h})
39			\end{minipage} }
40
41
42			\subsection{Advection of momentum}
43
44			The advective operator is second order accurate in space:
45			\begin{eqnarray}
46			{\cal A}_w \Delta r_f h_w G_u^{adv} & = &
47			\delta_i \overline{ U }^i \overline{ u }^i
48			+ \delta_j \overline{ V }^i \overline{ u }^j
49			+ \delta_k \overline{ W }^i \overline{ u }^k \\
50			{\cal A}_s \Delta r_f h_s G_v^{adv} & = &
51			\delta_i \overline{ U }^j \overline{ v }^i
52			+ \delta_j \overline{ V }^j \overline{ v }^j
53			+ \delta_k \overline{ W }^j \overline{ v }^k \\
54			{\cal A}_c \Delta r_c G_w^{adv} & = &
55			\delta_i \overline{ U }^k \overline{ w }^i
56			+ \delta_j \overline{ V }^k \overline{ w }^j
57			+ \delta_k \overline{ W }^k \overline{ w }^k \\
58			\end{eqnarray}
59			and because of the flux form does not contribute to the global budget
60			of linear momentum. The quantities $U$, $V$ and $W$ are volume fluxes
61			defined:
62			\marginpar{$U$: {\bf uTrans} }
63			\marginpar{$V$: {\bf vTrans} }
64			\marginpar{$W$: {\bf rTrans} }
65			\begin{eqnarray}
66			U & = & \Delta y_g \Delta r_f h_w u \\
67			V & = & \Delta x_g \Delta r_f h_s v \\
68			W & = & {\cal A}_c w
69			\end{eqnarray}
70			The advection of momentum takes the same form as the advection of
71			tracers but by a translated advective flow. Consequently, the
72			conservation of second moments, derived for tracers later, applies to
73			$u^2$ and $v^2$ and $w^2$ so that advection of momentum correctly
74			conserves kinetic energy.
75
76			\fbox{ \begin{minipage}{4.75in}
77			{\em S/R MOM\_U\_ADV\_UU} ({\em mom\_u\_adv\_uu.F})
78
79			{\em S/R MOM\_U\_ADV\_VU} ({\em mom\_u\_adv\_vu.F})
80
81			{\em S/R MOM\_U\_ADV\_WU} ({\em mom\_u\_adv\_wu.F})
82
83			{\em S/R MOM\_U\_ADV\_UV} ({\em mom\_u\_adv\_uv.F})
84
85			{\em S/R MOM\_U\_ADV\_VV} ({\em mom\_u\_adv\_vv.F})
86
87			{\em S/R MOM\_U\_ADV\_WV} ({\em mom\_u\_adv\_wv.F})
88
89			$uu$, $uv$, $vu$, $vv$: {\bf aF} (local to {\em calc\_mom\_rhs.F})
90			\end{minipage} }
91
92
93
94			\subsection{Coriolis terms}
95
96			The ``pure C grid'' Coriolis terms (i.e. in absence of C-D scheme) are
97			discretized:
98			\begin{eqnarray}
99			{\cal A}_w \Delta r_f h_w G_u^{Cor} & = &
100			\overline{ f {\cal A}_c \Delta r_f h_c \overline{ v }^j }^i
101			- \epsilon_{nh} \overline{ f' {\cal A}_c \Delta r_f h_c \overline{ w }^k }^i \\
102			{\cal A}_s \Delta r_f h_s G_v^{Cor} & = &
103			- \overline{ f {\cal A}_c \Delta r_f h_c \overline{ u }^i }^j \\
104			{\cal A}_c \Delta r_c G_w^{Cor} & = &
105			\epsilon_{nh} \overline{ f' {\cal A}_c \Delta r_f h_c \overline{ u }^i }^k
106			\end{eqnarray}
107			where the Coriolis parameters $f$ and $f'$ are defined:
108			\begin{eqnarray}
109			f & = & 2 \Omega \sin{\phi} \\
110			f' & = & 2 \Omega \cos{\phi}
111			\end{eqnarray}
112			when using spherical geometry, otherwise the $\beta$-plane definition is used:
113			\begin{eqnarray}
114			f & = & f_o + \beta y \\
115			f' & = & 0
116			\end{eqnarray}
117
118			This discretization globally conserves kinetic energy. It should be
119			noted that despite the use of this discretization in former
120			publications, all calculations to date have used the following
121			different discretization:
122			\begin{eqnarray}
123			G_u^{Cor} & = &
124			f_u \overline{ v }^{ji}
125			- \epsilon_{nh} f_u' \overline{ w }^{ik} \\
126			G_v^{Cor} & = &
127			- f_v \overline{ u }^{ij} \\
128			G_w^{Cor} & = &
129			\epsilon_{nh} f_w' \overline{ u }^{ik}
130			\end{eqnarray}
131			\marginpar{Need to change the default in code to match this}
132			where the subscripts on $f$ and $f'$ indicate evaluation of the
133			Coriolis parameters at the appropriate points in space. The above
134			discretization does {\em not} conserve anything, especially energy. An
135			option to recover this discretization has been retained for backward
136			compatibility testing (set run-time logical {\bf
137			useNonconservingCoriolis} to {\em true} which otherwise defaults to
138			{\em false}).
139
140			\fbox{ \begin{minipage}{4.75in}
141			{\em S/R MOM\_CDSCHEME} ({\em mom\_cdscheme.F})
142
143			{\em S/R MOM\_U\_CORIOLIS} ({\em mom\_u\_coriolis.F})
144
145			{\em S/R MOM\_V\_CORIOLIS} ({\em mom\_v\_coriolis.F})
146
147			$G_u^{Cor}$, $G_v^{Cor}$: {\bf cF} (local to {\em calc\_mom\_rhs.F})
148			\end{minipage} }
149
150
151			\subsection{Curvature metric terms}
152
153			The most commonly used coordinate system on the sphere is the
154			geographic system $(\lambda,\phi)$. The curvilinear nature of these
155			coordinates on the sphere lead to some ``metric'' terms in the
156			component momentum equations. Under the thin-atmosphere and
157			hydrostatic approximations these terms are discretized:
158			\begin{eqnarray}
159			{\cal A}_w \Delta r_f h_w G_u^{metric} & = &
160			\overline{ \frac{ \overline{u}^i }{a} \tan{\phi} {\cal A}_c \Delta r_f h_c \overline{ v }^j }^i \\
161			{\cal A}_s \Delta r_f h_s G_v^{metric} & = &
162			- \overline{ \frac{ \overline{u}^i }{a} \tan{\phi} {\cal A}_c \Delta r_f h_c \overline{ u }^i }^j \\
163			G_w^{metric} & = & 0
164			\end{eqnarray}
165			where $a$ is the radius of the planet (sphericity is assumed) or the
166			radial distance of the particle (i.e. a function of height). It is
167			easy to see that this discretization satisfies all the properties of
168			the discrete Coriolis terms since the metric factor $\frac{u}{a}
169			\tan{\phi}$ can be viewed as a modification of the vertical Coriolis
170			parameter: $f \rightarrow f+\frac{u}{a} \tan{\phi}$.
171
172			However, as for the Coriolis terms, a non-energy conserving form has
173			exclusively been used to date:
174			\begin{eqnarray}
175			G_u^{metric} & = & \frac{u \overline{v}^{ij} }{a} \tan{\phi} \\
176			G_v^{metric} & = & \frac{ \overline{u}^{ij} \overline{u}^{ij}}{a} \tan{\phi}
177			\end{eqnarray}
178			where $\tan{\phi}$ is evaluated at the $u$ and $v$ points
179			respectively.
180
181			\fbox{ \begin{minipage}{4.75in}
182			{\em S/R MOM\_U\_METRIC\_SPHERE} ({\em mom\_u\_metric\_sphere.F})
183
184			{\em S/R MOM\_V\_METRIC\_SPHERE} ({\em mom\_v\_metric\_sphere.F})
185
186			$G_u^{metric}$, $G_v^{metric}$: {\bf mT} (local to {\em calc\_mom\_rhs.F})
187			\end{minipage} }
188
189
190
191			\subsection{Non-hydrostatic metric terms}
192
193			For the non-hydrostatic equations, dropping the thin-atmosphere
194			approximation re-introduces metric terms involving $w$ and are
195			required to conserve anglular momentum:
196			\begin{eqnarray}
197			{\cal A}_w \Delta r_f h_w G_u^{metric} & = &
198			- \overline{ \frac{ \overline{u}^i \overline{w}^k }{a} {\cal A}_c \Delta r_f h_c }^i \\
199			{\cal A}_s \Delta r_f h_s G_v^{metric} & = &
200			- \overline{ \frac{ \overline{v}^j \overline{w}^k }{a} {\cal A}_c \Delta r_f h_c}^j \\
201			{\cal A}_c \Delta r_c G_w^{metric} & = &
202			\overline{ \frac{ {\overline{u}^i}^2 + {\overline{v}^j}^2}{a} {\cal A}_c \Delta r_f h_c }^k
203			\end{eqnarray}
204
205			Because we are always consistent, even if consistently wrong, we have,
206			in the past, used a different discretization in the model which is:
207			\begin{eqnarray}
208			G_u^{metric} & = &
209			- \frac{u}{a} \overline{w}^{ik} \\
210			G_v^{metric} & = &
211			- \frac{v}{a} \overline{w}^{jk} \\
212			G_w^{metric} & = &
213			\frac{1}{a} ( {\overline{u}^{ik}}^2 + {\overline{v}^{jk}}^2 )
214			\end{eqnarray}
215
216			\fbox{ \begin{minipage}{4.75in}
217			{\em S/R MOM\_U\_METRIC\_NH} ({\em mom\_u\_metric\_nh.F})
218
219			{\em S/R MOM\_V\_METRIC\_NH} ({\em mom\_v\_metric\_nh.F})
220
221			$G_u^{metric}$, $G_v^{metric}$: {\bf mT} (local to {\em calc\_mom\_rhs.F})
222			\end{minipage} }
223
224
225			\subsection{Lateral dissipation}
226
227			Historically, we have represented the SGS Reynolds stresses as simply
228			down gradient momentum fluxes, ignoring constraints on the stress
229			tensor such as symmetry.
230			\begin{eqnarray}
231			{\cal A}_w \Delta r_f h_w G_u^{h-diss} & = &
232			\delta_i \Delta y_f \Delta r_f h_c \tau_{11}
233			+ \delta_j \Delta x_v \Delta r_f h_\zeta \tau_{12} \\
234			{\cal A}_s \Delta r_f h_s G_v^{h-diss} & = &
235			\delta_i \Delta y_u \Delta r_f h_\zeta \tau_{21}
236			+ \delta_j \Delta x_f \Delta r_f h_c \tau_{22}
237			\end{eqnarray}
238			\marginpar{Check signs of stress definitions}
239
240			The lateral viscous stresses are discretized:
241			\begin{eqnarray}
242			\tau_{11} & = & A_h c_{11\Delta}(\phi) \frac{1}{\Delta x_f} \delta_i u
243			-A_4 c_{11\Delta^2}(\phi) \frac{1}{\Delta x_f} \delta_i \nabla^2 u \\
244			\tau_{12} & = & A_h c_{12\Delta}(\phi) \frac{1}{\Delta y_u} \delta_j u
245			-A_4 c_{12\Delta^2}(\phi)\frac{1}{\Delta y_u} \delta_j \nabla^2 u \\
246			\tau_{21} & = & A_h c_{21\Delta}(\phi) \frac{1}{\Delta x_v} \delta_i v
247			-A_4 c_{21\Delta^2}(\phi) \frac{1}{\Delta x_v} \delta_i \nabla^2 v \\
248			\tau_{22} & = & A_h c_{22\Delta}(\phi) \frac{1}{\Delta y_f} \delta_j v
249			-A_4 c_{22\Delta^2}(\phi) \frac{1}{\Delta y_f} \delta_j \nabla^2 v
250			\end{eqnarray}
251			where the non-dimensional factors $c_{lm\Delta^n}(\phi), \{l,m,n\} \in
252			\{1,2\}$ define the ``cosine'' scaling with latitude which can be
253			applied in various ad-hoc ways. For instance, $c_{11\Delta} =
254			c_{21\Delta} = (\cos{\phi})^{3/2}$, $c_{12\Delta}=c_{22\Delta}=0$ would
255			represent the an-isotropic cosine scaling typically used on the
256			``lat-lon'' grid for Laplacian viscosity.
257			\marginpar{Need to tidy up method for controlling this in code}
258
259			It should be noted that dispite the ad-hoc nature of the scaling, some
260			scaling must be done since on a lat-lon grid the converging meridians
261			make it very unlikely that a stable viscosity parameter exists across
262			the entire model domain.
263
264			The Laplacian viscosity coefficient, $A_h$ ({\bf viscAh}), has units
265			of $m^2 s^{-1}$. The bi-harmonic viscosity coefficient, $A_4$ ({\bf
266			viscA4}), has units of $m^4 s^{-1}$.
267
268			\fbox{ \begin{minipage}{4.75in}
269			{\em S/R MOM\_U\_XVISCFLUX} ({\em mom\_u\_xviscflux.F})
270
271			{\em S/R MOM\_U\_YVISCFLUX} ({\em mom\_u\_yviscflux.F})
272
273			{\em S/R MOM\_V\_XVISCFLUX} ({\em mom\_v\_xviscflux.F})
274
275			{\em S/R MOM\_V\_YVISCFLUX} ({\em mom\_v\_yviscflux.F})
276
277			$\tau_{11}$, $\tau_{12}$, $\tau_{22}$, $\tau_{22}$: {\bf vF}, {\bf
278			v4F} (local to {\em calc\_mom\_rhs.F})
279			\end{minipage} }
280
281			Two types of lateral boundary condition exist for the lateral viscous
282			terms, no-slip and free-slip.
283
284			The free-slip condition is most convenient to code since it is
285			equivalent to zero-stress on boundaries. Simple masking of the stress
286			components sets them to zero. The fractional open stress is properly
287			handled using the lopped cells.
288
289			The no-slip condition defines the normal gradient of a tangential flow
290			such that the flow is zero on the boundary. Rather than modify the
291			stresses by using complicated functions of the masks and ``ghost''
292			points (see \cite{Adcroft+Marshall98}) we add the boundary stresses as
293			an additional source term in cells next to solid boundaries. This has
294			the advantage of being able to cope with ``thin walls'' and also makes
295			the interior stress calculation (code) independent of the boundary
296			conditions. The ``body'' force takes the form:
297			\begin{eqnarray}
298			G_u^{side-drag} & = &
299			\frac{4}{\Delta z_f} \overline{ (1-h_\zeta) \frac{\Delta x_v}{\Delta y_u} }^j
300			\left( A_h c_{12\Delta}(\phi) u - A_4 c_{12\Delta^2}(\phi) \nabla^2 u \right)
301			\\
302			G_v^{side-drag} & = &
303			\frac{4}{\Delta z_f} \overline{ (1-h_\zeta) \frac{\Delta y_u}{\Delta x_v} }^i
304			\left( A_h c_{21\Delta}(\phi) v - A_4 c_{21\Delta^2}(\phi) \nabla^2 v \right)
305			\end{eqnarray}
306
307			In fact, the above discretization is not quite complete because it
308			assumes that the bathymetry at velocity points is deeper than at
309			neighbouring vorticity points, e.g. $1-h_w < 1-h_\zeta$
310
311			\fbox{ \begin{minipage}{4.75in}
312			{\em S/R MOM\_U\_SIDEDRAG} ({\em mom\_u\_sidedrag.F})
313
314			{\em S/R MOM\_V\_SIDEDRAG} ({\em mom\_v\_sidedrag.F})
315
316			$G_u^{side-drag}$, $G_v^{side-drag}$: {\bf vF} (local to {\em calc\_mom\_rhs.F})
317			\end{minipage} }
318
319
320			\subsection{Vertical dissipation}
321
322			Vertical viscosity terms are discretized with only partial adherence
323			to the variable grid lengths introduced by the finite volume
324			formulation. This reduces the formal accuracy of these terms to just
325			first order but only next to boundaries; exactly where other terms
326			appear such as linar and quadratic bottom drag.
327			\begin{eqnarray}
328			G_u^{v-diss} & = &
329			\frac{1}{\Delta r_f h_w} \delta_k \tau_{13} \\
330			G_v^{v-diss} & = &
331			\frac{1}{\Delta r_f h_s} \delta_k \tau_{23} \\
332			G_w^{v-diss} & = & \epsilon_{nh}
333			\frac{1}{\Delta r_f h_d} \delta_k \tau_{33}
334			\end{eqnarray}
335			represents the general discrete form of the vertical dissipation terms.
336
337			In the interior the vertical stresses are discretized:
338			\begin{eqnarray}
339			\tau_{13} & = & A_v \frac{1}{\Delta r_c} \delta_k u \\
340			\tau_{23} & = & A_v \frac{1}{\Delta r_c} \delta_k v \\
341			\tau_{33} & = & A_v \frac{1}{\Delta r_f} \delta_k w
342			\end{eqnarray}
343			It should be noted that in the non-hydrostatic form, the stress tensor
344			is even less consistent than for the hydrostatic (see Wazjowicz
345			\cite{Waojz}). It is well known how to do this properly (see Griffies
346			\cite{Griffies}) and is on the list of to-do's.
347
348			\fbox{ \begin{minipage}{4.75in}
349			{\em S/R MOM\_U\_RVISCLFUX} ({\em mom\_u\_rviscflux.F})
350
351			{\em S/R MOM\_V\_RVISCLFUX} ({\em mom\_v\_rviscflux.F})
352
353			$\tau_{13}$: {\bf urf} (local to {\em calc\_mom\_rhs.F})
354
355			$\tau_{23}$: {\bf vrf} (local to {\em calc\_mom\_rhs.F})
356			\end{minipage} }
357
358
359			As for the lateral viscous terms, the free-slip condition is
360			equivalent to simply setting the stress to zero on boundaries. The
361			no-slip condition is implemented as an additional term acting on top
362			of the interior and free-slip stresses. Bottom drag represents
363			additional friction, in addition to that imposed by the no-slip
364			condition at the bottom. The drag is cast as a stress expressed as a
365			linear or quadratic function of the mean flow in the layer above the
366			topography:
367			\begin{eqnarray}
368			\tau_{13}^{bottom-drag} & = &
369			\left(
370			2 A_v \frac{1}{\Delta r_c}
371			+ r_b
372			+ C_d \sqrt{ \overline{2 KE}^i }
373			\right) u \\
374			\tau_{23}^{bottom-drag} & = &
375			\left(
376			2 A_v \frac{1}{\Delta r_c}
377			+ r_b
378			+ C_d \sqrt{ \overline{2 KE}^j }
379			\right) v
380			\end{eqnarray}
381			where these terms are only evaluated immediately above topography.
382			$r_b$ ({\bf bottomDragLinear}) has units of $m s^{-1}$ and a typical value
383			of the order 0.0002 $m s^{-1}$. $C_d$ ({\bf bottomDragQuadratic}) is
384			dimensionless with typical values in the range 0.001--0.003.
385
386			\fbox{ \begin{minipage}{4.75in}
387			{\em S/R MOM\_U\_BOTTOMDRAG} ({\em mom\_u\_bottomdrag.F})
388
389			{\em S/R MOM\_V\_BOTTOMDRAG} ({\em mom\_v\_bottomdrag.F})
390
391			$\tau_{13}^{bottom-drag}$, $\tau_{23}^{bottom-drag}$: {\bf vf} (local to {\em calc\_mom\_rhs.F})
392			\end{minipage} }
393
394			\subsection{Derivation of discrete energy conservation}
395
396			These discrete equations conserve kinetic plus potential energy using the
397			following definitions:
398			\begin{equation}
399			KE = \frac{1}{2} \left( \overline{ u^2 }^i + \overline{ v^2 }^j +
400			\epsilon_{nh} \overline{ w^2 }^k \right)
401			\end{equation}
402