| 526 |
atmospheric/oceanic form is selected based on the string variable {\bf |
atmospheric/oceanic form is selected based on the string variable {\bf |
| 527 |
buoyancyRelation}. |
buoyancyRelation}. |
| 528 |
|
|
|
\subsection{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}) |
|
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|
|
|
$G_w$: {\bf Gw} ({\em DYNVARS.h}) |
|
|
\end{minipage} } |
|
|
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|
|
|
|
|
\subsubsection{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} } |
|
|
|
|
|
|
|
|
|
|
|
\subsubsection{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} } |
|
|
|
|
|
|
|
|
\subsubsection{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} } |
|
|
|
|
|
|
|
|
|
|
|
\subsubsection{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} } |
|
|
|
|
|
|
|
|
\subsubsection{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} } |
|
|
|
|
|
|
|
|
\subsubsection{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 |
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|
of the interior and free-slip stresses. Bottom drag represents |
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|
additional friction, in addition to that imposed by the no-slip |
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|
condition at the bottom. The drag is cast as a stress expressed as a |
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|
linear or quadratic function of the mean flow in the layer above the |
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|
topography: |
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|
\begin{eqnarray} |
|
|
\tau_{13}^{bottom-drag} & = & |
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|
\left( |
|
|
2 A_v \frac{1}{\Delta r_c} |
|
|
+ r_b |
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|
+ C_d \sqrt{ \overline{2 KE}^i } |
|
|
\right) u \\ |
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|
\tau_{23}^{bottom-drag} & = & |
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|
\left( |
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|
2 A_v \frac{1}{\Delta r_c} |
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|
+ r_b |
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|
+ C_d \sqrt{ \overline{2 KE}^j } |
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|
\right) v |
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|
\end{eqnarray} |
|
|
where these terms are only evaluated immediately above topography. |
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|
$r_b$ ({\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. |
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\fbox{ \begin{minipage}{4.75in} |
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{\em S/R MOM\_U\_BOTTOMDRAG} ({\em mom\_u\_bottomdrag.F}) |
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{\em S/R MOM\_V\_BOTTOMDRAG} ({\em mom\_v\_bottomdrag.F}) |
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$\tau_{13}^{bottom-drag}$, $\tau_{23}^{bottom-drag}$: {\bf vf} (local to {\em calc\_mom\_rhs.F}) |
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|
\end{minipage} } |
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\subsection{Tracer equations} |
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|
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The tracer equations are discretized consistantly with the continuity |
|
|
equation to facilitate conservation properties analogous to the |
|
|
continuum: |
|
|
\begin{equation} |
|
|
{\cal A}_c \Delta r_f h_c \partial_\theta |
|
|
+ \delta_i U \overline{ \theta }^i |
|
|
+ \delta_j V \overline{ \theta }^j |
|
|
+ \delta_k W \overline{ \theta }^k |
|
|
= {\cal A}_c \Delta r_f h_c {\cal S}_\theta + \theta {\cal A}_c \delta_k (P-E)_{r=0} |
|
|
\end{equation} |
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|
The quantities $U$, $V$ and $W$ are volume fluxes defined: |
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|
\marginpar{$U$: {\bf uTrans} } |
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|
\marginpar{$V$: {\bf vTrans} } |
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|
\marginpar{$W$: {\bf rTrans} } |
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|
\begin{eqnarray} |
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|
U & = & \Delta y_g \Delta r_f h_w u \\ |
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V & = & \Delta x_g \Delta r_f h_s v \\ |
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W & = & {\cal A}_c w |
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|
\end{eqnarray} |
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|
${\cal S}$ represents the ``parameterized'' SGS processes and |
|
|
physics associated with the tracer. For instance, potential |
|
|
temperature equation in the ocean has is forced by surface and |
|
|
partially penetrating heat fluxes: |
|
|
\begin{equation} |
|
|
{\cal A}_c \Delta r_f h_c {\cal S}_\theta = \frac{1}{c_p \rho_o} \delta_k {\cal A}_c {\cal Q} |
|
|
\end{equation} |
|
|
while the salt equation has no real sources, ${\cal S}=0$, which |
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|
leaves just the $P-E$ term. |
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|
The continuity equation can be recovered by setting ${\cal Q}=0$ and |
|
|
$\theta=1$. The term $\theta (P-E)_{r=0}$ is required to retain local |
|
|
conservation of $\theta$. Global conservation is not possible using |
|
|
the flux-form (as here) and a linearized free-surface |
|
|
(\cite{Griffies00,Campin02}). |
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\subsection{Derivation of discrete energy conservation} |
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|
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|
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) |
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|
\end{equation} |
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|
\subsection{Vector invariant momentum equations} |
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|
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|
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. |
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|
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|
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. |
|
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|
|
|
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}) |
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|
|
$G_u$: {\bf Gu} ({\em DYNVARS.h}) |
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|
|
$G_v$: {\bf Gv} ({\em DYNVARS.h}) |
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|
$G_w$: {\bf Gw} ({\em DYNVARS.h}) |
|
|
\end{minipage} } |
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|
|
|
\subsubsection{Relative vorticity} |
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|
|
|
|
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} } |
|
|
|
|
|
|
|
|
\subsubsection{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} } |
|
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|
|
|
\subsubsection{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} } |
|
|
|
|
|
|
|
|
\subsubsection{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} } |
|
|
|
|
|
|
|
|
|
|
|
\subsubsection{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} } |
|
|
|
|
|
|
|
|
|
|
|
\subsubsection{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} } |
|
|
|
|
|
|
|
|
\subsubsection{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} } |
|
|
|
|
|
|
|
|
\subsubsection{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} } |
|
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