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% $Name$ |
% $Name$ |
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\section{Flux-form momentum equations} |
\section{Flux-form momentum equations} |
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\label{sec:flux-form_momentum_eqautions} |
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The original finite volume model was based on the Eulerian flux form |
The original finite volume model was based on the Eulerian flux form |
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momentum equations. This is the default though the vector invariant |
momentum equations. This is the default though the vector invariant |
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\marginpar{$G_w$: {\bf Gw} } |
\marginpar{$G_w$: {\bf Gw} } |
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\begin{eqnarray} |
\begin{eqnarray} |
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G_u & = & G_u^{adv} + G_u^{cor} + G_u^{h-diss} + G_u^{v-diss} + |
G_u & = & G_u^{adv} + G_u^{cor} + G_u^{h-diss} + G_u^{v-diss} + |
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G_u^{metric} + G_u^{nh-metric} \\ |
G_u^{metric} + G_u^{nh-metric} \label{eq:gsplit_momu} \\ |
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G_v & = & G_v^{adv} + G_v^{cor} + G_v^{h-diss} + G_v^{v-diss} + |
G_v & = & G_v^{adv} + G_v^{cor} + G_v^{h-diss} + G_v^{v-diss} + |
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G_v^{metric} + G_v^{nh-metric} \\ |
G_v^{metric} + G_v^{nh-metric} \label{eq:gsplit_momv} \\ |
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G_w & = & G_w^{adv} + G_w^{cor} + G_w^{h-diss} + G_w^{v-diss} + |
G_w & = & G_w^{adv} + G_w^{cor} + G_w^{h-diss} + G_w^{v-diss} + |
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G_w^{metric} + G_w^{nh-metric} |
G_w^{metric} + G_w^{nh-metric} \label{eq:gsplit_momw} |
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\end{eqnarray} |
\end{eqnarray} |
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In the hydrostatic limit, $G_w=0$ and $\epsilon_{nh}=0$, reducing the |
In the hydrostatic limit, $G_w=0$ and $\epsilon_{nh}=0$, reducing the |
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vertical momentum to hydrostatic balance. |
vertical momentum to hydrostatic balance. |
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{\cal A}_w \Delta r_f h_w G_u^{adv} & = & |
{\cal A}_w \Delta r_f h_w G_u^{adv} & = & |
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\delta_i \overline{ U }^i \overline{ u }^i |
\delta_i \overline{ U }^i \overline{ u }^i |
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+ \delta_j \overline{ V }^i \overline{ u }^j |
+ \delta_j \overline{ V }^i \overline{ u }^j |
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+ \delta_k \overline{ W }^i \overline{ u }^k \\ |
+ \delta_k \overline{ W }^i \overline{ u }^k \label{eq:discrete-momadvu} \\ |
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{\cal A}_s \Delta r_f h_s G_v^{adv} & = & |
{\cal A}_s \Delta r_f h_s G_v^{adv} & = & |
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\delta_i \overline{ U }^j \overline{ v }^i |
\delta_i \overline{ U }^j \overline{ v }^i |
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+ \delta_j \overline{ V }^j \overline{ v }^j |
+ \delta_j \overline{ V }^j \overline{ v }^j |
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+ \delta_k \overline{ W }^j \overline{ v }^k \\ |
+ \delta_k \overline{ W }^j \overline{ v }^k \label{eq:discrete-momadvv} \\ |
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{\cal A}_c \Delta r_c G_w^{adv} & = & |
{\cal A}_c \Delta r_c G_w^{adv} & = & |
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\delta_i \overline{ U }^k \overline{ w }^i |
\delta_i \overline{ U }^k \overline{ w }^i |
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+ \delta_j \overline{ V }^k \overline{ w }^j |
+ \delta_j \overline{ V }^k \overline{ w }^j |
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+ \delta_k \overline{ W }^k \overline{ w }^k \\ |
+ \delta_k \overline{ W }^k \overline{ w }^k \label{eq:discrete-momadvw} |
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\end{eqnarray} |
\end{eqnarray} |
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and because of the flux form does not contribute to the global budget |
and because of the flux form does not contribute to the global budget |
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of linear momentum. The quantities $U$, $V$ and $W$ are volume fluxes |
of linear momentum. The quantities $U$, $V$ and $W$ are volume fluxes |
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\marginpar{$V$: {\bf vTrans} } |
\marginpar{$V$: {\bf vTrans} } |
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\marginpar{$W$: {\bf rTrans} } |
\marginpar{$W$: {\bf rTrans} } |
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\begin{eqnarray} |
\begin{eqnarray} |
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U & = & \Delta y_g \Delta r_f h_w u \\ |
U & = & \Delta y_g \Delta r_f h_w u \label{eq:utrans} \\ |
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V & = & \Delta x_g \Delta r_f h_s v \\ |
V & = & \Delta x_g \Delta r_f h_s v \label{eq:vtrans} \\ |
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W & = & {\cal A}_c w |
W & = & {\cal A}_c w \label{eq:rtrans} |
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\end{eqnarray} |
\end{eqnarray} |
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The advection of momentum takes the same form as the advection of |
The advection of momentum takes the same form as the advection of |
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tracers but by a translated advective flow. Consequently, the |
tracers but by a translated advective flow. Consequently, the |
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\end{eqnarray} |
\end{eqnarray} |
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where the Coriolis parameters $f$ and $f'$ are defined: |
where the Coriolis parameters $f$ and $f'$ are defined: |
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\begin{eqnarray} |
\begin{eqnarray} |
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f & = & 2 \Omega \sin{\phi} \\ |
f & = & 2 \Omega \sin{\varphi} \\ |
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f' & = & 2 \Omega \cos{\phi} |
f' & = & 2 \Omega \cos{\varphi} |
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\end{eqnarray} |
\end{eqnarray} |
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when using spherical geometry, otherwise the $\beta$-plane definition is used: |
where $\varphi$ is geographic latitude when using spherical geometry, |
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|
otherwise the $\beta$-plane definition is used: |
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\begin{eqnarray} |
\begin{eqnarray} |
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f & = & f_o + \beta y \\ |
f & = & f_o + \beta y \\ |
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f' & = & 0 |
f' & = & 0 |
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\marginpar{Need to change the default in code to match this} |
\marginpar{Need to change the default in code to match this} |
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where the subscripts on $f$ and $f'$ indicate evaluation of the |
where the subscripts on $f$ and $f'$ indicate evaluation of the |
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Coriolis parameters at the appropriate points in space. The above |
Coriolis parameters at the appropriate points in space. The above |
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discretization does {\em not} conserve anything, especially energy. An |
discretization does {\em not} conserve anything, especially energy and |
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option to recover this discretization has been retained for backward |
for historical reasons is the default for the code. A |
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compatibility testing (set run-time logical {\bf |
flag controls this discretization: set run-time logical {\bf |
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useNonconservingCoriolis} to {\em true} which otherwise defaults to |
useEnergyConservingCoriolis} to {\em true} which otherwise defaults to |
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{\em false}). |
{\em false}. |
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\fbox{ \begin{minipage}{4.75in} |
\fbox{ \begin{minipage}{4.75in} |
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{\em S/R MOM\_CDSCHEME} ({\em mom\_cdscheme.F}) |
{\em S/R MOM\_CDSCHEME} ({\em mom\_cdscheme.F}) |
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\subsection{Curvature metric terms} |
\subsection{Curvature metric terms} |
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|
|
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The most commonly used coordinate system on the sphere is the |
The most commonly used coordinate system on the sphere is the |
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geographic system $(\lambda,\phi)$. The curvilinear nature of these |
geographic system $(\lambda,\varphi)$. The curvilinear nature of these |
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coordinates on the sphere lead to some ``metric'' terms in the |
coordinates on the sphere lead to some ``metric'' terms in the |
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component momentum equations. Under the thin-atmosphere and |
component momentum equations. Under the thin-atmosphere and |
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hydrostatic approximations these terms are discretized: |
hydrostatic approximations these terms are discretized: |
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\begin{eqnarray} |
\begin{eqnarray} |
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{\cal A}_w \Delta r_f h_w G_u^{metric} & = & |
{\cal A}_w \Delta r_f h_w G_u^{metric} & = & |
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\overline{ \frac{ \overline{u}^i }{a} \tan{\phi} {\cal A}_c \Delta r_f h_c \overline{ v }^j }^i \\ |
\overline{ \frac{ \overline{u}^i }{a} \tan{\varphi} {\cal A}_c \Delta r_f h_c \overline{ v }^j }^i \\ |
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{\cal A}_s \Delta r_f h_s G_v^{metric} & = & |
{\cal A}_s \Delta r_f h_s G_v^{metric} & = & |
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- \overline{ \frac{ \overline{u}^i }{a} \tan{\phi} {\cal A}_c \Delta r_f h_c \overline{ u }^i }^j \\ |
- \overline{ \frac{ \overline{u}^i }{a} \tan{\varphi} {\cal A}_c \Delta r_f h_c \overline{ u }^i }^j \\ |
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G_w^{metric} & = & 0 |
G_w^{metric} & = & 0 |
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\end{eqnarray} |
\end{eqnarray} |
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where $a$ is the radius of the planet (sphericity is assumed) or the |
where $a$ is the radius of the planet (sphericity is assumed) or the |
| 168 |
radial distance of the particle (i.e. a function of height). It is |
radial distance of the particle (i.e. a function of height). It is |
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easy to see that this discretization satisfies all the properties of |
easy to see that this discretization satisfies all the properties of |
| 170 |
the discrete Coriolis terms since the metric factor $\frac{u}{a} |
the discrete Coriolis terms since the metric factor $\frac{u}{a} |
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\tan{\phi}$ can be viewed as a modification of the vertical Coriolis |
\tan{\varphi}$ can be viewed as a modification of the vertical Coriolis |
| 172 |
parameter: $f \rightarrow f+\frac{u}{a} \tan{\phi}$. |
parameter: $f \rightarrow f+\frac{u}{a} \tan{\varphi}$. |
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|
|
| 174 |
However, as for the Coriolis terms, a non-energy conserving form has |
However, as for the Coriolis terms, a non-energy conserving form has |
| 175 |
exclusively been used to date: |
exclusively been used to date: |
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\begin{eqnarray} |
\begin{eqnarray} |
| 177 |
G_u^{metric} & = & \frac{u \overline{v}^{ij} }{a} \tan{\phi} \\ |
G_u^{metric} & = & \frac{u \overline{v}^{ij} }{a} \tan{\varphi} \\ |
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G_v^{metric} & = & \frac{ \overline{u}^{ij} \overline{u}^{ij}}{a} \tan{\phi} |
G_v^{metric} & = & \frac{ \overline{u}^{ij} \overline{u}^{ij}}{a} \tan{\varphi} |
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\end{eqnarray} |
\end{eqnarray} |
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where $\tan{\phi}$ is evaluated at the $u$ and $v$ points |
where $\tan{\varphi}$ is evaluated at the $u$ and $v$ points |
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respectively. |
respectively. |
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|
|
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\fbox{ \begin{minipage}{4.75in} |
\fbox{ \begin{minipage}{4.75in} |
| 194 |
|
|
| 195 |
For the non-hydrostatic equations, dropping the thin-atmosphere |
For the non-hydrostatic equations, dropping the thin-atmosphere |
| 196 |
approximation re-introduces metric terms involving $w$ and are |
approximation re-introduces metric terms involving $w$ and are |
| 197 |
required to conserve anglular momentum: |
required to conserve angular momentum: |
| 198 |
\begin{eqnarray} |
\begin{eqnarray} |
| 199 |
{\cal A}_w \Delta r_f h_w G_u^{metric} & = & |
{\cal A}_w \Delta r_f h_w G_u^{metric} & = & |
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- \overline{ \frac{ \overline{u}^i \overline{w}^k }{a} {\cal A}_c \Delta r_f h_c }^i \\ |
- \overline{ \frac{ \overline{u}^i \overline{w}^k }{a} {\cal A}_c \Delta r_f h_c }^i \\ |
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|
|
| 242 |
The lateral viscous stresses are discretized: |
The lateral viscous stresses are discretized: |
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\begin{eqnarray} |
\begin{eqnarray} |
| 244 |
\tau_{11} & = & A_h c_{11\Delta}(\phi) \frac{1}{\Delta x_f} \delta_i u |
\tau_{11} & = & A_h c_{11\Delta}(\varphi) \frac{1}{\Delta x_f} \delta_i u |
| 245 |
-A_4 c_{11\Delta^2}(\phi) \frac{1}{\Delta x_f} \delta_i \nabla^2 u \\ |
-A_4 c_{11\Delta^2}(\varphi) \frac{1}{\Delta x_f} \delta_i \nabla^2 u \\ |
| 246 |
\tau_{12} & = & A_h c_{12\Delta}(\phi) \frac{1}{\Delta y_u} \delta_j u |
\tau_{12} & = & A_h c_{12\Delta}(\varphi) \frac{1}{\Delta y_u} \delta_j u |
| 247 |
-A_4 c_{12\Delta^2}(\phi)\frac{1}{\Delta y_u} \delta_j \nabla^2 u \\ |
-A_4 c_{12\Delta^2}(\varphi)\frac{1}{\Delta y_u} \delta_j \nabla^2 u \\ |
| 248 |
\tau_{21} & = & A_h c_{21\Delta}(\phi) \frac{1}{\Delta x_v} \delta_i v |
\tau_{21} & = & A_h c_{21\Delta}(\varphi) \frac{1}{\Delta x_v} \delta_i v |
| 249 |
-A_4 c_{21\Delta^2}(\phi) \frac{1}{\Delta x_v} \delta_i \nabla^2 v \\ |
-A_4 c_{21\Delta^2}(\varphi) \frac{1}{\Delta x_v} \delta_i \nabla^2 v \\ |
| 250 |
\tau_{22} & = & A_h c_{22\Delta}(\phi) \frac{1}{\Delta y_f} \delta_j v |
\tau_{22} & = & A_h c_{22\Delta}(\varphi) \frac{1}{\Delta y_f} \delta_j v |
| 251 |
-A_4 c_{22\Delta^2}(\phi) \frac{1}{\Delta y_f} \delta_j \nabla^2 v |
-A_4 c_{22\Delta^2}(\varphi) \frac{1}{\Delta y_f} \delta_j \nabla^2 v |
| 252 |
\end{eqnarray} |
\end{eqnarray} |
| 253 |
where the non-dimensional factors $c_{lm\Delta^n}(\phi), \{l,m,n\} \in |
where the non-dimensional factors $c_{lm\Delta^n}(\varphi), \{l,m,n\} \in |
| 254 |
\{1,2\}$ define the ``cosine'' scaling with latitude which can be |
\{1,2\}$ define the ``cosine'' scaling with latitude which can be |
| 255 |
applied in various ad-hoc ways. For instance, $c_{11\Delta} = |
applied in various ad-hoc ways. For instance, $c_{11\Delta} = |
| 256 |
c_{21\Delta} = (\cos{\phi})^{3/2}$, $c_{12\Delta}=c_{22\Delta}=0$ would |
c_{21\Delta} = (\cos{\varphi})^{3/2}$, $c_{12\Delta}=c_{22\Delta}=0$ would |
| 257 |
represent the an-isotropic cosine scaling typically used on the |
represent the an-isotropic cosine scaling typically used on the |
| 258 |
``lat-lon'' grid for Laplacian viscosity. |
``lat-lon'' grid for Laplacian viscosity. |
| 259 |
\marginpar{Need to tidy up method for controlling this in code} |
\marginpar{Need to tidy up method for controlling this in code} |
| 260 |
|
|
| 261 |
It should be noted that dispite the ad-hoc nature of the scaling, some |
It should be noted that despite the ad-hoc nature of the scaling, some |
| 262 |
scaling must be done since on a lat-lon grid the converging meridians |
scaling must be done since on a lat-lon grid the converging meridians |
| 263 |
make it very unlikely that a stable viscosity parameter exists across |
make it very unlikely that a stable viscosity parameter exists across |
| 264 |
the entire model domain. |
the entire model domain. |
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\begin{eqnarray} |
\begin{eqnarray} |
| 300 |
G_u^{side-drag} & = & |
G_u^{side-drag} & = & |
| 301 |
\frac{4}{\Delta z_f} \overline{ (1-h_\zeta) \frac{\Delta x_v}{\Delta y_u} }^j |
\frac{4}{\Delta z_f} \overline{ (1-h_\zeta) \frac{\Delta x_v}{\Delta y_u} }^j |
| 302 |
\left( A_h c_{12\Delta}(\phi) u - A_4 c_{12\Delta^2}(\phi) \nabla^2 u \right) |
\left( A_h c_{12\Delta}(\varphi) u - A_4 c_{12\Delta^2}(\varphi) \nabla^2 u \right) |
| 303 |
\\ |
\\ |
| 304 |
G_v^{side-drag} & = & |
G_v^{side-drag} & = & |
| 305 |
\frac{4}{\Delta z_f} \overline{ (1-h_\zeta) \frac{\Delta y_u}{\Delta x_v} }^i |
\frac{4}{\Delta z_f} \overline{ (1-h_\zeta) \frac{\Delta y_u}{\Delta x_v} }^i |
| 306 |
\left( A_h c_{21\Delta}(\phi) v - A_4 c_{21\Delta^2}(\phi) \nabla^2 v \right) |
\left( A_h c_{21\Delta}(\varphi) v - A_4 c_{21\Delta^2}(\varphi) \nabla^2 v \right) |
| 307 |
\end{eqnarray} |
\end{eqnarray} |
| 308 |
|
|
| 309 |
In fact, the above discretization is not quite complete because it |
In fact, the above discretization is not quite complete because it |
| 310 |
assumes that the bathymetry at velocity points is deeper than at |
assumes that the bathymetry at velocity points is deeper than at |
| 311 |
neighbouring vorticity points, e.g. $1-h_w < 1-h_\zeta$ |
neighboring vorticity points, e.g. $1-h_w < 1-h_\zeta$ |
| 312 |
|
|
| 313 |
\fbox{ \begin{minipage}{4.75in} |
\fbox{ \begin{minipage}{4.75in} |
| 314 |
{\em S/R MOM\_U\_SIDEDRAG} ({\em mom\_u\_sidedrag.F}) |
{\em S/R MOM\_U\_SIDEDRAG} ({\em mom\_u\_sidedrag.F}) |
| 325 |
to the variable grid lengths introduced by the finite volume |
to the variable grid lengths introduced by the finite volume |
| 326 |
formulation. This reduces the formal accuracy of these terms to just |
formulation. This reduces the formal accuracy of these terms to just |
| 327 |
first order but only next to boundaries; exactly where other terms |
first order but only next to boundaries; exactly where other terms |
| 328 |
appear such as linar and quadratic bottom drag. |
appear such as linear and quadratic bottom drag. |
| 329 |
\begin{eqnarray} |
\begin{eqnarray} |
| 330 |
G_u^{v-diss} & = & |
G_u^{v-diss} & = & |
| 331 |
\frac{1}{\Delta r_f h_w} \delta_k \tau_{13} \\ |
\frac{1}{\Delta r_f h_w} \delta_k \tau_{13} \\ |
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