| 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 |
|
Parent Directory
| 
Revision Log
| 
Revision Graph