/[MITgcm]/manual/s_algorithm/text/mom_fluxform.tex

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-revision 1.1 by adcroft,
Thu Aug  9 19:48:39 2001 UTC
+revision 1.9 by jmc,
Tue Aug  2 20:54:44 2005 UTC
 Line 2
  % $Name$
  \section{Flux-form momentum equations}
+ \label{sect:flux-form_momentum_equations}
+ \begin{rawhtml}
+ <!-- CMIREDIR:flux-form_momentum_eqautions: -->
+ \end{rawhtml}
  The original finite volume model was based on the Eulerian flux form
  momentum equations. This is the default though the vector invariant
-Line 15 
 dissipation and metric forces:
+Line 19 
 dissipation and metric forces:
  \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_u^{metric} + G_u^{nh-metric} \label{eq:gsplit_momu} \\
  G_v & = & G_v^{adv} + G_v^{cor} + G_v^{h-diss} + G_v^{v-diss} +
- G_v^{metric} + G_v^{nh-metric} \\
+ G_v^{metric} + G_v^{nh-metric} \label{eq:gsplit_momv} \\
  G_w & = & G_w^{adv} + G_w^{cor} + G_w^{h-diss} + G_w^{v-diss} +
- G_w^{metric} + G_w^{nh-metric}
+ G_w^{metric} + G_w^{nh-metric} \label{eq:gsplit_momw}
  \end{eqnarray}
  In the hydrostatic limit, $G_w=0$ and $\epsilon_{nh}=0$, reducing the
  vertical momentum to hydrostatic balance.
-Line 46 
 The advective operator is second order a
+Line 50 
 The advective operator is second order a
  {\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 \\
+ + \delta_k \overline{ W }^i \overline{ u }^k \label{eq:discrete-momadvu} \\
  {\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 \\
+ + \delta_k \overline{ W }^j \overline{ v }^k \label{eq:discrete-momadvv} \\
  {\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 \\
+ + \delta_k \overline{ W }^k \overline{ w }^k \label{eq:discrete-momadvw}
  \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
-Line 63 
 defined:
+Line 67 
 defined:
  \marginpar{$V$: {\bf vTrans} }
  \marginpar{$W$: {\bf rTrans} }
  \begin{eqnarray}
- U & = & \Delta y_g \Delta r_f h_w u \\
+ U & = & \Delta y_g \Delta r_f h_w u \label{eq:utrans} \\
- V & = & \Delta x_g \Delta r_f h_s v \\
+ V & = & \Delta x_g \Delta r_f h_s v \label{eq:vtrans} \\
- W & = & {\cal A}_c w
+ W & = & {\cal A}_c w \label{eq:rtrans}
  \end{eqnarray}
  The advection of momentum takes the same form as the advection of
  tracers but by a translated advective flow. Consequently, the
-Line 106 
 discretized:
+Line 110 
 discretized:
  \end{eqnarray}
  where the Coriolis parameters $f$ and $f'$ are defined:
  \begin{eqnarray}
- f & = & 2 \Omega \sin{\phi} \\
+ f & = & 2 \Omega \sin{\varphi} \\
- f' & = & 2 \Omega \cos{\phi}
+ f' & = & 2 \Omega \cos{\varphi}
  \end{eqnarray}
- when using spherical geometry, otherwise the $\beta$-plane definition is used:
+ where $\varphi$ is geographic latitude when using spherical geometry,
+ otherwise the $\beta$-plane definition is used:
  \begin{eqnarray}
  f & = & f_o + \beta y \\
  f' & = & 0
-Line 131 
 G_w^{Cor} & = &
+Line 136 
 G_w^{Cor} & = &
  \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
+ discretization does {\em not} conserve anything, especially energy and
- option to recover this discretization has been retained for backward
+ for historical reasons is the default for the code. A
- compatibility testing (set run-time logical {\bf
+ flag controls this discretization: set run-time logical {\bf
- useNonconservingCoriolis} to {\em true} which otherwise defaults to
+ useEnergyConservingCoriolis} to {\em true} which otherwise defaults to
- {\em false}).
+ {\em false}.
  \fbox{ \begin{minipage}{4.75in}
  {\em S/R MOM\_CDSCHEME} ({\em mom\_cdscheme.F})
-Line 151 
 $G_u^{Cor}$, $G_v^{Cor}$: {\bf cF} (loca
+Line 156 
 $G_u^{Cor}$, $G_v^{Cor}$: {\bf cF} (loca
  \subsection{Curvature metric terms}
  The most commonly used coordinate system on the sphere is the
- geographic system $(\lambda,\phi)$. The curvilinear nature of these
+ geographic system $(\lambda,\varphi)$. 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 \\
+   \overline{ \frac{ \overline{u}^i }{a} \tan{\varphi} {\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 \\
+ - \overline{ \frac{ \overline{u}^i }{a} \tan{\varphi} {\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
+ \tan{\varphi}$ can be viewed as a modification of the vertical Coriolis
- parameter: $f \rightarrow f+\frac{u}{a} \tan{\phi}$.
+ parameter: $f \rightarrow f+\frac{u}{a} \tan{\varphi}$.
  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_u^{metric} & = & \frac{u \overline{v}^{ij} }{a} \tan{\varphi} \\
- 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}
  \end{eqnarray}
- where $\tan{\phi}$ is evaluated at the $u$ and $v$ points
+ where $\tan{\varphi}$ is evaluated at the $u$ and $v$ points
  respectively.
  \fbox{ \begin{minipage}{4.75in}
-Line 192 
 $G_u^{metric}$, $G_v^{metric}$: {\bf mT}
+Line 197 
 $G_u^{metric}$, $G_v^{metric}$: {\bf mT}
  For the non-hydrostatic equations, dropping the thin-atmosphere
  approximation re-introduces metric terms involving $w$ and are
- required to conserve anglular momentum:
+ required to conserve angular 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 \\
-Line 239 
 tensor such as symmetry.
+Line 244 
 tensor such as symmetry.
  The lateral viscous stresses are discretized:
  \begin{eqnarray}
- \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
-                -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 \\
- \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
-                -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 \\
- \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
-                -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 \\
- \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
-                -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
  \end{eqnarray}
- where the non-dimensional factors $c_{lm\Delta^n}(\phi), \{l,m,n\} \in
+ where the non-dimensional factors $c_{lm\Delta^n}(\varphi), \{l,m,n\} \in
  \{1,2\}$ 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
+ c_{21\Delta} = (\cos{\varphi})^{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
+ It should be noted that despite 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.
-Line 289 
 handled using the lopped cells.
+Line 294 
 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
+ points (see \cite{adcroft:98}) 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
-Line 297 
 conditions. The ``body'' force takes the
+Line 302 
 conditions. The ``body'' force takes the
  \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)
+ \left( A_h c_{12\Delta}(\varphi) u - A_4 c_{12\Delta^2}(\varphi) \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)
+ \left( A_h c_{21\Delta}(\varphi) v - A_4 c_{21\Delta^2}(\varphi) \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. $1-h_w < 1-h_\zeta$
+ neighboring 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})
-Line 323 
 Vertical viscosity terms are discretized
+Line 328 
 Vertical viscosity terms are discretized
  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.
+ appear such as linear and quadratic bottom drag.
  \begin{eqnarray}
  G_u^{v-diss} & = &
  \frac{1}{\Delta r_f h_w} \delta_k \tau_{13} \\
-Line 341 
 In the interior the vertical stresses ar
+Line 346 
 In the interior the vertical stresses ar
  \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
+ is even less consistent than for the hydrostatic (see
- \cite{Waojz}). It is well known how to do this properly (see Griffies
+ \cite{wajsowicz:93}). It is well known how to do this properly (see
- \cite{Griffies}) and is on the list of to-do's.
+ \cite{griffies:00}) 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})

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