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% $Header: /u/u0/gcmpack/mitgcmdoc/part2/mom_fluxform.tex,v 1.4 2001/10/25 00:55:28 cnh Exp $ |
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% $Name: $ |
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|
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\section{Flux-form momentum equations} |
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\label{sec:flux-form_momentum_eqautions} |
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|
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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 |
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form is optionally available (and recommended in some cases). |
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|
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The ``G's'' (our colloquial name for all terms on rhs!) are broken |
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into the various advective, Coriolis, horizontal dissipation, vertical |
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dissipation and metric forces: |
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\marginpar{$G_u$: {\bf Gu} } |
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\marginpar{$G_v$: {\bf Gv} } |
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\marginpar{$G_w$: {\bf Gw} } |
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\begin{eqnarray} |
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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} \label{eq:gsplit_momu} \\ |
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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} \label{eq:gsplit_momv} \\ |
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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} \label{eq:gsplit_momw} |
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\end{eqnarray} |
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In the hydrostatic limit, $G_w=0$ and $\epsilon_{nh}=0$, reducing the |
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vertical momentum to hydrostatic balance. |
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|
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These terms are calculated in routines called from subroutine {\em |
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CALC\_MOM\_RHS} a collected into the global arrays {\bf Gu}, {\bf Gv}, |
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and {\bf Gw}. |
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|
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\fbox{ \begin{minipage}{4.75in} |
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{\em S/R CALC\_MOM\_RHS} ({\em pkg/mom\_fluxform/calc\_mom\_rhs.F}) |
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|
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$G_u$: {\bf Gu} ({\em DYNVARS.h}) |
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|
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$G_v$: {\bf Gv} ({\em DYNVARS.h}) |
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|
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$G_w$: {\bf Gw} ({\em DYNVARS.h}) |
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\end{minipage} } |
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|
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|
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\subsection{Advection of momentum} |
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|
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The advective operator is second order accurate in space: |
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\begin{eqnarray} |
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{\cal A}_w \Delta r_f h_w G_u^{adv} & = & |
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\delta_i \overline{ U }^i \overline{ u }^i |
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+ \delta_j \overline{ V }^i \overline{ u }^j |
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+ \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} & = & |
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\delta_i \overline{ U }^j \overline{ v }^i |
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+ \delta_j \overline{ V }^j \overline{ v }^j |
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+ \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} & = & |
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\delta_i \overline{ U }^k \overline{ w }^i |
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+ \delta_j \overline{ V }^k \overline{ w }^j |
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+ \delta_k \overline{ W }^k \overline{ w }^k \label{eq:discrete-momadvw} |
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\end{eqnarray} |
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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 |
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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 \label{eq:utrans} \\ |
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V & = & \Delta x_g \Delta r_f h_s v \label{eq:vtrans} \\ |
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W & = & {\cal A}_c w \label{eq:rtrans} |
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\end{eqnarray} |
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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 |
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conservation of second moments, derived for tracers later, applies to |
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$u^2$ and $v^2$ and $w^2$ so that advection of momentum correctly |
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conserves kinetic energy. |
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|
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\fbox{ \begin{minipage}{4.75in} |
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{\em S/R MOM\_U\_ADV\_UU} ({\em mom\_u\_adv\_uu.F}) |
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|
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{\em S/R MOM\_U\_ADV\_VU} ({\em mom\_u\_adv\_vu.F}) |
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|
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{\em S/R MOM\_U\_ADV\_WU} ({\em mom\_u\_adv\_wu.F}) |
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|
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{\em S/R MOM\_U\_ADV\_UV} ({\em mom\_u\_adv\_uv.F}) |
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|
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{\em S/R MOM\_U\_ADV\_VV} ({\em mom\_u\_adv\_vv.F}) |
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|
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{\em S/R MOM\_U\_ADV\_WV} ({\em mom\_u\_adv\_wv.F}) |
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|
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$uu$, $uv$, $vu$, $vv$: {\bf aF} (local to {\em calc\_mom\_rhs.F}) |
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\end{minipage} } |
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|
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|
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|
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\subsection{Coriolis terms} |
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|
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The ``pure C grid'' Coriolis terms (i.e. in absence of C-D scheme) are |
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discretized: |
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\begin{eqnarray} |
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{\cal A}_w \Delta r_f h_w G_u^{Cor} & = & |
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\overline{ f {\cal A}_c \Delta r_f h_c \overline{ v }^j }^i |
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- \epsilon_{nh} \overline{ f' {\cal A}_c \Delta r_f h_c \overline{ w }^k }^i \\ |
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{\cal A}_s \Delta r_f h_s G_v^{Cor} & = & |
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- \overline{ f {\cal A}_c \Delta r_f h_c \overline{ u }^i }^j \\ |
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{\cal A}_c \Delta r_c G_w^{Cor} & = & |
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\epsilon_{nh} \overline{ f' {\cal A}_c \Delta r_f h_c \overline{ u }^i }^k |
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\end{eqnarray} |
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where the Coriolis parameters $f$ and $f'$ are defined: |
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\begin{eqnarray} |
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f & = & 2 \Omega \sin{\varphi} \\ |
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f' & = & 2 \Omega \cos{\varphi} |
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\end{eqnarray} |
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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} |
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f & = & f_o + \beta y \\ |
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f' & = & 0 |
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\end{eqnarray} |
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|
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This discretization globally conserves kinetic energy. It should be |
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noted that despite the use of this discretization in former |
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publications, all calculations to date have used the following |
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different discretization: |
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\begin{eqnarray} |
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G_u^{Cor} & = & |
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f_u \overline{ v }^{ji} |
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- \epsilon_{nh} f_u' \overline{ w }^{ik} \\ |
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G_v^{Cor} & = & |
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- f_v \overline{ u }^{ij} \\ |
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G_w^{Cor} & = & |
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\epsilon_{nh} f_w' \overline{ u }^{ik} |
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\end{eqnarray} |
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\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 |
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Coriolis parameters at the appropriate points in space. The above |
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discretization does {\em not} conserve anything, especially energy and |
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for historical reasons is the default for the code. A |
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flag controls this discretization: set run-time logical {\bf |
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useEnergyConservingCoriolis} to {\em true} which otherwise defaults to |
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{\em false}. |
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|
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\fbox{ \begin{minipage}{4.75in} |
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{\em S/R MOM\_CDSCHEME} ({\em mom\_cdscheme.F}) |
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|
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{\em S/R MOM\_U\_CORIOLIS} ({\em mom\_u\_coriolis.F}) |
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|
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{\em S/R MOM\_V\_CORIOLIS} ({\em mom\_v\_coriolis.F}) |
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|
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$G_u^{Cor}$, $G_v^{Cor}$: {\bf cF} (local to {\em calc\_mom\_rhs.F}) |
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\end{minipage} } |
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|
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|
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\subsection{Curvature metric terms} |
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|
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The most commonly used coordinate system on the sphere is the |
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geographic system $(\lambda,\varphi)$. The curvilinear nature of these |
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coordinates on the sphere lead to some ``metric'' terms in the |
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component momentum equations. Under the thin-atmosphere and |
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hydrostatic approximations these terms are discretized: |
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\begin{eqnarray} |
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{\cal A}_w \Delta r_f h_w G_u^{metric} & = & |
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\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} & = & |
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- \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 |
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\end{eqnarray} |
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where $a$ is the radius of the planet (sphericity is assumed) or the |
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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 |
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the discrete Coriolis terms since the metric factor $\frac{u}{a} |
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\tan{\varphi}$ can be viewed as a modification of the vertical Coriolis |
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parameter: $f \rightarrow f+\frac{u}{a} \tan{\varphi}$. |
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|
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However, as for the Coriolis terms, a non-energy conserving form has |
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exclusively been used to date: |
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\begin{eqnarray} |
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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{\varphi} |
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\end{eqnarray} |
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where $\tan{\varphi}$ is evaluated at the $u$ and $v$ points |
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respectively. |
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|
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\fbox{ \begin{minipage}{4.75in} |
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{\em S/R MOM\_U\_METRIC\_SPHERE} ({\em mom\_u\_metric\_sphere.F}) |
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|
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{\em S/R MOM\_V\_METRIC\_SPHERE} ({\em mom\_v\_metric\_sphere.F}) |
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|
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$G_u^{metric}$, $G_v^{metric}$: {\bf mT} (local to {\em calc\_mom\_rhs.F}) |
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\end{minipage} } |
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|
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|
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|
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\subsection{Non-hydrostatic metric terms} |
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|
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For the non-hydrostatic equations, dropping the thin-atmosphere |
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approximation re-introduces metric terms involving $w$ and are |
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required to conserve anglular momentum: |
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\begin{eqnarray} |
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{\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 \\ |
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{\cal A}_s \Delta r_f h_s G_v^{metric} & = & |
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- \overline{ \frac{ \overline{v}^j \overline{w}^k }{a} {\cal A}_c \Delta r_f h_c}^j \\ |
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{\cal A}_c \Delta r_c G_w^{metric} & = & |
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\overline{ \frac{ {\overline{u}^i}^2 + {\overline{v}^j}^2}{a} {\cal A}_c \Delta r_f h_c }^k |
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\end{eqnarray} |
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|
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Because we are always consistent, even if consistently wrong, we have, |
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in the past, used a different discretization in the model which is: |
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\begin{eqnarray} |
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G_u^{metric} & = & |
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- \frac{u}{a} \overline{w}^{ik} \\ |
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G_v^{metric} & = & |
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- \frac{v}{a} \overline{w}^{jk} \\ |
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G_w^{metric} & = & |
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\frac{1}{a} ( {\overline{u}^{ik}}^2 + {\overline{v}^{jk}}^2 ) |
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\end{eqnarray} |
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|
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\fbox{ \begin{minipage}{4.75in} |
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{\em S/R MOM\_U\_METRIC\_NH} ({\em mom\_u\_metric\_nh.F}) |
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|
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{\em S/R MOM\_V\_METRIC\_NH} ({\em mom\_v\_metric\_nh.F}) |
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|
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$G_u^{metric}$, $G_v^{metric}$: {\bf mT} (local to {\em calc\_mom\_rhs.F}) |
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\end{minipage} } |
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|
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|
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\subsection{Lateral dissipation} |
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|
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Historically, we have represented the SGS Reynolds stresses as simply |
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down gradient momentum fluxes, ignoring constraints on the stress |
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tensor such as symmetry. |
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\begin{eqnarray} |
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{\cal A}_w \Delta r_f h_w G_u^{h-diss} & = & |
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\delta_i \Delta y_f \Delta r_f h_c \tau_{11} |
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+ \delta_j \Delta x_v \Delta r_f h_\zeta \tau_{12} \\ |
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{\cal A}_s \Delta r_f h_s G_v^{h-diss} & = & |
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\delta_i \Delta y_u \Delta r_f h_\zeta \tau_{21} |
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+ \delta_j \Delta x_f \Delta r_f h_c \tau_{22} |
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\end{eqnarray} |
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\marginpar{Check signs of stress definitions} |
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|
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The lateral viscous stresses are discretized: |
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\begin{eqnarray} |
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\tau_{11} & = & A_h c_{11\Delta}(\varphi) \frac{1}{\Delta x_f} \delta_i u |
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-A_4 c_{11\Delta^2}(\varphi) \frac{1}{\Delta x_f} \delta_i \nabla^2 u \\ |
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\tau_{12} & = & A_h c_{12\Delta}(\varphi) \frac{1}{\Delta y_u} \delta_j u |
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-A_4 c_{12\Delta^2}(\varphi)\frac{1}{\Delta y_u} \delta_j \nabla^2 u \\ |
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\tau_{21} & = & A_h c_{21\Delta}(\varphi) \frac{1}{\Delta x_v} \delta_i v |
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-A_4 c_{21\Delta^2}(\varphi) \frac{1}{\Delta x_v} \delta_i \nabla^2 v \\ |
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\tau_{22} & = & A_h c_{22\Delta}(\varphi) \frac{1}{\Delta y_f} \delta_j v |
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-A_4 c_{22\Delta^2}(\varphi) \frac{1}{\Delta y_f} \delta_j \nabla^2 v |
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\end{eqnarray} |
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where the non-dimensional factors $c_{lm\Delta^n}(\varphi), \{l,m,n\} \in |
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\{1,2\}$ define the ``cosine'' scaling with latitude which can be |
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applied in various ad-hoc ways. For instance, $c_{11\Delta} = |
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c_{21\Delta} = (\cos{\varphi})^{3/2}$, $c_{12\Delta}=c_{22\Delta}=0$ would |
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represent the an-isotropic cosine scaling typically used on the |
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``lat-lon'' grid for Laplacian viscosity. |
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\marginpar{Need to tidy up method for controlling this in code} |
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|
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It should be noted that dispite the ad-hoc nature of the scaling, some |
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scaling must be done since on a lat-lon grid the converging meridians |
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make it very unlikely that a stable viscosity parameter exists across |
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the entire model domain. |
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|
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The Laplacian viscosity coefficient, $A_h$ ({\bf viscAh}), has units |
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of $m^2 s^{-1}$. The bi-harmonic viscosity coefficient, $A_4$ ({\bf |
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viscA4}), has units of $m^4 s^{-1}$. |
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|
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\fbox{ \begin{minipage}{4.75in} |
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{\em S/R MOM\_U\_XVISCFLUX} ({\em mom\_u\_xviscflux.F}) |
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|
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{\em S/R MOM\_U\_YVISCFLUX} ({\em mom\_u\_yviscflux.F}) |
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|
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{\em S/R MOM\_V\_XVISCFLUX} ({\em mom\_v\_xviscflux.F}) |
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|
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{\em S/R MOM\_V\_YVISCFLUX} ({\em mom\_v\_yviscflux.F}) |
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|
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$\tau_{11}$, $\tau_{12}$, $\tau_{22}$, $\tau_{22}$: {\bf vF}, {\bf |
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v4F} (local to {\em calc\_mom\_rhs.F}) |
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\end{minipage} } |
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|
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Two types of lateral boundary condition exist for the lateral viscous |
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terms, no-slip and free-slip. |
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|
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The free-slip condition is most convenient to code since it is |
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equivalent to zero-stress on boundaries. Simple masking of the stress |
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components sets them to zero. The fractional open stress is properly |
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handled using the lopped cells. |
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|
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The no-slip condition defines the normal gradient of a tangential flow |
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such that the flow is zero on the boundary. Rather than modify the |
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stresses by using complicated functions of the masks and ``ghost'' |
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points (see \cite{Adcroft+Marshall98}) we add the boundary stresses as |
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an additional source term in cells next to solid boundaries. This has |
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the advantage of being able to cope with ``thin walls'' and also makes |
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the interior stress calculation (code) independent of the boundary |
| 298 |
conditions. The ``body'' force takes the form: |
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\begin{eqnarray} |
| 300 |
G_u^{side-drag} & = & |
| 301 |
\frac{4}{\Delta z_f} \overline{ (1-h_\zeta) \frac{\Delta x_v}{\Delta y_u} }^j |
| 302 |
\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} & = & |
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\frac{4}{\Delta z_f} \overline{ (1-h_\zeta) \frac{\Delta y_u}{\Delta x_v} }^i |
| 306 |
\left( A_h c_{21\Delta}(\varphi) v - A_4 c_{21\Delta^2}(\varphi) \nabla^2 v \right) |
| 307 |
\end{eqnarray} |
| 308 |
|
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In fact, the above discretization is not quite complete because it |
| 310 |
assumes that the bathymetry at velocity points is deeper than at |
| 311 |
neighbouring vorticity points, e.g. $1-h_w < 1-h_\zeta$ |
| 312 |
|
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\fbox{ \begin{minipage}{4.75in} |
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{\em S/R MOM\_U\_SIDEDRAG} ({\em mom\_u\_sidedrag.F}) |
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|
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{\em S/R MOM\_V\_SIDEDRAG} ({\em mom\_v\_sidedrag.F}) |
| 317 |
|
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$G_u^{side-drag}$, $G_v^{side-drag}$: {\bf vF} (local to {\em calc\_mom\_rhs.F}) |
| 319 |
\end{minipage} } |
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|
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|
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\subsection{Vertical dissipation} |
| 323 |
|
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Vertical viscosity terms are discretized with only partial adherence |
| 325 |
to the variable grid lengths introduced by the finite volume |
| 326 |
formulation. This reduces the formal accuracy of these terms to just |
| 327 |
first order but only next to boundaries; exactly where other terms |
| 328 |
appear such as linar and quadratic bottom drag. |
| 329 |
\begin{eqnarray} |
| 330 |
G_u^{v-diss} & = & |
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\frac{1}{\Delta r_f h_w} \delta_k \tau_{13} \\ |
| 332 |
G_v^{v-diss} & = & |
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\frac{1}{\Delta r_f h_s} \delta_k \tau_{23} \\ |
| 334 |
G_w^{v-diss} & = & \epsilon_{nh} |
| 335 |
\frac{1}{\Delta r_f h_d} \delta_k \tau_{33} |
| 336 |
\end{eqnarray} |
| 337 |
represents the general discrete form of the vertical dissipation terms. |
| 338 |
|
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In the interior the vertical stresses are discretized: |
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\begin{eqnarray} |
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\tau_{13} & = & A_v \frac{1}{\Delta r_c} \delta_k u \\ |
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\tau_{23} & = & A_v \frac{1}{\Delta r_c} \delta_k v \\ |
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\tau_{33} & = & A_v \frac{1}{\Delta r_f} \delta_k w |
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\end{eqnarray} |
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It should be noted that in the non-hydrostatic form, the stress tensor |
| 346 |
is even less consistent than for the hydrostatic (see Wazjowicz |
| 347 |
\cite{Waojz}). It is well known how to do this properly (see Griffies |
| 348 |
\cite{Griffies}) and is on the list of to-do's. |
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|
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\fbox{ \begin{minipage}{4.75in} |
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{\em S/R MOM\_U\_RVISCLFUX} ({\em mom\_u\_rviscflux.F}) |
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|
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{\em S/R MOM\_V\_RVISCLFUX} ({\em mom\_v\_rviscflux.F}) |
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|
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$\tau_{13}$: {\bf urf} (local to {\em calc\_mom\_rhs.F}) |
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|
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$\tau_{23}$: {\bf vrf} (local to {\em calc\_mom\_rhs.F}) |
| 358 |
\end{minipage} } |
| 359 |
|
| 360 |
|
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As for the lateral viscous terms, the free-slip condition is |
| 362 |
equivalent to simply setting the stress to zero on boundaries. The |
| 363 |
no-slip condition is implemented as an additional term acting on top |
| 364 |
of the interior and free-slip stresses. Bottom drag represents |
| 365 |
additional friction, in addition to that imposed by the no-slip |
| 366 |
condition at the bottom. The drag is cast as a stress expressed as a |
| 367 |
linear or quadratic function of the mean flow in the layer above the |
| 368 |
topography: |
| 369 |
\begin{eqnarray} |
| 370 |
\tau_{13}^{bottom-drag} & = & |
| 371 |
\left( |
| 372 |
2 A_v \frac{1}{\Delta r_c} |
| 373 |
+ r_b |
| 374 |
+ C_d \sqrt{ \overline{2 KE}^i } |
| 375 |
\right) u \\ |
| 376 |
\tau_{23}^{bottom-drag} & = & |
| 377 |
\left( |
| 378 |
2 A_v \frac{1}{\Delta r_c} |
| 379 |
+ r_b |
| 380 |
+ C_d \sqrt{ \overline{2 KE}^j } |
| 381 |
\right) v |
| 382 |
\end{eqnarray} |
| 383 |
where these terms are only evaluated immediately above topography. |
| 384 |
$r_b$ ({\bf bottomDragLinear}) has units of $m s^{-1}$ and a typical value |
| 385 |
of the order 0.0002 $m s^{-1}$. $C_d$ ({\bf bottomDragQuadratic}) is |
| 386 |
dimensionless with typical values in the range 0.001--0.003. |
| 387 |
|
| 388 |
\fbox{ \begin{minipage}{4.75in} |
| 389 |
{\em S/R MOM\_U\_BOTTOMDRAG} ({\em mom\_u\_bottomdrag.F}) |
| 390 |
|
| 391 |
{\em S/R MOM\_V\_BOTTOMDRAG} ({\em mom\_v\_bottomdrag.F}) |
| 392 |
|
| 393 |
$\tau_{13}^{bottom-drag}$, $\tau_{23}^{bottom-drag}$: {\bf vf} (local to {\em calc\_mom\_rhs.F}) |
| 394 |
\end{minipage} } |
| 395 |
|
| 396 |
\subsection{Derivation of discrete energy conservation} |
| 397 |
|
| 398 |
These discrete equations conserve kinetic plus potential energy using the |
| 399 |
following definitions: |
| 400 |
\begin{equation} |
| 401 |
KE = \frac{1}{2} \left( \overline{ u^2 }^i + \overline{ v^2 }^j + |
| 402 |
\epsilon_{nh} \overline{ w^2 }^k \right) |
| 403 |
\end{equation} |
| 404 |
|
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