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% $Header: /u/gcmpack/mitgcmdoc/part2/mom_vecinv.tex,v 1.2 2001/10/25 18:36:53 cnh Exp $ |
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% $Name: $ |
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|
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\section{Vector invariant momentum equations} |
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|
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The finite volume method lends itself to describing the continuity and |
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tracer equations in curvilinear coordinate systems. However, in |
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curvilinear coordinates many new metric terms appear in the momentum |
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equations (written in Lagrangian or flux-form) making generalization |
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far from elegant. Fortunately, an alternative form of the equations, |
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the vector invariant equations are exactly that; invariant under |
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coordinate transformations so that they can be applied uniformly in |
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any orthogonal curvilinear coordinate system such as spherical |
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coordinates, boundary following or the conformal spherical cube |
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system. |
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|
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The non-hydrostatic vector invariant equations read: |
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\begin{equation} |
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\partial_t \vec{v} + ( 2\vec{\Omega} + \vec{\zeta}) \wedge \vec{v} |
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- b \hat{r} |
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+ \vec{\nabla} B = \vec{\nabla} \cdot \vec{\bf \tau} |
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\end{equation} |
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which describe motions in any orthogonal curvilinear coordinate |
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system. Here, $B$ is the Bernoulli function and $\vec{\zeta}=\nabla |
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\wedge \vec{v}$ is the vorticity vector. We can take advantage of the |
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elegance of these equations when discretizing them and use the |
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discrete definitions of the grad, curl and divergence operators to |
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satisfy constraints. We can also consider the analogy to forming |
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derived equations, such as the vorticity equation, and examine how the |
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discretization can be adjusted to give suitable vorticity advection |
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among other things. |
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|
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The underlying algorithm is the same as for the flux form |
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equations. All that has changed is the contents of the ``G's''. For |
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the time-being, only the hydrostatic terms have been coded but we will |
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indicate the points where non-hydrostatic contributions will enter: |
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\begin{eqnarray} |
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G_u & = & G_u^{fv} + G_u^{\zeta_3 v} + G_u^{\zeta_2 w} + G_u^{\partial_x B} |
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+ G_u^{\partial_z \tau^x} + G_u^{h-dissip} + G_u^{v-dissip} \\ |
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G_v & = & G_v^{fu} + G_v^{\zeta_3 u} + G_v^{\zeta_1 w} + G_v^{\partial_y B} |
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+ G_v^{\partial_z \tau^y} + G_v^{h-dissip} + G_v^{v-dissip} \\ |
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G_w & = & G_w^{fu} + G_w^{\zeta_1 v} + G_w^{\zeta_2 u} + G_w^{\partial_z B} |
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+ G_w^{h-dissip} + G_w^{v-dissip} |
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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 CALC\_MOM\_RHS} ({\em pkg/mom\_vecinv/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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\subsection{Relative vorticity} |
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|
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The vertical component of relative vorticity is explicitly calculated |
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and use in the discretization. The particular form is crucial for |
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numerical stability; alternative definitions break the conservation |
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properties of the discrete equations. |
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|
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Relative vorticity is defined: |
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\begin{equation} |
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\zeta_3 = \frac{\Gamma}{A_\zeta} |
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= \frac{1}{{\cal A}_\zeta} ( \delta_i \Delta y_c v - \delta_j \Delta x_c u ) |
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\end{equation} |
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where ${\cal A}_\zeta$ is the area of the vorticity cell presented in |
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the vertical and $\Gamma$ is the circulation about that cell. |
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|
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\fbox{ \begin{minipage}{4.75in} |
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{\em S/R MOM\_VI\_CALC\_RELVORT3} ({\em mom\_vi\_calc\_relvort3.F}) |
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|
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$\zeta_3$: {\bf vort3} (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{Kinetic energy} |
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|
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The kinetic energy, denoted $KE$, is defined: |
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\begin{equation} |
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KE = \frac{1}{2} ( \overline{ u^2 }^i + \overline{ v^2 }^j |
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+ \epsilon_{nh} \overline{ w^2 }^k ) |
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\end{equation} |
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|
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\fbox{ \begin{minipage}{4.75in} |
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{\em S/R MOM\_VI\_CALC\_KE} ({\em mom\_vi\_calc\_ke.F}) |
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|
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$KE$: {\bf KE} (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{Coriolis terms} |
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|
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The potential enstrophy conserving form of the linear Coriolis terms |
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are written: |
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\begin{eqnarray} |
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G_u^{fv} & = & |
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\frac{1}{\Delta x_c} |
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\overline{ \frac{f}{h_\zeta} }^j \overline{ \overline{ \Delta x_g h_s v }^j }^i \\ |
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G_v^{fu} & = & - |
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\frac{1}{\Delta y_c} |
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\overline{ \frac{f}{h_\zeta} }^i \overline{ \overline{ \Delta y_g h_w u }^i }^j |
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\end{eqnarray} |
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Here, the Coriolis parameter $f$ is defined at vorticity (corner) |
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points. |
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\marginpar{$f$: {\bf fCoriG}} |
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\marginpar{$h_\zeta$: {\bf hFacZ}} |
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|
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The potential enstrophy conserving form of the non-linear Coriolis |
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terms are written: |
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\begin{eqnarray} |
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G_u^{\zeta_3 v} & = & |
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\frac{1}{\Delta x_c} |
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\overline{ \frac{\zeta_3}{h_\zeta} }^j \overline{ \overline{ \Delta x_g h_s v }^j }^i \\ |
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G_v^{\zeta_3 u} & = & - |
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\frac{1}{\Delta y_c} |
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\overline{ \frac{\zeta_3}{h_\zeta} }^i \overline{ \overline{ \Delta y_g h_w u }^i }^j |
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\end{eqnarray} |
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\marginpar{$\zeta_3$: {\bf vort3}} |
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|
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The Coriolis terms can also be evaluated together and expressed in |
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terms of absolute vorticity $f+\zeta_3$. The potential enstrophy |
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conserving form using the absolute vorticity is written: |
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\begin{eqnarray} |
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G_u^{fv} + G_u^{\zeta_3 v} & = & |
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\frac{1}{\Delta x_c} |
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\overline{ \frac{f + \zeta_3}{h_\zeta} }^j \overline{ \overline{ \Delta x_g h_s v }^j }^i \\ |
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G_v^{fu} + G_v^{\zeta_3 u} & = & - |
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\frac{1}{\Delta y_c} |
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\overline{ \frac{f + \zeta_3}{h_\zeta} }^i \overline{ \overline{ \Delta y_g h_w u }^i }^j |
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\end{eqnarray} |
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|
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\marginpar{Run-time control needs to be added for these options} The |
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distinction between using absolute vorticity or relative vorticity is |
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useful when constructing higher order advection schemes; monotone |
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advection of relative vorticity behaves differently to monotone |
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advection of absolute vorticity. Currently the choice of |
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relative/absolute vorticity, centered/upwind/high order advection is |
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available only through commented subroutine calls. |
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|
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\fbox{ \begin{minipage}{4.75in} |
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{\em S/R MOM\_VI\_CORIOLIS} ({\em mom\_vi\_coriolis.F}) |
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|
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{\em S/R MOM\_VI\_U\_CORIOLIS} ({\em mom\_vi\_u\_coriolis.F}) |
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|
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{\em S/R MOM\_VI\_V\_CORIOLIS} ({\em mom\_vi\_v\_coriolis.F}) |
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|
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$G_u^{fv}$, $G_u^{\zeta_3 v}$: {\bf uCf} (local to {\em calc\_mom\_rhs.F}) |
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|
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$G_v^{fu}$, $G_v^{\zeta_3 u}$: {\bf vCf} (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{Shear terms} |
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|
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The shear terms ($\zeta_2w$ and $\zeta_1w$) are are discretized to |
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guarantee that no spurious generation of kinetic energy is possible; |
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the horizontal gradient of Bernoulli function has to be consistent |
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with the vertical advection of shear: |
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\marginpar{N-H terms have not been tried!} |
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\begin{eqnarray} |
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G_u^{\zeta_2 w} & = & |
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\frac{1}{ {\cal A}_w \Delta r_f h_w } \overline{ |
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\overline{ {\cal A}_c w }^i ( \delta_k u - \epsilon_{nh} \delta_j w ) |
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}^k \\ |
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G_v^{\zeta_1 w} & = & |
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\frac{1}{ {\cal A}_s \Delta r_f h_s } \overline{ |
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\overline{ {\cal A}_c w }^i ( \delta_k u - \epsilon_{nh} \delta_j w ) |
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}^k |
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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\_VI\_U\_VERTSHEAR} ({\em mom\_vi\_u\_vertshear.F}) |
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|
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{\em S/R MOM\_VI\_V\_VERTSHEAR} ({\em mom\_vi\_v\_vertshear.F}) |
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|
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$G_u^{\zeta_2 w}$: {\bf uCf} (local to {\em calc\_mom\_rhs.F}) |
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|
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$G_v^{\zeta_1 w}$: {\bf vCf} (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{Gradient of Bernoulli function} |
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|
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\begin{eqnarray} |
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G_u^{\partial_x B} & = & |
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\frac{1}{\Delta x_c} \delta_i ( \phi' + KE ) \\ |
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G_v^{\partial_y B} & = & |
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\frac{1}{\Delta x_y} \delta_j ( \phi' + KE ) |
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%G_w^{\partial_z B} & = & |
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%\frac{1}{\Delta r_c} h_c \delta_k ( \phi' + KE ) |
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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\_VI\_U\_GRAD\_KE} ({\em mom\_vi\_u\_grad\_ke.F}) |
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|
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{\em S/R MOM\_VI\_V\_GRAD\_KE} ({\em mom\_vi\_v\_grad\_ke.F}) |
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|
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$G_u^{\partial_x KE}$: {\bf uCf} (local to {\em calc\_mom\_rhs.F}) |
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|
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$G_v^{\partial_y KE}$: {\bf vCf} (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{Horizontal dissipation} |
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|
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The horizontal divergence, a complimentary quantity to relative |
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vorticity, is used in parameterizing the Reynolds stresses and is |
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discretized: |
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\begin{equation} |
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D = \frac{1}{{\cal A}_c h_c} ( |
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\delta_i \Delta y_g h_w u |
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+ \delta_j \Delta x_g h_s v ) |
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\end{equation} |
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|
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\fbox{ \begin{minipage}{4.75in} |
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{\em S/R MOM\_VI\_CALC\_HDIV} ({\em mom\_vi\_calc\_hdiv.F}) |
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|
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$D$: {\bf hDiv} (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{Horizontal dissipation} |
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|
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The following discretization of horizontal dissipation conserves |
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potential vorticity (thickness weighted relative vorticity) and |
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divergence and dissipates energy, enstrophy and divergence squared: |
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\begin{eqnarray} |
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G_u^{h-dissip} & = & |
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\frac{1}{\Delta x_c} \delta_i ( A_D D - A_{D4} D^*) |
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- \frac{1}{\Delta y_u h_w} \delta_j h_\zeta ( A_\zeta \zeta - A_{\zeta4} \zeta^* ) |
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\\ |
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G_v^{h-dissip} & = & |
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\frac{1}{\Delta x_v h_s} \delta_i h_\zeta ( A_\zeta \zeta - A_\zeta \zeta^* ) |
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+ \frac{1}{\Delta y_c} \delta_j ( A_D D - A_{D4} D^* ) |
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\end{eqnarray} |
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where |
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\begin{eqnarray} |
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D^* & = & \frac{1}{{\cal A}_c h_c} ( |
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\delta_i \Delta y_g h_w \nabla^2 u |
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+ \delta_j \Delta x_g h_s \nabla^2 v ) \\ |
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\zeta^* & = & \frac{1}{{\cal A}_\zeta} ( |
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\delta_i \Delta y_c \nabla^2 v |
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- \delta_j \Delta x_c \nabla^2 u ) |
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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\_VI\_HDISSIP} ({\em mom\_vi\_hdissip.F}) |
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|
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$G_u^{h-dissip}$: {\bf uDiss} (local to {\em calc\_mom\_rhs.F}) |
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|
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$G_v^{h-dissip}$: {\bf vDiss} (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{Vertical dissipation} |
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|
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Currently, this is exactly the same code as the flux form equations. |
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\begin{eqnarray} |
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G_u^{v-diss} & = & |
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\frac{1}{\Delta r_f h_w} \delta_k \tau_{13} \\ |
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G_v^{v-diss} & = & |
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\frac{1}{\Delta r_f h_s} \delta_k \tau_{23} |
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\end{eqnarray} |
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represents the general discrete form of the vertical dissipation terms. |
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|
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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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\end{eqnarray} |
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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}) |
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\end{minipage} } |
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