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% $Header: /u/gcmpack/mitgcmdoc/part2/time_stepping.tex,v 1.6 2001/09/26 20:19:52 adcroft Exp $ |
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% $Name: $ |
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|
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The convention used in this section is as follows: |
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Time is ``discretized'' using a time step $\Delta t$ |
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and $\Phi^n$ refers to the variable $\Phi$ |
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at time $t = n \Delta t$ . We use the notation $\Phi^{(n)}$ |
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when time interpolation is required to estimate the value of $\phi$ |
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at the time $n \Delta t$. |
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|
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\section{Time integration} |
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|
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The discretization in time of the model equations (cf section I ) |
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does not depend of the discretization in space of each |
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term and so this section can be read independently. |
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|
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The continuous form of the model equations is: |
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|
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\begin{eqnarray} |
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\partial_t \theta & = & G_\theta |
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\label{eq-tCsC-theta} |
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\\ |
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\partial_t S & = & G_s |
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\label{eq-tCsC-salt} |
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\\ |
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b' & = & b'(\theta,S,r) |
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\\ |
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\partial_r \phi'_{hyd} & = & -b' |
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\label{eq-tCsC-hyd} |
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\\ |
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\partial_t \vec{\bf v} |
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+ {\bf \nabla}_h b_s \eta |
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+ \epsilon_{nh} {\bf \nabla}_h \phi'_{nh} |
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& = & \vec{\bf G}_{\vec{\bf v}} |
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- {\bf \nabla}_h \phi'_{hyd} |
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\label{eq-tCsC-Hmom} |
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\\ |
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\epsilon_{nh} \frac {\partial{\dot{r}}}{\partial{t}} |
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+ \epsilon_{nh} \partial_r \phi'_{nh} |
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& = & \epsilon_{nh} G_{\dot{r}} |
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\label{eq-tCsC-Vmom} |
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\\ |
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{\bf \nabla}_h \cdot \vec{\bf v} + \partial_r \dot{r} |
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& = & 0 |
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\label{eq-tCsC-cont} |
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\end{eqnarray} |
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where |
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\begin{eqnarray*} |
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G_\theta & = & |
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- \vec{\bf v} \cdot {\bf \nabla} \theta + {\cal Q}_\theta |
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\\ |
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G_S & = & |
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- \vec{\bf v} \cdot {\bf \nabla} S + {\cal Q}_S |
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\\ |
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\vec{\bf G}_{\vec{\bf v}} |
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& = & |
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- \vec{\bf v} \cdot {\bf \nabla} \vec{\bf v} |
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- f \hat{\bf k} \wedge \vec{\bf v} |
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+ \vec{\cal F}_{\vec{\bf v}} |
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\\ |
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G_{\dot{r}} |
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& = & |
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- \vec{\bf v} \cdot {\bf \nabla} \dot{r} |
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+ {\cal F}_{\dot{r}} |
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\end{eqnarray*} |
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The exact form of all the ``{\it G}''s terms is described in the next |
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section \ref{sect:discrete}. Here its sufficient to mention that they contains |
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all the RHS terms except the pressure/geo-potential terms. |
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|
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The switch $\epsilon_{nh}$ allows one to activate the non-hydrostatic |
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mode ($\epsilon_{nh}=1$) for the ocean model. Otherwise, in the |
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hydrostatic limit $\epsilon_{nh} = 0$ and equation \ref{eq-tCsC-Vmom} |
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is not used. |
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|
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As discussed in section \ref{sect:1.3.6.2}, the equation for $\eta$ is |
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obtained by integrating the continuity equation over the entire depth |
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of the fluid, from $R_{fixed}(x,y)$ up to $R_o(x,y)$. The linear free |
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surface evolves according to: |
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\begin{eqnarray} |
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\epsilon_{fs} \partial_t \eta = |
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\left. \dot{r} \right|_{r=r_{surf}} + \epsilon_{fw} (P-E) = |
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- {\bf \nabla} \cdot \int_{R_{fixed}}^{R_o} \vec{\bf v} dr |
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+ \epsilon_{fw} (P-E) |
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\label{eq-tCsC-eta} |
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\end{eqnarray} |
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|
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Here, $\epsilon_{fs}$ is a flag to switch between the free-surface, |
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$\epsilon_{fs}=1$, and a rigid-lid, $\epsilon_{fs}=0$. The flag |
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$\epsilon_{fw}$ determines whether an exchange of fresh water is |
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included at the ocean surface (natural BC) ($\epsilon_{fw} = 1$) or |
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not ($\epsilon_{fw} = 0$). |
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|
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The hydrostatic potential is found by integrating (equation |
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\ref{eq-tCsC-hyd}) with the boundary condition that |
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$\phi'_{hyd}(r=R_o) = 0$: |
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\begin{eqnarray*} |
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& & |
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\int_{r'}^{R_o} \partial_r \phi'_{hyd} dr = |
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\left[ \phi'_{hyd} \right]_{r'}^{R_o} = |
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\int_{r'}^{R_o} - b' dr |
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\\ |
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\Rightarrow & & |
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\phi'_{hyd}(x,y,r') = \int_{r'}^{R_o} b' dr |
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\end{eqnarray*} |
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|
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\subsection{General method} |
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|
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An overview of the general method is now presented with explicit |
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references to the Fortran code. We often refer to the discretized |
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equations of the model that are detailed in the following sections. |
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|
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The general algorithm consist of a ``predictor step'' that computes |
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the forward tendencies ("G" terms") comprising of explicit-in-time |
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terms and the ``first guess'' values (star notation): $\theta^*, S^*, |
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\vec{\bf v}^*$ (and $\dot{r}^*$ in non-hydrostatic mode). This is done |
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in the two routines {\it THERMODYNAMICS} and {\it DYNAMICS}. |
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|
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Terms that are integrated implicitly in time are handled at the end of |
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the {\it THERMODYNAMICS} and {\it DYNAMICS} routines. Then the |
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surface pressure and non hydrostatic pressure are solved for in ({\it |
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SOLVE\_FOR\_PRESSURE}). |
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|
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Finally, in the ``corrector step'', (routine {\it |
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THE\_CORRECTION\_STEP}) the new values of $u,v,w,\theta,S$ are |
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determined (see details in \ref{sect:II.1.3}). |
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|
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At this point, the regular time stepping process is complete. However, |
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after the correction step there are optional adjustments such as |
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convective adjustment or filters (zonal FFT filter, shapiro filter) |
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that can be applied on both momentum and tracer fields, just prior to |
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incrementing the time level to $n+1$. |
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|
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Since the pressure solver precision is of the order of the ``target |
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residual'' and can be lower than the the computer truncation error, |
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and also because some filters might alter the divergence part of the |
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flow field, a final evaluation of the surface r anomaly $\eta^{n+1}$ |
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is performed in {\it CALC\_EXACT\_ETA}. This ensures exact volume |
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conservation. Note that there is no need for an equivalent |
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non-hydrostatic ``exact conservation'' step, since by default $w$ is |
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already computed after the filters are applied. |
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|
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Optional forcing terms (usually part of a physics ``package''), that |
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account for a specific source or sink process (e.g. condensation as a |
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sink of water vapor Q) are generally incorporated in the main |
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algorithm as follows: at the the beginning of the time step, the |
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additional tendencies are computed as a function of the present state |
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(time step $n$) and external forcing; then within the main part of |
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model, only those new tendencies are added to the model variables. |
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|
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[more details needed]\\ |
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|
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The atmospheric physics follows this general scheme. |
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|
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[more about C\_grid, A\_grid conversion \& drag term]\\ |
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|
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|
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|
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\subsection{Standard synchronous time stepping} |
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|
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In the standard formulation, the surface pressure is evaluated at time |
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step n+1 (an implicit method). Equations \ref{eq-tCsC-theta} to |
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\ref{eq-tCsC-cont} are then discretized in time as follows: |
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\begin{eqnarray} |
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\left[ 1 - \partial_r \kappa_v^\theta \partial_r \right] |
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\theta^{n+1} & = & \theta^* |
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\label{eq-tDsC-theta} |
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\\ |
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\left[ 1 - \partial_r \kappa_v^S \partial_r \right] |
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S^{n+1} & = & S^* |
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\label{eq-tDsC-salt} |
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\\ |
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%{b'}^{n} & = & b'(\theta^{n},S^{n},r) |
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%\partial_r {\phi'_{hyd}}^{n} & = & {-b'}^{n} |
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%\\ |
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{\phi'_{hyd}}^{n} & = & \int_{r'}^{R_o} b'(\theta^{n},S^{n},r) dr |
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\label{eq-tDsC-hyd} |
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\\ |
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\vec{\bf v} ^{n+1} |
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+ \Delta t {\bf \nabla}_h b_s {\eta}^{n+1} |
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+ \epsilon_{nh} \Delta t {\bf \nabla} {\phi'_{nh}}^{n+1} |
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- \partial_r A_v \partial_r \vec{\bf v}^{n+1} |
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& = & |
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\vec{\bf v}^* |
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\label{eq-tDsC-Hmom} |
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\\ |
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\epsilon_{fs} {\eta}^{n+1} + \Delta t |
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{\bf \nabla}_h \cdot \int_{R_{fixed}}^{R_o} \vec{\bf v}^{n+1} dr |
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& = & |
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\epsilon_{fs} {\eta}^{n} + \epsilon_{fw} \Delta_t (P-E)^{n} |
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\nonumber |
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\\ |
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% = \epsilon_{fs} {\eta}^{n} & + & \epsilon_{fw} \Delta_t (P-E)^{n} |
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\label{eq-tDsC-eta} |
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\\ |
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\epsilon_{nh} \left( \dot{r} ^{n+1} |
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+ \Delta t \partial_r {\phi'_{nh}} ^{n+1} |
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\right) |
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& = & \epsilon_{nh} \dot{r}^* |
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\label{eq-tDsC-Vmom} |
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\\ |
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{\bf \nabla}_h \cdot \vec{\bf v}^{n+1} + \partial_r \dot{r}^{n+1} |
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& = & 0 |
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\label{eq-tDsC-cont} |
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\end{eqnarray} |
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where |
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\begin{eqnarray} |
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\theta^* & = & |
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\theta ^{n} + \Delta t G_{\theta} ^{(n+1/2)} |
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\\ |
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S^* & = & |
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S ^{n} + \Delta t G_{S} ^{(n+1/2)} |
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\\ |
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\vec{\bf v}^* & = & |
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\vec{\bf v}^{n} + \Delta t \vec{\bf G}_{\vec{\bf v}} ^{(n+1/2)} |
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+ \Delta t {\bf \nabla}_h {\phi'_{hyd}}^{(n+1/2)} |
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\\ |
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\dot{r}^* & = & |
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\dot{r} ^{n} + \Delta t G_{\dot{r}} ^{(n+1/2)} |
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\end{eqnarray} |
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|
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Note that implicit vertical viscosity and diffusivity terms are not |
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considered as part of the ``{\it G}'' terms, but are written |
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separately here. |
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|
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The default time-stepping method is the Adams-Bashforth quasi-second |
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order scheme in which the ``G'' terms are extrapolated forward in time |
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from time-levels $n-1$ and $n$ to time-level $n+1/2$ and provides a |
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simple but stable algorithm: |
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\begin{equation} |
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G^{(n+1/2)} = G^n + (1/2+\epsilon_{AB}) (G^n - G^{n-1}) |
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\end{equation} |
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where $\epsilon_{AB}$ is a small number used to stabilize the time |
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stepping. |
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|
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In the standard non-staggered formulation, the Adams-Bashforth time |
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stepping is also applied to the hydrostatic pressure/geo-potential |
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gradient, $\nabla_h \Phi'_{hyd}$. Note that presently, this term is in |
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fact incorporated to the $\vec{\bf G}_{\vec{\bf v}}$ arrays ({\bf |
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gU,gV}). |
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\marginpar{JMC: Clarify ``this term''?} |
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|
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\fbox{ \begin{minipage}{4.75in} |
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{\em S/R TIMESTEP} ({\em timestep.F}) |
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|
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$G_u^n$: {\bf Gu} ({\em DYNVARS.h}) |
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|
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$G_u^{n-1}, u^*$: {\bf GuNm1} ({\em DYNVARS.h}) |
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|
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$G_v^n$: {\bf Gv} ({\em DYNVARS.h}) |
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|
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$G_v^{n-1}, v^*$: {\bf GvNm1} ({\em DYNVARS.h}) |
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|
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$G_u^{(n+1/2)}$: {\bf GuTmp} (local) |
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|
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$G_v^{(n+1/2)}$: {\bf GvTmp} (local) |
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|
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\end{minipage} } |
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|
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|
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|
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|
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|
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\subsection{Stagger baroclinic time stepping} |
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|
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An alternative to synchronous time-stepping is to stagger the momentum |
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and tracer fields in time. This allows the evaluation and gradient of |
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$\phi'_{hyd}$ to be centered in time with out needing to use the |
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Adams-Bashforth extrapoltion. This option is known as staggered |
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baroclinic time stepping because tracer and momentum are stepped |
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forward-in-time one after the other. It can be activated by turning |
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on a run-time parameter {\bf staggerTimeStep} in namelist ``{\it |
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PARM01}''. |
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|
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The main advantage of staggered time-stepping compared to synchronous, |
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is improved stability to internal gravity wave motions and a very |
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natural implementation of a 2nd order in time hydrostatic |
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pressure/geo-potential gradient term. However, synchronous |
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time-stepping might be better for convection problems, time dependent |
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forcing and diagnosing the model are also easier and it allows a more |
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efficient parallel decomposition. |
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|
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The staggered baroclinic time-stepping scheme is equations |
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\ref{eq-tDsC-theta} to \ref{eq-tDsC-cont} except that \ref{eq-tDsC-hyd} is replaced with: |
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\begin{equation} |
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{\phi'_{hyd}}^{n+1/2} = \int_{r'}^{R_o} b'(\theta^{n+1/2},S^{n+1/2},r) |
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dr |
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\end{equation} |
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and the time-level for tracers has been shifted back by half: |
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\begin{eqnarray*} |
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\theta^* & = & |
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\theta ^{(n-1/2)} + \Delta t G_{\theta} ^{(n)} |
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\\ |
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S^* & = & |
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S ^{(n-1/2)} + \Delta t G_{S} ^{(n)} |
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\\ |
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\left[ 1 - \partial_r \kappa_v^\theta \partial_r \right] |
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\theta^{n+1/2} & = & \theta^* |
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\\ |
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\left[ 1 - \partial_r \kappa_v^S \partial_r \right] |
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S^{n+1/2} & = & S^* |
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\end{eqnarray*} |
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|
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|
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\subsection{Surface pressure} |
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|
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Substituting \ref{eq-tDsC-Hmom} into \ref{eq-tDsC-cont}, assuming |
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$\epsilon_{nh} = 0$ yields a Helmholtz equation for ${\eta}^{n+1}$: |
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\begin{eqnarray} |
| 309 |
\epsilon_{fs} {\eta}^{n+1} - |
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{\bf \nabla}_h \cdot \Delta t^2 (R_o-R_{fixed}) |
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{\bf \nabla}_h b_s {\eta}^{n+1} |
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= {\eta}^* |
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\label{eq-solve2D} |
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\end{eqnarray} |
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where |
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\begin{eqnarray} |
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{\eta}^* = \epsilon_{fs} \: {\eta}^{n} - |
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\Delta t {\bf \nabla}_h \cdot \int_{R_{fixed}}^{R_o} \vec{\bf v}^* dr |
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\: + \: \epsilon_{fw} \Delta_t (P-E)^{n} |
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\label{eq-solve2D_rhs} |
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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 SOLVE\_FOR\_PRESSURE} ({\em solve\_for\_pressure.F}) |
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|
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$u^*$: {\bf GuNm1} ({\em DYNVARS.h}) |
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|
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$v^*$: {\bf GvNm1} ({\em DYNVARS.h}) |
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|
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$\eta^*$: {\bf cg2d\_b} (\em SOLVE\_FOR\_PRESSURE.h) |
| 331 |
|
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$\eta^{n+1}$: {\bf etaN} (\em DYNVARS.h) |
| 333 |
|
| 334 |
\end{minipage} } |
| 335 |
|
| 336 |
|
| 337 |
Once ${\eta}^{n+1}$ has been found, substituting into |
| 338 |
\ref{eq-tDsC-Hmom} yields $\vec{\bf v}^{n+1}$ if the model is |
| 339 |
hydrostatic ($\epsilon_{nh}=0$): |
| 340 |
$$ |
| 341 |
\vec{\bf v}^{n+1} = \vec{\bf v}^{*} |
| 342 |
- \Delta t {\bf \nabla}_h b_s {\eta}^{n+1} |
| 343 |
$$ |
| 344 |
|
| 345 |
This is known as the correction step. However, when the model is |
| 346 |
non-hydrostatic ($\epsilon_{nh}=1$) we need an additional step and an |
| 347 |
additional equation for $\phi'_{nh}$. This is obtained by substituting |
| 348 |
\ref{eq-tDsC-Hmom} and \ref{eq-tDsC-Vmom} into |
| 349 |
\ref{eq-tDsC-cont}: |
| 350 |
\begin{equation} |
| 351 |
\left[ {\bf \nabla}_h^2 + \partial_{rr} \right] {\phi'_{nh}}^{n+1} |
| 352 |
= \frac{1}{\Delta t} \left( |
| 353 |
{\bf \nabla}_h \cdot \vec{\bf v}^{**} + \partial_r \dot{r}^* \right) |
| 354 |
\end{equation} |
| 355 |
where |
| 356 |
\begin{displaymath} |
| 357 |
\vec{\bf v}^{**} = \vec{\bf v}^* - \Delta t {\bf \nabla}_h b_s {\eta}^{n+1} |
| 358 |
\end{displaymath} |
| 359 |
Note that $\eta^{n+1}$ is also used to update the second RHS term |
| 360 |
$\partial_r \dot{r}^* $ since |
| 361 |
the vertical velocity at the surface ($\dot{r}_{surf}$) |
| 362 |
is evaluated as $(\eta^{n+1} - \eta^n) / \Delta t$. |
| 363 |
|
| 364 |
Finally, the horizontal velocities at the new time level are found by: |
| 365 |
\begin{equation} |
| 366 |
\vec{\bf v}^{n+1} = \vec{\bf v}^{**} |
| 367 |
- \epsilon_{nh} \Delta t {\bf \nabla}_h {\phi'_{nh}}^{n+1} |
| 368 |
\end{equation} |
| 369 |
and the vertical velocity is found by integrating the continuity |
| 370 |
equation vertically. Note that, for the convenience of the restart |
| 371 |
procedure, the vertical integration of the continuity equation has |
| 372 |
been moved to the beginning of the time step (instead of at the end), |
| 373 |
without any consequence on the solution. |
| 374 |
|
| 375 |
\fbox{ \begin{minipage}{4.75in} |
| 376 |
{\em S/R CORRECTION\_STEP} ({\em correction\_step.F}) |
| 377 |
|
| 378 |
$\eta^{n+1}$: {\bf etaN} (\em DYNVARS.h) |
| 379 |
|
| 380 |
$\phi_{nh}^{n+1}$: {\bf phi\_nh} (\em DYNVARS.h) |
| 381 |
|
| 382 |
$u^*$: {\bf GuNm1} ({\em DYNVARS.h}) |
| 383 |
|
| 384 |
$v^*$: {\bf GvNm1} ({\em DYNVARS.h}) |
| 385 |
|
| 386 |
$u^{n+1}$: {\bf uVel} ({\em DYNVARS.h}) |
| 387 |
|
| 388 |
$v^{n+1}$: {\bf vVel} ({\em DYNVARS.h}) |
| 389 |
|
| 390 |
\end{minipage} } |
| 391 |
|
| 392 |
|
| 393 |
|
| 394 |
Regarding the implementation of the surface pressure solver, all |
| 395 |
computation are done within the routine {\it SOLVE\_FOR\_PRESSURE} and |
| 396 |
its dependent calls. The standard method to solve the 2D elliptic |
| 397 |
problem (\ref{eq-solve2D}) uses the conjugate gradient method (routine |
| 398 |
{\it CG2D}); the solver matrix and conjugate gradient operator are |
| 399 |
only function of the discretized domain and are therefore evaluated |
| 400 |
separately, before the time iteration loop, within {\it INI\_CG2D}. |
| 401 |
The computation of the RHS $\eta^*$ is partly done in {\it |
| 402 |
CALC\_DIV\_GHAT} and in {\it SOLVE\_FOR\_PRESSURE}. |
| 403 |
|
| 404 |
The same method is applied for the non hydrostatic part, using a |
| 405 |
conjugate gradient 3D solver ({\it CG3D}) that is initialized in {\it |
| 406 |
INI\_CG3D}. The RHS terms of 2D and 3D problems are computed together |
| 407 |
at the same point in the code. |
| 408 |
|
| 409 |
|
| 410 |
|
| 411 |
\subsection{Crank-Nickelson barotropic time stepping} |
| 412 |
|
| 413 |
The full implicit time stepping described previously is |
| 414 |
unconditionally stable but damps the fast gravity waves, resulting in |
| 415 |
a loss of potential energy. The modification presented now allows one |
| 416 |
to combine an implicit part ($\beta,\gamma$) and an explicit part |
| 417 |
($1-\beta,1-\gamma$) for the surface pressure gradient ($\beta$) and |
| 418 |
for the barotropic flow divergence ($\gamma$). |
| 419 |
\\ |
| 420 |
For instance, $\beta=\gamma=1$ is the previous fully implicit scheme; |
| 421 |
$\beta=\gamma=1/2$ is the non damping (energy conserving), unconditionally |
| 422 |
stable, Crank-Nickelson scheme; $(\beta,\gamma)=(1,0)$ or $=(0,1)$ |
| 423 |
corresponds to the forward - backward scheme that conserves energy but is |
| 424 |
only stable for small time steps.\\ |
| 425 |
In the code, $\beta,\gamma$ are defined as parameters, respectively |
| 426 |
{\it implicSurfPress}, {\it implicDiv2DFlow}. They are read from |
| 427 |
the main data file "{\it data}" and are set by default to 1,1. |
| 428 |
|
| 429 |
Equations \ref{eq-tDsC-Hmom} and \ref{eq-tDsC-eta} are modified as follows: |
| 430 |
$$ |
| 431 |
\frac{ \vec{\bf v}^{n+1} }{ \Delta t } |
| 432 |
+ {\bf \nabla}_h b_s [ \beta {\eta}^{n+1} + (1-\beta) {\eta}^{n} ] |
| 433 |
+ \epsilon_{nh} {\bf \nabla}_h {\phi'_{nh}}^{n+1} |
| 434 |
= \frac{ \vec{\bf v}^* }{ \Delta t } |
| 435 |
$$ |
| 436 |
$$ |
| 437 |
\epsilon_{fs} \frac{ {\eta}^{n+1} - {\eta}^{n} }{ \Delta t} |
| 438 |
+ {\bf \nabla}_h \cdot \int_{R_{fixed}}^{R_o} |
| 439 |
[ \gamma \vec{\bf v}^{n+1} + (1-\gamma) \vec{\bf v}^{n}] dr |
| 440 |
= \epsilon_{fw} (P-E) |
| 441 |
$$ |
| 442 |
where: |
| 443 |
\begin{eqnarray*} |
| 444 |
\vec{\bf v}^* & = & |
| 445 |
\vec{\bf v} ^{n} + \Delta t \vec{\bf G}_{\vec{\bf v}} ^{(n+1/2)} |
| 446 |
+ (\beta-1) \Delta t {\bf \nabla}_h b_s {\eta}^{n} |
| 447 |
+ \Delta t {\bf \nabla}_h {\phi'_{hyd}}^{(n+1/2)} |
| 448 |
\\ |
| 449 |
{\eta}^* & = & |
| 450 |
\epsilon_{fs} {\eta}^{n} + \epsilon_{fw} \Delta t (P-E) |
| 451 |
- \Delta t {\bf \nabla}_h \cdot \int_{R_{fixed}}^{R_o} |
| 452 |
[ \gamma \vec{\bf v}^* + (1-\gamma) \vec{\bf v}^{n}] dr |
| 453 |
\end{eqnarray*} |
| 454 |
\\ |
| 455 |
In the hydrostatic case ($\epsilon_{nh}=0$), allowing us to find |
| 456 |
${\eta}^{n+1}$, thus: |
| 457 |
$$ |
| 458 |
\epsilon_{fs} {\eta}^{n+1} - |
| 459 |
{\bf \nabla}_h \cdot \beta\gamma \Delta t^2 b_s (R_o - R_{fixed}) |
| 460 |
{\bf \nabla}_h {\eta}^{n+1} |
| 461 |
= {\eta}^* |
| 462 |
$$ |
| 463 |
and then to compute (correction step): |
| 464 |
$$ |
| 465 |
\vec{\bf v}^{n+1} = \vec{\bf v}^{*} |
| 466 |
- \beta \Delta t {\bf \nabla}_h b_s {\eta}^{n+1} |
| 467 |
$$ |
| 468 |
|
| 469 |
The non-hydrostatic part is solved as described previously. |
| 470 |
|
| 471 |
Note that: |
| 472 |
\begin{enumerate} |
| 473 |
\item The non-hydrostatic part of the code has not yet been |
| 474 |
updated, so that this option cannot be used with $(\beta,\gamma) \neq (1,1)$. |
| 475 |
\item The stability criteria with Crank-Nickelson time stepping |
| 476 |
for the pure linear gravity wave problem in cartesian coordinates is: |
| 477 |
\begin{itemize} |
| 478 |
\item $\beta + \gamma < 1$ : unstable |
| 479 |
\item $\beta \geq 1/2$ and $ \gamma \geq 1/2$ : stable |
| 480 |
\item $\beta + \gamma \geq 1$ : stable if |
| 481 |
$$ |
| 482 |
c_{max}^2 (\beta - 1/2)(\gamma - 1/2) + 1 \geq 0 |
| 483 |
$$ |
| 484 |
$$ |
| 485 |
\mbox{with }~ |
| 486 |
%c^2 = 2 g H {\Delta t}^2 |
| 487 |
%(\frac{1-cos 2 \pi / k}{\Delta x^2} |
| 488 |
%+\frac{1-cos 2 \pi / l}{\Delta y^2}) |
| 489 |
%$$ |
| 490 |
%Practically, the most stringent condition is obtained with $k=l=2$ : |
| 491 |
%$$ |
| 492 |
c_{max} = 2 \Delta t \: \sqrt{g H} \: |
| 493 |
\sqrt{ \frac{1}{\Delta x^2} + \frac{1}{\Delta y^2} } |
| 494 |
$$ |
| 495 |
\end{itemize} |
| 496 |
\end{enumerate} |
| 497 |
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