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This chapter lays out the numerical schemes that are |
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employed in the core MITgcm algorithm. Whenever possible |
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links are made to actual program code in the MITgcm implementation. |
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The chapter begins with a discussion of the temporal discretization |
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used in MITgcm. This discussion is followed by sections that |
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describe the spatial discretization. The schemes employed for momentum |
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terms are described first, afterwards the schemes that apply to |
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passive and dynamically active tracers are described. |
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\section{Time-stepping} |
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The equations of motion integrated by the model involve four |
The equations of motion integrated by the model involve four |
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prognostic equations for flow, $u$ and $v$, temperature, $\theta$, and |
prognostic equations for flow, $u$ and $v$, temperature, $\theta$, and |
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salt/moisture, $S$, and three diagnostic equations for vertical flow, |
salt/moisture, $S$, and three diagnostic equations for vertical flow, |
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\begin{figure} |
\begin{figure} |
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\begin{center} \fbox{ \begin{minipage}{4.5in} \begin{tabbing} |
\begin{center} \fbox{ \begin{minipage}{4.5in} \begin{tabbing} |
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aaa \= aaa \= aaa \= aaa \= aaa \= aaa \kill |
aaa \= aaa \= aaa \= aaa \= aaa \= aaa \kill |
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FORWARD\_STEP \\ |
\proclink{FORWARD\_STEP}{../code/._model_src_forward_step.F} \\ |
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\> DYNAMICS \\ |
\> DYNAMICS \\ |
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\>\> TIMESTEP \` $u^*$,$v^*$ (\ref{eq:ustar-rigid-lid},\ref{eq:vstar-rigid-lid}) \\ |
\>\> TIMESTEP \` $u^*$,$v^*$ (\ref{eq:ustar-rigid-lid},\ref{eq:vstar-rigid-lid}) \\ |
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\> SOLVE\_FOR\_PRESSURE \\ |
\> SOLVE\_FOR\_PRESSURE \\ |
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\item |
\item |
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the prognostic phase, equations \ref{eq:ustar-rigid-lid} and \ref{eq:vstar-rigid-lid}, |
the prognostic phase, equations \ref{eq:ustar-rigid-lid} and \ref{eq:vstar-rigid-lid}, |
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stepping forward $u^n$ and $v^n$ to $u^{*}$ and $v^{*}$ is coded in |
stepping forward $u^n$ and $v^n$ to $u^{*}$ and $v^{*}$ is coded in |
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{\em TIMESTEP.F} |
\proclink{TIMESTEP}{../code/._model_src_timestep.F} |
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\item |
\item |
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the vertical integration, $H \widehat{u^*}$ and $H |
the vertical integration, $H \widehat{u^*}$ and $H |
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\widehat{v^*}$, divergence and inversion of the elliptic operator in |
\widehat{v^*}$, divergence and inversion of the elliptic operator in |
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equation \ref{eq:elliptic} is coded in {\em |
equation \ref{eq:elliptic} is coded in |
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SOLVE\_FOR\_PRESSURE.F} |
\proclink{SOLVE\_FOR\_PRESSURE}{../code/._model_src_solve_for_pressure.F} |
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\item |
\item |
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finally, the new flow field at time level $n+1$ given by equations |
finally, the new flow field at time level $n+1$ given by equations |
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\ref{eq:un+1-rigid-lid} and \ref{eq:vn+1-rigid-lid} is calculated in {\em CORRECTION\_STEP.F}. |
\ref{eq:un+1-rigid-lid} and \ref{eq:vn+1-rigid-lid} is calculated in |
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\proclink{CORRECTION\_STEP}{../code/._model_src_correction_step.F}. |
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\end{itemize} |
\end{itemize} |
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The calling tree for these routines is given in |
The calling tree for these routines is given in |
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Fig.~\ref{fig:call-tree-pressure-method}. |
Fig.~\ref{fig:call-tree-pressure-method}. |
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connecting $u,v^n$ with $G_\theta^{n-1/2}$. The flow used to advect |
connecting $u,v^n$ with $G_\theta^{n-1/2}$. The flow used to advect |
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tracers around is not naturally located in time. This could be avoided |
tracers around is not naturally located in time. This could be avoided |
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by applying the Adams-Bashforth extrapolation to the tracer field |
by applying the Adams-Bashforth extrapolation to the tracer field |
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itself and advection that around but this is not yet available. We're |
itself and advecting that around but this approach is not yet |
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not aware of any detrimental effect of this feature. The difficulty |
available. We're not aware of any detrimental effect of this |
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lies mainly in interpretation of what time-level variables and terms |
feature. The difficulty lies mainly in interpretation of what |
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correspond to. |
time-level variables and terms correspond to. |
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\section{Non-hydrostatic formulation} |
\section{Non-hydrostatic formulation} |
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\label{sect:non-hydrostatic} |
\label{sect:non-hydrostatic} |
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[to be written...] |
The non-hydrostatic formulation re-introduces the full vertical |
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momentum equation and requires the solution of a 3-D elliptic |
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equations for non-hydrostatic pressure perturbation. We still |
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intergrate vertically for the hydrostatic pressure and solve a 2-D |
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elliptic equation for the surface pressure/elevation for this reduces |
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the amount of work needed to solve for the non-hydrostatic pressure. |
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The momentum equations are discretized in time as follows: |
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\begin{eqnarray} |
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\frac{1}{\Delta t} u^{n+1} + g \partial_x \eta^{n+1} + \partial_x \phi_{nh}^{n+1} |
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& = & \frac{1}{\Delta t} u^{n} + G_u^{(n+1/2)} \label{eq:discrete-time-u-nh} \\ |
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\frac{1}{\Delta t} v^{n+1} + g \partial_y \eta^{n+1} + \partial_y \phi_{nh}^{n+1} |
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& = & \frac{1}{\Delta t} v^{n} + G_v^{(n+1/2)} \label{eq:discrete-time-v-nh} \\ |
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\frac{1}{\Delta t} w^{n+1} + \partial_r \phi_{nh}^{n+1} |
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& = & \frac{1}{\Delta t} w^{n} + G_w^{(n+1/2)} \label{eq:discrete-time-w-nh} \\ |
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\end{eqnarray} |
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which must satisfy the discrete-in-time depth integrated continuity, |
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equation~\ref{eq:discrete-time-backward-free-surface} and the local continuity equation |
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\begin{equation} |
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\partial_x u^{n+1} + \partial_y v^{n+1} + \partial_r w^{n+1} = 0 |
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\label{eq:non-divergence-nh} |
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\end{equation} |
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As before, the explicit predictions for momentum are consolidated as: |
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\begin{eqnarray*} |
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u^* & = & u^n + \Delta t G_u^{(n+1/2)} \\ |
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v^* & = & v^n + \Delta t G_v^{(n+1/2)} \\ |
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w^* & = & w^n + \Delta t G_w^{(n+1/2)} |
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\end{eqnarray*} |
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but this time we introduce an intermediate step by splitting the |
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tendancy of the flow as follows: |
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\begin{eqnarray} |
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u^{n+1} = u^{**} - \Delta t \partial_x \phi_{nh}^{n+1} |
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& & |
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u^{**} = u^{*} - \Delta t g \partial_x \eta^{n+1} \\ |
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v^{n+1} = v^{**} - \Delta t \partial_y \phi_{nh}^{n+1} |
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& & |
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v^{**} = v^{*} - \Delta t g \partial_y \eta^{n+1} |
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\end{eqnarray} |
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Substituting into the depth integrated continuity |
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(equation~\ref{eq:discrete-time-backward-free-surface}) gives |
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\begin{equation} |
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\partial_x H \partial_x \left( g \eta^{n+1} + \widehat{\phi}_{nh}^{n+1} \right) |
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+ |
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\partial_y H \partial_y \left( g \eta^{n+1} + \widehat{\phi}_{nh}^{n+1} \right) |
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- \frac{\epsilon_{fs}\eta^*}{\Delta t^2} |
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= - \frac{\eta^*}{\Delta t^2} |
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\end{equation} |
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which is approximated by equation |
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\ref{eq:elliptic-backward-free-surface} on the basis that i) |
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$\phi_{nh}^{n+1}$ is not yet known and ii) $\nabla \widehat{\phi}_{nh} |
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<< g \nabla \eta$. If \ref{eq:elliptic-backward-free-surface} is |
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solved accurately then the implication is that $\widehat{\phi}_{nh} |
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\approx 0$ so that thet non-hydrostatic pressure field does not drive |
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barotropic motion. |
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The flow must satisfy non-divergence |
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(equation~\ref{eq:non-divergence-nh}) locally, as well as depth |
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integrated, and this constraint is used to form a 3-D elliptic |
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equations for $\phi_{nh}^{n+1}$: |
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\begin{equation} |
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\partial_{xx} \phi_{nh}^{n+1} + \partial_{yy} \phi_{nh}^{n+1} + |
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\partial_{rr} \phi_{nh}^{n+1} = |
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\partial_x u^{**} + \partial_y v^{**} + \partial_r w^{*} |
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\end{equation} |
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Equation for $w^{n+1}$ will be here as will 3-D elliptic equations. |
The entire algorithm can be summarized as the sequential solution of |
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\label{eq:discrete-time-w} |
the following equations: |
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\begin{eqnarray} |
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u^{*} & = & u^{n} + \Delta t G_u^{(n+1/2)} \label{eq:ustar-nh} \\ |
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v^{*} & = & v^{n} + \Delta t G_v^{(n+1/2)} \label{eq:vstar-nh} \\ |
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w^{*} & = & w^{n} + \Delta t G_w^{(n+1/2)} \label{eq:wstar-nh} \\ |
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\eta^* & = & \epsilon_{fs} \left( \eta^{n} +P-E+R \right)- \Delta t |
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\partial_x H \widehat{u^{*}} |
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+ \partial_y H \widehat{v^{*}} |
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\\ |
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\partial_x g H \partial_x \eta^{n+1} |
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+ \partial_y g H \partial_y \eta^{n+1} |
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- \frac{\epsilon_{fs} \eta^{n+1}}{\Delta t^2} |
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& = & |
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- \frac{\eta^*}{\Delta t^2} |
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\label{eq:elliptic-nh} |
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\\ |
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u^{**} & = & u^{*} - \Delta t g \partial_x \eta^{n+1} \label{eq:unx-nh}\\ |
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v^{**} & = & v^{*} - \Delta t g \partial_y \eta^{n+1} \label{eq:vnx-nh}\\ |
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\partial_{xx} \phi_{nh}^{n+1} + \partial_{yy} \phi_{nh}^{n+1} + |
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\partial_{rr} \phi_{nh}^{n+1} & = & |
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\partial_x u^{**} + \partial_y v^{**} + \partial_r w^{*} \\ |
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u^{n+1} & = & u^{**} - \Delta t \partial_x \phi_{nh}^{n+1} \label{eq:un+1-nh}\\ |
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v^{n+1} & = & v^{**} - \Delta t \partial_y \phi_{nh}^{n+1} \label{eq:vn+1-nh}\\ |
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\partial_r w^{n+1} & = & - \partial_x u^{n+1} - \partial_y v^{n+1} |
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\end{eqnarray} |
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where the last equation is solved by vertically integrating for |
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$w^{n+1}$. |
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This is known as the correction step. However, when the model is |
This is known as the correction step. However, when the model is |
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non-hydrostatic ($\epsilon_{nh}=1$) we need an additional step and an |
non-hydrostatic ($\epsilon_{nh}=1$) we need an additional step and an |
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additional equation for $\phi'_{nh}$. This is obtained by substituting |
additional equation for $\phi'_{nh}$. This is obtained by substituting |
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\ref{eq:discrete-time-u}, \ref{eq:discrete-time-v} and \ref{eq:discrete-time-w} |
\ref{eq:discrete-time-u-nh}, \ref{eq:discrete-time-v-nh} and \ref{eq:discrete-time-w-nh} |
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into continuity: |
into continuity: |
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\begin{equation} |
\begin{equation} |
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\left[ {\bf \nabla}_h^2 + \partial_{rr} \right] {\phi'_{nh}}^{n+1} |
\left[ {\bf \nabla}_h^2 + \partial_{rr} \right] {\phi'_{nh}}^{n+1} |
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