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% $Name$ |
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\section{Tracer equations} |
\section{Tracer equations} |
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\label{sec:tracer_equations} |
\label{sect:tracer_equations} |
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The basic discretization used for the tracer equations is the second |
The basic discretization used for the tracer equations is the second |
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order piece-wise constant finite volume form of the forced |
order piece-wise constant finite volume form of the forced |
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described here. |
described here. |
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\subsection{Time-stepping of tracers: ABII} |
\subsection{Time-stepping of tracers: ABII} |
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\label{sec:tracer_equations_abII} |
\label{sect:tracer_equations_abII} |
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The default advection scheme is the centered second order method which |
The default advection scheme is the centered second order method which |
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requires a second order or quasi-second order time-stepping scheme to |
requires a second order or quasi-second order time-stepping scheme to |
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everywhere else. This term is therefore referred to as the surface |
everywhere else. This term is therefore referred to as the surface |
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correction term. Global conservation is not possible using the |
correction term. Global conservation is not possible using the |
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flux-form (as here) and a linearized free-surface |
flux-form (as here) and a linearized free-surface |
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(\cite{Griffies00,Campin02}). |
(\cite{griffies:00,campin:02}). |
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The continuity equation can be recovered by setting |
The continuity equation can be recovered by setting |
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$G_{diff}=G_{forc}=0$ and $\tau=1$. |
$G_{diff}=G_{forc}=0$ and $\tau=1$. |
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\section{Linear advection schemes} |
\section{Linear advection schemes} |
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\label{sect:tracer-advection} |
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\begin{rawhtml} |
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<!-- CMIREDIR:linear_advection_schemes --> |
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\end{rawhtml} |
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\begin{figure} |
\begin{figure} |
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\resizebox{5.5in}{!}{\includegraphics{part2/advect-1d-lo.eps}} |
\resizebox{5.5in}{!}{\includegraphics{part2/advect-1d-lo.eps}} |
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For non-divergent flow, this discretization can be shown to conserve |
For non-divergent flow, this discretization can be shown to conserve |
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the tracer both locally and globally and to globally conserve tracer |
the tracer both locally and globally and to globally conserve tracer |
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variance, $\tau^2$. The proof is given in \cite{Adcroft95,Adcroft97}. |
variance, $\tau^2$. The proof is given in \cite{adcroft:95,adcroft:97}. |
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\fbox{ \begin{minipage}{4.75in} |
\fbox{ \begin{minipage}{4.75in} |
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{\em S/R GAD\_C2\_ADV\_X} ({\em gad\_c2\_adv\_x.F}) |
{\em S/R GAD\_C2\_ADV\_X} ({\em gad\_c2\_adv\_x.F}) |
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r = \frac{ \tau_{i+1} - \tau_{i} }{ \tau_{i} - \tau_{i-1} } & \forall & u < 0 |
r = \frac{ \tau_{i+1} - \tau_{i} }{ \tau_{i} - \tau_{i-1} } & \forall & u < 0 |
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\end{eqnarray} |
\end{eqnarray} |
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as it's argument. There are many choices of limiter function but we |
as it's argument. There are many choices of limiter function but we |
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only provide the Superbee limiter \cite{Roe85}: |
only provide the Superbee limiter \cite{roe:85}: |
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\begin{equation} |
\begin{equation} |
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\psi(r) = \max[0,\min[1,2r],\min[2,r]] |
\psi(r) = \max[0,\min[1,2r],\min[2,r]] |
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\end{equation} |
\end{equation} |
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The DST3 method described above must be used in a forward-in-time |
The DST3 method described above must be used in a forward-in-time |
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manner and is stable for $0 \le |c| \le 1$. Although the scheme |
manner and is stable for $0 \le |c| \le 1$. Although the scheme |
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appears to be forward-in-time, it is in fact second order in time and |
appears to be forward-in-time, it is in fact third order in time and |
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the accuracy increases with the Courant number! For low Courant |
the accuracy increases with the Courant number! For low Courant |
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number, DST3 produces very similar results (indistinguishable in |
number, DST3 produces very similar results (indistinguishable in |
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Fig.~\ref{fig:advect-1d-lo}) to the linear third order method but for |
Fig.~\ref{fig:advect-1d-lo}) to the linear third order method but for |
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phenomenon. |
phenomenon. |
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Finally, the bottom left and right panels use the same advection |
Finally, the bottom left and right panels use the same advection |
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scheme but the right does not use the mutli-dimensional method. At low |
scheme but the right does not use the multi-dimensional method. At low |
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Courant number this appears to not matter but for moderate Courant |
Courant number this appears to not matter but for moderate Courant |
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number severe distortion of the feature is apparent. Moreover, the |
number severe distortion of the feature is apparent. Moreover, the |
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stability of the multi-dimensional scheme is determined by the maximum |
stability of the multi-dimensional scheme is determined by the maximum |
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non-linear schemes have the most stability (up to Courant number 1). |
non-linear schemes have the most stability (up to Courant number 1). |
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\item If you need to know how much diffusion/dissipation has occurred you |
\item If you need to know how much diffusion/dissipation has occurred you |
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will have a lot of trouble figuring it out with a non-linear method. |
will have a lot of trouble figuring it out with a non-linear method. |
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\item The presence of false extrema is unphysical and this alone is the |
\item The presence of false extrema is non-physical and this alone is the |
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strongest argument for using a positive scheme. |
strongest argument for using a positive scheme. |
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\end{itemize} |
\end{itemize} |