/[MITgcm]/manual/s_algorithm/text/tracer.tex
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revision 1.9 by adcroft, Tue Nov 13 15:32:28 2001 UTC revision 1.22 by jmc, Wed Jun 28 17:01:34 2006 UTC
# Line 2  Line 2 
2  % $Name$  % $Name$
3    
4  \section{Tracer equations}  \section{Tracer equations}
5  \label{sec:tracer_equations}  \label{sect:tracer_equations}
6    \begin{rawhtml}
7    <!-- CMIREDIR:tracer_equations: -->
8    \end{rawhtml}
9    
10  The basic discretization used for the tracer equations is the second  The basic discretization used for the tracer equations is the second
11  order piece-wise constant finite volume form of the forced  order piece-wise constant finite volume form of the forced
# Line 15  part of the tracer equations and the var Line 18  part of the tracer equations and the var
18  described here.  described here.
19    
20  \subsection{Time-stepping of tracers: ABII}  \subsection{Time-stepping of tracers: ABII}
21  \label{sec:tracer_equations_abII}  \label{sect:tracer_equations_abII}
22    \begin{rawhtml}
23    <!-- CMIREDIR:tracer_equations_abII: -->
24    \end{rawhtml}
25    
26  The default advection scheme is the centered second order method which  The default advection scheme is the centered second order method which
27  requires a second order or quasi-second order time-stepping scheme to  requires a second order or quasi-second order time-stepping scheme to
# Line 43  only affects the surface layer since the Line 49  only affects the surface layer since the
49  everywhere else. This term is therefore referred to as the surface  everywhere else. This term is therefore referred to as the surface
50  correction term. Global conservation is not possible using the  correction term. Global conservation is not possible using the
51  flux-form (as here) and a linearized free-surface  flux-form (as here) and a linearized free-surface
52  (\cite{Griffies00,Campin02}).  (\cite{griffies:00,campin:02}).
53    
54  The continuity equation can be recovered by setting  The continuity equation can be recovered by setting
55  $G_{diff}=G_{forc}=0$ and $\tau=1$.  $G_{diff}=G_{forc}=0$ and $\tau=1$.
# Line 123  the forward method. Line 129  the forward method.
129    
130    
131  \section{Linear advection schemes}  \section{Linear advection schemes}
132    \label{sect:tracer-advection}
133    \begin{rawhtml}
134    <!-- CMIREDIR:linear_advection_schemes: -->
135    \end{rawhtml}
136    
137  \begin{figure}  \begin{figure}
138  \resizebox{5.5in}{!}{\includegraphics{part2/advect-1d-lo.eps}}  \resizebox{5.5in}{!}{\includegraphics{part2/advect-1d-lo.eps}}
# Line 200  W & = & {\cal A}_c w Line 210  W & = & {\cal A}_c w
210    
211  For non-divergent flow, this discretization can be shown to conserve  For non-divergent flow, this discretization can be shown to conserve
212  the tracer both locally and globally and to globally conserve tracer  the tracer both locally and globally and to globally conserve tracer
213  variance, $\tau^2$. The proof is given in \cite{Adcroft95,Adcroft97}.  variance, $\tau^2$. The proof is given in \cite{adcroft:95,adcroft:97}.
214    
215  \fbox{ \begin{minipage}{4.75in}  \fbox{ \begin{minipage}{4.75in}
216  {\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})
# Line 349  if the limiter is set to zero. Line 359  if the limiter is set to zero.
359    
360    
361  \section{Non-linear advection schemes}  \section{Non-linear advection schemes}
362    \begin{rawhtml}
363    <!-- CMIREDIR:non-linear_advection_schemes: -->
364    \end{rawhtml}
365    
366  Non-linear advection schemes invoke non-linear interpolation and are  Non-linear advection schemes invoke non-linear interpolation and are
367  widely used in computational fluid dynamics (non-linear does not refer  widely used in computational fluid dynamics (non-linear does not refer
# Line 387  r = \frac{ \tau_{i-1} - \tau_{i-2} }{ \t Line 400  r = \frac{ \tau_{i-1} - \tau_{i-2} }{ \t
400  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
401  \end{eqnarray}  \end{eqnarray}
402  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
403  only provide the Superbee limiter \cite{Roe85}:  only provide the Superbee limiter \cite{roe:85}:
404  \begin{equation}  \begin{equation}
405  \psi(r) = \max[0,\min[1,2r],\min[2,r]]  \psi(r) = \max[0,\min[1,2r],\min[2,r]]
406  \end{equation}  \end{equation}
# Line 449  to centered second order advection in th Line 462  to centered second order advection in th
462    
463  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
464  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
465  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
466  the accuracy increases with the Courant number! For low Courant  the accuracy increases with the Courant number! For low Courant
467  number, DST3 produces very similar results (indistinguishable in  number, DST3 produces very similar results (indistinguishable in
468  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
# Line 614  as if in one dimension: Line 627  as if in one dimension:
627  \tau^{n+1/3} & = & \tau^{n}  \tau^{n+1/3} & = & \tau^{n}
628  - \Delta t \left( \frac{1}{\Delta x} \delta_i F^x(\tau^{n})  - \Delta t \left( \frac{1}{\Delta x} \delta_i F^x(\tau^{n})
629             + \tau^{n} \frac{1}{\Delta x} \delta_i u \right) \\             + \tau^{n} \frac{1}{\Delta x} \delta_i u \right) \\
630  \tau^{n+2/3} & = & \tau^{n}  \tau^{n+2/3} & = & \tau^{n+1/3}
631  - \Delta t \left( \frac{1}{\Delta y} \delta_j F^y(\tau^{n+1/3})  - \Delta t \left( \frac{1}{\Delta y} \delta_j F^y(\tau^{n+1/3})
632             + \tau^{n} \frac{1}{\Delta y} \delta_i v \right) \\             + \tau^{n} \frac{1}{\Delta y} \delta_i v \right) \\
633  \tau^{n+3/3} & = & \tau^{n}  \tau^{n+3/3} & = & \tau^{n+2/3}
634  - \Delta t \left( \frac{1}{\Delta r} \delta_k F^x(\tau^{n+2/3})  - \Delta t \left( \frac{1}{\Delta r} \delta_k F^x(\tau^{n+2/3})
635             + \tau^{n} \frac{1}{\Delta r} \delta_i w \right)             + \tau^{n} \frac{1}{\Delta r} \delta_i w \right)
636  \end{eqnarray}  \end{eqnarray}
# Line 653  $W$: {\bf rTrans} (local) Line 666  $W$: {\bf rTrans} (local)
666    
667    
668  \section{Comparison of advection schemes}  \section{Comparison of advection schemes}
669    \label{sect:tracer_advection_schemes}
670    \begin{rawhtml}
671    <!-- CMIREDIR:comparison_of_advection_schemes: -->
672    \end{rawhtml}
673    
674    \begin{table}[htb]
675    \centering
676     \begin{tabular}[htb]{|l|c|c|c|c|l|}
677       \hline
678       Advection Scheme & code & use  & use Multi- & Stencil & comments \\
679                        &      & A.B. & dimension & (1 dim) & \\
680       \hline \hline
681       $1^{rst}$order upwind  & 1 &  No & Yes & 3 pts & linear/$\tau$, non-linear/v\\
682       \hline
683       centered $2^{nd}$order & 2 &  Yes & No & 3 pts & linear \\
684       \hline
685       $3^{rd}$order upwind   & 3 &  Yes & No & 5 pts & linear/$\tau$\\
686       \hline
687       centered $4^{th}$order & 4 &  Yes & No & 5 pts & linear \\
688       \hline \hline
689       $2^{nd}$order DST (Lax-Wendroff)  & 20 &
690                             No & Yes & 3 pts & linear/$\tau$, non-linear/v\\
691       \hline
692       $3^{rd}$order DST & 30 &  No & Yes & 5 pts & linear/$\tau$, non-linear/v\\
693       \hline \hline
694       $2^{nd}$order Flux Limiters & 77 &  No & Yes & 5 pts & non-linear \\
695       \hline
696       $3^{nd}$order DST Flux limiter & 33 &  No & Yes & 5 pts & non-linear \\
697       \hline
698     \end{tabular}
699     \caption{Summary of the different advection schemes available in MITgcm.
700              ``A.B.'' stands for Adams-Bashforth and ``DST'' for direct space time.
701              The code corresponds to the number used to select the corresponding
702              advection scheme in the parameter file (e.g., {\bf tempAdvScheme}=3 in
703              file {\em data} selects the $3^{rd}$ order upwind advection scheme
704              for temperature).
705       }
706     \label{tab:advectionShemes_summary}
707    \end{table}
708    
709    
710  Figs.~\ref{fig:advect-2d-lo-diag}, \ref{fig:advect-2d-mid-diag} and  Figs.~\ref{fig:advect-2d-lo-diag}, \ref{fig:advect-2d-mid-diag} and
711  \ref{fig:advect-2d-hi-diag} show solutions to a simple diagonal  \ref{fig:advect-2d-hi-diag} show solutions to a simple diagonal
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