/[MITgcm]/manual/s_algorithm/text/tracer.tex
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revision 1.17 by jmc, Thu Oct 14 19:53:04 2004 UTC revision 1.25 by jmc, Fri Aug 27 13:08:18 2010 UTC
# Line 3  Line 3 
3    
4  \section{Tracer equations}  \section{Tracer equations}
5  \label{sect: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 16  described here. Line 19  described here.
19    
20  \subsection{Time-stepping of tracers: ABII}  \subsection{Time-stepping of tracers: ABII}
21  \label{sect: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 129  the forward method. Line 135  the forward method.
135  \end{rawhtml}  \end{rawhtml}
136    
137  \begin{figure}  \begin{figure}
138  \resizebox{5.5in}{!}{\includegraphics{part2/advect-1d-lo.eps}}  \resizebox{5.5in}{!}{\includegraphics{s_algorithm/figs/advect-1d-lo.eps}}
139  \caption{  \caption{
140  Comparison of 1-D advection schemes. Courant number is 0.05 with 60  Comparison of 1-D advection schemes. Courant number is 0.05 with 60
141  points and solutions are shown for T=1 (one complete period).  points and solutions are shown for T=1 (one complete period).
# Line 147  $\mu=c/(1-c)$. Line 153  $\mu=c/(1-c)$.
153  \end{figure}  \end{figure}
154    
155  \begin{figure}  \begin{figure}
156  \resizebox{5.5in}{!}{\includegraphics{part2/advect-1d-hi.eps}}  \resizebox{5.5in}{!}{\includegraphics{s_algorithm/figs/advect-1d-hi.eps}}
157  \caption{  \caption{
158  Comparison of 1-D advection schemes. Courant number is 0.89 with 60  Comparison of 1-D advection schemes. Courant number is 0.89 with 60
159  points and solutions are shown for T=1 (one complete period).  points and solutions are shown for T=1 (one complete period).
# Line 353  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    \label{sect:non-linear_advection_schemes}
363  \begin{rawhtml}  \begin{rawhtml}
364  <!-- CMIREDIR:non-linear_advection_schemes: -->  <!-- CMIREDIR:non-linear_advection_schemes: -->
365  \end{rawhtml}  \end{rawhtml}
# Line 543  $\tau$: {\bf tracer} (argument) Line 550  $\tau$: {\bf tracer} (argument)
550  \subsection{Multi-dimensional advection}  \subsection{Multi-dimensional advection}
551    
552  \begin{figure}  \begin{figure}
553  \resizebox{5.5in}{!}{\includegraphics{part2/advect-2d-lo-diag.eps}}  \resizebox{5.5in}{!}{\includegraphics{s_algorithm/figs/advect-2d-lo-diag.eps}}
554  \caption{  \caption{
555  Comparison of advection schemes in two dimensions; diagonal advection  Comparison of advection schemes in two dimensions; diagonal advection
556  of a resolved Gaussian feature. Courant number is 0.01 with  of a resolved Gaussian feature. Courant number is 0.01 with
# Line 564  lines). Line 571  lines).
571  \end{figure}  \end{figure}
572    
573  \begin{figure}  \begin{figure}
574  \resizebox{5.5in}{!}{\includegraphics{part2/advect-2d-mid-diag.eps}}  \resizebox{5.5in}{!}{\includegraphics{s_algorithm/figs/advect-2d-mid-diag.eps}}
575  \caption{  \caption{
576  Comparison of advection schemes in two dimensions; diagonal advection  Comparison of advection schemes in two dimensions; diagonal advection
577  of a resolved Gaussian feature. Courant number is 0.27 with  of a resolved Gaussian feature. Courant number is 0.27 with
# Line 585  lines). Line 592  lines).
592  \end{figure}  \end{figure}
593    
594  \begin{figure}  \begin{figure}
595  \resizebox{5.5in}{!}{\includegraphics{part2/advect-2d-hi-diag.eps}}  \resizebox{5.5in}{!}{\includegraphics{s_algorithm/figs/advect-2d-hi-diag.eps}}
596  \caption{  \caption{
597  Comparison of advection schemes in two dimensions; diagonal advection  Comparison of advection schemes in two dimensions; diagonal advection
598  of a resolved Gaussian feature. Courant number is 0.47 with  of a resolved Gaussian feature. Courant number is 0.47 with
# Line 621  as if in one dimension: Line 628  as if in one dimension:
628  \tau^{n+1/3} & = & \tau^{n}  \tau^{n+1/3} & = & \tau^{n}
629  - \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})
630             + \tau^{n} \frac{1}{\Delta x} \delta_i u \right) \\             + \tau^{n} \frac{1}{\Delta x} \delta_i u \right) \\
631  \tau^{n+2/3} & = & \tau^{n}  \tau^{n+2/3} & = & \tau^{n+1/3}
632  - \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})
633             + \tau^{n} \frac{1}{\Delta y} \delta_i v \right) \\             + \tau^{n} \frac{1}{\Delta y} \delta_i v \right) \\
634  \tau^{n+3/3} & = & \tau^{n}  \tau^{n+3/3} & = & \tau^{n+2/3}
635  - \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})
636             + \tau^{n} \frac{1}{\Delta r} \delta_i w \right)             + \tau^{n} \frac{1}{\Delta r} \delta_i w \right)
637  \end{eqnarray}  \end{eqnarray}
# Line 658  $W$: {\bf rTrans} (local) Line 665  $W$: {\bf rTrans} (local)
665    
666  \end{minipage} }  \end{minipage} }
667    
668    \begin{figure}
669    \resizebox{3.5in}{!}{\includegraphics{s_algorithm/figs/multiDim_CS.eps}}
670    \caption{Muti-dimensional advection time-stepping with Cubed-Sphere topology
671    \label{fig:advect-multidim_cs}
672    }
673    \end{figure}
674    
675  \section{Comparison of advection schemes}  \section{Comparison of advection schemes}
676    \label{sect:tracer_advection_schemes}
677    \begin{rawhtml}
678    <!-- CMIREDIR:comparison_of_advection_schemes: -->
679    \end{rawhtml}
680    
681  \begin{table}[htb]  \begin{table}[htb]
682  \centering  \centering
# Line 668  $W$: {\bf rTrans} (local) Line 685  $W$: {\bf rTrans} (local)
685     Advection Scheme & code & use  & use Multi- & Stencil & comments \\     Advection Scheme & code & use  & use Multi- & Stencil & comments \\
686                      &      & A.B. & dimension & (1 dim) & \\                      &      & A.B. & dimension & (1 dim) & \\
687     \hline \hline     \hline \hline
688       $1^{rst}$order upwind  & 1 &  No & Yes & 3 pts & linear/$\tau$, non-linear/v\\
689       \hline
690     centered $2^{nd}$order & 2 &  Yes & No & 3 pts & linear \\     centered $2^{nd}$order & 2 &  Yes & No & 3 pts & linear \\
691     \hline     \hline
692     $3^{rd}$order upwind   & 3 &  Yes & No & 5 pts & linear/tracer\\     $3^{rd}$order upwind   & 3 &  Yes & No & 5 pts & linear/$\tau$\\
693     \hline     \hline
694     centered $4^{th}$order & 4 &  Yes & No & 5 pts & linear \\     centered $4^{th}$order & 4 &  Yes & No & 5 pts & linear \\
695     \hline \hline     \hline \hline
696  %  Lax-Wendroff       & 10 &  No & Yes & 3 pts & linear/tracer, non-linear/flow\\     $2^{nd}$order DST (Lax-Wendroff)  & 20 &
697  %  \hline                           No & Yes & 3 pts & linear/$\tau$, non-linear/v\\
698     $3^{rd}$order DST & 30 &  No & Yes & 5 pts & linear/tracer, non-linear/flow\\     \hline
699       $3^{rd}$order DST & 30 &  No & Yes & 5 pts & linear/$\tau$, non-linear/v\\
700     \hline \hline     \hline \hline
701     $2^{nd}$order Flux Limiters & 77 &  No & Yes & 5 pts & non-linear \\     $2^{nd}$order Flux Limiters & 77 &  No & Yes & 5 pts & non-linear \\
702     \hline     \hline
# Line 686  $W$: {\bf rTrans} (local) Line 706  $W$: {\bf rTrans} (local)
706   \caption{Summary of the different advection schemes available in MITgcm.   \caption{Summary of the different advection schemes available in MITgcm.
707            ``A.B.'' stands for Adams-Bashforth and ``DST'' for direct space time.            ``A.B.'' stands for Adams-Bashforth and ``DST'' for direct space time.
708            The code corresponds to the number used to select the corresponding            The code corresponds to the number used to select the corresponding
709            advection scheme in the parameter file (e.g., {\em tempAdvScheme=3} in            advection scheme in the parameter file (e.g., {\bf tempAdvScheme}=3 in
710            file {\em data} selects the $3^{rd}$ order upwind advection scheme            file {\em data} selects the $3^{rd}$ order upwind advection scheme
711            for temperature).            for temperature).
712     }     }
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