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revision 1.15 by afe, Tue Mar 23 16:47:04 2004 UTC revision 1.26 by jmc, Mon Aug 30 23:09:19 2010 UTC
# Line 2  Line 2 
2  % $Name$  % $Name$
3    
4  \section{Tracer equations}  \section{Tracer equations}
5  \label{sect:tracer_equations}  \label{sec: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{sect:tracer_equations_abII}  \label{sec: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 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}  \label{sec:tracer-advection}
133  \begin{rawhtml}  \begin{rawhtml}
134  <!-- CMIREDIR:linear_advection_schemes: -->  <!-- CMIREDIR:linear_advection_schemes: -->
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{sec:non-linear_advection_schemes}
363    \begin{rawhtml}
364    <!-- CMIREDIR:non-linear_advection_schemes: -->
365    \end{rawhtml}
366    
367  Non-linear advection schemes invoke non-linear interpolation and are  Non-linear advection schemes invoke non-linear interpolation and are
368  widely used in computational fluid dynamics (non-linear does not refer  widely used in computational fluid dynamics (non-linear does not refer
# Line 540  $\tau$: {\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 561  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 582  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 618  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 655  $W$: {\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{sec:tracer_advection_schemes}
677    \begin{rawhtml}
678    <!-- CMIREDIR:comparison_of_advection_schemes: -->
679    \end{rawhtml}
680    
681    \begin{table}[htb]
682    \centering
683     \begin{tabular}[htb]{|l|c|c|c|c|l|}
684       \hline
685       Advection Scheme & code & use  & use Multi- & Stencil & comments \\
686                        &      & A.B. & dimension & (1 dim) & \\
687       \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 \\
691       \hline
692       $3^{rd}$order upwind   & 3 &  Yes & No & 5 pts & linear/$\tau$\\
693       \hline
694       centered $4^{th}$order & 4 &  Yes & No & 5 pts & linear \\
695       \hline \hline
696       $2^{nd}$order DST (Lax-Wendroff)  & 20 &
697                             No & Yes & 3 pts & linear/$\tau$, non-linear/v\\
698       \hline
699       $3^{rd}$order DST & 30 &  No & Yes & 5 pts & linear/$\tau$, non-linear/v\\
700       \hline \hline
701       $2^{nd}$order Flux Limiters & 77 &  No & Yes & 5 pts & non-linear \\
702       \hline
703       $3^{nd}$order DST Flux limiter & 33 &  No & Yes & 5 pts & non-linear \\
704       \hline
705     \end{tabular}
706     \caption{Summary of the different advection schemes available in MITgcm.
707              ``A.B.'' stands for Adams-Bashforth and ``DST'' for direct space time.
708              The code corresponds to the number used to select the corresponding
709              advection scheme in the parameter file (e.g., {\bf tempAdvScheme}=3 in
710              file {\em data} selects the $3^{rd}$ order upwind advection scheme
711              for temperature).
712       }
713     \label{tab:advectionShemes_summary}
714    \end{table}
715    
716    
717  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
718  \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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