/[MITgcm]/manual/s_algorithm/text/tracer.tex

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-revision 1.8 by cnh,
Thu Oct 25 18:36:53 2001 UTC
+revision 1.27 by heimbach,
Wed Oct 26 17:30:23 2016 UTC
 Line 3
  \section{Tracer equations}
  \label{sec:tracer_equations}
+ \begin{rawhtml}
+ <!-- CMIREDIR:tracer_equations: -->
+ \end{rawhtml}
  The basic discretization used for the tracer equations is the second
  order piece-wise constant finite volume form of the forced
-Line 16 
 described here.
+Line 19 
 described here.
  \subsection{Time-stepping of tracers: ABII}
  \label{sec:tracer_equations_abII}
+ \begin{rawhtml}
+ <!-- CMIREDIR:tracer_equations_abII: -->
+ \end{rawhtml}
  The default advection scheme is the centered second order method which
  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
  everywhere else. This term is therefore referred to as the surface
  correction term. Global conservation is not possible using the
  flux-form (as here) and a linearized free-surface
- (\cite{Griffies00,Campin02}).
+ (\cite{griffies:00,campin:02}).
  The continuity equation can be recovered by setting
  $G_{diff}=G_{forc}=0$ and $\tau=1$.
-Line 123 
 the forward method.
+Line 129 
 the forward method.
  \section{Linear advection schemes}
+ \label{sec:tracer-advection}
+ \begin{rawhtml}
+ <!-- CMIREDIR:linear_advection_schemes: -->
+ \end{rawhtml}
  \begin{figure}
- \resizebox{5.5in}{!}{\includegraphics{part2/advect-1d-lo.eps}}
+ \resizebox{5.5in}{!}{\includegraphics{s_algorithm/figs/advect-1d-lo.eps}}
  \caption{
  Comparison of 1-D advection schemes. Courant number is 0.05 with 60
  points and solutions are shown for T=1 (one complete period).
-Line 143 
 $\mu=c/(1-c)$.
+Line 153 
 $\mu=c/(1-c)$.
  \end{figure}
  \begin{figure}
- \resizebox{5.5in}{!}{\includegraphics{part2/advect-1d-hi.eps}}
+ \resizebox{5.5in}{!}{\includegraphics{s_algorithm/figs/advect-1d-hi.eps}}
  \caption{
  Comparison of 1-D advection schemes. Courant number is 0.89 with 60
  points and solutions are shown for T=1 (one complete period).
-Line 200 
 W & = & {\cal A}_c w
+Line 210 
 W & = & {\cal A}_c w
  For non-divergent flow, this discretization can be shown to conserve
  the tracer both locally and globally and to globally conserve tracer
- variance, $\tau^2$. The proof is given in \cite{Adcroft95,Adcroft97}.
+ variance, $\tau^2$. The proof is given in \cite{adcroft:95,adcroft:97}.
  \fbox{ \begin{minipage}{4.75in}
  {\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.
  \section{Non-linear advection schemes}
+ \label{sec:non-linear_advection_schemes}
+ \begin{rawhtml}
+ <!-- CMIREDIR:non-linear_advection_schemes: -->
+ \end{rawhtml}
  Non-linear advection schemes invoke non-linear interpolation and are
  widely used in computational fluid dynamics (non-linear does not refer
-Line 387 
 r = \frac{ \tau_{i-1} - \tau_{i-2} }{ \t
+Line 401 
 r = \frac{ \tau_{i-1} - \tau_{i-2} }{ \t
  r = \frac{ \tau_{i+1} - \tau_{i} }{ \tau_{i} - \tau_{i-1} } & \forall & u < 0
  \end{eqnarray}
  as it's argument. There are many choices of limiter function but we
- only provide the Superbee limiter \cite{Roe85}:
+ only provide the Superbee limiter \cite{roe:85}:
  \begin{equation}
  \psi(r) = \max[0,\min[1,2r],\min[2,r]]
  \end{equation}
-Line 449 
 to centered second order advection in th
+Line 463 
 to centered second order advection in th
  The DST3 method described above must be used in a forward-in-time
  manner and is stable for $0 \le |c| \le 1$. Although the scheme
- 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
  the accuracy increases with the Courant number! For low Courant
  number, DST3 produces very similar results (indistinguishable in
  Fig.~\ref{fig:advect-1d-lo}) to the linear third order method but for
-Line 536 
 $\tau$: {\bf tracer} (argument)
+Line 550 
 $\tau$: {\bf tracer} (argument)
  \subsection{Multi-dimensional advection}
  \begin{figure}
- \resizebox{5.5in}{!}{\includegraphics{part2/advect-2d-lo-diag.eps}}
+ \resizebox{5.5in}{!}{\includegraphics{s_algorithm/figs/advect-2d-lo-diag.eps}}
  \caption{
  Comparison of advection schemes in two dimensions; diagonal advection
  of a resolved Gaussian feature. Courant number is 0.01 with
-Line 557 
 lines).
+Line 571 
 lines).
  \end{figure}
  \begin{figure}
- \resizebox{5.5in}{!}{\includegraphics{part2/advect-2d-mid-diag.eps}}
+ \resizebox{5.5in}{!}{\includegraphics{s_algorithm/figs/advect-2d-mid-diag.eps}}
  \caption{
  Comparison of advection schemes in two dimensions; diagonal advection
  of a resolved Gaussian feature. Courant number is 0.27 with
-Line 578 
 lines).
+Line 592 
 lines).
  \end{figure}
  \begin{figure}
- \resizebox{5.5in}{!}{\includegraphics{part2/advect-2d-hi-diag.eps}}
+ \resizebox{5.5in}{!}{\includegraphics{s_algorithm/figs/advect-2d-hi-diag.eps}}
  \caption{
  Comparison of advection schemes in two dimensions; diagonal advection
  of a resolved Gaussian feature. Courant number is 0.47 with
-Line 614 
 as if in one dimension:
+Line 628 
 as if in one dimension:
  \tau^{n+1/3} & = & \tau^{n}
  - \Delta t \left( \frac{1}{\Delta x} \delta_i F^x(\tau^{n})
             + \tau^{n} \frac{1}{\Delta x} \delta_i u \right) \\
- \tau^{n+2/3} & = & \tau^{n}
+ \tau^{n+2/3} & = & \tau^{n+1/3}
  - \Delta t \left( \frac{1}{\Delta y} \delta_j F^y(\tau^{n+1/3})
             + \tau^{n} \frac{1}{\Delta y} \delta_i v \right) \\
- \tau^{n+3/3} & = & \tau^{n}
+ \tau^{n+3/3} & = & \tau^{n+2/3}
  - \Delta t \left( \frac{1}{\Delta r} \delta_k F^x(\tau^{n+2/3})
             + \tau^{n} \frac{1}{\Delta r} \delta_i w \right)
  \end{eqnarray}
-Line 651 
 $W$: {\bf rTrans} (local)
+Line 665 
 $W$: {\bf rTrans} (local)
  \end{minipage} }
+ \begin{figure}
+ \resizebox{3.5in}{!}{\includegraphics{s_algorithm/figs/multiDim_CS.eps}}
+ \caption{Muti-dimensional advection time-stepping with Cubed-Sphere topology
+ \label{fig:advect-multidim_cs}
+ }
+ \end{figure}
  \section{Comparison of advection schemes}
+ \label{sec:tracer_advection_schemes}
+ \begin{rawhtml}
+ <!-- CMIREDIR:comparison_of_advection_schemes: -->
+ \end{rawhtml}
+ \begin{table}[htb]
+ \centering
+ {\small
+  \begin{tabular}[htb]{|l|c|c|c|c|l|}
+    \hline
+    Advection Scheme & code & use  & use Multi- & Stencil & comments \\
+                     &      & A.B. & dimension & (1 dim) & \\
+    \hline \hline
+    $1^{rst}$order upwind  & 1 &  No & Yes & 3 pts & linear/$\tau$, non-linear/v\\
+    \hline
+    centered $2^{nd}$order & 2 &  Yes & No & 3 pts & linear \\
+    \hline
+    $3^{rd}$order upwind   & 3 &  Yes & No & 5 pts & linear/$\tau$\\
+    \hline
+    centered $4^{th}$order & 4 &  Yes & No & 5 pts & linear \\
+    \hline \hline
+    $2^{nd}$order DST (Lax-Wendroff)  & 20 &
+                          No & Yes & 3 pts & linear/$\tau$, non-linear/v\\
+    \hline
+    $3^{rd}$order DST & 30 &  No & Yes & 5 pts & linear/$\tau$, non-linear/v\\
+    \hline
+    $2^{nd}$order-moment Prather & 80 & No & Yes & ~ & ~ \\
+    \hline \hline
+    $2^{nd}$order Flux Limiters & 77 &  No & Yes & 5 pts & non-linear \\
+    \hline
+    $3^{nd}$order DST Flux limiter & 33 &  No & Yes & 5 pts & non-linear \\
+    \hline
+    $2^{nd}$order-moment Prather w. limiter & 81 & No & Yes & ~ & ~ \\
+    \hline
+    piecewise parabolic w. ``null'' limiter & 40 & No & Yes & ~ & ~ \\
+    \hline
+    piecewise parabolic w. ``mono'' limiter & 41 & No & Yes & ~ & ~ \\
+    \hline
+    piecewise quartic w. ``null'' limiter & 50 & No & Yes & ~ & ~ \\
+    \hline
+    piecewise quartic w. ``mono'' limiter & 51 & No & Yes & ~ & ~ \\
+    \hline
+    piecewise quartic w. ``weno'' limiter & 52 & No & Yes & ~ & ~ \\
+    \hline
+    $7^{nd}$order one-step method & 7 & No & Yes & ~ & ~ \\
+    with Monotonicity Preserving Limiter & ~ & ~ & ~ & ~ & ~ \\
+    \hline
+  \end{tabular}
+  }
+  \caption{Summary of the different advection schemes available in MITgcm.
+           ``A.B.'' stands for Adams-Bashforth and ``DST'' for direct space time.
+           The code corresponds to the number used to select the corresponding
+           advection scheme in the parameter file (e.g., {\bf tempAdvScheme}=3 in
+           file {\em data} selects the $3^{rd}$ order upwind advection scheme
+           for temperature).
+    }
+  \label{tab:advectionShemes_summary}
+ \end{table}
  Figs.~\ref{fig:advect-2d-lo-diag}, \ref{fig:advect-2d-mid-diag} and
  \ref{fig:advect-2d-hi-diag} show solutions to a simple diagonal
-Line 675 
 Figs.~\ref{fig:advect-1d-lo} and \ref{fi
+Line 755 
 Figs.~\ref{fig:advect-1d-lo} and \ref{fi
  phenomenon.
  Finally, the bottom left and right panels use the same advection
- 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
  Courant number this appears to not matter but for moderate Courant
  number severe distortion of the feature is apparent. Moreover, the
  stability of the multi-dimensional scheme is determined by the maximum
-Line 704 
 flux limited scheme is almost essential.
+Line 784 
 flux limited scheme is almost essential.
  non-linear schemes have the most stability (up to Courant number 1).
  \item If you need to know how much diffusion/dissipation has occurred you
  will have a lot of trouble figuring it out with a non-linear method.
- \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
  strongest argument for using a positive scheme.
  \end{itemize}

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