--- manual/s_algorithm/text/tracer.tex 2001/10/25 18:36:53 1.8 +++ manual/s_algorithm/text/tracer.tex 2008/01/15 23:52:12 1.23 @@ -1,8 +1,11 @@ -% $Header: /home/ubuntu/mnt/e9_copy/manual/s_algorithm/text/tracer.tex,v 1.8 2001/10/25 18:36:53 cnh Exp $ +% $Header: /home/ubuntu/mnt/e9_copy/manual/s_algorithm/text/tracer.tex,v 1.23 2008/01/15 23:52:12 jmc Exp $ % $Name: $ \section{Tracer equations} -\label{sec:tracer_equations} +\label{sect:tracer_equations} +\begin{rawhtml} + +\end{rawhtml} The basic discretization used for the tracer equations is the second order piece-wise constant finite volume form of the forced @@ -15,7 +18,10 @@ described here. \subsection{Time-stepping of tracers: ABII} -\label{sec:tracer_equations_abII} +\label{sect:tracer_equations_abII} +\begin{rawhtml} + +\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 @@ -43,7 +49,7 @@ 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 $\tau=1$. @@ -123,6 +129,10 @@ \section{Linear advection schemes} +\label{sect:tracer-advection} +\begin{rawhtml} + +\end{rawhtml} \begin{figure} \resizebox{5.5in}{!}{\includegraphics{part2/advect-1d-lo.eps}} @@ -200,7 +210,7 @@ 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}) @@ -349,6 +359,9 @@ \section{Non-linear advection schemes} +\begin{rawhtml} + +\end{rawhtml} Non-linear advection schemes invoke non-linear interpolation and are widely used in computational fluid dynamics (non-linear does not refer @@ -387,7 +400,7 @@ 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} @@ -449,7 +462,7 @@ 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 @@ -614,10 +627,10 @@ \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} @@ -651,8 +664,54 @@ \end{minipage} } +\begin{figure} +\resizebox{3.5in}{!}{\includegraphics{part2/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{sect:tracer_advection_schemes} +\begin{rawhtml} + +\end{rawhtml} + +\begin{table}[htb] +\centering + \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 \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 + \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 @@ -675,7 +734,7 @@ 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 @@ -704,6 +763,6 @@ 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}