| 3 |
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| 4 |
\section{Tracer equations} |
\section{Tracer equations} |
| 5 |
\label{sect:tracer_equations} |
\label{sect:tracer_equations} |
| 6 |
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\begin{rawhtml} |
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<!-- CMIREDIR:tracer_equations: --> |
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\end{rawhtml} |
| 9 |
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| 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 |
| 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 |
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\begin{rawhtml} |
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<!-- CMIREDIR:tracer_equations_abII: --> |
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\end{rawhtml} |
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| 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 |
| 359 |
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| 360 |
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|
| 361 |
\section{Non-linear advection schemes} |
\section{Non-linear advection schemes} |
| 362 |
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\label{sect:non-linear_advection_schemes} |
| 363 |
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\begin{rawhtml} |
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<!-- CMIREDIR:non-linear_advection_schemes: --> |
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\end{rawhtml} |
| 366 |
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| 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 |
| 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} |
| 665 |
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|
| 666 |
\end{minipage} } |
\end{minipage} } |
| 667 |
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|
| 668 |
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\begin{figure} |
| 669 |
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\resizebox{3.5in}{!}{\includegraphics{part2/multiDim_CS.eps}} |
| 670 |
|
\caption{Muti-dimensional advection time-stepping with Cubed-Sphere topology |
| 671 |
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\label{fig:advect-multidim_cs} |
| 672 |
|
} |
| 673 |
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\end{figure} |
| 674 |
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|
| 675 |
\section{Comparison of advection schemes} |
\section{Comparison of advection schemes} |
| 676 |
|
\label{sect:tracer_advection_schemes} |
| 677 |
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\begin{rawhtml} |
| 678 |
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<!-- CMIREDIR:comparison_of_advection_schemes: --> |
| 679 |
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\end{rawhtml} |
| 680 |
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| 681 |
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\begin{table}[htb] |
| 682 |
|
\centering |
| 683 |
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\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 |
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$2^{nd}$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 |
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\hline |
| 705 |
|
\end{tabular} |
| 706 |
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\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 |
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\label{tab:advectionShemes_summary} |
| 714 |
|
\end{table} |
| 715 |
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| 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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