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1.23 |
% $Header: /u/gcmpack/manual/part2/tracer.tex,v 1.22 2006/06/28 17:01:34 jmc Exp $ |
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1.2 |
% $Name: $ |
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1.1 |
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\section{Tracer equations} |
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1.12 |
\label{sect:tracer_equations} |
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1.18 |
\begin{rawhtml} |
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<!-- CMIREDIR:tracer_equations: --> |
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\end{rawhtml} |
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1.1 |
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1.2 |
The basic discretization used for the tracer equations is the second |
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order piece-wise constant finite volume form of the forced |
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1.8 |
advection-diffusion equations. There are many alternatives to second |
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1.2 |
order method for advection and alternative parameterizations for the |
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sub-grid scale processes. The Gent-McWilliams eddy parameterization, |
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KPP mixing scheme and PV flux parameterization are all dealt with in |
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separate sections. The basic discretization of the advection-diffusion |
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part of the tracer equations and the various advection schemes will be |
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described here. |
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1.3 |
\subsection{Time-stepping of tracers: ABII} |
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1.12 |
\label{sect:tracer_equations_abII} |
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1.18 |
\begin{rawhtml} |
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<!-- CMIREDIR:tracer_equations_abII: --> |
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\end{rawhtml} |
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1.3 |
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The default advection scheme is the centered second order method which |
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requires a second order or quasi-second order time-stepping scheme to |
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be stable. Historically this has been the quasi-second order |
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Adams-Bashforth method (ABII) and applied to all terms. For an |
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arbitrary tracer, $\tau$, the forced advection-diffusion equation |
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reads: |
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\begin{equation} |
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\partial_t \tau + G_{adv}^\tau = G_{diff}^\tau + G_{forc}^\tau |
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\end{equation} |
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where $G_{adv}^\tau$, $G_{diff}^\tau$ and $G_{forc}^\tau$ are the |
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tendencies due to advection, diffusion and forcing, respectively, |
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namely: |
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\begin{eqnarray} |
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G_{adv}^\tau & = & \partial_x u \tau + \partial_y v \tau + \partial_r w \tau |
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- \tau \nabla \cdot {\bf v} \\ |
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G_{diff}^\tau & = & \nabla \cdot {\bf K} \nabla \tau |
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\end{eqnarray} |
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and the forcing can be some arbitrary function of state, time and |
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space. |
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The term, $\tau \nabla \cdot {\bf v}$, is required to retain local |
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conservation in conjunction with the linear implicit free-surface. It |
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only affects the surface layer since the flow is non-divergent |
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everywhere else. This term is therefore referred to as the surface |
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correction term. Global conservation is not possible using the |
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flux-form (as here) and a linearized free-surface |
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1.10 |
(\cite{griffies:00,campin:02}). |
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1.3 |
|
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The continuity equation can be recovered by setting |
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$G_{diff}=G_{forc}=0$ and $\tau=1$. |
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1.8 |
The driver routine that calls the routines to calculate tendencies are |
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1.3 |
{\em S/R CALC\_GT} and {\em S/R CALC\_GS} for temperature and salt |
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(moisture), respectively. These in turn call a generic advection |
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diffusion routine {\em S/R GAD\_CALC\_RHS} that is called with the |
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1.8 |
flow field and relevant tracer as arguments and returns the collective |
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tendency due to advection and diffusion. Forcing is add subsequently |
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in {\em S/R CALC\_GT} or {\em S/R CALC\_GS} to the same tendency |
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1.3 |
array. |
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\fbox{ \begin{minipage}{4.75in} |
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{\em S/R GAD\_CALC\_RHS} ({\em pkg/generic\_advdiff/gad\_calc\_rhs.F}) |
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$\tau$: {\bf tracer} (argument) |
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$G^{(n)}$: {\bf gTracer} (argument) |
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$F_r$: {\bf fVerT} (argument) |
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\end{minipage} } |
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1.8 |
The space and time discretization are treated separately (method of |
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lines). Tendencies are calculated at time levels $n$ and $n-1$ and |
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1.4 |
extrapolated to $n+1/2$ using the Adams-Bashforth method: |
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1.3 |
\marginpar{$\epsilon$: {\bf AB\_eps}} |
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\begin{equation} |
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1.4 |
G^{(n+1/2)} = |
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1.3 |
(\frac{3}{2} + \epsilon) G^{(n)} - (\frac{1}{2} + \epsilon) G^{(n-1)} |
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\end{equation} |
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where $G^{(n)} = G_{adv}^\tau + G_{diff}^\tau + G_{src}^\tau$ at time |
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1.8 |
step $n$. The tendency at $n-1$ is not re-calculated but rather the |
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tendency at $n$ is stored in a global array for later re-use. |
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1.4 |
|
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\fbox{ \begin{minipage}{4.75in} |
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{\em S/R ADAMS\_BASHFORTH2} ({\em model/src/adams\_bashforth2.F}) |
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$G^{(n+1/2)}$: {\bf gTracer} (argument on exit) |
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$G^{(n)}$: {\bf gTracer} (argument on entry) |
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$G^{(n-1)}$: {\bf gTrNm1} (argument) |
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$\epsilon$: {\bf ABeps} (PARAMS.h) |
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\end{minipage} } |
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1.8 |
The tracers are stepped forward in time using the extrapolated tendency: |
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1.4 |
\begin{equation} |
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\tau^{(n+1)} = \tau^{(n)} + \Delta t G^{(n+1/2)} |
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\end{equation} |
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\marginpar{$\Delta t$: {\bf deltaTtracer}} |
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\fbox{ \begin{minipage}{4.75in} |
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{\em S/R TIMESTEP\_TRACER} ({\em model/src/timestep\_tracer.F}) |
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$\tau^{(n+1)}$: {\bf gTracer} (argument on exit) |
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$\tau^{(n)}$: {\bf tracer} (argument on entry) |
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$G^{(n+1/2)}$: {\bf gTracer} (argument) |
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$\Delta t$: {\bf deltaTtracer} (PARAMS.h) |
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\end{minipage} } |
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1.3 |
|
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Strictly speaking the ABII scheme should be applied only to the |
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cnh |
1.8 |
advection terms. However, this scheme is only used in conjunction with |
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1.3 |
the standard second, third and fourth order advection |
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schemes. Selection of any other advection scheme disables |
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Adams-Bashforth for tracers so that explicit diffusion and forcing use |
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the forward method. |
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adcroft |
1.4 |
\section{Linear advection schemes} |
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adcroft |
1.12 |
\label{sect:tracer-advection} |
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afe |
1.14 |
\begin{rawhtml} |
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afe |
1.15 |
<!-- CMIREDIR:linear_advection_schemes: --> |
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afe |
1.14 |
\end{rawhtml} |
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1.3 |
|
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\begin{figure} |
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\resizebox{5.5in}{!}{\includegraphics{part2/advect-1d-lo.eps}} |
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\caption{ |
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Comparison of 1-D advection schemes. Courant number is 0.05 with 60 |
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points and solutions are shown for T=1 (one complete period). |
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a) Shows the upwind biased schemes; first order upwind, DST3, |
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third order upwind and second order upwind. |
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b) Shows the centered schemes; Lax-Wendroff, DST4, centered second order, |
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centered fourth order and finite volume fourth order. |
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c) Shows the second order flux limiters: minmod, Superbee, |
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MC limiter and the van Leer limiter. |
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d) Shows the DST3 method with flux limiters due to Sweby with |
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$\mu=1$, $\mu=c/(1-c)$ and a fourth order DST method with Sweby limiter, |
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$\mu=c/(1-c)$. |
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\label{fig:advect-1d-lo} |
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} |
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\end{figure} |
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\begin{figure} |
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\resizebox{5.5in}{!}{\includegraphics{part2/advect-1d-hi.eps}} |
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\caption{ |
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Comparison of 1-D advection schemes. Courant number is 0.89 with 60 |
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points and solutions are shown for T=1 (one complete period). |
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a) Shows the upwind biased schemes; first order upwind and DST3. |
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Third order upwind and second order upwind are unstable at this Courant number. |
162 |
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b) Shows the centered schemes; Lax-Wendroff, DST4. Centered second order, |
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centered fourth order and finite volume fourth order and unstable at this |
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Courant number. |
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c) Shows the second order flux limiters: minmod, Superbee, |
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MC limiter and the van Leer limiter. |
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d) Shows the DST3 method with flux limiters due to Sweby with |
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$\mu=1$, $\mu=c/(1-c)$ and a fourth order DST method with Sweby limiter, |
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$\mu=c/(1-c)$. |
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\label{fig:advect-1d-hi} |
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} |
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\end{figure} |
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The advection schemes known as centered second order, centered fourth |
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order, first order upwind and upwind biased third order are known as |
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linear advection schemes because the coefficient for interpolation of |
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the advected tracer are linear and a function only of the flow, not |
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the tracer field it self. We discuss these first since they are most |
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commonly used in the field and most familiar. |
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|
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adcroft |
1.2 |
\subsection{Centered second order advection-diffusion} |
182 |
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The basic discretization, centered second order, is the default. It is |
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cnh |
1.8 |
designed to be consistent with the continuity equation to facilitate |
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adcroft |
1.3 |
conservation properties analogous to the continuum. However, centered |
186 |
cnh |
1.8 |
second order advection is notoriously noisy and must be used in |
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conjunction with some finite amount of diffusion to produce a sensible |
188 |
adcroft |
1.3 |
solution. |
189 |
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The advection operator is discretized: |
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adcroft |
1.1 |
\begin{equation} |
192 |
adcroft |
1.3 |
{\cal A}_c \Delta r_f h_c G_{adv}^\tau = |
193 |
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\delta_i F_x + \delta_j F_y + \delta_k F_r |
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adcroft |
1.1 |
\end{equation} |
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adcroft |
1.2 |
where the area integrated fluxes are given by: |
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\begin{eqnarray} |
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adcroft |
1.3 |
F_x & = & U \overline{ \tau }^i \\ |
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F_y & = & V \overline{ \tau }^j \\ |
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F_r & = & W \overline{ \tau }^k |
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1.2 |
\end{eqnarray} |
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1.1 |
The quantities $U$, $V$ and $W$ are volume fluxes defined: |
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\marginpar{$U$: {\bf uTrans} } |
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\marginpar{$V$: {\bf vTrans} } |
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\marginpar{$W$: {\bf rTrans} } |
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\begin{eqnarray} |
206 |
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U & = & \Delta y_g \Delta r_f h_w u \\ |
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V & = & \Delta x_g \Delta r_f h_s v \\ |
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W & = & {\cal A}_c w |
209 |
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\end{eqnarray} |
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|
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adcroft |
1.3 |
For non-divergent flow, this discretization can be shown to conserve |
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the tracer both locally and globally and to globally conserve tracer |
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adcroft |
1.10 |
variance, $\tau^2$. The proof is given in \cite{adcroft:95,adcroft:97}. |
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adcroft |
1.3 |
|
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\fbox{ \begin{minipage}{4.75in} |
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{\em S/R GAD\_C2\_ADV\_X} ({\em gad\_c2\_adv\_x.F}) |
217 |
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218 |
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$F_x$: {\bf uT} (argument) |
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220 |
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$U$: {\bf uTrans} (argument) |
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222 |
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$\tau$: {\bf tracer} (argument) |
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{\em S/R GAD\_C2\_ADV\_Y} ({\em gad\_c2\_adv\_y.F}) |
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$F_y$: {\bf vT} (argument) |
227 |
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228 |
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$V$: {\bf vTrans} (argument) |
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$\tau$: {\bf tracer} (argument) |
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{\em S/R GAD\_C2\_ADV\_R} ({\em gad\_c2\_adv\_r.F}) |
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$F_r$: {\bf wT} (argument) |
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$W$: {\bf rTrans} (argument) |
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$\tau$: {\bf tracer} (argument) |
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\end{minipage} } |
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\subsection{Third order upwind bias advection} |
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Upwind biased third order advection offers a relatively good |
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compromise between accuracy and smoothness. It is not a ``positive'' |
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scheme meaning false extrema are permitted but the amplitude of such |
248 |
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are significantly reduced over the centered second order method. |
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The third order upwind fluxes are discretized: |
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\begin{eqnarray} |
252 |
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F_x & = & U \overline{\tau - \frac{1}{6} \delta_{ii} \tau}^i |
253 |
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+ \frac{1}{2} |U| \delta_i \frac{1}{6} \delta_{ii} \tau \\ |
254 |
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F_y & = & V \overline{\tau - \frac{1}{6} \delta_{ii} \tau}^j |
255 |
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+ \frac{1}{2} |V| \delta_j \frac{1}{6} \delta_{jj} \tau \\ |
256 |
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F_r & = & W \overline{\tau - \frac{1}{6} \delta_{ii} \tau}^k |
257 |
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+ \frac{1}{2} |W| \delta_k \frac{1}{6} \delta_{kk} \tau |
258 |
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\end{eqnarray} |
259 |
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260 |
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At boundaries, $\delta_{\hat{n}} \tau$ is set to zero allowing |
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cnh |
1.8 |
$\delta_{nn}$ to be evaluated. We are currently examine the accuracy |
262 |
adcroft |
1.3 |
of this boundary condition and the effect on the solution. |
263 |
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adcroft |
1.4 |
\fbox{ \begin{minipage}{4.75in} |
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{\em S/R GAD\_U3\_ADV\_X} ({\em gad\_u3\_adv\_x.F}) |
266 |
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267 |
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$F_x$: {\bf uT} (argument) |
268 |
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269 |
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$U$: {\bf uTrans} (argument) |
270 |
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271 |
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$\tau$: {\bf tracer} (argument) |
272 |
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273 |
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{\em S/R GAD\_U3\_ADV\_Y} ({\em gad\_u3\_adv\_y.F}) |
274 |
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275 |
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$F_y$: {\bf vT} (argument) |
276 |
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277 |
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$V$: {\bf vTrans} (argument) |
278 |
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279 |
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$\tau$: {\bf tracer} (argument) |
280 |
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281 |
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{\em S/R GAD\_U3\_ADV\_R} ({\em gad\_u3\_adv\_r.F}) |
282 |
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283 |
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$F_r$: {\bf wT} (argument) |
284 |
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285 |
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$W$: {\bf rTrans} (argument) |
286 |
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287 |
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$\tau$: {\bf tracer} (argument) |
288 |
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|
289 |
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\end{minipage} } |
290 |
adcroft |
1.3 |
|
291 |
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\subsection{Centered fourth order advection} |
292 |
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293 |
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Centered fourth order advection is formally the most accurate scheme |
294 |
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we have implemented and can be used to great effect in high resolution |
295 |
cnh |
1.8 |
simulation where dynamical scales are well resolved. However, the |
296 |
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scheme is noisy like the centered second order method and so must be |
297 |
adcroft |
1.3 |
used with some finite amount of diffusion. Bi-harmonic is recommended |
298 |
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since it is more scale selective and less likely to diffuse away the |
299 |
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well resolved gradient the fourth order scheme worked so hard to |
300 |
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create. |
301 |
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302 |
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The centered fourth order fluxes are discretized: |
303 |
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\begin{eqnarray} |
304 |
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F_x & = & U \overline{\tau - \frac{1}{6} \delta_{ii} \tau}^i \\ |
305 |
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F_y & = & V \overline{\tau - \frac{1}{6} \delta_{ii} \tau}^j \\ |
306 |
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F_r & = & W \overline{\tau - \frac{1}{6} \delta_{ii} \tau}^k |
307 |
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\end{eqnarray} |
308 |
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309 |
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As for the third order scheme, the best discretization near boundaries |
310 |
cnh |
1.8 |
is under investigation but currently $\delta_i \tau=0$ on a boundary. |
311 |
adcroft |
1.3 |
|
312 |
adcroft |
1.4 |
\fbox{ \begin{minipage}{4.75in} |
313 |
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{\em S/R GAD\_C4\_ADV\_X} ({\em gad\_c4\_adv\_x.F}) |
314 |
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315 |
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$F_x$: {\bf uT} (argument) |
316 |
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317 |
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$U$: {\bf uTrans} (argument) |
318 |
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319 |
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$\tau$: {\bf tracer} (argument) |
320 |
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321 |
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{\em S/R GAD\_C4\_ADV\_Y} ({\em gad\_c4\_adv\_y.F}) |
322 |
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323 |
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$F_y$: {\bf vT} (argument) |
324 |
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325 |
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$V$: {\bf vTrans} (argument) |
326 |
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327 |
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$\tau$: {\bf tracer} (argument) |
328 |
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329 |
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{\em S/R GAD\_C4\_ADV\_R} ({\em gad\_c4\_adv\_r.F}) |
330 |
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331 |
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$F_r$: {\bf wT} (argument) |
332 |
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333 |
|
|
$W$: {\bf rTrans} (argument) |
334 |
|
|
|
335 |
|
|
$\tau$: {\bf tracer} (argument) |
336 |
|
|
|
337 |
|
|
\end{minipage} } |
338 |
|
|
|
339 |
|
|
|
340 |
adcroft |
1.3 |
\subsection{First order upwind advection} |
341 |
|
|
|
342 |
|
|
Although the upwind scheme is the underlying scheme for the robust or |
343 |
|
|
non-linear methods given later, we haven't actually supplied this |
344 |
|
|
method for general use. It would be very diffusive and it is unlikely |
345 |
|
|
that it could ever produce more useful results than the positive |
346 |
|
|
higher order schemes. |
347 |
|
|
|
348 |
|
|
Upwind bias is introduced into many schemes using the {\em abs} |
349 |
|
|
function and is allows the first order upwind flux to be written: |
350 |
|
|
\begin{eqnarray} |
351 |
|
|
F_x & = & U \overline{ \tau }^i - \frac{1}{2} |U| \delta_i \tau \\ |
352 |
|
|
F_y & = & V \overline{ \tau }^j - \frac{1}{2} |V| \delta_j \tau \\ |
353 |
|
|
F_r & = & W \overline{ \tau }^k - \frac{1}{2} |W| \delta_k \tau |
354 |
|
|
\end{eqnarray} |
355 |
|
|
|
356 |
|
|
If for some reason, the above method is required, then the second |
357 |
|
|
order flux limiter scheme described later reduces to the above scheme |
358 |
|
|
if the limiter is set to zero. |
359 |
|
|
|
360 |
|
|
|
361 |
|
|
\section{Non-linear advection schemes} |
362 |
edhill |
1.16 |
\begin{rawhtml} |
363 |
|
|
<!-- CMIREDIR:non-linear_advection_schemes: --> |
364 |
|
|
\end{rawhtml} |
365 |
adcroft |
1.3 |
|
366 |
|
|
Non-linear advection schemes invoke non-linear interpolation and are |
367 |
|
|
widely used in computational fluid dynamics (non-linear does not refer |
368 |
|
|
to the non-linearity of the advection operator). The flux limited |
369 |
|
|
advection schemes belong to the class of finite volume methods which |
370 |
|
|
neatly ties into the spatial discretization of the model. |
371 |
|
|
|
372 |
|
|
When employing the flux limited schemes, first order upwind or |
373 |
|
|
direct-space-time method the time-stepping is switched to forward in |
374 |
|
|
time. |
375 |
|
|
|
376 |
|
|
\subsection{Second order flux limiters} |
377 |
|
|
|
378 |
|
|
The second order flux limiter method can be cast in several ways but |
379 |
|
|
is generally expressed in terms of other flux approximations. For |
380 |
|
|
example, in terms of a first order upwind flux and second order |
381 |
|
|
Lax-Wendroff flux, the limited flux is given as: |
382 |
|
|
\begin{equation} |
383 |
|
|
F = F_1 + \psi(r) F_{LW} |
384 |
|
|
\end{equation} |
385 |
|
|
where $\psi(r)$ is the limiter function, |
386 |
|
|
\begin{equation} |
387 |
|
|
F_1 = u \overline{\tau}^i - \frac{1}{2} |u| \delta_i \tau |
388 |
|
|
\end{equation} |
389 |
|
|
is the upwind flux, |
390 |
|
|
\begin{equation} |
391 |
|
|
F_{LW} = F_1 + \frac{|u|}{2} (1-c) \delta_i \tau |
392 |
|
|
\end{equation} |
393 |
|
|
is the Lax-Wendroff flux and $c = \frac{u \Delta t}{\Delta x}$ is the |
394 |
|
|
Courant (CFL) number. |
395 |
|
|
|
396 |
|
|
The limiter function, $\psi(r)$, takes the slope ratio |
397 |
|
|
\begin{eqnarray} |
398 |
|
|
r = \frac{ \tau_{i-1} - \tau_{i-2} }{ \tau_{i} - \tau_{i-1} } & \forall & u > 0 |
399 |
|
|
\\ |
400 |
|
|
r = \frac{ \tau_{i+1} - \tau_{i} }{ \tau_{i} - \tau_{i-1} } & \forall & u < 0 |
401 |
|
|
\end{eqnarray} |
402 |
|
|
as it's argument. There are many choices of limiter function but we |
403 |
adcroft |
1.11 |
only provide the Superbee limiter \cite{roe:85}: |
404 |
adcroft |
1.3 |
\begin{equation} |
405 |
|
|
\psi(r) = \max[0,\min[1,2r],\min[2,r]] |
406 |
|
|
\end{equation} |
407 |
|
|
|
408 |
adcroft |
1.4 |
\fbox{ \begin{minipage}{4.75in} |
409 |
|
|
{\em S/R GAD\_FLUXLIMIT\_ADV\_X} ({\em gad\_fluxlimit\_adv\_x.F}) |
410 |
|
|
|
411 |
|
|
$F_x$: {\bf uT} (argument) |
412 |
|
|
|
413 |
|
|
$U$: {\bf uTrans} (argument) |
414 |
|
|
|
415 |
|
|
$\tau$: {\bf tracer} (argument) |
416 |
|
|
|
417 |
|
|
{\em S/R GAD\_FLUXLIMIT\_ADV\_Y} ({\em gad\_fluxlimit\_adv\_y.F}) |
418 |
|
|
|
419 |
|
|
$F_y$: {\bf vT} (argument) |
420 |
|
|
|
421 |
|
|
$V$: {\bf vTrans} (argument) |
422 |
|
|
|
423 |
|
|
$\tau$: {\bf tracer} (argument) |
424 |
|
|
|
425 |
|
|
{\em S/R GAD\_FLUXLIMIT\_ADV\_R} ({\em gad\_fluxlimit\_adv\_r.F}) |
426 |
|
|
|
427 |
|
|
$F_r$: {\bf wT} (argument) |
428 |
|
|
|
429 |
|
|
$W$: {\bf rTrans} (argument) |
430 |
|
|
|
431 |
|
|
$\tau$: {\bf tracer} (argument) |
432 |
|
|
|
433 |
|
|
\end{minipage} } |
434 |
|
|
|
435 |
adcroft |
1.3 |
|
436 |
|
|
\subsection{Third order direct space time} |
437 |
|
|
|
438 |
|
|
The direct-space-time method deals with space and time discretization |
439 |
cnh |
1.8 |
together (other methods that treat space and time separately are known |
440 |
adcroft |
1.3 |
collectively as the ``Method of Lines''). The Lax-Wendroff scheme |
441 |
|
|
falls into this category; it adds sufficient diffusion to a second |
442 |
|
|
order flux that the forward-in-time method is stable. The upwind |
443 |
|
|
biased third order DST scheme is: |
444 |
|
|
\begin{eqnarray} |
445 |
|
|
F = u \left( \tau_{i-1} |
446 |
|
|
+ d_0 (\tau_{i}-\tau_{i-1}) + d_1 (\tau_{i-1}-\tau_{i-2}) \right) |
447 |
|
|
& \forall & u > 0 \\ |
448 |
|
|
F = u \left( \tau_{i} |
449 |
|
|
- d_0 (\tau_{i}-\tau_{i-1}) - d_1 (\tau_{i+1}-\tau_{i}) \right) |
450 |
|
|
& \forall & u < 0 |
451 |
|
|
\end{eqnarray} |
452 |
|
|
where |
453 |
|
|
\begin{eqnarray} |
454 |
|
|
d_1 & = & \frac{1}{6} ( 2 - |c| ) ( 1 - |c| ) \\ |
455 |
|
|
d_2 & = & \frac{1}{6} ( 1 - |c| ) ( 1 + |c| ) |
456 |
|
|
\end{eqnarray} |
457 |
cnh |
1.8 |
The coefficients $d_0$ and $d_1$ approach $1/3$ and $1/6$ respectively |
458 |
adcroft |
1.3 |
as the Courant number, $c$, vanishes. In this limit, the conventional |
459 |
|
|
third order upwind method is recovered. For finite Courant number, the |
460 |
|
|
deviations from the linear method are analogous to the diffusion added |
461 |
|
|
to centered second order advection in the Lax-Wendroff scheme. |
462 |
|
|
|
463 |
|
|
The DST3 method described above must be used in a forward-in-time |
464 |
|
|
manner and is stable for $0 \le |c| \le 1$. Although the scheme |
465 |
adcroft |
1.13 |
appears to be forward-in-time, it is in fact third order in time and |
466 |
adcroft |
1.3 |
the accuracy increases with the Courant number! For low Courant |
467 |
|
|
number, DST3 produces very similar results (indistinguishable in |
468 |
|
|
Fig.~\ref{fig:advect-1d-lo}) to the linear third order method but for |
469 |
|
|
large Courant number, where the linear upwind third order method is |
470 |
|
|
unstable, the scheme is extremely accurate |
471 |
|
|
(Fig.~\ref{fig:advect-1d-hi}) with only minor overshoots. |
472 |
|
|
|
473 |
adcroft |
1.4 |
\fbox{ \begin{minipage}{4.75in} |
474 |
|
|
{\em S/R GAD\_DST3\_ADV\_X} ({\em gad\_dst3\_adv\_x.F}) |
475 |
|
|
|
476 |
|
|
$F_x$: {\bf uT} (argument) |
477 |
|
|
|
478 |
|
|
$U$: {\bf uTrans} (argument) |
479 |
|
|
|
480 |
|
|
$\tau$: {\bf tracer} (argument) |
481 |
|
|
|
482 |
|
|
{\em S/R GAD\_DST3\_ADV\_Y} ({\em gad\_dst3\_adv\_y.F}) |
483 |
|
|
|
484 |
|
|
$F_y$: {\bf vT} (argument) |
485 |
|
|
|
486 |
|
|
$V$: {\bf vTrans} (argument) |
487 |
|
|
|
488 |
|
|
$\tau$: {\bf tracer} (argument) |
489 |
|
|
|
490 |
|
|
{\em S/R GAD\_DST3\_ADV\_R} ({\em gad\_dst3\_adv\_r.F}) |
491 |
|
|
|
492 |
|
|
$F_r$: {\bf wT} (argument) |
493 |
|
|
|
494 |
|
|
$W$: {\bf rTrans} (argument) |
495 |
|
|
|
496 |
|
|
$\tau$: {\bf tracer} (argument) |
497 |
|
|
|
498 |
|
|
\end{minipage} } |
499 |
|
|
|
500 |
|
|
|
501 |
adcroft |
1.3 |
\subsection{Third order direct space time with flux limiting} |
502 |
|
|
|
503 |
|
|
The overshoots in the DST3 method can be controlled with a flux limiter. |
504 |
|
|
The limited flux is written: |
505 |
|
|
\begin{equation} |
506 |
|
|
F = |
507 |
|
|
\frac{1}{2}(u+|u|)\left( \tau_{i-1} + \psi(r^+)(\tau_{i} - \tau_{i-1} )\right) |
508 |
|
|
+ |
509 |
|
|
\frac{1}{2}(u-|u|)\left( \tau_{i-1} + \psi(r^-)(\tau_{i} - \tau_{i-1} )\right) |
510 |
|
|
\end{equation} |
511 |
|
|
where |
512 |
|
|
\begin{eqnarray} |
513 |
|
|
r^+ & = & \frac{\tau_{i-1} - \tau_{i-2}}{\tau_{i} - \tau_{i-1}} \\ |
514 |
|
|
r^- & = & \frac{\tau_{i+1} - \tau_{i}}{\tau_{i} - \tau_{i-1}} |
515 |
|
|
\end{eqnarray} |
516 |
|
|
and the limiter is the Sweby limiter: |
517 |
|
|
\begin{equation} |
518 |
|
|
\psi(r) = \max[0, \min[\min(1,d_0+d_1r],\frac{1-c}{c}r ]] |
519 |
|
|
\end{equation} |
520 |
adcroft |
1.1 |
|
521 |
adcroft |
1.4 |
\fbox{ \begin{minipage}{4.75in} |
522 |
|
|
{\em S/R GAD\_DST3FL\_ADV\_X} ({\em gad\_dst3\_adv\_x.F}) |
523 |
|
|
|
524 |
|
|
$F_x$: {\bf uT} (argument) |
525 |
|
|
|
526 |
|
|
$U$: {\bf uTrans} (argument) |
527 |
|
|
|
528 |
|
|
$\tau$: {\bf tracer} (argument) |
529 |
|
|
|
530 |
|
|
{\em S/R GAD\_DST3FL\_ADV\_Y} ({\em gad\_dst3\_adv\_y.F}) |
531 |
|
|
|
532 |
|
|
$F_y$: {\bf vT} (argument) |
533 |
|
|
|
534 |
|
|
$V$: {\bf vTrans} (argument) |
535 |
|
|
|
536 |
|
|
$\tau$: {\bf tracer} (argument) |
537 |
|
|
|
538 |
|
|
{\em S/R GAD\_DST3FL\_ADV\_R} ({\em gad\_dst3\_adv\_r.F}) |
539 |
|
|
|
540 |
|
|
$F_r$: {\bf wT} (argument) |
541 |
|
|
|
542 |
|
|
$W$: {\bf rTrans} (argument) |
543 |
|
|
|
544 |
|
|
$\tau$: {\bf tracer} (argument) |
545 |
|
|
|
546 |
|
|
\end{minipage} } |
547 |
|
|
|
548 |
|
|
|
549 |
adcroft |
1.3 |
\subsection{Multi-dimensional advection} |
550 |
adcroft |
1.1 |
|
551 |
adcroft |
1.4 |
\begin{figure} |
552 |
|
|
\resizebox{5.5in}{!}{\includegraphics{part2/advect-2d-lo-diag.eps}} |
553 |
|
|
\caption{ |
554 |
|
|
Comparison of advection schemes in two dimensions; diagonal advection |
555 |
cnh |
1.8 |
of a resolved Gaussian feature. Courant number is 0.01 with |
556 |
adcroft |
1.4 |
30$\times$30 points and solutions are shown for T=1/2. White lines |
557 |
|
|
indicate zero crossing (ie. the presence of false minima). The left |
558 |
|
|
column shows the second order schemes; top) centered second order with |
559 |
|
|
Adams-Bashforth, middle) Lax-Wendroff and bottom) Superbee flux |
560 |
|
|
limited. The middle column shows the third order schemes; top) upwind |
561 |
|
|
biased third order with Adams-Bashforth, middle) third order direct |
562 |
|
|
space-time method and bottom) the same with flux limiting. The top |
563 |
|
|
right panel shows the centered fourth order scheme with |
564 |
|
|
Adams-Bashforth and right middle panel shows a fourth order variant on |
565 |
|
|
the DST method. Bottom right panel shows the Superbee flux limiter |
566 |
cnh |
1.8 |
(second order) applied independently in each direction (method of |
567 |
adcroft |
1.4 |
lines). |
568 |
|
|
\label{fig:advect-2d-lo-diag} |
569 |
|
|
} |
570 |
|
|
\end{figure} |
571 |
|
|
|
572 |
|
|
\begin{figure} |
573 |
|
|
\resizebox{5.5in}{!}{\includegraphics{part2/advect-2d-mid-diag.eps}} |
574 |
|
|
\caption{ |
575 |
|
|
Comparison of advection schemes in two dimensions; diagonal advection |
576 |
cnh |
1.8 |
of a resolved Gaussian feature. Courant number is 0.27 with |
577 |
adcroft |
1.4 |
30$\times$30 points and solutions are shown for T=1/2. White lines |
578 |
|
|
indicate zero crossing (ie. the presence of false minima). The left |
579 |
|
|
column shows the second order schemes; top) centered second order with |
580 |
|
|
Adams-Bashforth, middle) Lax-Wendroff and bottom) Superbee flux |
581 |
|
|
limited. The middle column shows the third order schemes; top) upwind |
582 |
|
|
biased third order with Adams-Bashforth, middle) third order direct |
583 |
|
|
space-time method and bottom) the same with flux limiting. The top |
584 |
|
|
right panel shows the centered fourth order scheme with |
585 |
|
|
Adams-Bashforth and right middle panel shows a fourth order variant on |
586 |
|
|
the DST method. Bottom right panel shows the Superbee flux limiter |
587 |
cnh |
1.8 |
(second order) applied independently in each direction (method of |
588 |
adcroft |
1.4 |
lines). |
589 |
|
|
\label{fig:advect-2d-mid-diag} |
590 |
|
|
} |
591 |
|
|
\end{figure} |
592 |
|
|
|
593 |
|
|
\begin{figure} |
594 |
|
|
\resizebox{5.5in}{!}{\includegraphics{part2/advect-2d-hi-diag.eps}} |
595 |
|
|
\caption{ |
596 |
|
|
Comparison of advection schemes in two dimensions; diagonal advection |
597 |
cnh |
1.8 |
of a resolved Gaussian feature. Courant number is 0.47 with |
598 |
adcroft |
1.4 |
30$\times$30 points and solutions are shown for T=1/2. White lines |
599 |
|
|
indicate zero crossings and initial maximum values (ie. the presence |
600 |
|
|
of false extrema). The left column shows the second order schemes; |
601 |
|
|
top) centered second order with Adams-Bashforth, middle) Lax-Wendroff |
602 |
|
|
and bottom) Superbee flux limited. The middle column shows the third |
603 |
|
|
order schemes; top) upwind biased third order with Adams-Bashforth, |
604 |
|
|
middle) third order direct space-time method and bottom) the same with |
605 |
|
|
flux limiting. The top right panel shows the centered fourth order |
606 |
|
|
scheme with Adams-Bashforth and right middle panel shows a fourth |
607 |
|
|
order variant on the DST method. Bottom right panel shows the Superbee |
608 |
cnh |
1.8 |
flux limiter (second order) applied independently in each direction |
609 |
adcroft |
1.4 |
(method of lines). |
610 |
|
|
\label{fig:advect-2d-hi-diag} |
611 |
|
|
} |
612 |
|
|
\end{figure} |
613 |
|
|
|
614 |
|
|
|
615 |
|
|
|
616 |
cnh |
1.8 |
In many of the aforementioned advection schemes the behavior in |
617 |
adcroft |
1.3 |
multiple dimensions is not necessarily as good as the one dimensional |
618 |
cnh |
1.8 |
behavior. For instance, a shape preserving monotonic scheme in one |
619 |
adcroft |
1.3 |
dimension can have severe shape distortion in two dimensions if the |
620 |
|
|
two components of horizontal fluxes are treated independently. There |
621 |
|
|
is a large body of literature on the subject dealing with this problem |
622 |
|
|
and among the fixes are operator and flux splitting methods, corner |
623 |
|
|
flux methods and more. We have adopted a variant on the standard |
624 |
|
|
splitting methods that allows the flux calculations to be implemented |
625 |
|
|
as if in one dimension: |
626 |
|
|
\begin{eqnarray} |
627 |
|
|
\tau^{n+1/3} & = & \tau^{n} |
628 |
|
|
- \Delta t \left( \frac{1}{\Delta x} \delta_i F^x(\tau^{n}) |
629 |
|
|
+ \tau^{n} \frac{1}{\Delta x} \delta_i u \right) \\ |
630 |
jmc |
1.21 |
\tau^{n+2/3} & = & \tau^{n+1/3} |
631 |
adcroft |
1.3 |
- \Delta t \left( \frac{1}{\Delta y} \delta_j F^y(\tau^{n+1/3}) |
632 |
|
|
+ \tau^{n} \frac{1}{\Delta y} \delta_i v \right) \\ |
633 |
jmc |
1.21 |
\tau^{n+3/3} & = & \tau^{n+2/3} |
634 |
adcroft |
1.3 |
- \Delta t \left( \frac{1}{\Delta r} \delta_k F^x(\tau^{n+2/3}) |
635 |
|
|
+ \tau^{n} \frac{1}{\Delta r} \delta_i w \right) |
636 |
|
|
\end{eqnarray} |
637 |
adcroft |
1.1 |
|
638 |
adcroft |
1.3 |
In order to incorporate this method into the general model algorithm, |
639 |
cnh |
1.8 |
we compute the effective tendency rather than update the tracer so |
640 |
adcroft |
1.3 |
that other terms such as diffusion are using the $n$ time-level and |
641 |
|
|
not the updated $n+3/3$ quantities: |
642 |
|
|
\begin{equation} |
643 |
|
|
G^{n+1/2}_{adv} = \frac{1}{\Delta t} ( \tau^{n+3/3} - \tau^{n} ) |
644 |
|
|
\end{equation} |
645 |
|
|
So that the over all time-stepping looks likes: |
646 |
|
|
\begin{equation} |
647 |
|
|
\tau^{n+1} = \tau^{n} + \Delta t \left( G^{n+1/2}_{adv} + G_{diff}(\tau^{n}) + G^{n}_{forcing} \right) |
648 |
|
|
\end{equation} |
649 |
adcroft |
1.4 |
|
650 |
|
|
\fbox{ \begin{minipage}{4.75in} |
651 |
|
|
{\em S/R GAD\_ADVECTION} ({\em gad\_advection.F}) |
652 |
|
|
|
653 |
|
|
$\tau$: {\bf Tracer} (argument) |
654 |
|
|
|
655 |
|
|
$G^{n+1/2}_{adv}$: {\bf Gtracer} (argument) |
656 |
|
|
|
657 |
|
|
$F_x, F_y, F_r$: {\bf af} (local) |
658 |
|
|
|
659 |
|
|
$U$: {\bf uTrans} (local) |
660 |
|
|
|
661 |
|
|
$V$: {\bf vTrans} (local) |
662 |
|
|
|
663 |
|
|
$W$: {\bf rTrans} (local) |
664 |
|
|
|
665 |
|
|
\end{minipage} } |
666 |
|
|
|
667 |
jmc |
1.23 |
\begin{figure} |
668 |
|
|
\resizebox{3.5in}{!}{\includegraphics{part2/multiDim_CS.eps}} |
669 |
|
|
\caption{Muti-dimensional advection time-stepping with Cubed-Sphere topology |
670 |
|
|
\label{fig:advect-multidim_cs} |
671 |
|
|
} |
672 |
|
|
\end{figure} |
673 |
adcroft |
1.4 |
|
674 |
|
|
\section{Comparison of advection schemes} |
675 |
jmc |
1.20 |
\label{sect:tracer_advection_schemes} |
676 |
edhill |
1.18 |
\begin{rawhtml} |
677 |
|
|
<!-- CMIREDIR:comparison_of_advection_schemes: --> |
678 |
|
|
\end{rawhtml} |
679 |
adcroft |
1.4 |
|
680 |
jmc |
1.17 |
\begin{table}[htb] |
681 |
|
|
\centering |
682 |
|
|
\begin{tabular}[htb]{|l|c|c|c|c|l|} |
683 |
|
|
\hline |
684 |
|
|
Advection Scheme & code & use & use Multi- & Stencil & comments \\ |
685 |
|
|
& & A.B. & dimension & (1 dim) & \\ |
686 |
|
|
\hline \hline |
687 |
jmc |
1.21 |
$1^{rst}$order upwind & 1 & No & Yes & 3 pts & linear/$\tau$, non-linear/v\\ |
688 |
|
|
\hline |
689 |
jmc |
1.17 |
centered $2^{nd}$order & 2 & Yes & No & 3 pts & linear \\ |
690 |
|
|
\hline |
691 |
jmc |
1.19 |
$3^{rd}$order upwind & 3 & Yes & No & 5 pts & linear/$\tau$\\ |
692 |
jmc |
1.17 |
\hline |
693 |
|
|
centered $4^{th}$order & 4 & Yes & No & 5 pts & linear \\ |
694 |
|
|
\hline \hline |
695 |
jmc |
1.21 |
$2^{nd}$order DST (Lax-Wendroff) & 20 & |
696 |
jmc |
1.22 |
No & Yes & 3 pts & linear/$\tau$, non-linear/v\\ |
697 |
jmc |
1.21 |
\hline |
698 |
jmc |
1.19 |
$3^{rd}$order DST & 30 & No & Yes & 5 pts & linear/$\tau$, non-linear/v\\ |
699 |
jmc |
1.17 |
\hline \hline |
700 |
|
|
$2^{nd}$order Flux Limiters & 77 & No & Yes & 5 pts & non-linear \\ |
701 |
|
|
\hline |
702 |
|
|
$3^{nd}$order DST Flux limiter & 33 & No & Yes & 5 pts & non-linear \\ |
703 |
|
|
\hline |
704 |
|
|
\end{tabular} |
705 |
|
|
\caption{Summary of the different advection schemes available in MITgcm. |
706 |
|
|
``A.B.'' stands for Adams-Bashforth and ``DST'' for direct space time. |
707 |
|
|
The code corresponds to the number used to select the corresponding |
708 |
jmc |
1.19 |
advection scheme in the parameter file (e.g., {\bf tempAdvScheme}=3 in |
709 |
jmc |
1.17 |
file {\em data} selects the $3^{rd}$ order upwind advection scheme |
710 |
|
|
for temperature). |
711 |
|
|
} |
712 |
|
|
\label{tab:advectionShemes_summary} |
713 |
|
|
\end{table} |
714 |
|
|
|
715 |
|
|
|
716 |
adcroft |
1.4 |
Figs.~\ref{fig:advect-2d-lo-diag}, \ref{fig:advect-2d-mid-diag} and |
717 |
|
|
\ref{fig:advect-2d-hi-diag} show solutions to a simple diagonal |
718 |
|
|
advection problem using a selection of schemes for low, moderate and |
719 |
|
|
high Courant numbers, respectively. The top row shows the linear |
720 |
|
|
schemes, integrated with the Adams-Bashforth method. Theses schemes |
721 |
|
|
are clearly unstable for the high Courant number and weakly unstable |
722 |
|
|
for the moderate Courant number. The presence of false extrema is very |
723 |
|
|
apparent for all Courant numbers. The middle row shows solutions |
724 |
|
|
obtained with the unlimited but multi-dimensional schemes. These |
725 |
|
|
solutions also exhibit false extrema though the pattern now shows |
726 |
|
|
symmetry due to the multi-dimensional scheme. Also, the schemes are |
727 |
|
|
stable at high Courant number where the linear schemes weren't. The |
728 |
|
|
bottom row (left and middle) shows the limited schemes and most |
729 |
|
|
obvious is the absence of false extrema. The accuracy and stability of |
730 |
|
|
the unlimited non-linear schemes is retained at high Courant number |
731 |
cnh |
1.8 |
but at low Courant number the tendency is to loose amplitude in sharp |
732 |
adcroft |
1.4 |
peaks due to diffusion. The one dimensional tests shown in |
733 |
|
|
Figs.~\ref{fig:advect-1d-lo} and \ref{fig:advect-1d-hi} showed this |
734 |
cnh |
1.8 |
phenomenon. |
735 |
adcroft |
1.4 |
|
736 |
|
|
Finally, the bottom left and right panels use the same advection |
737 |
adcroft |
1.9 |
scheme but the right does not use the multi-dimensional method. At low |
738 |
adcroft |
1.4 |
Courant number this appears to not matter but for moderate Courant |
739 |
cnh |
1.8 |
number severe distortion of the feature is apparent. Moreover, the |
740 |
adcroft |
1.4 |
stability of the multi-dimensional scheme is determined by the maximum |
741 |
|
|
Courant number applied of each dimension while the stability of the |
742 |
|
|
method of lines is determined by the sum. Hence, in the high Courant |
743 |
|
|
number plot, the scheme is unstable. |
744 |
|
|
|
745 |
|
|
With many advection schemes implemented in the code two questions |
746 |
|
|
arise: ``Which scheme is best?'' and ``Why don't you just offer the |
747 |
|
|
best advection scheme?''. Unfortunately, no one advection scheme is |
748 |
|
|
``the best'' for all particular applications and for new applications |
749 |
|
|
it is often a matter of trial to determine which is most |
750 |
|
|
suitable. Here are some guidelines but these are not the rule; |
751 |
|
|
\begin{itemize} |
752 |
|
|
\item If you have a coarsely resolved model, using a |
753 |
|
|
positive or upwind biased scheme will introduce significant diffusion |
754 |
|
|
to the solution and using a centered higher order scheme will |
755 |
|
|
introduce more noise. In this case, simplest may be best. |
756 |
|
|
\item If you have a high resolution model, using a higher order |
757 |
|
|
scheme will give a more accurate solution but scale-selective |
758 |
|
|
diffusion might need to be employed. The flux limited methods |
759 |
|
|
offer similar accuracy in this regime. |
760 |
cnh |
1.8 |
\item If your solution has shocks or propagating fronts then a |
761 |
adcroft |
1.4 |
flux limited scheme is almost essential. |
762 |
|
|
\item If your time-step is limited by advection, the multi-dimensional |
763 |
cnh |
1.8 |
non-linear schemes have the most stability (up to Courant number 1). |
764 |
|
|
\item If you need to know how much diffusion/dissipation has occurred you |
765 |
adcroft |
1.4 |
will have a lot of trouble figuring it out with a non-linear method. |
766 |
adcroft |
1.9 |
\item The presence of false extrema is non-physical and this alone is the |
767 |
adcroft |
1.4 |
strongest argument for using a positive scheme. |
768 |
|
|
\end{itemize} |