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% $Header: /u/gcmpack/mitgcmdoc/part2/tracer.tex,v 1.2 2001/08/09 20:45:27 adcroft Exp $ |
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% $Name: $ |
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|
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\section{Tracer equations} |
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|
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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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advection-diussion equations. There are many alternatives to second |
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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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|
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\subsection{Time-stepping of tracers: ABII} |
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|
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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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|
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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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(\cite{Griffies00,Campin02}). |
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|
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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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|
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The driver routine that calls the routines to calculate tendancies are |
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{\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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flow field and relevent tracer as arguments and returns the collective |
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tendancy 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 tendancy |
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array. |
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|
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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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|
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$\tau$: {\bf tracer} (argument) |
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|
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$G^{(n)}$: {\bf gTracer} (argument) |
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|
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$F_r$: {\bf fVerT} (argument) |
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|
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\end{minipage} } |
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|
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|
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The space and time discretizations are treated seperately (method of |
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lines). The Adams-Bashforth time discretization reads: |
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\marginpar{$\epsilon$: {\bf AB\_eps}} |
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\marginpar{$\Delta t$: {\bf deltaTtracer}} |
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\begin{equation} |
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\tau^{(n+1)} = \tau^{(n)} + \Delta t \left( |
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(\frac{3}{2} + \epsilon) G^{(n)} - (\frac{1}{2} + \epsilon) G^{(n-1)} |
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\right) |
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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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step $n$. |
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|
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Strictly speaking the ABII scheme should be applied only to the |
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advection terms. However, this scheme is only used in conjuction with |
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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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|
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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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|
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$\tau$: {\bf tracer} (argument) |
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|
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$G^{(n)}$: {\bf gTracer} (argument) |
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|
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$G^{(n-1)}$: {\bf gTrNm1} (argument) |
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|
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$\Delta t$: {\bf deltaTtracer} (PARAMS.h) |
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|
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\end{minipage} } |
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|
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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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|
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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. |
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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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|
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\section{Linear advection schemes} |
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|
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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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\subsection{Centered second order advection-diffusion} |
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|
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The basic discretization, centered second order, is the default. It is |
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designed to be consistant with the continuity equation to facilitate |
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conservation properties analogous to the continuum. However, centered |
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second order advection is notoriously noisey and must be used in |
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conjuction with some finite amount of diffusion to produce a sensible |
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solution. |
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|
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The advection operator is discretized: |
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\begin{equation} |
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{\cal A}_c \Delta r_f h_c G_{adv}^\tau = |
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\delta_i F_x + \delta_j F_y + \delta_k F_r |
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\end{equation} |
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where the area integrated fluxes are given by: |
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\begin{eqnarray} |
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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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\end{eqnarray} |
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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} |
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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 |
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\end{eqnarray} |
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|
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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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variance, $\tau^2$. The proof is given in \cite{Adcroft95,Adcroft97}. |
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|
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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}) |
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|
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$F_x$: {\bf uT} (argument) |
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|
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$U$: {\bf uTrans} (argument) |
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|
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$\tau$: {\bf tracer} (argument) |
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|
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{\em S/R GAD\_C2\_ADV\_Y} ({\em gad\_c2\_adv\_y.F}) |
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|
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$F_y$: {\bf vT} (argument) |
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|
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$V$: {\bf vTrans} (argument) |
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|
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$\tau$: {\bf tracer} (argument) |
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|
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{\em S/R GAD\_C2\_ADV\_R} ({\em gad\_c2\_adv\_r.F}) |
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|
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$F_r$: {\bf wT} (argument) |
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|
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$W$: {\bf rTrans} (argument) |
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|
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$\tau$: {\bf tracer} (argument) |
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|
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\end{minipage} } |
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|
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|
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\subsection{Third order upwind bias advection} |
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|
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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 |
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are significantly reduced over the centered second order method. |
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|
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The third order upwind fluxes are discretized: |
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\begin{eqnarray} |
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F_x & = & U \overline{\tau - \frac{1}{6} \delta_{ii} \tau}^i |
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+ \frac{1}{2} |U| \delta_i \frac{1}{6} \delta_{ii} \tau \\ |
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F_y & = & V \overline{\tau - \frac{1}{6} \delta_{ii} \tau}^j |
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+ \frac{1}{2} |V| \delta_j \frac{1}{6} \delta_{jj} \tau \\ |
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F_r & = & W \overline{\tau - \frac{1}{6} \delta_{ii} \tau}^k |
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+ \frac{1}{2} |W| \delta_k \frac{1}{6} \delta_{kk} \tau |
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\end{eqnarray} |
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|
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At boundaries, $\delta_{\hat{n}} \tau$ is set to zero allowing |
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$\delta_{nn}$ to be evaluated. We are currently examing the accuracy |
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of this boundary condition and the effect on the solution. |
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|
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|
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\subsection{Centered fourth order advection} |
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|
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Centered fourth order advection is formally the most accurate scheme |
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we have implemented and can be used to great effect in high resolution |
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simultation where dynamical scales are well resolved. However, the |
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scheme is noisey like the centered second order method and so must be |
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used with some finite amount of diffusion. Bi-harmonic is recommended |
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since it is more scale selective and less likely to diffuse away the |
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well resolved gradient the fourth order scheme worked so hard to |
| 241 |
create. |
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|
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The centered fourth order fluxes are discretized: |
| 244 |
\begin{eqnarray} |
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F_x & = & U \overline{\tau - \frac{1}{6} \delta_{ii} \tau}^i \\ |
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F_y & = & V \overline{\tau - \frac{1}{6} \delta_{ii} \tau}^j \\ |
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F_r & = & W \overline{\tau - \frac{1}{6} \delta_{ii} \tau}^k |
| 248 |
\end{eqnarray} |
| 249 |
|
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As for the third order scheme, the best discretization near boundaries |
| 251 |
is under investigation but currenlty $\delta_i \tau=0$ on a boundary. |
| 252 |
|
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\subsection{First order upwind advection} |
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|
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Although the upwind scheme is the underlying scheme for the robust or |
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non-linear methods given later, we haven't actually supplied this |
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method for general use. It would be very diffusive and it is unlikely |
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that it could ever produce more useful results than the positive |
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higher order schemes. |
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|
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Upwind bias is introduced into many schemes using the {\em abs} |
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function and is allows the first order upwind flux to be written: |
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\begin{eqnarray} |
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F_x & = & U \overline{ \tau }^i - \frac{1}{2} |U| \delta_i \tau \\ |
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F_y & = & V \overline{ \tau }^j - \frac{1}{2} |V| \delta_j \tau \\ |
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F_r & = & W \overline{ \tau }^k - \frac{1}{2} |W| \delta_k \tau |
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\end{eqnarray} |
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|
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If for some reason, the above method is required, then the second |
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order flux limiter scheme described later reduces to the above scheme |
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if the limiter is set to zero. |
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|
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|
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\section{Non-linear advection schemes} |
| 275 |
|
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Non-linear advection schemes invoke non-linear interpolation and are |
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widely used in computational fluid dynamics (non-linear does not refer |
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to the non-linearity of the advection operator). The flux limited |
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advection schemes belong to the class of finite volume methods which |
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neatly ties into the spatial discretization of the model. |
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|
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When employing the flux limited schemes, first order upwind or |
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direct-space-time method the time-stepping is switched to forward in |
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time. |
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|
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\subsection{Second order flux limiters} |
| 287 |
|
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The second order flux limiter method can be cast in several ways but |
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is generally expressed in terms of other flux approximations. For |
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example, in terms of a first order upwind flux and second order |
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Lax-Wendroff flux, the limited flux is given as: |
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\begin{equation} |
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F = F_1 + \psi(r) F_{LW} |
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\end{equation} |
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where $\psi(r)$ is the limiter function, |
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\begin{equation} |
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F_1 = u \overline{\tau}^i - \frac{1}{2} |u| \delta_i \tau |
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\end{equation} |
| 299 |
is the upwind flux, |
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\begin{equation} |
| 301 |
F_{LW} = F_1 + \frac{|u|}{2} (1-c) \delta_i \tau |
| 302 |
\end{equation} |
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is the Lax-Wendroff flux and $c = \frac{u \Delta t}{\Delta x}$ is the |
| 304 |
Courant (CFL) number. |
| 305 |
|
| 306 |
The limiter function, $\psi(r)$, takes the slope ratio |
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\begin{eqnarray} |
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r = \frac{ \tau_{i-1} - \tau_{i-2} }{ \tau_{i} - \tau_{i-1} } & \forall & u > 0 |
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\\ |
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r = \frac{ \tau_{i+1} - \tau_{i} }{ \tau_{i} - \tau_{i-1} } & \forall & u < 0 |
| 311 |
\end{eqnarray} |
| 312 |
as it's argument. There are many choices of limiter function but we |
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only provide the Superbee limiter \cite{Roe85}: |
| 314 |
\begin{equation} |
| 315 |
\psi(r) = \max[0,\min[1,2r],\min[2,r]] |
| 316 |
\end{equation} |
| 317 |
|
| 318 |
|
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\subsection{Third order direct space time} |
| 320 |
|
| 321 |
The direct-space-time method deals with space and time discretization |
| 322 |
together (other methods that treat space and time seperately are known |
| 323 |
collectively as the ``Method of Lines''). The Lax-Wendroff scheme |
| 324 |
falls into this category; it adds sufficient diffusion to a second |
| 325 |
order flux that the forward-in-time method is stable. The upwind |
| 326 |
biased third order DST scheme is: |
| 327 |
\begin{eqnarray} |
| 328 |
F = u \left( \tau_{i-1} |
| 329 |
+ d_0 (\tau_{i}-\tau_{i-1}) + d_1 (\tau_{i-1}-\tau_{i-2}) \right) |
| 330 |
& \forall & u > 0 \\ |
| 331 |
F = u \left( \tau_{i} |
| 332 |
- d_0 (\tau_{i}-\tau_{i-1}) - d_1 (\tau_{i+1}-\tau_{i}) \right) |
| 333 |
& \forall & u < 0 |
| 334 |
\end{eqnarray} |
| 335 |
where |
| 336 |
\begin{eqnarray} |
| 337 |
d_1 & = & \frac{1}{6} ( 2 - |c| ) ( 1 - |c| ) \\ |
| 338 |
d_2 & = & \frac{1}{6} ( 1 - |c| ) ( 1 + |c| ) |
| 339 |
\end{eqnarray} |
| 340 |
The coefficients $d_0$ and $d_1$ approach $1/3$ and $1/6$ repectively |
| 341 |
as the Courant number, $c$, vanishes. In this limit, the conventional |
| 342 |
third order upwind method is recovered. For finite Courant number, the |
| 343 |
deviations from the linear method are analogous to the diffusion added |
| 344 |
to centered second order advection in the Lax-Wendroff scheme. |
| 345 |
|
| 346 |
The DST3 method described above must be used in a forward-in-time |
| 347 |
manner and is stable for $0 \le |c| \le 1$. Although the scheme |
| 348 |
appears to be forward-in-time, it is in fact second order in time and |
| 349 |
the accuracy increases with the Courant number! For low Courant |
| 350 |
number, DST3 produces very similar results (indistinguishable in |
| 351 |
Fig.~\ref{fig:advect-1d-lo}) to the linear third order method but for |
| 352 |
large Courant number, where the linear upwind third order method is |
| 353 |
unstable, the scheme is extremely accurate |
| 354 |
(Fig.~\ref{fig:advect-1d-hi}) with only minor overshoots. |
| 355 |
|
| 356 |
\subsection{Third order direct space time with flux limiting} |
| 357 |
|
| 358 |
The overshoots in the DST3 method can be controlled with a flux limiter. |
| 359 |
The limited flux is written: |
| 360 |
\begin{equation} |
| 361 |
F = |
| 362 |
\frac{1}{2}(u+|u|)\left( \tau_{i-1} + \psi(r^+)(\tau_{i} - \tau_{i-1} )\right) |
| 363 |
+ |
| 364 |
\frac{1}{2}(u-|u|)\left( \tau_{i-1} + \psi(r^-)(\tau_{i} - \tau_{i-1} )\right) |
| 365 |
\end{equation} |
| 366 |
where |
| 367 |
\begin{eqnarray} |
| 368 |
r^+ & = & \frac{\tau_{i-1} - \tau_{i-2}}{\tau_{i} - \tau_{i-1}} \\ |
| 369 |
r^- & = & \frac{\tau_{i+1} - \tau_{i}}{\tau_{i} - \tau_{i-1}} |
| 370 |
\end{eqnarray} |
| 371 |
and the limiter is the Sweby limiter: |
| 372 |
\begin{equation} |
| 373 |
\psi(r) = \max[0, \min[\min(1,d_0+d_1r],\frac{1-c}{c}r ]] |
| 374 |
\end{equation} |
| 375 |
|
| 376 |
\subsection{Multi-dimensional advection} |
| 377 |
|
| 378 |
In many of the aforementioned advection schemes the behaviour in |
| 379 |
multiple dimensions is not necessarily as good as the one dimensional |
| 380 |
behaviour. For instance, a shape preserving monotonic scheme in one |
| 381 |
dimension can have severe shape distortion in two dimensions if the |
| 382 |
two components of horizontal fluxes are treated independently. There |
| 383 |
is a large body of literature on the subject dealing with this problem |
| 384 |
and among the fixes are operator and flux splitting methods, corner |
| 385 |
flux methods and more. We have adopted a variant on the standard |
| 386 |
splitting methods that allows the flux calculations to be implemented |
| 387 |
as if in one dimension: |
| 388 |
\begin{eqnarray} |
| 389 |
\tau^{n+1/3} & = & \tau^{n} |
| 390 |
- \Delta t \left( \frac{1}{\Delta x} \delta_i F^x(\tau^{n}) |
| 391 |
+ \tau^{n} \frac{1}{\Delta x} \delta_i u \right) \\ |
| 392 |
\tau^{n+2/3} & = & \tau^{n} |
| 393 |
- \Delta t \left( \frac{1}{\Delta y} \delta_j F^y(\tau^{n+1/3}) |
| 394 |
+ \tau^{n} \frac{1}{\Delta y} \delta_i v \right) \\ |
| 395 |
\tau^{n+3/3} & = & \tau^{n} |
| 396 |
- \Delta t \left( \frac{1}{\Delta r} \delta_k F^x(\tau^{n+2/3}) |
| 397 |
+ \tau^{n} \frac{1}{\Delta r} \delta_i w \right) |
| 398 |
\end{eqnarray} |
| 399 |
|
| 400 |
In order to incorporate this method into the general model algorithm, |
| 401 |
we compute the effective tendancy rather than update the tracer so |
| 402 |
that other terms such as diffusion are using the $n$ time-level and |
| 403 |
not the updated $n+3/3$ quantities: |
| 404 |
\begin{equation} |
| 405 |
G^{n+1/2}_{adv} = \frac{1}{\Delta t} ( \tau^{n+3/3} - \tau^{n} ) |
| 406 |
\end{equation} |
| 407 |
So that the over all time-stepping looks likes: |
| 408 |
\begin{equation} |
| 409 |
\tau^{n+1} = \tau^{n} + \Delta t \left( G^{n+1/2}_{adv} + G_{diff}(\tau^{n}) + G^{n}_{forcing} \right) |
| 410 |
\end{equation} |
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