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revision 1.20 by jmc, Sun Oct 17 04:14:21 2004 UTC revision 1.27 by jmc, Fri Aug 27 13:08:18 2010 UTC
# Line 10  describe the spatial discretization. The Line 10  describe the spatial discretization. The
10  terms are described first, afterwards the schemes that apply to  terms are described first, afterwards the schemes that apply to
11  passive and dynamically active tracers are described.  passive and dynamically active tracers are described.
12    
13    \input{s_algorithm/text/notation}
14    
15  \section{Time-stepping}  \section{Time-stepping}
16  \begin{rawhtml}  \begin{rawhtml}
# Line 73  described in section \ref{sect:nonlinear Line 74  described in section \ref{sect:nonlinear
74    
75  \begin{figure}  \begin{figure}
76  \begin{center}  \begin{center}
77  \resizebox{4.0in}{!}{\includegraphics{part2/pressure-method-rigid-lid.eps}}  \resizebox{4.0in}{!}{\includegraphics{s_algorithm/figs/pressure-method-rigid-lid.eps}}
78  \end{center}  \end{center}
79  \caption{  \caption{
80  A schematic of the evolution in time of the pressure method  A schematic of the evolution in time of the pressure method
# Line 197  The calling tree for these routines is g Line 198  The calling tree for these routines is g
198  Fig.~\ref{fig:call-tree-pressure-method}.  Fig.~\ref{fig:call-tree-pressure-method}.
199    
200    
201    %\paragraph{Need to discuss implicit viscosity somewhere:}
202  \paragraph{Need to discuss implicit viscosity somewhere:}  In general, the horizontal momentum time-stepping can contain some terms
203    that are treated implicitly in time,
204    such as the vertical viscosity when using the backward time-stepping scheme
205    (\varlink{implicitViscosity}{implicitViscosity} {\it =.TRUE.}).
206    The method used to solve those implicit terms is provided in
207    section \ref{sect:implicit-backward-stepping}, and modifies
208    equations \ref{eq:discrete-time-u} and \ref{eq:discrete-time-v} to
209    give:
210  \begin{eqnarray}  \begin{eqnarray}
211  \frac{1}{\Delta t} u^{n+1} - \partial_z A_v \partial_z u^{n+1}  u^{n+1} - \Delta t \partial_z A_v \partial_z u^{n+1}
212  + g \partial_x \eta^{n+1} & = & \frac{1}{\Delta t} u^{n} +  + \Delta t g \partial_x \eta^{n+1} & = & u^{n} + \Delta t G_u^{(n+1/2)}
 G_u^{(n+1/2)}  
213  \\  \\
214  \frac{1}{\Delta t} v^{n+1} - \partial_z A_v \partial_z v^{n+1}  v^{n+1} - \Delta t \partial_z A_v \partial_z v^{n+1}
215  + g \partial_y \eta^{n+1} & = & \frac{1}{\Delta t} v^{n} + G_v^{(n+1/2)}  + \Delta t g \partial_y \eta^{n+1} & = & v^{n} + \Delta t G_v^{(n+1/2)}
216  \end{eqnarray}  \end{eqnarray}
217    
218    
# Line 244  pressure method is then replaced by the Line 251  pressure method is then replaced by the
251  \label{eq:discrete-time-backward-free-surface}  \label{eq:discrete-time-backward-free-surface}
252  \end{equation}  \end{equation}
253  where the use of flow at time level $n+1$ makes the method implicit  where the use of flow at time level $n+1$ makes the method implicit
254  and backward in time. The is the preferred scheme since it still  and backward in time. This is the preferred scheme since it still
255  filters the fast unresolved wave motions by damping them. A centered  filters the fast unresolved wave motions by damping them. A centered
256  scheme, such as Crank-Nicholson, would alias the energy of the fast  scheme, such as Crank-Nicholson (see section \ref{sect:freesurf-CrankNick}),
257  modes onto slower modes of motion.  would alias the energy of the fast modes onto slower modes of motion.
258    
259  As for the rigid-lid pressure method, equations  As for the rigid-lid pressure method, equations
260  \ref{eq:discrete-time-u}, \ref{eq:discrete-time-v} and  \ref{eq:discrete-time-u}, \ref{eq:discrete-time-v} and
# Line 278  recovered. However, the implicit treatme Line 285  recovered. However, the implicit treatme
285  the flow to be divergent and for the surface pressure/elevation to  the flow to be divergent and for the surface pressure/elevation to
286  respond on a finite time-scale (as opposed to instantly). To recover  respond on a finite time-scale (as opposed to instantly). To recover
287  the rigid-lid formulation, we introduced a switch-like parameter,  the rigid-lid formulation, we introduced a switch-like parameter,
288  $\epsilon_{fs}$, which selects between the free-surface and rigid-lid;  $\epsilon_{fs}$ (\varlink{freesurfFac}{freesurfFac}),
289    which selects between the free-surface and rigid-lid;
290  $\epsilon_{fs}=1$ allows the free-surface to evolve; $\epsilon_{fs}=0$  $\epsilon_{fs}=1$ allows the free-surface to evolve; $\epsilon_{fs}=0$
291  imposes the rigid-lid. The evolution in time and location of variables  imposes the rigid-lid. The evolution in time and location of variables
292  is exactly as it was for the rigid-lid model so that  is exactly as it was for the rigid-lid model so that
# Line 316  FORWARD\_STEP \\ Line 324  FORWARD\_STEP \\
324  \end{tabbing} \end{minipage} } \end{center}  \end{tabbing} \end{minipage} } \end{center}
325  \caption{  \caption{
326  Calling tree for the Adams-Bashforth time-stepping of temperature with  Calling tree for the Adams-Bashforth time-stepping of temperature with
327  implicit diffusion.}  implicit diffusion.
328      (\filelink{THERMODYNAMICS}{model-src-thermodynamics.F},
329       \filelink{ADAMS\_BASHFORTH2}{model-src-adams_bashforth2.F})}
330  \label{fig:call-tree-adams-bashforth}  \label{fig:call-tree-adams-bashforth}
331  \end{figure}  \end{figure}
332    
# Line 347  simpler to include these terms and this Line 357  simpler to include these terms and this
357  and forcing evolves smoothly. Problems can, and do, arise when forcing  and forcing evolves smoothly. Problems can, and do, arise when forcing
358  or motions are high frequency and this corresponds to a reduced  or motions are high frequency and this corresponds to a reduced
359  stability compared to a simple forward time-stepping of such terms.  stability compared to a simple forward time-stepping of such terms.
360  The model offers the possibility to leave the forcing term outside the  The model offers the possibility to leave the tracer and momentum
361  Adams-Bashforth extrapolation, by turning off the logical flag  forcing terms and the dissipation terms outside the
362  {\bf forcing\_In\_AB } (parameter file {\em data}, namelist {\em PARM01},  Adams-Bashforth extrapolation, by turning off the logical flags
363  default value = True).  \varlink{forcing\_In\_AB}{forcing_In_AB}
364    (parameter file {\em data}, namelist {\em PARM01}, default value = True).
365    (\varlink{tracForcingOutAB}{tracForcingOutAB}, default=0,
366    \varlink{momForcingOutAB}{momForcingOutAB}, default=0)
367    and \varlink{momDissip\_In\_AB}{momDissip_In_AB}
368    (parameter file {\em data}, namelist {\em PARM01}, default value = True).
369    respectively.
370    
371  A stability analysis for an oscillation equation should be given at this point.  A stability analysis for an oscillation equation should be given at this point.
372  \marginpar{AJA needs to find his notes on this...}  \marginpar{AJA needs to find his notes on this...}
# Line 358  A stability analysis for an oscillation Line 374  A stability analysis for an oscillation
374  A stability analysis for a relaxation equation should be given at this point.  A stability analysis for a relaxation equation should be given at this point.
375  \marginpar{...and for this too.}  \marginpar{...and for this too.}
376    
377    \begin{figure}
378    \begin{center}
379    \resizebox{5.5in}{!}{\includegraphics{s_algorithm/figs/oscil+damp_AB2.eps}}
380    \end{center}
381    \caption{
382    Oscillatory and damping response of
383    quasi-second order Adams-Bashforth scheme for different values
384    of the $\epsilon_{AB}$ parameter (0., 0.1, 0.25, from top to bottom)
385    The analytical solution (in black), the physical mode (in blue)
386    and the numerical mode (in red) are represented with a CFL
387    step of 0.1.
388    The left column represents the oscillatory response
389    on the complex plane for CFL ranging from 0.1 up to 0.9.
390    The right column represents the damping response amplitude
391    (y-axis) function of the CFL (x-axis).
392    }
393    \label{fig:adams-bashforth-respons}
394    \end{figure}
395    
396    
397    
398  \section{Implicit time-stepping: backward method}  \section{Implicit time-stepping: backward method}
399    \label{sect:implicit-backward-stepping}
400  \begin{rawhtml}  \begin{rawhtml}
401  <!-- CMIREDIR:implicit_time-stepping_backward: -->  <!-- CMIREDIR:implicit_time-stepping_backward: -->
402  \end{rawhtml}  \end{rawhtml}
# Line 414  implicit and are thus cast as a an expli Line 451  implicit and are thus cast as a an expli
451    
452  \begin{figure}  \begin{figure}
453  \begin{center}  \begin{center}
454  \resizebox{5.0in}{!}{\includegraphics{part2/adams-bashforth-sync.eps}}  \resizebox{5.0in}{!}{\includegraphics{s_algorithm/figs/adams-bashforth-sync.eps}}
455  \end{center}  \end{center}
456  \caption{  \caption{
457  A schematic of the explicit Adams-Bashforth and implicit time-stepping  A schematic of the explicit Adams-Bashforth and implicit time-stepping
# Line 534  time-step. The corresponding calling tre Line 571  time-step. The corresponding calling tre
571    
572  \begin{figure}  \begin{figure}
573  \begin{center}  \begin{center}
574  \resizebox{5.5in}{!}{\includegraphics{part2/adams-bashforth-staggered.eps}}  \resizebox{5.5in}{!}{\includegraphics{s_algorithm/figs/adams-bashforth-staggered.eps}}
575  \end{center}  \end{center}
576  \caption{  \caption{
577  A schematic of the explicit Adams-Bashforth and implicit time-stepping  A schematic of the explicit Adams-Bashforth and implicit time-stepping
# Line 608  G_{\theta,S}^{(n+1/2)} & = & (3/2+\epsil Line 645  G_{\theta,S}^{(n+1/2)} & = & (3/2+\epsil
645  (\theta^*,S^*) & = & (\theta^{n},S^{n}) + \Delta t G_{\theta,S}^{(n+1/2)}  (\theta^*,S^*) & = & (\theta^{n},S^{n}) + \Delta t G_{\theta,S}^{(n+1/2)}
646  \label{eq:tstar-staggered} \\  \label{eq:tstar-staggered} \\
647  (\theta^{n+1},S^{n+1}) & = & {\cal L}^{-1}_{\theta,S} (\theta^*,S^*)  (\theta^{n+1},S^{n+1}) & = & {\cal L}^{-1}_{\theta,S} (\theta^*,S^*)
648  \label{eq:t-n+1-staggered} \\  \label{eq:t-n+1-staggered}
649  \end{eqnarray}  \end{eqnarray}
650  The corresponding calling tree is given in  The corresponding calling tree is given in
651  \ref{fig:call-tree-adams-bashforth-staggered}.  \ref{fig:call-tree-adams-bashforth-staggered}.
# Line 691  The momentum equations are discretized i Line 728  The momentum equations are discretized i
728  \frac{1}{\Delta t} v^{n+1} + g \partial_y \eta^{n+1} + \partial_y \phi_{nh}^{n+1}  \frac{1}{\Delta t} v^{n+1} + g \partial_y \eta^{n+1} + \partial_y \phi_{nh}^{n+1}
729  & = & \frac{1}{\Delta t} v^{n} + G_v^{(n+1/2)} \label{eq:discrete-time-v-nh} \\  & = & \frac{1}{\Delta t} v^{n} + G_v^{(n+1/2)} \label{eq:discrete-time-v-nh} \\
730  \frac{1}{\Delta t} w^{n+1} + \partial_r \phi_{nh}^{n+1}  \frac{1}{\Delta t} w^{n+1} + \partial_r \phi_{nh}^{n+1}
731  & = & \frac{1}{\Delta t} w^{n} + G_w^{(n+1/2)} \label{eq:discrete-time-w-nh} \\  & = & \frac{1}{\Delta t} w^{n} + G_w^{(n+1/2)} \label{eq:discrete-time-w-nh}
732  \end{eqnarray}  \end{eqnarray}
733  which must satisfy the discrete-in-time depth integrated continuity,  which must satisfy the discrete-in-time depth integrated continuity,
734  equation~\ref{eq:discrete-time-backward-free-surface} and the local continuity equation  equation~\ref{eq:discrete-time-backward-free-surface} and the local continuity equation
# Line 795  where Line 832  where
832  \begin{eqnarray}  \begin{eqnarray}
833  {\eta}^* = \epsilon_{fs} \: {\eta}^{n} -  {\eta}^* = \epsilon_{fs} \: {\eta}^{n} -
834  \Delta t {\bf \nabla}_h \cdot \int_{R_{fixed}}^{R_o} \vec{\bf v}^* dr  \Delta t {\bf \nabla}_h \cdot \int_{R_{fixed}}^{R_o} \vec{\bf v}^* dr
835  \: + \: \epsilon_{fw} \Delta_t (P-E)^{n}  \: + \: \epsilon_{fw} \Delta t (P-E)^{n}
836  \label{eq-solve2D_rhs}  \label{eq-solve2D_rhs}
837  \end{eqnarray}  \end{eqnarray}
838    
839  \fbox{ \begin{minipage}{4.75in}  \fbox{ \begin{minipage}{4.75in}
840  {\em S/R SOLVE\_FOR\_PRESSURE} ({\em solve\_for\_pressure.F})  {\em S/R SOLVE\_FOR\_PRESSURE} ({\em solve\_for\_pressure.F})
841    
842  $u^*$: {\bf GuNm1} ({\em DYNVARS.h})  $u^*$: {\bf gU} ({\em DYNVARS.h})
843    
844  $v^*$: {\bf GvNm1} ({\em DYNVARS.h})  $v^*$: {\bf gV} ({\em DYNVARS.h})
845    
846  $\eta^*$: {\bf cg2d\_b} (\em SOLVE\_FOR\_PRESSURE.h)  $\eta^*$: {\bf cg2d\_b} (\em SOLVE\_FOR\_PRESSURE.h)
847    
# Line 814  $\eta^{n+1}$: {\bf etaN} (\em DYNVARS.h) Line 851  $\eta^{n+1}$: {\bf etaN} (\em DYNVARS.h)
851    
852    
853  Once ${\eta}^{n+1}$ has been found, substituting into  Once ${\eta}^{n+1}$ has been found, substituting into
854  \ref{eq:discrete-time-u,eq:discrete-time-v} yields $\vec{\bf v}^{n+1}$ if the model is  \ref{eq:discrete-time-u}, \ref{eq:discrete-time-v} yields $\vec{\bf v}^{n+1}$
855  hydrostatic ($\epsilon_{nh}=0$):  if the model is hydrostatic ($\epsilon_{nh}=0$):
856  $$  $$
857  \vec{\bf v}^{n+1} = \vec{\bf v}^{*}  \vec{\bf v}^{n+1} = \vec{\bf v}^{*}
858  - \Delta t {\bf \nabla}_h b_s {\eta}^{n+1}  - \Delta t {\bf \nabla}_h b_s {\eta}^{n+1}
# Line 856  without any consequence on the solution. Line 893  without any consequence on the solution.
893    
894  $\eta^{n+1}$: {\bf etaN} (\em DYNVARS.h)  $\eta^{n+1}$: {\bf etaN} (\em DYNVARS.h)
895    
896  $\phi_{nh}^{n+1}$: {\bf phi\_nh} (\em DYNVARS.h)  $\phi_{nh}^{n+1}$: {\bf phi\_nh} (\em NH\_VARS.h)
897    
898  $u^*$: {\bf GuNm1} ({\em DYNVARS.h})  $u^*$: {\bf gU} ({\em DYNVARS.h})
899    
900  $v^*$: {\bf GvNm1} ({\em DYNVARS.h})  $v^*$: {\bf gV} ({\em DYNVARS.h})
901    
902  $u^{n+1}$: {\bf uVel} ({\em DYNVARS.h})  $u^{n+1}$: {\bf uVel} ({\em DYNVARS.h})
903    
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