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revision 1.18 by jmc, Thu Oct 14 22:22:30 2004 UTC revision 1.21 by jmc, Tue Apr 4 20:16:39 2006 UTC
# Line 12  passive and dynamically active tracers a Line 12  passive and dynamically active tracers a
12    
13    
14  \section{Time-stepping}  \section{Time-stepping}
15    \begin{rawhtml}
16    <!-- CMIREDIR:time-stepping: -->
17    \end{rawhtml}
18    
19  The equations of motion integrated by the model involve four  The equations of motion integrated by the model involve four
20  prognostic equations for flow, $u$ and $v$, temperature, $\theta$, and  prognostic equations for flow, $u$ and $v$, temperature, $\theta$, and
21  salt/moisture, $S$, and three diagnostic equations for vertical flow,  salt/moisture, $S$, and three diagnostic equations for vertical flow,
# Line 61  treated more exactly, including non-line Line 65  treated more exactly, including non-line
65  described in section \ref{sect:nonlinear-freesurface}.  described in section \ref{sect:nonlinear-freesurface}.
66    
67    
68  \section{Pressure method with rigid-lid} \label{sect:pressure-method-rigid-lid}  \section{Pressure method with rigid-lid}
69    \label{sect:pressure-method-rigid-lid}
70    \begin{rawhtml}
71    <!-- CMIREDIR:pressure_method_rigid_lid: -->
72    \end{rawhtml}
73    
74  \begin{figure}  \begin{figure}
75  \begin{center}  \begin{center}
# Line 203  G_u^{(n+1/2)} Line 211  G_u^{(n+1/2)}
211    
212  \section{Pressure method with implicit linear free-surface}  \section{Pressure method with implicit linear free-surface}
213  \label{sect:pressure-method-linear-backward}  \label{sect:pressure-method-linear-backward}
214    \begin{rawhtml}
215    <!-- CMIREDIR:pressure_method_linear_backward: -->
216    \end{rawhtml}
217    
218  The rigid-lid approximation filters out external gravity waves  The rigid-lid approximation filters out external gravity waves
219  subsequently modifying the dispersion relation of barotropic Rossby  subsequently modifying the dispersion relation of barotropic Rossby
# Line 214  The rigid-lid approximation can be easil Line 225  The rigid-lid approximation can be easil
225  of the free-surface equation which can be written:  of the free-surface equation which can be written:
226  \begin{equation}  \begin{equation}
227  \partial_t \eta + \partial_x H \widehat{u} + \partial_y H \widehat{v} = P-E+R  \partial_t \eta + \partial_x H \widehat{u} + \partial_y H \widehat{v} = P-E+R
228  \label{eq:linear-free-surface=P-E+R}  \label{eq:linear-free-surface=P-E}
229  \end{equation}  \end{equation}
230  which differs from the depth integrated continuity equation with  which differs from the depth integrated continuity equation with
231  rigid-lid (\ref{eq:rigid-lid-continuity}) by the time-dependent term  rigid-lid (\ref{eq:rigid-lid-continuity}) by the time-dependent term
# Line 222  and fresh-water source term. Line 233  and fresh-water source term.
233    
234  Equation \ref{eq:discrete-time-cont-rigid-lid} in the rigid-lid  Equation \ref{eq:discrete-time-cont-rigid-lid} in the rigid-lid
235  pressure method is then replaced by the time discretization of  pressure method is then replaced by the time discretization of
236  \ref{eq:linear-free-surface=P-E+R} which is:  \ref{eq:linear-free-surface=P-E} which is:
237  \begin{equation}  \begin{equation}
238  \eta^{n+1}  \eta^{n+1}
239  + \Delta t \partial_x H \widehat{u^{n+1}}  + \Delta t \partial_x H \widehat{u^{n+1}}
240  + \Delta t \partial_y H \widehat{v^{n+1}}  + \Delta t \partial_y H \widehat{v^{n+1}}
241  =  =
242  \eta^{n}  \eta^{n}
243  + \Delta t ( P - E + R )  + \Delta t ( P - E )
244  \label{eq:discrete-time-backward-free-surface}  \label{eq:discrete-time-backward-free-surface}
245  \end{equation}  \end{equation}
246  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
# Line 244  As for the rigid-lid pressure method, eq Line 255  As for the rigid-lid pressure method, eq
255  \begin{eqnarray}  \begin{eqnarray}
256  u^{*} & = & u^{n} + \Delta t G_u^{(n+1/2)} \label{eq:ustar-backward-free-surface} \\  u^{*} & = & u^{n} + \Delta t G_u^{(n+1/2)} \label{eq:ustar-backward-free-surface} \\
257  v^{*} & = & v^{n} + \Delta t G_v^{(n+1/2)} \label{eq:vstar-backward-free-surface} \\  v^{*} & = & v^{n} + \Delta t G_v^{(n+1/2)} \label{eq:vstar-backward-free-surface} \\
258  \eta^* & = & \epsilon_{fs} \left( \eta^{n} +P-E+R \right)- \Delta t  \eta^* & = & \epsilon_{fs} \left( \eta^{n} + \Delta t (P-E) \right)- \Delta t
259    \partial_x H \widehat{u^{*}}    \partial_x H \widehat{u^{*}}
260  + \partial_y H \widehat{v^{*}}  + \partial_y H \widehat{v^{*}}
261  \\  \\
262    \partial_x g H \partial_x \eta^{n+1}    \partial_x g H \partial_x \eta^{n+1}
263  + \partial_y g H \partial_y \eta^{n+1}  & + & \partial_y g H \partial_y \eta^{n+1}
264  - \frac{\epsilon_{fs} \eta^{n+1}}{\Delta t^2}   - \frac{\epsilon_{fs} \eta^{n+1}}{\Delta t^2}
265  & = &   =
266  - \frac{\eta^*}{\Delta t^2}  - \frac{\eta^*}{\Delta t^2}
267  \label{eq:elliptic-backward-free-surface}  \label{eq:elliptic-backward-free-surface}
268  \\  \\
# Line 279  pressure-method. Line 290  pressure-method.
290    
291  \section{Explicit time-stepping: Adams-Bashforth}  \section{Explicit time-stepping: Adams-Bashforth}
292  \label{sect:adams-bashforth}  \label{sect:adams-bashforth}
293    \begin{rawhtml}
294    <!-- CMIREDIR:adams_bashforth: -->
295    \end{rawhtml}
296    
297  In describing the the pressure method above we deferred describing the  In describing the the pressure method above we deferred describing the
298  time discretization of the explicit terms. We have historically used  time discretization of the explicit terms. We have historically used
# Line 346  A stability analysis for a relaxation eq Line 360  A stability analysis for a relaxation eq
360    
361    
362  \section{Implicit time-stepping: backward method}  \section{Implicit time-stepping: backward method}
363    \begin{rawhtml}
364    <!-- CMIREDIR:implicit_time-stepping_backward: -->
365    \end{rawhtml}
366    
367  Vertical diffusion and viscosity can be treated implicitly in time  Vertical diffusion and viscosity can be treated implicitly in time
368  using the backward method which is an intrinsic scheme.  using the backward method which is an intrinsic scheme.
# Line 391  implicit and are thus cast as a an expli Line 408  implicit and are thus cast as a an expli
408    
409  \section{Synchronous time-stepping: variables co-located in time}  \section{Synchronous time-stepping: variables co-located in time}
410  \label{sect:adams-bashforth-sync}  \label{sect:adams-bashforth-sync}
411    \begin{rawhtml}
412    <!-- CMIREDIR:adams_bashforth_sync: -->
413    \end{rawhtml}
414    
415  \begin{figure}  \begin{figure}
416  \begin{center}  \begin{center}
# Line 422  FORWARD\_STEP \\ Line 442  FORWARD\_STEP \\
442  \>\>\> GAD\_CALC\_RHS \` $G_\theta^n = G_\theta( u, \theta^n )$ (\ref{eq:Gt-n-sync})\\  \>\>\> GAD\_CALC\_RHS \` $G_\theta^n = G_\theta( u, \theta^n )$ (\ref{eq:Gt-n-sync})\\
443  \>\>\> EXTERNAL\_FORCING \` $G_\theta^n = G_\theta^n + {\cal Q}$ \\  \>\>\> EXTERNAL\_FORCING \` $G_\theta^n = G_\theta^n + {\cal Q}$ \\
444  \>\>\> ADAMS\_BASHFORTH2 \` $G_\theta^{(n+1/2)}$ (\ref{eq:Gt-n+5-sync}) \\  \>\>\> ADAMS\_BASHFORTH2 \` $G_\theta^{(n+1/2)}$ (\ref{eq:Gt-n+5-sync}) \\
445  \>\> TIMESTEP\_TRACER \` $\tau^*$ (\ref{eq:tstar-sync}) \\  \>\> TIMESTEP\_TRACER \` $\theta^*$ (\ref{eq:tstar-sync}) \\
446  \>\> IMPLDIFF \` $\tau^{(n+1)}$ (\ref{eq:t-n+1-sync}) \\  \>\> IMPLDIFF \` $\theta^{(n+1)}$ (\ref{eq:t-n+1-sync}) \\
447  \> DYNAMICS \\  \> DYNAMICS \\
448  \>\> CALC\_PHI\_HYD \` $\phi_{hyd}^n$ (\ref{eq:phi-hyd-sync}) \\  \>\> CALC\_PHI\_HYD \` $\phi_{hyd}^n$ (\ref{eq:phi-hyd-sync}) \\
449  \>\> MOM\_FLUXFORM or MOM\_VECINV \` $G_{\vec{\bf v}}^n$ (\ref{eq:Gv-n-sync})\\  \>\> MOM\_FLUXFORM or MOM\_VECINV \` $G_{\vec{\bf v}}^n$ (\ref{eq:Gv-n-sync})\\
# Line 445  FORWARD\_STEP \\ Line 465  FORWARD\_STEP \\
465  Calling tree for the overall synchronous algorithm using  Calling tree for the overall synchronous algorithm using
466  Adams-Bashforth time-stepping.  Adams-Bashforth time-stepping.
467  The place where the model geometry  The place where the model geometry
468  ({\em hFac} factors) is updated is added here but is only relevant  ({\bf hFac} factors) is updated is added here but is only relevant
469  for the non-linear free-surface algorithm.  for the non-linear free-surface algorithm.
470  For completeness, the external forcing,  For completeness, the external forcing,
471  ocean and atmospheric physics have been added, although they are mainly  ocean and atmospheric physics have been added, although they are mainly
# Line 508  time-step. The corresponding calling tre Line 528  time-step. The corresponding calling tre
528    
529  \section{Staggered baroclinic time-stepping}  \section{Staggered baroclinic time-stepping}
530  \label{sect:adams-bashforth-staggered}  \label{sect:adams-bashforth-staggered}
531    \begin{rawhtml}
532    <!-- CMIREDIR:adams_bashforth_staggered: -->
533    \end{rawhtml}
534    
535  \begin{figure}  \begin{figure}
536  \begin{center}  \begin{center}
# Line 524  extrapolate tendencies to $n$ (dashed ar Line 547  extrapolate tendencies to $n$ (dashed ar
547  The hydrostatic pressure/geo-potential $\phi_{hyd}$ is evaluated directly  The hydrostatic pressure/geo-potential $\phi_{hyd}$ is evaluated directly
548  at time level $n$ (vertical arrows) and used with the extrapolated tendencies  at time level $n$ (vertical arrows) and used with the extrapolated tendencies
549  to step forward the flow variables from $n-1/2$ to $n+1/2$ (solid arc-arrow).  to step forward the flow variables from $n-1/2$ to $n+1/2$ (solid arc-arrow).
550  The implicit-in-time operator ${\cal L_{u,v}}$ (vertical arrows) is  The implicit-in-time operator ${\cal L}_{\bf u,v}$ (vertical arrows) is
551  then applied to the previous estimation of the the flow field ($*$-variables)  then applied to the previous estimation of the the flow field ($*$-variables)
552  and yields to the two velocity components $u,v$ at time level $n+1/2$.  and yields to the two velocity components $u,v$ at time level $n+1/2$.
553  These are then used to calculate the advection term (dashed arc-arrow)  These are then used to calculate the advection term (dashed arc-arrow)
# Line 552  thermodynamics solver is delayed from ha Line 575  thermodynamics solver is delayed from ha
575  allowing the use of the most recent velocities to compute  allowing the use of the most recent velocities to compute
576  the advection terms. Once the thermodynamics fields are  the advection terms. Once the thermodynamics fields are
577  updated, the hydrostatic pressure is computed  updated, the hydrostatic pressure is computed
578  to step frowrad the dynamics  to step forwrad the dynamics.
579  Note that the pressure gradient must also be taken out of the  Note that the pressure gradient must also be taken out of the
580  Adams-Bashforth extrapolation. Also, retaining the integer time-levels,  Adams-Bashforth extrapolation. Also, retaining the integer time-levels,
581  $n$ and $n+1$, does not give a user the sense of where variables are  $n$ and $n+1$, does not give a user the sense of where variables are
# Line 560  located in time.  Instead, we re-write t Line 583  located in time.  Instead, we re-write t
583  \ref{eq:Gt-n-sync} to \ref{eq:v-n+1-sync}, annotating the  \ref{eq:Gt-n-sync} to \ref{eq:v-n+1-sync}, annotating the
584  position in time of variables appropriately:  position in time of variables appropriately:
585  \begin{eqnarray}  \begin{eqnarray}
586    \phi^{n}_{hyd} & = & \int b(\theta^{n},S^{n}) dr
587    \label{eq:phi-hyd-staggered} \\
588  \vec{\bf G}_{\vec{\bf v}}^{n-1/2} & = & \vec{\bf G}_{\vec{\bf v}} ( \vec{\bf v}^{n-1/2} )  \vec{\bf G}_{\vec{\bf v}}^{n-1/2} & = & \vec{\bf G}_{\vec{\bf v}} ( \vec{\bf v}^{n-1/2} )
589  \label{eq:Gv-n-staggered} \\  \label{eq:Gv-n-staggered} \\
590  \vec{\bf G}_{\vec{\bf v}}^{(n)} & = & (3/2 + \epsilon_{AB} ) \vec{\bf G}_{\vec{\bf v}}^{n-1/2} - (1/2 + \epsilon_{AB} ) \vec{\bf G}_{\vec{\bf v}}^{n-3/2}  \vec{\bf G}_{\vec{\bf v}}^{(n)} & = & (3/2 + \epsilon_{AB} ) \vec{\bf G}_{\vec{\bf v}}^{n-1/2} - (1/2 + \epsilon_{AB} ) \vec{\bf G}_{\vec{\bf v}}^{n-3/2}
591  \label{eq:Gv-n+5-staggered} \\  \label{eq:Gv-n+5-staggered} \\
 \phi^{n}_{hyd} & = & \int b(\theta^{n},S^{n}) dr  
 \label{eq:phi-hyd-staggered} \\  
592  \vec{\bf v}^{*} & = & \vec{\bf v}^{n-1/2} + \Delta t \left( \vec{\bf G}_{\vec{\bf v}}^{(n)} - \nabla \phi_{hyd}^{n} \right)  \vec{\bf v}^{*} & = & \vec{\bf v}^{n-1/2} + \Delta t \left( \vec{\bf G}_{\vec{\bf v}}^{(n)} - \nabla \phi_{hyd}^{n} \right)
593  \label{eq:vstar-staggered} \\  \label{eq:vstar-staggered} \\
594  \vec{\bf v}^{**} & = & {\cal L}_{\vec{\bf v}}^{-1} ( \vec{\bf v}^* )  \vec{\bf v}^{**} & = & {\cal L}_{\vec{\bf v}}^{-1} ( \vec{\bf v}^* )
# Line 590  G_{\theta,S}^{(n+1/2)} & = & (3/2+\epsil Line 613  G_{\theta,S}^{(n+1/2)} & = & (3/2+\epsil
613  The corresponding calling tree is given in  The corresponding calling tree is given in
614  \ref{fig:call-tree-adams-bashforth-staggered}.  \ref{fig:call-tree-adams-bashforth-staggered}.
615  The staggered algorithm is activated with the run-time flag  The staggered algorithm is activated with the run-time flag
616  {\bf staggerTimeStep=.TRUE.} in parameter file {\em data},  {\bf staggerTimeStep}{\em=.TRUE.} in parameter file {\em data},
617  namelist {\em PARM01}.  namelist {\em PARM01}.
618    
619  \begin{figure}  \begin{figure}
# Line 620  FORWARD\_STEP \\ Line 643  FORWARD\_STEP \\
643       (\ref{eq:Gt-n-staggered})\\       (\ref{eq:Gt-n-staggered})\\
644  \>\>\> EXTERNAL\_FORCING \` $G_\theta^n = G_\theta^n + {\cal Q}$ \\  \>\>\> EXTERNAL\_FORCING \` $G_\theta^n = G_\theta^n + {\cal Q}$ \\
645  \>\>\> ADAMS\_BASHFORTH2 \` $G_\theta^{(n+1/2)}$ (\ref{eq:Gt-n+5-staggered}) \\  \>\>\> ADAMS\_BASHFORTH2 \` $G_\theta^{(n+1/2)}$ (\ref{eq:Gt-n+5-staggered}) \\
646  \>\> TIMESTEP\_TRACER \` $\tau^*$ (\ref{eq:tstar-staggered}) \\  \>\> TIMESTEP\_TRACER \` $\theta^*$ (\ref{eq:tstar-staggered}) \\
647  \>\> IMPLDIFF \` $\tau^{(n+1)}$ (\ref{eq:t-n+1-staggered}) \\  \>\> IMPLDIFF \` $\theta^{(n+1)}$ (\ref{eq:t-n+1-staggered}) \\
648  \> TRACERS\_CORRECTION\_STEP  \\  \> TRACERS\_CORRECTION\_STEP  \\
649  \>\> CYCLE\_TRACER \` $\theta^{n+1}$ \\  \>\> CYCLE\_TRACER \` $\theta^{n+1}$ \\
650  \>\> FILTER \` Shapiro Filter, Zonal Filter (FFT) \\  \>\> FILTER \` Shapiro Filter, Zonal Filter (FFT) \\
# Line 631  FORWARD\_STEP \\ Line 654  FORWARD\_STEP \\
654  Calling tree for the overall staggered algorithm using  Calling tree for the overall staggered algorithm using
655  Adams-Bashforth time-stepping.  Adams-Bashforth time-stepping.
656  The place where the model geometry  The place where the model geometry
657  ({\em hFac} factors) is updated is added here but is only relevant  ({\bf hFac} factors) is updated is added here but is only relevant
658  for the non-linear free-surface algorithm.  for the non-linear free-surface algorithm.
659  }  }
660  \label{fig:call-tree-adams-bashforth-staggered}  \label{fig:call-tree-adams-bashforth-staggered}
# Line 650  time-level variables and terms correspon Line 673  time-level variables and terms correspon
673    
674  \section{Non-hydrostatic formulation}  \section{Non-hydrostatic formulation}
675  \label{sect:non-hydrostatic}  \label{sect:non-hydrostatic}
676    \begin{rawhtml}
677    <!-- CMIREDIR:non-hydrostatic_formulation: -->
678    \end{rawhtml}
679    
680  The non-hydrostatic formulation re-introduces the full vertical  The non-hydrostatic formulation re-introduces the full vertical
681  momentum equation and requires the solution of a 3-D elliptic  momentum equation and requires the solution of a 3-D elliptic
# Line 749  $w^{n+1}$. Line 775  $w^{n+1}$.
775    
776    
777  \section{Variants on the Free Surface}  \section{Variants on the Free Surface}
778    \label{sect:free-surface}
779    
780  We now describe the various formulations of the free-surface that  We now describe the various formulations of the free-surface that
781  include non-linear forms, implicit in time using Crank-Nicholson,  include non-linear forms, implicit in time using Crank-Nicholson,
# Line 787  $\eta^{n+1}$: {\bf etaN} (\em DYNVARS.h) Line 814  $\eta^{n+1}$: {\bf etaN} (\em DYNVARS.h)
814    
815    
816  Once ${\eta}^{n+1}$ has been found, substituting into  Once ${\eta}^{n+1}$ has been found, substituting into
817  \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 $\vec{\bf v}^{n+1}$
818  hydrostatic ($\epsilon_{nh}=0$):  if the model is hydrostatic ($\epsilon_{nh}=0$):
819  $$  $$
820  \vec{\bf v}^{n+1} = \vec{\bf v}^{*}  \vec{\bf v}^{n+1} = \vec{\bf v}^{*}
821  - \Delta t {\bf \nabla}_h b_s {\eta}^{n+1}  - \Delta t {\bf \nabla}_h b_s {\eta}^{n+1}
# Line 861  at the same point in the code. Line 888  at the same point in the code.
888    
889    
890  \subsection{Crank-Nickelson barotropic time stepping}  \subsection{Crank-Nickelson barotropic time stepping}
891    \label{sect:freesurf-CrankNick}
892    
893  The full implicit time stepping described previously is  The full implicit time stepping described previously is
894  unconditionally stable but damps the fast gravity waves, resulting in  unconditionally stable but damps the fast gravity waves, resulting in
# Line 875  stable, Crank-Nickelson scheme; $(\beta, Line 903  stable, Crank-Nickelson scheme; $(\beta,
903  corresponds to the forward - backward scheme that conserves energy but is  corresponds to the forward - backward scheme that conserves energy but is
904  only stable for small time steps.\\  only stable for small time steps.\\
905  In the code, $\beta,\gamma$ are defined as parameters, respectively  In the code, $\beta,\gamma$ are defined as parameters, respectively
906  {\it implicSurfPress}, {\it implicDiv2DFlow}. They are read from  {\bf implicSurfPress}, {\bf implicDiv2DFlow}. They are read from
907  the main data file "{\it data}" and are set by default to 1,1.  the main parameter file "{\em data}" and are set by default to 1,1.
908    
909  Equations \ref{eq:ustar-backward-free-surface} --  Equations \ref{eq:ustar-backward-free-surface} --
910  \ref{eq:vn+1-backward-free-surface} are modified as follows:  \ref{eq:vn+1-backward-free-surface} are modified as follows:
911  $$  \begin{eqnarray*}
912  \frac{ \vec{\bf v}^{n+1} }{ \Delta t }  \frac{ \vec{\bf v}^{n+1} }{ \Delta t }
913  + {\bf \nabla}_h b_s [ \beta {\eta}^{n+1} + (1-\beta) {\eta}^{n} ]  + {\bf \nabla}_h b_s [ \beta {\eta}^{n+1} + (1-\beta) {\eta}^{n} ]
914  + \epsilon_{nh} {\bf \nabla}_h {\phi'_{nh}}^{n+1}  + \epsilon_{nh} {\bf \nabla}_h {\phi'_{nh}}^{n+1}
915   = \frac{ \vec{\bf v}^* }{ \Delta t }   = \frac{ \vec{\bf v}^* }{ \Delta t }
916  $$  \end{eqnarray*}
917  $$  \begin{eqnarray}
918  \epsilon_{fs} \frac{ {\eta}^{n+1} - {\eta}^{n} }{ \Delta t}  \epsilon_{fs} \frac{ {\eta}^{n+1} - {\eta}^{n} }{ \Delta t}
919  + {\bf \nabla}_h \cdot \int_{R_{fixed}}^{R_o}  + {\bf \nabla}_h \cdot \int_{R_{fixed}}^{R_o}
920  [ \gamma \vec{\bf v}^{n+1} + (1-\gamma) \vec{\bf v}^{n}] dr  [ \gamma \vec{\bf v}^{n+1} + (1-\gamma) \vec{\bf v}^{n}] dr
921  = \epsilon_{fw} (P-E)  = \epsilon_{fw} (P-E)
922  $$  \label{eq:eta-n+1-CrankNick}
923    \end{eqnarray}
924  where:  where:
925  \begin{eqnarray*}  \begin{eqnarray*}
926  \vec{\bf v}^* & = &  \vec{\bf v}^* & = &
# Line 913  $$ Line 942  $$
942  {\bf \nabla}_h {\eta}^{n+1}  {\bf \nabla}_h {\eta}^{n+1}
943  = {\eta}^*  = {\eta}^*
944  $$  $$
945  and then to compute (correction step):  and then to compute ({\em CORRECTION\_STEP}):
946  $$  $$
947  \vec{\bf v}^{n+1} = \vec{\bf v}^{*}  \vec{\bf v}^{n+1} = \vec{\bf v}^{*}
948  - \beta \Delta t {\bf \nabla}_h b_s {\eta}^{n+1}  - \beta \Delta t {\bf \nabla}_h b_s {\eta}^{n+1}
949  $$  $$
950    
951  The non-hydrostatic part is solved as described previously.  %The non-hydrostatic part is solved as described previously.
952    
953  Note that:  \noindent
954    Notes:
955  \begin{enumerate}  \begin{enumerate}
956    \item The RHS term of equation \ref{eq:eta-n+1-CrankNick}
957    corresponds the contribution of fresh water flux (P-E)
958    to the free-surface variations ($\epsilon_{fw}=1$,
959    {\bf useRealFreshWater}{\em=TRUE} in parameter file {\em data}).
960    In order to remain consistent with the tracer equation, specially in
961    the non-linear free-surface formulation, this term is also
962    affected by the Crank-Nickelson time stepping. The RHS reads:
963    $\epsilon_{fw} ( \gamma (P-E)^{n+1/2} + (1-\gamma) (P-E)^{n-1/2} )$
964  \item The non-hydrostatic part of the code has not yet been  \item The non-hydrostatic part of the code has not yet been
965  updated, so that this option cannot be used with $(\beta,\gamma) \neq (1,1)$.  updated, and therefore cannot be used with $(\beta,\gamma) \neq (1,1)$.
966  \item The stability criteria with Crank-Nickelson time stepping  \item The stability criteria with Crank-Nickelson time stepping
967  for the pure linear gravity wave problem in cartesian coordinates is:  for the pure linear gravity wave problem in cartesian coordinates is:
968  \begin{itemize}  \begin{itemize}
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