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revision 1.18 by jmc, Thu Oct 14 22:22:30 2004 UTC revision 1.22 by jmc, Tue Jun 27 19:10:32 2006 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{part2/notation}
14    
15  \section{Time-stepping}  \section{Time-stepping}
16    \begin{rawhtml}
17    <!-- CMIREDIR:time-stepping: -->
18    \end{rawhtml}
19    
20  The equations of motion integrated by the model involve four  The equations of motion integrated by the model involve four
21  prognostic equations for flow, $u$ and $v$, temperature, $\theta$, and  prognostic equations for flow, $u$ and $v$, temperature, $\theta$, and
22  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 66  treated more exactly, including non-line
66  described in section \ref{sect:nonlinear-freesurface}.  described in section \ref{sect:nonlinear-freesurface}.
67    
68    
69  \section{Pressure method with rigid-lid} \label{sect:pressure-method-rigid-lid}  \section{Pressure method with rigid-lid}
70    \label{sect:pressure-method-rigid-lid}
71    \begin{rawhtml}
72    <!-- CMIREDIR:pressure_method_rigid_lid: -->
73    \end{rawhtml}
74    
75  \begin{figure}  \begin{figure}
76  \begin{center}  \begin{center}
# Line 203  G_u^{(n+1/2)} Line 212  G_u^{(n+1/2)}
212    
213  \section{Pressure method with implicit linear free-surface}  \section{Pressure method with implicit linear free-surface}
214  \label{sect:pressure-method-linear-backward}  \label{sect:pressure-method-linear-backward}
215    \begin{rawhtml}
216    <!-- CMIREDIR:pressure_method_linear_backward: -->
217    \end{rawhtml}
218    
219  The rigid-lid approximation filters out external gravity waves  The rigid-lid approximation filters out external gravity waves
220  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 226  The rigid-lid approximation can be easil
226  of the free-surface equation which can be written:  of the free-surface equation which can be written:
227  \begin{equation}  \begin{equation}
228  \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
229  \label{eq:linear-free-surface=P-E+R}  \label{eq:linear-free-surface=P-E}
230  \end{equation}  \end{equation}
231  which differs from the depth integrated continuity equation with  which differs from the depth integrated continuity equation with
232  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 234  and fresh-water source term.
234    
235  Equation \ref{eq:discrete-time-cont-rigid-lid} in the rigid-lid  Equation \ref{eq:discrete-time-cont-rigid-lid} in the rigid-lid
236  pressure method is then replaced by the time discretization of  pressure method is then replaced by the time discretization of
237  \ref{eq:linear-free-surface=P-E+R} which is:  \ref{eq:linear-free-surface=P-E} which is:
238  \begin{equation}  \begin{equation}
239  \eta^{n+1}  \eta^{n+1}
240  + \Delta t \partial_x H \widehat{u^{n+1}}  + \Delta t \partial_x H \widehat{u^{n+1}}
241  + \Delta t \partial_y H \widehat{v^{n+1}}  + \Delta t \partial_y H \widehat{v^{n+1}}
242  =  =
243  \eta^{n}  \eta^{n}
244  + \Delta t ( P - E + R )  + \Delta t ( P - E )
245  \label{eq:discrete-time-backward-free-surface}  \label{eq:discrete-time-backward-free-surface}
246  \end{equation}  \end{equation}
247  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 256  As for the rigid-lid pressure method, eq
256  \begin{eqnarray}  \begin{eqnarray}
257  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} \\
258  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} \\
259  \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
260    \partial_x H \widehat{u^{*}}    \partial_x H \widehat{u^{*}}
261  + \partial_y H \widehat{v^{*}}  + \partial_y H \widehat{v^{*}}
262  \\  \\
263    \partial_x g H \partial_x \eta^{n+1}    \partial_x g H \partial_x \eta^{n+1}
264  + \partial_y g H \partial_y \eta^{n+1}  & + & \partial_y g H \partial_y \eta^{n+1}
265  - \frac{\epsilon_{fs} \eta^{n+1}}{\Delta t^2}   - \frac{\epsilon_{fs} \eta^{n+1}}{\Delta t^2}
266  & = &   =
267  - \frac{\eta^*}{\Delta t^2}  - \frac{\eta^*}{\Delta t^2}
268  \label{eq:elliptic-backward-free-surface}  \label{eq:elliptic-backward-free-surface}
269  \\  \\
# Line 279  pressure-method. Line 291  pressure-method.
291    
292  \section{Explicit time-stepping: Adams-Bashforth}  \section{Explicit time-stepping: Adams-Bashforth}
293  \label{sect:adams-bashforth}  \label{sect:adams-bashforth}
294    \begin{rawhtml}
295    <!-- CMIREDIR:adams_bashforth: -->
296    \end{rawhtml}
297    
298  In describing the the pressure method above we deferred describing the  In describing the the pressure method above we deferred describing the
299  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 361  A stability analysis for a relaxation eq
361    
362    
363  \section{Implicit time-stepping: backward method}  \section{Implicit time-stepping: backward method}
364    \begin{rawhtml}
365    <!-- CMIREDIR:implicit_time-stepping_backward: -->
366    \end{rawhtml}
367    
368  Vertical diffusion and viscosity can be treated implicitly in time  Vertical diffusion and viscosity can be treated implicitly in time
369  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 409  implicit and are thus cast as a an expli
409    
410  \section{Synchronous time-stepping: variables co-located in time}  \section{Synchronous time-stepping: variables co-located in time}
411  \label{sect:adams-bashforth-sync}  \label{sect:adams-bashforth-sync}
412    \begin{rawhtml}
413    <!-- CMIREDIR:adams_bashforth_sync: -->
414    \end{rawhtml}
415    
416  \begin{figure}  \begin{figure}
417  \begin{center}  \begin{center}
# Line 422  FORWARD\_STEP \\ Line 443  FORWARD\_STEP \\
443  \>\>\> 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})\\
444  \>\>\> EXTERNAL\_FORCING \` $G_\theta^n = G_\theta^n + {\cal Q}$ \\  \>\>\> EXTERNAL\_FORCING \` $G_\theta^n = G_\theta^n + {\cal Q}$ \\
445  \>\>\> 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}) \\
446  \>\> TIMESTEP\_TRACER \` $\tau^*$ (\ref{eq:tstar-sync}) \\  \>\> TIMESTEP\_TRACER \` $\theta^*$ (\ref{eq:tstar-sync}) \\
447  \>\> IMPLDIFF \` $\tau^{(n+1)}$ (\ref{eq:t-n+1-sync}) \\  \>\> IMPLDIFF \` $\theta^{(n+1)}$ (\ref{eq:t-n+1-sync}) \\
448  \> DYNAMICS \\  \> DYNAMICS \\
449  \>\> CALC\_PHI\_HYD \` $\phi_{hyd}^n$ (\ref{eq:phi-hyd-sync}) \\  \>\> CALC\_PHI\_HYD \` $\phi_{hyd}^n$ (\ref{eq:phi-hyd-sync}) \\
450  \>\> 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 466  FORWARD\_STEP \\
466  Calling tree for the overall synchronous algorithm using  Calling tree for the overall synchronous algorithm using
467  Adams-Bashforth time-stepping.  Adams-Bashforth time-stepping.
468  The place where the model geometry  The place where the model geometry
469  ({\em hFac} factors) is updated is added here but is only relevant  ({\bf hFac} factors) is updated is added here but is only relevant
470  for the non-linear free-surface algorithm.  for the non-linear free-surface algorithm.
471  For completeness, the external forcing,  For completeness, the external forcing,
472  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 529  time-step. The corresponding calling tre
529    
530  \section{Staggered baroclinic time-stepping}  \section{Staggered baroclinic time-stepping}
531  \label{sect:adams-bashforth-staggered}  \label{sect:adams-bashforth-staggered}
532    \begin{rawhtml}
533    <!-- CMIREDIR:adams_bashforth_staggered: -->
534    \end{rawhtml}
535    
536  \begin{figure}  \begin{figure}
537  \begin{center}  \begin{center}
# Line 524  extrapolate tendencies to $n$ (dashed ar Line 548  extrapolate tendencies to $n$ (dashed ar
548  The hydrostatic pressure/geo-potential $\phi_{hyd}$ is evaluated directly  The hydrostatic pressure/geo-potential $\phi_{hyd}$ is evaluated directly
549  at time level $n$ (vertical arrows) and used with the extrapolated tendencies  at time level $n$ (vertical arrows) and used with the extrapolated tendencies
550  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).
551  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
552  then applied to the previous estimation of the the flow field ($*$-variables)  then applied to the previous estimation of the the flow field ($*$-variables)
553  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 $n+1/2$.
554  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 576  thermodynamics solver is delayed from ha
576  allowing the use of the most recent velocities to compute  allowing the use of the most recent velocities to compute
577  the advection terms. Once the thermodynamics fields are  the advection terms. Once the thermodynamics fields are
578  updated, the hydrostatic pressure is computed  updated, the hydrostatic pressure is computed
579  to step frowrad the dynamics  to step forwrad the dynamics.
580  Note that the pressure gradient must also be taken out of the  Note that the pressure gradient must also be taken out of the
581  Adams-Bashforth extrapolation. Also, retaining the integer time-levels,  Adams-Bashforth extrapolation. Also, retaining the integer time-levels,
582  $n$ 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 584  located in time.  Instead, we re-write t
584  \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
585  position in time of variables appropriately:  position in time of variables appropriately:
586  \begin{eqnarray}  \begin{eqnarray}
587    \phi^{n}_{hyd} & = & \int b(\theta^{n},S^{n}) dr
588    \label{eq:phi-hyd-staggered} \\
589  \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} )
590  \label{eq:Gv-n-staggered} \\  \label{eq:Gv-n-staggered} \\
591  \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}
592  \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} \\  
593  \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)
594  \label{eq:vstar-staggered} \\  \label{eq:vstar-staggered} \\
595  \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 614  G_{\theta,S}^{(n+1/2)} & = & (3/2+\epsil
614  The corresponding calling tree is given in  The corresponding calling tree is given in
615  \ref{fig:call-tree-adams-bashforth-staggered}.  \ref{fig:call-tree-adams-bashforth-staggered}.
616  The staggered algorithm is activated with the run-time flag  The staggered algorithm is activated with the run-time flag
617  {\bf staggerTimeStep=.TRUE.} in parameter file {\em data},  {\bf staggerTimeStep}{\em=.TRUE.} in parameter file {\em data},
618  namelist {\em PARM01}.  namelist {\em PARM01}.
619    
620  \begin{figure}  \begin{figure}
# Line 620  FORWARD\_STEP \\ Line 644  FORWARD\_STEP \\
644       (\ref{eq:Gt-n-staggered})\\       (\ref{eq:Gt-n-staggered})\\
645  \>\>\> EXTERNAL\_FORCING \` $G_\theta^n = G_\theta^n + {\cal Q}$ \\  \>\>\> EXTERNAL\_FORCING \` $G_\theta^n = G_\theta^n + {\cal Q}$ \\
646  \>\>\> 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}) \\
647  \>\> TIMESTEP\_TRACER \` $\tau^*$ (\ref{eq:tstar-staggered}) \\  \>\> TIMESTEP\_TRACER \` $\theta^*$ (\ref{eq:tstar-staggered}) \\
648  \>\> IMPLDIFF \` $\tau^{(n+1)}$ (\ref{eq:t-n+1-staggered}) \\  \>\> IMPLDIFF \` $\theta^{(n+1)}$ (\ref{eq:t-n+1-staggered}) \\
649  \> TRACERS\_CORRECTION\_STEP  \\  \> TRACERS\_CORRECTION\_STEP  \\
650  \>\> CYCLE\_TRACER \` $\theta^{n+1}$ \\  \>\> CYCLE\_TRACER \` $\theta^{n+1}$ \\
651  \>\> FILTER \` Shapiro Filter, Zonal Filter (FFT) \\  \>\> FILTER \` Shapiro Filter, Zonal Filter (FFT) \\
# Line 631  FORWARD\_STEP \\ Line 655  FORWARD\_STEP \\
655  Calling tree for the overall staggered algorithm using  Calling tree for the overall staggered algorithm using
656  Adams-Bashforth time-stepping.  Adams-Bashforth time-stepping.
657  The place where the model geometry  The place where the model geometry
658  ({\em hFac} factors) is updated is added here but is only relevant  ({\bf hFac} factors) is updated is added here but is only relevant
659  for the non-linear free-surface algorithm.  for the non-linear free-surface algorithm.
660  }  }
661  \label{fig:call-tree-adams-bashforth-staggered}  \label{fig:call-tree-adams-bashforth-staggered}
# Line 650  time-level variables and terms correspon Line 674  time-level variables and terms correspon
674    
675  \section{Non-hydrostatic formulation}  \section{Non-hydrostatic formulation}
676  \label{sect:non-hydrostatic}  \label{sect:non-hydrostatic}
677    \begin{rawhtml}
678    <!-- CMIREDIR:non-hydrostatic_formulation: -->
679    \end{rawhtml}
680    
681  The non-hydrostatic formulation re-introduces the full vertical  The non-hydrostatic formulation re-introduces the full vertical
682  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 776  $w^{n+1}$.
776    
777    
778  \section{Variants on the Free Surface}  \section{Variants on the Free Surface}
779    \label{sect:free-surface}
780    
781  We now describe the various formulations of the free-surface that  We now describe the various formulations of the free-surface that
782  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 815  $\eta^{n+1}$: {\bf etaN} (\em DYNVARS.h)
815    
816    
817  Once ${\eta}^{n+1}$ has been found, substituting into  Once ${\eta}^{n+1}$ has been found, substituting into
818  \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 $\vec{\bf v}^{n+1}$
819  hydrostatic ($\epsilon_{nh}=0$):  if the model is hydrostatic ($\epsilon_{nh}=0$):
820  $$  $$
821  \vec{\bf v}^{n+1} = \vec{\bf v}^{*}  \vec{\bf v}^{n+1} = \vec{\bf v}^{*}
822  - \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 889  at the same point in the code.
889    
890    
891  \subsection{Crank-Nickelson barotropic time stepping}  \subsection{Crank-Nickelson barotropic time stepping}
892    \label{sect:freesurf-CrankNick}
893    
894  The full implicit time stepping described previously is  The full implicit time stepping described previously is
895  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 904  stable, Crank-Nickelson scheme; $(\beta,
904  corresponds to the forward - backward scheme that conserves energy but is  corresponds to the forward - backward scheme that conserves energy but is
905  only stable for small time steps.\\  only stable for small time steps.\\
906  In the code, $\beta,\gamma$ are defined as parameters, respectively  In the code, $\beta,\gamma$ are defined as parameters, respectively
907  {\it implicSurfPress}, {\it implicDiv2DFlow}. They are read from  {\bf implicSurfPress}, {\bf implicDiv2DFlow}. They are read from
908  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.
909    
910  Equations \ref{eq:ustar-backward-free-surface} --  Equations \ref{eq:ustar-backward-free-surface} --
911  \ref{eq:vn+1-backward-free-surface} are modified as follows:  \ref{eq:vn+1-backward-free-surface} are modified as follows:
912  $$  \begin{eqnarray*}
913  \frac{ \vec{\bf v}^{n+1} }{ \Delta t }  \frac{ \vec{\bf v}^{n+1} }{ \Delta t }
914  + {\bf \nabla}_h b_s [ \beta {\eta}^{n+1} + (1-\beta) {\eta}^{n} ]  + {\bf \nabla}_h b_s [ \beta {\eta}^{n+1} + (1-\beta) {\eta}^{n} ]
915  + \epsilon_{nh} {\bf \nabla}_h {\phi'_{nh}}^{n+1}  + \epsilon_{nh} {\bf \nabla}_h {\phi'_{nh}}^{n+1}
916   = \frac{ \vec{\bf v}^* }{ \Delta t }   = \frac{ \vec{\bf v}^* }{ \Delta t }
917  $$  \end{eqnarray*}
918  $$  \begin{eqnarray}
919  \epsilon_{fs} \frac{ {\eta}^{n+1} - {\eta}^{n} }{ \Delta t}  \epsilon_{fs} \frac{ {\eta}^{n+1} - {\eta}^{n} }{ \Delta t}
920  + {\bf \nabla}_h \cdot \int_{R_{fixed}}^{R_o}  + {\bf \nabla}_h \cdot \int_{R_{fixed}}^{R_o}
921  [ \gamma \vec{\bf v}^{n+1} + (1-\gamma) \vec{\bf v}^{n}] dr  [ \gamma \vec{\bf v}^{n+1} + (1-\gamma) \vec{\bf v}^{n}] dr
922  = \epsilon_{fw} (P-E)  = \epsilon_{fw} (P-E)
923  $$  \label{eq:eta-n+1-CrankNick}
924    \end{eqnarray}
925  where:  where:
926  \begin{eqnarray*}  \begin{eqnarray*}
927  \vec{\bf v}^* & = &  \vec{\bf v}^* & = &
# Line 913  $$ Line 943  $$
943  {\bf \nabla}_h {\eta}^{n+1}  {\bf \nabla}_h {\eta}^{n+1}
944  = {\eta}^*  = {\eta}^*
945  $$  $$
946  and then to compute (correction step):  and then to compute ({\em CORRECTION\_STEP}):
947  $$  $$
948  \vec{\bf v}^{n+1} = \vec{\bf v}^{*}  \vec{\bf v}^{n+1} = \vec{\bf v}^{*}
949  - \beta \Delta t {\bf \nabla}_h b_s {\eta}^{n+1}  - \beta \Delta t {\bf \nabla}_h b_s {\eta}^{n+1}
950  $$  $$
951    
952  The non-hydrostatic part is solved as described previously.  %The non-hydrostatic part is solved as described previously.
953    
954  Note that:  \noindent
955    Notes:
956  \begin{enumerate}  \begin{enumerate}
957    \item The RHS term of equation \ref{eq:eta-n+1-CrankNick}
958    corresponds the contribution of fresh water flux (P-E)
959    to the free-surface variations ($\epsilon_{fw}=1$,
960    {\bf useRealFreshWater}{\em=TRUE} in parameter file {\em data}).
961    In order to remain consistent with the tracer equation, specially in
962    the non-linear free-surface formulation, this term is also
963    affected by the Crank-Nickelson time stepping. The RHS reads:
964    $\epsilon_{fw} ( \gamma (P-E)^{n+1/2} + (1-\gamma) (P-E)^{n-1/2} )$
965  \item The non-hydrostatic part of the code has not yet been  \item The non-hydrostatic part of the code has not yet been
966  updated, so that this option cannot be used with $(\beta,\gamma) \neq (1,1)$.  updated, and therefore cannot be used with $(\beta,\gamma) \neq (1,1)$.
967  \item The stability criteria with Crank-Nickelson time stepping  \item The stability criteria with Crank-Nickelson time stepping
968  for the pure linear gravity wave problem in cartesian coordinates is:  for the pure linear gravity wave problem in cartesian coordinates is:
969  \begin{itemize}  \begin{itemize}
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