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-revision 1.17 by jmc,
Wed Oct 13 18:50:54 2004 UTC
+revision 1.26 by jmc,
Thu Jun 29 01:45:32 2006 UTC
 Line 10 
 describe the spatial discretization. The
  terms are described first, afterwards the schemes that apply to
  passive and dynamically active tracers are described.
+ \input{part2/notation}
  \section{Time-stepping}
+ \begin{rawhtml}
+ <!-- CMIREDIR:time-stepping: -->
+ \end{rawhtml}
  The equations of motion integrated by the model involve four
  prognostic equations for flow, $u$ and $v$, temperature, $\theta$, and
  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
  described in section \ref{sect:nonlinear-freesurface}.
- \section{Pressure method with rigid-lid} \label{sect:pressure-method-rigid-lid}
+ \section{Pressure method with rigid-lid}
+ \label{sect:pressure-method-rigid-lid}
+ \begin{rawhtml}
+ <!-- CMIREDIR:pressure_method_rigid_lid: -->
+ \end{rawhtml}
  \begin{figure}
  \begin{center}
-Line 189 
 The calling tree for these routines is g
+Line 198 
 The calling tree for these routines is g
  Fig.~\ref{fig:call-tree-pressure-method}.
+ %\paragraph{Need to discuss implicit viscosity somewhere:}
- \paragraph{Need to discuss implicit viscosity somewhere:}
+ In general, the horizontal momentum time-stepping can contain some terms
+ that are treated implicitly in time,
+ such as the vertical viscosity when using the backward time-stepping scheme
+ (\varlink{implicitViscosity}{implicitViscosity} {\it =.TRUE.}).
+ The method used to solve those implicit terms is provided in
+ section \ref{sect:implicit-backward-stepping}, and modifies
+ equations \ref{eq:discrete-time-u} and \ref{eq:discrete-time-v} to
+ give:
  \begin{eqnarray}
- \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}
- + 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)}
  \\
- \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}
- + 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)}
  \end{eqnarray}
  \section{Pressure method with implicit linear free-surface}
  \label{sect:pressure-method-linear-backward}
+ \begin{rawhtml}
+ <!-- CMIREDIR:pressure_method_linear_backward: -->
+ \end{rawhtml}
  The rigid-lid approximation filters out external gravity waves
  subsequently modifying the dispersion relation of barotropic Rossby
-Line 214 
 The rigid-lid approximation can be easil
+Line 232 
 The rigid-lid approximation can be easil
  of the free-surface equation which can be written:
  \begin{equation}
  \partial_t \eta + \partial_x H \widehat{u} + \partial_y H \widehat{v} = P-E+R
- \label{eq:linear-free-surface=P-E+R}
+ \label{eq:linear-free-surface=P-E}
  \end{equation}
  which differs from the depth integrated continuity equation with
  rigid-lid (\ref{eq:rigid-lid-continuity}) by the time-dependent term
-Line 222 
 and fresh-water source term.
+Line 240 
 and fresh-water source term.
  Equation \ref{eq:discrete-time-cont-rigid-lid} in the rigid-lid
  pressure method is then replaced by the time discretization of
- \ref{eq:linear-free-surface=P-E+R} which is:
+ \ref{eq:linear-free-surface=P-E} which is:
  \begin{equation}
  \eta^{n+1}
  + \Delta t \partial_x H \widehat{u^{n+1}}
  + \Delta t \partial_y H \widehat{v^{n+1}}
  =
  \eta^{n}
- + \Delta t ( P - E + R )
+ + \Delta t ( P - E )
  \label{eq:discrete-time-backward-free-surface}
  \end{equation}
  where the use of flow at time level $n+1$ makes the method implicit
- and backward in time. The is the preferred scheme since it still
+ and backward in time. This is the preferred scheme since it still
  filters the fast unresolved wave motions by damping them. A centered
- scheme, such as Crank-Nicholson, would alias the energy of the fast
+ scheme, such as Crank-Nicholson (see section \ref{sect:freesurf-CrankNick}),
- modes onto slower modes of motion.
+ would alias the energy of the fast modes onto slower modes of motion.
  As for the rigid-lid pressure method, equations
  \ref{eq:discrete-time-u}, \ref{eq:discrete-time-v} and
-Line 244 
 As for the rigid-lid pressure method, eq
+Line 262 
 As for the rigid-lid pressure method, eq
  \begin{eqnarray}
  u^{*} & = & u^{n} + \Delta t G_u^{(n+1/2)} \label{eq:ustar-backward-free-surface} \\
  v^{*} & = & v^{n} + \Delta t G_v^{(n+1/2)} \label{eq:vstar-backward-free-surface} \\
- \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
    \partial_x H \widehat{u^{*}}
  + \partial_y H \widehat{v^{*}}
  \\
    \partial_x g H \partial_x \eta^{n+1}
- + \partial_y g H \partial_y \eta^{n+1}
+ & + & \partial_y g H \partial_y \eta^{n+1}
- - \frac{\epsilon_{fs} \eta^{n+1}}{\Delta t^2}
+  - \frac{\epsilon_{fs} \eta^{n+1}}{\Delta t^2}
- & = &
+  =
  - \frac{\eta^*}{\Delta t^2}
  \label{eq:elliptic-backward-free-surface}
  \\
-Line 267 
 recovered. However, the implicit treatme
+Line 285 
 recovered. However, the implicit treatme
  the flow to be divergent and for the surface pressure/elevation to
  respond on a finite time-scale (as opposed to instantly). To recover
  the rigid-lid formulation, we introduced a switch-like parameter,
- $\epsilon_{fs}$, which selects between the free-surface and rigid-lid;
+ $\epsilon_{fs}$ (\varlink{freesurfFac}{freesurfFac}),
+ which selects between the free-surface and rigid-lid;
  $\epsilon_{fs}=1$ allows the free-surface to evolve; $\epsilon_{fs}=0$
  imposes the rigid-lid. The evolution in time and location of variables
  is exactly as it was for the rigid-lid model so that
-Line 279 
 pressure-method.
+Line 298 
 pressure-method.
  \section{Explicit time-stepping: Adams-Bashforth}
  \label{sect:adams-bashforth}
+ \begin{rawhtml}
+ <!-- CMIREDIR:adams_bashforth: -->
+ \end{rawhtml}
  In describing the the pressure method above we deferred describing the
  time discretization of the explicit terms. We have historically used
-Line 302 
 FORWARD\_STEP \\
+Line 324 
 FORWARD\_STEP \\
  \end{tabbing} \end{minipage} } \end{center}
  \caption{
  Calling tree for the Adams-Bashforth time-stepping of temperature with
- implicit diffusion.}
+ implicit diffusion.
+   (\filelink{THERMODYNAMICS}{model-src-thermodynamics.F},
+    \filelink{ADAMS\_BASHFORTH2}{model-src-adams_bashforth2.F})}
  \label{fig:call-tree-adams-bashforth}
  \end{figure}
-Line 333 
 simpler to include these terms and this
+Line 357 
 simpler to include these terms and this
  and forcing evolves smoothly. Problems can, and do, arise when forcing
  or motions are high frequency and this corresponds to a reduced
  stability compared to a simple forward time-stepping of such terms.
- The model offers the possibility to leave the forcing term outside the
+ The model offers the possibility to leave the tracer and momentum
- Adams-Bashforth extrapolation, by turning off the logical flag
+ forcing terms and the dissipation terms outside the
- {\bf forcing\_In\_AB } (parameter file {\em data}, namelist {\em PARM01},
+ Adams-Bashforth extrapolation, by turning off the logical flags
- default value = True).
+ \varlink{forcing\_In\_AB}{forcing_In_AB}
+ (parameter file {\em data}, namelist {\em PARM01}, default value = True).
+ (\varlink{tracForcingOutAB}{tracForcingOutAB}, default=0,
+ \varlink{momForcingOutAB}{momForcingOutAB}, default=0)
+ and \varlink{momDissip\_In\_AB}{momDissip_In_AB}
+ (parameter file {\em data}, namelist {\em PARM01}, default value = True).
+ respectively.
  A stability analysis for an oscillation equation should be given at this point.
  \marginpar{AJA needs to find his notes on this...}
-Line 344 
 A stability analysis for an oscillation
+Line 374 
 A stability analysis for an oscillation
  A stability analysis for a relaxation equation should be given at this point.
  \marginpar{...and for this too.}
+ \begin{figure}
+ \begin{center}
+ \resizebox{5.5in}{!}{\includegraphics{part2/oscil+damp_AB2.eps}}
+ \end{center}
+ \caption{
+ Oscillatory and damping response of
+ quasi-second order Adams-Bashforth scheme for different values
+ of the $\epsilon_{AB}$ parameter (0., 0.1, 0.25, from top to bottom)
+ The analytical solution (in black), the physical mode (in blue)
+ and the numerical mode (in red) are represented with a CFL
+ step of 0.1.
+ The left column represents the oscillatory response
+ on the complex plane for CFL ranging from 0.1 up to 0.9.
+ The right column represents the damping response amplitude
+ (y-axis) function of the CFL (x-axis).
+ }
+ \label{fig:adams-bashforth-respons}
+ \end{figure}
  \section{Implicit time-stepping: backward method}
+ \label{sect:implicit-backward-stepping}
+ \begin{rawhtml}
+ <!-- CMIREDIR:implicit_time-stepping_backward: -->
+ \end{rawhtml}
  Vertical diffusion and viscosity can be treated implicitly in time
  using the backward method which is an intrinsic scheme.
-Line 391 
 implicit and are thus cast as a an expli
+Line 445 
 implicit and are thus cast as a an expli
  \section{Synchronous time-stepping: variables co-located in time}
  \label{sect:adams-bashforth-sync}
+ \begin{rawhtml}
+ <!-- CMIREDIR:adams_bashforth_sync: -->
+ \end{rawhtml}
  \begin{figure}
  \begin{center}
-Line 422 
 FORWARD\_STEP \\
+Line 479 
 FORWARD\_STEP \\
  \>\>\> GAD\_CALC\_RHS \` $G_\theta^n = G_\theta( u, \theta^n )$ (\ref{eq:Gt-n-sync})\\
  \>\>\> EXTERNAL\_FORCING \` $G_\theta^n = G_\theta^n + {\cal Q}$ \\
  \>\>\> ADAMS\_BASHFORTH2 \` $G_\theta^{(n+1/2)}$ (\ref{eq:Gt-n+5-sync}) \\
- \>\> TIMESTEP\_TRACER \` $\tau^*$ (\ref{eq:tstar-sync}) \\
+ \>\> TIMESTEP\_TRACER \` $\theta^*$ (\ref{eq:tstar-sync}) \\
- \>\> IMPLDIFF \` $\tau^{(n+1)}$ (\ref{eq:t-n+1-sync}) \\
+ \>\> IMPLDIFF \` $\theta^{(n+1)}$ (\ref{eq:t-n+1-sync}) \\
  \> DYNAMICS \\
  \>\> CALC\_PHI\_HYD \` $\phi_{hyd}^n$ (\ref{eq:phi-hyd-sync}) \\
  \>\> MOM\_FLUXFORM or MOM\_VECINV \` $G_{\vec{\bf v}}^n$ (\ref{eq:Gv-n-sync})\\
-Line 445 
 FORWARD\_STEP \\
+Line 502 
 FORWARD\_STEP \\
  Calling tree for the overall synchronous algorithm using
  Adams-Bashforth time-stepping.
  The place where the model geometry
- ({\em hFac} factors) is updated is added here but is only relevant
+ ({\bf hFac} factors) is updated is added here but is only relevant
  for the non-linear free-surface algorithm.
  For completeness, the external forcing,
  ocean and atmospheric physics have been added, although they are mainly
-Line 508 
 time-step. The corresponding calling tre
+Line 565 
 time-step. The corresponding calling tre
  \section{Staggered baroclinic time-stepping}
  \label{sect:adams-bashforth-staggered}
+ \begin{rawhtml}
+ <!-- CMIREDIR:adams_bashforth_staggered: -->
+ \end{rawhtml}
  \begin{figure}
  \begin{center}
-Line 516 
 time-step. The corresponding calling tre
+Line 576 
 time-step. The corresponding calling tre
  \caption{
  A schematic of the explicit Adams-Bashforth and implicit time-stepping
  phases of the algorithm but with staggering in time of thermodynamic
- variables with the flow. Explicit thermodynamics tendencies are
+ variables with the flow.
- evaluated at time level $n$ as a function of the thermodynamics
+ Explicit momentum tendencies are evaluated at time level $n-1/2$ as a
- state at that time level $n$ and flow at time $n+1/2$ (dotted arrow). The
+ function of the flow field at that time level $n-1/2$.
- explicit tendency from the previous time level, $n-1$, is used to
+ The explicit tendency from the previous time level, $n-3/2$, is used to
- extrapolate tendencies to $n+1/2$ (dashed arrow). This extrapolated
+ extrapolate tendencies to $n$ (dashed arrow).
- tendency allows thermo-dynamics variables to be stably integrated
+ The hydrostatic pressure/geo-potential $\phi_{hyd}$ is evaluated directly
- forward-in-time to render an estimate ($*$-variables) at the $n+1$
+ at time level $n$ (vertical arrows) and used with the extrapolated tendencies
- time level (solid arc-arrow). The implicit-in-time operator ${\cal
+ to step forward the flow variables from $n-1/2$ to $n+1/2$ (solid arc-arrow).
- L_{\theta,S}}$ is solved to yield the thermodynamic variables at time
+ The implicit-in-time operator ${\cal L}_{\bf u,v}$ (vertical arrows) is
- level $n+1$. These are then used to calculate the hydrostatic
+ then applied to the previous estimation of the the flow field ($*$-variables)
- pressure/geo-potential, $\phi_{hyd}$ (vertical arrows). The
+ and yields to the two velocity components $u,v$ at time level $n+1/2$.
- hydrostatic pressure gradient is evaluated directly at time level
+ These are then used to calculate the advection term (dashed arc-arrow)
- $n+1$ in stepping forward the flow variables from $n+1/2$ to $n+3/2$
+ of the thermo-dynamics tendencies at time step $n$.
- (solid arc-arrow). }
+ The extrapolated thermodynamics tendency, from time level $n-1$ and $n$
+ to $n+1/2$, allows thermodynamic variables to be stably integrated
+ forward-in-time (solid arc-arrow) up to time level $n+1$.
+ }
  \label{fig:adams-bashforth-staggered}
  \end{figure}
-Line 549 
 thermodynamics solver is delayed from ha
+Line 612 
 thermodynamics solver is delayed from ha
  allowing the use of the most recent velocities to compute
  the advection terms. Once the thermodynamics fields are
  updated, the hydrostatic pressure is computed
- to step frowrad the dynamics
+ to step forwrad the dynamics.
  Note that the pressure gradient must also be taken out of the
  Adams-Bashforth extrapolation. Also, retaining the integer time-levels,
  $n$ and $n+1$, does not give a user the sense of where variables are
-Line 557 
 located in time.  Instead, we re-write t
+Line 620 
 located in time.  Instead, we re-write t
  \ref{eq:Gt-n-sync} to \ref{eq:v-n+1-sync}, annotating the
  position in time of variables appropriately:
  \begin{eqnarray}
+ \phi^{n}_{hyd} & = & \int b(\theta^{n},S^{n}) dr
+ \label{eq:phi-hyd-staggered} \\
  \vec{\bf G}_{\vec{\bf v}}^{n-1/2} & = & \vec{\bf G}_{\vec{\bf v}} ( \vec{\bf v}^{n-1/2} )
  \label{eq:Gv-n-staggered} \\
  \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}
  \label{eq:Gv-n+5-staggered} \\
- \phi^{n}_{hyd} & = & \int b(\theta^{n},S^{n}) dr
- \label{eq:phi-hyd-staggered} \\
  \vec{\bf v}^{*} & = & \vec{\bf v}^{n-1/2} + \Delta t \left( \vec{\bf G}_{\vec{\bf v}}^{(n)} - \nabla \phi_{hyd}^{n} \right)
  \label{eq:vstar-staggered} \\
  \vec{\bf v}^{**} & = & {\cal L}_{\vec{\bf v}}^{-1} ( \vec{\bf v}^* )
-Line 582 
 G_{\theta,S}^{(n+1/2)} & = & (3/2+\epsil
+Line 645 
 G_{\theta,S}^{(n+1/2)} & = & (3/2+\epsil
  (\theta^*,S^*) & = & (\theta^{n},S^{n}) + \Delta t G_{\theta,S}^{(n+1/2)}
  \label{eq:tstar-staggered} \\
  (\theta^{n+1},S^{n+1}) & = & {\cal L}^{-1}_{\theta,S} (\theta^*,S^*)
- \label{eq:t-n+1-staggered} \\
+ \label{eq:t-n+1-staggered}
  \end{eqnarray}
  The corresponding calling tree is given in
  \ref{fig:call-tree-adams-bashforth-staggered}.
  The staggered algorithm is activated with the run-time flag
- {\bf staggerTimeStep=.TRUE.} in parameter file {\em data},
+ {\bf staggerTimeStep}{\em=.TRUE.} in parameter file {\em data},
  namelist {\em PARM01}.
  \begin{figure}
-Line 617 
 FORWARD\_STEP \\
+Line 680 
 FORWARD\_STEP \\
       (\ref{eq:Gt-n-staggered})\\
  \>\>\> EXTERNAL\_FORCING \` $G_\theta^n = G_\theta^n + {\cal Q}$ \\
  \>\>\> ADAMS\_BASHFORTH2 \` $G_\theta^{(n+1/2)}$ (\ref{eq:Gt-n+5-staggered}) \\
- \>\> TIMESTEP\_TRACER \` $\tau^*$ (\ref{eq:tstar-staggered}) \\
+ \>\> TIMESTEP\_TRACER \` $\theta^*$ (\ref{eq:tstar-staggered}) \\
- \>\> IMPLDIFF \` $\tau^{(n+1)}$ (\ref{eq:t-n+1-staggered}) \\
+ \>\> IMPLDIFF \` $\theta^{(n+1)}$ (\ref{eq:t-n+1-staggered}) \\
  \> TRACERS\_CORRECTION\_STEP  \\
  \>\> CYCLE\_TRACER \` $\theta^{n+1}$ \\
  \>\> FILTER \` Shapiro Filter, Zonal Filter (FFT) \\
-Line 628 
 FORWARD\_STEP \\
+Line 691 
 FORWARD\_STEP \\
  Calling tree for the overall staggered algorithm using
  Adams-Bashforth time-stepping.
  The place where the model geometry
- ({\em hFac} factors) is updated is added here but is only relevant
+ ({\bf hFac} factors) is updated is added here but is only relevant
  for the non-linear free-surface algorithm.
  }
  \label{fig:call-tree-adams-bashforth-staggered}
-Line 636 
 for the non-linear free-surface algorith
+Line 699 
 for the non-linear free-surface algorith
  The only difficulty with this approach is apparent in equation
  \ref{eq:Gt-n-staggered} and illustrated by the dotted arrow
- connecting $u,v^n$ with $G_\theta^{n-1/2}$. The flow used to advect
+ connecting $u,v^{n+1/2}$ with $G_\theta^{n}$. The flow used to advect
  tracers around is not naturally located in time. This could be avoided
  by applying the Adams-Bashforth extrapolation to the tracer field
  itself and advecting that around but this approach is not yet
-Line 647 
 time-level variables and terms correspon
+Line 710 
 time-level variables and terms correspon
  \section{Non-hydrostatic formulation}
  \label{sect:non-hydrostatic}
+ \begin{rawhtml}
+ <!-- CMIREDIR:non-hydrostatic_formulation: -->
+ \end{rawhtml}
  The non-hydrostatic formulation re-introduces the full vertical
  momentum equation and requires the solution of a 3-D elliptic
-Line 662 
 The momentum equations are discretized i
+Line 728 
 The momentum equations are discretized i
  \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} + G_v^{(n+1/2)} \label{eq:discrete-time-v-nh} \\
  \frac{1}{\Delta t} w^{n+1} + \partial_r \phi_{nh}^{n+1}
- & = & \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}
  \end{eqnarray}
  which must satisfy the discrete-in-time depth integrated continuity,
  equation~\ref{eq:discrete-time-backward-free-surface} and the local continuity equation
-Line 746 
 $w^{n+1}$.
+Line 812 
 $w^{n+1}$.
  \section{Variants on the Free Surface}
+ \label{sect:free-surface}
  We now describe the various formulations of the free-surface that
  include non-linear forms, implicit in time using Crank-Nicholson,
-Line 765 
 where
+Line 832 
 where
  \begin{eqnarray}
  {\eta}^* = \epsilon_{fs} \: {\eta}^{n} -
  \Delta t {\bf \nabla}_h \cdot \int_{R_{fixed}}^{R_o} \vec{\bf v}^* dr
- \: + \: \epsilon_{fw} \Delta_t (P-E)^{n}
+ \: + \: \epsilon_{fw} \Delta t (P-E)^{n}
  \label{eq-solve2D_rhs}
  \end{eqnarray}
  \fbox{ \begin{minipage}{4.75in}
  {\em S/R SOLVE\_FOR\_PRESSURE} ({\em solve\_for\_pressure.F})
- $u^*$: {\bf GuNm1} ({\em DYNVARS.h})
+ $u^*$: {\bf gU} ({\em DYNVARS.h})
- $v^*$: {\bf GvNm1} ({\em DYNVARS.h})
+ $v^*$: {\bf gV} ({\em DYNVARS.h})
  $\eta^*$: {\bf cg2d\_b} (\em SOLVE\_FOR\_PRESSURE.h)
-Line 784 
 $\eta^{n+1}$: {\bf etaN} (\em DYNVARS.h)
+Line 851 
 $\eta^{n+1}$: {\bf etaN} (\em DYNVARS.h)
  Once ${\eta}^{n+1}$ has been found, substituting into
- \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 $\vec{\bf v}^{n+1}$
- hydrostatic ($\epsilon_{nh}=0$):
+ if the model is hydrostatic ($\epsilon_{nh}=0$):
  $$
  \vec{\bf v}^{n+1} = \vec{\bf v}^{*}
  - \Delta t {\bf \nabla}_h b_s {\eta}^{n+1}
-Line 826 
 without any consequence on the solution.
+Line 893 
 without any consequence on the solution.
  $\eta^{n+1}$: {\bf etaN} (\em DYNVARS.h)
- $\phi_{nh}^{n+1}$: {\bf phi\_nh} (\em DYNVARS.h)
+ $\phi_{nh}^{n+1}$: {\bf phi\_nh} (\em NH\_VARS.h)
- $u^*$: {\bf GuNm1} ({\em DYNVARS.h})
+ $u^*$: {\bf gU} ({\em DYNVARS.h})
- $v^*$: {\bf GvNm1} ({\em DYNVARS.h})
+ $v^*$: {\bf gV} ({\em DYNVARS.h})
  $u^{n+1}$: {\bf uVel} ({\em DYNVARS.h})
-Line 858 
 at the same point in the code.
+Line 925 
 at the same point in the code.
  \subsection{Crank-Nickelson barotropic time stepping}
+ \label{sect:freesurf-CrankNick}
  The full implicit time stepping described previously is
  unconditionally stable but damps the fast gravity waves, resulting in
-Line 872 
 stable, Crank-Nickelson scheme; $(\beta,
+Line 940 
 stable, Crank-Nickelson scheme; $(\beta,
  corresponds to the forward - backward scheme that conserves energy but is
  only stable for small time steps.\\
  In the code, $\beta,\gamma$ are defined as parameters, respectively
- {\it implicSurfPress}, {\it implicDiv2DFlow}. They are read from
+ {\bf implicSurfPress}, {\bf implicDiv2DFlow}. They are read from
- the main data file "{\it data}" and are set by default to 1,1.
+ the main parameter file "{\em data}" and are set by default to 1,1.
  Equations \ref{eq:ustar-backward-free-surface} --
  \ref{eq:vn+1-backward-free-surface} are modified as follows:
- $$
+ \begin{eqnarray*}
  \frac{ \vec{\bf v}^{n+1} }{ \Delta t }
  + {\bf \nabla}_h b_s [ \beta {\eta}^{n+1} + (1-\beta) {\eta}^{n} ]
  + \epsilon_{nh} {\bf \nabla}_h {\phi'_{nh}}^{n+1}
   = \frac{ \vec{\bf v}^* }{ \Delta t }
- $$
+ \end{eqnarray*}
- $$
+ \begin{eqnarray}
  \epsilon_{fs} \frac{ {\eta}^{n+1} - {\eta}^{n} }{ \Delta t}
  + {\bf \nabla}_h \cdot \int_{R_{fixed}}^{R_o}
  [ \gamma \vec{\bf v}^{n+1} + (1-\gamma) \vec{\bf v}^{n}] dr
  = \epsilon_{fw} (P-E)
- $$
+ \label{eq:eta-n+1-CrankNick}
+ \end{eqnarray}
  where:
  \begin{eqnarray*}
  \vec{\bf v}^* & = &
-Line 910 
 $$
+Line 979 
 $$
  {\bf \nabla}_h {\eta}^{n+1}
  = {\eta}^*
  $$
- and then to compute (correction step):
+ and then to compute ({\em CORRECTION\_STEP}):
  $$
  \vec{\bf v}^{n+1} = \vec{\bf v}^{*}
  - \beta \Delta t {\bf \nabla}_h b_s {\eta}^{n+1}
  $$
- The non-hydrostatic part is solved as described previously.
+ %The non-hydrostatic part is solved as described previously.
- Note that:
+ \noindent
+ Notes:
  \begin{enumerate}
+ \item The RHS term of equation \ref{eq:eta-n+1-CrankNick}
+ corresponds the contribution of fresh water flux (P-E)
+ to the free-surface variations ($\epsilon_{fw}=1$,
+ {\bf useRealFreshWater}{\em=TRUE} in parameter file {\em data}).
+ In order to remain consistent with the tracer equation, specially in
+ the non-linear free-surface formulation, this term is also
+ affected by the Crank-Nickelson time stepping. The RHS reads:
+ $\epsilon_{fw} ( \gamma (P-E)^{n+1/2} + (1-\gamma) (P-E)^{n-1/2} )$
  \item The non-hydrostatic part of the code has not yet been
- updated, so that this option cannot be used with $(\beta,\gamma) \neq (1,1)$.
+ updated, and therefore cannot be used with $(\beta,\gamma) \neq (1,1)$.
  \item The stability criteria with Crank-Nickelson time stepping
  for the pure linear gravity wave problem in cartesian coordinates is:
  \begin{itemize}

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