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-revision 1.8 by jmc,
Wed Oct 13 18:56:52 2004 UTC
+revision 1.16 by jmc,
Thu Apr 18 03:36:16 2013 UTC
 Line 3
- \subsection{Non-linear free surface}
+ \subsection{Non-linear free-surface}
- \label{sect:nonlinear-freesurface}
+ \label{sec:nonlinear-freesurface}
  Recently, new options have been added to the model
  that concern the free surface formulation.
- \subsubsection{Non-uniform linear-relation for the surface potential}
+ \subsubsection{pressure/geo-potential and free surface}
+ \label{sec:phi-freesurface}
- The linear relation between surface pressure/geo-potential
+ For the atmosphere, since $\phi = \phi_{topo} - \int^p_{p_s} \alpha dp$,
- ($\Phi_{surf}$) and surface displacement ($\eta$) could be considered
+ subtracting the reference state defined in section
- to be a constant ($b_s=$ constant)
+ \ref{sec:hpe-p-geo-potential-split}~:\\
- \marginpar{add a reference to part.1 here}
+ $$
- but is in fact dependent on the position ($x,y,r$)
+ \phi_o = \phi_{topo} - \int^p_{p_o} \alpha_o dp
- since we linearize:
+ \hspace{5mm}\mathrm{with}\hspace{3mm} \phi_o(p_o)=\phi_{topo}
- $$\Phi_{surf}=\int_{R_o}^{R_o+\eta} b dr \simeq b_s \eta
+ $$
- ~\mathrm{with}~ b_s = b(\theta,S,r)_{r=R_o}
+ we get:
- \simeq b_s(\theta_{ref}(R_o),S_{ref}(R_o),R_o)$$
+ $$
- Note that, for convenience, the effect on $b_s$ of the local surface
+ \phi' = \phi - \phi_o = \int^{p_s}_p \alpha dp - \int^{p_o}_p \alpha_o dp
- $\theta,S$ are not considered here, but are incorporated in to
+ $$
- $\Phi'_{hyd}$.
+ For the ocean, the reference state is simpler since $\rho_c$ does not dependent
+ on $z$ ($b_o=g$) and the surface reference position is uniformly $z=0$ ($R_o=0$),
- For the ocean, $b_s = g \rho_{surf} / \rho_o \simeq g$ is a very good
+ and the same subtraction leads to a similar relation.
- approximation since the relative difference in surface density are
+ For both fluid, using the isomorphic notations, we can write:
- usually small and only due to local $\theta,S$ gradients (because the
+ $$
- upper surface, $R_o = 0$, is essentially flat). Therefore, they can
+ \phi' = \int^{r_{surf}}_r b~ dr - \int^{R_o}_r b_o dr
- easily be incorporated in $\Phi'_{hyd}$.
+ $$
+ and re-write as:
+ \begin{equation}
+ \phi' = \int^{r_{surf}}_{R_o} b~ dr + \int^{R_o}_r (b - b_o) dr
+ \label{eq:split-phi-Ro}
+ \end{equation}
+ or:
+ \begin{equation}
+ \phi' = \int^{r_{surf}}_{R_o} b_o dr + \int^{r_{surf}}_r (b - b_o) dr
+ \label{eq:split-phi-bo}
+ \end{equation}
+ In section \ref{sec:finding_the_pressure_field}, following eq.\ref{eq:split-phi-Ro},
+ the pressure/geo-potential $\phi'$ has been separated into surface ($\phi_s$),
+ and hydrostatic anomaly ($\phi'_{hyd}$).
+ In this section, the split between $\phi_s$ and $\phi'_{hyd}$ is
+ made according to equation \ref{eq:split-phi-bo}. This slightly
+ different definition reflects the actual implementation in the code
+ and is valid for both linear and non-linear
+ free-surface formulation, in both r-coordinate and r*-coordinate.
+ Because the linear free-surface approximation ignore the tracer content
+ of the fluid parcel between $R_o$ and $r_{surf}=R_o+\eta$,
+ for consistency reasons, this part is also neglected in $\phi'_{hyd}$~:
+ $$
+ \phi'_{hyd} = \int^{r_{surf}}_r (b - b_o) dr \simeq \int^{R_o}_r (b - b_o) dr
+ $$
+ Note that in this case, the two definitions of $\phi_s$ and $\phi'_{hyd}$
+ from equation \ref{eq:split-phi-Ro} and \ref{eq:split-phi-bo} converge toward
+ the same (approximated) expressions: $\phi_s = \int^{r_{surf}}_{R_o} b_o dr$
+ and $\phi'_{hyd}=\int^{R_o}_r b' dr$.\\
+ On the contrary, the unapproximated formulation ("non-linear free-surface",
+ see the next section) retains the full expression:
+ $\phi'_{hyd} = \int^{r_{surf}}_r (b - b_o) dr $~.
+ This is obtained by selecting {\bf nonlinFreeSurf}=4 in parameter
+ file {\em data}.\\
+ Regarding the surface potential:
+ $$\phi_s = \int_{R_o}^{R_o+\eta} b_o dr = b_s \eta
+ \hspace{5mm}\mathrm{with}\hspace{5mm}
+ b_s = \frac{1}{\eta} \int_{R_o}^{R_o+\eta} b_o dr $$
+ $b_s \simeq b_o(R_o)$ is an excellent approximation (better than
+ the usual numerical truncation, since generally $|\eta|$ is smaller
+ than the vertical grid increment).
+ For the ocean, $\phi_s = g \eta$ and $b_s = g$ is uniform.
  For the atmosphere, however, because of topographic effects, the
- reference surface pressure $R_o$ has large spatial variations that
+ reference surface pressure $R_o=p_o$ has large spatial variations that
  are responsible for significant $b_s$ variations (from 0.8 to 1.2
- $[m^3/kg]$). For this reason, we use a non-uniform linear coefficient
+ $[m^3/kg]$). For this reason, when {\bf uniformLin\_PhiSurf} {\em=.FALSE.}
- $b_s$.
+ (parameter file {\em data}, namelist {\em PARAM01})
+ a non-uniform linear coefficient $b_s$ is used and computed
- In practice, in an oceanic configuration or when the default value
- (TRUE) of the parameter {\bf uniformLin\_PhiSurf} is used, then $b_s$
- is simply set to $g$ for the ocean and $1.$ for the atmosphere.
- Turning {\bf uniformLin\_PhiSurf} to "FALSE", tells the code to
- evaluate $b_s$ from the reference vertical profile $\theta_{ref}$
  ({\it S/R INI\_LINEAR\_PHISURF}) according to the reference surface
- pressure $P_o$ ($=R_o$): $b_s = c_p \kappa (P_o / Pc)^{(\kappa - 1)}
+ pressure $p_o$:
- \theta_{ref}(P_o)$
+ $b_s = b_o(R_o) = c_p \kappa (p_o / P^o_{SL})^{(\kappa - 1)} \theta_{ref}(p_o)$.
+ with $P^o_{SL}$ the mean sea-level pressure.
  \subsubsection{Free surface effect on column total thickness
- (Non-linear free surface)}
+ (Non-linear free-surface)}
  The total thickness of the fluid column is $r_{surf} - R_{fixed} =
  \eta + R_o - R_{fixed}$. In most applications, the free surface
  displacements are small compared to the total thickness
- $\eta << H_o = R_o - R_{fixed}$.
+ $\eta \ll H_o = R_o - R_{fixed}$.
  In the previous sections and in older version of the model,
  the linearized free-surface approximation was made, assuming
- $r_{surf} - R_{fixed} \simeq H_o$ when the horizontal transport is
+ $r_{surf} - R_{fixed} \simeq H_o$ when computing horizontal transports,
- computed, either in the continuity equation or in tracer and momentum
+ either in the continuity equation or in tracer and momentum
  advection terms.
- This approximation is dropped when using the non-linear free surface
+ This approximation is dropped when using the non-linear free-surface
  formulation and the total thickness, including the time varying part
- $\eta$, is consisdered when computing horizontal transport.
+ $\eta$, is considered when computing horizontal transports.
  Implications for the barotropic part are presented hereafter.
- In sections \ref{sect:freesurf-tracer-advection} and
+ In section \ref{sec:freesurf-tracer-advection} consequences for
- \ref{sect:freesurf-momentum-advection}, consequences for tracer
+ tracer conservation is briefly discussed (more details can be
- and momentum are brifly discussed. a more detailed description
+ found in \cite{campin:02})~; the general time-stepping is presented
- is available in \cite{campin:02}.
+ in section \ref{sec:nonlin-freesurf-timestepping} with some
+ limitations regarding the vertical resolution in section
+ \ref{sec:nonlin-freesurf-dz_surf}.
- In the non-linear formulation, the continuous form of the model equations
+ In the non-linear formulation, the continuous form of the model
- remains unchanged, except for the 2D continuity equation
+ equations remains unchanged, except for the 2D continuity equation
  (\ref{eq:discrete-time-backward-free-surface}) which is now
  integrated from $R_{fixed}(x,y)$ up to $r_{surf}=R_o+\eta$ :
-Line 100 
 where
+Line 142 
 where
  {\eta}^* = \epsilon_{fs} \: {\eta}^{n} -
  \Delta t {\bf \nabla}_h \cdot \int_{R_{fixed}}^{R_o+\eta^n} \vec{\bf v}^* dr
  \: + \: \epsilon_{fw} \Delta_t (P-E)^{n}
  \end{eqnarray*}
  This method requires us to update the solver matrix at each time step.
  Alternatively, the non-linear contribution can be evaluated fully
-Line 112 
 explicitly:
+Line 154 
 explicitly:
  = {\eta}^*
  +{\bf \nabla}_h \cdot \Delta t^2 (\eta^{n})
  {\bf \nabla}_h b_s {\eta}^{n}
  \end{eqnarray*}
  This formulation allows one to keep the initial solver matrix
  unchanged though throughout the integration, since the non-linear free
  surface only affects the RHS.
-Line 122 
 column thickness appears only in the int
+Line 164 
 column thickness appears only in the int
  (\ref{eq-solve2D_rhs}) but not directly in the equation
  (\ref{eq-solve2D}).
- Those different options (see tab.?? for the one still available)
+ Those different options (see Table \ref{tab:nonLinFreeSurf_flags})
- have been tested and show litle differences. However, we recommand
+ have been tested and show little differences. However, we recommend
  the use of the most precise method (the 1rst one) since the
- computation cost involved in the solver matrix update are negligeable.
+ computation cost involved in the solver matrix update is negligible.
- \begin{center}
+ \begin{table}[htb]
+ %\begin{center}
+ \centering
   \begin{tabular}[htb]{|l|c|l|}
     \hline
     parameter & value & description \\
     \hline
                     & -1 & linear free-surface, restart from a pickup file \\
                     &    & produced with \#undef EXACT\_CONSERV code\\
     \cline{2-3}
-                    & 0 & Linear free-Surface \\
+                    & 0 & Linear free-surface \\
     \cline{2-3}
      nonlinFreeSurf & 4 & Non-linear free-surface \\
     \cline{2-3}
-Line 154 
 computation cost involved in the solver
+Line 198 
 computation cost involved in the solver
                    &   & slope of the coordinate in $\nabla \Phi$ \\
     \hline
    \end{tabular}
- \end{center}
+   \caption{Non-linear free-surface flags}
+   \label{tab:nonLinFreeSurf_flags}
+ %\end{center}
+ \end{table}
- \subsubsection{Free surface effect on the surface level thickness
+ \subsubsection{Tracer conservation with non-linear free-surface}
- (Non-linear free surface): Tracer advection}
+ \label{sec:freesurf-tracer-advection}
- \label{sect:freesurf-tracer-advection}
  To ensure global tracer conservation (i.e., the total amount) as well
  as local conservation, the change in the surface level thickness must
-Line 167 
 be consistent with the way the continuit
+Line 213 
 be consistent with the way the continuit
  in the barotropic part (to find $\eta$) and baroclinic part (to find
  $w = \dot{r}$).
- To illustrate this, consider the shallow water model, with uniform
+ To illustrate this, consider the shallow water model, with a source
- Cartesian horizontal grid:
+ of fresh water (P):
  $$
- \partial_t h + \nabla \cdot h \vec{\bf v} = 0
+ \partial_t h + \nabla \cdot h \vec{\bf v} = P
  $$
  where $h$ is the total thickness of the water column.
  To conserve the tracer $\theta$ we have to discretize:
  $$
- \partial_t (h \theta) + \nabla \cdot ( h \theta \vec{\bf v})= 0
+ \partial_t (h \theta) + \nabla \cdot ( h \theta \vec{\bf v})
+   = P \theta_{\mathrm{rain}}
  $$
  Using the implicit (non-linear) free surface described above (section
- \ref{sect:pressure-method-linear-backward}) we have:
+ \ref{sec:pressure-method-linear-backward}) we have:
  \begin{eqnarray*}
- h^{n+1} = h^{n} - \Delta_t \nabla \cdot (h^n \, \vec{\bf v}^{n+1} ) \\
+ h^{n+1} = h^{n} - \Delta t \nabla \cdot (h^n \, \vec{\bf v}^{n+1} ) + \Delta t P \\
  \end{eqnarray*}
  The discretized form of the tracer equation must adopt the same
  ``form'' in the computation of tracer fluxes, that is, the same value
  of $h$, as used in the continuity equation:
  \begin{eqnarray*}
  h^{n+1} \, \theta^{n+1} = h^n \, \theta^n
-         - \Delta_t \nabla \cdot (h^n \, \theta^n \, \vec{\bf v}^{n+1})
+         - \Delta t \nabla \cdot (h^n \, \theta^n \, \vec{\bf v}^{n+1})
+         + \Delta t P \theta_{rain}
  \end{eqnarray*}
- The use of a 3 time-levels timestepping scheme such as the Adams-Bashforth
+ The use of a 3 time-levels time-stepping scheme such as the Adams-Bashforth
- make the conservation less straitforward.
+ make the conservation sightly tricky.
  The current implementation with the Adams-Bashforth time-stepping
  provides an exact local conservation and prevents any drift in
  the global tracer content (\cite{campin:02}).
-Line 202 
 model uses the $G_\theta$ terms (first s
+Line 250 
 model uses the $G_\theta$ terms (first s
  surface formulation (with the "{\it surface correction}" turned "on",
  see tracer section):
  $$
  G_\theta^n = \left(- \nabla \cdot (h^n \, \theta^n \, \vec{\bf v}^{n+1})
           - \dot{r}_{surf}^{n+1} \theta^n \right) / h^n
  $$
  Then, in a second step, the thickness variation (expansion/reduction)
  is taken into account:
  $$
- \theta^{n+1} = \theta^n + \Delta_t \frac{h^n}{h^{n+1}} G_\theta^{(n+1/2)}
+ \theta^{n+1} = \theta^n + \Delta t \frac{h^n}{h^{n+1}}
+    \left( G_\theta^{(n+1/2)} + P (\theta_{\mathrm{rain}} - \theta^n )/h^n \right)
+ %\theta^{n+1} = \theta^n + \frac{\Delta t}{h^{n+1}}
+ %   \left( h^n G_\theta^{(n+1/2)} + P (\theta_{\mathrm{rain}} - \theta^n ) \right)
  $$
  Note that with a simple forward time step (no Adams-Bashforth),
+ these two formulations are equivalent,
  since
  $
- \dot{r}_{surf}^{n+1}
+ (h^{n+1} - h^{n})/ \Delta t =
- = - \nabla \cdot (h^n \, \vec{\bf v}^{n+1} ) = (h^{n+1} - h^{n})
+ P - \nabla \cdot (h^n \, \vec{\bf v}^{n+1} ) = P + \dot{r}_{surf}^{n+1}
- / \Delta_t
  $
- these two formulations are equivalent.
- Implementation in the MITgcm is as follows.  The model ``geometry''
+ \subsubsection{Time stepping implementation of the
+ non-linear free-surface}
+ \label{sec:nonlin-freesurf-timestepping}
+ The grid cell thickness was hold constant with the linear
+ free-surface~; with the non-linear free-surface, it is now varying
+ in time, at least at the surface level.
+ This implies some modifications of the general algorithm described
+ earlier in sections \ref{sec:adams-bashforth-sync} and
+ \ref{sec:adams-bashforth-staggered}.
+ A simplified version of the staggered in time, non-linear
+ free-surface algorithm is detailed hereafter, and can be compared
+ to the equivalent linear free-surface case (eq. \ref{eq:Gv-n-staggered}
+ to \ref{eq:t-n+1-staggered}) and can also be easily transposed
+ to the synchronous time-stepping case.
+ Among the simplifications, salinity equation, implicit operator
+ and detailed elliptic equation are omitted. Surface forcing is
+ explicitly written as fluxes of temperature, fresh water and
+ momentum, $Q^{n+1/2}, P^{n+1/2}, F_{\bf v}^n$ respectively.
+ $h^n$ and $dh^n$ are the column and grid box thickness in r-coordinate.
+ %-------------------------------------------------------------
+ \begin{eqnarray}
+ \phi^{n}_{hyd} & = & \int b(\theta^{n},S^{n},r) dr
+ \label{eq:phi-hyd-nlfs} \\
+ \vec{\bf G}_{\vec{\bf v}}^{n-1/2}\hspace{-2mm} & = &
+ \vec{\bf G}_{\vec{\bf v}} (dh^{n-1},\vec{\bf v}^{n-1/2})
+ \hspace{+2mm};\hspace{+2mm}
+ \vec{\bf G}_{\vec{\bf v}}^{(n)} =
+    \frac{3}{2} \vec{\bf G}_{\vec{\bf v}}^{n-1/2}
+ -  \frac{1}{2} \vec{\bf G}_{\vec{\bf v}}^{n-3/2}
+ \label{eq:Gv-n-nlfs} \\
+ %\vec{\bf G}_{\vec{\bf v}}^{(n)} & = &
+ %   \frac{3}{2} \vec{\bf G}_{\vec{\bf v}}^{n-1/2}
+ %-  \frac{1}{2} \vec{\bf G}_{\vec{\bf v}}^{n-3/2}
+ %\label{eq:Gv-n+5-nlfs} \\
+ %\vec{\bf v}^{*} & = & \vec{\bf v}^{n-1/2} + \frac{\Delta t}{dh^{n}} \left(
+ %dh^{n-1}\vec{\bf G}_{\vec{\bf v}}^{(n)} + F_{\vec{\bf v}}^{n} \right)
+ \vec{\bf v}^{*} & = & \vec{\bf v}^{n-1/2} + \Delta t \frac{dh^{n-1}}{dh^{n}} \left(
+ \vec{\bf G}_{\vec{\bf v}}^{(n)} + F_{\vec{\bf v}}^{n}/dh^{n-1} \right)
+ - \Delta t \nabla \phi_{hyd}^{n}
+ \label{eq:vstar-nlfs}
+ \end{eqnarray}
+ \hspace{3cm}$\longrightarrow$~~{\it update model~geometry~:~}${\bf hFac}(dh^n)$\\
+ \begin{eqnarray}
+ \eta^{n+1/2} \hspace{-2mm} & = &
+ \eta^{n-1/2} + \Delta t P^{n+1/2} - \Delta t
+   \nabla \cdot \int \vec{\bf v}^{n+1/2} dh^{n} \nonumber \\
+              & = & \eta^{n-1/2} + \Delta t P^{n+1/2} - \Delta t
+   \nabla \cdot \int \!\!\! \left( \vec{\bf v}^* - g \Delta t \nabla \eta^{n+1/2} \right) dh^{n}
+ \label{eq:nstar-nlfs} \\
+ \vec{\bf v}^{n+1/2}\hspace{-2mm} & = &
+ \vec{\bf v}^{*} - g \Delta t \nabla \eta^{n+1/2}
+ \label{eq:v-n+1-nlfs} \\
+ h^{n+1} & = & h^{n} + \Delta t P^{n+1/2} - \Delta t
+   \nabla \cdot \int \vec{\bf v}^{n+1/2} dh^{n}
+ \label{eq:h-n+1-nlfs} \\
+ G_{\theta}^{n} & = & G_{\theta} ( dh^{n}, u^{n+1/2}, \theta^{n} )
+ \hspace{+2mm};\hspace{+2mm}
+ G_{\theta}^{(n+1/2)} = \frac{3}{2} G_{\theta}^{n} - \frac{1}{2} G_{\theta}^{n-1}
+ \label{eq:Gt-n-nlfs} \\
+ %\theta^{n+1} & = &\theta^{n} + \frac{\Delta t}{dh^{n+1}} \left( dh^n
+ %G_{\theta}^{(n+1/2)} + Q^{n+1/2} + P^{n+1/2} (\theta_{\mathrm{rain}}-\theta^n) \right)
+ \theta^{n+1} & = &\theta^{n} + \Delta t \frac{dh^n}{dh^{n+1}} \left(
+ G_{\theta}^{(n+1/2)}
+ +( P^{n+1/2} (\theta_{\mathrm{rain}}-\theta^n) + Q^{n+1/2})/dh^n \right)
+ \nonumber \\
+ & & \label{eq:t-n+1-nlfs}
+ \end{eqnarray}
+ %-------------------------------------------------------------
+ Two steps have been added to linear free-surface algorithm
+ (eq. \ref{eq:Gv-n-staggered} to \ref{eq:t-n+1-staggered}):
+ Firstly, the model ``geometry''
  (here the {\bf hFacC,W,S}) is updated just before entering {\it
- SOLVE\_FOR\_PRESSURE}, using the current $\eta$ field.  Then, at the
+ SOLVE\_FOR\_PRESSURE}, using the current $dh^{n}$ field.
- end of the time step, the variables are advanced in time, so that
+ Secondly, the vertically integrated continuity equation
- $\eta^n$ replaces $\eta^{n-1}$.  At the next time step, the tracer
+ (eq.\ref{eq:h-n+1-nlfs}) has been added ({\bf exactConserv}{\em =TRUE},
- tendency ($G_\theta$) is computed using the same geometry, which is
+ in parameter file {\em data}, namelist {\em PARM01})
- now consistent with $\eta^{n-1}$.  Finally, in S/R {\it
+ just before computing the vertical velocity, in subroutine
- TIMESTEP\_TRACER}, the expansion/reduction ratio is applied to the
+ {\em INTEGR\_CONTINUITY}.
- surface level to compute the new tracer field.
+ %This ensures that tracer and continuity equation discretization a
+ Although this equation might appear redundant with eq.\ref{eq:nstar-nlfs},
+ the integrated column thickness $h^{n+1}$ will be different from
- \subsubsection{Free surface effect on the surface level thickness
+ $\eta^{n+1/2} + H$~ in the following cases:
- (Non-linear free surface): Momentum advection}
+ \begin{itemize}
- \label{sect:freesurf-momentum-advection}
+ \item when Crank-Nicolson time-stepping is used (see section
+ \ref{sec:freesurf-CrankNick}).
- With the flux form formulation, advection of momentum
+ \item when filters are applied to the flow field, after
- can be treated exactly as the tracer advection is.
+ (\ref{eq:v-n+1-nlfs}) and alter the divergence of the flow.
- Here the expansion/reduction factors ($hFacW^{n+1}/hFacW^n$ for $u$
+ \item when the solver does not iterate until convergence~;
- and $hFacS^{n+1}/hFacS^n$ for $v$) are simply applied in the
+  for example, because a too large residual target was set
- subroutine {\it TIMESTEP}.
+  ({\bf cg2dTargetResidual}, parameter file {\em data}, namelist
+  {\em PARM02}).
- Regarding momentum advection,
+ \end{itemize}\noindent
- the vector invariant formulation is similar to the
+ In this staggered time-stepping algorithm, the momentum tendencies
- advective form (as opposed to the flux form) and therefore
+ are computed using $dh^{n-1}$ geometry factors.
- does not need specific modification to include the
+ (eq.\ref{eq:Gv-n-nlfs}) and then rescaled in subroutine {\it TIMESTEP},
- free surface effect on the surface level thickness.
+ (eq.\ref{eq:vstar-nlfs}), similarly to tracer tendencies (see section
- Updating the {\bf hFacC,W,S} and the {\bf recip\_hFac}(s)
+ \ref{sec:freesurf-tracer-advection}).
- at one given place (like describe before) is sufficient.
+ The tracers are stepped forward later, using the recently updated
+ flow field ${\bf v}^{n+1/2}$ and the corresponding model geometry
+ $dh^{n}$ to compute the tendencies (eq.\ref{eq:Gt-n-nlfs});
+ Then the tendencies are rescaled by $dh^n/dh^{n+1}$ to derive
+ the new tracers values $(\theta,S)^{n+1}$ (eq.\ref{eq:t-n+1-nlfs},
+ in subroutine {\em CALC\_GT, CALC\_GS}).
+ Note that the fresh-water input is added in a consistent way in the
+ continuity equation and in the tracer equation, taking into account
+ the fresh-water temperature $\theta_{\mathrm{rain}}$.
+ Regarding the restart procedure,
+ two 2.D fields $h^{n-1}$ and $(h^n-h^{n-1})/\Delta t$
+ in addition to the standard
+ state variables and tendencies ($\eta^{n-1/2}$, ${\bf v}^{n-1/2}$,
+ $\theta^n$, $S^n$, ${\bf G}_{\bf v}^{n-3/2}$, $G_{\theta,S}^{n-1}$)
+ are stored in a "{\em pickup}" file.
+ The model restarts reading this "{\em pickup}" file,
+ then update the model geometry according to $h^{n-1}$,
+ and compute $h^n$ and the vertical velocity
+ %$h^n=h^{n-1} + \Delta t [(h^n-h^{n-1})/\Delta t]$,
+ before starting the main calling sequence (eq.\ref{eq:phi-hyd-nlfs}
+ to \ref{eq:t-n+1-nlfs}, {\em S/R FORWARD\_STEP}).
+ \\
+ \fbox{ \begin{minipage}{4.75in}
+ {\em INTEGR\_CONTINUITY} ({\em integr\_continuity.F})
+ $h^{n+1} -H_o$: {\bf etaH} ({\em DYNVARS.h})
+ $h^{n} -H_o$: {\bf etaHnm1} ({\em SURFACE.h})
+ $(h^{n+1}-h^{n})/\Delta t$: {\bf dEtaHdt} ({\em SURFACE.h})
+ \end{minipage} }
- \subsubsection{Non-linear free surface and vertical resolution}
+ \subsubsection{Non-linear free-surface and vertical resolution}
- \label{sect:nonlin-freesurf-dz_surf}
+ \label{sec:nonlin-freesurf-dz_surf}
  When the amplitude of the free-surface variations becomes
  as large as the vertical resolution near the surface,
  the surface layer thickness can decrease to nearly zero or
- can even vanishe completly.
+ can even vanish completely.
  This later possibility has not been implemented, and a
  minimum relative thickness is imposed ({\bf hFacInf},
  parameter file {\em data}, namelist {\em PARM01}) to prevent
  numerical instabilities caused by very thin surface level.
- A better atlternative to the vanishing level problem has been
+ A better alternative to the vanishing level problem has been
  found and implemented recently, and rely on a different
  vertical coordinate $r^*$~:
  The time variation ot the total column thickness becomes
  part of the r* coordinate motion, as in a $\sigma_{z},\sigma_{p}$
  model, but the fixed part related to topography is treated
  as in a height or pressure coordinate model.
- A complete description is given in \cite{adcroft:04}.
+ A complete description is given in \cite{adcroft:04a}.
  The time-stepping implementation of the $r^*$ coordinate is
  identical to the non-linear free-surface in $r$ coordinate,
- and differences appear only in the spacial discretisation.
+ and differences appear only in the spacial discretization.
  \marginpar{needs a subsection ref. here}

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