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revision 1.7 by adcroft, Tue Nov 13 20:27:54 2001 UTC revision 1.8 by jmc, Wed Oct 13 18:56:52 2004 UTC
# Line 6  Line 6 
6  \subsection{Non-linear free surface}  \subsection{Non-linear free surface}
7  \label{sect:nonlinear-freesurface}  \label{sect:nonlinear-freesurface}
8    
9  Recently, two options have been added to the model (and have not yet  Recently, new options have been added to the model
10  been extensively tested) that concern the free surface formulation.  that concern the free surface formulation.
11    
12    
13  \subsubsection{Non-uniform linear-relation for the surface potential}  \subsubsection{Non-uniform linear-relation for the surface potential}
# Line 51  pressure $P_o$ ($=R_o$): $b_s = c_p \kap Line 51  pressure $P_o$ ($=R_o$): $b_s = c_p \kap
51  (Non-linear free surface)}  (Non-linear free surface)}
52    
53  The total thickness of the fluid column is $r_{surf} - R_{fixed} =  The total thickness of the fluid column is $r_{surf} - R_{fixed} =
54  \eta + R_o - R_{fixed}$ In the linear free surface approximation  \eta + R_o - R_{fixed}$. In most applications, the free surface
55  (detailed before), only the fixed part of it ($R_o - R_{fixed})$ is  displacements are small compared to the total thickness
56  considered when we integrate the continuity equation or compute tracer  $\eta << H_o = R_o - R_{fixed}$.
57  and momentum advection term.  In the previous sections and in older version of the model,
58    the linearized free-surface approximation was made, assuming
59    $r_{surf} - R_{fixed} \simeq H_o$ when the horizontal transport is
60    computed, either in the continuity equation or in tracer and momentum
61    advection terms.
62  This approximation is dropped when using the non-linear free surface  This approximation is dropped when using the non-linear free surface
63  formulation.  Here we discuss sections the barotropic part. In  formulation and the total thickness, including the time varying part
64  sections \ref{sect:freesurf-tracer-advection} and  $\eta$, is consisdered when computing horizontal transport.
65  \ref{sect:freesurf-momentum-advection} we consider the baroclinic  Implications for the barotropic part are presented hereafter.
66  component.  In sections \ref{sect:freesurf-tracer-advection} and
67    \ref{sect:freesurf-momentum-advection}, consequences for tracer
68    and momentum are brifly discussed. a more detailed description
69    is available in \cite{campin:02}.
70    
71    
72  The continuous form of the model equations remains unchanged, except  In the non-linear formulation, the continuous form of the model equations
73  for the 2D continuity equation (\ref{eq:discrete-time-backward-free-surface}) which is now  remains unchanged, except for the 2D continuity equation
74    (\ref{eq:discrete-time-backward-free-surface}) which is now
75  integrated from $R_{fixed}(x,y)$ up to $r_{surf}=R_o+\eta$ :  integrated from $R_{fixed}(x,y)$ up to $r_{surf}=R_o+\eta$ :
76    
77  \begin{displaymath}  \begin{displaymath}
# Line 77  integrated from $R_{fixed}(x,y)$ up to $ Line 84  integrated from $R_{fixed}(x,y)$ up to $
84  Since $\eta$ has a direct effect on the horizontal velocity (through  Since $\eta$ has a direct effect on the horizontal velocity (through
85  $\nabla_h \Phi_{surf}$), this adds a non-linear term to the free  $\nabla_h \Phi_{surf}$), this adds a non-linear term to the free
86  surface equation. Several options for the time discretization of this  surface equation. Several options for the time discretization of this
87  non-linear part have been tested.  non-linear part can be considered, as detailed below.
88    
89  If the column thickness is evaluated at time step $n$, and with  If the column thickness is evaluated at time step $n$, and with
90  implicit treatment of the surface potential gradient, equations  implicit treatment of the surface potential gradient, equations
# Line 115  column thickness appears only in the int Line 122  column thickness appears only in the int
122  (\ref{eq-solve2D_rhs}) but not directly in the equation  (\ref{eq-solve2D_rhs}) but not directly in the equation
123  (\ref{eq-solve2D}).  (\ref{eq-solve2D}).
124    
125    Those different options (see tab.?? for the one still available)
126    have been tested and show litle differences. However, we recommand
127    the use of the most precise method (the 1rst one) since the
128    computation cost involved in the solver matrix update are negligeable.
129    
130    \begin{center}
131     \begin{tabular}[htb]{|l|c|l|}
132       \hline
133       parameter & value & description \\
134       \hline
135                       & -1 & linear free-surface, restart from a pickup file \\
136                       &    & produced with \#undef EXACT\_CONSERV code\\
137       \cline{2-3}
138                       & 0 & Linear free-Surface \\
139       \cline{2-3}
140        nonlinFreeSurf & 4 & Non-linear free-surface \\
141       \cline{2-3}
142                       & 3 & same as 4 but neglecting
143                               $\int_{R_o}^{R_o+\eta} b' dr $ in $\Phi'_{hyd}$ \\
144       \cline{2-3}
145                       & 2 & same as 3 but do not update cg2d solver matrix \\
146       \cline{2-3}
147                      & 1 & same as 2 but treat momentum as in Linear FS \\
148       \hline
149                      & 0 & do not use $r*$ vertical coordinate (= default)\\
150       \cline{2-3}
151        select\_rStar & 2 & use $r^*$ vertical coordinate \\
152       \cline{2-3}
153                      & 1 & same as 2 but without the contribution of the\\
154                      &   & slope of the coordinate in $\nabla \Phi$ \\
155       \hline
156      \end{tabular}
157    \end{center}
158    
159    
160  \subsubsection{Free surface effect on the surface level thickness  \subsubsection{Free surface effect on the surface level thickness
161  (Non-linear free surface): Tracer advection}  (Non-linear free surface): Tracer advection}
# Line 149  h^{n+1} \, \theta^{n+1} = h^n \, \theta^ Line 190  h^{n+1} \, \theta^{n+1} = h^n \, \theta^
190          - \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})
191  \end{eqnarray*}  \end{eqnarray*}
192    
193  For Adams-Bashforth time-stepping, we implement this scheme slightly  The use of a 3 time-levels timestepping scheme such as the Adams-Bashforth
194  differently from the linear free-surface method, using two steps: the  make the conservation less straitforward.
195  variation of the water column thickness (from $h^n$ to $h^{n+1}$) is  The current implementation with the Adams-Bashforth time-stepping
196    provides an exact local conservation and prevents any drift in
197    the global tracer content (\cite{campin:02}).
198    Compared to the linear free-surface method, an additional step is required:
199    the variation of the water column thickness (from $h^n$ to $h^{n+1}$) is
200  not incorporated directly into the tracer equation.  Instead, the  not incorporated directly into the tracer equation.  Instead, the
201  model uses the $G_\theta$ terms (first step) as in the linear free  model uses the $G_\theta$ terms (first step) as in the linear free
202  surface formulation (with the "{\it surface correction}" turned "on",  surface formulation (with the "{\it surface correction}" turned "on",
# Line 189  surface level to compute the new tracer Line 234  surface level to compute the new tracer
234  (Non-linear free surface): Momentum advection}      (Non-linear free surface): Momentum advection}    
235  \label{sect:freesurf-momentum-advection}  \label{sect:freesurf-momentum-advection}
236    
237    With the flux form formulation, advection of momentum
238    can be treated exactly as the tracer advection is.
239    Here the expansion/reduction factors ($hFacW^{n+1}/hFacW^n$ for $u$
240    and $hFacS^{n+1}/hFacS^n$ for $v$) are simply applied in the
241    subroutine {\it TIMESTEP}.
242    
243  Regarding momentum advection,  Regarding momentum advection,
244  the vector invariant formulation is similar to the  the vector invariant formulation is similar to the
245  advective form (as opposed to the flux form) and therefore  advective form (as opposed to the flux form) and therefore
# Line 197  free surface effect on the surface level Line 248  free surface effect on the surface level
248  Updating the {\bf hFacC,W,S} and the {\bf recip\_hFac}(s)  Updating the {\bf hFacC,W,S} and the {\bf recip\_hFac}(s)
249  at one given place (like describe before) is sufficient.  at one given place (like describe before) is sufficient.
250    
251  With the flux form formulation, advection of momentum  \subsubsection{Non-linear free surface and vertical resolution}
252  can be treated exactly as the tracer advection is.  \label{sect:nonlin-freesurf-dz_surf}
253  Here the expansion/reduction factors ($hFacW^{n+1}/hFacW^n$ for $u$  
254  and $hFacS^{n+1}/hFacS^n$ for $v$) are simply applied in the  When the amplitude of the free-surface variations becomes
255  subroutine {\it TIMESTEP}.  as large as the vertical resolution near the surface,
256    the surface layer thickness can decrease to nearly zero or
257    can even vanishe completly.
258    This later possibility has not been implemented, and a
259    minimum relative thickness is imposed ({\bf hFacInf},
260    parameter file {\em data}, namelist {\em PARM01}) to prevent
261    numerical instabilities caused by very thin surface level.
262    
263    A better atlternative to the vanishing level problem has been
264    found and implemented recently, and rely on a different
265    vertical coordinate $r^*$~:
266    The time variation ot the total column thickness becomes
267    part of the r* coordinate motion, as in a $\sigma_{z},\sigma_{p}$
268    model, but the fixed part related to topography is treated
269    as in a height or pressure coordinate model.
270    A complete description is given in \cite{adcroft:04}.
271    
272    The time-stepping implementation of the $r^*$ coordinate is
273    identical to the non-linear free-surface in $r$ coordinate,
274    and differences appear only in the spacial discretisation.
275    \marginpar{needs a subsection ref. here}
276    
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