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Subsections
2.10.2 Nonlinear freesurface
Recently, new options have been added to the model
that concern the free surface formulation.
2.10.2.1 pressure/geopotential and free surface
For the atmosphere, since
,
subtracting the reference state defined in section
1.4.1.2 :
we get:
For the ocean, the reference state is simpler since
does not dependent
on
(
) and the surface reference position is uniformly
(
),
and the same subtraction leads to a similar relation.
For both fluid, using the isomorphic notations, we can write:
and rewrite as:

(2.77) 
or:

(2.78) 
In section 1.3.6, following eq.2.77,
the pressure/geopotential
has been separated into surface (
),
and hydrostatic anomaly (
).
In this section, the split between
and
is
made according to equation 2.78. This slightly
different definition reflects the actual implementation in the code
and is valid for both linear and nonlinear
freesurface formulation, in both rcoordinate and r*coordinate.
Because the linear freesurface approximation ignore the tracer content
of the fluid parcel between
and
,
for consistency reasons, this part is also neglected in
:
Note that in this case, the two definitions of
and
from equation 2.77 and 2.78 converge toward
the same (approximated) expressions:
and
.
On the contrary, the unapproximated formulation ("nonlinear freesurface",
see the next section) retains the full expression:
.
This is obtained by selecting nonlinFreeSurf=4 in parameter
file data.
Regarding the surface potential:
is an excellent approximation (better than
the usual numerical truncation, since generally
is smaller
than the vertical grid increment).
For the ocean,
and
is uniform.
For the atmosphere, however, because of topographic effects, the
reference surface pressure
has large spatial variations that
are responsible for significant
variations (from 0.8 to 1.2
). For this reason, when uniformLin_PhiSurf =.FALSE.
(parameter file data, namelist PARAM01)
a nonuniform linear coefficient
is used and computed
(S/R INI_LINEAR_PHISURF) according to the reference surface
pressure
:
.
with
the mean sealevel pressure.
The total thickness of the fluid column is
. In most applications, the free surface
displacements are small compared to the total thickness
.
In the previous sections and in older version of the model,
the linearized freesurface approximation was made, assuming
when computing horizontal transports,
either in the continuity equation or in tracer and momentum
advection terms.
This approximation is dropped when using the nonlinear freesurface
formulation and the total thickness, including the time varying part
, is considered when computing horizontal transports.
Implications for the barotropic part are presented hereafter.
In section 2.10.2.3 consequences for
tracer conservation is briefly discussed (more details can be
found in Campin et al. [2004]) ; the general timestepping is presented
in section 2.10.2.4 with some
limitations regarding the vertical resolution in section
2.10.2.5.
In the nonlinear formulation, the continuous form of the model
equations remains unchanged, except for the 2D continuity equation
(2.16) which is now
integrated from
up to
:
Since
has a direct effect on the horizontal velocity (through
), this adds a nonlinear term to the free
surface equation. Several options for the time discretization of this
nonlinear part can be considered, as detailed below.
If the column thickness is evaluated at time step
, and with
implicit treatment of the surface potential gradient, equations
(2.72 and 2.73) becomes:
where
This method requires us to update the solver matrix at each time step.
Alternatively, the nonlinear contribution can be evaluated fully
explicitly:
This formulation allows one to keep the initial solver matrix
unchanged though throughout the integration, since the nonlinear free
surface only affects the RHS.
Finally, another option is a "linearized" formulation where the total
column thickness appears only in the integral term of the RHS
(2.73) but not directly in the equation
(2.72).
Those different options (see Table 2.1)
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 is negligible.
Table 2.1:
Nonlinear freesurface flags
parameter 
value 
description 

1 
linear freesurface, restart from a pickup file 


produced with #undef EXACT_CONSERV code 

0 
Linear freesurface 
nonlinFreeSurf 
4 
Nonlinear freesurface 

3 
same as 4 but neglecting
in


2 
same as 3 but do not update cg2d solver matrix 

1 
same as 2 but treat momentum as in Linear FS 

0 
do not use
vertical coordinate (= default) 
select_rStar 
2 
use
vertical coordinate 

1 
same as 2 but without the contribution of the 


slope of the coordinate in


2.10.2.3 Tracer conservation with nonlinear freesurface
To ensure global tracer conservation (i.e., the total amount) as well
as local conservation, the change in the surface level thickness must
be consistent with the way the continuity equation is integrated, both
in the barotropic part (to find
) and baroclinic part (to find
).
To illustrate this, consider the shallow water model, with a source
of fresh water (P):
where
is the total thickness of the water column.
To conserve the tracer
we have to discretize:
Using the implicit (nonlinear) free surface described above (section
2.4) we have:
The discretized form of the tracer equation must adopt the same
``form'' in the computation of tracer fluxes, that is, the same value
of
, as used in the continuity equation:
The use of a 3 timelevels timestepping scheme such as the AdamsBashforth
make the conservation sightly tricky.
The current implementation with the AdamsBashforth timestepping
provides an exact local conservation and prevents any drift in
the global tracer content (Campin et al. [2004]).
Compared to the linear freesurface method, an additional step is required:
the variation of the water column thickness (from
to
) is
not incorporated directly into the tracer equation. Instead, the
model uses the
terms (first step) as in the linear free
surface formulation (with the "surface correction" turned "on",
see tracer section):
Then, in a second step, the thickness variation (expansion/reduction)
is taken into account:
Note that with a simple forward time step (no AdamsBashforth),
these two formulations are equivalent,
since
2.10.2.4 Time stepping implementation of the
nonlinear freesurface
The grid cell thickness was hold constant with the linear
freesurface ; with the nonlinear freesurface, it is now varying
in time, at least at the surface level.
This implies some modifications of the general algorithm described
earlier in sections 2.7 and
2.8.
A simplified version of the staggered in time, nonlinear
freesurface algorithm is detailed hereafter, and can be compared
to the equivalent linear freesurface case (eq. 2.42
to 2.52) and can also be easily transposed
to the synchronous timestepping 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,
respectively.
and
are the column and grid box thickness in rcoordinate.
update model geometry :







(2.82) 



(2.83) 



(2.84) 



(2.85) 







(2.86) 
Two steps have been added to linear freesurface algorithm
(eq. 2.42 to 2.52):
Firstly, the model ``geometry''
(here the hFacC,W,S) is updated just before entering SOLVE_FOR_PRESSURE, using the current
field.
Secondly, the vertically integrated continuity equation
(eq.2.84) has been added (exactConserv=TRUE,
in parameter file data, namelist PARM01)
just before computing the vertical velocity, in subroutine
INTEGR_CONTINUITY.
Although this equation might appear redundant with eq.2.82,
the integrated column thickness
will be different from
in the following cases:
 when CrankNicolson timestepping is used (see section
2.10.1).
 when filters are applied to the flow field, after
(2.83) and alter the divergence of the flow.
 when the solver does not iterate until convergence ;
for example, because a too large residual target was set
(cg2dTargetResidual, parameter file data, namelist
PARM02).
In this staggered timestepping algorithm, the momentum tendencies
are computed using
geometry factors.
(eq.2.80) and then rescaled in subroutine TIMESTEP,
(eq.2.81), similarly to tracer tendencies (see section
2.10.2.3).
The tracers are stepped forward later, using the recently updated
flow field
and the corresponding model geometry
to compute the tendencies (eq.2.85);
Then the tendencies are rescaled by
to derive
the new tracers values
(eq.2.86,
in subroutine CALC_GT, CALC_GS).
Note that the freshwater input is added in a consistent way in the
continuity equation and in the tracer equation, taking into account
the freshwater temperature
.
Regarding the restart procedure,
two 2.D fields
and
in addition to the standard
state variables and tendencies (
,
,
,
,
,
)
are stored in a "pickup" file.
The model restarts reading this "pickup" file,
then update the model geometry according to
,
and compute
and the vertical velocity
before starting the main calling sequence (eq.2.79
to 2.86, S/R FORWARD_STEP).
2.10.2.5 Nonlinear freesurface and vertical resolution
When the amplitude of the freesurface variations becomes
as large as the vertical resolution near the surface,
the surface layer thickness can decrease to nearly zero or
can even vanish completely.
This later possibility has not been implemented, and a
minimum relative thickness is imposed (hFacInf,
parameter file data, namelist PARM01) to prevent
numerical instabilities caused by very thin surface level.
A better alternative to the vanishing level problem has been
found and implemented recently, and rely on a different
vertical coordinate
:
The time variation ot the total column thickness becomes
part of the r* coordinate motion, as in a
model, but the fixed part related to topography is treated
as in a height or pressure coordinate model.
A complete description is given in Adcroft and Campin [2004].
The timestepping implementation of the
coordinate is
identical to the nonlinear freesurface in
coordinate,
and differences appear only in the spacial discretization.
Next: 2.11 Spatial discretization of
Up: 2.10 Variants on the
Previous: 2.10.1 CrankNicolson barotropic time
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Massachusetts Institute of Technology 
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