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A linear analytical solution can be found for the winddriven problem in a
circular domain with Coriolis forces and damped by a linear bottom
friction. No viscosity is included. The boundary condition is simply the
nonormal flow condition at the model boundary. The steady state
linearized shallow water equations in cylindrical coordinates for this
problem are

(3.6) 
where the wind forcing is given in
cylindrical coordinates by the relationship

(3.7) 
where R is the radius of the circular domain. From
(3.63.8), we derive an equation for

(3.8) 
with the boundary condition of nonormal flow

(3.9) 
This leads to the solution without Coriolis force,

(3.10) 
and with Coriolis force

(3.11) 
With or without the Coriolis terms, the velocity components take the
simple form of

(3.12) 
which translate in the Cartesian coordinate system to

(3.13) 
We perform a one year spinup for all models with
W=10^{4}m^{2}s^{2}, f=10^{4}s^{1} or zero and
s^{1}. This is enough to converge to a steady state
accurate at six digits for the kinetic energy. The normalized error is computed
in the same manner as in (3.5) but using the elevation field. We
focus on the elevation this time because, for the HT and LLS FE
models, the pressure basis functions are different from the basis
functions used to represent the velocity. Furthermore, the previous test
case does not allow for an interesting comparison of the elevation fields
(the elevation is imposed at initial time), whereas this one does.
We first analyze the results from the Cgrid model. Because of the
presence of steps (Fig. 3.8), it is not clear which opposing
effect is dominant when the resolution is increased: an increased accuracy
in the interior and a more accurate representation of the boundary, or a
lower accuracy because of the increased number of steps. For brevity, we
only show the results for one case, at f=0, since convergence properties
are not significantly different than those at .
Figure 3.9 shows the convergence of the normalized error in
with increasing resolution. It appears that the convergence order
of the Cgrid FD model is closer to one (1.1 when f=0 and 1.3 when
f=10^{4}s^{1}) than two, the maximum for this second order FD
formulation.
Therefore, the steps have a direct influence on the order of the FD
model. The order is reduced compared to the previous testcase with
straight walls. The perturbation due to the singular steps on the flow
does degrade the accuracy, although not to the point that the errors
increases with increasing resolution.
Figure 3.8:
Grids for the circular domain for the FD models.
,
and
points
for domain on the
left, center and right respectively.

We now compare the solution from the Cgrid FD model with the OFDM4 and
RFDM4 models.
Figure 3.9 shows that the order of the Agrid
model is actually less than two in presence of steplike walls.
Furthermore, there is no longer a difference, in term of truncation order,
between the second order Cgrid and the 4th order Agrid models unlike
the
case with straight walls. Therefore, the presence of steps along irregular
boundaries has a detrimental effect on the accuracy of high order FD
formulations if the flow is allowed to slip along the walls.
Figure 3.9:
Convergence with resolution of the
normalized elevation error for the second order Cgrid FD,
OFDM4 and RFDM4 models in a circular domain.

We now compare the FE models to the Cgrid model. In this circular
geometry, all FE models have the advantage that the representation of the
boundary is improving as the resolution is increased. Therefore, it
should be possible to observe convergence order close or even exceeding
two. Figure 3.10 and Table 3.2
show that all FE models have a convergence
rate close to second order except for the LLS model. The LLS model also
shows the largest errors. The reasons for the poor performance of this
model are as follows. Firstly, the geometry is resolved by the
macroelements. Thus the representation of the boundary suffers from being
half sampled compared to the permitted resolution. Second, we focus here
on the elevation errors which are always larger for the LLS model because
the piecewise constant basis functions are not as accurate as those of the
other models. For the HT model, the improvement in the error
compared to the previous testcase is probably due to the basis function
for
being continuous. In fact, all FE models used this basis
function for the elevation except for the LLS model. Hence in terms of
accuracy, all FE models appear to perform better than FD models in
nonrectangular geometries for linear problems, except for the LLS
model. In terms of cost, the equalorder FE models are the most effective.
However, we still need to demonstrate the efficiency of FE models for
nonlinear problems before concluding on the general effectiveness of FE
models in irregular domains.
Figure 3.10:
Normalized elevation error in a circular domain for an inviscid
linear solution.
The four FE models (LW, HT, PZM, LLS)
are tested against the
analytical solution with increasing resolution. The
error for the FD model (FDM) is given for comparison.

For the SE model, the results are given in Fig. 3.11 where we
compare the solutions from the Cgrid FD, LW FE and SE models. The results
for the SE model shows a surprising feature. The 3rd order SE model has a
better accuracy than the FE model but the errors for the 5th and 7th order
SE are larger than expected. The convergence order is also affected (see
Table 3.2). In this particular example, the main source of
errors comes from the discretization of the circular geometry by piecewise
parabolas. A quadratic spline description of the circular boundary allows
for (at least) a 3rd convergence order. This explains why the convergence
order for the 3rd order SE model appears optimal but less optimal for the
5th and 7th order SE model. The order of the solution improves in the
interior but the error along the boundary being larger leads and causes a
overall loss in the convergence order.
One solution would be to implement more complex curved elements along the
boundary (using cubic or more splines), but as explained in
Section 2.4.4, increasing the order of the piecewise curves
along the curved elements is not always practical.
Figure 3.11:
Normalized elevation error for the Cgrid FD, LWFE and SE models for
a circular domain.
The curve for the SE model at n_{c}=7 (SPOC 7)
is on the right of
that for n_{c}=5 (SPOC 5) presenting some kind of ``saturation''
effect.

Table 3.2:
Convergence order in elevation,
for the different models for the linear
winddriven experiment in a circular domain without
Coriolis terms.
Model 
convergence order for
the error in

Cgrid FD

1.15 
OFDM 4 
1.51 
RFDM 4 
1.24 
LW

2.40 
HT 
1.91 
LLS 
0.98 
PZM 
1.94 
SPOC 3

3.33 
SPOC 5 
4.09 
SPOC 7 
4.64 

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Next: Conservative Properties of the
Up: Testing the Different Numerical
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Frederic Dupont
20010911