Etudes et évaluation de processus océaniques par des hiérarchies ...

4.6. ESTIMATION OF FRICTION PARAMETERS AND LAWS IN 1.5D SHALLOW-WATER GRAVI
Ocean Dynamics
tel-00545911, version 1 - 13 Dec 2010
when considering the estimation of the parameter values
(see below).
5.2 Discriminating between laws
In the previous subsection, we demonstrated that the
assimilation procedure manages to estimate the right
parameter values in the case of linear and quadratic
friction. The type of the friction law was, however,
imposed before the estimation procedure, a feature that
we relax now. That is, we still use the same data but
the assimilation procedure does not know with which
kind of friction law the data were produced and what
the corresponding parameter value is. We investigate
in experiments G00001 and G00010 if the assimilation
scheme obtains the parameter values and, in this way,
manages to determine the friction law. We emphasise
that the difference to experiments A00001 and A00010
is that, now, the assimilation scheme does not know
that the other parameter is vanishing, but it has to
establish it.
The results are presented in Fig 4. The initial ensemble
of the parameter values is the same for both
experiments. In both cases, we see a good convergence
to the right values and, thus, a clear determination of
the right friction law.
5.3 Estimating both parameters
The next step is to consider dynamics which include
both friction laws. The procedure is identical to that
of the previous subsection, only that now, both parameter
values are non-vanishing. There are indeed a
large number of examples where the friction law passes
from linear to quadratic and where both laws coexist
(see Schlichting and Gertsen 2000 pp. chap 1.3). In
cases where one friction law is established, the friction
coefficient (weakly) depends on the Reynolds number.
By allowing for two, or more, friction laws, not only
the optimal value of the friction parameter is estimated
but also its variation with the Reynolds number (see
Section 6). The case of estimating both τ and c D at the
same time is more challenging, as they both include
friction in the model dynamics and have, to the first
order, the same effect on the gravity current, that is,
make it move down-slope. Furthermore, we choose,
in experiments G0021, G0121, and G0221, values for
τ and c D such that they have a similar magnitude of
the friction force, that is τ ≈ c D |u| and |u| = √ u 2 + v 2 ,
which is the most challenging case. A large number
of experiments have been performed with different
values of the friction parameters, the results show no
qualitative differences. In Fig. 5, the convergences of
the parameter values are shown. In general, one notices
a good convergence in τ-c D -space of all members of
the ensemble towards the true values. A better convergence
for runs with lower perturbations σ (noise level)
is noticed. A further reduction of the variance of the
noise added to the observations (necessary to make
the EnKF consistent; see Burgers et al. 1998) leads to
a divergence of the EnKF. A conspicuous feature of
Fig. 5 is the fact that the ensemble is aligned along
a straight line, which corresponds to a space-timemean
absolute velocity of the gravity current of
Fig. 4 Distribution of the ensemble in τ (horizontal direction)
and c D (vertical direction) space for the two experiments: G0001
(black dots) and G0011 (red dots), at different times: Initial distribution
(left) (some initial values lie outside the area shown), after
7 days (middle) and after seven iterations of the 7-day dynamics
(right). The values of τ and c D are normalised by the true value
used in the control run (see Table 1). The experiment shows a
successful convergence towards the true values. The black dots
converge to (τ, c D ) = (τ 0 , 0) and the red dots to (τ, c D ) = (0, c D0 )