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

152 CHAPITRE 4. ETUDES DE PROCESSUS OCÉANOGRAPHIQUES
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on the results. The term involving the viscosity/diffusivity ν represents horizontal dissipative
processes, its value is chosen to provide numerical stability of the calculations
(see subsection 2.5).
Clearly, the geostrophic velocity:
ũ = 0, ṽ = g′ (∂xh + tan α), (10)
f
is a solution of eqs. (4 – 6) when D u = D v = 0 and ν = 0. The estimation of D or more
precisely of the parameters τ, r and c D are the subject of the present work. By using
data assimilation I plan to obtain the friction constants τ and c D and thus determine
if the friction acting on the vein is dominated by a linear or a quadratic law.
2.5 Numerical Implementation of the Shallow Water Model
The shallow water model is implemented with a first order finite difference scheme in
space and time. There are 500 points in the x-direction, leading to a resolution of 200m,
the time step is 5s. The value of the horizontal viscosity/diffusivity is a function of the
resolution and provides for the numerical stability of the code. I verified that the here
presented results show only a negligible dependence when the value of the horizontal
viscosity/diffusivity was halved and doubled, the actual value used is ν H = 5m 2 s −1 , it
is idential to the corresponding value ν h in the non-hydrostatic model.
tel-00545911, version 1 - 13 Dec 2010
3 Ensemble Kalman Filter and its Implementation
For the self containedness of this publication the implementation of the Ensemble
Kalman filter is explained here, the reader familiar with Wirth & Verron (2008) is
invited to skip the section.
The ensemble Kalman filter (EnKF) is the main tool of our experiments performing
the parameter estimation and providing us with the actual parameter values. The
EnKF was introduced by Evensen (1994) and is used in data assimilation and parameter
estimation experiments (see Evensen (2003) and Brusdal et al. 2003). I refer the
reader not familiar with the EnKF and the employed notation to the above mentioned
publications.
Every hour in time and every 1 km in the x-direction, the vertical extension of
the gravity current, that is h(x, t), is assimilated. Choosing a horizontal resolution
for the assimilation 5 times sparser than the dynamical model does not only reduce
the size of the assimilation experiment, but is also consistent with the fact, that the
grid-scale dynamics of the numerical model is dominated by dissipation and has only
negligible dynamical information. I am only assimilating the vertical extension of the
gravity current as it is the variable most easily measured in the ocean and in laboratory
experiments. The measurement of the vertically integrated velocity within the gravity
current, the other dynamical variable of the shallow water model, is more difficult to
measure in the ocean and in laboratory experiments.
The assimilation is performed on the augmented state vector consisting of the
vertical extension, the two velocity components and the three constant-in-time friction
parameters:
x(¯x, ¯t) = (h(¯x, ¯t), ũ(¯x, ¯t), ṽ(¯x, ¯t), τ, r, c D ) t . (11)