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In this chapter, we review the three numerical methods and the different
models we will use in this thesis. In particular, we stress the
limitations of each as it relates to the discretization of irregular
domains. In the case of the spectral element method, we contribute to the
development of the method by proposing our own adaptive technique.
Furthermore, we present our own implementation of curved spectral
elements. Although curved elements are quite natural to the spectral
element method, we found very little information in the literature with
respect to their implementation.
The idealized equations we propose to solve are the shallow water
(SW) equations. These equations are grossly simplified compared to the
primitive equations. Nonetheless, the dynamical processes involved in the
formation of winddriven circulations and the interactions with irregular
coastlines are similar enough that we can restrict ourselves to these
equations as an introductory study. The equations are



(2.1) 



(2.2) 
where symbols are defined in Table 1. These equations correspond to a
Boussinesq, hydrostatic, homogeneous ocean in which we assume that there
is no vertical structure, reducing the real threedimensional (3D) problem
to a simple twodimensional (2D) problem. One remark concerns the
treatment of the gravity waves in these equations. The natural speed of
barotropic gravity waves is
where g=9.81m s^{2} is the
acceleration due to gravity and H is the typical oceanic depth. Since a
reasonable value for H is about 4000 m, the phase speed for barotropic
gravity waves is about 200 m/s. In order to use reasonable timesteps and
be able to perform long time simulations, these modes have to be slowed
down by using a ``reduced'' gravity. This approach is not inconsistent
with the actual physics of the ocean. In fact, in the presence of a
thermocline and a deep layer at rest below the thermocline, the SW
equations with reduced gravity represent, in some sense, the first
baroclinic mode dynamics, i.e., the dynamics of the upper layer. Indeed,
this upper layer happens to be the location of the most intense dynamical
events. The reduced gravity is defined as
where
is the jump in density through the pycnocline and
the average value for the density of the ocean. For example, the
Kelvin waves observed in the equatorial Pacific and along the western
American coast have phase speed of 23 m/s [Boulanger and Fu, 1996,Ramp et al. , 1997], close
to the phase speed of 3.16 m/s obtained in a SW reduced gravity model
where the reduced gravity is fixed at 10^{2}m s^{2} and the depth
above the thermocline is taken to be 1000 m. Hence, these equations are
consistent with a first order approximation of the physical processes
involved in the layer above the thermocline.
Table 2.1:
List of variables in (2.12.2)

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Frederic Dupont
20010911