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

110 CHAPITRE 4. ETUDES DE PROCESSUS OCÉANOGRAPHIQUES
APRIL 2008 W I R T H A N D B A R N I E R 809
FIG. 2. Isosurfaces of vertical velocity w 0.05 m s 1 ( red, blue) looking upward from the ocean floor (x
to the right, y downward) at the end of the experiments at t 168 h: (left) E04 and (right) E34. The elongation
of structures in the y direction is conspicuous in experiment E34.
tel-00545911, version 1 - 13 Dec 2010
front descends into the unstratified ocean until it
reaches the ocean floor after about 18–30 h, depending
on the strength of the forcing and rotation.
The subject of this paper is the subsequent statistically
stationary stage of convective dynamics in the entire
water column. During this stage descending plumes
surrounded by rising water mix the water column as can
be seen in Fig. 2. A shallow boundary layer develops at
the ocean surface. As in the case of a single plume, the
influence of the (no slip) bottom slows down the front
propagation 500 m (the typical horizontal plume scale)
above the bottom, emphasizing once more the importance
of the Ekman layer dynamics at the ocean floor.
Figure 2 exposes that the differences in the horizontally
averaged temperature structure between the tilted and
nontilted dynamics are small.
a. The signature of the Taylor–Proudman–Poincaré
theorem
The analysis presented in this section is based on the
discussion of the derivation, generality, and consequences
of the Taylor–Proudman–Poincaré theorem by
Colin de Verdière (2002).
When friction and nonlinearity are negligible the dynamics
of a stratified Boussinesq fluid, subject to rotation,
is governed by the TPP theorem:
2 y y z z u y
w g 0 x
. 0 7 The TPP theorem states that in the direction of the
axis of rotation the velocity component aligned with
gravity (w) does not change. While the constraint in the
direction of the axis of rotation on the other two components
of the velocity vector is influenced by changes
in the density structure, the constraint of the TPP theorem
on the horizontal components of the velocity vector
is in fact the classical thermal wind relation in the
case of a vertical rotation vector, and we will call it the
generalized thermal wind relation for arbitrary directions
of the rotation vector.
There are numerous discussions of the TPP theorem
when applied to both large- and mesoscale ocean dynamics,
that is, when the Rossby number associated
with the dynamics is (very) small. In our case, however,
the Rossby number Ro w rms /(fL) is an order of unity
or larger, and the dynamics are clearly in a three-dimensional
regime. The TPP theorem nevertheless
leaves an imprint in the turbulent dynamics. To investigate
this imprint we calculate the correlation of a component
of the velocity vector in two parallel planes that
are separated in the y direction by a distance y. The
southward plane, which spans the entire (periodic) domain
in the x direction and 1 km in the z direction
(from a depth of 2250 to 1250 m), is kept fixed, while
the northward plane, which is of equal size, is shifted in
the x and z direction by the amount (x, z). The correlation
between a velocity component in the two
planes is then calculated. In mathematical terms (applied
to the U component), we calculate the following
correlations: