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

142 CHAPITRE 4. ETUDES DE PROCESSUS OCÉANOGRAPHIQUES
562 Ocean Dynamics (2009) 59:551–563
tel-00545911, version 1 - 13 Dec 2010
downslope transport in his simulations using a Mellor–
Yamada parametrisation of vertical mixing.
The vertical resolution in today’s OGCMs is too
sparse to explicitly resolve the two-layer structure of
the gravity current dynamics. The friction layer is, at
best, a few metres thick, and its explicit representation
asks for a resolution of the order of 1 m at the bottom,
a resolution that is a few hundred times smaller than
the actually employed vertical resolution at depth in
OGCMs. General circulation models are, thus, not able
to correctly predict the two-part structure of gravity
currents, underestimate the detrainment and overestimate
the descent of the vein (when neglecting other
difficulties which inhibit the descent of the vein). Or,
even worse, the detrained water will be found diluted
in the grid-box downstream of the vein; as the thickness
of the grid-box is much larger than that of the Ekman
layer, the water will continue to move with the vein, and
the diluted water will be counted as entrained rather
than detrained. Following these arguments, it seems
difficult to correctly represent the dynamics of a gravity
current when the vertical extension of the bottom gridbox
is smaller than the Ekman layer thickness.
Large-scale instabilities are explicitly suppressed
in the research presented here by choosing a 2.5-
dimensional geometry. Our results are, however, key
to the research concerning the evolution of such instability
as it gives the basic state from which large-scale
instabilities will grow. So far, in the research on the
instability of the vein of a gravity current, the basic state
was conjectured or chosen to simplify the calculations
(Meacham and Stephens 2001). The stable law of the
shape of a gravity current, presented in Subsection 3.3,
can now be used as the basis of analytical or numerical
three-dimensional stability analysis.
In the experiments presented here, the Ekman number
(the vertical viscosity ν v ) was fixed, leading to a
laminar Ekman layer dynamics in our experiments,
when a low value of the reduced gravity is considered. I
exploited this dynamics and obtained that the evolution
of the shape of the vein is governed by the heat equation
with a diffusion coefficient, which can be calculated
from the external parameters of the problem. For
higher values of the reduced gravity, or lower values
of the Richardson number, the vertical velocity within
the Ekman layer leads to a change in its dynamics, and
the linear friction laws are no longer applicable but are
replaced by a quadratic drag law. I showed that, when
a quadratic drag law applies, the dynamics is governed
by a nonlinear heat equation.
It would be desirable to perform the 2.5-dimensional
Boussinesq simulations with an explicitly resolved turbulent
Ekman layer, which requires resolutions in the
centimetre scale in the vertical and horizontal direction,
but this is beyond our actual computer resources. In the
ocean, the vertical turbulent viscosity or even the friction
law depends on unobserved quantities as the bottom
roughness. Ongoing research is directed towards
the determination of the friction laws and coefficients
by data assimilation (Wirth and Verron 2008); by doing
so, I hope to be able to avoid the explicit resolution and
determination of small-scale processes and, nevertheless,
obtain solid estimates for the small-scale turbulent
fluxes, which will allow us to concentrate on the study
of large-scale features.
This work is complementary to the theories of
Killworth (2001) and others, who suppose that the
thickness of the gravity current is determined locally by
the entrainment or detrainment at the upper interface
through a local Froude number criteria. The theory
presented here determines the thickness non-locally by
the convergences and divergences of the Ekman fluxes
at the lower boundary and determines non-locally the
evolution of the overall shape of the gravity current, using
a (non-local) heat equation. This proves that there
cannot be a purely local parametrisation of gravity
current thickness. In realistic gravity currents, the two
mechanisms are likely to act simultaneously. Furthermore,
it was found in laboratory experiments on nonrotating
gravity currents that the dissipative processes
at the interface have a negligible role compared to
those due to bottom friction (Ermanyuk and Gavrilov
2007). In oceanic gravity currents, the mixing and entrainment
is enhanced by the roughness of the ocean
floor, the change of slope and large-scale instabilities,
features which are not considered here and which are
the subjects of future research.
Acknowledgements I am grateful to Bernard Barnier, Yves
Morel, Joel Sommeria and Jacques Verron for discussion and to
two anonymous reviewers for their remarks which have greatly
improved the paper. This work is part of the COUGAR project
funded by ANR-06-JCJC-0031-01.
Appendix: Force balance in a rotating gravity current
In the x direction (upslope), the dominant force balance
is between the Coriolis force and reduced gravity.
In the y direction, the reduced gravity vanishes and
the dominant force balance is between friction and
the Coriolis force (Fig. 7). Using linear Ekman layer
theory, I obtain:
f H
v
cosθ sin θ − ν√ 2 v
cos(θ + π/4) = 0. (8)
δ cos θ