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

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Chapter 10
Penetration of Surface Fluxes
tel-00545911, version 1 - 13 Dec 2010
The ocean is mostly driven by the fluxes of heat, fresh water and momentum at its surface.
The influence of these fluxes are, however, not only felt in a thin layer at the ocean surface, but
influence the dynamics of the entire ocean. In this chapter we discuss how the forcing applied
at the surface of the ocean penetrates into the depth of the ocean.
For the processes of vertical penetration, it is clear that we can no longer neglect the
dynamics in the vertical direction, and the shallow water equations are not adapted for the
processes studied here (with the exception of gravity currents). We thus have to look for other
simplifications of the the full three-dimensional Navier-Stokes equations. A first guess might
be to neglect the dynamics all together and pretend that the transport to the interior is due
to molecular motions, that is viscosity and diffusivities (for heat and salt). This possibility is
discussed and refuted in section 10.1.
In section 10.2 we show, using the Navier-Stokes equations and some “hand-waving” that
the three dimensional dynamics at small scales creates some viscous and diffusive behavior at
large scales. This idea is the basis of all realistic calculations not only in ocean dynamics but
in fluid dynamics in general.
10.1 Molecular Transport
The molecular thermal diffusivity of sea water is κ ≈ 10 −7 m 2 s −1 . The diffusion equation in the
vertical is given by,
∂ t T = ∂ z (κ∂ z T). (10.1)
We further suppose that there is a periodic heat flux of magnitude Q at the surface (boundary
condition), that is:
∂ z T | z=0 = Q cos(2πt/τ + π/4). (10.2)
c p ρκ
The linear equation (10.1) with the boundary conditions (10.2) has the solution:
with:
T(z,t) = T A e −z/L cos(2πt/τ − z/L), (10.3)
T A = Q √ √ τ τκ
c p ρ 2πκ and L = π . (10.4)
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