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

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42 CHAPTER 6. GYRE CIRCULATION
From what we learned in section 6.2 it seems, at first sight, unlikely that a fluid layer, that is
forced by a wind stress at the surface will develop a velocity independent of depth. It seems
much more likely that a substantial shear will develop in the upper-part (Ekman-layer) of the
fluid, and that the main body of the fluid rests motionless. This is however not the case, the
wind-stress is indeed transferred to the deep layers. How this happens is the subject of this
section.
As we have seen in section 6.2 the transport in the Ekman layer (H Ek ≈ 30m) is given by,
u Ek H Ek = τ y /f (6.33)
v Ek H Ek = −τ x /f (6.34)
using the zero divergence we see that the Ekman dynamics leads to a vertical velocity
w Ek = −∂ x (τ y /f) + ∂ y (τ x /f), (6.35)
tel-00545911, version 1 - 13 Dec 2010
In the geostrophic interior no direct action of the wind-stress is felt and eqs. 6.1 – 6.3 give,
βv = f∂ z w, (6.36)
which is called the Sverdrup relation. On the surface w Ek has to be compensated by a vertical
“geostrophic” velocity w G = −w Ek . Using eq. (6.4) we get,
βHv G = fw G = −fw Ek = fH Ek (∂ x u Ek + ∂ y v Ek ) = f [∂ x (τ y /f) − ∂ y (τ x /f)]. (6.37)
The total zonal (Sverdrup) transport is,
Hv S = Hv G + H Ek v Ek = f/β [∂ x (τ y /f) − ∂ y (τ x /f)] − τ x /f (6.38)
which is identical to 6.4!
What do all this beautiful calculations tell us?
= (∂ x τ y − ∂ y τ x )/β (6.39)
• The Sverdrup transport can be split up between an Ekman transport and a geostrophic
interior transport.
• The Ekman transport is directly set into motion by the by the wind stress through (eddy)
viscous friction.
• The interior dynamics is set up by the vertical velocity induced by the divergence of the
Ekman transport
• The interior dynamics is put into motion by stretching of the water column and the
conservation of planetary potential vorticity (f/H).