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

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Chapter 6
Gyre Circulation
tel-00545911, version 1 - 13 Dec 2010
The ocean is forced at its surface by a wind-stress ˜τ which is measured in Newton/m 2 . A
typical value for the ocean is in the order of 0.1N/m 2 . In the present manuscript we work with
τ = ˜τ/ρ which has units of m 2 /s 2 .
6.1 Sverdrup Dynamics in the SW Model (the math)
In all the ocean basins an almost stationary large scale gyre circulation is observed. We suppose
that this circulation is a consequence of the wind shear at the ocean surface. We thus add some
(wind) forcing to the linearized stationary shallow water equations on the β-plane.
Adding −∂ y (6.1) and ∂ x (6.2) leads to:
−fv + g∂ x η = τ x /H (6.1)
+fu + g∂ y η = τ y /H (6.2)
H(∂ x u + ∂ y v) = 0 (6.3)
+boundary conditions.
Hβv = (∂ x τ y − ∂ y τ x ) (6.4)
So at every point the meridional component of the fluid transport (vH) is completely determined
by the vorticity of the surface stress! Equation (6.4) is called the Sverdrup relation. It says
that the if vorticity is injected into the fluid parcel it can not increase its vorticity as this
would contradict stationarity, so it moves northward where planetary potential vorticity (f/H)
is larger. So the Sverdrup relation is a statement of conservation of potential vorticity in a
forced and stationary situation.
When knowing the wind field, the Sverdrup relation gives v, using the zero divergence of
geostrophic flow we can calculate ∂ x u. If we know u at one point in a ocean basin at every
latitude we can determine u in the whole basin by integrating in the zonal direction,
u(x1,y1) = −
∫ x1
x0
∂ y v(x,y1)dx + u(x0,y1). (6.5)
But u is prescribed at the two boundaries of the ocean basin (as the velocity vector at the
boundary is directed parallel to the boundary), which makes u over-determined. What does
this mean in “physical terms?” Take a look at fig. 6.1, where a caricature of the North
Atlantic with a simplified wind-stress (independent of longitude) is given. The corresponding
v component of the velocity is also given. If we start by imposing a vanishing zonal velocity at
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