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

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6.1. SVERDRUP DYNAMICS IN THE SW MODEL (THE MATH) 37
and the dominant balance is then,
βv B = ν eddy ∂ xxx v B , (6.7)
which has solutions of the form,
v B = C 1 exp(2x/r) + exp(−x/r) (C 2 cos(−x/˜r) + C 3 sin(−x/˜r)), (6.8)
tel-00545911, version 1 - 13 Dec 2010
with r = (β/ν eddy ) 1/3 /2 and ˜r = r/ √ 3. One condition of the boundary solution is, that it
has to decrease away from the boundary, which means that C 1 = 0 if the boundary current
develops on the western boundary and C 2 = C 3 = 0 if the boundary current develops on the
eastern boundary. The boundary dynamics is there to insure that u = v = 0 on the boundary,
these are two conditions. If the boundary current is on the eastern boundary we have only one
constant to adjust, so it is usually not possible. So the frictional boundary current can do its
job (satisfy the boundary conditions) only if it is on the western boundary.
There are still other dynamical arguments why the boundary current can not be on the
eastern boundary: (i) in the situation in fig. 6.1 the wind injects negative vorticity in the flow,
vorticity is conserved by the fluid column moving with the flow, not subject to any forcing.
In a stationary state the vorticity extracted has to be re-injected during the cyclical path of
a fluid column. A boundary layer at the western border does exactly this. A boundary layer
at the eastern border would drain even more vorticity, which leads to a contradiction in terms
of the vorticity balance. (ii) The dynamical adjustment in the ocean is performed by Rossby
waves, which have a westward group velocity. This means that the dynamics at a point adjusts
to the dynamics to its eastern side. That’s what the boundary current does, so it has to be to
the extreme western part of the basin to adjust to the entire interior dynamics.
On the southern hemisphere the boundary current is also on the western boundary as β
(unlike f) has the same sign on both hemispheres! In the above derivation of the Sverdrup
transport only β but not f was involved.
So the big picture is: (i) the ocean interior is well described by Sverdrup dynamics, (ii) which
is complemented at the western boundary by a thin boundary current, which is dominated by
friction.
Comment 1: The wind stress induces a transport (uH,vH) rather than a velocity (u,v).
Exercise 36: which dynamics would we expect in fig. 6.1 when rotation vanishes?
Exercise 37: in the above calculations we have neglected the non-linear terms. This is only
valid when the Rossby numbers are small. What is the Rossby number of the interior flow at
mid latitudes when v = .1m/s, L = 5000km. What is the Rossby number of the boundary
layer flow at mid latitudes when v = 1.0m/s, L = 100km.
Comment 2: For the Sverdrup relation to apply, it is not so much the Rossby number that
has to be small but the two terms neglected, (i) the time derivative of the relative vorticity
∂ t ζ and (ii) the non-linear term u∇ζ, have to be small compared to the transport of planetary
vorticity vβ. Observations show that the mean wind forcing and thus the mean circulation
changes only slightly during several years in large parts of the worlds ocean. The total vorticity,
measured from an inertial frame, of the fluid motion on our planet can be decomposed in the
relative part, measured from a frame moving (rotating) with the surface of the earth, and
the planetary part given by the Coriolis parameter f. In the boundary layer, however, the
non-linear term is not smaller than the transport of planetary vorticity and there are nonlinear
phenomena in the western boundary currents, as for example the Gulf-Stream and the
Kuroshio, which are not well explained by the above theory.