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

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36 CHAPTER 6. GYRE CIRCULATION
the eastern boundary the stream lines will look as in fig. 6.1 (if we impose it at the western
boundary the picture will flipped with respect to a vertical line). It is clearly seen that stream
lines intersect the western boundary, which means, that there is flow through the western
boundary. This is contrary to the concept of a boundary.
tel-00545911, version 1 - 13 Dec 2010
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wind stress
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wind stress
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North Atlantic
❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄
❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄
❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄
Figure 6.1: Sverdrup Balance: v-component only
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Figure 6.2: Sverdrup Balance: Stream function with u=0 on western boundary
How can we solve the problem? Why not, for a change, look at the real ocean. Measurements
of the ocean indicate that the circulation is, after all, well represented by (6.1) with the only
difference of a strong western boundary currents which we do not have in (6.1). The western
boundary current, which closes the Sverdrup circulation, is dominated by friction (eddy viscosity
effects). From a conceptual view point it is clear that such an area is necessary and that each
stream line has to pass by such an area, as the wind constantly injects (negative) vorticity
(and energy) in the ocean, that has to be dissipated somewhere. But lets be more quantitative.
When including friction at the western or eastern boundary we have to change eq. (6.2) to:
fu + g∂ y η = τ y /H + ν eddy ∂ xx v. (6.6)
The term ν eddy ∂ xx v represents the dominant contribution of friction as it is the meridional
velocity component v that changes fastest in the zonal x-direction. The eddy viscosity ν eddy
is many orders of magnitude larger than the molecular viscosity of sea water. The concept of
eddy viscosity is explained in section 10.2. Near the boundary we can neglect the wind forcing