## Flow in gyres

Suppose that the wind blows zonally across the ocean, with a stronger eastward wind to the north, as in figure 4.3. Away from coastal regions (where friction may be important) the forces present are the zonal wind force (which here we simply denote F W), the Coriolis force (fv and fu) and the pressure gradient force (9]/9x and 9]/9y, where ] = pip), and we represent their balance mathematically as fv 2r + FU fu f-' (4.5a, b)

dx w dy in the zonal and meridional directions, respectively. If there were no wind, the flow would be in geostrophic balance, and indeed the flow is in geostrophic balance at depths greater than 100 m or so, below the level at which the winds' effects are directly felt. Conservation of mass also gives a relation between u and v, namely f + f (4^)

If we cross-differentiate equation 4.5 (i.e., differentiate equation 4.5a with respect to x and equation 4.5b with respect to y and subtract), then the divergence terms vanish using equation 4.6 and the pressure gradient terms cancel, and we obtain a rjx b V, a7)

where b = df/dy is the rate at which the Coriolis parameter increases northward. The balance between the varying wind and the meridional flow embodied in equation 4.7 is known as Sverdrup balance, and the effect of differential rotation is called the beta effect. If the wind stress has a positive curl, that is, if 9FW/2y > 0, then, because b is also positive, v must be negative and the interior flow must be equatorward. There must be a poleward return flow in a boundary current at either the western or the eastern edge of the ocean basin, where the effects of friction conceivably can be such as to balance the Coriolis and wind stress curl terms. But only if the flow returns in the western boundary current can the frictional effects balance the wind stress curl overall, for then the flow overall has the same sense as the wind.

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