Sverdrup Balance (1) The large-scale, upper ocean flow in the ...

Sverdrup Balance(1) The large-scale, upper ocean flow in the ocean interior is broad, slow,and extends over thousands of kilometers. This is true for all of the majorocean basins.Typically(1)so thatFrom this, we can conclude that the center of the large-scale ocean gyres,far from boundaries, is in geostrophic balance.(2) Recall the plane, f = f o + y.For this to be a useful approximation, we needso for geostrophic phenomena on scales smaller than a few thousandkilometers, we can use the plane approximation.(2)(3)1

(3) Geostrophic and Ekman effectsLet u=(u, v) be the horizontal velocity, composed of a geostrophic and anEkman part. Thus, u = u g + u e . The equations of motion then become(4)This can be separated into Ekman and geostrophic parts. The geostrophicpart is(5)The Ekman part is(6)The first set of equations can be solved for v g in terms of w to find thatThis is a simple vorticity equation for the geostrophic, mid-ocean flow. Theleft side is the planetary vorticity; the right side is the vortex stretching.This can be integrated to find thatwhere V is the meridional velocity averaged between the surface and z= D and w E is the Ekman pumping or suction velocity.. The verticalvelocity w is assumed to fall to zero at z = D.(7)(8)2

Using the Ekman equations, and incorporating the -plane approximation,it can be shown that (see the notes on Ekman pumping and suction)(9)Putting the Ekman and geostrophic parts together, we find that(10)¡ ¡ ¡Thus the total meridional transport (left side) is equal to the geostrophictransport (first term on right, usually referred to as the Sverdrup relation)plus Ekman transport (second term on right).Questions:(1) Does this work?(2) Can it be tested?(3) What modifications might be necessary?3