Question If Wind Stress Curl Is Reversed What Will Happen to Layer

0) Before finish up with the derivation of the conservation of potential vorticity— matt suggested that I recap some of my comments on Ekman Transport.

 A) the units of are m2/s—or velocity times depth. It can be called the transport per f unit width. The depth is the Ekman Depth ( De   2A v / f  A vT ) where T is the local inertial period. Note that for a constant wind stress the Ekman transport increases with decreasing f and the Ekman depth increases with decreasing f.

LAT F Transport Ekman Depth Draw arrows on board

However, since the transport goes with 1/f while the depth goes with the square root of 1/f, the transport increases more rapidly than the Ekman depth, and consequently the for a constant wind speed depth average current speed in the Ekman layer is larger towards the equator than it is at the pole by the square root of 1/f.

Ekman Pumping.

Start with schematic of negative wind stress curl—driving convergences—thicken upper layer. Show how upper layer converges—and drives down lower layer.

Question—If wind stress curl is reversed what will happen to layer.

A: Upper layer thins.

Q: Now what other forces are at play?

A: Baroclinic pressure gradient arises—can shut down Ekman pumping.

See Box 6.2 in Knauss for the calculation for Ekman pumping.

Draw example of geostrophic flow. See if class can figure out what would happen in bottom frictional layer.

This is called bottom Ekman Layer. Responsible for Coastal Upwelling.

Draw schematic of Eddy. See if class can figure out what friction will do to eddy. This is why surface winds have component down gradient—because of friction. In a frictionless world (sometimes we call this invicid) if you stood with your back to the wind (in the northern hemisphere) the low pressure would be to your left. However, since you’re standing in the Bottom Ekman Layer the flow is not purely geostrophic and the wind has a component towards the low. Subsequently the low is in your forward left quarter.

1) Finish up with derivation of the conservation of potential vorticity. Note that I f forgot to include term v on thursday. y

In the final analysis we end up with the simple equation:

D  f       0 Dt  H 

f   which means that the term is a constant. This is called Potential Vorticity and the H above equation shows that it is conserved (Recall that this calculation was done without surface wind stress or bottom stress—which can add or remove vorticity—but in the absence of friction potential vorticity is conserved.

This is a powerful statement – perhaps the most powerful in the field of geophysical fluid dynamics.

We call the relative vorticity because it is the vorticity relative to the rotating earth.

The term (+f) is the absolute vorticity.

The absolute vorticity divided by the depth is the potential vorticity and this is what is conserved.

This means if any one of the three variables change (f,H, one of the other two must change to compensate in such a way that the potential vorticity remains constant.

For example:

Consider a column of fluid on the North Pole where f=1.5 *10-4 1/s. If it were to move to the south where f=1.0 *10-4 it would acquire a relative vorticity of  0.5 *10-4 if the depth of the fluid (H) remains the same. A positive relative vorticity would be a counter clockwise rotation.

Similarly a column of fluid can thin. This can happen because of topography or due to a converging layer above. Consider the latter in context with the wind forcing in the ocean. Ekman transports tend to drive a convergence in the upper layer in the middle of the ocean’s gyre. This tends to thicken the upper layer and subsequently squash the layer beneath it. The layer beneath it—which we can assume is frictionless because it is not in contact with either the surface and bottom and interfacial friction is small must reduce it’s relative vorticity. So as the layer is squashed H decreases which requires the relative vorticity,  to decrease to retain potential vorticity—corresponding to a clockwise rotation of the fluid. Of course once you’ve generated relative vorticity you have fluid in motion. For the clockwise rotation associated with the squashing of the lower layer the fluid on the east side of the gyre is moving to latitudes of lower f—thus this also will also contribute to balance the “vortex squashing” associated with the wind and thus does not require as large a reduction in the relative vorticity. On the otherside, however, the fluid is moving to the north and thus fluid there is moving into regions of increased f, thus both the increase in f and reduction in H both tend to increase the potential vorticity and this requires an increase in the relative vorticity on the western side of the basin. This is accomplished by increased lateral shears and produces an asymmetrical gyre in the ocean basins.

Sverdrup

Western intensification

Interestingly, one can also invoke friction to explain the existence of western intensification. This was one of Henry Stommel’s many contributions to oceanography.

Wind forcing puts negative vorticity into the ocean.

Columns of fluid heading south on the eastern side of the ocean move to lower latitudes where f is lower and thus pick up positive vorticity which tends to counter act the negative vorticity put in by the wind.

However, columns of fluid moving north on the western side of the basin are moving to regions of increasing f and thus pick up negative vorticity which augments the vorticity put in by the wind. Stommel added friction, which we can assume is proportional to current speed.

But how does friction generate vorticity? If we assume that all of the friction occurs where the fluid rubs agaist the

If the wind field puts in one unit of vorticity