**Notes:****The last lecture we saw how conservation of potential vorticity governs the evolution of flows in the ocean. What can we learn from vorticity about the steady, large-scale circulation using vorticity?**

**Sverdrup combined geostrophy with Ekman dynamics to find a powerful relation for the size and pattern of ocean's circulation in response to the winds. We are going to derive Sverdrup's balance by first considering the geostrophic interior and the frictional surface boundary layer (Ekman layer) separately, in order to highlight how these two regimes couple to form an ocean gyre.**

**Initially, we cross differentiate the u and v momentum equations as we did in last class, but this time we assume a steady state:**

**Cross-differentiating:**

So, the divergence of the flow is proportional (at any latitude) to its northward or meridional velocity. How can that be? Remember last class, if you change the thickness of a water column over a rotating sphere, it will need to change its spin to conserve PV. If ζ << f, then the water will move north/south!

If we vertically integrate the geostrophic velocity balance we find:

Then:

Thus, at any latitude, the meridional transport is proportional to the vertical velocity at the top of the geostrophic ocean.

**Again we cross-differentiate these equations:**

Vertically integrate over the depth of the Ekman layer, using as boundary conditions w(z

The second term on the left hand side is small except near the Equator, where β > f. The equation then becomes:

**We can combine the Ekman and geostrophic layers very simply by looking at the two vorticity equations we have.**

Equating w

This is a pointwise balance, true for anywhere in the ocean where the following assumptions are valid:

- Flow is steady - d/dt = 0;

- Flow is frictionless below the Ekman layer;

- w

So, the wind-driven ocean gyres can be though of as two layers:

(1) A surface layer where the curl of the wind field drives convergences and divergences which drive vertical flows (or changes in layer depth).

(2) An interior, geostrophic layer driven by the vertical flows (Ekman pumping/suction) that result from divergent/convergent Ekman transports.

Sverdrup did the entire problem at once, by taking the curl of the momentum equations with the wind stress terms included:

This depth-integrated balance drops the details of the Ekman layer + geostrophic interior physics, to give us the (deceptively) simple result that the WIND STRESS CURL DRIVES THE MERIDIONAL TRANSPORT.