Oct 28: Ekman pumping and wind stress curl

 

Balance between Coriolis and the vertical friction mixing of momentum, or stress, leads to Ekman currents. Integrating this balance vertically (in practice over the top few tens of meters) gives us the Ekman transport:

 

 

These components of Ekman transport describe depth integrated flow (in m2s-1) (i.e. velocity times depth) that is 900 to the right (left) of the wind stress in the northern (southern) hemisphere.

 

The units of m2 s-1 can be thought of as the total transport (m3 s-1) per meter perpendicular to the current.

 

Conveniently, the details of the eddy viscosity profile (the vertical rate of momentum mixing) are irrelevant to this transport result.

 

 

In an infinite ocean, uniform winds would generate uniform Ekman transports and the ocean currents would be the same everywhere.

 

But the ocean is not infinite, and winds are variable, so Ekman transports are not spatially uniform, which leads to converge and divergence of the surface currents.

 

Upwelling

 

This effect is dramatic at the coast where winds parallel to the coast cannot drive Ekman transports across the coastline. The details of the ocean circulation response in this case are complicated, but the dominant features of the transport patterns can be deduced from mass balance concepts.

 

 

Assuming the upwelling pattern is 2-dimensional and uniform alongshore, the Ekman transport offshore must be balanced by water uplifted from below.

 

The zone of active upwelling can be seen as a band of cold water adjacent to the coast, and this has a characteristic with determined by the Rossby radius which depends

 

 

In coastal NJ waters the scales are roughly h = 10m, density difference of 2 kg m-3, and f=10-4 s-1. g =9.81 m s-2, so

 

R ~ (10 x 10 x 2/1025)1/2 104 = 14 km

 

The vertical transport due to upwelling occurs over this horizontal distance next to the coast, so an average vertical velocity can be estimated from mass conservation.

 

Mass conservation also demands that flow feed the upwelling, and this would come from offshore in the 2-dimensional idealized case, or possibly from a divergence of the along shelf flow.

 

There must be an alongshelf flow because of the pressure gradient set up by horizontal density pattern (geostrophy) which will come to later.

 

 

 

Ekman pumping.

 

Q: Recalling what you know about the global patterns of winds, what latitudes would you expect to be characterized by converging Ekman transports and therefore downward Ekman pumping?

 

A: The region between the Trades and Westerlies

 

The net influence of converging or diverging horizontal Ekman transports can be quantified by considering the conservation of mass equation:

 

 

Figure 3.24 Ocean Circulation: Ekman pumping (convergence and divergence)

 

Recall that the Ekman transport components are related to the wind stress:

 

 

Where the Ekman pumping velocity wE is negative, i.e. there is a convergence of Ekman transports that pump water downward into the ocean interior.

 

This downward Ekman pumping between the Trades and Westerlies generates a depressed thermocline in the center of the subtropical gyres.

 

This is the case in both hemispheres, because the sign of curl wind stress and f both differ, so we < 0.

 

The baroclinic pressure gradients associated with this drive the large scale gyre circulations, and conservation of mass closes the gyre circulations with intense poleward western boundary currents.

 

Suggested reading:

 

        Chapters 3.2, 3.3, 3.4 of Ocean Circulation

        Section 9.4 of Pond and Pickard

        Chapter 9 of Stewart