Why is ocean surface velocity around 3% of wind velocity?

Question

Many introductory level textbooks suggest that, as a rule of thumb, the velocity of the ocean surface is around 3% the wind velocity at 10 meters above the sea surface.

For instance, Ocean Circulation by the Open University Press, page 39:

Another useful empirical observation is that the surface current speed is typically 3% of the wind speed, so that a 10 m/s wind might be expected to give rise to a surface current of about 0.3 m/s

What are the mechanisms that result in this ratio? How universal is this relationship?

Answer 1

A very good analysis looking at the 3% relationship is an article by Weber (1983). His main result is that:

We assume a fully developed sea, and take the eddy viscosity to be proportional to the friction velocity times a characteristic depth. Hence the total current (Ekman current plus wave-induced current) can be expressed as functions of the wind speed. The results show that the magnitude of the total surface current lies between 3.1 and 3.4% of the wind speed at 10 m height for winds between 5 and 30 m s⁻¹.

The idea is that by including both the Ekman current (induced by the mean wind stress at the surface) and the wave-induced current (Stokes drift), the resulting total surface velocity can be easily represented as a function of wind velocity at 10 meters. Both Ekman and wave-induced currents are about the same size and add up to be around 3% of the wind speed. The assumptions include the full range for Ekman dynamics (homogeneous, not affected by boundaries), plus some about waves (non-decaying, small amplitude). In the real ocean, eddy viscosity is vertically variable, but, as it is the case with Ekman dynamics, the approximation is quite close to what is observed in the ocean. The approximations that Weber used are valid for a large range of wind values (5-30 m s⁻¹) and they break down in weak wind (not fully developed waves) and strong wind conditions (whitecapping, wave breaking).

Weber, J. E., 1983: Steady Wind- and Wave-Induced Currents in the Open Ocean. J. Phys. Oceanogr., 13, 524–530.