Oct 25: Ekman Transport

The details of the Ekman spiral velocity profile as a function of depth are more of theoretical interest that practical importance. Current profiles closely resembling the theoretical result are seldom, if ever, observed.

The details of the spiral profile depend on the assumed eddy viscosity, and A_v = constant is not a particularly good assumption. Recall that the size of turbulent eddies tends to scale with distance from the boundary so that A_v is generally proportional to z which leads to the log-layer dependence. In Ekman dynamics, the log-layer structure is modified by Coriolis.

The fact that the surface current is to the right of the wind (f > 0) is a key result, but the magnitude of the angle will depend on the details.

In practical applications such as oil-spill tracking and search-and-rescue, empirical values for the angle of motion with respect to the wind direction are used based on experience and observation.

However, a robust and important result that is independent of these details is obtained if we integrate the equations over a depth large enough to encompass the whole Ekman layer (in practice, just a few times the Ekman scale depth).

Start with the Ekman equations expressed in terms of the stresses rather than eddy viscosity

u-momentum equation:

We need not actually integrate to –∞ because the Ekman currents decay exponentially fast.

In practice, it is sufficient to integrate from the surface to some depth z=-D, where D is a few times the Ekman depth, at which depth

Similarly, for the v-momentum equation:

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

The details of the eddy viscosity profile have no influence on this result, and the calculation is very robust.

This Ekman balance between wind stress and Coriolis is established over several inertial periods, i.e. the balance is not established instantly when the wind starts blowing.

The ocean response to suddenly imposed winds is a set of inertial oscillations. The inertial oscillations decay over a period that is several times their natural oscillatory timescale f^-1 leaving steady Ekman transports in their wake.

Important concepts:

· Ekman currents are stronger at the surface, and decay approximately exponentially over a depth scale given by . Typical Ekman depths are of order 10 to 30 meters.

· Regardless of the details of the eddy viscosity profile, the Ekman transports are:

which are directed perpendicular to the wind stress direction; right (left) in the northern (southern) hemisphere

· Ekman transports are fully established after several inertial periods, i.e. 1 to 2 days

Ekman dynamics is a very practical way to estimate the oceanic response to winds on time scales of a few days.

Objects floating near the surface within the Ekman layer will be transported by Ekman currents, and their drift can be predicted with considerable skill using these simple equations.

However, Ekman transports have a far more significant impact on the entire upper ocean circulation (over much greater depths than the Ekman layer) through a rather subtle interaction with the oceanic pressure field.

Where Ekman transports converge and diverge they generate pressure gradients that are in turn balanced by the Coriolis force, and the resulting geostrophically balanced currents form the upper ocean pattern of gyres and western boundary currents.