## Dynamics V: rotation and wind stress (Ekman layers) and other mixed layer topics

Lynne Talley, Fall, 2015

### Link to powerpoint

Watch powerpoint presentation.

DPO: Section 7.5

Additional, with more dynamics: R. Stewart's online book, chapter 9.2, 9.3, 9.5.

1. Ekman velocity and transport. The wind acts directly and frictionally, through vertical eddy viscosity, on the top 50 to 100 meters of the ocean, in the "Ekman layer". In the northern hemisphere, the frictional surface flow is at an angle to the right of the wind (45 degrees if viscosity is uniform with depth). This frictional surface flow then acts frictionally on the water slightly beneath it, which then is slightly more to the right, etc etc downward with the tips of the vectors tracing a spiral. The frictionally-forced flows become weaker and weaker with depth (exponentially weaker), and die out around 50 to 100 meters down. This spiral is called the "Ekman spiral". The exact details (angle of each successive layer as we move downward through the spiral) of how it spirals depend on the strength and vertical distribution of the vertical eddy viscosity.

If the "transports" are all added up, that is, integrate the velocity over depth at each location, from the bottom to the top of the Ekman layer, the total "Ekman transport" is exactly at right angles to the wind - to the right in the northern hemisphere and left in the southern hemisphere. This direction of the "Ekman transport" is independent of the exact details of the spiral, hence exact details of the vertical eddy viscosity.

The Ekman transport components in the x and y directions (east and north) are proportional to the wind stress tauy and taux, in the y and x directions:
(UEk,VEk) = (1/rho*f)*(tauy, - taux).
The units are: m2/sec, since this is actually just a velocity integrated in the vertical direction, and not over an area. Total Ekman transport across, for instance, a vertical section or line or curve across the ocean, or around a box, would then be integrated along the horizontal curve, yielding a complete transport in m3/sec.

This Ekman effect has been demonstrated by Ralph and Niiler (1999) using surface drifter data from the Pacific (drogues at 15 m). (Figure 7.8 in DPO 6th)

2. Other wind effects in addition to surface waves and Ekman flow: Langmuir circulation
(See DPO section 7.5.2, with much more explanation in the supplementary chapter S7.5.2.)

3. Surface mixed layers: buoyancy and turbulent mixing
(See DPO section 7.4)

### Study questions:

1. If the eddy viscosity were to double, explain if and by how much the (a) Ekman transport, (b) Ekman layer thickness, and (c) surface Ekman velocity would change.

2. Draw a schematic of an open ocean wind field that produces Ekman convergence in the southern hemisphere.

3. Compare the Ekman layer thickness and the observed global surface mixed layer thickness. Where would Ekman layers be contained within the winter surface mixed layer and where would they extend below it?

4. Compare the timescales of surface waves, Langmuir circulation, inertial circulation and a fully-developed Ekman layer.

Study calculation
A steady wind field is blowing on a rectangular ocean basin 8000 km wide. At 25oN, the wind blows from the east at a speed of u10=8 m/s. At 40oN, the wind blows from the west at a speed of u10=8 m/s. What is the rate of wind-driven Ekman mass convergence between 25oN and 40oN (in kg/s)? What is the average Ekman pumping (in cm/s) between 25oN and 40oN? [1 degree of latitude = 111.12 km].

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