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

Lynne Talley, Fall, 2016

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Reading:

DPO: Section 7.5

**
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 tau_{y} and
tau_{x}, in the y and x
directions:

(U_{Ek},V_{Ek}) =
(1/rho*f)*(tau_{y}, - tau_{x}).

The units are: m^{2}/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
m^{3}/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)

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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 25^{o}N, the wind blows from the east at a
speed of u_{10}=8 m/s. At 40^{o}N, the wind blows
from the west at a speed of u_{10}=8 m/s. What is the rate
of wind-driven Ekman mass convergence between 25^{o}N and
40^{o}N (in kg/s)? What is the average Ekman pumping (in cm/s) between
25^{o}N and 40^{o}N? [1 degree of latitude = 111.12
km].

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Last modified: Oct. 24, 2016