9.2 Ekman Layer at the Sea Surface
Steady winds blowing on the sea surface produce a thin, horizontal boundary layer, the Ekman layer. By thin, I mean a layer that is at most a few-hundred meters thick, which is thin compared with the depth of the water in the deep ocean. A similar boundary layer exists at the bottom of the ocean, the bottom Ekman layer, and at the bottom of the atmosphere just above the sea surface, the planetary boundary layer or frictional layer described in §4.3. The Ekman layer is named after Professor Walfrid Ekman, who worked out its dynamics for his doctoral thesis.
Ekman's work was the first of a remarkable series of studies conducted during the first half of the twentieth century that led to an understanding of how winds drive the ocean's circulation (Table 9.1). In this chapter we consider Nansen and Ekman's work. The rest of the story is given in chapters 11 and 13.
Nansen's Qualitative Arguments
Nansen argued that three forces must be important:
Nansen argued further that the forces must have the following attributes:
Ekman assumed a steady, homogeneous, horizontal flow with friction on a rotating Earth. Thus horizontal and temporal derivatives are zero:
Ekman further assumed a constant vertical eddy viscosity of the form (8:12):
where Txz, Tyz are the components of the wind stress in the x, y directions, and ρw is the density of sea water.
With these assumptions, and using (9.7) in (8.14), the x and y components of the momentum equation have the simple form:
where f is the Coriolis parameter. It is easy to verify that the equations (9.9) have solutions:
To evaluate the constants, let's assume that the wind blows toward the north. There is nothing special about this choice of wind direction. We just need to pick a direction, and north is convenient. When the wind is blowing to the north (T = Tyz ). The constants are
and V0 is the velocity of the current at the sea surface.
Now let's look at the form of the solutions. At the sea surface z = 0, exp(z = 0) = 1, and
The current has a speed of V0 to the northeast. In general, the surface current is 45° to the right of the wind when looking downwind in the northern hemisphere. The current is 45° to the left of the wind in the southern hemisphere. Below the surface, the velocity decays exponentially with depth (Figure 9.3):
Values for Ekman's Constants
The wind stress is well known, and Ekman used the bulk formula (4.2):
where ρair is the density of air, CD is the drag coefficient, and U10 is the wind speed at 10m above the sea. Ekman turned to the literature to obtain values for V0 as a function of wind speed. He found:
With this information, he could then calculate the velocity as a function of depth knowing the wind speed U10 and wind direction.
Ekman Layer Depth
Using (9.13) in (9.10) , dividing by U10, and using (9.14) and (9.15) gives:
in SI units; wind in meters per second gives depth in meters. The constant in (9.16) is based on ρw = 1027 kg/m3, ρair = 1.25 kg/m3, and Ekman's value of CD = 2.6×10-3 for the drag coefficient.
Using (9.16) with typical winds, the depth of the Ekman layer varies from about 45m to 300m (Table 9.3), and the velocity of the surface current varies from 2.5% to 1.1% of the wind speed depending on latitude.
The Ekman Number: Coriolis and Frictional Forces The depth of the Ekman layer is closely related to the depth at which frictional force is equal to the Coriolis force in the momentum equation (9.9) . The Coriolis force is fu, and the frictional force is Az ∂2U/∂z2. The ratio of the forces, which is non dimensional, is called the Ekman Number Ez:
where we have approximated the terms using typical velocities u, and typical depths d. The subscript z is needed because the ocean is stratified and mixing in the vertical is much less than mixing in the horizontal. Note that as depth increases, friction becomes small, and eventually, only the Coriolis force remains.
Solving (9.17) for d gives
which agrees with the functional form (9.15) proposed by Ekman. Equating (9.18) and (9.15) requires Ez = 1/(2π2) = 0.05 at the Ekman depth. Thus Ekman chose a depth at which frictional forces are much smaller than the Coriolis force.
Bottom Ekman Layer
The velocity goes to zero at the boundary, u = v = 0 at z = 0. The direction of the flow close to the boundary is 45° to the left of the flow U outside the boundary layer in the northern hemisphere; and the direction of the flow rotates with distance above the boundary (Figure 9.4). The direction of rotation is anti-cyclonic with distance above the bottom.
Winds above the planetary boundary layer are perpendicular to the pressure gradient in the atmosphere and parallel to lines of constant surface pressure. Winds at the surface are 45° to the left of the winds aloft, and surface currents are 45° to the right of the wind at the surface. Therefore we expect currents at the sea surface to be nearly in the direction of winds above the planetary boundary layer and parallel to lines of constant pressure. Observations of surface drifters in the Pacific tend to confirm the hypothesis (Figure 9.5).
Examining Ekman's Assumptions
Observations of Flow Near the Sea Surface
Weller and Plueddmann (1996) measured currents from 2 m to 132 m using 14 vector-measuring current meters deployed from the Floating Instrument Platform FLIP in February and March 1990 500 km west of point Conception, California. This was the last of a remarkable series of experiments coordinated by Weller using instruments on FLIP.
Davis, DeSzoeke, and Niiler (1981) measured currents from 2m to 175m using 19 vector-measuring current meters deployed from a mooring for 19 days in August and September 1977 at 50°N, 145°W in the northeast Pacific.
Ralph and Niiler (2000) tracked 1503 drifters drogued to 15m depth in the Pacific from March 1987 to December 1994. Wind velocity was obtained every 6 hours from the European Centre for Medium-Range Weather Forecasts ECMWF.
The results of the experiments indicate that:
Influence of Stability in the Ekman Layer
where N is the stability frequence defined by (8.360. Furthermore
Notice that (9.22) and (9.23) are now dimensionally correct. The equations used earlier, (9.14), (9.16), (9.20), and (9.21) all required a dimensional coefficient.
|Department of Oceanography, Texas A&M University
Robert H. Stewart, email@example.com
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Updated on October 1, 2008