Chapter 9 - Response of the
Upper Ocean to Winds

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
Fridtjof Nansen noticed that wind tended to blow ice at an angle of 20°-40° to the right of the wind in the Arctic, by which he meant that the track of the iceberg was to the right of the wind looking downwind (See Figure 9.2) . He later worked out the balance of forces that must exist when wind tried to push icebergs downwind on a rotating Earth.

Figure 9.2 The balance of forces acting on an iceberg in a wind on a rotating Earth.

Nansen argued that three forces must be important:

1. Wind Stress, W;
2. Friction F (otherwise the iceberg would move as fast as the wind);
3. Coriolis Force, C.

Nansen argued further that the forces must have the following attributes:

1. Drag must be opposite the direction of the ice's velocity;
2. Coriolis force must be perpendicular to the velocity;
3. The forces must balance for steady flow.

Ekman's Solution
Nansen asked Vilhelm Bjerknes to let one of Bjerknes students make a theoretical study of the influence of Earth's rotation on wind-driven currents. Walfrid Ekman was chosen, and he presented the results in his thesis at Uppsala. Ekman later expanded the study to include the influence of continents and differences of density of water (Ekman, 1905). The following follows Ekman's line of reasoning in that paper.

Ekman assumed a steady, homogeneous, horizontal flow with friction on a rotating Earth. Thus horizontal and temporal derivatives are zero:

(9.6)

Ekman further assumed a constant vertical eddy viscosity of the form (8:12):

(9.7)

where T_xz, T_yz are the components of the wind stress in the x, y directions, and ρ_w 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:

	(9.8a)
	(9.8b)

where f is the Coriolis parameter. It is easy to verify that the equations (9.9) have solutions:

	(9.9a)
	(9.9b)

when the wind is blowing to the north (T = T_yz ). The constants are

(9.10)

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

	(9.11a)
	(9.11b)

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):

(9.12)

Figure 9.3. Ekman current generated by a 10m/s wind at 35°N.

Values for Ekman's Constants
To proceed further, we need values for any two of the free parameters: the velocity at the surface, V₀; the coefficient of eddy viscosity, A_z; or the wind stress T.

The wind stress is well known, and Ekman used the bulk formula (4.2):

(9.13)

where ρ_air is the density of air, C_D 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:

(9.14)

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
The thickness of the Ekman layer is arbitrary because the Ekman currents decrease exponentially with depth. Ekman proposed that the thickness be the depth D_E at which the current velocity is opposite the velocity at the surface, which occurs at a depth D_E = π/a, and the Ekman layer depth is:

(9.15)

Using (9.13) in (9.10) , dividing by U₁₀, and using (9.14) and (9.15) gives:

(9.16)

in SI units; wind in meters per second gives depth in meters. The constant in (9.16) is based on ρ_w = 1027 kg/m³, ρ_air = 1.25 kg/m³, and Ekman's value of C_D = 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 A_z ∂²U/∂z². The ratio of the forces, which is non dimensional, is called the Ekman Number E_z:

(9.17)

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

(9.18)

which agrees with the functional form (9.15) proposed by Ekman. Equating (9.18) and (9.15) requires E_z = 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 Ekman layer at the bottom of the ocean and the atmosphere differs from the layer at the ocean surface. The solution for a bottom layer below a fluid with velocity U in the x-direction is:

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.

Figure 9.4 Ekman layer for the lowest kilometer in the atmosphere (solid line), together with wind velocity measured by Dobson (1914) - - - . The numbers give height above the surface in meters. The boundary layer at the bottom of the ocean has a similar shape. From Houghton (1977).

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).

Figure 9.5 Trajectories of surface drifters in April 1978 together with surface pressure in the atmosphere averaged for the month. Note that drifters tend to follow lines of constant pressure except in the Kuroshio where ocean currents are fast compared with velocities in the Ekman layer in the ocean. From McNally, et al. (1983).

Examining Ekman's Assumptions
Before considering the validity of Ekman's theory for describing flow in the surface boundary layer of the ocean, let's first examine the validity of Ekman's assumptions. He assumed:

1. No boundaries. This is valid away from coasts.
2. Deep water. This is valid if depth >> 200m.
3. f-plane. This is valid.
4. Steady state. This is valid if wind blows for longer than a pendulum day. Note however that Ekman also calculated a time-dependent solution, as did Hasselmann (1970).
5. A_z is a function of U ²₁₀ only. It is assumed to be independent of depth. This is not a good assumption. The mixed layer may be thinner than the Ekman depth, and A_z will change rapidly at the bottom of the mixed layer because mixing is a function of stability. Mixing across a stable layer is much less than mixing through a layer of a neutral stability. More realistic profiles for the coefficient of eddy viscosity as a function of depth change the shape of the calculated velocity profile. We reconsider this problem below.
6. Homogeneous density. This is probably good, except as it effects stability.

Observations of Flow Near the Sea Surface
Does the flow close to the sea surface agree with Ekman's theory? Measurements of currents made during several, very careful experiments indicate that Ekman's theory is remarkably good. The theory accurately describes the flow averaged over many days. The measurements also point out the limitations of the theory.

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:

1. Inertial currents are the largest component of the flow.
2. The flow is nearly independent of depth within the mixed layer for periods near the inertial period. Thus the mixed layer moves like a slab at the inertial period. Current shear is concentrated at the top of the thermocline.
3. The flow averaged over many inertial periods is almost exactly that calculated from Ekman's theory. The shear of the Ekman currents extends through the averaged mixed layer and into the thermocline. Ralph and Niiler found:

   	(9.20)
   	(9.21)

   The Ekman-layer depth D_E is almost exactly that proposed by Ekman (9.16), but the surface current V₀ is half his value (9.14).
4. The transport is 90° to the right of the wind in the northern hemisphere. The transport direction agrees well with Ekman's theory.

Influence of Stability in the Ekman Layer
Ralph and Niiler (1999) point out that Ekman's choice of an equation for surface currents (9.14), which leads to (9.16), is consistent with theories that include the influence of stability in the upper ocean. Currents with periods near the inertial period produce shear in the thermocline. The shear mixes the surface layers when the Richardson number falls below the critical value (Pollard et al., 1973). This idea, when included in mixed-layer theories, leads to a surface current V₀ that is proportional to O(N/f)

(9.22)

furthermore

(9.23)

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.

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