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 Section 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 \(\PageIndex{1}\)). In this chapter we consider Nansen and Ekman’s work. The rest of the story is given in Chapters 11 and 13.

Table \(\PageIndex{1}\): Inertial Oscillations

Latitude \((\varphi)\)	\(T_{i} \ (\text{hr})\)	\(D \ (\text{km})\)
	for \(V = 20 \ \text{cm/s}\)
\(90^{\circ}\)	11.97	2.7
\(35^{\circ}\)	20.87	4.8
\(10^{\circ}\)	68.93	15.8

Nansen’s Qualitative Arguments

Fridtjof Nansen noticed that wind tended to blow ice at an angle of 20\(^{\circ}\)–40\(^{\circ}\) 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 \(\PageIndex{1}\)). He later worked out the balance of forces that must exist when wind tried to push icebergs downwind on a rotating Earth.

Figure \(\PageIndex{1}\): 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, \(\mathbf{W}\);
2. Friction, \(\mathbf{F}\) (otherwise the iceberg would move as fast as the wind);
3. Coriolis Force, \(\mathbf{C}\).

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: \(\mathbf{W} + \mathbf{F} + \mathbf{C} = 0\)

Table \(\PageIndex{2}\). Contributions to the Theory of the Wind-Driven Circulation.

Fridtjof Nansen	(1898)	Qualitative theory, currents transport water at an angle to the wind.
Vagn Walfrid Ekman	(1902)	Quantitative theory for wind-driven transport at the sea surface
Harald Sverdrup	(1947)	Theory for wind-driven circulation in the eastern Pacific.
Henry Stommel	(1948)	Theory for westward intensification of wind-driven circulation (western boundary currents).
Walter Munk	(1950)	Quantitative theory for main features of the winddriven circulation.
Kirk Bryan	(1963)	Numerical models of the oceanic circulation.
Bert Semtner and Robert Chervin	(1988)	Global, eddy-resolving, realistic model of the ocean’s circulation.

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 (Kullenberg, 1954). 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: \[\frac{\partial}{\partial t} = \frac{\partial}{\partial x} = \frac{\partial}{\partial y} = 0 \nonumber \]

For flow on a rotating Earth, this leaves a balance between frictional and Coriolis forces \((8.3.4)-(8.3.5)\). Ekman further assumed a constant vertical eddy viscosity of the form \((8.3.2)\): \[T_{xz} = \rho A_{z} \frac{\partial u}{\partial z}, \quad T_{yz} = \rho A_{z} \frac{\partial v}{\partial z} \nonumber \]

where \(T_{xz}, T_{yz}\) are the components of the wind stress in the \(x, y\) directions, and \(\rho\) is the density of sea water.

Using \((\PageIndex{2})\) in \((8.3.4)-(8.3.5)\), the \(x\) and \(y\) momentum equations are: \[\begin{align} fv + A_{z} \frac{\partial^{2} u}{\partial z^{2}} &= 0 \\ -fu + A_{z} \frac{\partial^{2} v}{\partial z^{2}} &= 0 \end{align} \nonumber \]

where \(f\) is the Coriolis parameter.

It is easy to verify that the equations \((\PageIndex{3}-\PageIndex{4})\) have solutions: \[\begin{align} u &= V_{0} \exp (az) \cos (\pi/4 + az) \\ v &= V_{0} \exp (az) \sin (\pi/4 + az) \end{align} \nonumber \]

when the wind is blowing to the north \((T = T_{yz})\). The constants are \[V_{0} = \frac{T}{\sqrt{\rho^{2}_{w} f A_{z}}} \quad \text{and} \quad a = \sqrt{\frac{f}{2A_{z}}} \nonumber \]

and \(V_{0}\) 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 \[\begin{align} u(0) &= V_{0} \cos(\pi/4) \\ v(0) &= V_{0} \sin(\pi/4) \end{align} \nonumber \]

The current has a speed of \(V_{0}\) to the northeast. In general, the surface current is 45\(^{\circ}\) to the right of the wind when looking downwind in the northern hemisphere. The current is 45\(^{\circ}\) to the left of the wind in the southern hemisphere. Below the surface, the velocity decays exponentially with depth (figure \(\PageIndex{2}\)): \[\left[u^{2} (z) + v^{2} (z)\right]^{1/2} = V_{0} \exp(az) \nonumber \]

Figure \(\PageIndex{2}\): Ekman current generated by a 10 m/s wind at 35\(^{\circ}\)N.

Values for Ekman’s Constants

To proceed further, we need values for any two of the free parameters: the velocity at the surface, \(V_{0}\); the coefficient of eddy viscosity, \(A_{z}\); or the wind stress \(T\).

The wind stress is well known, and Ekman used the bulk formula \((4.6.1)\): \[T_{yz} = T = \rho_{air} C_{D} U_{10}^{2} \nonumber \]

where \(\rho_{air}\) is the density of air, \(C_{D}\) is the drag coefficient, and \(U_{10}\) is the wind speed at 10 m above the sea. Ekman turned to the literature to obtain values for \(V_{0}\) as a function of wind speed. He found: \[V_{0} = \frac{0.0127}{\sqrt{\sin |\varphi|}} U_{10}, \quad\quad |\varphi| \geq 10 \nonumber \]

With this information, he could then calculate the velocity as a function of depth knowing the wind speed \(U_{10}\) 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 \(D_{E} = \pi/a\), and the Ekman layer depth is: \[\boxed{D_{E} = \sqrt{\frac{2 \pi^{2} A_{z}}{f}}} \nonumber \]

Using \((\PageIndex{10})\) in \((\PageIndex{7})\), dividing by \(U_{10}\), and using \((\PageIndex{12})\) and \((\PageIndex{13})\) gives: \[D_{E} = \frac{7.6}{\sqrt{\sin |\varphi|}} U_{10} \nonumber \]

in SI units. Wind in meters per second gives depth in meters. The constant in \((\PageIndex{14})\) is based on \(\rho = 1027 \ \text{kg/m}^{3}\), \(\rho_{air} = 1.25 \ \text{kg/m}^{3}\), and Ekman’s value of \(C_{D} = 2.6 \times 10^{-3}\) for the drag coefficient.

Using \((\PageIndex{14})\) with typical winds, the depth of the Ekman layer varies from about 45 to 300 meters (Table \(\PageIndex{3}\)), and the velocity of the surface current varies from 2.5% to 1.1% of the wind speed depending on latitude.

Table \(\PageIndex{3}\). Typical Ekman Depths.

	Latitude
\(U_{10} \ [\text{m/s}]\)	\(15^{\circ}\)	\(45^{\circ}\)
\(5\)	\(75 \ \text{m}\)	\(45 \ \text{m}\)
\(10\)	\(150 \ \text{m}\)	\(90 \ \text{m}\)
\(20\)	\(300 \ \text{m}\)	\(180 \ \text{m}\)

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 \((\PageIndex{5})-(\PageIndex{6})\). The Coriolis force is \(fu\), and the frictional force is \(A_{z} \ \partial^{2}U/\partial z^{2}\). The ratio of the forces, which is nondimensional, is called the Ekman Number \(E_{z}\):

\[E_{z} = \frac{\text{Friction Force}}{\text{Coriolis Force}} = \frac{A_{z} \dfrac{\partial^{2} u}{\partial z^{2}}}{fu} = \frac{A_{z} \dfrac{u}{d^{2}}}{fu} \nonumber \]\[\boxed{E_{z} = \frac{A_{z}}{f d^{2}}} \nonumber \]

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 \((\PageIndex{15})\) for \(d\) gives \[d = \sqrt{\frac{A_{z}}{f E_{z}}} \nonumber \]

which agrees with the functional form \((\PageIndex{13})\) proposed by Ekman. Equating \((\PageIndex{16})\) and \((\PageIndex{13})\) requires \(E_{z} = 1/(2\pi^{2}) = 0.05\) 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: \[\begin{align} u &= U[1 - \exp (-az) \cos az] \\ v &= U \exp (-az) \sin az \end{align} \nonumber \]

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\(^{\circ}\) 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 \(\PageIndex{3}\)). The direction of rotation is anticyclonic with distance above the bottom.

Figure \(\PageIndex{3}\): Ekman layer in the lowest kilometer of the atmosphere (solid line), with wind velocity measured by Dobson (1914) (dotted line). The numbers give height above the surface in meters. The Ekman layer at the sea floor has a similar shape. After Houghton (1977: 107).

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\(^{\circ}\) to the left of the winds aloft, and surface currents are 45\(^{\circ}\) 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 \(\PageIndex{4}\)).

Figure \(\PageIndex{4}\): 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. After 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 \(>> 200 \ \text{m}\).
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 \(U_{10}^{2}\) 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. I 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.

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 kilometers 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 2 m to 175 m using 19 vector-measuring current meters deployed from a mooring for 19 days in August and September 1977 at 50\(^{\circ}\)N, 145\(^{\circ}\)W in the northeast Pacific.

Ralph and Niiler (2000) tracked 1503 drifters drogued to 15 m 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 (using SI units, \(U\) in m/s): \[D_{E} = \frac{7.12}{\sqrt{\sin |\varphi|}} U_{10} \nonumber \] \[V_{0} = \frac{0.0068}{\sqrt{\sin |\varphi|}} U_{10} \nonumber \]The Ekman-layer depth \(D_{E}\) is almost exactly that proposed by Ekman \((\PageIndex{14})\), but the surface current \(V_{0}\) is half his value \((\PageIndex{12})\).
4. The angle between the wind and the flow at the surface depends on latitude, and it is near 45\(^{\circ}\) at mid latitudes (figure \(\PageIndex{5}\)).
5. The transport is 90\(^{\circ}\) 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 (2000) point out that Ekman’s choice of an equation for surface currents \((\PageIndex{12})\), which leads to \((\PageIndex{14})\), 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 \(V_{0}\) that is proportional to \(\sqrt{N/f}\): \[V_{0} \sim U_{10} \sqrt{N/f} \nonumber \]

where \(N\) is the stability frequency defined by \((8.5.6)\). Furthermore

\[A_{z} \sim U_{10}^{2}/N \quad\quad \text{and} \quad\quad D_{E} \sim U_{10} \sqrt{N f} \nonumber \]

Notice that \((\PageIndex{21})\) and \((\PageIndex{22})\) are now dimensionally correct. The equations used earlier, \((\PageIndex{12})\), \((\PageIndex{14})\), \((\PageIndex{19})\), and \((\PageIndex{20})\) all required a dimensional coefficient.

Figure \(\PageIndex{5}\): Angle between the wind and flow at the surface calculated by Maximenko and Niiler using positions from drifters drogued at 15 m with satellite-altimeter, gravity, and grace data and winds from the NCAR/NCEP reanalysis.