Ekman theory explains the theoretical state of circulation if water currents were driven only by the transfer of momentum from the wind. In the physical world, this is difficult to observe because of the influences of many simultaneous
current driving forces (for example,
pressure and
density gradients). Though the following theory technically applies to the idealized situation involving only wind forces, Ekman motion describes the wind-driven portion of circulation seen in the surface layer. Surface currents flow at a 45° angle to the wind due to a balance between the Coriolis force and the
drags generated by the wind and the water. If the ocean is divided vertically into thin layers, the magnitude of the velocity (the speed) decreases from a maximum at the surface until it dissipates. The direction also shifts slightly across each subsequent layer (right in the Northern Hemisphere and left in the Southern Hemisphere). This is called an
Ekman spiral. The layer of water from the surface to the point of dissipation of this spiral is known as the
Ekman layer. If all flow over the Ekman layer is integrated, the net transportation is at 90° to the right (left) of the surface wind in the Northern (Southern) Hemisphere. The oceanic wind driven Ekman spiral is the result of a force balance created by a
shear stress force,
Coriolis force and the water drag. This force balance gives a resulting current of the water different from the winds. In the ocean, there are two places where the Ekman spiral can be observed. At the surface of the ocean, the shear stress force corresponds with the
wind stress force. At the bottom of the ocean, the
shear stress force is created by
friction with the ocean floor. This phenomenon was first observed at the surface by the Norwegian oceanographer
Fridtjof Nansen during his
Fram expedition. He noticed that icebergs did not drift in the same direction as the wind. His student, the Swedish oceanographer
Vagn Walfrid Ekman, was the first person to physically explain this process.
Bottom Ekman spiral In order to derive the properties of an Ekman spiral a look is taken at a uniform, horizontal
geostrophic interior flow in a homogeneous fluid. This flow will be denoted by \vec{u} = (\bar{u},\bar{v}), where the two components are constant because of uniformity. Another result of this property is that the horizontal gradients will equal zero. As a result, the
continuity equation will yield, \frac{\partial w}{\partial z} = 0. Note that the concerning interior flow is horizontal, so w = 0 at all depths, even in the boundary layers. In this case, the
Navier-Stokes momentum equations, governing geophysical motion can now be reduced to: : \begin{align} -fv &= -\frac{1}{\rho_0}\frac{\partial p}{\partial x} + \nu_E \frac{\partial^2 u}{\partial z^2}, \\[5pt] fu &= -\frac{1}{\rho_0}\frac{\partial p}{\partial y} + \nu_E \frac{\partial^2 v}{\partial z^2}, \\[5pt] 0 &= -\frac{1}{\rho_0}\frac{\partial p}{\partial z}, \end{align} Where f is the
Coriolis parameter, \rho_0 the fluid
density and \nu_E the
eddy viscosity, which are all taken as a constant here for simplicity. These parameters have a small variance on the scale of an Ekman spiral, thus this approximation will hold. A uniform flow requires a uniformly varying
pressure gradient. When substituting the flow components of the interior flow, u = \bar{u} and v = \bar{v}, in the equations above, the following is obtained: : \begin{align} -f\bar{v} &= -\frac{1}{\rho_0}\frac{\partial p}{\partial x} = \text{constant} \\[5pt] f\bar{u} &= -\frac{1}{\rho_0}\frac{\partial p}{\partial y} = \text{constant} \end{align} Using the last of the three equations at the top of this section, yields that the pressure is independent of depth. : \begin{align} -f(v - \bar{v}) &= \nu_E \frac{\partial^2 u}{\partial z^2} \\[5pt] f(u - \bar{u}) &= \nu_E \frac{\partial^2 v}{\partial z^2} \end{align} u = \bar{u} + A e^{\lambda z} and v = \bar{v} + B e^{\lambda z} will suffice as a solution to the differential equations above. After substitution of these possible solutions in the same equations, \nu_E^2\lambda^4 + f^2 = 0 will follow. Now, \lambda has the following possible outcomes: : \lambda = \pm (1 \pm i)\sqrt{\frac{f}{2\nu_E}} Because of the
no-slip condition at the bottom and the constant interior flow for z \gg d, coefficients A and B can be determined. In the end, this will lead to the following solution for \vec{u}(z): and that the Ekman layer depth and thus the Ekman spiral is less deep than expected. There are three main factors which contribute to the reason why this is,
stratification,
turbulence and horizontal gradients.
waves and the
Stokes-Coriolis force.
Ekman layer The
Ekman layer is the layer in a
fluid where there is a
force balance between
pressure gradient force,
Coriolis force and
turbulent drag. It was first described by
Vagn Walfrid Ekman. Ekman layers occur both in the atmosphere and in the ocean. There are two types of Ekman layers. The first type occurs at the surface of the ocean and is forced by surface winds, which act as a drag on the surface of the ocean. The second type occurs at the
bottom of the atmosphere and ocean, where frictional forces are associated with flow over rough surfaces.
Mathematical formulation The mathematical formulation of the Ekman layer begins by assuming a neutrally stratified fluid, a balance between the forces of pressure gradient, Coriolis and turbulent drag. : \begin{align} -fv &= -\frac{1}{\rho_o} \frac{\partial p}{\partial x}+K_m \frac{\partial^2 u}{\partial z^2}, \\[5pt] fu &= -\frac{1}{\rho_o} \frac{\partial p}{\partial y}+K_m \frac{\partial^2 v}{\partial z^2}, \\[5pt] 0 &= -\frac{1}{\rho_o} \frac{\partial p}{\partial z}, \end{align} where \ u and \ v are the velocities in the \ x and \ y directions, respectively, \ f is the local
Coriolis parameter, and \ K_m is the diffusive eddy viscosity, which can be derived using
mixing length theory. Note that p is a
modified pressure: we have incorporated the
hydrostatic of the pressure, to take account of gravity. There are many regions where an Ekman layer is theoretically plausible; they include the bottom of the atmosphere, near the surface of the earth and ocean, the bottom of the ocean, near the
sea floor and at the top of the ocean, near the air-water interface. Different
boundary conditions are appropriate for each of these different situations. Each of these situations can be accounted for through the boundary conditions applied to the resulting system of ordinary differential equations. The separate cases of top and bottom boundary layers are shown below.
Ekman layer at the ocean (or free) surface We will consider boundary conditions of the Ekman layer in the upper ocean: : \text{at } z = 0 : \quad A \frac{\partial u}{\partial z} = \tau^x \quad \text{and} \quad A \frac{\partial v}{\partial z} = \tau^y, where \ \tau^x and \ \tau^y are the components of the surface stress, \ \tau , of the wind field or ice layer at the top of the ocean, and \ A \equiv \rho K_m is the dynamic viscosity. For the boundary condition on the other side, as \ z \to -\infty : u \to u_g, v \to v_g, where \ u_g and \ v_g are the
geostrophic flows in the \ x and \ y directions.
Solution These differential equations can be solved to find: : \begin{align} u &= u_g + \frac{\sqrt{2}}{\rho_ofd}e^{z/d}\left [\tau^x \cos(z/d - \pi/4) - \tau^y \sin(z/d - \pi/4)\right ], \\[5pt] v &= v_g + \frac{\sqrt{2}}{\rho_ofd}e^{z/d}\left [\tau^x \sin(z/d - \pi/4) + \tau^y \cos(z/d - \pi/4)\right ], \\[5pt] d &= \sqrt{2 K_m/|f|}. \end{align} The value d is called the Ekman layer depth, and gives an indication of the penetration depth of wind-induced turbulent mixing in the ocean. Note that it varies on two parameters: the turbulent diffusivity K_m, and the latitude, as encapsulated by f. For a typical K_m=0.1 m^2/s, and at 45° latitude (f=10^{-4} s^{-1}), then d is approximately 45 meters. This Ekman depth prediction does not always agree precisely with observations. This variation of horizontal velocity with depth (-z) is referred to as the
Ekman spiral, diagrammed above and at right. By applying the continuity equation we can have the vertical velocity as following :w = \frac{1}{f\rho_o}\left [-\left (\frac{\partial \tau^x}{\partial x} + \frac{\partial \tau^y}{\partial y} \right )e^{z/d}\sin(z/d) + \left (\frac{\partial \tau^y}{\partial x} - \frac{\partial \tau^x}{\partial y} \right )(1-e^{z/d}\cos(z/d))\right ]. Note that when vertically-integrated, the volume transport associated with the Ekman spiral is to the right of the wind direction in the Northern Hemisphere.
Ekman layer at the bottom of the ocean and atmosphere The traditional development of Ekman layers bounded below by a surface utilizes two boundary conditions: • A
no-slip condition at the surface; • The Ekman velocities approaching the geostrophic velocities as z goes to infinity.
Experimental observations of the Ekman layer There is much difficulty associated with observing the Ekman layer for two main reasons: the theory is too simplistic as it assumes a constant eddy viscosity, which Ekman himself anticipated, saying and because it is difficult to design instruments with great enough sensitivity to observe the velocity profile in the ocean.
Laboratory demonstrations The bottom Ekman layer can readily be observed in a rotating cylindrical tank of water by dropping in dye and changing the rotation rate slightly. Surface Ekman layers can also be observed in rotating tanks.
In the atmosphere In the atmosphere, the Ekman solution generally overstates the magnitude of the horizontal wind field because it does not account for the velocity shear in the
surface layer. Splitting the
planetary boundary layer into the surface layer and the Ekman layer generally yields more accurate results.
In the ocean The Ekman layer, with its distinguishing feature the Ekman spiral, is rarely observed in the ocean. The Ekman layer near the surface of the ocean extends only about 10 – 20 meters deep,
Instrumentation Observations of the Ekman layer have only been possible since the development of robust surface moorings and sensitive current meters. Ekman himself developed a current meter to observe the spiral that bears his name, but was not successful. The Vector Measuring Current Meter and the
Acoustic Doppler Current Profiler are both used to measure current.
Observations The first documented observations of an Ekman-like spiral in the ocean were made in the Arctic Ocean from a drifting ice floe in 1958. More recent observations include (not an exhaustive list): • The 1980
mixed layer experiment • Within the Sargasso Sea during the 1982 Long Term Upper Ocean Study • Within the California Current during the 1993 Eastern Boundary Current experiment • Within the Drake Passage region of the Southern Ocean • In the eastern tropical Pacific, at 2°N, 140°W, using 5 current meters between 5 and 25 meters depth. This study noted that the geostrophic shear associated with tropical stability waves modified the Ekman spiral relative to what is expected with horizontally uniform density. • North of the Kerguelen Plateau during the 2008 SOFINE experiment Common to several of these observations spirals were found to be "compressed", displaying larger estimates of eddy viscosity when considering the rate of rotation with depth than the eddy viscosity derived from considering the rate of decay of speed. ==See also==