Ekman layer

We will consider boundary conditions of the Ekman layer in the upper ocean:[2]

${\displaystyle {\text{at }}z=0:\quad A{\frac {\partial u}{\partial z}}=\tau ^{x}\quad {\text{and}}\quad A{\frac {\partial v}{\partial z}}=\tau ^{y},}$

where ${\displaystyle \ \tau ^{x}}$ and ${\displaystyle \ \tau ^{y}}$ are the components of the surface stress, ${\displaystyle \ \tau }$, of the wind field or ice layer at the top of the ocean, and ${\displaystyle \ A\equiv \rho K_{m}}$ is the dynamic viscosity.

For the boundary condition on the other side, as ${\displaystyle \ z\to -\infty :u\to u_{g},v\to v_{g}}$, where ${\displaystyle \ u_{g}}$ and ${\displaystyle \ v_{g}}$ are the geostrophic flows in the ${\displaystyle \ x}$ and ${\displaystyle \ y}$ directions.

Three views of the wind-driven Ekman layer at the surface of the ocean in the Northern Hemisphere. The geostrophic velocity is zero in this example.

These differential equations can be solved to find:

${\displaystyle {\begin{aligned}u&=u_{g}+{\frac {\sqrt {2}}{\rho _{o}fd}}e^{z/d}\left[\tau ^{x}\cos(z/d-\pi /4)-\tau ^{y}\sin(z/d-\pi /4)\right],\\[5pt]v&=v_{g}+{\frac {\sqrt {2}}{\rho _{o}fd}}e^{z/d}\left[\tau ^{x}\sin(z/d-\pi /4)+\tau ^{y}\cos(z/d-\pi /4)\right],\\[5pt]d&={\sqrt {2K_{m}/|f|}}.\end{aligned}}}$

The value ${\displaystyle d}$ 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 ${\displaystyle K_{m}}$, and the latitude, as encapsulated by ${\displaystyle f}$. For a typical ${\displaystyle K_{m}=0.1}$ m${\displaystyle ^{2}}$/s, and at 45° latitude (${\displaystyle f=10^{-4}}$ s${\displaystyle ^{-1}}$), then ${\displaystyle d}$ is approximately 45 meters. This Ekman depth prediction does not always agree precisely with observations.

This variation of horizontal velocity with depth (${\displaystyle -z}$) 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

${\displaystyle w={\frac {1}{f\rho _{o}}}\left[-\left({\frac {\partial \tau ^{x}}{\partial x}}+{\frac {\partial \tau ^{y}}{\partial y}}\right)e^{z/d}\sin(z/d)+\left({\frac {\partial \tau ^{y}}{\partial x}}-{\frac {\partial \tau ^{x}}{\partial y}}\right)(1-e^{z/d}\cos(z/d))\right].}$

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.

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 ${\displaystyle z}$ goes to infinity.

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, 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.[4]

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,[4] and instrumentation sensitive enough to observe a velocity profile in such a shallow depth has only been available since around 1980.[2] Also, wind waves modify the flow near the surface, and make observations close to the surface rather difficult.[5]

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.[6] The Vector Measuring Current Meter [7] and the Acoustic Doppler Current Profiler are both used to measure current.

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.[8] More recent observations include (not an exhaustive list):

• The 1980 mixed layer experiment[9]
• Within the Sargasso Sea during the 1982 Long Term Upper Ocean Study [10]
• Within the California Current during the 1993 Eastern Boundary Current experiment [11]
• Within the Drake Passage region of the Southern Ocean [12]
• In the eastern tropical Pacific, at 2°N, 140°W, using 5 current meters between 5 and 25 meters depth.[13] 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 [14]

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.[10][11][12][14]