# Ekman velocity

In oceanography, Ekman velocity – also referred as a kind of the residual ageostropic velocity as it derivates from geostrophy – is part of the total horizontal velocity (u) in the upper layer of water of the open ocean. This velocity, caused by winds blowing over the surface of the ocean, is such that the Coriolis force on this layer is balanced by the force of the wind.

Typically, it takes about two days for the Ekman velocity to develop before it is directed at right angles to the wind. The Ekman velocity is named after the Swedish oceanographer Vagn Walfrid Ekman (1874–1954).

## Theory

Through vertical eddy viscosity, winds act directly and frictionally on the Ekman layer, which typically is the upper 50–100m in the ocean. The frictional surface flow (u) is at an angle to the right of the wind (45 degrees if viscosity is uniform in the vertical z-direction). This surface flow then modifies the flow slightly beneath it, which then is slightly more to the right, and finally the exponentially-weaker-with-depth flow vectors die down at around 50–100 meters, and finally form a spiral, called the Ekman spiral. The angle of each successive layer as we move downward through the spiral depends on the strength and vertical distribution of the vertical eddy viscosity.

When the contributions from all the vertical layers are added up – the integration of the velocity over depth, from the bottom to the top of the Ekman layer – the total "Ekman transport" is exactly 90 degrees to the right of the wind direction in the Northern Hemisphere and left in the Southern Hemisphere.

## Mathematical formulation

Suppose geostrophic balance is achieved in the Ekman layer, and wind stress is applied at the water surface:

${\displaystyle f{\hat {\boldsymbol {z}}}\times {\boldsymbol {u}}=-\nabla \phi +{\frac {\partial {\boldsymbol {\tau }}}{\partial z}},}$ (1)

where ${\displaystyle \phi =p/\rho _{0},\,}$ ${\displaystyle {\boldsymbol {\tau }}}$ is the applied stress divided by ${\displaystyle \rho _{0}\,}$ (the mean density of water in the Ekman layer)；${\displaystyle {\hat {\boldsymbol {z}}}}$ is the unit vector in the vertical direction (opposing the direction of gravity).

The definition of Ekman velocity is the difference between the total horizontal velocity (${\displaystyle {\boldsymbol {u}}}$) and the geostrophic velocity (${\displaystyle {\boldsymbol {u}}_{g}}$):

${\displaystyle {\boldsymbol {u}}_{e}={\boldsymbol {u}}-{\boldsymbol {u}}_{g}.}$ (2)

As the geostropic velocity (${\displaystyle {\boldsymbol {u}}_{g}}$) is defined as

${\displaystyle {\boldsymbol {u}}_{g}={\frac {1}{f}}{\hat {\boldsymbol {z}}}\times \nabla \phi ,}$ (3)

therefore

${\displaystyle f{\hat {\boldsymbol {z}}}\times {\boldsymbol {u}}_{e}={\frac {\partial {\boldsymbol {\tau }}}{\partial z}}}$ (4)

or

${\displaystyle {\boldsymbol {u}}_{e}=-{\hat {\boldsymbol {z}}}\times {\Big (}{\frac {1}{f}}{\frac {\partial {\boldsymbol {\tau }}}{\partial z}}{\Big )}.}$ (5)

Next, the Ekman transport is obtained by integrating the Ekman velocity from the bottom level (${\displaystyle z=z_{b}\,}$) – at which the Ekman velocity vanishes – to the surface (${\displaystyle z=z_{t}\,}$).

${\displaystyle {\boldsymbol {U}}_{e}=\int _{z_{b}}^{z_{t}}{\boldsymbol {u}}_{e}\;dz=-{\hat {\boldsymbol {z}}}\times {\Big (}{\frac {{\boldsymbol {\tau }}_{t}-{\boldsymbol {\tau }}_{b}}{f}}{\Big )}.}$ (6)

The SI unit of Ekman transport is: m2·s−1, which is the horizontal velocity integrated in the vertical direction.

## Usage

Based on Ekman theory and geostrophic dynamics, the analysis of near-surface currents, i.e. tropical Pacific near-surface currents, can be generated by using high resolution data of wind and altimeter sea level. The surface velocity is defined as the motion of a standard World Ocean Circulation Experiment/Tropical Ocean-Global Atmosphere (WOCE/TOGA) 15m drogue drifter. Near-surface Ekman velocity can be estimated with variables which best represent the ageostrophic motion of the WOCE/TOGA 15m drogue drifters relative to the surface wind stress. Geostrophic velocities are calculated with sea level gradients which are derived from TOPEX/Poseidon sea surface height analyses (TOPEX/Poseidon altimeter sea level anomalies from along-track data is used here, interpolated to 1°X1°grid, spanning the domain of 25°N-25°S,90°E-290°E, during Oct1992-Sep1998).[1] Geostrophic and Ekman velocity are assumed to satisfy the lowest-order dynamics of the surface velocity, and they can be obtained independently from surface height and wind stress data. Standard f plan satisfies geostrophic balance, the lowest-order balance for quasi-steady circulation at higher latitudes.[2] However, coriolis parameter f is close to zero near the equator, the geostropic balance is not satisfied as the velocity is proportional to the height gradient divided by the coriolis parameter f. It has been shown in many studies that beta plane geostrophic approximation involving the second derivative of surface height is in good agreement with the observed velocities in the equatorial undercurrent,[3][4] as a result, geostrophic currents which are near equator are obtained with a weighted blend of the equatorial beta-plane and conventional f-plane geostrophic equations.[5]

A negative sea surface temperature (SST) anomaly prevails in the eastern equatorial Pacific from October to January. A zone of strong easterly Ekman flow propagates westward into the central Pacific basin near the date line during December–February. Relaxation of trade winds in the eastern coincided with the eastward propagation of the geostrophic flow east of 240°E (particularly in February), while westward currents dominated in the central and western equatorial region, which then reversal in the east, with weak local trade winds and weak upwelling along the coast, coincided with the onset of warm SST anomaly.(This anomaly first appeared off South America in March and April). Geostrophic current anomaly, like a Kelvin wave signature propagating eastward to South America between December and April can be easily discerned, and this arrival to South America also coincided with the above-mentioned SST anomaly onset. The geostrophic reversed in April to a strong eastward jet spanning the whole equatorial Pacific. As the El Niño SST anomaly developed during May and June, this eastward geostropic flow persisted.[6]

## Footnotes

1. ^ Fu, L., E. J. Christensen, C. A. Yamarone, M. Lefebvfe, Y. Menard, M. Dorer, and P. Escudier, 1994: TOPEX/POSEIDON mission overview, J. Geophys. Res., 99, 24,369-24,382.
2. ^ Pedlosky,J., Geophysical Fluid Dynamics, 624 pp., Springer-Verlag,New York, 1979.
3. ^ Lukas, R., and E. Firing, 1984: The geostrophic balance of the Pacific Equatorial Undercurrent, Deep Sea Res., Part A, 31, 61-66.
4. ^ Picaut, J., S. P. Hayes, and M. J. McPhaden, 1989: Use of the geostrophic approximation to estimate time-varying zonal currents at the equator, J. Geophys Res., 94, 3228-323.
5. ^ Lagerloef,G. S. E., G. Mitchum, R. Lukas and P. Niiler, 1999: Tropical Pacific near-surface currents estimated from altimeter, wind and drifter data, J. Geophys. Res., 104, 23,313-23,326.
6. ^ El Nino Tropical Pacific Currents

## References

• Pedlosky, J. (1996), Ocean Circulation Theory (2nd ed.), Springer, 453 pp., ISBN 3-540-60489-8
• Salmon, R. (1998), Lectures on Geophysical Fluid Dynamics, Oxford University Press, 378 pp., ISBN 0-19-510808-6
• Vallis, G. K. (2006), Atmospheric and Oceanic Fluid Dynamics, Cambridge University Press, 745 pp., ISBN 0-521-84969-1
• Lagerloef, G. S. E.; Mitchum, G.; Lukas, R.; Niiler, P. (1999), "Tropical Pacific near-surface currents estimated from altimeter, wind and drifter data", J. Geophys. Res., 104 (C10): 23,313–23,326, doi:10.1029/1999JC900197
• Fu, L., E. J. Christensen, C. A. Yamarone, M. Lefebvfe, Y. Menard, M. Dorer, and P. Escudier, 1994: TOPEX/POSEIDON mission overview, J. Geophys. Res., 99, 24,369-24,382，doi:10.1029/94JC01761
• Pedlosky,J., Geophysical Fluid Dynamics, 624 pp., Springer-Verlag,New York, 1979.
• Lukas, R., and E. Firing, 1984：The geostrophic balance of the Pacific Equatorial Undercurrent, Deep Sea Res., Part A, 31, 61-66. doi:10.1016/0198-0149(84)90072-4
• Picaut, J., S. P. Hayes, and M. J. McPhaden, 1989：Use of the geostrophic approximation to estimate time-varying zonal currents at the equator, J. Geophys Res., 94, 3228-323. doi:10.1029/JC094iC03p03228