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.

Ekman layer
The Ekman layer is the layer in a fluid where the flow is the result of a balance between pressure gradient, Coriolis and turbulent drag forces. In the picture above, the wind blowing North creates a surface stress and a resulting Ekman spiral is found below it in the column of water.

History

Ekman developed the theory of the Ekman layer after Fridtjof Nansen observed that ice drifts at an angle of 20°-40° to the right of the prevailing wind direction while on an Arctic expedition aboard the Fram. Nansen asked his colleague, Vilhelm Bjerknes to set one of his students upon study of the problem. Bjerknes tapped Ekman, who presented his results in 1902 as his doctoral thesis.[1]

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.

where and are the velocities in the and directions, respectively, is the local Coriolis parameter, and is the diffusive eddy viscosity, which can be derived using mixing length theory. Note that 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:[2]

where and are the components of the surface stress, , of the wind field or ice layer at the top of the ocean.

For the boundary condition on the other side, as , where and are the geostrophic flows in the and directions.

Solution

EkmanSpiral
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:

The value 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 , and the latitude, as encapsulated by . For a typical , and at 45° latitude (), then is approximately 45 meters. This Ekman depth prediction does not always agree precisely with observations.

This variation of horizontal velocity with depth () 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

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 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,[3] saying

It is obvious that cannot generally be regarded as a constant when the density of water is not uniform within the region considered

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.[1] Surface Ekman layers can also be observed in rotating tanks.[2]

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 boundary layer into the surface layer and the Ekman layer generally yields more accurate results.[4]

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,[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]

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.[6] The Vector Measuring Current Meter [7] 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 flow 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 Kerguelan 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]

See also

References

  1. ^ Cushman-Roisin, Benoit (1994). "Chapter 5 - The Ekman Layer". Introduction to Geophysical Fluid Dynamics (1st ed.). Prentice Hall. pp. 76–77. ISBN 0-13-353301-8.
  2. ^ a b Vallis, Geoffrey K. (2006). "Chapter 2 – Effects of Rotation and Stratification". Atmospheric and Oceanic Fluid Dynamics (1st ed.). Cambridge, UK: Cambridge University Press. pp. 112–113. ISBN 0-521-84969-1.
  3. ^ Ekman, V.W. (1905). "On the influence of the earth's rotation on ocean currents". Ark. Mat. Astron. Fys. 2 (11): 1–52.
  4. ^ a b Holton, James R. (2004). "Chapter 5 – The Planetary Boundary Layer". Dynamic Meteorology. International Geophysics Series. 88 (4th ed.). Burlington, MA: Elsevier Academic Press. pp. 129–130. ISBN 0-12-354015-1.
  5. ^ Santala, M. J.; Terray, E. A. (1992). "A technique for making unbiased estimates of current shear from a wave-follower". Deep-Sea Research. 39 (3–4): 607–622. Bibcode:1992DSRI...39..607S. doi:10.1016/0198-0149(92)90091-7.
  6. ^ Rudnick, Daniel (2003). "Observations of Momentum Transfer in the Upper Ocean: Did Ekman Get It Right?". Near-Boundary Processes and their Parameterization. Manoa, Hawaii: School of Ocean and Earth Science and Technology.
  7. ^ Weller, R.A.; Davis, R.E. (1980). "A vector-measuring current meter". Deep-Sea Research. 27 (7): 565–582. Bibcode:1980DSRI...27..565W. doi:10.1016/0198-0149(80)90041-2.
  8. ^ Hunkins, K. (1966). "Ekman drift currents in the Arctic Ocean". Deep-Sea Research. 13: 607–620. Bibcode:1966DSROA..13..607H. doi:10.1016/0011-7471(66)90592-4.
  9. ^ Davis, R.E.; de Szoeke, R.; Niiler., P. (1981). "Part II: Modelling the mixed layer response". Deep-Sea Research. 28 (12): 1453–1475. Bibcode:1981DSRI...28.1453D. doi:10.1016/0198-0149(81)90092-3.
  10. ^ a b Price, J.F.; Weller, R.A.; Schudlich, R.R. (1987). "Wind-Driven Ocean Currents and Ekman Transport". Science. 238: 1534–1538. Bibcode:1987Sci...238.1534P. doi:10.1126/science.238.4833.1534. PMID 17784291.
  11. ^ a b Chereskin, T.K. (1995). "Direct evidence for an Ekman balance in the California Current". Journal of Geophysical Research. 100: 18261–18269. Bibcode:1995JGR...10018261C. doi:10.1029/95JC02182.
  12. ^ a b Lenn, Y; Chereskin, T.K. (2009). "Observation of Ekman Currents in the Southern Ocean". Journal of Physical Oceanography. 39: 768–779. Bibcode:2009JPO....39..768L. doi:10.1175/2008jpo3943.1.
  13. ^ Cronin, M.F.; Kessler, W.S. (2009). "Near-Surface Shear Flow in the Tropical Pacific Cold Tongue Front". Journal of Physical Oceanography. 39: 1200–1215. Bibcode:2009JPO....39.1200C. doi:10.1175/2008JPO4064.1.
  14. ^ a b Roach, C.J.; Phillips, H.E.; Bindoff, N.L.; Rintoul, S.R. (2015). "Detecting and Characterizing Ekman Currents in the Southern Ocean". Journal of Physical Oceanography. 45: 1205–1223. Bibcode:2015JPO....45.1205R. doi:10.1175/JPO-D-14-0115.1.

External links

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Boundary current

Boundary currents are ocean currents with dynamics determined by the presence of a coastline, and fall into two distinct categories: western boundary currents and eastern boundary currents.

Boundary layer

In physics and fluid mechanics, a boundary layer is an important concept and refers to the layer of fluid in the immediate vicinity of a bounding surface where the effects of viscosity are significant.

In the Earth's atmosphere, the atmospheric boundary layer is the air layer near the ground affected by diurnal heat, moisture or momentum transfer to or from the surface. On an aircraft wing the boundary layer is the part of the flow close to the wing, where viscous forces distort the surrounding non-viscous flow.

Coriolis–Stokes force

In fluid dynamics, the Coriolis–Stokes force is a forcing of the mean flow in a rotating fluid due to interaction of the Coriolis effect and wave-induced Stokes drift. This force acts on water independently of the wind stress.

This force is named after Gaspard-Gustave Coriolis and George Gabriel Stokes, two nineteenth-century scientists. Important initial studies into the effects of the Earth's rotation on the wave motion – and the resulting forcing effects on the mean ocean circulation – were done by Ursell & Deacon (1950), Hasselmann (1970) and Pollard (1970).

The Coriolis–Stokes forcing on the mean circulation in an Eulerian reference frame was first given by Hasselmann (1970):

to be added to the common Coriolis forcing Here is the mean flow velocity in an Eulerian reference frame and is the Stokes drift velocity – provided both are horizontal velocities (perpendicular to ). Further is the fluid density, is the cross product operator, where is the Coriolis parameter (with the Earth's rotation angular speed and the sine of the latitude) and is the unit vector in the vertical upward direction (opposing the Earth's gravity).

Since the Stokes drift velocity is in the wave propagation direction, and is in the vertical direction, the Coriolis–Stokes forcing is perpendicular to the wave propagation direction (i.e. in the direction parallel to the wave crests). In deep water the Stokes drift velocity is with the wave's phase velocity, the wavenumber, the wave amplitude and the vertical coordinate (positive in the upward direction opposing the gravitational acceleration).

Ekman

Ekman is a surname of Swedish origin which may refer to

Ekman number

The Ekman number (Ek) is a dimensionless number used in fluid dynamics to describe the ratio of viscous forces to Coriolis forces. It is frequently used in describing geophysical phenomena in the oceans and atmosphere in order to characterise the ratio of viscous forces to the Coriolis forces arising from planetary rotation. It is named after the Swedish oceanographer Vagn Walfrid Ekman.

When the Ekman number is small, disturbances are able to propagate before decaying owing to low frictional effects. The Ekman number also describes the order of magnitude for the thickness of an Ekman layer, a boundary layer in which viscous diffusion is balanced by Coriolis effects, rather than the usual convective inertia.

Ekman spiral

The Ekman spiral is a structure of currents or winds near a horizontal boundary in which the flow direction rotates as one moves away from the boundary. It derives its name from the Swedish oceanographer Vagn Walfrid Ekman. The deflection of surface currents was first noticed by the Norwegian oceanographer Fridtjof Nansen during the Fram expedition (1893–1896) and the effect was first physically explained by Vagn Walfrid Ekman.

Ekman transport

Ekman transport, part of Ekman motion theory first investigated in 1902 by Vagn Walfrid Ekman, refers to the wind-driven net transport of the surface layer of a fluid that, due to the Coriolis effect, occurs at 90° to the direction of the surface wind. This phenomenon was first noted by Fridtjof Nansen, who recorded that ice transport appeared to occur at an angle to the wind direction during his Arctic expedition during the 1890s. The direction of transport is dependent on the hemisphere: in the northern hemisphere, transport occurs at 90° clockwise from wind direction, while in the southern hemisphere it occurs at a 90° counterclockwise.

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).

Geophysical fluid dynamics

Geophysical fluid dynamics, in its broadest meaning, refers to the fluid dynamics of naturally occurring flows, such as lava flows, oceans, and planetary atmospheres, on Earth and other planets.Two physical features that are common to many of the phenomena studied in geophysical fluid dynamics are rotation of the fluid due to the planetary rotation and stratification (layering). The applications of geophysical fluid dynamics do not generally include the circulation of the mantle, which is the subject of geodynamics, or fluid phenomena in the magnetosphere.

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