The Ekman layer is the layer in a
fluid
In physics, a fluid is a liquid, gas, or other material that continuously deforms (''flows'') under an applied shear stress, or external force. They have zero shear modulus, or, in simpler terms, are substances which cannot resist any shear ...
where there is a
force
In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a p ...
balance between
pressure gradient force
In fluid mechanics, the pressure-gradient force is the force that results when there is a difference in pressure across a surface. In general, a pressure is a force per unit area, across a surface. A difference in pressure across a surface t ...
,
Coriolis force and
turbulent drag. It was first described by
Vagn Walfrid Ekman
Vagn Walfrid Ekman (3 May 1874 – 9 March 1954) was a Swedish oceanographer.
Born in Stockholm to Fredrik Laurentz Ekman, himself an oceanographer, he became committed to oceanography while studying physics at the University of Uppsala and ...
. 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.
History
Ekman developed the theory of the Ekman layer after
Fridtjof Nansen observed that
ice
Ice is water frozen into a solid state, typically forming at or below temperatures of 0 degrees Celsius or Depending on the presence of impurities such as particles of soil or bubbles of air, it can appear transparent or a more or less opaqu ...
drifts at an angle of 20°–40° to the right of the
prevailing wind
In meteorology, prevailing wind in a region of the Earth's surface is a surface wind that blows predominantly from a particular direction. The dominant winds are the trends in direction of wind with the highest speed over a particular point o ...
direction while on an
Arctic
The Arctic ( or ) is a polar regions of Earth, polar region located at the northernmost part of Earth. The Arctic consists of the Arctic Ocean, adjacent seas, and parts of Canada (Yukon, Northwest Territories, Nunavut), Danish Realm (Greenla ...
expedition aboard the
Fram
Fram may refer to:
Ships
* ''Fram'' (ship), an arctic exploration vessel from Norway
* MS ''Fram'', expedition cruise ship owned by Hurtigruten Group
Places and geography
* Fram, Paraguay, a town in Itapúa, Paraguay
* Fram Formation, a se ...
. Nansen asked his colleague,
Vilhelm Bjerknes
Vilhelm Friman Koren Bjerknes ( , ; 14 March 1862 – 9 April 1951) was a Norwegian physicist and meteorologist who did much to found the modern practice of weather forecasting. He formulated the primitive equations that are still in use in num ...
to set one of his students upon study of the problem. Bjerknes tapped Ekman, who presented his results in 1902 as his
doctoral thesis
A thesis ( : theses), or dissertation (abbreviated diss.), is a document submitted in support of candidature for an academic degree or professional qualification presenting the author's research and findings.International Standard ISO 7144: ...
.
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
The Coriolis frequency ''ƒ'', also called the Coriolis parameter or Coriolis coefficient, is equal to twice the rotation rate ''Ω'' of the Earth multiplied by the sine of the latitude \varphi.
:f = 2 \Omega \sin \varphi.\,
The rotation rate ...
, and
is the diffusive eddy viscosity, which can be derived using
mixing length theory
In fluid dynamics, the mixing length model is a method attempting to describe momentum transfer by turbulence Reynolds stresses within a Newtonian fluid boundary layer by means of an eddy viscosity. The model was developed by Ludwig Prandtl ...
. Note that
is a
modified pressure
Some systems in fluid dynamics involve a fluid being subject to conservative body forces. Since a conservative body force is the gradient of some potential function, it has the same effect as a gradient in fluid pressure. It is often convenient to ...
: we have incorporated the
hydrostatic
Fluid statics or hydrostatics is the branch of fluid mechanics that studies the condition of the equilibrium of a floating body and submerged body "fluids at hydrostatic equilibrium and the pressure in a fluid, or exerted by a fluid, on an imme ...
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:
:
where
and
are the components of the surface stress,
, of the wind field or ice layer at the top of the ocean, and
is the dynamic viscosity.
For the boundary condition on the other side, as
, where
and
are the
geostrophic
A geostrophic current is an oceanic current in which the pressure gradient force is balanced by the Coriolis effect. The direction of geostrophic flow is parallel to the isobars, with the high pressure to the right of the flow in the Northern ...
flows in the
and
directions.
Solution
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
m
/s, and at 45° latitude (
s
), 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
In fluid dynamics, the no-slip condition for viscous fluids assumes that at a solid boundary, the fluid will have zero velocity relative to the boundary.
The fluid velocity at all fluid–solid boundaries is equal to that of the solid boundary. C ...
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,
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 slightl
Surface Ekman layers can also be observed in rotating tank
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
In meteorology, the planetary boundary layer (PBL), also known as the atmospheric boundary layer (ABL) or peplosphere, is the lowest part of the atmosphere and its behaviour is directly influenced by its contact with a planetary surface. On Ear ...
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,
and instrumentation sensitive enough to observe a velocity profile in such a shallow depth has only been available since around 1980.
Also,
wind waves
In fluid dynamics, a wind wave, water wave, or wind-generated water wave, is a surface wave that occurs on the free surface of bodies of water as a result from the wind blowing over the water surface. The contact distance in the direction of t ...
modify the flow near the surface, and make observations close to the surface rather difficult.
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
An acoustic Doppler current profiler (ADCP) is a hydroacoustic current meter similar to a sonar, used to measure water current velocities over a depth range using the Doppler effect of sound waves scattered back from particles within the water ...
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
Mixed is the past tense of ''mix''.
Mixed may refer to:
* Mixed (United Kingdom ethnicity category), an ethnicity category that has been used by the United Kingdom's Office for National Statistics since the 1991 Census
* ''Mixed'' (album), a co ...
* 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
*
*
*
References
External links
Bottom Ekman layer lab demonstration
{{Authority control
Boundary layer meteorology
Oceanography
fr:Spirale d'Ekman#Couche d'Ekman