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 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.
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.
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.
For the boundary condition on the other side, as , where and are the geostrophic flows in the and directions.
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.
The traditional development of Ekman layers bounded below by a surface utilizes two boundary conditions:
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.
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 boundary layer into the surface layer and the Ekman layer generally yields more accurate results.
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 modify the flow near the surface, and make observations close to the surface rather difficult.
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.
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. More recent observations include (not an exhaustive list):
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.
The Bahama Banks are the submerged carbonate platforms that make up much of the Bahama Archipelago. The term is usually applied in referring to either the Great Bahama Bank around Andros Island, or the Little Bahama Bank of Grand Bahama Island and Great Abaco, which are the largest of the platforms, and the Cay Sal Bank north of Cuba. The islands of these banks are politically part of the Bahamas. Other banks are the three banks of the Turks and Caicos Islands, namely the Caicos Bank of the Caicos Islands, the bank of the Turks Islands, and wholly submerged Mouchoir Bank. Further southeast are the equally wholly submerged Silver Bank and Navidad Bank north of the Dominican Republic.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 toEkman 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.Index of wave articles
This is a list of Wave topics.Ocean dynamics
Ocean dynamics define and describe the motion of water within the oceans. Ocean temperature and motion fields can be separated into three distinct layers: mixed (surface) layer, upper ocean (above the thermocline), and deep ocean.
Ocean dynamics has traditionally been investigated by sampling from instruments in situ.The mixed layer is nearest to the surface and can vary in thickness from 10 to 500 meters. This layer has properties such as temperature, salinity and dissolved oxygen which are uniform with depth reflecting a history of active turbulence (the atmosphere has an analogous planetary boundary layer). Turbulence is high in the mixed layer. However, it becomes zero at the base of the mixed layer. Turbulence again increases below the base of the mixed layer due to shear instabilities. At extratropical latitudes this layer is deepest in late winter as a result of surface cooling and winter storms and quite shallow in summer. Its dynamics is governed by turbulent mixing as well as Ekman pumping, exchanges with the overlying atmosphere, and horizontal advection.The upper ocean, characterized by warm temperatures and active motion, varies in depth from 100 m or less in the tropics and eastern oceans to in excess of 800 meters in the western subtropical oceans. This layer exchanges properties such as heat and freshwater with the atmosphere on timescales of a few years. Below the mixed layer the upper ocean is generally governed by the hydrostatic and geostrophic relationships. Exceptions include the deep tropics and coastal regions.
The deep ocean is both cold and dark with generally weak velocities (although limited areas of the deep ocean are known to have significant recirculations). The deep ocean is supplied with water from the upper ocean in only a few limited geographical regions: the subpolar North Atlantic and several sinking regions around the Antarctic. Because of the weak supply of water to the deep ocean the average residence time of water in the deep ocean is measured in hundreds of years. In this layer as well the hydrostatic and geostrophic relationships are generally valid and mixing is generally quite weak.Oceanic plateau
An oceanic or submarine plateau is a large, relatively flat elevation that is higher than the surrounding relief with one or more relatively steep sides.There are 184 oceanic plateaus covering an area of 18,486,600 km2 (7,137,700 sq mi), or about 5.11% of the oceans. The South Pacific region around Australia and New Zealand contains the greatest number of oceanic plateaus (see map).
Oceanic plateaus produced by large igneous provinces are often associated with hotspots, mantle plumes, and volcanic islands — such as Iceland, Hawaii, Cape Verde, and Kerguelen. The three largest plateaus, the Caribbean, Ontong Java, and Mid-Pacific Mountains, are located on thermal swells. Other oceanic plateaus, however, are made of rifted continental crust, for example Falkland Plateau, Lord Howe Rise, and parts of Kerguelen, Seychelles, and Arctic ridges.
Plateaus formed by large igneous provinces were formed by the equivalent of continental flood basalts such as the Deccan Traps in India and the Snake River Plain in the United States.
In contrast to continental flood basalts, most igneous oceanic plateaus erupt through young and thin (6–7 km (3.7–4.3 mi)) mafic or ultra-mafic crust and are therefore uncontaminated by felsic crust and representative for their mantle sources.
These plateaus often rise 2–3 km (1.2–1.9 mi) above the surrounding ocean floor and are more buoyant than oceanic crust. They therefore tend to withstand subduction, more-so when thick and when reaching subduction zones shortly after their formations. As a consequence, they tend to "dock" to continental margins and be preserved as accreted terranes. Such terranes are often better preserved than the exposed parts of continental flood basalts and are therefore a better record of large-scale volcanic eruptions throughout Earth's history. This "docking" also means that oceanic plateaus are important contributors to the growth of continental crust. Their formations often had a dramatic impact on global climate, such as the most recent plateaus formed, the three, large, Cretaceous oceanic plateaus in the Pacific and Indian Ocean: Ontong Java, Kerguelen, and Caribbean.Sverdrup balance
The Sverdrup balance, or Sverdrup relation, is a theoretical relationship between the wind stress exerted on the surface of the open ocean and the vertically integrated meridional (north-south) transport of ocean water.Tea leaf paradox
The tea leaf paradox describes a phenomenon where tea leaves in a cup of tea migrate to the center and bottom of the cup after being stirred rather than being forced to the edges of the cup, as would be expected in a spiral centrifuge. The correct physical explanation of the paradox was for the first time given by James Thomson in 1857. He correctly connected the appearance of secondary flow (in both Earth atmosphere and tea cup) with ″friction on the bottom″ . The formation of secondary flows in an annular channel was theoretically treated by Boussinesq as early as in 1868. The migration of near-bottom particles in river-bend flows was experimentally investigated by A. Ya. Milovich in 1913. The solution first came from Albert Einstein in a 1926 paper in which he explained the erosion of river banks, and repudiated Baer's law.Undersea mountain range
Undersea mountain ranges are mountain ranges that are mostly or entirely underwater, and specifically under the surface of an ocean. If originated from current tectonic forces, they are often referred to as a mid-ocean ridge. In contrast, if formed by past above-water volcanism, they are known as a seamount chain. The largest and best known undersea mountain range is a mid-ocean ridge, the Mid-Atlantic Ridge. It has been observed that, "similar to those on land, the undersea mountain ranges are the loci of frequent volcanic and earthquake activity".Wave base
The wave base, in physical oceanography, is the maximum depth at which a water wave's passage causes significant water motion. For water depths deeper than the wave base, bottom sediments and the seafloor are no longer stirred by the wave motion above.Weddell Polynya
The Weddell Polynya, or Weddell Sea Polynya, is a polynya or irregular area of open water surrounded by sea ice in the Weddell Sea of the Southern Ocean off Antarctica and near the Maud Rise.Wind shear
Wind shear (or windshear), sometimes referred to as wind gradient, is a difference in wind speed and/or direction over a relatively short distance in the atmosphere. Atmospheric wind shear is normally described as either vertical or horizontal wind shear. Vertical wind shear is a change in wind speed or direction with change in altitude. Horizontal wind shear is a change in wind speed with change in lateral position for a given altitude.Wind shear is a microscale meteorological phenomenon occurring over a very small distance, but it can be associated with mesoscale or synoptic scale weather features such as squall lines and cold fronts. It is commonly observed near microbursts and downbursts caused by thunderstorms, fronts, areas of locally higher low-level winds referred to as low level jets, near mountains, radiation inversions that occur due to clear skies and calm winds, buildings, wind turbines, and sailboats. Wind shear has significant effects on control of an aircraft, and it has been a sole or contributing cause of many aircraft accidents.
Wind shear is sometimes experienced by pedestrians at ground level when walking across a plaza towards a tower block and suddenly encountering a strong wind stream that is flowing around the base of the tower.
Sound movement through the atmosphere is affected by wind shear, which can bend the wave front, causing sounds to be heard where they normally would not, or vice versa. Strong vertical wind shear within the troposphere also inhibits tropical cyclone development, but helps to organize individual thunderstorms into longer life cycles which can then produce severe weather. The thermal wind concept explains how differences in wind speed at different heights are dependent on horizontal temperature differences, and explains the existence of the jet stream.