Ekman-Transport

The Ekman transport is vertically above a turbulent boundary layer of the atmosphere and the ocean as a result of Earth's rotation integrated flow. The transport mechanism is named after the Swedish oceanographer Vagn Walfrid Ekman, who has set up the first realistic theory of wind-driven flow ( Ekman, 1905). The Ekman transport is determined by the balance between the induced by the moving water column Coriolis force and the difference of the turbulent shear stress between the upper and lower boundary of the water column in the turbulent boundary layer. The characteristic time for the adjustment of this equilibrium is the inertial period.

• 6.1 buoyancy in the open ocean
• 6.2 Ekman transport in the surface layer of a bounded sea
• 6.3 Ekman transport at the equator and equatorial buoyancy
• 6.4 Ekman transport in the bottom boundary layer of a bounded sea
• 6.5 Literature

Turbulent boundary layers

The atmosphere has a pronounced turbulent boundary layer at its lower boundary, which is formed by the solid earth and the surfaces of lakes, seas and oceans. The intense turbulence in the atmospheric boundary layer is produced by vertical Stromscherung by flow around the roughness elements as well as by thermal convection. The ocean has both a turbulent boundary layer below the sea surface ( top layer called ) directly, as well as the atmosphere of a turbulent boundary layer on the seabed, the benthic boundary layer is called. The causes of turbulence in the benthic boundary layer are largely similar to that in the atmospheric boundary layer. Mixed layer in the average flow, the injection of turbulence by breaking seas in the uppermost meter of the coating layer and its distribution by vertical Langmuir- circulation over the entire top layer makes the generation of the turbulence of the vertical shear a significant role. Turbulent boundary layers are well mixed throughout its thickness, while outside its borders, the stable stratification of the atmosphere and ocean turbulence largely suppressed.

Turbulent shear stress

The turbulent wind that blows over the earth's surface exerts on its base, be it the solid earth or the sea surface, a turbulent shear stress from. This shear stress is a noticeable inhibitory force for atmospheric movements and at the same time an important driving force for oceanic motions dar. analog to the atmosphere, the shear stress on the seabed a restraining force for the ocean currents dar. The vector of the horizontal turbulent shear stress at the surface, the force per unit area is that of the surface of the earth immediately adjacent turbulent air layers and the solid or liquid is applied between the surface of the earth.

Linearized equations of motion of a fluid on the rotating earth

To integrate the horizontal shear stresses in the equations of motion for the mean flow, one imagines the atmosphere and the ocean of thin layers consisting ago against each other like the cards in a stack can move playing cards themselves. Then the resultant force per unit area. To a layer, the difference vectors between the shear stress of the top and bottom of the layer The shear stress caused by the force per unit mass is then. The reason for neglecting the horizontal derivatives of the shear stress, is that the vertical scale of the turbulent boundary layer are substantially smaller than the scales which occur within which horizontal variations in the shear stress. The linearized equations of motion for a fluid to the rotating at a constant angular speed are then ground in the light of the horizontal shear stress

In the above equations are as follows:

• T: time
• X, y, z: coordinates of a rectangular coordinate system with the origin at the sea level, on the latitude, for example, positively directed to the east, north positively and positively against the force of gravity.
• U, v: the horizontal component of the velocity vector in the direction of x-and y -axis.
• P: the pressure disturbance, that is, the deviation from the hydrostatic pressure.
• : The density of the liquid; In this case, air or water.
• The components of the turbulent shear stress in the x -and y- axis.

Turbulence models of the turbulent boundary layer of the atmosphere and the ocean are currently not in a form that would allow the vertical profile of the turbulent shear stress within the boundary layers as a function of the averaged state variables velocity and density, and the momentum and buoyancy fluxes at the edges of the boundary layers express exactly.

Integral properties of turbulent boundary layers

It appears, however, that quite simple models can be used to investigate some integral properties of the boundary layers and to determine their impact on the flow outside the boundary layers. Here, it is assumed that the horizontal flow can be in one driven by the pressure gradient part that exists throughout the liquid, and in a generator driven by the shear stress component, which only exists in the boundary layer Ekman flow, disassemble, namely

The Ekman flow satisfies the equations

And results of the sub - integrated to the ceiling,

Here is the vector of the Ekman transport.

For the atmospheric boundary layer and the benthic boundary layer of the ocean, one can assume that the turbulent shear stress above disappears because the turbulence outside the boundary layers due to the stable density stratification is very small. It thus results for the Ekman transport within these boundary layers

Oceanic surface layer

For the oceanic surface layer can be assumed that the turbulent shear stress can be below the top layer off, also because of the strong density stratification, neglected. Thus follows for the Ekman transport of the surface layer

The lower edge of the atmospheric boundary layer over the sea at identical with the upper edge of the ocean surface layer, at, namely, the sea surface.

Transient processes in the turbulent boundary layer

The behavior of the Ekman transport in the turbulent boundary layer in the transition from a state of rest to a state of equilibrium between the Coriolis force and the shear stress at the edge of the boundary layer can be good for examining the oceanic surface layer. It is assumed that the wind shear stress at the sea surface at the time t = 0 suddenly begins and then remains constant, ie. Here is the Heaviside function. The constancy of the wind shear stress can assume one if its horizontal variation is on the scale that is much larger than the Rossbyradius in the ocean. This is in the open ocean is often the case. A solution to this problem is obtained relatively easily, if the above equation for the meridional component of the Ekman transport by i, the imaginary unit, multiplied and added both equations. This gives then

This equation has the solution

After switching on the voltage of the wind thrust Ekman is transported in the direction of the wind shear stress and increases linearly with time. Over time, the Ekman transport starts to turn away under the influence of the Coriolis force on the north ( south ) hemisphere in the (counter ) clockwise from the direction of the wind stress. After a period of inertia of the Ekman transport is at right angles in a clockwise direction to wind stress with the constant amount. This constant proportion of the Ekman transport, resulting from the balance of wind stress on the sea surface and the Coriolis force to inertial oscillations superimposed with the period, resulting from the balance between the inertia of the water particles and their Coriolis acceleration. The transition from a state of dynamic equilibrium of the turbulent surface layer to another is. The results obtained depend only on the existence of a turbulent shear stress at the top of the turbulent boundary layer or of its disappearance at the bottom and not on the characteristics of turbulence inside the boundary layer.

The characteristics of the transient phenomena after switching a shear stress in the near-surface boundary layer of the atmosphere and in the benthic barrier layer in the ocean are the same as those in the outer layer of the sea. However, the equilibrium between the Coriolis force and shear stress at the bottom of the near-surface boundary layers is different from that in the outer layer of the sea. For those

In these near-surface boundary layer of the Ekman - transport in the northern ( southern) is hemisphere rotated 90 ° counterclockwise ( in the ) counterclockwise with respect to the shear stress at the bottom of the boundary layer, and is thus opposite to that in the outer layer of the sea. It is interesting that the Ekman mass transport in the atmospheric boundary layer over the sea and in the top layer of the sea are the same with the opposite direction, so that the integrated mass transport through both layers is equal to zero.

Detection and significance of the Ekman transport

The costs associated with the Ekman transport velocities are relatively small compared to those driven by the pressure gradient flows. Moreover, especially the on the sea surface induced by the swell frequency flow fluctuations are much stronger than the Ekman flow. This poor signal / noise ratio represented a particular challenge to the experimental proof of the Ekman transport in the ocean, which could only be solved by the available flow measurement in the 1990s. Through careful simultaneous current and wind measurements in the open ocean has been demonstrated that the observed near-surface volume transport is consistent with the Ekman transport, and Plueddemann Weller (1996 ), Schudlich and Price (1998).

If the Ekman transport in a turbulent boundary layer spatially constant, so its effects are confined to this layer. Contributes significantly to the horizontal mixing of the dissolved and particulate material in this layer.

Of great importance for the overall dynamics of the ocean and the atmosphere of the Ekman transport is when its divergence in the turbulent boundary layer is different from zero. Vertical velocities related produce pressure disturbances outside the boundary layers created by the geostrophic after adjustment geostrophic currents in all of air or water column. By integration of the equation of continuity of the thickness of the turbulent boundary layer is obtained the relation between the divergence of the Ekman - transport and vertical velocity at the edges of the turbulent boundary layers.

Buoyancy in the open ocean

About the real ocean the wind is not blowing up everywhere equally and not everywhere in the same direction. Thus more water is removed by Ekman transport as is nachgeschoben in some areas. The Ekman transport in the surface layer, in this case to a divergence. For reasons of conservation of mass of water must flow from the bottom. This buoyancy is also called Ekman - suction. In other areas, is brought by the kovergenten Ekman transport in the surface layer of water transported several pages. There decreases surface water. This is called output or Ekman pumping. This is done by attaching to the high - and low-pressure wind fields at the sea surface. Taking a deep the cyclonic wind stress causes buoyancy. under a high causes the anticyclonic wind stress output.

The formation of the divergence of the Ekman transport as a function of wind shear stress results after the decay of the inertial oscillations

At the bottom of a turbulent outer layer, a vertical velocity that is proportional to the rotation of the Coriolis divided by the horizontal wind shear stress at the sea surface. This process is of fundamental importance for the excitation of wind generated ocean currents. The rotation of the wind stress is emerging over the ocean between the various branches of the Planetary circulation, for example, between the westerly wind belts and the trade zones. Between the latter, the Ekman transport accumulates a growing mountain of water, pushing the thermocline deep into the ocean inside. After the geostrophic adjustment process of these forms in the respective ocean at the core of the subtropical eddy (English gyre ). The growth of the mountain water is cut off by the arrival of the long oceanic Rossby waves from the eastern edge of the front of the ocean, see, eg, Gill ( 1982). Behind the front, a steady state is established, wherein the divergence of the Ekman - transport is compensated for by the planetary divergence of the meridional flow. This steady state is referred to as Sverdrup regime. Since the propagating westward Rossby waves the growth of water mountain in the eastern part of the ocean rather stop than in the western part, the height of the water mountain rises in the subtropical eddy ( gyre ) slowly from the eastern to the western shore of the ocean in the order of 1 m at.

Ekman transport in the surface layer of a bounded sea

For the turbulent surface layer of the sea, the boundary condition. Thus results for the vertical velocity at the bottom of the top layer of the sea

We assume that the wind at the surface of a sea of width W is blowing parallel to its shores in the positive x - direction. For the Ekman transport in the surface layer of the sea. The Ekman transport is thus divergent only at the banks and we obtain for the vertical velocity at the bottom of the top layer approximately

Namely buoyancy on the left bank and Down Welling on the right bank when looking downwind. In reality, a coastal boundary layer of the width forms a Rossby radius at each bank, on spread the vertical velocities. In addition, a compensation current is set to the Ekman transport below the surface layer by radiating barotropic Poincaré waves behind the front.

The directed into the interior of the sea on the left bank of the canal Ekman transport where it leads to a growing with time lowering of the sea level within the coastal boundary layer and the buoyancy to a bulging of the thermocline. After the geostrophic adjustment to the pressure disturbance caused thereby is directed in the cover layer within the coastal boundary layer of a horizontally bundled, accelerating geostrophic flow in wind direction, the coastal jet stream or English Coastal Jet is called. On the opposite bank of the Down Welling leads process together with the geostrophic adjustment to a flowing in the same direction coastal jet stream.

The coastal jet streams together with the Ekman transport in the surface layer and located below the top layer compensation current in a bounded sea on the northern hemisphere circulation in the form of a right-handed screw, the tip of which points in the direction of the wind vector.

Ekman transport at the equator and equatorial buoyancy

Analog dynamic processes, such as in a confined sea produces a spatially constant zonal wind shear stress above the equator. The change of sign of the Coriolis f at the equator with the result that, in dynamic terms, the equator is a virtual coast. Facing east wind shear stress generated in the equatorial surface layer by the change of sign of f Coriolisparameters one of the two hemispheres to the equator directed Ekman transport, there has Down Welling with an eastward equatorial jet current. Westward wind stress a directed to the poles Ekman transport results in the equatorial upwelling and a westward jet stream has generated. The meridional width of the respective buoyancy zones and jet streams is determined by the equatorial Rossby radius.

Ekman transport in the bottom boundary layer of a bounded sea

Flows driven by a pressure gradient flow over a solid surface with a certain roughness, thus forming in the immediate vicinity of the solid wall turbulent boundary layer. For the turbulent ground layer of the atmosphere and the sea, the boundary condition. Thus results for the vertical velocity at the top of the atmospheric boundary layer or benthic after the formation of the Ekman transport

Consider an infinitely long channel with its major axis parallel to the x - direction and assume that the main flow and thus the shear stress is directed in the positive x - direction at the bottom so the Ekman transport in the benthic bottom layer toward the Gradientströmung is looking, directed to 90 ° in the counterclockwise direction. The transversely adjusting to the channel axis Ekman transport must vanish on the banks at. It applies to the Ekman transport in the channel. The Ekman transport in the bottom boundary layer is thus divergent only at the banks and we obtain for the vertical velocity at the top of the bottom friction layer approximately

Here the derivation is from. In the direction of the shear stress at the bottom looking, the vertical velocity of the upper edge of the bottom friction layer on the left bank upstream and on the right bank is on the north hemisphere, analogous to the wind-driven buoyancy in the restricted sea, downward. In the nature of the transition from the fully developed Ekman transport inside the channel is done to his disappearance at the shore within a coastal boundary layer of the width of a Rossby radius by the radiation of barotropic Poincaré waves from the shore into the interior of the channel. Behind the front of the Poincaré waves compensation flow is perpendicular to the channel axis, which compensates the friction of the base transportation Ekman - layer in the form that the mass transfer disappears transversely to the channel axis. This raises a secondary circulation, which is characterized by the Ekman transport in the bottom boundary layer and the oppositely directed compensation current in the overlying layers. The circulation has on the northern hemisphere, as in the wind-driven case, the shape of a right-handed screw, the tip pointing towards the Gradientenströmung. Buoyancy and output take place within the coastal boundary layers. They lead to a bulging of the density layers above the bottom boundary layer on the left bank and to a lowering on the right bank. After the geostrophic adjustment were raised in the two coastal boundary layers baroclinic currents, which are superimposed on the barotropic channel flow and cause vertical Stromscherungen.