Ekman transport

The Ekman transport is a flow that is vertically integrated across a turbulent boundary layer of the atmosphere and the ocean as a result of the earth's rotation . It is determined by the balance between the Coriolis force , which is induced by the moving water column , and the difference in the turbulent shear stresses between the upper and lower edges of this water column in the turbulent boundary layer. The characteristic time for establishing this equilibrium is the period of inertia .

The Ekman transport is named after the Swedish oceanographer Vagn Walfrid Ekman , who together with him in 1905 put forward the first realistic theory of a wind-driven flow .

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 this atmospheric boundary layer is generated by vertical current shear, by flow around elements of the roughness of the ground and by thermal convection .

The ocean has turbulent boundary layers both immediately below its surface (called the cover layer) and - like the atmosphere - on the sea floor , the benthic boundary layer . For the generation of turbulence in the oceanic cover layer, in addition to the vertical shear of the mean current, the injection of turbulence through breaking seas into the top meters of the cover layer and its vertical distribution over the entire cover layer by the Langmuir circulation play an essential role . The causes of turbulence in the benthic boundary layer are largely similar to those in the atmospheric boundary layer.

Turbulent boundary layers are well mixed over their entire thickness, while outside their boundaries the stable stratification of the atmosphere and the ocean largely suppresses the turbulence.

Turbulent shear stress

The turbulent wind that blows over the earth's surface exerts a turbulent shear stress on its support, be it the solid earth or the sea surface. This shear stress represents a noticeable inhibiting force for atmospheric movements and at the same time an important driving force for oceanic movement processes. Analogous to the atmosphere, the shear stress on the sea floor represents an inhibiting force for the ocean currents . The vector of the horizontal turbulent shear stress on the earth's surface represents the force pro The unit of area that is exerted between the turbulent air layers immediately adjacent to the earth's surface and the solid or liquid earth's surface. ${\ displaystyle {\ vec {\ tau}}}$

Linearized equation of motion of a liquid on the rotating earth

In order to include the horizontal shear stresses in the equations of motion for the mean current , imagine the atmosphere and the ocean as consisting of thin layers that can move against each other like the cards in a deck of playing cards. Then the resulting force per unit area on a layer is precisely the difference in the shear stress vectors between the top and bottom of the layer. The force per unit of mass caused by the shear stress is then:

{\ displaystyle {\ frac {1} {\ rho}} \ cdot {\ frac {\ partial {\ vec {\ tau}}} {\ partial z}}}

The reason for neglecting horizontal derivatives of the shear stress is that the vertical scales of the turbulent boundary layers are much smaller than the scales within which horizontal variations of the shear stress occur.

The linearized equations of motion for a liquid on the earth rotating with constant angular velocity are then, taking into account the horizontal shear stress: ${\ displaystyle {\ vec {\ omega}}}$

{\ displaystyle {\ frac {\ partial u} {\ partial t}} - fv = - {\ frac {1} {\ rho}} {\ frac {\ partial p} {\ partial x}} + {\ frac {1} {\ rho}} {\ frac {\ partial \ tau _ {x}} {\ partial z}}}

,

{\ displaystyle {\ frac {\ partial v} {\ partial t}} + fu = - {\ frac {1} {\ rho}} {\ frac {\ partial p} {\ partial y}} + {\ frac {1} {\ rho}} {\ frac {\ partial \ tau _ {y}} {\ partial z}}}

.

With:

{\ displaystyle t:}

the time

${\ displaystyle x, y, z:}$ the coordinates of a right-angled coordinate system with the zero point at sea level on the geographical latitude , e.g. B. positive to the east, positive to the north and positive directed against gravity . ${\ displaystyle \ phi}$

{\ displaystyle u, v:}

the horizontal components of the velocity vector in the direction of the x and y axes.

{\ displaystyle p:}

the pressure disorder, d. H. the deviation from hydrostatic pressure .

{\ displaystyle \ rho}

: the density of the liquid; in this case air or water.

{\ displaystyle f = 2 \ cdot | {\ vec {\ omega}} | \ cdot \ sin \ phi}

the Coriolis parameter

{\ displaystyle \ tau _ {x}, \ tau _ {y}}

: the components of the turbulent shear stress in the direction of the x and y axes.

Turbulence models of the turbulent boundary layers of the atmosphere and the ocean are not yet available in a form that would allow the vertical course of the turbulent shear stress within the boundary layers as a function of the averaged state variables speed and density, as well as the momentum and buoyancy flows at the edges of the Express boundary layers exactly.

Integral properties of turbulent boundary layers

It turns out, however, that fairly simple models can be used to study some integral properties of the boundary layers and to determine their effects on the flow outside the boundary layers. It is assumed that the horizontal flow can be broken down into a part driven by the pressure gradient , which exists in the entire liquid, and a part driven by the shear stress , the Ekman flow that only exists in the boundary layer, namely ${\ displaystyle {\ vec {u}} _ {p}}$ ${\ displaystyle {\ vec {u}} _ {E}}$

{\ displaystyle {\ vec {u}} = {\ vec {u}} _ {p} + {\ vec {u}} _ {E}}

.

The Ekman flow satisfies the equations

{\ displaystyle {\ frac {\ partial u_ {E}} {\ partial t}} - fv_ {E} = {\ frac {1} {\ rho}} {\ frac {\ partial \ tau _ {x}} {\ partial z}}}

,

{\ displaystyle {\ frac {\ partial v_ {E}} {\ partial t}} + fu_ {E} = {\ frac {1} {\ rho}} {\ frac {\ partial \ tau _ {y}} {\ partial z}}}

,

and results in integrated from the lower to the upper limit , ${\ displaystyle z_ {u}}$ ${\ displaystyle z_ {o}}$

{\ displaystyle {\ frac {\ partial U_ {E}} {\ partial t}} - fV_ {E} = {\ frac {1} {\ rho}} \ left (\ tau _ {x} (z_ {o }) - \ tau _ {x} (z_ {u}) \ right)}

,

{\ displaystyle {\ frac {\ partial V_ {E}} {\ partial t}} + fU_ {E} = {\ frac {1} {\ rho}} \ left (\ tau _ {y} (z_ {o }) - \ tau _ {y} (z_ {u}) \ right)}

.

Where is the vector of Ekman transport. ${\ displaystyle {\ vec {U}} _ {E} = \ int _ {z_ {u}} ^ {z_ {o}} {\ vec {u}} _ {E} dz = \ int _ {z_ { u}} ^ {z_ {o}} \ left ({\ vec {u}} - {\ vec {u}} _ {p} \ right) dz}$

Boundary layers close to the ground

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 is very small due to the stable density stratification. It thus arises for the Ekman transport within these boundary layers ${\ displaystyle z_ {o}}$

{\ displaystyle {\ frac {\ partial U_ {E}} {\ partial t}} - fV_ {E} = - {\ frac {1} {\ rho}} \ tau _ {x} (z_ {u}) }

,

{\ displaystyle {\ frac {\ partial V_ {E}} {\ partial t}} + fU_ {E} = - {\ frac {1} {\ rho}} \ tau _ {y} (z_ {u}) }

.

Oceanic top layer

For the oceanic cover layer, it can be assumed that the turbulent shear stress below this cover layer can be neglected, also because of the strong density stratification. For the Ekman transport of the surface layer follows ${\ displaystyle z_ {u}}$

{\ displaystyle {\ frac {\ partial U_ {E}} {\ partial t}} - fV_ {E} = {\ frac {1} {\ rho}} \ tau _ {x} (z_ {o})}

,

{\ displaystyle {\ frac {\ partial V_ {E}} {\ partial t}} + fU_ {E} = {\ frac {1} {\ rho}} \ tau _ {y} (z_ {o})}

,

where the lower edge of the atmospheric boundary layer over the sea at is identical to the upper edge of the oceanic cover layer at , namely the sea surface. ${\ displaystyle z_ {u}}$ ${\ displaystyle z_ {o}}$

Transient processes in the turbulent boundary layer

The behavior of Ekman transport in the turbulent boundary layer during 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 investigated well for the oceanic cover layer. It is assumed that the wind shear stress on the sea surface starts suddenly at the time and then remains constant, i.e. H. . Here is the Heaviside function . The constancy of the wind shear stress can be assumed if its horizontal variation occurs on scales that are considerably larger than the Rossby radius in the ocean. This is often the case in the open ocean. A solution to this problem can be obtained relatively easily if the above equation for the meridional component of the Ekman transport is multiplied by i, the imaginary unit, and both equations are added. You then get ${\ displaystyle t = 0}$ ${\ displaystyle {\ vec {\ tau}} \ left (t \ right) = \ Theta \ left (t \ right) {\ vec {\ tau}} (z = 0)}$ ${\ displaystyle \ Theta (t)}$

{\ displaystyle {\ frac {\ partial} {\ partial t}} \ left (U_ {E} + iV_ {E} \ right) + if \ left (U_ {E} + iV_ {E} \ right) = { \ frac {\ Theta \ left (t \ right)} {\ rho}} \ left (\ tau _ {x} (z_ {o}) + i \ tau _ {y} (z_ {o}) \ right) }

.

This equation has the solution

{\ displaystyle U_ {E} + iV_ {E} = - {\ frac {i \ Theta \ left (t \ right)} {\ rho f}} \ left (1-e ^ {- ift} \ right) \ left (\ tau _ {x} (z_ {o}) + i \ tau _ {y} (z_ {o}) \ right)}

.

After switching on the wind shear voltage, the Ekman transport takes place in the direction of the wind shear voltage and increases linearly with time. In the course of time, the Ekman transport begins to rotate under the influence of the Coriolis force in the north (south) hemisphere clockwise (counter-clockwise) away from the direction of the wind shear stress. After a period of inertia , the Ekman transport takes place at a right angle clockwise to the wind shear stress with the constant amount . This constant part of the Ekman transport, which results from the equilibrium of the wind shear stress on the sea surface and the Coriolis force , is superimposed on inertial oscillations with the period resulting from the equilibrium between the inertia of the water particles and their Coriolis acceleration. The transition time from one dynamic state of equilibrium of the turbulent surface layer to another is . The results obtained depend only on the existence of a turbulent shear stress at the upper edge of the turbulent boundary layer or on its disappearance at the lower edge and not on the properties of the turbulence inside the boundary layer. ${\ displaystyle T_ {i} = {\ frac {2 \ pi} {f}}}$ ${\ displaystyle {\ frac {\ tau (z_ {o})} {\ rho f}}}$ ${\ displaystyle T_ {i}}$ ${\ displaystyle T_ {i}}$

The properties of the transient processes after switching on a shear stress in the surface boundary layer of the atmosphere and in the benthic boundary layer in the ocean are the same as those in the surface layer of the sea. In contrast, the state of equilibrium between Coriolis force and shear stress at the lower edge of the two boundary layers close to the ground is different from that in the surface layer of the sea. For those applies

{\ displaystyle U_ {E} + iV_ {E} = {\ frac {i \ left (\ tau _ {x} (z_ {u}) + i \ tau _ {y} (z_ {u}) \ right) } {\ rho f}}}

.

In these near-surface boundary layers, the Ekman transport in the north (south) hemisphere is rotated 90 ° counterclockwise (clockwise) with respect to the shear stress at the bottom of the boundary layer, and is thus opposite to that in the surface layer of the sea. It is interesting that the Ekman mass transport in the atmospheric boundary layer above the sea and that in the surface layer of the sea are of the same size and in opposite directions, so that the mass transport integrated over both layers is zero.

Evidence and importance of the Ekman transport

The velocities associated with Ekman transport are relatively small compared to those of flows driven by pressure gradients. In addition, the high-frequency flow fluctuations induced on the sea surface by the swell are much stronger than the Ekman flow. This poor signal / noise ratio posed a particular challenge to the experimental verification of Ekman transport in the ocean, which could only be solved by the flow measurement technology available in the 1990s. Careful simultaneous current and wind measurements in the open ocean have shown that the observed near-surface volume transport is consistent with the Ekman transport, Weller and Plueddemann (1996), Schudlich and Price (1998).

If the Ekman transport is spatially constant in a turbulent boundary layer, its effects on this layer remain limited. It contributes significantly to the horizontal mixing of dissolved and particulate material in this layer.

The Ekman transport becomes of great importance for the overall dynamics of the ocean and the atmosphere when its divergence in the turbulent boundary layer is different from zero. The associated vertical velocities generate pressure disturbances outside the boundary layers, through which geostrophic currents arise in the entire air or water column after the geostrophic adjustment . By integrating the continuity equation over the layer thickness of the turbulent boundary layer, one obtains the connection between the divergence of the Ekman transport and the vertical speed at the edges of the turbulent boundary layers.

{\ displaystyle {\ frac {\ partial} {\ partial x}} \ int _ {z_ {u}} ^ {z_ {o}} u_ {E} dz + {\ frac {\ partial} {\ partial y}} \ int _ {z_ {u}} ^ {z_ {o}} v_ {E} dz + w \ left (z_ {o} \ right) -w \ left (z_ {u} \ right) = 0}

.

Buoyancy in the open ocean

Over the real ocean, the wind is not equally strong everywhere and does not blow in the same direction everywhere. As a result, in some areas more water is removed by Ekman transport than is pushed in. In this case, the Ekman transport in the top layer exhibits a divergence. For reasons of mass conservation, water must flow in from below. This buoyancy is also known as Ekman suction. In other areas, the convergent Ekman transport in the surface layer transports water from several sides. Surface water sinks there. One then speaks of downforce or Ekman pumping. This happens through the wind fields on the sea surface associated with the high and low pressure areas. Below a low, the cyclonic wind shear stress causes lift. below a high, the anti-cyclonic wind shear stress causes downforce.

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

{\ displaystyle w \ left (z_ {u} \ right) = {\ frac {\ partial} {\ partial x}} {\ frac {\ tau _ {y} (z_ {o})} {\ rho f} } - {\ frac {\ partial} {\ partial y}} {\ frac {\ tau _ {x} (z_ {o})} {\ rho f}}}

,

a vertical speed at the bottom of the turbulent cover layer that is proportional to the rotation of the horizontal wind shear stress on the sea surface divided by the Coriolis parameter. This process is of fundamental importance for the stimulation of the wind-generated ocean currents . The rotation of the wind shear stress forms over the ocean between the various branches of the planetary circulation , e.g. B. out between the westerly wind belts and the passport zones. Between the latter, the Ekman transport accumulates a growing mountain of water and pushes the thermocline deep into the ocean. After the geostrophic adjustment , this process forms the core for the subtropical eddy (gyre) in the respective ocean . The growth of the water mountain is switched off by the arrival of the front of long oceanic Rossby waves from the eastern edge of the ocean, see e.g. B. Gill (1982). A steady state is established behind the front, in which the divergence of the Ekman transport is compensated for by the planetary divergence of the meridional flow. This steady state is known as the Sverdrup regime. Since the Rossby waves propagating westward stop the growth of the water mountain in the eastern part of the ocean rather than in the western part, the height of the water mountain in the subtropical vortex (gyre) increases slowly from the east to the west bank of the ocean in the order of 1 m.

Ekman transport in the top layer of a marginal sea

The boundary condition applies to the turbulent surface layer of the sea . This results in the vertical speed at the lower edge of the surface layer of the sea ${\ displaystyle w \ left (z_ {o} \ right) \ approx 0}$

{\ displaystyle w \ left (z_ {u} \ right) = {\ frac {\ partial U_ {E}} {\ partial x}} + {\ frac {\ partial V_ {E}} {\ partial y}} }

.

We assume that the wind on the surface of a sea of width W blows parallel to its coasts in the positive x-direction. The following applies to Ekman transport in the surface layer of the sea . The Ekman transport is therefore only divergent on the banks and the vertical speed at the lower edge of the surface layer is approximately obtained ${\ displaystyle V_ {E} = - \ Theta \ left (W / 2- | y | \ right) {\ frac {\ tau _ {o} ^ {x}} {f \ rho}}}$

{\ displaystyle w \ left (z_ {u} \ right) = {\ frac {\ tau _ {u} ^ {x}} {f \ rho}} \ left (\ delta \ left (yW / 2 \ right) - \ delta \ left (y + W / 2 \ right) \ right)}

,

namely, buoyancy on the left bank and downwelling on the right bank when looking downwind. In reality, a coastal boundary layer the width of a Rossby radius is formed on the respective bank, over which the vertical speeds are distributed. In addition, the radiation of barotropic Poincaré waves behind their front creates a compensation current for Ekman transport below the surface layer.

The Ekman transport, which is directed towards the interior of the sea on the left bank of the canal, leads to an increasing lowering of the sea level within the coastal boundary layer and the upwelling to a bulging of the thermocline. After the geostrophic adjustment to the pressure disturbances caused by this, a horizontally bundled, accelerating geostrophic flow in the direction of the wind sets up in the cover layer within the coastal boundary layer, which is called the coastal jet flow . On the opposite bank, the downwelling process together with the geostrophic adjustment leads to a coastal jet current flowing in the same direction.

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

Ekman transport at the equator and equatorial buoyancy

Similar dynamic processes as in a limited sea generate a spatially constant zonal wind shear stress over the equator. The change in sign of the Coriolis parameter f at the equator has the consequence that in dynamic terms the equator represents a virtual coast. Wind shear stress directed to the east in the equatorial cover layer creates an Ekman transport from both hemispheres to the equator due to the change in sign of the Coriolis parameter f, which results in downwelling there with an equatorial jet current directed to the east. Westward wind shear stress results in an Ekman transport directed towards the poles, which creates equatorial lift and a westward jet stream. The meridional width of the respective buoyancy zones and jet streams is determined by the equatorial Rossby radius.

Ekman transport in the soil boundary layer of a bordered sea

If a flow driven by pressure gradients flows over a solid base with a certain roughness, a turbulent boundary layer forms in the immediate vicinity of the solid wall. The boundary condition applies to the turbulent soil layer of the atmosphere and the sea . This results in the vertical speed at the upper edge of the atmospheric or benthic boundary layer after the formation of the Ekman transport ${\ displaystyle w \ left (z_ {u} \ right) = 0}$

{\ displaystyle w \ left (z_ {o} \ right) = - {\ frac {\ partial U_ {E}} {\ partial x}} - {\ frac {\ partial V_ {E}} {\ partial y} }}

.

If we consider an infinitely long canal with its main axis parallel to the x-direction and assume that the main flow and thus the shear stress on the ground is directed in the positive x-direction, the Ekman transport in the benthic soil layer is looking in the direction of the gradient flow , directed 90 ° counterclockwise. The Ekman transport, which occurs transversely to the canal axis, must disappear from the banks at . It applies to Ekman transport in the canal . The Ekman transport in the ground friction layer is therefore only divergent on the banks and the vertical speed at the upper edge of the ground friction layer is approximately obtained ${\ displaystyle y = \ pm W / 2}$ ${\ displaystyle V_ {E} = \ Theta \ left (W / 2- | y | \ right) {\ frac {\ tau _ {u} ^ {x}} {f \ rho}}}$

{\ displaystyle w \ left (z_ {o} \ right) = {\ frac {\ tau _ {u} ^ {x}} {f \ rho}} \ left (\ delta \ left (yW / 2 \ right) - \ delta \ left (y + W / 2 \ right) \ right)}

.

Here is the derivation of . Looking in the direction of the shear stress on the ground, in the northern hemisphere, analogous to wind-driven buoyancy in the confined sea, the vertical speed of the upper edge of the ground friction layer is directed upwards on the left bank and downwards on the right bank. In nature, the transition from the fully developed Ekman transport inside the canal to its disappearance on the bank within a coastal boundary layer the width of a Rossby radius takes place through the radiation of barotropic Poincaré waves from the bank into the interior of the canal. Behind the front of the Poincaré waves, there is a compensation flow perpendicular to the canal axis, which compensates for the Ekman transport of the soil friction layer in such a way that the mass transport across the canal axis disappears. A secondary circulation is established, which is characterized by the Ekman transport in the soil friction layer and the oppositely directed compensation current in the layers above. In the northern hemisphere, as in the wind-driven case, the circulation has the form of a clockwise screw, the tip of which points in the direction of the gradient flow . Upwelling and downforce take place within the coastal boundary layers. They lead to a bulging of the density layers above the soil boundary layer on the left bank and to a subsidence on the right bank. After the geostrophic adjustment, baroclinic currents appear in the two coastal boundary layers, which are superimposed on the barotropic canal currents and lead to vertical shear currents there. ${\ displaystyle \ delta (x)}$ ${\ displaystyle \ Theta (x)}$

literature

Ekman, VW, 1905. On the influence of the earth's rotation on ocean currents. Arch. Math. Astron. Phys. 2, No. 11
Gill, AE (1982). Atmosphere-Ocean Dynamics. Academic Press Inc. New York, London, Tokyo, ISBN 0-12-283520-4
Fennel, W. and H.-U. Lass, 1989. Analytical Theory of Forced Ocean Waves. Akademie-Verlag-Berlin, ISBN 3-05-500421-3
Weller, RA, Plueddemann, AJ, 1996. Observations of the vertical structure of the oceanic boundary layer. J. Geophys. Res., 101, C4, 8789-8806
Schudlich, RR, Price, JF, 1998. Observations of Seasonal Variation in the Ekman Layer. J. Phys. Oceanogr., 28, 6, 1187-1204

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