Introduction to the Transition (or Ekman) Layer

This experimental study focuses on the effect of horizontal boundaries with pyramid-shaped roughness elements on the heat transfer in rotating Rayleigh–Bénard convection. It is shown that the Ekman pumping mechanism, which is responsible for the heat transfer enhancement under rotation in the case of smooth top and bottom surfaces, is unaffected by the roughness as long as the Ekman layer thickness $\unicode[STIX]{x1D6FF}_{E}$ is significantly larger than the roughness height $k$. As the rotation rate increases, and thus $\unicode[STIX]{x1D6FF}_{E}$ decreases, the roughness elements penetrate the radially inward flow in the interior of the Ekman boundary layer that feeds the columnar Ekman vortices. This perturbation generates additional thermal disturbances which are found to increase the heat transfer efficiency even further. However, when $\unicode[STIX]{x1D6FF}_{E}\approx k$, the Ekman boundary layer is strongly perturbed by the roughness elements and the Ekman pumping mechanism is suppressed. The results suggest that the Ekman pumping is re-established for $\unicode[STIX]{x1D6FF}_{E}\ll k$ as the faces of the pyramidal roughness elements then act locally as a sloping boundary on which an Ekman layer can be formed.

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Comparison of different methods to estimate the exchange of momentum in the Ekman layer of the sea

Deep Sea Research Part B Oceanographic Literature Review ◽

10.1016/0198-0254(80)95740-4 ◽

1980 ◽

Vol 27 (12) ◽

pp. 847

Keyword(s):

Ekman Layer

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Direct Numerical Simulation of the Turbulent Ekman Layer: Turbulent Energy Budgets

48th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition ◽

10.2514/6.2010-1254 ◽

2010 ◽

Author(s):

Stuart Marlatt ◽

Scottr Waggy ◽

Sedat Biringen

Keyword(s):

Numerical Simulation ◽

Direct Numerical Simulation ◽

Turbulent Energy ◽

Ekman Layer ◽

Energy Budgets

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Spin-up from rest of a compressible fluid in a rapidly rotating cylinder

Journal of Fluid Mechanics ◽

10.1017/s0022112092003471 ◽

1992 ◽

Vol 237 ◽

pp. 413-434 ◽

Cited By ~ 7

Author(s):

Jae Min Hyun ◽

Jun Sang Park

Keyword(s):

Mach Number ◽

Compressible Fluid ◽

Numerical Solutions ◽

Stokes Equations ◽

Axial Direction ◽

Ekman Layer ◽

Infinite Cylinder ◽

Full Time ◽

Viscous Effects ◽

Initial State

Spin-up flows of a compressible gas in a finite, closed cylinder from an initial state of rest are studied, The flow is characterized by small reference Ekman numbers, and the peripheral Mach number is O(1). Comprehensive numerical solutions have been obtained for the full, time-dependent compressible Navier-Stokes equations. The details of the flow, temperature, and density evolution are described. In the early phase of spin-up, owing to the thermoacoustic disturbances caused by the compressible Rayleigh effect, the flows are oscillatory, and this oscillatory behaviour is pronounced at higher Mach numbers. The principal dynamical role of the Ekman layer is dominant over moderate times of orders of the homogeneous spin-up timescales. Owing to the density stratification in the radial direction, the Ekman layer is thicker in the central region of the interior. The interior azimuthal flows are mainly uniform in the axial direction. As the Mach number increases, the rate of spin-up in the interior becomes slower, and the propagating shear front is more diffusive. Explicit comparisons with the results for an infinite cylinder are made to ascertain the contributions of the endwall disks. In contrast to the usual incompressible spin-up from rest, the viscous effects are relatively more important for the case of a compressible fluid.

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Destruction of Potential Vorticity by Winds

Journal of Physical Oceanography ◽

10.1175/jpo2830.1 ◽

2005 ◽

Vol 35 (12) ◽

pp. 2457-2466 ◽

Cited By ~ 145

Author(s):

Leif N. Thomas

Keyword(s):

Boundary Layer ◽

Wind Stress ◽

Potential Vorticity ◽

Ekman Layer ◽

Buoyancy Flux ◽

Coriolis Parameter ◽

Mode Water ◽

Ocean Fronts ◽

Secondary Circulations ◽

Frictional Forces

Abstract The destruction of potential vorticity (PV) at ocean fronts by wind stress–driven frictional forces is examined using PV flux formalism and numerical simulations. When a front is forced by “downfront” winds, that is, winds blowing in the direction of the frontal jet, a nonadvective frictional PV flux that is upward at the sea surface is induced. The flux extracts PV out of the ocean, leading to the formation of a boundary layer thicker than the Ekman layer, with nearly zero PV and nonzero stratification. The PV reduction is not only active in the Ekman layer but is transmitted through the boundary layer via secondary circulations that exchange low PV from the Ekman layer with high PV from the pycnocline. Extraction of PV from the pycnocline by the secondary circulations results in an upward advective PV flux at the base of the boundary layer that scales with the surface, nonadvective, frictional PV flux and that leads to the deepening of the layer. At fronts forced by both downfront winds and a destabilizing atmospheric buoyancy flux FBatm, the critical parameter that determines whether the wind or the buoyancy flux is the dominant cause for PV destruction is (H/δe)(FBwind/FBatm), where H and δe are the mixed layer and Ekman layer depths, FBwind = S2τo/(ρof ), S2 is the magnitude of the lateral buoyancy gradient of the front, τo is the downfront component of the wind stress, ρo is a reference density, and f is the Coriolis parameter. When this parameter is greater than 1, PV destruction by winds dominates and may play an important role in the formation of mode water.

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Eddy formation by dense flows on slopes in a rotating fluid

Journal of Fluid Mechanics ◽

10.1017/s0022112098001013 ◽

1998 ◽

Vol 363 ◽

pp. 229-252 ◽

Cited By ~ 65

Author(s):

GREGORY F. LANE-SERFF ◽

PETER G. BAINES

Keyword(s):

Ocean Circulation ◽

Laboratory Experiments ◽

Lower Layer ◽

Ekman Layer ◽

Global Ocean ◽

Circulation System ◽

Geostrophic Adjustment ◽

Viscous Effects ◽

Dense Fluid ◽

Global Ocean Circulation

Properties of the flow generated by a continuous source of dense fluid on a slope in a rotating system are investigated with a variety of laboratory experiments. The dense fluid may initially flow down the slope but it turns (under the influence of rotation) to flow along the slope, and initial geostrophic adjustment gives it an anticyclonic velocity profile. Some of the dense fluid drains downslope in a viscous Ekman layer, which may become unstable to growing waves. Provided that the viscous draining is not too strong, cyclonic vortices form periodically in the upper layer and the dense flow breaks up into a series of domes. Three processes may contribute to the formation of these eddies. First, initial downslope flow of the dense current may stretch columns of ambient fluid by the ‘Taylor column’ process (which we term ‘capture’). Secondly, the initial geostrophic adjustment implies lower-layer collapse which may stretch the fluid column, and thirdly, viscous drainage will progressively stretch and spin up a captured water column. Overall this last process may be the most significant, but viscous drainage has contradictory effects, in that it progressively removes dense lower-layer fluid which terminates the process when the layer thickness approaches that of the Ekman layer. The eddies produced propagate along the slope owing to the combined effects of buoyancy–Coriolis balance and ‘beta-gyres’. This removes fluid from the vicinity of the source and causes the cycle to repeat. The vorticity of the upper-layer cyclones increases linearly with Γ=Lα/D (where L is the Rossby deformation radius, α the bottom slope and D the total depth), reaching approximately 2f in the experiments presented here. The frequency at which the eddy/dome structures are produced also increases with Γ, while the speed at which the structures propagate along the slope is reduced by viscous effects. The flow of dense fluid on slopes is a very important part of the global ocean circulation system and the implications of the laboratory experiments for oceanographic flows are discussed.

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