Top-to-bottom Ekman layer and its implications for shallow rotating flows


The analytical solution is derived for rotational frictional flow in a shallow layer of fluid in which the top and bottom Ekman layers join without leaving a frictionless interior. This vertical structure has significant implications for the horizontal flow. In particular, for a layer of water subjected to both a surface wind stress and bottom friction, the vorticity of the horizontal flow is a function not only of the curl of the wind stress (the classical result for deep water known as Ekman pumping) but also of its divergence. The importance of this divergence term peaks for a water depth around 3 times the Ekman layer thickness. This means that a curl-free but non-uniform wind stress on a shallow sea or lake can, through the dual action of rotation and friction, generate vorticity in the wind-driven currents. We also find that the reduction of three-dimensional dynamics to a two-dimensional model is more subtle than one could have anticipated and needs to be approached with utmost care. Taking the bottom stress as dependent solely on the depth-averaged flow, even with some veering, is not appropriate. The bottom stress ought to include a component proportional to the surface stress, which is negligible for large depths but increases with decreasing water depth.

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    Note that this demands a flat horizontal bottom, which we had assumed from the start.

  2. 2.

    Not even in the asymptotic cases of very small or very large values.

  3. 3.

    There is some subjectivity as to where the top layer ends and the bottom one begins, obviously, but this does not need to be resolved for the sake of the argument made here.


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The first author expresses his gratitude to Prof. GertJan van Heijst for having organized the 2017 Symposium on Shallow Flows in Eindhoven. The second author wishes to acknowledge past support from the Belgian Fund for Scientific Research (F.R.S. − FNRS) in recognition of the fact that some elements of the present paper originated while he served as Research Associate of the FNRS earlier in his career.

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Correspondence to Benoit Cushman-Roisin.

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Cushman-Roisin, B., Deleersnijder, E. Top-to-bottom Ekman layer and its implications for shallow rotating flows. Environ Fluid Mech 19, 1105–1119 (2019).

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  • Coriolis force
  • Ekman layer
  • Ekman pumping
  • Shallow water
  • Two-dimensional modeling