Advertisement

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

  • Benoit Cushman-RoisinEmail author
  • Eric Deleersnijder
Original Article

Abstract

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.

Keywords

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

Notes

Acknowledgements

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.

References

  1. 1.
    Bennetts DA, Hocking LM (1973) On nonlinear Ekman and Stewartson layers in a rotating fluid. Proc R Soc Lond A 333:469–489CrossRefGoogle Scholar
  2. 2.
    Cheng RT, Powell TM, Dillon TM (1976) Numerical models of wind-driven circulation in lakes. Appl Math Model 1:141–159CrossRefGoogle Scholar
  3. 3.
    Cushman-Roisin B, Beckers J-M (2011) Introduction to geophysical fluid dynamics–physical and numerical aspects, 2nd edn. Academic Press, New YorkGoogle Scholar
  4. 4.
    Ekman VW (1923) Über Horizontalzirkulation bei winderzeugten Meeresströmungen. Arkiv Mat Astr Fysik 17:26Google Scholar
  5. 5.
    Forristall GZ (1974) Three-dimensional structure of storm-generated currents. J Geophys Res 79(18):2721–2729CrossRefGoogle Scholar
  6. 6.
    Heaps NS (1969) A two-dimensional numerical sea model. Philos Trans R Soc A 265:93–137CrossRefGoogle Scholar
  7. 7.
    Jelesnianski CP (1970) Bottom stress time-history in linearized equations of motion for storm surges. Mon Weather Rev 98(6):462–478CrossRefGoogle Scholar
  8. 8.
    Józsa J (2014) On the internal boundary layer related wind stress curl and its role in generating shallow lake circulations. J Hydrol Hydromech 62(1):16–23.  https://doi.org/10.2478/johh-2014-0004 CrossRefGoogle Scholar
  9. 9.
    Lentz S, Guza RT, Elgar S, Feddersen F, Herbers THC (1999) Momentum balances on the North Carolina inner shelf. J Geophys Res Oceans 104:18205–18226CrossRefGoogle Scholar
  10. 10.
    Lick W (1976) Numerical modeling of lake currents. Ann Rev Earth Planet Sci 4:49–74CrossRefGoogle Scholar
  11. 11.
    Lynch DR, Officer CB (1985) Analytic test cases for three-dimensional hydrodynamic models. Int J Numer Methods Fluids 5:529–543CrossRefGoogle Scholar
  12. 12.
    Nihoul JCJ (1977) Three-dimensional model of tides and storm surges in a shallow well-mixed continental sea. Dyn Atmos Oceans 2:29–47CrossRefGoogle Scholar
  13. 13.
    Price JJ, Weller RA, Schudlich RR (1987) Wind-driven ocean currents and Ekman transport. Science 238(4833):1534–1538CrossRefGoogle Scholar
  14. 14.
    Schwab DJ (1983) Numerical simulation of low-frequency current fluctuations in Lake Michigan. J Phys Oceanogr 13:2213–2224CrossRefGoogle Scholar
  15. 15.
    Weisberg RH, Zhang L (2008) Hurricane storm surge simulations comparing three-dimensional with two-dimensional formulations based on an Ivan-like storm over the Tampa Bay, Florida region. J Geophys Res Oceans.  https://doi.org/10.1029/2008JC005115
  16. 16.
    Welander P (1957) Wind action on a shallow sea: some generalizations of Ekman’s theory. Tellus 9:45–52Google Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Thayer School of EngineeringDartmouth CollegeHanoverUSA
  2. 2.Institute of Mechanics, Materials and Civil Engineering (IMMC) and Earth and Life Institute (ELI)Université catholique de LouvainLouvain-la-NeuveBelgium
  3. 3.Delft Institute of Applied Mathematics (DIAM)Delft University of TechnologyDelftThe Netherlands

Personalised recommendations