Hostname: page-component-848d4c4894-m9kch Total loading time: 0 Render date: 2024-05-11T09:25:58.527Z Has data issue: false hasContentIssue false
  • English
  • Français

Secondary instability of the stably stratified Ekman layer

Published online by Cambridge University Press:  01 July 2013

Nadia Mkhinini ,
Thomas Dubos  and
Philippe Drobinski
Nadia Mkhinini*
Affiliation:
Laboratoire de Météorologie Dynamique, École Polytechnique, 91120 Palaiseau, France
Thomas Dubos
Affiliation:
Laboratoire de Météorologie Dynamique, École Polytechnique, 91120 Palaiseau, France
Philippe Drobinski
Affiliation:
Laboratoire de Météorologie Dynamique, École Polytechnique, 91120 Palaiseau, France
*
†Email address for correspondence: nadia.mkhinini@doc.polytechnique.org
Get access Rights & Permissions [Opens in a new window]

Abstract

The Ekman flow, an exact solution of the Boussinesq equations with rotation, is a prototype flow for both atmospheric and oceanic boundary layers. The effect of stratification on the finite-amplitude longitudinal rolls developing in the Ekman flow and their three-dimensional stability is studied by means of linearized and nonlinear numerical simulations. Similarities and differences with respect to billows developing in the Kelvin–Helmholtz (KH) unidirectional stratified shear flow are discussed. Prandtl number effects are investigated as well as the role played by the buoyant-convective instability. For low Prandtl number, the amplitude of the saturated rolls vanishes at the critical bulk Richardson number, while at high Prandtl number, finite-amplitude rolls are found. The Prandtl number also affects how the growth rate of the secondary instability evolves as the Richardson number is increased. For low Prandtl number, the growth rate decreases as the Richardson number increases while it remains significant for large Prandtl number over the range of stratification studied. This behaviour is likely a result of the differing amplitudes of the roll vortices. Furthermore, the most unstable wave vector is much lower than for the secondary instability of KH billows. Examination of the energetics of the secondary instability shows that buoyant-convective instability is present locally at high Reynolds and Prandtl numbers but plays an overall minor role despite the presence in the base flow of statically unstable regions characterized by a high Richardson number.

JFM classification

Instability: Instability Geophysical and Geological Flows: Stratified flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 728 , 10 August 2013 , pp. 29 - 57
Copyright
©2013 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Brown, R. A. 1970 A secondary flow model for the planetary boundary layer. J. Atmos. Sci 27, 742–757.Google Scholar
Brown, R. A. 1972 On the inflection point instability of a stratified Ekman boundary layer. J. Atmos. Sci 29, 850–859.Google Scholar
Caldwell, D. R. & Van Atta, C. W. 1970 Characteristics of Ekman boundary layer instabilities. J. Fluid Mech. 44, 79–95.CrossRefGoogle Scholar
Caulfield, C. P. & Peltier, W. R. 2000 The anatomy of the mixing transition in homogeneous and stratified free shear layers. J. Fluid Mech. 413, 1–47.Google Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Clarendon.Google Scholar
Chlond, A. 1987 A numerical study of horizontal roll vortices in neutral and unstable boundary layers. Beitr. Phys. Atmos. 60, 144–169.Google Scholar
Coleman, G. N. 1999 Similarity statistics from a direct numerical simulation of the neutrally stratified planetary boundary layer. J. Atmos. Sci. 56, 891–900.2.0.CO;2>CrossRefGoogle Scholar
Coleman, G. N., Ferziger, J. H. & Spalart, P. R. 1990 A numerical study of the turbulent Ekman layer. J. Fluid Mech. 213, 313–348.Google Scholar
Coleman, G. N., Ferziger, J. H. & Spalart, P. R. 1992 Direct simulation of the stably stratified turbulent Ekman layer. J. Fluid Mech. 244, 677–712.Google Scholar
Dubos, T., Barthlott, C. & Drobinski, P. 2008 Emergence and secondary instability of Ekman layer rolls. J. Atmos. Sci 65, 2326–2342.Google Scholar
Ekman, V. W. 1905 On the influence of the Earth’s rotation on ocean currents. Arkiv. Matem. Atr. Fysik 2 (11), 1–54.Google Scholar
Etling, D. & Brown, R. A. 1993 Roll vortices in the planetary boundary layer: a review. Boundary-Layer Meteorol. 21, 215–248.Google Scholar
Faller, A. J. & Kaylor, R. E. 1966 A numerical study of the instability of the laminar Ekman boundary layer. J. Atmos. Sci. 23, 466–480.2.0.CO;2>CrossRefGoogle Scholar
Foster, R. C. 1996 An analytic model for planetary boundary layer roll vortices. PhD thesis, University of Washington.Google Scholar
Haeusser, T. M. & Leibovich, S. 2003 Pattern formation in the marginally unstable Ekman layer. J. Fluid Mech. 479, 125–144.Google Scholar
Howard, L. N. 1961 Note on a paper of John W. Miles. J. Fluid Mech. 10, 509–512.CrossRefGoogle Scholar
Julien, S., Ortiz, S. & Chomaz, J.-M. 2004 Secondary instability mechanisms in the wake of a flat plate. Eur. J. Mech. (B/Fluids) 23, 157–165.Google Scholar
Kaylor, R. & Faller, A. J. 1972 Instability of the stratified Ekman boundary layer and the generation of internal waves. J. Atmos. Sci. 29, 497–509.2.0.CO;2>CrossRefGoogle Scholar
Klaassen, G. P. & Peltier, W. R. 1985 The effect of Prandtl number on the evolution and stability of Kelvin–Helmholtz billows. Geophys. Astrophys. Fluid Dyn. 32, 23–60.Google Scholar
Leibovich, S. & Lele, S. K. 1985 The influence of the horizontal component of Earth’s angular velocity on the instability of the Ekman layer. J. Fluid Mech. 150, 41–87.Google Scholar
Lilly, D. K. 1966 On the instability of Ekman boundary flow. J. Atmos. Sci. 23, 481–494.Google Scholar
Mahrt, L. 1999 Stratified atmospheric boundary layers. Boundary-Layer Meteorol. 90, 375–396.CrossRefGoogle Scholar
Mashayek, A. & Peltier, W. R. 2012 The ‘zoo’ of secondary instabilities precursory to stratified shear flow transition. Part 1. Shear aligned convection, pairing, and braid instabilities. J. Fluid Mech. 708, 5–44.CrossRefGoogle Scholar
Miles, J. W. 1961 On the stability of heterogeneous shear flows. J. Fluid Mech. 10, 496–508.CrossRefGoogle Scholar
Mkhinini, N., Dubos, T & Drobinski, P. 2013 On the non-linear destabilization of stably stratified shear flow. J. Fluid Mech. (submitted).CrossRefGoogle Scholar
Peltier, W. R. & Caulfield, C. P. 2003 Mixing efficiency in stratified shear flows. Annu. Rev. Fluid Mech. 35, 135–167.Google Scholar
Saad, Y. & Schultz, M. H. 1986 GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Comput. 7, 856–869.Google Scholar
Spalart, P. R. 1989 Theoretical and numerical study of a three-dimensional turbulent boundary layer. J. Fluid Mech. 205, 319–340.Google Scholar
Spalart, P. R., Coleman, G. N. & Johnstone, R. 2008 Direct numerical simulation of the Ekman layer: a step in Reynolds number, and cautious support for a log law with a shifted origin. Phys. Fluids 20 (10), 101507.Google Scholar
Spalart, P. R., Coleman, G. N. & Johnstone, R. 2009 Retraction: ‘direct numerical simulation of the Ekman layer: a step in Reynolds number, and cautious support for a log law with a shifted origin’. Phys. Fluids 21 (10), 109901.Google Scholar
Young, G. S., Kristovich, D. A. R., Hjelmfelt, M. R. & Foster, R. C. 2002 Rolls, streets, waves, and more: a review of quasi-two-dimensional structures in the atmospheric boundary layer. Bull. Am. Meteorol. Soc. 83, 997–1001.Google Scholar