Skip to main content Accessibility help

The linear instability of the stratified plane Couette flow

  • Giulio Facchini (a1), Benjamin Favier (a1), Patrice Le Gal (a1), Meng Wang (a2) and Michael Le Bars (a1)...


We present the stability analysis of a plane Couette flow which is stably stratified in the vertical direction orthogonal to the horizontal shear. Interest in such a flow comes from geophysical and astrophysical applications where background shear and vertical stable stratification commonly coexist. We perform the linear stability analysis of the flow in a domain which is periodic in the streamwise and vertical directions and confined in the cross-stream direction. The stability diagram is constructed as a function of the Reynolds number $Re$ and the Froude number $Fr$ , which compares the importance of shear and stratification. We find that the flow becomes unstable when shear and stratification are of the same order (i.e.  $Fr\sim 1$ ) and above a moderate value of the Reynolds number $Re\gtrsim 700$ . The instability results from a wave resonance mechanism already known in the context of channel flows – for instance, unstratified plane Couette flow in the shallow-water approximation. The result is confirmed by fully nonlinear direct numerical simulations and, to the best of our knowledge, constitutes the first evidence of linear instability in a vertically stratified plane Couette flow. We also report the study of a laboratory flow generated by a transparent belt entrained by two vertical cylinders and immersed in a tank filled with salty water, linearly stratified in density. We observe the emergence of a robust spatio-temporal pattern close to the threshold values of $Fr$ and $Re$ indicated by linear analysis, and explore the accessible part of the stability diagram. With the support of numerical simulations we conclude that the observed pattern is a signature of the same instability predicted by the linear theory, although slightly modified due to streamwise confinement.


Corresponding author

Email address for correspondence:


Hide All
Acheson, D. J. 1990 Elementary Fluid Dynamics. Oxford University Press.
Arratia, C.2011 Non-modal instability mechanisms in stratified and homogeneous shear flow. Theses, Ecole Polytechnique X.
Baines, P. G. & Mitsudera, H. 1994 On the mechanism of shear flow instabilities. J. Fluid Mech. 276, 327342.
Bakas, N. A. & Farrell, B. F. 2009 Gravity waves in a horizontal shear flow. Part ii: interaction between gravity waves and potential vorticity perturbations. J. Phys. Oceanogr. 39 (3), 497511.
Barkley, D. & Tuckerman, L. S. 2005 Computational study of turbulent laminar patterns in Couette flow. Phys. Rev. Lett. 94, 014502.
Bayly, B. J., Orszag, A. & Herbert, T. 1988 Instability mechanisms in shear-flow transition. Annu. Rev. Fluid Mech. 20 (1), 359391.
Boulanger, N., Meunier, P. & Le Dizès, S. 2008 Tilt-induced instability of a stratified vortex. J. Fluid Mech. 596, 120.
Candelier, J., Le Dizès, S. & Millet, C. 2011 Shear instability in a stratified fluid when shear and stratification are not aligned. J. Fluid Mech. 685, 191201.
Caulfield, C.-C. P. 1994 Multiple linear instability of layered stratified shear flow. J. Fluid Mech. 258, 255285.
Chen, J.2016 Stabilité d’un écoulement stratifié sur une paroi et dans un canal. PhD thesis, École Centrale Marseille.
Chen, J., Bai, Y. & Le Dizès, S. 2016 Instability of a boundary layer flow on a vertical wall in a stably stratified fluid. J. Fluid Mech. 795, 262277.
Davey, A. 1973 On the stability of plane Couette flow to infinitesimal disturbances. J. Fluid Mech. 57 (2), 369380.
Deloncle, A., Chomaz, J.-M. & Billant, P. 2007 Three-dimensional stability of a horizontally sheared flow in a stably stratified fluid. J. Fluid Mech. 570, 297305.
Dengler, M. & Quadfasel, D. 2002 Equatorial deep jets and abyssal mixing in the Indian Ocean. J. Phys. Oceanogr. 32 (4), 11651180.
d’Orgeville, M., Hua, B. L., Schopp, R. & Bunge, L. 2004 Extended deep equatorial layering as a possible imprint of inertial instability. Geophys. Res. Lett. 31 (22), l22303.
Dubrulle, B., Marié, L., Normand, C., Richard, D., Hersant, F. & Zahn, J.-P. 2005 An hydrodynamic shear instability in stratified disks. J. Astron. Astrophys. 429, 113.
Dunkerton, T. J. 1981 On the inertial stability of the equatorial middle atmosphere. J. Atmos. Sci. 38 (11), 23542364.
Fischer, P. F. 1997 An overlapping Schwarz method for spectral element solution of the incompressible Navier–Stokes equations. J. Comput. Phys. 133 (1), 84101.
Fischer, P. F., Loth, F., Lee, S. E., Lee, S.-W., Smith, D. S. & Bassiouny, H. S. 2007 Simulation of high-Reynolds number vascular flows. Comput. Meth. Appl. Mech. Engng 196 (31), 30493060.
Heisenberg, W. 1924 Über Stabilität und Turbulenz von Flüssigkeitsströmen. Ann. Phys. 379, 577627.
Helmholtz, H. L. F. v. 1868 XLIII. On discontinuous movements of fluids. Phil. Mag. 36 (244), 337346.
Holmboe, J. 1962 On the behaviour of symmetric waves in stratified shear layers. Geofys. Publ. 24, 67113.
Howard, L. N. 1961 Note on a paper of John W. Miles. J. Fluid Mech. 10 (4), 509512.
Ibanez, R., Swinney, H. L. & Rodenborn, B. 2016 Observations of the stratorotational instability in rotating concentric cylinders. Phys. Rev. Fluids 1, 053601.
Lord Kelvin 1871 XLVI. Hydrokinetic solutions and observations. Phil. Mag. 42 (281), 362377.
Kushner, P. J., McIntyre, M. E. & Shepherd, T. G. 1998 Coupled Kelvin-wave and mirage-wave instabilities in semigeostrophic dynamics. J. Phys. Oceanogr. 28 (3), 513518.
Le Bars, M. & Le Gal, P. 2007 Experimental analysis of the stratorotational instability in a cylindrical Couette flow. Phys. Rev. Lett. 99, 064502.
Lin, C. 1966 The Theory of Hydrodynamic Stability. Cambridge University Press, xi, 155 pp.
Lucas, D. & Caulfield, C. P. 2017 Irreversible mixing by unstable periodic orbits in buoyancy dominated stratified turbulence. J. Fluid Mech. 832, R1.
Lucas, D., Caulfield, C. P. & Kerswell, R. R. 2017 Layer formation in horizontally forced stratified turbulence: connecting exact coherent structures to linear instabilities. J. Fluid Mech. 832, 409437.
Marcus, P. S., Pei, S., Jiang, C.-H. & Hassanzadeh, P. 2013 Three-dimensional vortices generated by self-replication in stably stratified rotating shear flows. Phys. Rev. Lett. 111, 084501.
Meunier, P. & Leweke, T. 2003 Analysis and treatment of errors due to high velocity gradients in particle image velocimetry. Exp. Fluids 35 (5), 408421.
Miles, J. W. 1961 On the stability of heterogeneous shear flows. J. Fluid Mech. 10 (4), 496508.
Molemaker, M. J., McWilliams, J. C. & Yavneh, Irad 2001 Instability and equilibration of centrifugally stable stratified Taylor–Couette flow. Phys. Rev. Lett. 86, 52705273.
Oglethorpe, R. L. F., Caulfield, C. P. & Woods, A. W. 2013 Spontaneous layering in stratified turbulent Taylor–Couette flow. J. Fluid Mech. 721, R3.
Orr, W. M’F. 1907 The stability or instability of the steady motions of a perfect liquid and of a viscous liquid. Part I: a perfect liquid. Proc. R. Irish Acad. A 27, 968.
Orszag, S. A. 1971 Accurate solution of the Orr–Sommerfeld stability equation. J. Fluid Mech. 50 (4), 689703.
Oster, G. 1965 Density gradients. Sci. Am. 213, 7076.
Park, J. & Billant, P. 2013 The stably stratified Taylor–Couette flow is always unstable except for solid-body rotation. J. Fluid Mech. 725, 262280.
Lord Rayleigh 1879 On the stability, or instability, of certain fluid motions. Proc. Lond. Math. Soc. s1–11 (1), 5772.
Lord Rayleigh 1917 On the dynamics of revolving fluids. Proc. R. Soc. Lond. A 93 (648), 148154.
Reynolds, O. 1883 An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous, and of the law of resistance in parallel channels. Phil. Trans. R. Soc. Lond. 174, 935982.
Romanov, V. A. 1973 Stability of plane-parallel Couette flow. Funct. Anal. Applics. 7 (2), 137146.
Satomura, T. 1981 An investigation of shear instability in a shallow water. J. Met. Soc. Japan. II 59 (1), 148167.
Schlichting, H. 1933 Zur Enstehung der Turbulenz bei der Plattenstrmung. Nachr. Ges. Wiss. Göttingen 1933, 181208.
Taylor, G. I. 1931 Effect of variation in density on the stability of superposed streams of fluid. Proc. R. Soc. Lond. A 132 (820), 499523.
Thorpe, S. A. 2016 Layers and internal waves in uniformly stratified fluids stirred by vertical grids. J. Fluid Mech. 793, 380413.
Vanneste, J. & Yavneh, I. 2007 Unbalanced instabilities of rapidly rotating stratified shear flows. J. Fluid Mech. 584, 373396.
Woods, A. W., Caulfield, C. P., Landel, J. R. & Kuesters, A. 2010 Non-invasive turbulent mixing across a density interface in a turbulent Taylor–Couette flow. J. Fluid Mech. 663, 347357.
Yavneh, I., McWilliams, J. C. & Molemaker, M. J. 2001 Non-axisymmetric instability of centrifugally stable stratified Taylor–Couette flow. J. Fluid Mech. 448, 121.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *

JFM classification


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed