Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-29T14:18:03.004Z Has data issue: false hasContentIssue false
  • English
  • Français

On the stability of heterogeneous shear flows. Part 2

Published online by Cambridge University Press:  28 March 2006

John W. Miles
John W. Miles
Affiliation:
Department of Mathematics, Institute of Advanced Studies, Australian National University, Canberra
Get access Rights & Permissions [Opens in a new window]

Abstract

Small disturbances relative to a horizontally stratified shear flow are considered on the assumptions that the velocity and density gradients in the undisturbed flow are non-negative and possess analytic continuations into a complex velocity plane. It is shown that the existence of a singular neutral mode (for which the wave speed is equal to the mean speed at some point in the flow) implies the existence of a contiguous, unstable mode in a wave-number (α), Richardson-number (J) plane. Explicit results are obtained for the rate of growth of nearly neutral disturbances relative to Hølmboe's shear flow, in which the velocity and the logarithm of the density are proportional to tanh (y/h). The neutral curve for this configuration, J = J0(α), is shown to be single-valued. Finally, it is shown that a relatively simple generalization of Hølmboe's density profile leads to a configuration having multiple-valued neutral curves, such that increasing J may be destabilizing for some range (s) of α.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 16 , Issue 2 , June 1963 , pp. 209 - 227
Copyright
© 1963 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

Drazin, P. G. 1958 J. Fluid Mech. 4, 214.
Drazin, P. G. & Howard, L. N. 1961 Proc. ASCE, Eng. Mech. Div., 87, 101.
Eliassen, A., Høiland, E. & Riis, E. 1953 Two-Dimensional Perturbation of a Flow with Constant Shear of a Stratified Fluid. Institute for Weather and Climate Research, Norwegian Academy of Sciences and Letters, Publ. no. 1.
Erdelyi, A., Magnus, W., Oberhettinger, F. & Tricomi, F. 1953 Higher Transcendental Functions. New York: McGraw-Hill.
Garcia, R. V. 1961 Unpublished Lecture, University of California, Los Angeles, 23 Feb. 1961; see also Hølmboe (1962).
Goldstein, S. 1931 Proc. Roy. Soc. A, 132, 524.
Heine, E. 1878 Handbuch der Kugelfunctionen. Berlin: G. Reimer..
Heun, K. 1889 Math. Ann. 33, 161.
HøLmboe, J. 1860 Unpublished Lecture Notes, University of California, Los Angeles.
HøLmboe, J. 1962 Geophysica Norvegica, 24, 67.
Howard, L. N. 1961 J. Fluid Mech. 10, 509.
Ince, E. L. 1944 Ordinary Differential Equations. New York: Dover.
Lin, C. C. 1945 Quart. Appl. Math. 3, 218.
Miles, J. W. 1961 J. Fluid Mech. 10, 496.
Taylor, G. I. 1931 Proc. Roy. Soc. A, 132, 499.