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The nonlinear critical layer resulting from the spatial or temporal evolution of weakly unstable disturbances in shear flows

Published online by Cambridge University Press:  26 April 2006

S. M. Churilov  and
I. G. Shukhman
S. M. Churilov
Affiliation:
Institute of Solar–Terrestrial Physics (ISTP), Siberian Department of Russian Academy of, Sciences, Irkutsk 33, PO Box 4026, 664033, Russia
I. G. Shukhman
Affiliation:
Institute of Solar–Terrestrial Physics (ISTP), Siberian Department of Russian Academy of, Sciences, Irkutsk 33, PO Box 4026, 664033, Russia
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Abstract

A study is made of the formation of a nonlinear critical layer (first found by Benney & Bergeron 1969 and Davis 1969) in homogeneous and weakly stratified incompressible shear flows as initially small unstable disturbances develop, whose growth rate is so small that all the evolution proceeds in the ‘quasi-steady’ regime. It is shown that such an evolution can be described from start to finish analytically using a pair of evolution equations for the wave amplitude and phase which involve universal functions of the familiar Haberman (1972) parameter, λ, that characterizes the relative importance of the dissipation and nonlinearity.

In addition to the function of a ‘logarithmic phase jump’ that was introduced and investigated by Haberman (1972), the evolution equations generally also contain other functions of λ. In this paper we introduce and study (numerically and analytically) another three such functions.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 318 , 10 July 1996 , pp. 189 - 221
Copyright
© 1996 Cambridge University Press

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References

Benney, D. J. & Bergeron, R. F. 1969 A new class of nonlinear waves in parallel flows. Stud. Appl. Maths 48, 181–204.Google Scholar
Benney, D. J. & Maslowe, S. A. 1975 The evolution in space and time of nonlinear waves in parallel shear flows. Stud. Appl. Maths 54, 181–205.Google Scholar
Brown, S. N. & Stewartson, K. 1978 The evolution of the critical layer of a Rossby wave. Part II. Geophys. Astrophys. Fluid Dyn. 10, 1–24.Google Scholar
Churilov, S. M. 1989 The nonlinear stabilization of a zonal shear flow instability. Geophys. Astrophys. Fluid Dyn. 46, 159–175.Google Scholar
Churilov, S. M. & Shukhman, I. G. 1987 The nonlinear development of disturbances in a zonal shear flow. Geophys. Astrophys. Fluid Dyn. 38, 145–175.Google Scholar
Churilov, S. M. & Shukhman, I. G. 1988 Nonlinear stability of a stratified shear flow in the regime with an unsteady critical layer. J. Fluid Mech. 194, 187–216.Google Scholar
Churilov, S. M. & Shukhman, I. G. 1992 Critical layer and nonlinear evolution of disturbances in weakly supercritical shear layer. XVIIIth Intl Congress of Theor. and Appl. Mech., Haifa, Israel. Abstracts, pp. 39–40; Preprint of Inst. Solar—Terrestrial Physics 4-93, Irkutsk. Also: Izv. RAN Fiz. Atmos. i Oceana 1995, 31 (4) 557–569 (in Russian).
Churilov, S. M. & Shukhman, I. G. 1994 Nonlinear spatial evolution of helical disturbances to an axial jet. J. Fluid. Mech. 281, 371–402.Google Scholar
Churilov, S. M. & Shukhman, I. G. 1995 Three-dimensional disturbances to a mixing layer in the nonlinear critical-layer regime. J. Fluid Mech. 291, 57–81.Google Scholar
Davis, R. E. 1969 On the high Reynolds number flow over a wavy boundary. J. Fluid Mech. 36, 337–346.Google Scholar
Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability. Cambridge University Press.
Goldstein, M. E. & Choi, S.-W. 1989 Nonlinear evolution of interacting oblique waves on twodimensional shear layers. J. Fluid Mech. 207, 97–120.Google Scholar
Goldstein, M. E. & Hultgren, L. S. 1988 Nonlinear spatial evolution of an externally excited instability wave in a free shear layer. J. Fluid Mech. 197, 295–330.Google Scholar
Goldstein, M. E. & Leib, S. J. 1989 Nonlinear evolution of oblique waves on compressible shear layer. J. Fluid Mech. 207, 73–96.Google Scholar
Haberman, R. 1972 Critical layers in parallel flows. Stud. Appl. Maths 51, 139–161.Google Scholar
Haberman, R. 1973 Wave-induced distortion of slightly stratified shear flow: a nonlinear critical-layer effect. J. Fluid Mech. 58, 727–735.Google Scholar
Huerre P. & Scott, J. F. 1980 Effects of critical layer structure on the nonlinear evolution of waves in free shear layers. Proc. R. Soc. Lond. A 371, 509–524.Google Scholar
Hultgren, L. S. 1992 Nonlinear spatial equilibration of an externally excited instability wave in a free shear layer. J. Fluid Mech. 236, 635–664.Google Scholar
Kelly, R. E. & Maslowe, S. A. 1970 The nonlinear critical layer in a slightly stratified shear flow. Stud. Appl. Maths 49, 301–326.Google Scholar
Shukhman, I. G. 1989 Nonlinear stability of a weakly supercritical mixing layer in a rotating fluid. J. Fluid Mech. 200, 425–450.Google Scholar
Shukhman, I. G. 1991 Nonlinear evolution of spiral density waves generated by the instability of the shear layer in a rotating compressible fluid. J. Fluid Mech. 233, 587–612.Google Scholar
Wu. X., Lee, S. S. & Cowley, S. J. 1993 On the weakly nonlinear three-dimensional instability of shear layers to pairs of oblique waves: the Stokes layer as a paradigm. J. Fluid Mech. 253, 681–721.Google Scholar