Hostname: page-component-7bb8b95d7b-wpx69 Total loading time: 0 Render date: 2024-10-06T14:14:45.931Z Has data issue: false hasContentIssue false
  • English
  • Français

Time-dependent critical layers in shear flows on the beta-plane

Published online by Cambridge University Press:  20 April 2006

Fred J. Hickernell
Fred J. Hickernell
Affiliation:
Department of Mathematics, University of Southern California, University Park, Los Angeles
Get access Rights & Permissions [Opens in a new window]

Abstract

The problem of a finite-amplitude free disturbance of an inviscid shear flow on the beta-plane is studied. Perturbation theory and matched asymptotics are used to derive an evolution equation for the amplitude of a singular neutral mode of the Kuo equation. The effects of time-dependence, nonlinearity and viscosity are included in the analysis of the critical-layer flow. Nonlinear effects inside the critical layer rather than outside the critical layer determine the evolution of the disturbance. The nonlinear term in the evolution equation is some type of convolution integral rather than a simple polynomial. This makes the evolution equation significantly different from those commonly encountered in fluid wave and stability problems.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 142 , May 1984 , pp. 431 - 449
Copyright
© 1984 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

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. 1978a The evolution of the critical layer of a Rossby wave. Part II. Geophys. Astrophys. Fluid Dyn. 10, 1–24.Google Scholar
Brown, S. N. & Stewartson, K. 1978b The evolution of a small inviscid disturbance to a marginally unstable stratified shear flow; stage two. Proc. R. Soc. Lond. A 363, 174–194.Google Scholar
Brown, S. N. & Stewartson, K. 1980 On the nonlinear reflexion of a gravity wave at a critical level. Part 1. J. Fluid Mech. 100, 577–595.Google Scholar
Brown, S. N. & Stewartson, K. 1982a On the nonlinear reflexion of a gravity wave at a critical level. Part 2. J. Fluid Mech. 115, 217–230.Google Scholar
Brown, S. N. & Stewartson, K. 1982b On the nonlinear reflexion of a gravity wave at a critical level. Part 3. J. Fluid Mech. 115, 231–250.Google Scholar
Burns, A. G. & Maslowe, S. A. 1983 Finite-amplitude stability of a zonal shear flow. J. Atmos. Sci. 40, 3–9.Google Scholar
Davis, R. E. 1969 On high Reynolds number flow over a wavy boundary. J. Fluid Mech. 36, 337–346.Google Scholar
Dickinson, R. E. & Clare, F. J. 1973 Numerical study of the unstable modes of a barotropic shear flow. J. Atmos. Sci. 30, 1035–1049.Google Scholar
Drazin, P. G. 1970 Kelvin—Helmholtz instability of finite amplitude. J. Fluid Mech. 42, 321–335.Google Scholar
Drazin, P. G. & Howard, L. N. 1966 Hydrodynamic stability of parallel flow of inviscid fluid. Adv. Appl. Mech. 9, 1–89.Google Scholar
Haberman, R. 1972 Critical layers in parallel flows. Stud. Appl. Math. 51, 139–161.Google Scholar
Hickernell, F. J. 1983 The evolution of large-horizontal-scale disturbances in marginally stable, inviscid, shear flows. I. Derivation of the amplitude evolution equations. Stud. Appl. Maths 69, 1–21.Google Scholar
Howard, L. N. & Drazin, P. G. 1964 On instability of parallel flow of inviscid fluid in a rotating system with variable coriolis parameter. J. Maths & Phys. 43, 83–99.Google Scholar
Huerre, P. 1980 The nonlinear stability of a free shear layer in the viscous critical layer regime. Phil. Trans. R. Soc. Lond. A 293, 643–675.Google Scholar
Huerre, P. & Scott, J. F. 1980 Effects of critical layer structure on the evolution of waves in free shear layers. Proc. R. Soc. Lond. A 371, 509–524.Google Scholar
Kelly, R. A. & Maslowe, S. A. 1970 The nonlinear critical layer in a slightly stratified shear flow. Stud. Appl. Math. 49, 301–326.Google Scholar
Kuo, H. L. 1949 Dynamic instability of two-dimensional nondivergent flow in a barotropic atmosphere. J. Met. 6, 105–122.Google Scholar
Maslowe, S. A. 1972 The generation of clear air turbulence by nonlinear waves. Stud. Appl. Maths 51, 1–16.Google Scholar
Maslowe, S. A. & Kelly, R. E. 1970 Finite-amplitude oscillations in a Kelvin—Helmholtz flow. Int. J. Non-linear Mech. 5, 427–435.Google Scholar
Maslowe, S. A. & Redekopp, L. G. 1979 Solitary waves in stratified shear flows. Geophys. Astrophys. Fluid Dyn. 13, 185–196.Google Scholar
Maslowe, S. A. & Redekopp, L. G. 1980 Long nonlinear waves in stratified shear flows. J. Fluid Mech. 101, 321–348.Google Scholar
Rayleigh, Lord 1880 On the stability or instability of certain fluid motions. Proc. Lond. Math. Soc. (Ser. 1) 11, 57–70.Google Scholar
Redekopp, L. G. 1977 Theory of solitary Rossby waves. J. Fluid Mech. 82, 725–745.Google Scholar
Rossby, C.-G. et al. 1939 Relation between variations in the intensity of the zonal circulation of the atmosphere and the displacement of the semi-permanent centers of action. J. Mar. Res. 2, 38–55.Google Scholar
Schade, H. 1964 Contribution to the nonlinear stability theory of inviscid shear layers. Phys. Fluids 7, 623–628.Google Scholar
Stakgold, I. 1979 Green's Functions and Boundary Value Problems. Wiley.
Stewartson, K. 1978 The evolution of the critical layer of a Rossby wave. Geophys. Astrophys. Fluid Dyn. 9, 185–200.Google Scholar
Stewartson, K. 1981 Marginally stable inviscid flows with critical layers. IMA J. Appl. Math. 27, 133–175.Google Scholar
Warn, T. & Warn, H. 1978 The evolution of a nonlinear critical level. Stud. Appl. Math. 59, 37–71.Google Scholar
Weissman, M. A. 1979 Nonlinear wave packets in the Kelvin—Helmholtz instability. Phil. Trans. R. Soc. Lond. A 290, 639–681.Google Scholar