Hostname: page-component-76fb5796d-zzh7m Total loading time: 0 Render date: 2024-04-26T05:22:08.682Z Has data issue: false hasContentIssue false
  • English
  • Français

The effect of barotropic instability on the nonlinear evolution of a Rossby-wave critical layer

Published online by Cambridge University Press:  26 April 2006

Peter H. Haynes
Peter H. Haynes
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge, CB3 9EW, UK
Get access Rights & Permissions [Opens in a new window]

Abstract

A study of the flow within the critical layer of a forced Rossby-wave is made, using a high-resolution numerical model. The possibility of growth of disturbances through barotropic instability and the extent to which these disturbances modify the subsequent time evolution is of particular interest. The flow is characterized by a parameter μ, equal to the cross-stream lengthscale divided by a downstream wavelength. In the long-wavelength case, μ [Lt ] 1, where there is a clear conceptual division between the instability and the basic flow, the results of the simulation confirm the importance of the growing and saturating disturbances in rearranging the vorticity within the critical layer. When the wavelength is not so long, the distinction between the instability and the straightforward time evolution of the basic flow is less clear. Nonetheless for μ < 0.25 the ultimate evolution is still sensitive to the details of the initial perturbations and in this sense the flow may be regarded as being unstable. The time-integrated absorptivity of the critical layer may be considerably increased by the effects of the instability, sometimes to three or four times that predicted by the Stewartson-Warn-Warn solution. The nature of the flow, at least during the period in which the dynamics are essentially inviscid, does not seem to change when higher harmonics to the forced wave are resonant. The behaviour seen in Béland's (1976) numerical model is re-examined in the light of these findings. A simple model of the redistribution of vorticity by the unstable disturbances is formulated, and its predictions are shown to agree well with the numerical simulations.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 207 , October 1989 , pp. 231 - 266
Copyright
© 1989 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

Abramowitz, M. & Stegun, I. A. 1965 A Handbook of Mathematical Functions. Dover, pp. 1046.
Béland, M. 1976 Numerical study of the nonlinear Rossby-wave critical layer development in a barotropic zonal flow. J. Atmos. Sci. 33, 2066–2078.Google Scholar
Béland, M. 1978 The evolution of a nonlinear Rossby-wave critical layer: effects of viscosity. J. Atmos. Sci. 35, 1802–1815.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
Dickinson, R. E. 1970 Development of a Rossby wave critical level. J. Atmos. Sci. 27, 627–633.Google Scholar
Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability. Cambridge University Press, pp. 525.
Geisler, J. E. & Dickinson, R. E. 1974 Numerical study of an interacting Rossby-wave and barotropic zonal flow near a critical level. J. Atmos. Sci. 31, 946–955.Google Scholar
Haberman, R. 1972 Critical layers in parallel flows. Stud. Appl. Maths. 51, 139–161.Google Scholar
Haynes, P. H. 1985 Nonlinear instability of a Rossby-wave critical layer. J. Fluid Mech. 161, 493–511.Google Scholar
Haynes, P. H. 1987 On the instability of sheared disturbances. J. Fluid Mech. 175, 463–478.Google Scholar
Haynes, P. H. 1988 Forced, dissipative generalizations of finite-amplitude wave-activity conservation relations for zonal and non-zonal basic flows. J. Atmos Sci. 45, 2352–2362.Google Scholar
Haynes, P. H. & McIntyre, M. E. 1987 On the representation of Rossby-wave critical layers and wave breaking in zonally truncated models. J. Atmos. Sci. 44, 2359–2382.Google Scholar
Held, I. M. & Philips, P. J. 1987 Linear and nonlinear barotropic decay on the sphere. J. Atmos. Sci. 44, 200–207.Google Scholar
Juckes, M. N. & McIntyre, M. E. 1987 A high-resolution one-layer model of breaking planetary waves in the stratosphere. Nature 328, 590–596.Google Scholar
Killworth, P. D. & McIntyre, M. E. 1985 Do Rossby-wave critical layers absorb, reflect or over-reflect? J. Fluid Mech. 161, 449–492.Google Scholar
Maslowe, S. A. 1986 Critical layers in shear flows. Ann. Rev. Fluid Mech. 18, 405–432.Google Scholar
McIntyre, M. E. & Palmer, T. N. 1985 A note on the general concept of wave breaking. Pure Appl. Geophys. 123, 964–975.Google Scholar
Ritchie, H. 1984 Amplification of forced Rossby waves in the presence of a nonlinear critical layer. J. Atmos. Sci. 41, 2012–2019.Google Scholar
Ritchie, H. 1985 Rossby-wave resonance in the presence of a nonlinear critical layer. Geophys. Astrophys. Fluid Dyn. 31, 49–92.Google Scholar
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. Maths 27, 133–175.Google Scholar
Temme, N. M. 1983 The numerical computation of the confluent hypergeometric function U(a,b,z). Numer. Maths 41, 63–82.Google Scholar
Warn, T. & Warn, H. 1976 On the development of a Rossby-wave critical level. J. Atmos. Sci. 33, 2021–2024.Google Scholar
Warn, T. & Warn, H. 1978 The evolution of a nonlinear critical level. Stud. Appl. Maths 59, 37–71.Google Scholar