Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-19T01:19:02.845Z Has data issue: false hasContentIssue false
  • English
  • Français

Steady-state nonlinear internal gravity-wave critical layers satisfying an upper radiation condition

Published online by Cambridge University Press:  26 April 2006

Kevin G. Lamb  and
Raymond T. Pierrehumbert
Kevin G. Lamb
Affiliation:
Department of Physics, Memorial University of Newfoundland, St. John's, Newfoundland, Canada A1B 3X7
Raymond T. Pierrehumbert
Affiliation:
Department of the Geophysical Sciences, University of Chicago, 5734 Ellis Avenue, Chicago, IL, 60637 USA
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider the behaviour of an internal gravity wave encountering a critical level in a stratified fluid, assuming the critical-level flow to be dominated by nonlinear effects. The background flow is a shear layer, and the stratification is sufficiently strong to support wave propagation everywhere. Incident and reflected waves are permitted below the critical level, and a radiation condition is imposed far above it. For this geometry we construct, by a combination of asymptotic and numerical means, steady, nonlinear solutions, and discuss the associated transmission coefficients, reflection coefficients, phase shifts, and resonance positions when the system is forced from below.

The inviscid solutions we exhibit have continuous density and velocity everywhere, and so do not require the introduction of internal viscous boundary layers. Further, the streamlines bounding the recirculating cat's-eye regions have corners, just as in the unstratified case. For weak stratification, the transmitted wave is nearly as strong as the incident wave, and there is accompanying strong over-reflection. As the stratification increases, the critical level becomes a nearly perfect reflector. The amount of transmission depends on wave amplitude, and the sensitivity increases with increasing stratification.

There are regions of parameter space for which steady solutions could not be found. The critical-layer structure appears to break down by unbounded thickening when the stratification becomes too strong, suggesting that in these cases some neglected physical process must intervene to limit growth of the recirculating region.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 238 , May 1992 , pp. 371 - 404
Copyright
© 1992 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

Bacmeister, J. T. & Pierrehumbert, R. T. 1988 On high drag states of nonlinear stratified flow over obstacles. J. Atmos. Sci. 45, 63–80.Google Scholar
Benney, D. J. & Bergeron, R. F. 1969 A new class of nonlinear waves in parallel flows. Stud. Appl. Maths 48, 181–204.Google Scholar
Booker, J. R. & Bretherton, F. P. 1967 The critical layer for internal gravity waves in a shear flow. J. Fluid Mech. 27, 513–539.Google Scholar
Bretherton, F. P. 1966 The propagation of groups of internal gravity waves in a shear flow. Q. J. R. Met. Soc. 92, 466–480.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
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. 1982 On the nonlinear reflection of a gravity wave at a critical level. Part 2. J. Fluid Mech. 115, 217–230.Google Scholar
Fritts, D. C. 1984 Gravity wave saturation in the middle atmosphere. A review of theory and observations. Rev. Geophys. Space Phys. 22, 275–308.Google Scholar
Graham, E. W. 1982 On the steady-state relations between disturbances above and below a critical level. J. Fluid Mech. 115, 395–410.Google Scholar
Haberman, R. 1972 Critical layers in parallel flows. Stud. Appl. Maths 51, 139–160.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
Killworth, P. & McIntyre, M. E. 1985 Do Rossby wave critical levels absorb, reflect or overreflect? J. Fluid Mech. 161, 449–492.Google Scholar
Lamb, K. G. 1989 Nonlinear internal gravity wave critical layers. Ph.D. thesis, Princeton University.
Maslowe, S. A. 1972 The generation of clear air turbulence by nonlinear waves. Stud. Appl. Maths 51, 1–16.Google Scholar
Maslowe, S. A. 1973 Finite-amplitude Kelvin-Helmholtz billows. Boundary-Layer Met. 5, 43–52.Google Scholar
Maslowe, S. A. 1986 Critical layers in shear flows. Ann. Rev. Fluid Mech. 18, 405–432.Google Scholar
Moore, D. W. & Saffman, P. G. 182 Finite-amplitude waves in inviscid shear flows. Proc. R. Soc. Lond. A 382, 389–410.
Palmer, T. N., Shutts, G. J. & Swinbank, R. 1986 Alleviation of a systematic westerly bias in general circulation and numerical weather prediction models through an orographic gravity wave drag parameterization. Q. J. R. Met. Soc. 112, 1001–1039.Google Scholar
Peltier, W. R. & Clark, T. L. 1983 Nonlinear mountain waves in two and three spatial dimensions. Q. J. R. Met. Soc. 109, 527–548.Google Scholar
Stewartson, K. 1981 Marginally stable inviscid flows with critical layers. IMA J. Appl. Maths 27, 133–175.Google Scholar
Winters, K. & D'Asaro, E. A. 1989 Two-dimensional instability of finite-amplitude internal gravity wave packets near a critical level. J. Geophys. Res. 94, 12709–12719.Google Scholar