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Do Rossby-wave critical layers absorb, reflect, or over-reflect?

Published online by Cambridge University Press:  21 April 2006

Peter D. Killworth  and
Michael E. McIntyre
Peter D. Killworth
Affiliation:
Department of Applied Mathematics & Theoretical Physics. University of Cambridge, Silver Street, Cambridge CB3 9EW, U.K. Present affiliation: Robert Hooke Institute for Atmospheric Research, Dept. of Atmospheric Physics, Clarendon Laboratory, Oxford OX1 3PU.
Michael E. McIntyre
Affiliation:
Department of Applied Mathematics & Theoretical Physics. University of Cambridge, Silver Street, Cambridge CB3 9EW, U.K.
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Abstract

The Stewartson–Warn–Warn (SWW) solution for the time evolution of an inviscid, nonlinear Rossby-wave critical layer, which predicts that the critical layer will alternate between absorbing and over-reflecting states as time goes on, is shown to be hydrodynamically unstable. The instability is a two-dimensional shear instability, owing its existence to a local reversal of the cross-stream absolute vorticity gradient within the long, thin Kelvin cat's eyes of the SWW streamline pattern. The unstable condition first develops while the critical layer is still an absorber, well before the first over-reflecting stage is reached. The exponentially growing modes have a two-scale cross-stream structure like that of the basic SWW solution. They are found analytically using the method of matched asymptotic expansions, enabling the problem to be reduced to a transcendental equation for the complex eigenvalue. Growth rates are of the order of the inner vorticity scale δq, i.e. the initial absolute vorticity gradient dq0/dy times the critical-layer width scale. This is much faster than the time evolution of the SWW solution itself, albeit much slower than the shear rate du0/dy of the basic flow. Nonlinear saturation of the growing instability is expected to take place in a central region of width comparable to the width of the SWW cat's-eye pattern, probably leading to chaotic motion there, with very large ‘eddy-viscosity’ values. Those values correspond to critical-layer Reynolds numbers δ−1 [Lt ] 1, suggesting that for most initial conditions the time evolution of the critical layer will depart drastically from that predicted by the SWW solution. A companion paper (Haynes 1985) establishes that the instability can, indeed, grow to large enough amplitudes for this to happen.

The simplest way in which the instability could affect the time evolution of the critical layer would be to prevent or reduce the oscillations between over-reflecting and absorbing states which, according to the SWW solution, follow the first onset of perfect reflection. The possibility that absorption (or over-reflection) might be prolonged indefinitely is ruled out, in many cases of interest (even if the ‘eddy viscosity’ is large), by the existence of a rigorous, general upper bound on the magnitude of the time-integrated absorptivity α(t). The bound is uniformly valid for all time t. The absorptivity α(t) is defined as the integral over all past t of the jump in the wave-induced Reynolds stress across the critical layer. In typical cases the bound implies that, no matter how large t may become, |α(t)| cannot greatly exceed the rate of absorption predicted by linear theory multiplied by the timescale on which linear theory breaks down, say the time for the cat's-eye flow to twist up the absolute vorticity contours by about half a turn. An alternative statement is that |α(t)| cannot greatly exceed the initial absolute vorticity gradient dq0/dy times the cube of the widthscale of the critical layer. In typical cases, therefore, a brief answer to the question posed in the title is that the critical layer absorbs at first, at a rate ∞ dq0/dy, whereas after linear theory breaks down the critical layer becomes a perfect reflector in the long-time average. If absolute vorticity gradients vanish throughout the critical layer then the bound is zero, implying perfect reflection for all t.

The general conditions for the bound to apply are that the wave amplitude and critical-layer width are uniformly bounded for all t, the motion is two-dimensional, and vorticity is neither created nor destroyed within the critical layer, nor transported into or out of it by diffusion, by advection, or by other means. Vorticity may, however, be diffused or turbulently transported within the critical layer, provided that the region within which the transport acts is of bounded width and the range of values of vorticity within that region remains bounded. There are no other restrictions on wave amplitude, none on wavelength, and no assumptions about flow details within the critical layer nor about the initial vorticity profile qo(y), apart from an assumption that q0(y) has singularities no worse than a finite number of jump discontinuities. The proof, in its most general form, makes use of anew finite-amplitude conservation theorem for disturbances to parallel shear flows, generalizing the classical results of Taylor, Eliassen & Palm, and others.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 161 , December 1985 , pp. 449 - 492
Copyright
© 1985 Cambridge University Press

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References

Abramowitz, M. & Stegun, I. A. 1965 Handbook of Mathematical Functions. Dover. 1046 pp.
Al-Ajmi, D. N., Harwood, R. S. & Miles, T. 1985 A sudden warming in the middle atmosphere of the southern hemisphere. Q. J. R. Met. Soc. 111, 359–389.Google Scholar
Arnol'd, V. I. 1966 On an a priori estimate in the theory of hydrodynamical stability. Izv. Vyssh. Uchebn. Zaved. Matematika 54, no. 5, 3–5. (English transl.: Amer. Math. Soc. Transl., Series 2, 79, 267–269 (1969).)Google Scholar
Béland, M. 1976 Numerical study of the nonlinear Rossby wave critical level development in a barotropic flow. J. Atmos. Sci. 33, 2066–2078.Google Scholar
Beéland, M. 1978 The evolution of a nonlinear Rossby wave critical level: effects of viscosity. J. Atmos. Sci. 35, 1802–1815.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
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 algebraic decay of disturbances in a stratified linear shear flow. J. Fluid Mech. 100, 811–816.Google Scholar
Charney, J. G. & Stern, M. E. 1962 On the stability of internal baroclinic jets in a rotating atmosphere. J. Atmos. Sci. 19, 159–172.Google Scholar
Clough, S. A., Grahame, N. S. & O'Neill, A. 1985 Potential vorticity in the stratosphere derived using data from satellites. Q. J. R. Met. Soc. 111, 335–358.Google Scholar
Davis, R. E. 1969 On the high Reynolds number flow over a wavy boundary. J. Fluid Mech. 36, 337–346.Google Scholar
Dickinson, R. E. 1970 Development of a Rossby wave critical level. J. Atmos. Sci. 27, 627–633.Google Scholar
Edmon, H. J., Hoskins, B. J. & McIntyre, M. E. 1980 Eliassen–Palm cross-sections for the troposphere. J. Atmos. Sci. 37, 2600–2616. (See also Corrigendum. J. Atmos. Sci. 38, 1115, especially second last item).Google Scholar
Eliassen, A. & Palm, E. 1961 On the transfer of energy in stationary mountain waves. Geofys. Publ. 22, no. 3, 1–23.Google Scholar
Fletcher, N. H. 1979 Air flow and sound generation in musical wind instruments. Ann. Rev. Fluid Mech. 11, 123–146.Google Scholar
Foote, J. R. & Lin, C. C. 1950 Some recent investigations in the theory of hydrodynamic stability. Q. Appl. Maths 8, 265–280.Google Scholar
Green, J. S. A. 1970 Transfer properties of the large-scale eddies and the general circulation of the atmosphere. Q. J. Met. Soc. 96, 157–185.Google Scholar
Haberman, R. 1972 Critical layers in parallel flows. Stud. Appl. Math. 51, 139–161.Google Scholar
Haidvogel, D. B. & Rhines, P. B. 1983 Waves and circulation driven by oscillatory winds in an idealized ocean basin. Geophys. Astrophys. Fluid Dyn. 25, 1–63.Google Scholar
Hamilton, K. 1982 Some features of the climatology of the northern hemisphere stratosphere revealed by NMC upper atmosphere analyses. J. Atmos. Sci. 39, 2737–2749.Google Scholar
Haynes, P. H. 1985 Nonlinear instability of a Rossby-wave critical layer. J. Fluid Mech. 161, 493–511.Google Scholar
Held, I. M. 1983 Stationary and quasi-stationary eddies in the extratropical troposphere: theory. In Large-scale dynamical processes in the atmosphere (ed. R. P. Pearce & B. J. Hoskins), pp. 127–168. Academic, 397 pp.
Holliday, D. & McIntyre, M. E. 1981 On potential energy density in an incompressible, stratified fluid. J. Fluid Mech. 107, 221–225.Google Scholar
Hoskins, B. J., McIntyre, M. E. & Robertson, A. W. 1985 On the use and significance of isentropic potential-vorticity maps. Q. J. R. Met. Soc. 111, 877–946.Google Scholar
Karoly, D. J. 1982 Eliassen–Palm cross sections for the northern and southern hemispheres. J. Atmos. Sci. 39, 178–182.Google Scholar
Kelvin, Lord (W. Thomson) 1887 Stability of fluid motion– rectilineal motion of viscous fluid between two parallel planes. Phil. Mag. 24, 188–196.Google Scholar
Leovy, C. B., Sun, C.-R., Hitchman, M. H., Remsberg, E. E., Russell, H. M., Gordley, L. L., Gille, J. C. & Lyjak, L. V. 1985 Transport of ozone in the middle stratosphere: evidence for planetary wave breaking. J. Atmos. Sci. 42, 230–244.Google Scholar
Lesser, M. B. & Crighton, D. J. 1975 Physical acoustics and the method of matched asymptotic expansions. In Physical Acoustics, vol. 11 (ed. W. P. Mason & R. N. Thurston), pp. 69–149. Academic.
Lin, C. C. 1955 The Theory of Hydrodynamic Stability. Cambridge University Press. 155 pp.
Maslowe, S. A. 1981 Shear flow instabilities and transition. In Hydrodynamic Instabilities and the Transition to Turbulence (ed. H. L. Swinney & J. P. Gollub), pp. 181–228. Springer. 292 pp.
McIntyre, M. E. 1982 How well do we understand the dynamics of stratospheric warmings? J. Met. Soc. Japan 60, 37–65.Google Scholar
McIntyre, M. E. & Palmer, T. N. 1983 Breaking planetary waves in the stratosphere. Nature 305, 593–600.Google Scholar
McIntyre, M. E. & Palmer, T. N. 1984 The ‘surf zone’ in the stratosphere. J. Atmos. Terr. Phys. 46, 825–849.Google Scholar
McIntyre, M. E., Schumacher, R. T. & Woodhouse, J. 1983 On the oscillations of musical instruments. J. Acoust Soc. Am. 74, 1325–1345.Google Scholar
McIntyre, M. E. & Shepherd, T. G. 1986 An exact conservation theorem for finite-amplitude disturbances to nonparallel shear flows, with remarks on Hamiltonian structure and on Arnol'd's stability theorems. J. Fluid Mech., submitted.Google Scholar
Nigam, S. & Held, I. M. 1983 The influence of a critical latitude on topographically forced stationary waves in a barotropic model. J. Atmos. Sci. 40, 2610–2622.Google Scholar
Rhines, P. B. 1977 The dynamics of unsteady currents. In The Sea: Ideas and observations on progress in the study of the seas (ed. E. D. Goldberg et al.), vol. 6, pp. 189–318. Wiley.
Rhines, P. B. 1979 Geostrophic turbulence. Ann. Rev. Fluid Mech. 11, 401–441.Google Scholar
Rhines, P. B. & Young, W. R. 1983 How rapidly is a passive scalar mixed within closed streamlines? J. Fluid Mech. 133, 133–145.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
Ruttenberg, S. 1980 collection of Extended Abstracts Presented at ICMUA Sessions and IUGG Symposium 18, pp. 199–204. ICMUA (Ruttenberg), NCAR, Box 3000, Boulder, CO 80307.
Shepherd, T. G. 1985 On the time development of small disturbances to plane Couette flow. J. Atmos. Sci. 42, 1868–1871.Google Scholar
Smith, F. T. & Bodonyi, R. J. 1982 Nonlinear critical layers and their development in streaming-flow stability. J. Fluid Mech. 118, 165–185.Google Scholar
Starr, V. P. 1968 Physics of Negative Viscosity Phenomena. McGraw-Hill. 256 pp.
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
Taylor, G. I. 1915 Eddy motion in the atmosphere. Phil. Trans. R. Soc. Lond. A 215, 1–26.Google Scholar
Tung, K. K. 1979 A theory of stationary long waves. Part III: Quasi-normal modes in a singular waveguide. Mon. Weather Rev. 107, 751–774.Google Scholar
Tung, K. K. & Lindzen, R. S. 1979a A theory of stationary long waves. Part I: A simple theory of blocking. Mon. Weather Rev. 107, 714–734.Google Scholar
Tung, K. K. & Lindzen, R. S. 1979b A theory of stationary long waves. Part II: Resonant Rossby waves in the presence of realistic vertical shears. Mon. Weather Rev. 107, 735–750.Google Scholar
Van Dyke, M. 1975 Perturbation Methods in Fluid Mechanics (annotated edition). Parabolic, 271 pp.
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. Math. 59, 37–71.Google Scholar
Whittaker, E. T. 1937 A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, 4th edn. Cambridge University Press, 456 pp.
Yamagata, T. 1976 On trajectories of Rossby wave-packets released in a lateral shear flow. J. Ocean. Soc. Japan. 32, 162–168.Google Scholar