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Dissipative wave-mean interactions and the transport of vorticity or potential vorticity

Published online by Cambridge University Press:  26 April 2006

M. E. McIntyre  and
W. A. Norton
M. E. McIntyre
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge, CB3 9EW, UK
W. A. Norton
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge, CB3 9EW, UK
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Abstract

Wave-mean interactions of the classical type, in which the effect of the waves on the mean motion depends on wave breaking or other types of wave dissipation, are to be sharply distinguished from other types of wave-mean interaction that have no such dependence on dissipation. Important cases arise both for unstratified (homentropic) flow and for stably stratified flow under gravity. A very general way of characterizing what is meant by the classical, dissipative type of wave-induced mean motion is to say that the wave-induced mean motions are balanced motions, in a sense to be discussed, and that the effective mean force corresponds to the wave-induced vorticity or potential vorticity transport that results from wave dissipation. For a stratified fluid, ‘potential vorticity’ is to be understood in the sense of Rossby and Ertel. ‘Balanced’ is to be understood in whatever sense is needed to imply the invertibility of the vorticity or potential vorticity field to give the other fields describing the mean motion. At first sight this appears to require that an appropriate Mach, Froude and/or Rossby number for the mean motion should be much smaller than unity, but the fundamental, and in practice less stringent, principal requirement appears to be that the spontaneous emission, or aerodynamic generation, of sound, gravity and/or inertio-gravity waves by the mean flow should be weak.

Three basic examples of dissipative wave-induced mean flow generation are presented and discussed. The first is the transport of vorticity by dissipating sound waves, which gives rise to classical acoustic streaming of the quartz-wind type. The transport or flux of vorticity can always be taken to be an exactly antisymmetric tensor; and in the case of a plane sound wave this tensor fluctuates about a mean value equal to $-\epsilon_{ijk}\dot{q}_k$, where $\dot{q}_k$ is the kth component of $\dot{\boldmath q}$, the rate of dissipation of the pseudomomentum or quasimomentum ${\boldmath q} \approx Ek/\overline{\rho}\omega$ per unit mass. Here $\overline{\rho}$ and E are the mean mass and wave-energy densities, ω the intrinsic frequency, and k the wavenumber. This is a succinct way of making evident why it is only the contribution q to the radiation stress convergence per unit mass that is significant for the generation of mean streaming. The second example is the transport of Rossby–Ertel potential vorticity (PV) by internal gravity waves that are either dissipating laminarly, or ‘breaking’ to produce inhomogeneous three-dimensional turbulence. This PV transport gives rise to mean streaming in much the same way as the vorticity transport in the acoustic example. The transport or flux of PV can always be taken to be directed exactly along the isentropic surfaces θ = constant of the stable stratification, where θ is potential temperature or potential density as appropriate; and in the case of a plane internal gravity wave the wave-induced PV transport fluctuates about a mean value G × q, where G is the basic gradient of θ associated with the stable stratification. This is a succinct way of making evident why it is only the projection of q onto the basic stratification surfaces that is significant. In both the acoustic and the internal-gravity examples the transport is non-advective, and often upgradient. The third example is the corresponding problem for Rossby waves, in which the typical effect of wave dissipation is a downgradient PV transport. This is brought about in an entirely different way, namely through advection of PV anomalies by the fluctuating velocity field of the wave motion, whether the dissipation be laminar or by breaking.

Processes of the sort idealized in the second and third examples are ubiquitous in the Earth's atmosphere and, for instance, largely control the strength of the global-scale middle atmospheric circulation and hence, for instance, the e-folding residence times (∼ 102 y) of man-made chlorofluorocarbons in the lower atmosphere.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 212 , March 1990 , pp. 403 - 435
Copyright
© 1990 Cambridge University Press

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References

Andrews, D. G., Holton, J. R. & Leovy, C. B., 1987 Middle Atmosphere Dynamics. Academic, 489 pp.
Andrews, D. G. & McIntyre, M. E., 1978 An exact theory of nonlinear waves on a Lagrangian-mean flow. J. Fluid Mech. 89, 609–646.Google Scholar
Batchelor, G. K.: 1952 The effect of homogeneous turbulence on material lines and surfaces. Proc. R. Soc. Lond. A 213, 349–366.Google Scholar
Batchelor, G. K.: 1953 The Theory of Homogeneous Turbulence. Cambridge University Press, 197 pp.
Batchelor, G. K.: 1959 Small-scale variation of convected quantities like temperature in turbulent fluid. Part 1. General discussion and the case of small conductivity. J. Fluid Mech. 5, 113–133.Google Scholar
Batchelor, G. K.: 1967 An Introduction to Fluid Dynamics. Cambridge University Press. 615 pp.
Bergmann, L.: 1954 Ultrasonics and their Scientific and Technical Applications. Wiley. [Original German edition: Stuttgart, Hirzel, 1938.]
Bretherton, F. P.: 1966 Critical layer instability in baroclinic flows. Q. J. R. Met. Soc. 92, 325–334.Google Scholar
Bretherton, F. P.: 1969 On the mean motion induced by internal gravity waves. J. Fluid Mech. 36, 785–803.Google Scholar
Bretherton, F. P.: 1971 The General Linearised Theory of Wave Propagation, §6. Lectures in Applied Mathematics, vol. 13, pp. 61–102. Amer. Math. Soc.
Brewer, A. W. & Wilson, A. W., 1968 The regions of formation of atmospheric ozone. Q. J. R. Met. Soc. 94, 249–265.Google Scholar
Brillouin, L.: 1925 On radiation stresses. Annales de Physique 4, 528–586 (in French).Google Scholar
Broutman, D. & Grimshaw, R., 1988 The energetics of the interaction between short small-amplitude internal waves and inertial waves. J. Fluid Mech. 196, 93–106.Google Scholar
Butchart, N.: 1987 Evidence for planetary wave breaking from satellite data: the relative roles of diabatic effects and irreversible mixing. In Transport Processes in the Middle Atmosphere (Proc. NATO Workshop, November 1986, Erice, Sicily (ed. G. Visconti & R. R. Garcia), pp. 121–136. 08, 475–502. Reidel.
Butchart, N. & Remsberg, E. E., 1986 Area of the stratospheric polar vortex as a diagnostic for tracer transport on an isentropic surface. J. Atmos. Sci. 43, 1319–1339.Google Scholar
Charney, J. G.: 1948 On the scale of atmospheric motions. Geofysiske Publ. 17 (2), 3–17.Google Scholar
Childress, S., Ierley, G. R., Spiegel, E. A. & Young, W. R., 1989 Blow-up of unsteady two-dimensional Euler and Navier–Stokes solutions having stagnation-point form. J. Fluid Mech. 203, 1–22.Google Scholar
Cocke, W. J.: 1969 Turbulent hydrodynamic line stretching: consequences of isotropy. Phys. Fluids 12, 2488–2492.Google Scholar
Craik, A. D. D.: 1977 The generation of Langmuir circulations by an instability mechanism. J. Fluid Mech. 81, 209–223.Google Scholar
Craik, A. D. D.: 1982a The drift velocity of water waves. J. Fluid Mech. 116, 187–205.Google Scholar
Craik, A. D. D.: 1982b Wave-induced longitudinal-vortex instability in shear flows. J. Fluid Mech. 125, 37–52.Google Scholar
Craik, A. D. D.: 1985 Wave Interactions and Mean Flows. Cambridge University Press, 322 pp.
Crighton, D. G.: 1975 Basic principles of aerodynamic noise generation. Prog. Aerospace Sci. 16, 31–96.Google Scholar
Crighton, D. G.: 1981 Acoustics as a branch of fluid mechanics. J. Fluid Mech. 106, 261–298.Google Scholar
Delisi, D. P. & Dunkerton, T. J., 1989 Laboratory observations of gravity wave critical-layer flows. Pure Appl. Geophys. 130, 445–461.Google Scholar
Dewar, R. L.: 1970 Interaction between hydromagnetic waves and a time-dependent in-homogeneous medium. Phys. Fluids 13, 2710–2720.Google Scholar
Dickinson, R. E.: 1969a Theory of planetary wave-zonal flow interaction. J. Atmos. Sci. 26, 73–81. (See especially p. 80).Google Scholar
Dickinson, R. E.: 1969b Vertical propagation of planetary Rossby waves through an atmosphere with Newtonian cooling. J. Geophys. Res. 74, 929–938.Google Scholar
Dunkerton, T. J.: 1983 Laterally-propagating Rossby waves in the easterly acceleration phase of the quasi-biennial oscillation. Atmos.-Ocean 21, 55–68.Google Scholar
Dunkerton, T. J.: 1989 Body force circulations in a compressible atmosphere: key concepts. Pure Appl. Geophys. 130, 243–262.Google Scholar
Dunkerton, T. J., Hsu, C.-P. & McIntyre, M. E. 1981 Some Eulerian and Lagrangian diagnostics for a model stratospheric warming. J. Atmos. Sci. 38, 819–843.Google Scholar
Dysthe, K. B. & Das, K. P., 1981 Coupling between a surface-wave spectrum and an internal wave: modulational interaction. J. Fluid Mech. 104, 483–503.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, 38, 1115, especially second last item.)Google Scholar
Fels, S. B.: 1985 Radiative-dynamical interactions in the middle atmosphere. In Issues in Atmospheric and Oceanic Modelling (ed. S. Manabe), Advances in Geophysics vol. 28 A. pp. 277–300. Academic.
Franklin, R. E., Price, M. & Williams, D. C., 1973 Acoustically driven water waves. J. Fluid Mech. 57, 257–268.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
Fritts, D. C.: 1987 Recent progress in gravity wave saturation studies. In Transport Processes in the Middle Atmosphere (Proc. NATO Workshop, November 1986, Erice, Sicily, (ed. G. Visconti & R. R. Garcia), pp. 31–46. Reidel.
Gill, A. E.: 1982 Atmosphere-Ocean Dynamics. Academic, 662 pp.
Goldstein, S. (ed.) 1938 Modern Developments in Fluid Mechanics vols. 1 and 2. Oxford University Press, 702 pp. (Reprinted New York, Dover, 1965).
Green, J. S. A.: 1970 Transfer properties of the large-scale eddies and the general circulation of the atmosphere. Q. J. R. Met. Soc. 96, 157–185.Google Scholar
Grimshaw, R.: 1977 The modulation of an internal gravity wave packet and the resonance with the mean motion. Stud. Appl. Maths, 56, 241–266.Google Scholar
Grimshaw, R.: 1979 Mean flows induced by internal gravity wave packets propagating in a shear flow. Phil. Trans. R. Soc. Lond. A 292, 391–417.Google Scholar
Hartmann, D. L., Heidt, L. E., Lowenstein, J. R., Podolske, J. R., Vedder, J., Starr, W. L. & Strahan, S. E., 1989 Transport into the south polar vortex in early spring. J. Geophys. Res., in press. [Special issue on the Airborne Antarctic Ozone Experiment.]Google Scholar
Haynes, P. H.: 1989 The effect of barotropic instability on the nonlinear evolution of a Rossby wave critical layer. J. Fluid Mech. 207, 231–266.Google Scholar
Haynes, P. H. & McIntyre, M. E., 1987 On the evolution of vorticity and potential vorticity in the presence of diabatic heating and frictional or other forces. J. Atmos. Sci. 44, 828–841 (referred to as HM).Google Scholar
Haynes, P. H. & McIntyre, M. E., 1990 On the conservation and impermeability theorems for potential vorticity. J. Atmos. Sci., submitted.Google Scholar
Haynes, P. H., Marks, C. J., McIntyre, M. E., Shepherd, T. G. & Shine, K. P., 1990 On the “downward control’ of extratropical diabatic circulations by eddy induced mean zonal forces. J. Atmos. Sci., to appear.Google Scholar
Haynes, P. H. & Norton, W. A., 1990 On mass and chemical tracer transports through the edge of a stratospheric polar vortex. Manuscript in preparation.
Hertz, G. & Mende, H., 1939 Der Schallstrahlungsdruck in Flüssigkeiten. Z. Phys. 114, 354–367.Google Scholar
Holton, J. R. & Dunkerton, T., 1978 On the role of wave transience and dissipation in stratospheric mean flow vacillations. J. Atmos. Sci. 35, 740–744.Google Scholar
Holton, J. R. & Lindzen, R. S., 1972 An updated theory of the quasi-biennial cycle of the tropical stratosphere. J. Atmos. Sci. 29, 1076–1080.Google Scholar
Hoskins, B. J. & Berrisford, P., 1988 A potential-vorticity perspective of the storm of 15–16 October 1987. Weather, 43, 122–129.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. Also 113, 402–404.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
Juckes, M. N., McIntyre, M. E. & Norton, W. A., 1990 High-resolution barotropic simulations of breaking stratospheric planetary waves and polar-vortex edge formation. Q. J. R. Met. Soc., to be submitted.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
Kleinschmidt, E.: 1950a Über Aufbau und Entstehung von Zyklonen (1. Teil) Met. Runds 3, 1–6.Google Scholar
Kleinschmidt, E.: 1950b Über Aufbau und Entstehung von Zyklonen (2. Teil) Met. Runds. 3, 54–61.Google Scholar
Kleinschmidt, E.: 1951 Über Aufbau und Entstehung von Zyklonen (3. Teil) Met. Runds. 4, 89–96.
Kraichnan, R. H.: 1974 Convection of a passive scalar by a quasi-uniform random straining field. J. Fluid Mech. 64, 737–762.Google Scholar
Kraichnan, R. H.: 1975 Statistical dynamics of two-dimensional flow. J. Fluid Mech. 67, 155–175.Google Scholar
Leibovich, S.: 1980 On wave-current interaction theories of Langmuir circulations. J. Fluid Mech. 99, 715–724.Google Scholar
Leibovich, S.: 1983 The form and dynamics of Langmuir circulations. Ann. Rev. Fluid Mech. 15, 391–427.Google Scholar
Lighthill, M. J.: 1978a Acoustic streaming. J. Sound Vib., 61, 391–418.Google Scholar
Lighthill, M. J.: 1978b Waves in Fluids. Cambridge University Press. 504 pp.
Lindzen, R. S.: 1981 Turbulence and stress owing to gravity wave and tidal breakdown. J. Geophys. Res. 86, 9707–9714.Google Scholar
Lindzen, R. S.: 1973 Wave-mean interactions in the upper atmosphere. Boundary-layer Met. 4, 327–343.Google Scholar
Lindzen, R. S. & Holton, J. R., 1968 A theory of the quasi-biennial oscillation. J. Atmos. Sci. 25, 1095–1107.Google Scholar
Longuet-Higgins, M. S.: 1953 Mass transport in water waves. Phil. Trans. R. Soc. Lond. A 245, 535–581.Google Scholar
Longuet-Higgins, M. S.: 1970a Longshore currents generated by obliquely incident sea waves 1. J. Geophys. Res. 75, 6778–6789.Google Scholar
Longuet-Higgins, M. S.: 1970b Longshore currents generated by obliquely incident sea waves 2. J. Geophys. Res. 75, 6790–6801.Google Scholar
Longuet-Higgins, M. S.: 1972 Recent progress in the study of longshore currents. In The Waves on Beaches and Resulting Sediment Transport (ed. R. E. Meyer), pp. 203–248. Academic.
Lorenz, E. N.: 1967 The Nature and Theory of the General Circulation of the Atmosphere. Geneva: World Met. Org., 161 pp.
Mahony, J. J. & Smith, R., 1972 On a model representation for certain spatial-resonance phenomena. J. Fluid Mech. 53, 193–207.Google Scholar
Mattocks, C. & Bleck, R., 1986 Jet streak dynamics and geostrophic adjustment processes during the initial stages of lee cyclogenesis. Mon. Wea. Rev. 114, 2033–2056.Google Scholar
McIntyre, M. E.: 1973 Mean motions and impulse of a guided internal gravity wave packet. J. Fluid Mech. 60, 801–811.Google Scholar
McIntyre, M. E.: 1980a An introduction to the generalized Lagrangian-mean description of wave, mean-flow interaction. Pure Appl. Geophys. 118, 152–176.Google Scholar
McIntyre, M. E.: 1980b Towards a Lagrangian-mean description of stratospheric circulations and chemical transports. Phil. Trans. R. Soc. Lond. A 296, 129–148. (Special Middle Atmosphere issue).Google Scholar
McIntyre, M. E.: 1981 On the ‘wave momentum’ myth. J. Fluid Mech. 106, 331–347.Google Scholar
McIntyre, M. E.: 1987 Dynamics and tracer transport in the middle atmosphere: an overview of some recent developments. In Transport Processes in the Middle Atmosphere, Proc. NATO Workshop, November 1986, Erice, Sicily (ed. G. Visconti & R. R. Garcia), pp. 267–296. Reidel.
McIntyre, M. E.: 1988 Numerical weather prediction: a vision of the future. Weather 43, 294–298.Google Scholar
McIntyre, M. E.: 1989a On the Antarctic ozone hole. J. Atmos. Terr. Phys. 51, 29–43.Google Scholar
McIntyre, M. E.: 1989b On dynamics and transport near the polar mesopause in summer. J. Geophys. Res., 94, 14617–14628. (Special Issue on Noctilucent Clouds.)Google Scholar
McIntyre, M. E. & Norton, W. A., 1990 Potential vorticity inversion on a hemisphere. J. Atmos. Sci., to be submitted (referred to as MN).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. & Palmer, T. N., 1985 A note on the general concept of wave breaking for Rossby and gravity waves. Pure Appl. Geophys. 123, 964–975.Google Scholar
Murphy, D. M., Tuck, A. F., Kelly, K. K., Chan, K. R., Loewenstein, M., Podolske, J. R., Proffitt, M. H. & Strahan, S. E., 1989 Indicators of transport and vertical motion from correlations between in-situ measurement in the Airborne Antarctic ozone experiment. J. Geophys. Res. 94, 11669–11685. (Special issue on the Airborne Antarctic Ozone Experiment.)Google Scholar
Obukhov, A. M.: 1962 On the dynamics of a stratified liquid. Dokl. Akad. Nauk SSSR 145 (6), 1239–1242. (English transl. in Sov. Phys. Dokl. 7, 682–684.)Google Scholar
O'Neill, A. & Pope, V. D. 1987 Seasonal evolution of the stratosphere in the northern hemisphere. In Transport Processes in the Middle Atmosphere. Proc. NATO Workshop, November 1986, Erice, Sicily (ed. G. Visconti & R. R. Garcia), pp. 57–69. Reidel.
Palmer, T. N. & Hsu, C.-P. F. 1983 Stratospheric sudden coolings and the role of nonlinear wave interactions in preconditioning the circumpolar flow. J. Atmos. Sci. 40, 909–928.Google Scholar
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 parametrization. Q. J. R. Met. Soc. 112, 1001–1039.Google Scholar
Pijpers, F. P. & Hearn, A. G., 1989 A model for a stellar wind driven by linear acoustic waves. Astron. Astrophys. 209, 198–210.Google Scholar
Plumb, R. A.: 1984 The quasi-biennial oscillation. In Dynamics of the Middle Atmosphere (ed. J. R. Holton & T. Matsuno), pp. 217–251. Reidel, Tokyo, Terra Scientific.
Plumb, R. A. & McEwan, A. D., 1978 The instability of a forced standing wave in a viscous stratified fluid. A laboratory analogue of the quasi-biennial oscillation. J. Atmos. Sci. 35, 1827–1839.Google Scholar
Press, W. H., Flannery, B. P., Teukolsky, S. A. & Vetterling, W. T., 1986 Numerical recipes: the Art of Scientific Computing. Cambridge University Press, 818 pp.
Proffitt, M. H., Kelly, K. K., Powell, J. A., Fahey, D. W., Schoeberl, M. R., Gary, B. L., Loewenstein, M., Podolske, J. R., Strahan, S.: 1989 Evidence for diabatic cooling and poleward transport within and around the 1987 Antarctic ozone hole. J. Geophys. Res., in press. (Special issue on the Airborne Antarctic Ozone Experiment.)Google Scholar
Rayleigh, Lord: 1896 The Theory of Sound, Vol. 2. Dover (reprinted 1945), 504 pp. (see § 352.
Rhines, P. B. & Holland, W. R., 1979 A theoretical discussion of eddy-driven mean flows. Dyn. Atmos. Oceans 3, 289–325.Google Scholar
Salby, M. L., Garcia, R. R., O'Sullivan, D. & Tribbia, J. 1990 Global transport calculations with an equivalent barotropic system. J. Atmos. Sci., in press.Google Scholar
Staquet, C. & Riley, J. J., 1989 On the velocity field associated with potential vorticity. Dyn. Atmos. Oceans, in press.Google Scholar
Stewart, R. W. & Thomson, R. E., 1977 Re-examination of vorticity transfer theory. Proc. R. Soc. Lond. A 354, 1–8.Google Scholar
Stewartson, K.: 1978 The evolution of the critical layer of a Rossby wave. Geophys. Astrophys. Fluid Dyn. 9, 185–200 (referred to as SWW).Google Scholar
Taylor, G. I.: 1915 Eddy motion in the atmosphere. Phil. Trans. R. Soc. Lond. A 215, 1–23.Google Scholar
Townsend, R. D. & Johnson, D. H., 1985 A diagnostic study of the isentropic zonally averaged mass circulation during the first GARP global experiment. J. Atmos. Sci. 42, 1565–1579.Google Scholar
Tuck, A. F.: 1989 Synoptic and chemical evolution of the Antarctic vortex in late winter and early spring, 1987. J. Geophys. Res. 94, 11687–11737. (Special issue on the Airborne Antarctic Ozone Experiment.)Google Scholar
Wallace, J. M. & Holton, J. R., 1968 A diagnostic numerical model of the quasi-biennial oscillation. J. Atmos. Sci. 25, 280–292.Google Scholar
Warn, T. & Warn, H. (1978) The evolution of a nonlinear critical level. Stud. Appl. Maths 59, 37–71 (referred to as SWW).Google Scholar
Westervelt, P. J.: 1963 Parametric acoustic array. J. Acoust. Soc. Am. 35, 535–537.Google Scholar
Westervelt, P. J.: 1977 Sound. In McGraw-Hill Yearbook on Science and Technology (ed. D. N. Lapides), vol. 16. pp. 389–390.
WMO 1985 Atmospheric ozone 1985: Assessment of our understanding of the processes controlling its present distribution and change. Geneva, World Meteorological Organization, Global Ozone Research & Monitoring Report No. 16. Available from Global Ozone Research and Monitoring Project. World Meteorological Organization, Case Postale 5. CH 1211, Geneva 20, Switzerland. In 3 volumes, 1095 pp. + 86 pp. refs. (See chapter 6.)Google Scholar
WMO 1989 Report of the International Ozone Trends Panel, 1989. Geneva. World Meteorological Organization, Global Ozone Research and Monitoring Report No. 18. Available from Global Ozone Research and Monitoring Project, World Meteorological Organization. Case Postale 5. CH 1211. Geneva 20. Switzerland.