Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-18T18:37:29.040Z Has data issue: false hasContentIssue false
  • English
  • Français

An exact theory of nonlinear waves on a Lagrangian-mean flow

Published online by Cambridge University Press:  19 April 2006

D. G. Andrews  and
M. E. Mcintyre
D. G. Andrews
Affiliation:
U.K. Universities’ Atmospheric Modelling Group, Reading, Berkshire, England Present address: Meteorology Department, Massachusetts Institute of Technology, Cambridge.
M. E. Mcintyre
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge
Get access Rights & Permissions [Opens in a new window]

Abstract

An exact and very general Lagrangian-mean description of the back effect of oscillatory disturbances upon the mean state is given. The basic formalism applies to any problem whose governing equations are given in the usual Eulerian form, and irrespective of whether spatial, temporal, ensemble, or ‘two-timing’ averages are appropriate. The generalized Lagrangian-mean velocity cannot be defined exactly as the ‘mean following a single fluid particle’, but in cases where spatial averages are taken can easily be visualized, for instance, as the motion of the centre of mass of a tube of fluid particles which lay along the direction of averaging in a hypothetical initial state of no disturbance.

The equations for the Lagrangian-mean flow are more useful than their Eulerian-mean counterparts in significant respects, for instance in explicitly representing the effect upon mean-flow evolution of wave dissipation or forcing. Applications to irrotational acoustic or water waves, and to astrogeophysical problems of waves on axisymmetric mean flows are discussed. In the latter context the equations embody generalizations of the Eliassen-Palm and Charney-Drazin theorems showing the effects on the mean flow of departures from steady, conservative waves, for arbitrary, finite-amplitude disturbances to a stratified, rotating fluid, with allowance for self-gravitation as well as for an external gravitational field.

The equations show generally how the pseudomomentum (or wave ‘momentum’) enters problems of mean-flow evolution. They also indicate the extent to which the net effect of the waves on the mean flow can be described by a ‘radiation stress’, and provide a general framework for explaining the asymmetry of radiation-stress tensors along the lines proposed by Jones (1973).

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 89 , Issue 4 , 27 December 1978 , pp. 609 - 646
Copyright
© 1978 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

Andrews, D. G. & Mcintyre, M. E. 1976a Planetary waves in horizontal and vertical shear: the generalized Eliassen-Palm relation and the mean zonal acceleration. J. Atmos. Sci. 33, 2031–2048.Google Scholar
Andrews, D. G. & Mcintyre, M. E. 1976b Planetary waves in horizontal and vertical shear: asymptotic theory for equatorial waves in weak shear. J. Atmos. Sci. 33, 2049–2053.Google Scholar
Andrews, D. G. & Mcintyre, M. E. 1978a Generalized Eliassen–Palm and Charney–Drazin theorems for waves on axisymmetric flows in compressible atmospheres. J. Atmos. Sci. 35, 175–185.Google Scholar
Andrews, D. G. & Mcintyre, M. E. 1978b On wave-action and its relatives. J. Fluid Mech. 89, 647–664.Google Scholar
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.
Boyd, J. 1976 The noninteraction of waves with the zonally-averaged flow on a spherical earth and the interrelationships of eddy fluxes of energy, heat and momentum. J. Atmos. Sci. 33, 2285–2291.2.0.CO;2>CrossRefGoogle Scholar
Braginskic, S. I. 1964 Self-excitation of a magnetic field during the motion of a highly conducting fluid. Zh. Eksp. Teor. Fiz. 47, 1084–1098. (English trans. 1965 Sov. Phys. J. Exp. Theor. Phys. 20, 726–735.)Google Scholar
Bretherton, F. P. 1969a Momentum transport by gravity waves. Quart. J. Roy. Met. Soc. 95, 213–243.Google Scholar
Bretherton, F. P. 1969b 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. Lectures Appl. Math. 13, 61–102. Am. Math. Soc. (See 6.)Google Scholar
Bretherton, F. P. 1979 Conservation of wave action and angular momentum in a spherical atmosphere. J. Fluid Mech. (to appear).Google Scholar
Bretherton, F. P. & Gabrett, C. J. R. 1968 Wavetrains in inhomogeneous moving media. Proc. Roy. Soc. A 302, 529–554.Google Scholar
Bretherton, F. P. & Haidvogel, D. B. 1976 Two-dimensional turbulence above topography. J. Fluid Mech. 78, 129–154, equation (53).Google Scholar
Brillouin, L. 1925 On radiation stresses (in French). Arm. Phys. 4, 528–586.Google Scholar
Brillouin, L. 1936 Radiation pressures and stresses (in French). Rev. Acoust. 5, 99–111. (See also Tensors in Mechanics and Elasticity. Academic, 1964.)Google Scholar
Charney, J. G. 1973 Planetary fluid dynamics, IV. In Dynamic Meteorology (ed. P. Morel), pp. 98–351. Reidel.
Chabney, J. G. & Dbazin, P. G. 1961 Propagation of planetary-scale disturbances from the lower into the upper atmosphere. J. Geophys. Res. 66, 83–109.Google Scholar
Courant, R. & Hilbert, D. 1962 Methods of Mathematical Physics, vol. 2. Wiley/Inter-science.
Dewab, R. L. 1970 Interaction between hydromagnetic waves and a time-dependent in-homogeneous medium. Phys. Fluids 13, 2710–2720.Google Scholar
Eckabt, C. 1963 Some transformations of the hydrodynamic equations. Phys. Fluids 6, 1037–1041.Google Scholar
Eliassen, A. & Kleinschmidt, E. 1957 Dynamic meteorology. In Encyclopaedia of Physics, vol. 48, pp. 1–154. Springer.
Eliassen, A. & Palm, E. 1961 On the transfer of energy in stationary mountain waves. Geofys. Publ. 22 (3), 1–23.Google Scholar
Fels, S. B. & Lindzen, R. S. 1974 The interaction of thermally excited gravity waves with mean flows. Geophys. Fluid Dyn. 6, 149–191.Google Scholar
Frieman, E. & Rotenberg, M. 1960 On hydromagnetic stability of stationary equilibria. Rev. Mod. Phys. 32, 898–902.Google Scholar
Garrett, C. J. R. 1968 On the interaction between internal gravity waves and a shear flow. J. Fluid Mech. 34, 711–720.Google Scholar
Geisler, J. E. 1974 A numerical model of the sudden stratospheric warming mechanism. J. Geophys. Res. 79, 4989–4999.Google Scholar
Grimshaw, R. H. J. 1975 Nonlinear internal gravity waves in a rotating fluid. J. Fluid Mech. 71, 497–512.Google Scholar
Grimshaw, R. H. J. 1977 The modulation of an internal gravity wave packet, and the resonance with the mean motion. Studies in Appl. Math. 56, 241–266.Google Scholar
Grimshaw, R. H. J. 1978 Mean flows induced by internal gravity wave packets propagatin in a shear flow. Submitted to Proc. Roy. Soc. A.Google Scholar
Holton, J. R. 1975 The Dynamic Meteorology of the Stratosphere and Mesosphere. Am. Met. Soc.
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. & Mass, C. 1976 Stratospheric vacillation cycles. J. Atmos. Sci. 33, 2218–2225.Google Scholar
Hoskins, B. J. 1974 The role of potential vorticity in symmetric stability and instability. Quart. J. Roy. Met. Soc. 100, 480–482.Google Scholar
Jones, W. L. 1973 Asymmetric wave-stress tensors and wave spin. J. Fluid Mech. 58, 737–747.Google Scholar
Landau, L. D. & Lifshitz, E. M. 1959 Fluid Mechanics. Pergamon.
Landau, L. D. & Lifshitz, E. M. 1975 The Classical Theory of Fields, 4th English edn. Pergamon.
Lighthill, M. J. 1978a Waves in Fluids. Cambridge University Press.
Lighthill, M. J. 1978b Acoustic streaming. Invited Lecture, Inst. Acoust. Spring Conf.Google Scholar
Lindzen, R. S. & Tsay, C. Y. 1975 Wave structure of the tropical stratosphere over the Marshall Islands area during 1 April–1 July 1958. J. Atmos. Sci. 32, 2008–2021.Google Scholar
Longuet-Higgins, M. S. & Stewart, R. W. 1964 Radiation stresses in water waves: a physical discussion, with applications. Deep-Sea Res. 11, 529–562.Google Scholar
Mccomas, C. H. & Bretherton, F. P. 1977 Resonant interaction of oceanic internal waves. J. Geophys. Res. 82, 1397–1412.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. 1977 Wave transport in stratified, rotating fluids. In Lecture Notes in Phys. vol. 71 (ed. E. A. Spiegel & J. P. Zahn), pp. 290–314. Springer. (The statement on p. 303, line 27 is wrong; but see equation (5.2) on p. 311.)
Mcintyre, M. E. 1978 An introduction to the generalized Lagrangian-mean description of wave, mean-flow interaction. Adv. Study Prog. Summer Lecture Notes, N.C.A.R.Google Scholar
Matsuno, T. 1971 A dynamical model of the stratospheric sudden warming. J. Atmos. Sci. 28, 1479–1494.Google Scholar
Moffatt, H. K. 1978 Magnetic Field Generation in Electrically Conducting Fluids. Cambridge University Press.
Moore, D. 1970 The mass transport velocity induced by free oscillations at a single frequency. Geophys. Fluid Dyn. 1, 237–247.Google Scholar
Müller, P. 1976 On the diffusion of momentum and mass by internal gravity waves. J. Fluid Mech. 77, 789–823.Google Scholar
Peierls, R. 1976 The momentum of light in a refracting medium. Proc. Roy. Soc. A 347, 475–491.Google Scholar
Plumb, R. A. 1975 Momentum transport by the thermal tide in the stratosphere of Venus. Quart. J. Roy. Met. Soc. 101, 763–776.Google Scholar
Plumb, R. A. 1977 The interaction of two internal waves with the mean flow: implications for the theory of the quasi-biennial oscillation. J. Atmos. Sci. 34, 1847–1858.Google Scholar
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, 1827–1839.2.0.CO;2>CrossRefGoogle Scholar
Rayleigh, Lord 1896 The Theory of Sound, vol. II, 352. Dover.
Riley, N. 1967 Oscillatory viscous flows, review and extension. J. Inst. Math. Appl. 3, 419–434.Google Scholar
Schlichting, H. 1932 Berechnung ebener periodischer Grenzschichtstromungen. Phys. Z. 33, 327.Google Scholar
Soward, A. M. 1972 A kinematic theory of large magnetic Reynolds number dynamos. Phil. Trans. Roy. Soc. A 272, 431–462.Google Scholar
Soward, A. M. & Roberts, P. H. 1976 Recent developments in the theory of the MHD dynamo (in Russian). Magnitiaya Gidrodynamica 1, 3–51.Google Scholar
Stuart, J. T. 1966 Double boundary layers in oscillatory viscous flow. J. Fluid Mech. 24, 673–687.Google Scholar
Uryu, M. 1974 Induction and transmission of mean zonal flow by quasi-geostrophic disturbances. J. Met. Soc. Japan 52, 341–363.Google Scholar
Watson, K. M., West, B. J. & Cohen, B. I. 1976 Coupling of surface and internal gravity waves: a mode coupling model. J. Fluid Mech. 77, 185–208.Google Scholar