Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-28T22:46:40.243Z Has data issue: false hasContentIssue false
  • English
  • Français

An exact local conservation theorem for finite-amplitude disturbances to non-parallel shear flows, with remarks on Hamiltonian structure and on Arnol'd's stability theorems

Published online by Cambridge University Press:  21 April 2006

M. E. Mcintyre  and
T. G. Shepherd
M. E. Mcintyre
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW, UK
T. G. Shepherd
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

Disturbances of arbitrary amplitude are superposed on a basic flow which is assumed to be steady and either (a) two-dimensional, homogeneous, and incompressible (rotating or non-rotating) or (b) stably stratified and quasi-geostrophic. Flow over shallow topography is allowed in either case. The basic flow, as well as the disturbance, is assumed to be subject neither to external forcing nor to dissipative processes like viscosity. An exact, local ‘wave-activity conservation theorem’ is derived in which the density A and flux F are second-order ‘wave properties’ or ‘disturbance properties’, meaning that they are O(a2) in magnitude as disturbance amplitude a → 0, and that they are evaluable correct to O(a2) from linear theory, to O(a3) from second-order theory, and so on to higher orders in a. For a disturbance in the form of a single, slowly varying, non-stationary Rossby wavetrain, $\overline{F}/\overline{A}$ reduces approximately to the Rossby-wave group velocity, where (${}^{-}$) is an appropriate averaging operator. F and A have the formal appearance of Eulerian quantities, but generally involve a multivalued function the correct branch of which requires a certain amount of Lagrangian information for its determination. It is shown that, in a certain sense, the construction of conservable, quasi-Eulerian wave properties like A is unique and that the multivaluedness is inescapable in general. The connection with the concepts of pseudoenergy (quasi-energy), pseudomomentum (quasi-momentum), and ‘Eliassen-Palm wave activity’ is noted.

The relationship of this and similar conservation theorems to dynamical fundamentals and to Arnol'd's nonlinear stability theorems is discussed in the light of recent advances in Hamiltonian dynamics. These show where such conservation theorems come from and how to construct them in other cases. An elementary proof of the Hamiltonian structure of two-dimensional Eulerian vortex dynamics is put on record, with explicit attention to the boundary conditions. The connection between Arnol'd's second stability theorem and the suppression of shear and self-tuning resonant instabilities by boundary constraints is discussed, and a finite-amplitude counterpart to Rayleigh's inflection-point theorem noted.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 181 , August 1987 , pp. 527 - 565
Copyright
© 1987 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

Abarbanel, H. D. I., Holm, D. D., Marsden, J. E. & Ratiu, T., 1984 Richardson number criterion for the nonlinear stability of three-dimensional flow. Phys. Rev. Lett. 52, 2352–2355.Google Scholar
Abarbanel, H. D. I., Holm, D. D., Marsden, J. E. & Ratiu, T. 1986 Nonlinear stability analysis of stratified fluid equilibria. Phil. Trans. R. Soc. Lond. A318, 349–409.Google Scholar
Abraham, R. & Marsden, J. E. 1978 Foundations of Mechanics, 2nd edn. Benjamin/Cummings, 806 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
Andrews, D. G. 1981 A note on potential energy density in a stratified compressible fluid. J. Fluid Mech. 107, 227–236.Google Scholar
Andrews, D. G. 1983 A conservation law for small-amplitude quasi-geostrophic disturbances on a zonally asymmetric basic flow. J. Atmos. Sci. 40, 85–90.Google Scholar
Andrews, D. G. & McIntyre, M. E. 1978a An exact theory of nonlinear waves on a Lagrangian-mean flow. J. Fluid Mech. 89, 609–646.Google Scholar
Andrews, D. G. & McIntyre, M. E. 1978b On wave-action and its relatives. J. Fluid Mech. 89, 647–664 (and Corrigenda 95, 796).Google Scholar
Arnol'D, V. I.1965 Conditions for nonlinear stability of stationary plane curvilinear flows of an ideal fluid. Dokl. Akad. Nauk. SSSR 162, 975–978. (English transl.: Soviet Math. 6, 773–777 (1965).)Google Scholar
Arnol'D, V. I.1966b On an a priori estimate in the theory of hydrodynamical stability. Izv. Vyssh. Uchebn. Zaved. Matematika 54, no. 5, 3–5. (English transl.: American Math. Soc. Transl., Series 2 79, 267–269 (1969).Google Scholar
Arnol'D, V. I.1966b Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits. Ann. Inst. Fourier (Grenoble) 16, 319–361.Google Scholar
Arnol'D, V. I.1969 The Hamiltonian nature of the Euler equations in the dynamics of a rigid body and of a perfect fluid. Usp. Mat. Nauk. 24, no. 3, 225–226. (In Russian.) (Rough translation available from authors.)Google Scholar
Arnol'D, V. I.1978 Mathematical Methods of Classical Mechanics. Springer, 462 pp. (Russian original: Nauka, 1974.)
Ball, J. M. & Marsden, J. E. 1984 Quasiconvexity at the boundary, positivity of the second variation, and elastic stability. Arch. Rat. Mech. Anal. 86, 251–277.Google Scholar
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press, 615 pp.
Benjamin, T. B. 1972 The stability of solitary waves. Proc. R. Soc. Lond. A 328, 153–183.Google Scholar
Benjamin, T. B. 1984 Impulse, flow force and variational principles. IMA J. Appl. Maths 32, 3–68.Google Scholar
Benjamin, T. B. 1986 On the Boussinesq model for two-dimensional wave motions in heterogeneous fluids. J. Fluid Mech. 165, 445–474.Google Scholar
Blumen, W. 1968 On the stability of quasi-geostrophic flow. J. Atmos. Sci. 25, 929–931.Google Scholar
Boyd, J. P. 1983 The continuous spectrum of linear Couette flow with the beta effect. J. Atmos. Sci. 40, 2304–2308.Google Scholar
Bretherton, F. P. 1966a Critical layer instability in baroclinic flows. Q. J. R. Met. Soc. 92, 325–334.Google Scholar
Bretherton, F. P. 1966b Baroclinic instability and the short wavelength cut-off in terms of potential vorticity. Q. J. R. Met. Soc. 92, 335–345.Google Scholar
Bretherton, F. P. 1970 A note on Hamilton's principle for perfect fluids. J. Fluid Mech. 44, 19–31.Google Scholar
Bretherton, F. P. & Garrett, C. J. R. 1968 Wavetrains in inhomogeneous moving media. Proc. R. Soc. Lond. A302, 529–554.Google Scholar
Butchart, N. & Remsberg, E. E. 1986 The 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. 1973 Planetary fluid dynamics. In Dynamic Meteorology (ed. P. Morel), pp. 97–351. Reidel.
Charney, J. G. & DeVore, J. G. 1979 Multiple flow equilibria in the atmosphere and blocking. J. Atmos. Sci. 36, 1205–1216.Google Scholar
Charney, J. G. & Flierl, G. R. 1981 Oceanic analogues of large-scale atmospheric motions. In Evolution of Physical Oceanography (ed. B. A. Warren & C. Wunsch), pp. 504–548. M.I.T. Press.
Charney, J. G. & Pedlosky, J. 1963 On the trapping of unstable planetary waves in the atmosphere. J. Geophys. Res. 68, 6441–6442.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
Deem, G. S. & Zabusky, N. J. 1978 Vortex waves: stationary ‘V states’, interactions, recurrence, and breaking. Phys. Rev. Lett. 40, 859–862.Google Scholar
Dikiy, L. A. & Kurganskiy, M. V. 1971 Integral conservation law for perturbations of zonal flow, and its application to stability studies. Izv. FAO 7, 939–945. (English transl.: Atmos. Ocean. Phys. 7, 623–626 (1971).)Google Scholar
Dirac, P. A. M. 1958 The Principles of Quantum Mechanics, 4th edn. Oxford University Press, 312pp.
Dritschel, D. G. 1986 The nonlinear evolution of rotating configurations of uniform vorticity. J. Fluid Mech. 172, 157–182.Google Scholar
Dunkerton, T. J. & Delisi, D. P. 1986 Evolution of potential vorticity in the winter stratosphere of January-February 1979. J. Geophys. Res. 91 (Dl), 1199–1208.Google Scholar
Farrell, B. F. 1982 The initial growth of disturbances in a baroclinic flow. J. Atmos. Sci. 39, 1663–1686.Google Scholar
FØtoft, R. 1953 On the changes in the spectral distribution of kinetic energy for two dimensional, nondivergent flow. Tellus 5, 225–230.Google Scholar
Goldstein, H. 1980 Classical Mechanics, 2nd edn. Addison-Wesley, 672 pp.
Grose, W. L. 1984 Recent advances in understanding stratospheric dynamics and transport processes: Application of satellite data to their interpretation. Adv. Space Res. 4, 19–28.CrossRefGoogle 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
Held, I. M. 1985 Pseudomomentum and the orthogonality of modes in shear flows. J. Atmos Sci. 42, 2280–2288.Google Scholar
Hirota, I. 1967 Dynamic stability of the stratospheric polar vortex. J. Met. Soc. Japan 45, 409–421.Google Scholar
Hirsch, M. W. & Smale, S. 1974 Differential Equations, Dynamical Systems, and Linear Algebra. London: Academic Press, 358 pp.
Holliday, D. & Mclntyre, M. E. 1981 On potential energy density in an incompressible, stratified fluid. J. Fluid Mech. 107, 221–225.Google Scholar
Holm, D. D. 1986 Hamiltonian formulation of the baroclinic quasigeostrophic fluid equations. Phys. Fluids 29, 7–8.Google Scholar
Holm, D. D., Marsden, J. E., Ratiu, T. & Weinstein, A. 1983 Nonlinear stability conditions and a priori estimates for barotropic hydrodynamics. Phys. Lett. 98A, 15–21.Google Scholar
Holm, D. D., Marsden, J. E., Ratiu, T. & Weinstein, A. 1985 Nonlinear stability of fluid and plasma equilibria. Phys. Rep. 123, 1–116.Google Scholar
Hoskins, B. J., Mclntyre, 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. & Mclntyre, M. E. 1987 A high resolution, one-layer model of breaking planetary waves in the stratosphere. Nature (to appear).
Killworth, P. D. & Mclntyre, M. E. 1985 Do Rossby-wave critical layers absorb, reflect or over-reflect?. J. Fluid Mech. 161, 449–492.Google Scholar
Kuznetsov, E. A. & Mikhailov, A. V. 1980 On the topological meaning of canonical Clebsch variables. Phys. Lett. 77A, 37–38.Google Scholar
Lamb, H. 1932 Hydrodynamics, 6th edn. Cambridge University Press, 738 pp.
Leovy, C. B., Sun, C.-R., Hitchman, M. H., Remsberg, E. E., Russell, J. 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
Lewis, D., Marsden, J. E., Montgomery, R. & Ratiu, T. 1986 The Hamiltonian structure for dynamic free boundary problems. Physica 18D, 391–404.Google Scholar
Lighthill, M. J. 1963 Introduction. Boundary layer theory. In Laminar Boundary Layers (ed. L. Rosenhead), pp. 46–113. Oxford University Press.
Longuet-Higgins, M. S. 1964 On group velocity and energy flux in planetary wave motions. Deep-Sea Res. 11, 35–42.Google Scholar
Mclntyre, M. E. 1987 The Lagrangian description in fluid mechanics. IMA J. Appl. Math., to be published.
Mclntyre, M. E. & Palmer, T. N. 1983 Breaking planetary waves in the stratosphere. Nature 305, 593–600.Google Scholar
Mclntyre, M. E. & Palmer, T. N. 1984 The 'surf zone’ in the stratosphere. J. Atmos. Terres. Phys. 46, 825–849.Google Scholar
Mclntyre, 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
Mclntyre, M. E. & Weissman, M. A. 1978 On radiating instabilities and resonant overreflection. J. Atmos. Sci. 35, 1190–1198.Google Scholar
Marsden, J. E. (ed.) 1984 Fluids and Plasmas: Geometry and Dynamics. Contemporary Mathematics, vol. 28, pp. vii–xiii. American Mathematical Society.
Marsden, J. E. & Weinstein, A. 1983 Coadjoint orbits, vortices, and Clebsch variables for incompressible fluids. Physica 7D, 305–323.Google Scholar
Matsuno, T. & Hirota, I. 1966 On the dynamical stability of polar vortex in wintertime. J. Met. Soc. Japan 44, 122–128.Google Scholar
Merilees, P. E. & Warn, H. 1975 On energy and enstrophy exchanges in two-dimensional non-divergent flow. J. Fluid Mech. 69, 625–630.Google Scholar
Morrison, P. J. 1982 Poisson brackets for fluids and plasmas. In Mathematical Methods in Hydrodynamics and Integrability in Dynamical Systems (ed. M. Tabor & Y. M. Treve), Amer. Inst. Phys. Conf. Proc. vol. 88, pp. 13–46.
Olver, P. J. 1980 On the Hamiltonian structure of evolution equations. Math. Proc. Camb. Phil. Soc. 88, 71–88.Google Scholar
Olver, P. J. 1982 A nonlinear Hamiltonian structure for the Euler equations. J. Math. Anal. Appl. 89, 233–250.
Olver, P. J. 1983 Conservation laws of free boundary problems and the classification of conservation laws for water waves. Trans. Am. Math. Soc. 277, 353–380.Google Scholar
Olver, P. J. 1984 Hamiltonian perturbation theory and water waves. In Fluids and Plasmas: Geometry and Dynamics (ed. J. E. Marsden). Contemporary Mathematics, vol. 28, pp. 231–249. American Mathematical Society.
Orr, W. M'F. 1907 The stability or instability of the steady motions of a perfect liquid and of a viscous liquid. Parts I and II. Proc. R. Irish Acad. A27, 9–68; 69–138.Google Scholar
Pedlosky, J. 1979 Geophysical Fluid Dynamics. Springer, 624 pp.
Penfield, P. 1966 Hamilton's principle for fluids. Phys. Fluids 9, 1184–1194.Google Scholar
Plumb, R. A. 1979 Forced waves in a baroclinic shear flow. Part 1: Undamped evolution near the baroclinic instability threshold. J. Atmos. Sci. 36, 205–216.Google Scholar
Plumb, R. A. 1981 Instability of the distorted polar night vortex: A theory of stratospheric warmings. J. Atmos. Sci. 38, 2514–2531.Google Scholar
Plumb, R. A. 1985a On the three-dimensional propagation of stationary waves. J. Atmos. Sci. 42, 217–229.Google Scholar
Plumb, R. A. 1985b An alternative form of Andrews’ conservation law for quasi-geostrophic waves on a steady, nonuniform flow. J. Atmos. Sci. 42, 298–300.Google Scholar
Ripa, P. 1981 Symmetries and conservation laws for internal gravity waves. In Nonlinear Properties of Internal Waves (ed. B. J. West), Amer. Inst. Phys. Conf. Proc. vol. 76, pp. 281–306.
Salmon, R. 1982 Hamilton's principle and Ertel's theorem. In Mathematical Methods in Hydrodynamics and Integrability in Dynamical Systems (ed. M. Tabor & Y. M. Treve), Amer. Inst. Phys. Conf. Proc. vol. 88, pp. 127–135.
Salmon, R. 1988 Hamiltonian fluid mechanics. Ann. Rev. Fluid Mech. (to appear).
Schiff, L. I. 1968 Quantum Mechanics, 3rd edn. McGraw-Hill, 544 pp.
Shepherd, T. G. 1985 Time development of small disturbances to plane Couette flow. J. Atmos. Sci. 42, 1868–1871.Google Scholar
Stewartson, K. 1978 The evolution of the critical layer of a Rossby wave. Geophys. Astrophys. Fluid Dyn. 9, 185–200.Google Scholar
Stuart, J. T. 1967 On finite amplitude oscillations in laminar mixing layers. J. Fluid Mech. 29, 417–440.Google Scholar
Sudarshan, E. C. G. & Mukunda, N. 1974 Classical Dynamics: A Modern Perspective. Willey - Interscience, 615 pp.
Swaters, G. E. 1986 A nonlinear stability theorem for baroclinic quasigeostrophic flow. Phys. Fluids 29, 5–6.Google Scholar
Thomson, W. 1887 Stability of fluid motion - rectilineal motion of viscous fluid between two parallel planes. Phil. Mag. 24, 188–196.Google Scholar
Wan, Y. H. & Pulvirenti, M. 1985 Nonlinear stability of circular vortex patches. Commun. Math. Phys. 99, 435–450.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.