Abstract
The Euler equations describing perfect-fluid motion represent a Hamiltonian dynamical system. The Hamiltonian framework implies, via Noether’s theorem, that symmetries are connected with extremal properties. For example, steady solutions are conditional extrema of the energy under smooth perturbations that preserve vortex topology, as pointed out originally by Arnol’d (1966a). In the three-dimensional context such isovortical perturbations consist of those that maintain the strength and topology of vortex tubes; in the two-dimensional context they consist of those that maintain the vorticity of each fluid element, which is a considerably stronger constraint. Just as with temporal symmetry, solutions that are independent of a spatial coordinate are conditional extrema of the associated momentum invariant under isovortical perturbations; and steadilytranslating solutions are conditional extrema of a linear combination of the energy and the momentum. This connection between symmetries and extremal properties is exploited to particular effect in Arnol’d’s (1966c) nonlinear stability theorems and their recent extensions and applications. Finally, a class of algorithms is described that can find true energy and energy-momenta extrema (under isovortical perturbations) of the two- and three-dimensional Euler equations, when they exist.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Abarbanel, H.D.I. and Holm, D.D., 1987 Nonlinear stability analysis of inviscid flows in three dimensions: incompressible fluids and barotropic fluids. Phys.Fluids,30 pp. 3369–3382.
Abraham, R. and Marsden,J.E., 1978 Foundations of Mechanics,2nd edn. Benjamin/Cummings, 806 pp.
Arnol’d, V.I., 1966e Sur un principe variationnel pour les écoulements stationnaires des liquides parfaits et ses applications aux problèmes de stabilité non linéaires. J.Méc, 5, pp. 29–43.
Arnol’d, V.I., 1966b Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications h l’hydrodynamique des fluides parfaits. Ann.Inst.Fourier (Grenoble), 16, pp. 319–361.
Arnol’d, V.I., 1966c On an a priori estimate in the theory of hydrodynamical stability, Izv.Vyssh.Uchebn.Zaved.Matematika,54, no.5, pp. 3–5. (English transi.: Amer.Math.Soc.Transl, Series 2,79, pp. 267–269 (1969)
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, pp. 225–226. (In Russian.)
Arnol’d, V.I., 1974 The asymptotic Hopf invariant and its applications. Proceed- ings of Summer School in Differential Equations, 1973, Erevan. (English transi.: Sel.Math.Sov, 5, pp. 327–345 (1986)
Arnol’d, V.I., 1989 Mathematical Methods of Classical Mechanics,2nd edn. Springer-Verlag, 511 pp.
Arnold, V.I. and Khesin, B., 1992 Topological methods in hydrodynamics. Ann.Rev.Fluid Mech, 24, pp. 145–166.
Benjamin, T.B., 1984 Impulse, flow force and variational principles. IMA J.Appl.Math, 32, pp. 3–68.
Carnevale, G.F. and Frederiksen, J.S., 1987 Nonlinear stability and statistical mechanics of flow over topography. J.Fluid Mech, 175, pp. 157–181.
Carnevale, G.F. and Shepherd, T.G., 1990 On the interpretation of Andrews’ theorem. Geophys.Astrophys.Fluid Dyn, 51, pp. 1–17.
Carnevale, G.F. and Vallis, G.K., 1990 Pseudo-advective relaxation to stable states of inviscid two-dimensional fluids. J.Fluid Mech, 213, pp. 549–571.
Cho, H.-R., Shepherd, T.G. and Vladimirov, V.A., 1991 Application of the direct Liapunov method to the problem of symmetric stability in the atmosphere. J.Atmos.Sci, submitted.
Ebin, D.G. and Marsden, J.E., 1970 Groups of diifeomorphisms and the motion of an incompressible fluid. Ann.Math., 92, pp. 102–163.
Goldstein, H., 1980 Classical Mechanics,2nd edn. Addison-Wesley, 672 pp.
Holm, D.D., Marsden, J.E., Ratiu, T. and Weinstein, A., 1985 Nonlinear stability of fluid and plasma equilibria. Phys.Rep, 123, pp. 1–116.
Kuznetsov, E.A. and Mikhauav, A.V., 1980 On the topological meaning of canonical Clebsch variables. Phys.Lett, 77A, pp. 37–38.
Lesieur, M., 1990 Turbulence in Fluids,2nd revised edn. Kluwer Academic, 412 pp.
Littlejohn, R.G., 1982 Singular Poisson tensors. In Mathematical Methods in Hydro-dynamics and Integrability in Dynamical Systems (ed. M. Tabor and Y.M. Treve ), Amer.Inst.Phys.Conf.Proc., vol. 88, pp. 47–66.
Marsden, J.E. (En.), 1984 Fluids and Plasmas: Geometry and Dynamics Contemporary Mathematics, vol.28, American Mathematical Society.
Marsden, J.E. and Weinstein, A., 1983 Coadjoint orbits, vortices, and Clebsch variables for incompressible fluids. Physica, 7D, pp. 305–323.
Mcintyre, M.E. and Shepherd, T.G., 1987 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. J.Fluid Mech, 181, pp. 527–565.
Moffatt, H.K., 1969 The degree of knottedness of tangled vortex lines. J.Fluid Mech, 35, pp. 117–129.
Moffatt, H.K., 1985 Magnetostatic equilibria and analogous Euler flows of arbitrarily complex topology. Part I. Fundamentals. J.Fluid Mech, 159, pp. 359–378.
Morrison, P.J., 1982 Poisson brackets for fluids and plasmas. In Mathematical Methods in
Hydrodynamics and Integrability in Dynamical Systems (ed. M. Tabor and Y.M. Treve), Amer.Inst.Phys.Conf.Proc., vol.88 pp.13–46.
Olver, P.J., 1982 A nonlinear Hamiltonian structure for the Euler equations. J.Math.Anal.Appl, 89, pp. 233–250.
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.
Ripa, P., 1983 General stability conditions for zonal flows in a one-layer model on the ß-plane or the sphere. J.Fluid Mech, 126, pp. 463–489.
Salmon, R., 1982 Hamilton’s principle and Ertel’s theorem. In Mathematical Methods in Hydrodynamics and Integrability in Dynamical Systems (ed. M. Tabor and Y.M. Treve), Amer.Inst.Phys.Conf.Proc., vol.88, pp. 127–135.
Shepherd, T.G., 1987 Non-ergodicity of inviscid two-dimensional flow on a beta-plane and on the surface of a rotating sphere. J.Fluid Mech,184 pp. 289–302.
Shepherd, T.G., 1988 Rigorous bounds on the nonlinear saturation of instabilities to parallel shear flows. J.Fluid Mech,196 pp. 291–322.
Shepherd, T.G., 1989 Nonlinear saturation of baroclinic instability. Part II: Continuously-stratified fluid. J.Atmos.Sci,46 pp. 888–907.
Shepherd, T.G., 1990a Symmetries, conservation laws, and Hamiltonian structure in geophysical fluid dynamics. Adv.Geophys,32 pp. 287–338.
Shepherd, T.G., 1990b A general method for finding extremal states of Hamiltonian dynamical systems, with applications to perfect fluids. J.Fluid Mech,213 pp. 573–587.
Shepherd, T.G., 1991a Nonlinear stability and the saturation of instabilities to axisymmetric vortices. Eur.J.Mech.B/Fluids,10 N° 2—Suppl., pp. 93–98.
Shepherd, T.G., 1991b Arnol’d stability applied to fluid flow: successes and failures. In Nonlinear Phenomena in Atmospheric and Oceanic Sciences (ed. G.F. Carnevale and R.T. Pierrehumbert), to appear. Springer-Verlag.
Swaters, G.E., 1986 A nonlinear stability theorem for baroclinic quasi-geostrophic flow. Phys.Fluids,29 pp. 5–6.
Saeri, A. and Holmes P., 1988 Nonlinear stability of axisymmetric swirling flows. Phil. Trans.R.Soc.Lond.A,326 pp. 327–354.
Tabor, M., 1989 Chaos and Integrability in Nonlinear Dynamics Wiley, 364 pp.
Vallis, G.K., Carnevale G.F. and Young W.R., 1989 Extremal energy properties and construction of stable solutions of the Euler equations. J.Fluid Mech,207 pp. 133–152.
Vladimirov, V.A., 1986 Conditions for nonlinear stability of flows of an ideal incompressible liquid. Zhurn.Prikl.Mekh.Tekh.Fiz,no.3, pp. 70–78. (English transl.: J.Appl.Mech.Tech.Phys,27 no.3, pp. 382–389.)
Vladiniirov,V.A., 1987 Integrals of two-dimensional motions of a perfect incompressible fluid of nonuniform density. Izv.Akad.Nauk.SSSR, Mekh.Zhid.Gaza,no.3, pp. 16–20. (English transl.: Fluid Dynamics,22, no.3, pp. 340–343.)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1992 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Shepherd, T.G. (1992). Extremal Properties and Hamiltonian Structure of the Euler Equations. In: Moffatt, H.K., Zaslavsky, G.M., Comte, P., Tabor, M. (eds) Topological Aspects of the Dynamics of Fluids and Plasmas. NATO ASI Series, vol 218. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-3550-6_15
Download citation
DOI: https://doi.org/10.1007/978-94-017-3550-6_15
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-4187-6
Online ISBN: 978-94-017-3550-6
eBook Packages: Springer Book Archive