Hostname: page-component-848d4c4894-sjtt6 Total loading time: 0 Render date: 2024-06-17T17:04:47.556Z Has data issue: false hasContentIssue false
  • English
  • Français

Extremal energy properties and construction of stable solutions of the Euler equations

Published online by Cambridge University Press:  26 April 2006

G. K. Vallis ,
G. F. Carnevale  and
W. R. Young
G. K. Vallis
Affiliation:
Scripps Institution of Oceanography, La Jolla, CA 92093, USA
G. F. Carnevale
Affiliation:
Scripps Institution of Oceanography, La Jolla, CA 92093, USA
W. R. Young
Affiliation:
Scripps Institution of Oceanography, La Jolla, CA 92093, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

Certain modifications of the Euler equations of fluid motion lead to systems in which the energy decays or grows monotonically, yet which preserve other dynamically important characteristics of the field. In particular, all topological invariants associated with the vorticity field are preserved. In cases where isolated energy extrema exist, a stable steady flow can be found. In two dimensions, highly constrained by vorticity invariants, it is shown that the modified dynamics will lead to at least one non-trivial stationary, generally stable, solution of the equations of motion from any initial conditions. Numerical implementation of the altered dynamics is straightforward, and thus provides a practical method for finding stable flows. The method is sufficiently general to be of use in other dynamical systems.

Insofar as three-dimensional turbulence is characterized by a cascade of energy, but not topological invariants, from large to small scales, the procedure has direct physical significance. It may be useful as a parameterization of the effects of small unresolved scales on those explicitly resolved in a calculation of turbulent flow.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 207 , October 1989 , pp. 133 - 152
Copyright
© 1989 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

Arnol'd, V. I. 1965a Conditions for nonlinear stability of stationary plane curvilinear flows of ideal fluid. Dokl. Akad. Nauk, SSSR 162, 975–978 [trans. in Soviet Math. 6, 773–777 (1965)].Google Scholar
Arnol'd, V. I. 1965b Variational principle for three-dimensional steady-state flows of an ideal fluid. Prikl. Math. I Mech. 29, 846–851 [transl. J. Appl. Math. Mech. 29, 1002–1008 (1965)].Google Scholar
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.
Dahlburg, J. P., Montgomery, D., Doolen, G. D. & Turner, L. 1987 Turbulent relaxation of a confined magnetofluid to a force-free state. J. Plasma Phys. 37, 299–321.Google Scholar
Kelvin, Lord 1887 On the stability of steady and of periodic fluid motion. Phil. Mag. 23, 459–464.Google Scholar
Matthaeus, W. H. & Montgomery, D. 1980 Selective decay hypotheses at high mechanical and magnetic Reynold's numbers. In Nonlinear Dynamics (ed. R. H. G. Hellerman), pp. 203–222. Ann. N.Y. Acad. Sci.
Moffatt, H. K. 1969 The degree of knottedness of tangled vortex lines. J. Fluid Mech. 35, 117–129.Google Scholar
Moffatt, H. K. 1985 Magnetostatic equilibria and analogous Euler flows of arbitrarily complex topology. Part I. Fundamentals. J. Fluid Mech. 159, 359–378.Google Scholar
Sadourny, R. & Basdevant, C. 1985 Parameterization of subgrid-scale barotropic and baroclinic eddies in quasi-geostrophic models: anticipated potential vorticity method. J. Atmos. Sci. 42, 1353–1363.Google Scholar