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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
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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

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