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Nonlinear stability of fluid and plasma equilibria

Holm, Darryl D. and Marsden, Jerrold E. and Ratiu, T. and Weinstein, Alan J. (1985) Nonlinear stability of fluid and plasma equilibria. Physics Reports, 123 (1-2). pp. 1-116. ISSN 0370-1573. doi:10.1016/0370-1573(85)90028-6.

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The Liapunov method for establishing stability has been used in a variety of fluid and plasma problems. For nondissipative systems, this stability method is related to well-known energy principles. A development of the Liapunov method for Hamiltonian systems due to Arnold uses the energy plus other conserved quantities, together with second variations and convexity estimates, to establish stability. For Hamiltonian systems, a useful class of these conserved quantities consists of the Casimir functionals, which Poisson-commute with all functionals of the given dynamical variables. Such conserved quantities, when added to the energy, help to provide convexity estimates bounding the growth of perturbations. These estimates enable one to prove nonlinear stability, whereas the commonly used second variation or spectral arguments only prove linearized stability. When combined with recent advances in the Hamiltonian structure of fluid and plasma systems, this convexity method proves to be widely and easily applicable. This paper obtains new nonlinear stability criteria for equilibria for MHD, multifluid plasmas and the Maxwell-Vlasov equations in two and three dimensions. Related systems, such as multilayer quasigeostrophic flow, adiabatic flow and the Poisson-Vlasov equation are also treated. Other related systems, such as stratified flow and reduced magnetohydrodynamic equilibria are mentioned where appropriate, but are treated in detail in other publications.

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Ratiu, T.0000-0003-1972-5768
Additional Information:© 1985 Published by Elsevier Science B.V. All rights reserved. Available online 5 September 2002. Research supported by DOE contract W-7405-ENG-36. Research partially supported by DOE contract DE-ATO3-82ER12097. Research supported by an NSF postdoctoral fellowship. Jerrold Marsden, Tudor Ratiu, and Alan Weinstein have enjoyed the hospitality of the Center for Nonlinear Studies in Los Alamos during various periods in 1983 when most of this research was carried out. Darryl Hoim and Tudor Ratiu also thank the Mathematics Department at Berkeley for facilitating this collaboration. We are grateful to Henry Abarbanel, Allan Kaufman, Richard Hazeltine, Philip Morrison, Philippe Similon, and Steve Wan for their interest and helpful comments. Correspondence with William Blumen, David Andrews, and Ettore Salusti is gratefully acknowledged.
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Department of Energy (DOE)W-7405-ENG-36
Department of Energy (DOE)DE-AT03-82ER12097
NSF Postdoctoral FellowshipUNSPECIFIED
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ID Code:19464
Deposited By: Ruth Sustaita
Deposited On:17 Aug 2010 18:16
Last Modified:08 Nov 2021 23:52

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