Mathematical Sciences: Equilibria Instabilities and Waves in Fluids and Plasmas

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9623033 Lipton The Principal Investigator (PI) will continue to study equilibria, instabilities and waves in fluids and plasmas with the emphasis on the comparative aspect of the problem. The PI is particularly interested in the following topics: (a) stationary states of magnetized plasmas with flow and their group-theoretical properties; (b) the stability and evolution of elliptical self-gravitating fluid masses (the Riemann ellipsoids) with applications to astrophysics; (c) the linear and nonlinear dynamics of fixed-boundary ellipsoids containing fluid. The PI is going to use various analytical techniques including the constrained minimization, `exact' Galerkin, group-theoretical, and geometrical optics methods in conjunction with numerical techniques such as the three-dimensional vortex method, large scale symbolic manipulations, and visualization, in order to achieve these goals. Further directions of attack include self-similar, time-dependent vortex rings, short wavelength instabilities of unmagnetized and magnetized accretion discs, global instabilities of collisionless gravitating systems,and spectral problems of hydrodynamics and magnetohydrodynamics. This research will advance current knowledge of the behavior of both laboratory and astrophysical fluids and plasmas. In particular, this research will contribute towards a better understanding of the properties of nonlinear PDEs of mixed elliptic/hyperbolic type describing symmetric stationary states of plasmas with flow; the stability of Riemann ellipsoids and fluid- and plasma-filled bodies; the evolution of nonaxisymmetric stars, and the validity of the fission theory of binary stars. %%% The dynamics of fluids and plasmas is of great interest to mathematicians and physicists alike. There are at least two reasons for this interest. First, fluid and plasma motions play a fundamental role in nature and technology with applications stretching from astrophysics, to oceanography, to weather predicti on, to controlled thermonuclear fusion. Second, a very rich and sophisticated mathematical apparatus is required in order to give an adequate description of these motions. The equilibrium theory studies the structure and properties of steady solutions of the nonlinear equations of fluid and plasma dynamics. These equilibrium solutions represent special states which do not change in time when left undisturbed. Usually it is rather difficult to describe equilibrium solutions, however, it is still much easier then to describe the general solutions. In addition, equilibrium solutions are the most interesting ones from a practical point of view. The stability theory studies the impact of initially small perturbations on a given steady fluid or plasma equilibrium. An equilibrium is called stable and can occur in nature if perturbations do not have a profound effect on its properties. An equilibrium is called unstable if under the influence of perturbations it either evolves into a different equilibrium, or loses its steady character altogether. Over a period of years many classical stability problems were solved, however, several important stability problems are still open. The wave theory studies the behavior of small perturbations in the vicinity of stable plasma equilibria. The main issues include the analysis of the oscillation frequencies, the structure of the corresponding eigenfunctions, and the asymptotic behavior of perturbations in time. The present research is aimed at advancing current understanding of equilibria, instabilities and waves. It blends theoretical and symbolic methods with high-performance computing and graphics. The outcome of this research will have both theoretical and practical value. Among other things, it will help to understand the dynamics of plasmas in the so-called tokamaks (installations which are used for controlled thermonuclear fusion), and the stability of stars. ***