a Study of Nonlinear Dynamical Models of Plasma Turbulence.
Abstract
The study of turbulent phenomena in plasma physics is clearly an important topic. Progress has, however, been very slow because of the complexity of the problem. In this dissertation, an exact nonlinear mode truncation method, i.e. the pole expansion method, is developed to study onedimensional models of plasma turbulence. The internal wave is magnetized plasma in which the magnetic field changes sharply over a small region obeys the modified BenjaminOno equation: ((PARDIFF)u/(PAR DIFF)t) + (mu)((PARDIFF)u/(PARDIFF)x) + (beta)H((PAR DIFF)('2)u/(PARDIFF)x('2)) + (nu)((PARDIFF)('2)u/(PAR DIFF)x('2)) + 2u((PARDIFF)u/(PARDIFF)x) = 0. This equation with periodic boundary conditions can be solved by a pole expansion technique, which changes the dissipative Benjamin Ono equation to a system of ordinary equations of poles. The pole expansion is an exact nonlinear mode truncation method in which a finite number of poles are used to represent the asymptotic behavior of the system. Numerical results and analytical proof show that the pole representation is complete. The equation of poles allows us to do linear and nonlinear stability analyses. Because the number of poles in the asymptotic state is proportional to the ratio of the growth to the damping, fully developed turbulence can be reached if the system has a very large number of poles. This is equivalent to having a large Reynolds number. An interesting aspect of the pole dynamics is that it exhibits a wide range of bifurcation phenomena with very complicated behavior. Almost all bifurcation phenomena discovered so far are observed in the system, such as period doubling, Hopf bifurcation, bifurcations on a 2torus, as well as crises. Additional research has been performed to study the statistical properties of turbulent flows, including correlation functions and energy spectra. The correlation functions suggest that the chaotic motion of poles may be associated with strongly turbulent behavior. The energy spectrum has the form E(k) (TURN) e('(alpha)(nu)k) in the high k region, where (nu) is the damping rate and (alpha) is a constant, while in the low k region E(k) (TURN) k(' 1). Finally, several similar systems are studied briefly and comparisons to the dissipative system are made.
 Publication:

Ph.D. Thesis
 Pub Date:
 1986
 Bibcode:
 1986PhDT.........5Q
 Keywords:

 Physics: Fluid and Plasma