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The statistical mechanics of vortex—acoustic ion wave turbulence

The equilibrium statistical mechanics of electrostatic ion wave turbulence is studied within the framework of a continuum ion flow with adiabatic electrons. Attention is drawn to the fact that the wave field consists in general of two components, namely ion-acoustic and ion vortex modes. It is shown that the latter can significantly affect the equilibria on accoant of their ability both to emit and to scatter ion sound. Exact equilibria for the vortex—acoustic wave field are given in terms of a canonical distribution and the correlation functions are expressed in terms of a generating functional. A nonlinear transformation of the wave field, which removes the vortex-acoustic interaction energy to lowest order in the strength of the coupling, while preserving the phase space volume element, is then introduced. This enables the Feynman-Hibbs variational principle to be used to obtain an approximate generating functional based on a ‘trial’ energy functional, which is a quadratic in the new variables. Detailed calculations are carried out for the case in which the dominant coupling is an indirect interaction of the vortex modes mediated by the sound field. An equation for the spectrum of the vortex modes is obtained for this case, which is shown to possess a simple exact solution. This solution shows that the spectrum of fluctuations changes considerably as the total energy increases. At low levels of excitation the solution reduces to equipartition of energy. At higher levels the indirect interaction becomes significant and the formation of a condensed vortex state is possible when the mean energy exceeds the electron thermal energy. It is suggested that condensed vortex states could occur in the plasma sheet of the earth's magnetosphere and it is shown that the predicted ratio of the mean ion energy to the mean electron energy is consistent with the trend of observed values.