Thermodynamics of reactive marginal instabilities

Abstract

The reactive instabilities are characterized by the absence of dissipation, algebraic dispersion relations (in slab approximation) and the existence of marginal (bifurcation) points of stability. In the vicinity of these points the reactive instabilities are amenable to a thermodynamic treatment based on the electrostatic entropy concept. After recalling the essential aspects of this concept and its connection with the linear description of the electrostatic instabilities, the theory is extended to the nonlinear domain near the marginal point. It is shownthat the marginal state is a maximum of the electrostatic entropy with respect to arbitrary variations of the bifurcation parameter (e.g. the temperature gradient). However, a nonlinear neighboring electrostatic structure is formed, which is the manifestation, on average, of large-amplitude fluctuations towards the linearly unstable side of the marginal point. A correspondence with numerical models of turbulence involving ion temperature gradient (ITG) saturation is noted. Compact analytical expressions for the mean square amplitude of the fluctuations valid in the range from the stable to the unstable side of the marginal point are presented. As an example, the theory is applied in detail to the flute modes.