Polytropes in plasmas described by kappa distributions – Application in atmospheric modelling

Abstract

The paper derives the complete formulations of polytropic behaviour for plasmas described by kappa distributions. This is achieved by employing both positive and negative types of phase-space kappa distributions of a Hamiltonian for a nonzero potential energy, while the cases of positive and negative potential energies are analysed separately. Then, we develop the general polytropic-barometric formula that describes the profiles of density, temperature, and thermal pressure. Furthermore, it is shown how the kappa and polytropic indices can be derived from observational measurements of the temperature altitude gradient. As an example, we calculated the kappa ≈ 3.35 and polytropic ≈ 0.74 indices of the terrestrial atmosphere at ∼100 km, revealing the existence of heating processes that add thermal energy to atmospheric particles. K E Y W O R D S kappa distributions, polytropic index, space plasmas 1 INTRODUCTION Gases and other classical particle systems are characterized by (a) weakly interacting particles, (b) short-range interactions-restricted to particles' first neighbours, (c) absence of any collective-behaviour among particles, and, most important, (d) absence of any local correlations among particles. This last property is certainly related to the former ones, but it is the cornerstone of the basic consideration of the classical Boltzmann-Gibbs statistical mechanics. [1] As a result, when these particle systems reside at thermal equilibrium, they have their phase-space described by the celebrated Maxwell-Boltzmann distribution. On the other hand, the kappa distributions were found to describe particle systems characterized by (a) weakly and/or strongly-interacting particles, (b) short-and/or long-range interactions, (c) collective-behaviour among particles, and again, most important, (d) existence of local correlations among particles. In fact, the kappa index that governs and parameterizes these distributions is inversely related to the correlation coefficient; thus, the higher the value of kappa, the closer we reach the classical limit of Maxwell-Boltzmann distributions, and the smaller the correlation coefficient is among particles. [2,3] Ideal particle systems with all the above properties are the collisionless and weakly coupled plasmas, where the Debye shielding induces local correlations among particles (Figure 1). [4-7] Especially, the space and astrophysical plasmas are such examples of particle systems with correlations described by kappa distributions [8,9] (also, see the book of kappa distributions: [10]). The equation-of-state and the velocity distribution function are two different topics and should not be confused. For instance, in the case of classical plasmas, both the equation-of-state and the velocity distribution function are