No abstract

One major characteristic associated with collisionless space plasmas is the development of non-Maxwellian velocity distribution that in many circumstances can be represented by the κ function characterized by the κ parameter. This paper discusses the mathematical character and physical origin of the κ function by first showing that the κ velocity function may be expressed in terms of exponential functions multiplied by the kinetic energy and its higher orders. The possible development of κ velocity distribution is illustrated by the problem of low-frequency waves and instabilities in uniform magnetized plasmas with bi-Maxwellian distribution. It is observed that the background and perturbed distribution functions bear the same forms as the zeroth- and higher-order terms of the κ function expanded in the limit of κ→∞. The consequence of assuming κ velocity distribution in inhomogeneous plasmas is illustrated by the Vlasov-Maxwell equilibrium problems that show the nonthermal equilibrium characteristic of nonuniform plasmas. A generalized Grad-Shafranov equation is proposed for two-dimensional Vlasov equilibria with κ velocity distribution.

An exact solution of the steady-state, one-dimensional Vlasov–Maxwell equations for a plasma current sheet with oppositely directed magnetic field was found by Harris in 1962. The so-called Harris magnetic field model assumes Maxwellian velocity distributions for oppositely drifting ions and electrons and has been widely used for plasma stability studies. This paper extends Harris solutions by using more general κ distribution functions that incorporate Maxwellian distribution in the limit of κ→∞. A new functional form for the plasma pressure as a function of the magnetic vector potential p(A) is found and the magnetic field is a modified tanhz function. In the extended solutions the effective temperature is no longer spatially uniform like in the Harris model and the thickness of the current layer decreases with decreasing κ.

No abstract

The comment questions the formulation of the κ velocity distribution function used in our paper as compared to a slightly different form used by the authors. The difference in the distribution function necessarily leads to different number densities, thermal pressures, etc. We show that the restriction with their distribution function is that the macroscopic temperature (or average kinetic energy) is the same for all spatially uniform systems with a family of κ distributions including the Maxwellian case. The distribution function used in our paper and widely adopted in various studies of nonthermal systems, however, does not impose such a constraint; in particular, the temperature has κ dependence reflecting the kinetic nature of different statistical systems. The points made in the comment are trivial and misleading.