The modified plasma dispersion function

TLDR

In plasma wave theory, the Maxwellian distribution gives rise to the plasma dispersion function Z(ξ), while many space‑physics distributions exhibit high‑energy tails modeled by kappa distributions, making the new function Z*κ(ξ) relevant for studying microinstabilities in Lorentzian‑type plasmas. The authors introduce a new special function Z*κ(ξ), analogous to Z(ξ), derived from the kappa distribution. This function, called the modified plasma dispersion function, is defined for the complex variable ξ. For integer κ, Z*κ(ξ) is expressed as a finite‑series closed form with explicit small‑ and large‑argument expansions, analytic expressions for κ=1–6, plotted real/imaginary parts, and it converges to the standard plasma dispersion function as κ→∞.

Abstract

In the linear theory of waves in a hot plasma if the zeroth-order velocity distribution function is taken to be Maxwellian, then there arises a special, complex-valued function of a complex variable ξ=x+iy, namely Z(ξ), known as the plasma dispersion function. In space physics many particle distributions possess a high-energy tail that can be well modeled by a generalized Lorentzian (or kappa) distribution function containing the spectral index κ. In this paper, as a natural analog to the definition of Z(ξ), a new special function Z*κ(ξ) is defined based on the kappa distribution function. Here, Z*κ(ξ) is called the modified plasma dispersion function. For any positive integral value of κ, Z*κ(ξ) is calculated in closed form as a finite series. General properties, including small-argument and large-argument expansions, of Z*κ(ξ) are given, and simple explicit forms are given for Z*1(ξ), Z*2(ξ), ..., Z*6(ξ). A comprehensive set of graphs of the real and imaginary parts of Z*κ(ξ) is presented. It is demonstrated how the modified plasma dispersion function approaches the plasma dispersion function in the limit as κ→∞, a result to be expected since the (appropriately defined) kappa distribution function formally approaches the Maxwellian as κ→∞. The function Z*κ(ξ) is expected to be instrumental in studying microinstabilities in plasmas when the particle distribution function is not only the standard generalized Lorentzian, but also of the Lorentzian type, including inter alia, the loss-cone, bi-Lorentzian, and product bi-Lorentzian distributions.