The modified plasma dispersion function

Journal Article · Thu Aug 01 00:00:00 EDT 1991 · Physics of Fluids B; (United States)

DOI:https://doi.org/10.1063/1.859653· OSTI ID:5666176

Summers, D ^[1]; Thorne, R M ^[2]

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1. Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John's, Newfoundland A1C5S7, Canada (CA)
2. Department of Atmospheric Sciences, University of California at Los Angeles, Los Angeles, California 90024-1565 (USA)

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 {xi}={ital x}+{ital iy}, namely {bold Z}({xi}), 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 {kappa}. In this paper, as a natural analog to the definition of {bold Z}({xi}), a new special function {bold Z}{sup *}{sub {kappa}}({xi}) is defined based on the kappa distribution function. Here, {bold Z}{sup *}{sub {kappa}}({xi}) is called the modified plasma dispersion function. For any positive integral value of {kappa}, {bold Z}{sup *}{sub {kappa}}({xi}) is calculated in closed form as a finite series. General properties, including small-argument and large-argument expansions, of {bold Z}{sup *}{sub {kappa}}({xi}) are given, and simple explicit forms are given for {bold Z}{sup *}{sub 1}({xi}), {bold Z}{sup *}{sub 2}({xi}), ..., {bold Z}{sup *}{sub 6}({xi}). A comprehensive set of graphs of the real and imaginary parts of {bold Z}{sup *}{sub {kappa}}({xi}) is presented. It is demonstrated how the modified plasma dispersion function approaches the plasma dispersion function in the limit as {kappa}{r arrow}{infinity}, a result to be expected since the (appropriately defined) kappa distribution function formally approaches the Maxwellian as {kappa}{r arrow}{infinity}. The function {bold Z}{sup *}{sub {kappa}}({xi}) 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 {ital inter} {ital alia}, the loss-cone, bi-Lorentzian, and product bi-Lorentzian distributions.

• OSTI ID: 5666176
• Journal Information: Physics of Fluids B; (United States), Vol. 3:8; ISSN 0899-8221
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
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