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

Related Subjects

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71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
BOLTZMANN STATISTICS
DISPERSION RELATIONS
DISTRIBUTION FUNCTIONS
FLUIDS
FULLY IONIZED GASES
FUNCTIONS
GASES
HOT PLASMA
INSTABILITY
IONIZED GASES
LORENTZ GAS
MODIFICATIONS
PLASMA
PLASMA INSTABILITY
PLASMA MICROINSTABILITIES
VELOCITY