Proton Cyclotron Instability in Two-Dimensional Magnetospheric Plasmas with Anisotropic Temperature

N.I. Grishanov
Kharkiv National University, Department of Physics and Technology, Kurchatov str., 31, Kharkiv, UKRAINE
Abstract. In this paper, we analyze the dispersion equation for field aligned ion-cyclotron waves in a twodimensional
(2D) magnetospheric plasma with anisotropic temperature. Magnetic field configuration with
circular magnetic field lines is considered. The steady-state bi-Maxwellian distribution function is used to model
the energetic protons in a hydrogen plasma.at the geostationary orbit. As in the uniform magnetic field, the
growth rate of the proton-cyclotron instability (PCI) in a 2D magnetospheric plasma is defined by the
contribution of the energetic ions/protons to the imaginary part of the transverse permittivity elements. We
demonstrate that the PCI growth rate in 2D axisymmetric magnetosphere can be significantly smaller than that is
for the straight magnetic field case with the same macroscopic bulk parameters.
Keywords: two-dimensional magnrtospheric plasma, trapped particles, ion-cyclotron waves, proton cyclotron instability
PACS: 94.30.cq, 94.30.Hn
1. INTRODUCTION
Cyclotron waves are an important constituent of plasmas in solar corona, solar wind and planetary
magnetospheres. As is well known, energetic particles with anisotropic temperature can excite a wide class of
cyclotron wave instabilities. Kinetic theory of such waves in the straight magnetic field plasmas is developed
very well, see e.g. Refs. [1-7] and bibliography therein. However, plasma models in the straight magnetic field
are quite rough for planetary magnetospheres which are three-dimensional in the general case. As more suitable,
the inner Earth’s magnetosphere can be considered as a two-dimensional (2D) plasma configuration with a point
dipole magnetic field lines. Another interesting 2D magnetosphere-like plasma model is a configuration with
circular magnetic field lines, which is artificial but simpler and helpful to describe the principal wave processes
in the Earth’s magnetosphere.
The dispersion equations for field aligned cyclotron waves in magnetospheric plasmas with dipole and
circular magnetic field lines were derived analytically in Ref. [8].
In this paper, we analyze the dispersion characteristics of the electromagnetic ion-cyclotron (EMIC)
waves in the hydrogen plasma confined in the last plasma model including the energetic protons with the bi-
Maxwellian distribution function.
2. DISPERSION EQUATION FOR FIELD-ALIGNED CYCLOTRON WAVES
As was shown in Ref. [8], the dispersion equations for field aligned cyclotron waves in 2D
magnetospheric plasmas with circular magnetic field lines can be rewritten by analogy with the straight uniform
magnetic field case in the form:

The PCI growth rates versus ω are present in Fig. 2a for EMIC waves in the straight magnetic field
plasma by Eqs. (5, 6), and in Fig. 2b for EMIC waves in the 2D magnetosphere-like plasma with circular
magnetic field lines by Eqs. (7, 8). The computations of γ
c are carried out in the interval 2Hz ≤ω ≤ 7Hz ,
whereas the minimal gyrofrequency of the protons at L=6.6 is closed to 0 11Hz Ω ≈ c . As shown in Fig. 2a and
Fig. 2b, the instability of EMIC waves is possible for the both plasma models in the frequency range c0 ω < Ω . It
should be noted that the proton-cyclotron instability is impossible for EMIC waves in the frequency range
Ωc0 <ω < Ωc0b(θ o ) , where Ωc0b(θ o ) is the maximal gyrofrequency of the protons at the given L-shell
magnetic field line.
As one can see, the dependence γ
s(ω) and γ
c(ω) on the wave frequency ω are similar; however,
γc
(ω)<< γ
s(ω) under the same bulk parameters. The ratio s / 4 10 ∝ ÷ c γ γ versus ω for considered
magnetospheric-like plasmas is present in Fig. 3. This dependence is not linear; the difference is very large (by
factor 10) for EMIC waves in the range of ω ~ 2Hz and is smaller (by factor 4) in the range of high frequencies
ω ~ 7Hz .

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[8] N.I. Grishanov, M.A. Raupp, A.F.D. Loula and J. Pereira Neto, 12th Int. Congress on Plasma Phys., Nice, France,
25-29 October, 2004, E-Proceedings of ICPP2004:

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