# Damping of Whistler Modes Guided by Cylindrical Plasma Structures

V. A. Es’kin and A.V.Kudrin

Department of Radiophysics, University of Nizhny Novgorod

23 Gagarin Ave., Nizhny Novgorod 603950, Russia

Abstract. We study the guided propagation of whistler waves along an axially magnetized plasma column surrounded by free

space or an unbounded background magnetoplasma. It is shown that the presence of collisional losses in a plasma can result

in simultaneous existence of weakly and strongly damped guided modes with close wavelengths but significantly different

field structures whose features are essentially determined by the relative contribution of small-scale quasi-electrostatic waves

to the total modal fields. The results obtained are useful for clarification of the power absorption of whistler-mode waves in

collisional cylindrical plasma structures.

Keywords: whistler mode waves, plasma waveguides, collisional damping

PACS: 52.35.Hr, 52.40.Fd, 52.80.Pi

INTRODUCTION

Over the past decade, there has been a substantial degree of interest in the guided propagation of whistler waves

which play an important role in many applications, ranging from space plasma research to radio-frequency (rf) plasma

sources used in general plasma physics experiments, plasma processing, etc. (see, e.g., [1–6] and references therein).

For the above-mentioned applications, the understanding of mechanisms of the power absorbtion in such modes is

crucial. The purpose of this paper is to analyze the influence of collisional losses on the dispersion properties, damping

features, and field structures of guided whistler-range modes in a cylindrical plasma column surrounded by free space

or an unbounded background magnetoplasma. Although it is usually thought that allowance for collisional losses in a

magnetoplasma can only result in attenuation of guided modes, it turns out that under certain conditions, the presence

of collisions leads to some unexpected behavior of the mode characteristics.

FORMULATION

Consider a cylindrical plasma column of radius a aligned with an external dc magnetic field B0 = B0 ˆz0 and located

in free space or a homogeneous background magnetoplasma. The plasma is described by the dielectric tensor ˆe =

e0(e ˆ r0 ˆ r0−ig ˆ r0

ˆ f0+ig ˆ f0 ˆ r0+e ˆ f0

ˆ f0+hˆz0ˆz0), where e0 is the permittivity of free space. We focus on waves whose

frequencies lie in the whistler range wLH |w −ine|wce wpe, where wLH is the lower hybrid frequency, ne is the

electron collision frequency, and wce and wpe are the electron cyclotron frequency and plasma frequency, respectively.

If the plasma column is uniform, then the axial field components in the region r < a are given by.

where Km is a modified Bessel function of the second kind of order m, C1,2 are constants, and T0 = k0(p2 −1)1/2. In

the case where the column is surrounded by a uniform magnetoplasma, the axial field components in the region r > a

may be represented in a form similar to Eq. (1) if the Bessel functions are replaced by appropriate Hankel functions

and the quantities entering the field expressions are replaced by those corresponding to the background plasma [1].

Application of the continuity conditions for the tangential field components Ef,z and Bf,z at r = a then yields a

dispersion relation, from which the axial wavenumbers p = p0 −i p00 of modes can be found. Here, p0 and p00 are the

propagation and damping constants, respectively.

DISPERSION PROPERTIES AND FIELD STRUCTURES OF MODES

Figures 1 and 2 show the propagation constants p0 and the damping constants p00 as functions of w for the m=1 modes

of a plasma column in free space. Note that the dimensionless parameters chosen for Figs. 1 and 2 are typical of rf

helicon discharges [4–6] and can be ensured if the plasma density N = 1013 cm−3, w/2p = 27.12 MHz, B0 = 800 G,

and a = 2.5 cm. It is seen in Fig. 1 that allowance for small collisional losses results in appearance of steeper parts

on the curves p0(w). Simultaneously, local minima of the damping constants p00(w) appear at the frequencies at

which the corresponding dependences p0(w) turn out to be steepest (see Fig. 1). With the further increase in the

electron collision frequency ne, the dispersion characteristics of modes undergo significant changes (see Fig. 2). It is

seen from comparison of Figs. 1 and 2 that beginning with a certain value of ne, the steeper parts of adjacent curves

p0(w) transform into a single curve corresponding to a mode with a relatively weak collisional damping. Accordingly,

a separate curve of the damping constant p00(w) forms for such a mode. In Fig. 2, the propagation and damping

constants of this mode are shown by the dark solid lines. The remaining parts of the curves p0(w) and p00(w) form the

corresponding dependences for modes with comparatively strong damping, shown by the faint solid lines in Fig. 2.

FIGURE 4. The same as in Fig. 3, but in the presence of small collisional losses in the plasma (ne = 0.15w). Other parameters

coincide with those for Fig. 3.

Thus, it is evident that with increasing electron-collision frequency, the modes are divided into two sets, namely,

weakly damped modes for which the relation p00/p0 ne/w holds, and strongly damped modes whose damping is

determined by an expression typical of quasi-electrostatic waves, i.e., p00/p0 ne/w.

Figures 5 and 6 show the field structure of one of the m = 1 mode of a plasma channel in free space for ne = 0

and ne = 6×10−3wce, respectively. In Fig. 5, one can see the presence of two significantly different spatial scales in

the field structure. These scales correspond to large-scale whistler (helicon) waves and small-scale quasi-electrostatic

waves (also known as Trivelpiece–Gould waves), which simultaneously contribute to the total modal fields. Allowance

for collisional losses leads to a drastic change in the field structure. It is seen in Fig. 6 that in the case of a collisional

plasma column, the small-scale quasi-electrostatic part of the total field of the weakly damped mode is localized in

proximity to the column boundary r = a. On the contrary, the fields of strongly damped modes are heavily affected

by quasi-electrostatic waves. The field structures of modes supported by an enhanced-density channel located in the

background magnetoplasma demonstrate similar behavior, and will not be presented here.

The above results refer to a sharp-walled plasma column. Analogous features in the mode dispersion properties and

field structures were revealed for the case of a smoother profile of plasma density. In this work, we do not give the

corresponding results for the sake of brevity.

CONCLUSION

In this paper, we studied the dispersion properties and field structures of whistler modes guided by an axially

magnetized cylindrical plasma column surrounded by either free space or the background plasma of lower density.

FIGURE 6. The same as in Fig. 5, but for ne = 6×10−3wce. Other parameters coincide with those for Fig. 5. The presented field

structure corresponds to the weakly damped mode referring to the dark solid lines in Fig. 2 and having the propagation constant p0

which is closest to that of the mode whose field structure is plotted in Fig. 5.

It follows from our analysis that for a lossy plasma, one can distinguish weakly and strongly damped modes whose

damping constants are determined by the relative contribution of quasi-electrostatic waves to the total modal fields.

Moreover, due to the presence of quasi-electrostatic waves, even the collisional damping of weakly damped modes

turns out to be notably greater than that of a pure electromagnetic whistler-mode wave propagating in a homogeneous

magnetoplasma along the external magnetic field. It is established that such behavior of the damping constants is

typical of modes guided by cylindrical plasma structures in the whistler range.

ACKNOWLEDGMENTS

This work was supported by the Russian Foundation for Basic Research (project No. 04–02–16506-a) and the Council

of the President of the Russian Federation for the State Support of the Leading Scientific Schools of the Russian

Federation (project No. NSh–1087.2006.2).

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