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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