Two-dimensional instability of fast ionization waves propagating
in an external electric ﬁeld
Departamento de Fisica, Universidade da Madeira, Largo do Municipio 9000 Funchal, Portugal
͑Received 10 March 1998; accepted 14 April 1998͒
Stability of fast planar ionization waves ͑IW͒ propagating in a cold electropositive gas in the
presence of constant applied electric ﬁeld E
has been investigated. The stationary IW are
one-dimensional ͑1D͒ nonlinear solutions depending on the variable
ϭxϪVt and connecting two
different steady states with constant parameters. The basic system of equations contains the rate
equations for the concentrations of electrons and positive ions and the Poisson equation for the
electric ﬁeld. There is a continuum of possible velocities of stationary anode-directed (E
Ͼ0) IW. All stationary IW are stable to small 1D perturbations. It has been
shown that in a wide range of the parameters E
/p and V/V
, where p is the gas pressure and V
is the electron drift velocity ahead of the wave front, both anode- and cathode-directed IW are
unstable to two-dimensional ͑2D͒ perturbations having the form f(
)exp(ikyϩt). The 2D
instability is displayed in the form of increasing corrugation of the planar IW. The integral
representation of the electric potential perturbation has been employed for calculation of the
eigenmodes and the growth rates in the long-wave limit k⌬Ӷ1 where ⌬ is the characteristic
thickness of a stationary IW. © 1998 American Institute of Physics. ͓S1070-664X͑98͒01208-7͔
Ionization waves ͑IW͒ propagating in an external ap-
plied electric ﬁeld provide an important mechanism of the
electrical breakdown of gases. A simple discharge cell in
which the IW can be observed is a gap between two parallel
electrodes ﬁlled with a nonionized gas subject to a rapidly
increasing voltage pulse applied to the electrodes. The main
kinds of IWs that exist in planar gaps are streamers
planar ionization waves.
The anode-directed ionization
waves ͑AIW͒ that propagate in the direction of the electron
drift velocity V
Ͻ0 is the electron mobility,
as well as cathode-directed ionization waves ͑CIW͒ propa-
gating in the opposite direction are observed in experiments.
Both AIW and CIW may propagate with velocities V that
exceed the drift velocity V
by 1–1.5 orders of magnitude.
Large velocities of IWs are explained by the presence of
some amount of secondary electrons ahead of the wave front.
In particular, the photoionization may provide long-scale
precursors of the electron density decaying in space and ex-
tending the leading edge of an IW.
Unlike streamers that
can be easily developed from single electron avalanches, the
planar ionization waves are created by simultaneously initi-
ating many avalanches uniformly over the cathode. Thus, in
the experiment of Koppitz ͑Ref. 4͒ an ultraviolet ﬂash was
used for a uniform generation of primary electrons in the
near-cathode region. Although IW initiated by many electron
avalanches that are overlapped in a short time are favorable
for one-dimensional ͑1D͒ modeling, the ﬁrst theoretical
models of planar IW
were intended for the modeling of
streamers that represent three-dimensional or, at least, two-
dimensional ͑2D͒ plasma conﬁgurations. The reason was that
the characteristic thickness ⌬ of the ionization front is much
less than the radius of curvature of the streamer tip and in a
small vicinity of the latter the 1D solutions are appropriate.
The 1D models of stationary IWs propagating with constant
velocities deal with the solutions of the basic equations that
depend on one variable
ϭxϪVt. Such models describe the
ionization growth and the electric ﬁeld screening within the
wave structure and they result in a reasonable relationship
between the quantities V and ⌬. The main distinction of the
ﬁeld distributions E(
) predicted by 1D models from the
axial ﬁeld distributions E
obtained for axially symmetric 2D
is the existence of a maximum of
close to the streamer tip.
There are a variety of applications of IW to plasma sci-
ence and technology. IW are employed for high-power pseu-
for pumping gaseous lasers
generation of volume
discharges for lasers.
The propagation of streamers is one of the main processes in
the streamer chambers that are designed for studying nuclear
reactions and cosmic rays.
Streamers occur also under
electrical breakdown of gaseous insulators and their devel-
opment can lead to damage of the high-voltage transmission
lines. Thus, the understanding of the main properties of
streamers is important also for high-voltage engineering.
Beyond various applications the interest in studying the
possible types of fast IWs and their structure stems from the
fact that IW extend the range of wave phenomena in active
and dissipative media which are known as auto-waves.
distinctive feature of such waves is stationary propagation
under the action of external energy sources. Some of the
properties of IWs are shared with such phenomena as ﬂame
waves of growth and migration of biological
Permanent address: Department of Mechanical Engineering, Ben-Gurion
University of the Negev, Beer Sheva 84105, Israel.
PHYSICS OF PLASMAS VOLUME 5, NUMBER 8 AUGUST 1998
30541070-664X/98/5(8)/3054/11/$15.00 © 1998 American Institute of Physics