J. Plasma
J.
Plasma Fusion
Fusion Res.
Res. SERIES,SERIES,Vol.
Vol.6 6(2004)(2004)421–424
000–000

Dust Acoustic and Dust Ion Acoustic Nonlinear Structures:
Theory and Experiments
POPEL Sergey I.
Institute for Dynamics of Geospheres, Leninsky pr. 38, Building. 1, 119334 Moscow, Russia(Received: 9 December 2003 / Accepted: 26 February 2004)

Abstract
Dust acoustic and dust ion acoustic nonlinear structures in complex (dusty) plasmas are described. The emphasis
is given to the new results. We describe the dust acoustic bow shock formed in the interaction of the solar wind with
cometary dusty coma. We discuss the possibility of observation of shocks related to the dust charging process inactive rocket experiments, which involve the release of some gaseous substance in near-Earth space, as well as thelaboratory experiments modeling active those. We present an explanation of the experiments in rf plasma discharge
where phase transition was occurred during dust acoustic shock propagation.
Keywords:complex (dusty) plasmas, nonlinear structures, shock waves, solitons

1. Introduction

Institute for Extraterrestrial Physics (Germany) [7].
The purpose of this paper is to reflect the most importantresults on dust ion acoustic and dust acoustic nonlinear
structures performed with the participation of researchers of
the Institute for Dynamics of Geospheres RAS (Russia). We
present both the result of the theoretical treatment of dust ion
acoustic shocks and the description of the dust acoustic bow
shock formed in the interaction of the solar wind with
cometary dusty coma. We discuss the possibility ofobservation of shocks related to the dust charging process inactive rocket experiments, which involve the release of somegaseous substance in near-Earth space, as well as thelaboratory experiments modeling active those. We present anexplanation of the experiments in rf plasma discharge wherephase transition was occurred during dust acoustic shockpropagation.

At present a major portion of the investigations ofplasmas is devoted to multicomponent plasmas containing
electrons, ions, charged microspheres or dust grains, andneutral particles. The term “complex plasmas” is finding
increasing use for such plasmas. The dust particle charging
process is a basis for the strong dissipativity of the complex
plasma system. The strong dissipativity results in a newphysics of nonlinear wave structures. First, anomalous
dissipation originating from dust grain charging appears.
Second, new kinds of nonlinearities take place. The
anomalous dissipation leads to preferential importance ofshock waves in complex plasmas, which have specific featuresthat distinguish them from ordinary collisional andcollisionless shock waves. That dust ion acoustic shock waves
associated with anomalous dissipation can actually exist wasproved analytically in [1]. Dust ion acoustic shock waves wereobserved for the first time in laboratory experiments at theInstitute of Space and Astronautical Science (Japan) [2] andat the University of Iowa (USA) [3]. Besides dust ion acoustic
solitons can propagate in complex plasmas. Theory
description of the dust ion acoustic solitons has been given in[4]. They were observed in laboratory experiments at theInstitute of Space and Astronautical Science [5].
At present the significant attention of researchers is paid
to the case of dust acoustic nonlinear waves. A threedimensional dust-acoustic shock was observed in a
microgravity experiment [6]. The experiments on dust
acoustic shocks in a laboratory two-dimensional (2D)
complex plasma have been performed at the Max-Planck-

2. Dust ion acoustic shocks
Let us formulate the main experimental results on shocks
in complex plasmas. In experiments [2], Nakamura et al.
revealed that the most important feature of dust ion acoustic
shocks in complex plasmas is the following.
(i) In the absence of dust, the effect of the electron and ion
charge separation gives rise to oscillations in the shock
wave profile in the vicinity of the shock front, while thepresence of dust suppresses these oscillations.
The experiments [3] showed that:(ii) Dust ion acoustic shocks are generated at sufficiently
high dust densities (under the experimental conditions©2004 by The Japan Society of Plasma
Science and Nuclear Fusion Research

Corresponding author’s e-mail: s_i_popel@mtu-net.ru
1421

Popel S.I., Dust Acoustic and Dust Ion Acoustic Nonlinear Structures: Theory and Experimentsof [3], at dust densities such that ε Zd0 ≡ nd0Zd0 /ni0 ≥ 0.75,
where q d = –Z de is the dust grain charge, –e is theelectron charge, nd(i) is the dust (ion) density, and thesubscript 0 stands for the unperturbed plasmaparameters). In [3], the conclusion about the formationof a shock wave was drawn from the fact that theperturbation front steepens as time elapses. At
sufficiently low dust densities, the perturbation front does
not steepen but instead widens.
(iii) When the shock wave structure has formed, the shock
front width ∆ξ ~ Mcs /νq is described by the theoreticalestimate, which is based on the model developed in [1],where Mcs is the shock-wave structure speed, M is theMach number, cs is the ion acoustic speed, νq is the graincharging rate.
(iv) The velocity of the dust ion acoustic waves increases
considerably with increasing ε Zd0. In this context, therequirement to the theoretical model is the adequate
description of the relevant experiments. We use the socalled ionization source model developed in [8,9].
We have tested [9] our theoretical model against theexperimental result (i), which was obtained in [2]. We haveshown that the electron and ion charge separation gives rise
to oscillations in the shock wave profile and that the dust
suppresses these oscillations, as is the case in the experimentsof [2]. The theoretically calculated rise time of the shock front
is about 5 µs, which corresponds to the experimental data.
We have studied the evolution problem under theconditions of the experiment [3]. We showed the widening of
the wave front at ε Zd0 = 0 and its steepening at ε Zd0 = 0.75.
This agrees with the experimental data from [3]. The extent
to which the shock front widens was calculated to be∆ξ ~ Mc s /νq ~ 0.3, which corresponds to that observedexperimentally (see Fig. 2b in [3]) and also to the estimate
obtained using the theoretical model of [1]. We demonstrated
that the dependence of the perturbation front velocity(normalized to its value in the absence of dust, ε = 0) on theparameter ε Z d0 is an a quite good agreement with theexperimental data (see Fig. 2 in [9]).
Thus, the theoretical ionization source model makes it
possible to describe all the main experimental results (i)–(iv)on dust ion acoustic shock waves.

observations is in the range 0.1–1 [12–14]. Assuming ten percent content of the dust in the cometary substance and bulk
density of about 1 g/cm 3, we obtain that the density of
micron-sized particles is of the order of 1011 cm-3 [15].
The crucial point of the investigation of the interactionof the solar wind with cometary comae is the description of
bow shock formed as a result of this interaction. The presenceof dust constituting tens per cent of the total mass of the coma
can modify the bow shock. The size distribution of coma
grains is such that the total area is dominated by the smallest
grains present, and the 1P/Halley in-situ results showed that
this continued to hold down to radii a ~ 0.01 µm. The size
distribution is well determined observationally [16] as an
approximate power-law function with index ~ – 4 in terms ofradius. We consider in all subsequent estimates andcalculations the characteristic dust grain size as a ~ 1 µm.
Due to the processes of photoionization the gas
surrounding the comet becomes ionized at the distances
exceeding several kilometers from the cometary nucleus [17].
It is the interaction between the cometary ions and the solar
wind protons that is considered as the strongest one in theinteraction between the cometary coma and the solar wind.
Because the main particles participating in the interaction areions, one may conclude that the bow shock is the ion acoustic
bow shock. The presence of dust in the coma leads to anotherimportant kind of interaction — the interaction between thesolar wind protons and charged dust of the coma. For theparameters of the solar radiation spectrum in the vicinity of
the Earth, electron and ion temperature in the solar wind
TeSW = TiSW = 15 eV, the ion temperature in the cometary coma
Tic = 0.03 eV, the electron and ion (proton) density in thesolar wind neSW ~ np = 20 cm-3, the ion density in the comanear the nucleus ni0c = 1.43·105 cm-3, the ion density in thecoma depending on the distance r from the nucleus as nic =n i0c(r 0 /r), the radius of the nucleus r 0 = 1 km, and H3O +cometary coma ions, we find that the bow shock is formed at
the distance R ~ 104 km and the dust particles acquire positive
charges reaching the values exceeding Zde = 2000e for r ≈ R.
The structure of the bow shock is described by the set ofequations which are a generalization of the ionization source
model to two kinds of ions and positive charges of dust grains.
For r0 = 1 km and dust densities nd > 106 cm-3 near the comet
nucleus, charged dust particles influence drastically thestructure of the bow shock front. There is a good agreement
between the estimate for the shock front width ∆ξ ~ cs /νq,
where cs is the ion acoustic speed in the solar and νq is thedust charging frequency. The bow shock is similar, by its
origin, to the shocks observed in [2,3] and those predicted
theoretically in [1].

3. Dust ion acoustic bow shock ininteraction of solar wind with cometarycoma
Studies of interaction of the solar wind with cometary
comae do not take into account usually the influence of dust.
Cometary nuclei are found to be, most likely, dominated byrefractories, very porous and fragile [10]. Under the action of
the solar light volatile components of the cometary nucleus
evaporate and vapor stream entrains dust particles.
The dust/gas ratio is introduced to characterize the ratioof masses of refractories to volatiles in the cometary nuclei.
In quantitative terms, an average value of dust/gas ratio
exceeds unity [11]. The dust/gas ratio obtained from the coma

4. Dust ion acoustic nonlinear structuresin active and laboratory experiments,
which use generator of high-speed
plasma jets
(1) The idea of the formation of shocks related to dust
charging in active rocket experiments, which involve the2422

Popel S.I., Dust Acoustic and Dust Ion Acoustic Nonlinear Structures: Theory and Experiments

release of some gaseous substance in near-Earth space, wasforwarded in [18]. The source for the charged particle releasein the ionosphere in these experiments is the generator ofhigh-speed plasma jets. The shock wave front is associatedwith the fore (border)-part of the jet propagating in the plasmaof the ionosphere. Nano- and micro- (dust) particles appear
as a result of condensation. Drops are charged due to theirinteraction with the ambient plasma and the photoelectriceffect. The optimum speeds of the jet are 10 km/s. The
optimum altitudes for such experiments are 500–600 km.
(2) The laboratory experiments modeling the active rocket
experiments have been carried out at the Institute forDynamics of Geospheres RAS. The main results are thefollowing: clusters consisting of 10–1,000 atoms appear at
the initial stage of jet expansion, significant part of substanceis condensed (approximately 50 %); appearance of anomalous
ionisation in the region in front of the propagating jet, that ischaracterized by irradiance in the environment of the jet;conservation of high ionisation degree for significant
expansion of the jet; significant radiation from the region of
the interaction of the jet and the ambient medium (the energyof the radiation reaches more than 60 % of the initial energyof the jet).

(1) The dust grain kinetic temperature and the defectfraction have a jump at the shock front, there is the density
compression; There is the mass transfer via the shock front.
The mass transfer is described well by the first Hugoniot
relationship n 2 (V – v 2 ) = n 1 V, where V is the shockpropagation velocity, m is the particle mass, n is the twodimensional particle number density, v is the component of
the particle velocity normal to the shock front, the indices 1
and 2 denote the condition ahead and behind the shock
respectively;(2) The width of the shock front is determined by several
mean free paths of dust grains in the region of the shock front.
This is related to the fact that in this region the grains are in
the “collisional regime”.
The observed shock is related to the melting of theplasma crystal. The solid-to-liquid (and possibly gaseous)phase transition for a plasma crystal appears to develop via
the growth of crystal defects, directed particle flows aroundcrystalline islands (floes), enhanced vibrations associated withgreater order and flow suspension, to a completely disorderedstate. The mechanism of the formation of the disordered stateis the increase in the dust grain temperature (up to 300 eV)
behind the shock front. For such a temperature the Coulomb
coupling parameter becomes of order unity. This seems
neither a straightforward first-order nor a classical secondorder transition.

5. Melting dust acoustic shock in a 2D
complex plasma
The experiments on shocks in a laboratory twodimensional complex plasma which suggest the melting
transition from “flat” plasma crystals to liquid (and possiblygaseous) phase were performed [7] in a setup using acapacitively coupled rf discharge. The discharge chamber had
a lower disk electrode and an upper ring electrode. The upper
electrode and the chamber were grounded. A rf power of
10 W was applied to the lower electrode. The working argon
gas pressure was 1.8 Pa in the chamber. Monodisperse plastic
microspheres of approximately 8.9 µm size were levitated in
the sheath above the lower electrode forming a monolayer
hexagonal lattice. They were confined radially in a bowl
shaped potential formed by a rim on the outer edge of theelectrode. A horizontal tungsten wire 0.1 mm in diameter wasplaced 4 mm below the particle layer and roughly half way
between the center and the edge of the electrode. The wire
was normally grounded so that it had little influence on theparticles. A short negative pulse (–100 V, 50 ms applied to
the wire at 0.07 s) pushed the particles away breaking thelattice above the wire and creating a pulsed one-dimensional
compressional disturbance propagating horizontallyperpendicular to the wire. The lattice also oscillated in thevertical direction with a small amplitude which caused aperiodic change of brightness of the particles. The disturbance
propagated to the other end of the crystal and melted it fromthe excitation edge to about the middle of the field of view.
A time interval of about 1 min allowed the lattice to come toan equilibrium between the experimental runs andrecrystallise.
The disturbance can be treated as a shock wave because:

6. SummaryThus, an anomalous dissipation originating from thecharging processes results in a possibility of the existence ofa new kind of shocks. The theoretical ionization source modelallows us to describe all the main results on the dust ion
acoustic shocks obtained in the laboratory experiments. The
dust ion acoustic nonlinear structures are important indifferent real and artificial objects of geophysical and spaceplasmas. In particular, for rather dense dusty coma the bow
shock formed as a result of the interaction of the solar windwith the coma is expected to be related to the anomalousdissipation due to the dust particle charging. Dust acoustic
shock propagation in plasma crystal can lead to the meltingof the latter.
This work was supported by INTAS (grant No. 20010391) and RFBR (grant No. 03-02-16664-a).

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