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WAVES AND INSTABILITIES IN
DUSTY PLASMAS
SUMMARY
SUBMITTED TO
M.J.P. ROHILKHAND UNIVERSITY, BAREILLY
FOR THE AWARD OF THE DEGREE OF
DOCTOR OF PHILOSOPHY
IN
MATHEMATICS
2007
SUPERVISOR
Research Scholar:
Dr. NAGENDRA KUMAR
VINOD KUMAR
Reader
Deptt. of Maths
K.G.K. (P.G.) College,
Moradabad -244001
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Summary
A dusty plasma is a plasma which consists of ions, electrons, and a significant density of dust grains. The grains suffer collisions
with electrons and protons and acquire electric charge in the process. The electrostatic potential due to the charged grains affects the
electron and proton density distributions which in turn modify the electron and proton fluxes impinging upon the grains. These dust
grains themselves can be anything from simple ice particulates, such as in cometary tails and in various space and astrophysical
plasmas, to almost anything capable of sustaining an electric charge. The study of dusty plasmas is now an active area of research in
the plasma physics community. The recent interest in dusty plasmas is due to their occurrence in the laboratory, in space and
astrophysics, and in industrial processes that involve plasmas. One of the most striking features, explained by the physics of dusty
plasmas are the spokes encountered in the B-ring of Saturn. These spokes are explained by dust grains that are levitated from large
boulders in the ring plane.
The subject of wave propagation and instabilities is important for the development of dusty plasma dynamics.
Wave
propagation provides a link between theory and experiment in the subject and allows us to probe the dusty plasma. The wave
behaviour of dusty plasmas differs from the behaviour of usual plasmas. Characteristic frequencies of the dust components are much
smaller than those corresponding with electrons or ions, and therefore the most interesting dusty plasma effects occur for low
frequencies. The mass of the dust grains is much higher than the mass of the plasma particles, and therefore in some cases the gravity
forces come into play. This is the case in the generalized Jeans instability. The number of free electrons is less than the number of
ions, because some of the electrons are captured by the grains, and are therefore immobilized by the high dust masses. The charge of
the dust grain depends on the local plasma conditions (temperature and plasma density), which will vary with the waves coming by
and therefore the dust grain charge has to be taken into account as an extra independent variable. In most space applications the grain
size is not fixed, but one encounters a power law for the grain size distribution. This induces a whole continuous range of different
charge over mass ratios, whereas these ratios are fixed for usual plasmas. Some basic properties of various waves in dusty plasmas
have been obtained from multifluid analyses (e.g. Shukla 1992; Shukla and Silin, 1992; D'Angelo 1990, 1994), which treat the dust
grains as a component of a three-component plasma comprising electrons, ions, and (negatively or positively) charged dust of uniform
mass and charge (and therefore of uniform size).
The presence of charged dust grains in many space environments such as planetary rings, comets can modify the behaviour
of standard instabilities in the ion frequency range, and leads to the appearance of new instabilities in lower frequency ranges
associated with dust grain dynamics. Since dust grains are subject to non-electromagnetic forces such as gravity, friction, or radiation
pressure, there can be new sources of free energy to drive instabilities, including relative drifts between the charged dust and the
lighter plasma particles (electrons and ions) in cosmic dusty plasma environments (e.g. Tsytovich et al. 1990). Several hydrodynamic
and kinetic instabilities in dusty plasmas have been investigated by several researchers ( see, e.g. D' Angelo, 1993; D' Angelo and
Song, 1990; Birk, 2002
Birk and Wiechen 2002)
The brief outlines of the research work presented in the thesis chapter wise are as fallows:
Chapter 2 deals with the study of fluid dynamical instabilities in magnetized partially ionized dense dusty plasma. We have
considered a magnetized partially ionized dense dusty plasma with dynamics governed by dust and neutral gas components. The
dynamics of dusty plasma are characterized by collective behaviour for the parameter regime of ordering a <<d<<λD, where a, d and
λD are the dust grain radius, the average intergrain distance and plasma Debye length. The electron dynamics is not considered as
electrons have no significant influence on the overall behaviour of dusty plasma. The plasma is considered quasi neutral. The
homogeneous magnetic field B0 is taken along the z-axis and the homogenous gravitational field is taken as g = -gey . We have
considered interface along the z-axis i.e. along the magnetic field and relative flow between dust and neutral gas. We derived
dispersion relations for two cases of flow. In first case, we have considered the relative flow between dust and neutral gas. To
discuss the instability of the system, we followed the Hurwitz criterion and constructed the test sequence. If there is no relative
velocity, our condition for instability is quite similar to the condition obtained by Birk (2002) except a slight error in one term. It is
also interesting to note that stabilization occurs for modes with Kz = 0 only due to dust neutral gas collisions. In second case, we have
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considered the equal initial inhomogeneous flow of dust and neutral components parallel to homogeneous magnetic field. We have
derived the instability condition using the same method as in first. Our result might be useful in many situations of astrophysical
magnetized dusty plasma namely comets and circumsteller dusty disk e.g. T-Tauri stars.
In chapter 3 "Ion acoustic waves in self gravitating dusty plasma" we have studied ion acoustic waves for two cases. In first
case, we have studied the propagation and damping of low frequency ion acoustic waves in magnetized self gravitating dusty plasma
taking into account the creation damping alone while in second case we have considered the creation damping together with Tromso
damping. In first case, self-gravitating contribution of electron is not considered. The electron inertia is neglected because we have
considered the effects of self gravitation in the low frequency range. We have obtained a dispersion relation, which has been solved
numerically. The figure shows that ratio k i/kr decreases sharply as angular frequency increases and becomes constant for higher values
of angular frequency. It is seen from the figure that ki/kr in case of creation and Tromso damping is much larger than that in case of
creation damping alone. It is found that ki is independent of frequency in case of creation damping . It can be seen from the figure
that as frequency increases an additional contribution to damping appears due to Tromso effect. The results of our investigations
should be useful in understanding the stability of dusty proto-stars and dusty dark molecular clouds.
Chapter 4 is devoted to the study of "Kelvin-Helmholtz instability in dusty plasma with sheared magnetic field. We
considered a multicomponent plasma consisting of electrons, ions and dust particles in a plane slab geometry where the equilibrium
density and magnetic shear vary in the x-direction . The density distribution is taken to be of the type n +0(x) = n+0eλx with n+0 =
constant. The sheared magnetic field is taken as B o(x) = (0, Boy, Boz(x)) . The electrostatic potential and velocity of positive ions
depend on the x, the ion equilibrium velocity vector is given by V+0 = (o, v+oy, v+oz(x)) with v+oy = constant. Potential gradient ∂ψ0/∂x
is also taken to be constant. Dispersion relation is obtained and discussed numerically. The instability has been discussed by
computing the growth rate numerically for different set of values of various parameters. System becomes unstable with increasing
value of shear parameter up to Zε = 0.8. It is also interesting to note that growth rate is always higher for larger values of shear
parameter. Our results might be applicable in the various situations occurring in astrophysical dusty plasmas such as cometary tails
and dark molecular clouds.
Chapter 5 is concerned with the study of K-H instability in a collisional dusty plasma. We considered dusty plasma
consisting of positive ions, electrons, dust grains and neutral gas molecules. The neutral gas is taken to be at rest. The density
distribution of the unperturbed state of ion is taken to be of the type n +0(x) = n+oe-λx where n+0 = constants. The ions flow along the
magnetic field B with a velocity that depends on x- coordinate. The ion zero order velocity vector V+0 is taken as (0, 0, v+ oz). We did
not consider the collisions of dust grains with neutral gas molecules because dust grains are so heavy that they do not participate in
plasma motion. The equilibrium electric field Eo is taken along the z-axis. We derived a third order dispersion relation which has been
solved numerically. The variation of growth rate as function of various parameters has been shown graphically. The results of our
problem might be useful in assessing the stability conditions of various regimes in the astrophysical and space plasmas.
References
Birk, G.T., 2002, “The onset of Rayleigh-Taylor instability in magnetized partially ionized dense dusty plasmas", Phys. Plasmas, 9,
745.
Birk, G.T. and Wiechen, H. 2002, “Shear flow instablities in magnetized partially ionized dense dusty plasmas", Phys. Plasmas, 9,
964.
D' Angelo, N. 1990, “Low frequency electrostatic waves in dusty plasmas", Planet. Space Sci., 38, 1143.
D' Angelo, N., 1993, “The Rayleigh-Taylor instabilities in dusty plasmas", Planet. Space Sci., 41, 469.
D' Angelo, N., 1994, “Ion-acoustic waves in dusty plasmas", Planetary Space Sc., 42, 507.
D' Angelo, N. and Song, B., 1990, “The Kelvin-Helmholtz instability in a dusty plasmas", Planet. Space. Sci., 38, 1577.
Shukla, P.K. 1992, “Low frequency modes in dusty plasmas", Physica Scripta , 45, 504.
Shukla, P.K. and Silin, V.P., 1992, “Dust ion-acoustic wave", Phys. Scripta, 45, 508.
Tsytovich, V.N., Morfill, G.E., Bingham, R. and de Angelis, U., 1990, Dusty Plasmas (Capri workshop, May 1989), Comm. Plasma
Phys. Contr. Fusion, 13, 153.
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