Analysis of Stability of a Plasma in Porous Medium
PARDEEP KUMAR
Department of Mathematics, ICDEOL, Himachal Pradesh University, Summerhill,
Shimla-5, INDIA
Abstract: The thermal convection of a plasma in porous medium is investigated in the presence of
finite Larmor radius (FLR) and Hall effects. Following linear stability theory and normal mode
analysis method, the dispersion relation is obtained. It is found that the presence of a magnetic field
(and hence the presence of FLR and Hall effects) introduces oscillatory modes in the system which
were, otherwise, non-existent in their absence. When the instability sets in as stationary convection,
the FLR may have a stabilizing or destabilizing effect, but a completely stabilizing one for a certain
wave-number range. Similarly, the Hall currents may have a stabilizing or destabilizing effect but a
completely stabilizing one for the same wave-number range under certain condition, whereas the
medium permeability always has a destabilizing effect for stationary convection. Also it is found that
the system is stable for
and under the condition
, the system becomes unstable.
Key-words: Finite Larmor Radius Effect, Hall Effects, Plasma, Porous Medium, Thermal Convection
Received: May 25, 2021. Revised: November 22, 2021. Accepted: December 18, 2021. Published: January 5, 2022.
1 Introduction
The theoretical and experimental results on
thermal convection in a fluid layer, under
varying assumptions of hydrodynamics and
hydromagnetics, have been discussed in a treatise
by Chandrasekhar [1]. The effects of the
finiteness of the ion Larmor radius which
exhibits itself in the form of ‘magnetic viscosity’
in the fluid equations have been studied by many
authors (Jukes [2]; Vandakurov [3]). Sharma and
Prakash [4] have studied the effect of finite
Larmor radius on the thermal instability of a
plasma. Melchior and Popowich [5] have
considered the finite Larmor radius effect on the
Kelvin-Helmholtz instability in a fully ionized
plasma while that on Rayleigh-Taylor instability
has been studied by Singh and Hans [6]). The
effect of finite Larmor radius on the thermal
instability of a plasma in the presence of a
vertical magnetic field has been studied by
Sharma [7].
The study of the breakdown of the stability of a
layer of fluid subject to a vertical temperature
gradient in porous medium and the possibility of
convective flow is of considerable interest in
recent years. The study of onset of convection in
a porous medium has attracted considerable
interest because of its natural occurrence and of
its intrinsic importance in many industrial
problems, particularly in petroleum-exploration,
chemical and nuclear industries. The derivation
of the basic equations of a layer of fluid heated
from below in porous medium, using Boussinesq
approximation, has been given by Joseph [8].
The study of a layer of fluid heated from below
in porous media is motivated both theoretically
and by its practical applications in engineering
disciplines. Among the applications in
engineering disciplines one can find the food
process industry, chemical process industry,
solidification and centrifugal casting of metals.
The development of geothermal power resources
has increased general interest in the properties of
convection in porous medium. Generally it is
accepted that comets consists of a dusty
‘snowball’ of a mixture of frozen gases which in
the process of their journey changes from solid to
gas and vice-versa. The physical properties of
comets, meteorites and interplanetary dust
strongly suggest the importance of porosity in the
astrophysical context (McDonnel [9]). The effect
of a magnetic field on the stability of such a flow
is of interest in geophysics, particularly in the
study of the Earth’s core where the Earth’s
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mantle, which consists of a conducting fluid,
behaves like a porous medium which can become
convectively unstable as a result of differential
diffusion. The other application of the results of
flow through a porous medium in the presence of
a magnetic field is in the study of the stability of
a convective flow in the geothermal region.
Lapwood [10] has studied the stability of
convective flow in a porous medium using
Rayleigh’s procedure. The Rayleigh instability of
a thermal boundary layer in flow through a
porous medium has been considered by Wooding
[11].
The properties of ionized space and laboratory
magnetic fluids (plasmas) have been intensively
investigated theoretically and experimentally in
the past sixty years. One of the key aspects
studied in this context is the stability of plasma
structures. Usually, instabilities can be divided
into two categories: macro- and micro-
instabilities. Macro-instabilities occur with low
frequencies compared to the plasma and
cyclotron frequency and they are studied within
the framework of magnetohydrodynamics
(MHD). Physicists have understood the
behaviour of macro-instabilities and they showed
how to avoid the most destructive of them, but
small-scale gradient driven micro-instabilities are
still a serious obstacle for having a stable plasma
for a large range of parameters. Micro-
instabilities are described by models which
include, e.g. finite Larmor radius (FLR) and
collision less dissipative effects in plasmas. Time
and length scales of micro-instabilities are
comparable to the turbulent length scales and the
length scales of transport coefficients. In general,
the FLR effect is neglected. However, when the
Larmor radius becomes comparable to the
hydromagnetic length of the problem (e.g.
wavelength) or the gyration frequency of ions in
the magnetic field is of the same order as the
wave frequency, finiteness of the Larmor radius
must be taken into account. Strictly speaking, the
space and time scale for the breakdown of
hydromagnetics are on the respective scales of
ion gyration about the field, and the ion Larmor
frequency. In the present paper, we explore the
effect of FLR and Hall effects on the thermal
instability of a plasma in porous medium. Finite
Larmor radius effect on plasma instabilities has
been the subject of many investigations. In many
astrophysical plasma situations such as in solar
corona, interstellar and interplanetary plasmas
the assumption of zero Larmor radius is not
valid. Roberts and Taylor [12] and Rosenbluth et
al. [13] have shown the stabilizing influence of
finite ion Larmor radius (FLR) effects on plasma
instabilities. Hernegger [14] investigated the
stabilizing effect of FLR on thermal instability
and showed that thermal criterion is changed by
FLR for wave propagation perpendicular to the
magnetic field. Sharma [15] investigated the
stabilizing effect of FLR on thermal instability of
rotating plasma. Ariel [16] discussed the
stabilizing effect of FLR on thermal instability of
conducting plasma layer of finite thickness
surrounded by a non-conducting matter. Vaghela
and Chhajlani [17] studied the stabilizing effect
of FLR on magneto-thermal stability of resistive
plasma through a porous medium with thermal
conduction. Bhatia and Chhonkar [18]
investigated the stabilizing effect of FLR on the
instability of a rotating layer of self-gravitating
plasma incorporating the effects of viscosity and
Hall current. Vyas and Chhajlani [19] pointed
out the stabilizing effect of FLR on the thermal
instability of magnetized rotating plasma
incorporating the effects of viscosity, finite
electrical conductivity, porosity and thermal
conductivity. Kaothekar and Chhajlani [20]
investigated the problem of Jeans instability of
self-gravitating rotating radiative plasma with
finite Larmor radius corrections. The frictional
effect of collisions of ionized with neutral atoms
on Rayleigh-Taylor instability of a composite
plasma in porous medium has been considered
by Kumar and Mohan [21]. Thus FLR effect is
an important factor in the discussion of thermal
convection and other hydromagnetic instabilities.
Keeping these in mind, an attempt is made to
study the effects of finite Larmor radius and Hall
effects on the thermal convection of
incompressible plasma in porous medium in the
present paper.
2 Formulation of the Problem and
Perturbation Equations
Here we consider an infinite horizontal layer of
viscous, heat conducting and finite electrically
conducting fluid of thickness bounded by the
planes and in an isotropic and
homogeneous medium of porosity and medium
permeability This layer is heated from below
such that a steady temperature gradient
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is maintained. Consider the cartesian
coordinates with origin on the lower
boundary and the -axis perpendicular to it
along the vertical. The fluid is acted on by a
horizontal magnetic field
and gravity
force .
Let
and denote
respectively the perturbations in fluid (filter)
velocity , magnetic field
, density and
temperature . Then the linearized
hydromagnetic perturbation equations relevant to
the problem are
where
Here
and stand for stress tensor taking
into account FLR effect, density at reference
level , viscosity, kinematic viscosity,
thermal conductivity, thermal diffusivity,
electrical resistivity, coefficient of volume
expansion, electron number density and charge
of an electron respectively. denote the
medium porosity, medium permeability while
stand for density and specific heat
of fluid and solid (porous matrix) material
respectively. In writing (1), use has been made of
the equation of state
where the suffix zero refers to values at the
reference level and so the change in
density caused by the perturbation in
temperature is given by
For the horizontal magnetic field
, the
stress tensor
, taking into account the finite ion
gyration (Vandakurov [3]), has the components
Here
, where and denote
respectively the number density, the ion
temperature and the ion gyration frequency.
3 Dispersion Relation
Analyzing the disturbances into normal modes,
we assume that the perturbation quantities are of
the form
where are the wave numbers in the
and directions respectively,
is the resultant wave number and
is, in general, a complex constant.
Expressing the coordinates in the new unit
of length and letting
and
; Equations (1) – (5)
using (8) and expression (9), give
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Eliminating and between equations
(10) - (14), we obtain
where
is the Rayleigh number,
is the modified Chandrasekhar number,
is a non-dimensional number
accounting for Hall current,
is a non-
dimensional number accounting for FLR effects
and
.
Here we consider the case of plasma layer with
two free boundaries and the adjoining medium to
be nonconducting. The boundaries are assumed
to be perfect conductors of heat. The case of two
free boundaries is slightly artificial, except in
stellar atmospheres (Spiegel [22]) and in certain
geophysical situations where it is most
appropriate. However, the case of two free
boundaries allows us to obtain analytical solution
without affecting the essential features of the
problem. The appropriate boundary conditions
for the present problem are
at and 1
and are continuous.
(16)
Using the boundary conditions (16), it can be
shown with the help of equations (10) – (14) that
all the even order derivatives of vanish for
and and hence the proper solution of
(15) characterizing the lowest mode is
,
(17)
where is a constant. Substituting (17) in (15)
and letting
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we obtain the dispersion relation
4 Important Theorems and
Discussion
Theorem 1: The system is stable or unstable.
Proof: Multiplying equation (10) by , the
complex conjugate of , integrating over the
range of and using equations (11) – (14)
together with boundary conditions (16), we
obtain
where
which are all positive definite. Putting
where are real and equating the real
and imaginary parts of equation (19), we obtain
and
It is evident from equation (21) that may be
positive or negative i.e. there may be instability
or stability in the system through porous medium
in the presence of finite Larmor radius, Hall
effects and magnetic field.
Theorem 2: The modes may be oscillatory or
non-oscillatory.
Proof: Equation (22) yields that may be zero
or non-zero, which means that the modes may be
non-oscillatory or oscillatory. In the absence of
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magnetic field (and hence absence of FLR and
Hall effects), equation (22) reduces to
and the terms in brackets are positive definite.
Thus which means that oscillatory modes
are not allowed and the principle of exchange of
stabilities is satisfied. The oscillatory modes are
introduced due to the presence of magnetic field
(and hence the presence of FLR and Hall effects)
which were non-existent in their absence.
Theorem 3: The system is stable for
and under the condition
, the system
becomes unstable.
Proof: From equation (22), it is clear that is
zero when the quantity multiplying it is not zero
and arbitrary when this quantity is zero.
If , then equation (22) gives
Substituting this in equation (21), we get
Equation (24) on using Rayleigh-Ritz inequality
gives
Therefore, it follows from relation (25) that
since minimum value of
with respect to
is
.
Now, let , we necessary have from (26)
that
Hence, if
then Therefore, the system is stable.
Thus, under condition (28), the system is stable
and under condition (27) the system becomes
unstable.
Theorem 4: For stationary convection
(I) FLR may have a stabilizing or
destabilizing effect, but a
completely stabilizing one for a
certain wave-number range
(II) In the absence of Hall currents, FLR
always has a stabilizing effect.
(III) The Hall currents may have a
stabilizing or destabilizing effect
but completely stabilizes the wave-
number range
if
.
(IV) The medium permeability always
has a destabilizing effect on the
system for stationary convection.
Proof: When the instability sets in as stationary
convection, the marginal state will be
characterized by Putting the
dispersion relation (18) reduces to
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which expresses the modified Rayleigh number
as a function of the dimensionless wave
number and the parameters and .
To study the effects of finite Larmor radius, Hall
currents and medium permeability, we examine
the nature of
analytically.
(I) It follows from equation (29),
which is positive for i.e. for
the wave number range satisfying
This shows that FLR has a stabilizing effect for
the wave number range (31). Thus, in the
presence of Hall effects on the thermal
instability in porous medium, the FLR may
have stabilizing or destabilizing effects but
completely stabilizes the wave number range
(31).
(II) In the absence of Hall currents
, equation (30) reduces to
which is always positive, hence in the absence
of Hall currents, FLR always has a stabilizing
effect.
(III) Equation (29) also yields
which is positive if and
i.e. if
In the presence of FLR and Hall effects on the
thermal instability in porous medium, the Hall
currents may have stabilizing or destabilizing
effect but completely stabilizes the wave
number range (31) if
Hence the result.
(IV) It is evident from equation (29) that
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which means that medium permeability has a
destabilizing effect.
5 Conclusions
The thermal convection of a plasma in a porous
medium in the presence of a finite Larmor radius
(FLR) and Hall effects is considered in the
present paper. The main conclusions from the
analysis of this paper are as follows:
It is found that magnetic field (and hence
the presence of FLR and Hall effects)
introduce oscillatory modes in the
system which were non-existent in their
absence.
It is observed that system is stable for
and under the condition
, the system becomes
unstable.
For the case of stationary convection:
FLR may have a stabilizing or
destabilizing effect, but a
completely stabilizing one for a
certain wave-number range
In the absence of Hall currents,
FLR always has a stabilizing
effect.
The Hall currents may have a
stabilizing or destabilizing effect
but completely stabilizes the
wave-number range
if
.
The medium permeability
always has a destabilizing effect
on the system for stationary
convection.
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[14] Hernegger, F., Effect of collisions and
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