Stability Analysis of a Compressible Plasma
PARDEEP KUMAR
Department of Mathematics, ICDEOL, Himachal Pradesh University,
Summer-Hill, Shimla-171005 (HP), INDIA
Abstract:- In the presence of a uniform horizontal magnetic field, thermal instability of a compressible
plasma is postulated to occur in the presence of effects of finite Larmor radius (FLR) and Hall
currents. The dispersion relation is found by using the linear stability theory, Boussinesq
approximation and the normal mode analysis approach, respectively. In the scenario of stationary
convection, it was discovered that the compressibility has a stabilizing effect whereas FLR and Hall
currents have stabilizing as well as destabilizing effects. For
, the system is stable. The
magnetic field, FLR and Hall currents introduce oscillatory modes in the system for
. In
addition to this, it has been discovered that the system is reliable for
andunder the
condition
the system goes into an unstable state.
Keywords: - Finite Larmor radius, Hall currents, compressibility, thermal instability, plasma.
Received: October 29, 2022. Revised: September 17, 2023. Accepted: October 19, 2023. Published: November 22, 2023.
1 Introduction
The thermal instability of a fluid layer heated
from below plays an important role in
geophysics, oceanography, atmospheric physics
etc., and has been investigated by many
authors, e.g. Be'nard [1], Rayleigh [2], Jeffreys
[3]. A detailed account of the theoretical and
experimental results of the onset of thermal
instability (Be'nard convection) in a fluid layer
under varying assumptions of hydrodynamics
and hydromagnetics has been given in a treatise
by Chandrasekhar [4]. The use of the
Boussinesq approximation has been made
throughout, which states that the variations of
density in the equations of motion can safely be
ignored everywhere except in its association
with the external force. The approximation is
well justified in the case of incompressible
fluids.
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
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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. 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.
The effects of finiteness of the ion Larmor
radius, showing up in the form of a magnetic
viscosity in the fluid equations, have been
studied by Rosenbluth et al. [5], Roberts and
Taylor [6], Vandakurov [7] and Jukes [8].
Melchior and Popowich [9] have considered the
finite Larmor radius (FLR) effect on the
Kelvin-Helmholtz instability of a fully ionized
plasma, while the effect on the Rayleigh-Taylor
instability has been studied by Singh and Hans
[10]. Sharma [11] has studied the effect of a
finite Larmor radius on the thermal instability
of a plasma.Hernegger [12] 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 [13] investigated the
stabilizing effect of FLR on thermal instability
of rotating plasma. Ariel [14] 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 [15] studied the
stabilizing effect of FLR on magneto-thermal
stability of resistive plasma through a porous
medium with thermal conduction. Bhatia and
Chhonkar [16] 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 [17] 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 [18] 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 [19]. Kumar et al. [20]
considered the Rayleigh-Taylor instability of an
infinitely conducting plasma in porous medium
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taking account the finiteness of ion Larmor
radius (FLR) in the presence of a horizontal
magnetic field. Kumar and Singh [21]
investigated the thermal convection of a plasma
in porous medium to include simultaneously
the effect of rotation and the finiteness of the
ion Larmor radius (FLR) in the presence of a
vertical magnetic field. The effect of finite
Larmor radius of the ions on thermal
convection of a plasma has been studied by
Kumar and Gupta [22]. Kumar [23]
investigated the thermal convection of a plasma
in porous medium in the presence of finite
Larmor radius (FLR) and Hall effects. Thus
FLR effect is an important factor in the
discussion of thermal convection and other
hydromagnetic instabilities.
When the fluids are compressible, the equations
governing the system become quite
complicated. To simplify the set of equations
governing the flow of compressible fluids,
Spiegel and Veronis [24] made the following
assumptions:
(i) The depth of fluid layer is much
smaller than the scale height as
defined by them and
(ii) The fluctuations in temperature,
pressure and density, introduced
due to motion, do not exceed their
static variations.
Under the above assumptions, the flow
equations are the same as those for
incompressible fluids except that the static
temperature gradient is replaced by its excess
over the adiabatic. The thermal instability in
compressible fluids in the presence of rotation
and magnetic field has been considered by
Sharma [25]. Sharma [26] also studied the
thermal instability of a compressible Hall
plasma. Sharma and Sharma [27] considered
the thermal instability of a partially ionized
plasma in the presence of compressibility and
collisional effects while the thermal instability
of a compressible plasma with FLR has been
studied by Sharma et al. [28]. Finite Larmor
radius (FLR) effects are likely to be important
in “weakly” unstable systems such as high beta
stellarator, mirror machines, slowly rotating
plasmas, large aspect tori etc. The Hall effects
are likely to be important in many astrophysical
situations as well as in flows of laboratory
plasma. Sherman and Sutton [29] considered
the effect of Hall currents on the efficiency of a
magneto-fluid dynamic generator while Sato
[30] and Tani [31] studied the incompressible
viscous flow of an ionized gas with tensor
conductivity in channels with consideration of
Hall effect. Sonnerup [32] and Uberoi and
Devanathan [33] investigated the effects of Hall
current on the propagation of small amplitude
waves taking compressibility into account.
Keeping in mind the importance in the physics
of atmosphere and astrophysics especially in
the case of ionosphere and outer layers of the
sun’s atmosphere, the present paper is devoted
to the study of thermal instability of a
compressible plasma under the effects of finite
Larmor radius (FLR) and Hall currents in the
presence of a uniform horizontal magnetic
field.
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2 Formulation of the Problem and
Perturbation Equations
Consider an infinite horizontal layer of
compressible, viscous, heat-conducting and
finite electrically conducting fluid of thickness
in which a uniform temperature gradient
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 fluid
and gravity force .
Following Spiegel and Veronis [24] and
Sharma et al. [28], the linearized
hydromagnetic perturbation equations
appropriate to the problem are
where
and
denote respectively the perturbations in
velocity, magnetic field
, pressure , density
and temperature Here
and stand for stress tensor perturbation,
constant space average of , viscosity,
kinematic viscosity, thermal conductivity,
thermal diffusivity, resistivity, adiabatic
gradient, acceleration due to gravity, coefficient
of thermal expansion, electron number density
and charge of an electron respectively.
For the horizontal magnetic field
, the
stress tensor
, taking into account the finite
ion gyration [Vandakurov [7]], has the
components
Here
where and
denote respectively the number density, the ion
temperature and the ion gyration frequency.
3 Dispersion Relation
We decompose the disturbances into normal
modes and assume that the perturbed quantities
are of the form
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where are the wave numbers along the
- and - directions respectively,
is the resultant wave number and
n is the frequency of oscillation. and stand
for the -components of vorticity and current
density respectively.
Let
and stand for the
coordinates in the new unit of length .
Equations (1) – (6), using expression (7), give
Eliminating and between equations
(8) – (12), we obtain
where
is the
Chandrsekhar number,
is the
Rayleigh number,
is the
non-dimensional number accounting for Hall
currents,
is a non-dimensional
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number accounting for FLR effect and
. Consider the case in which both the
boundaries are free and the medium adjoining
the fluid is non-conducting. The appropriate
boundary conditions for this case are
[Chandrasekhar [4]]
The case of two free boundaries, though little
artificial, is the most appropriate for stellar
atmospheres [Spiegel [34]]. Using the boundary
conditions (14), one can show that all the even
derivatives of must vanish for and
and hence the proper solution of (13)
characterizing the lowest mode is
where is a constant. Substituting (15) in
(13) and letting
and
we
obtain the dispersion relation
Equation (16) is the required dispersion relation
studying the effects of FLR and Hall currents
on thermal instability of a compressible plasma.
In the absence of Hall currents ,
equation (16) reduces to the dispersion relation
(Sharma et al. [28]).
4 Important Theorems and
Discussion
Theorem 1: The system is stable for .
Proof: Multiplying equation (8) by , the
complex conjugate of , integrating over the
range of , and making use of equations (9) –
(12) together with the boundary conditions
(14), we obtain
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where
The integrals are all positive definite.
Putting and equating the real and
imaginary parts of equation (17), we obtain
and
It is evident from equation (19) that is
negative if . The system is therefore
stable for .
Theorem 2: The modes may be oscillatory or
non-oscillatory in contrast to the case of no
magnetic field and in the absence of Hall
currents and finite Larmor radius where modes
are non-oscillatory, for .
Proof: It is clear from equation (20) that, for
, may be zero or non-zero, meaning
that the modes may be oscillatory or non-
oscillatory. The oscillatory modes are
introduced due to the presence of magnetic
field (and hence the presence of Hall currents
and FLR effects).
In the absence of a magnetic field and hence
absence of Hall currents and FLR effects,
equation (20) gives
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and the terms in brackets are positive when
Thus which means that
oscillatory modes are not allowed and the
principle of exchange of stabilities is satisfied,
but in the presence of Hall currents, magnetic
field and finite Larmor radius effects, the
oscillatory modes come into play.
Theorem 3: The system is stable for
and under the condition
, thesystem becomes
unstable.
Proof: From equation (20) it is clear that is
zero when the quantity multiplying it is not
zero and arbitrary when this quantity is zero.
If , equation (19) upon utilizing (20) and
the Rayleigh-Ritz inequality gives
since the minimum value of
with
respect to is
.
Now, let , we necessarily have from
inequality (22) that
Hence, if
then . Therefore, the system is stable.
Thus, under the condition (24), the system is
stable and under condition (23) the system
becomes unstable.
Theorem 4: For stationary convection case:
(I) In the absence of Hall currents,
finite Larmor radius has a
stabilizing effect and in the
presence of FLR and Hall currents,
the finite Larmor radius effect may
be both stabilizing as well as
destabilizing effects on the system.
(II) In the absence of FLR, Hall
currents always has a destabilizing
effect but in the presence of FLR
and Hall effects, the Hall currents
may have both destabilizing as well
as stabilizing effects on the system.
Proof: When the instability sets in as stationary
convection, the marginal state will be
characterized by . Putting, the
dispersion relation (16) reduces to
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which expresses the modified Rayleigh number
as a function of the dimensionless wave
number and the parameters and .
Let
and denote respectively the
critical Rayleigh numbers in the presence and
in the absence of compressibility. For fixed
values of and , let the non-dimensional
number accounting for the compressibility
effects be also kept fixed, then we find that
The effect of compressibility is thus to
postpone the onset of thermal instability.
Hence compressibility has a stabilizing
effect. is relevant here. The cases
and correspond to negative and infinite
values of critical Rayleigh numbers in the
presence of compressibility which are not
relevant in the present study.
To investigate the effects of finite Larmor
radius and Hall currents, we examine the
natures of
analytically.
(I) Equation (25) yields
which is positive if i.e.
the wave number range satisfying
This shows that FLR has a stabilizing effect for
the above wave-number range. In the absence
of Hall currents , FLR always has a
stabilizing effect. But in the presence of FLR
and Hall effects on thermal instability, the FLR
effect may be both stabilizing as well as
destabilizing but completely stabilizes the
above wave-number range.
(II) It is evident from equation (25) that
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which is positive if and
i.e. if
In the absence of FLR, equation (28) yields that
is always negative thus indicating the
destabilizing effect of Hall currents. In the
presence of FLR and Hall effects, the Hall
currents may have both destabilizing as well as
stabilizing effects and there is competition
between the destabilizing role of Hall currents
and stabilizing role of FLR but completely
stabilizes the above wave-number range if
.
5 Conclusions
An attempt has been made to investigate the
effects of compressibility, FLR and Hall
currents on the thermal instability of a plasma
in the presence of a uniform horizontal
magnetic field under the linear stability theory.
It has been shown by Sato [30] and Tani [31]
that inclusion of Hall currents gives rise to a
cross-flow i.e. a flow at right angles to the
primary flow in a channel in the presence of a
transverse magnetic field. Tani [31] found that
Hall effect produces a cross-flow of double-
swirl pattern in incompressible flow through a
straight channel with arbitrary cross-section.
This breakdown of the primary flow and the
formation of a secondary flow may be
attributed to the inherent instability of the
primary flow in the presence of Hall current.
Our stability analysis lends support to this
finding. The investigation of thermal instability
is motivated by its direct relevance to soil
sciences, groundwater hydrology, geophysical,
astrophysical and biometrics. The main
conclusions from the analysis of this paper are
as follows:
The system is found to be stable for
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Volume 2, 2023
The magnetic field, finite Larmor
radius and Hall currents introduce
oscillatory modes in the system for
It is observed that the system is stable
for
and under the
condition
, the
system becomes unstable.
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.
In the absence of FLR, the Hall
currents has a destabilizing effect.
In the presence of FLR and Hall
effects, the Hall currents may have both
destabilizing as well as stabilizing
effects and there is competition
between the destabilizing role of Hall
currents and stabilizing role of FLR but
completely stabilizes the above wave-
number range if
.
Acknowledgements:
The author is grateful to all three learned
referees for their useful technical comments
and valuable suggestions, which led to a
significant improvement of the paper.
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