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Neutral induced low frequency instability in a weakly ionized magnetized plasma
Planet. Space Sci.. Vol. 40, No. 9, pp. 1197~1201, 1992 Printed in Great bitam
0
00324633/92 f5.00c0.00 1992 Per$amon Press Ltd
NEUTRAL INDUCED LOW FREQUENCY INSTABILITY IN A WEAKLY IONIZED MAGNETIZED PLASMA C. B. DWNEDI
and A. C. DAS
Physical Research Laboratory, Navrangpura, Ahmedabad-380 009, India (Received
in final form
9
ApFZ?
1992)
Abstract-Using a three-fluid model, a linear local analysis of neutral induced low frequency (NILF) mode instability in an inhomogeneous weakly ionized plasma in a magnetic field has been carried out. It is concluded that this novel instability, electro-mechanical in nature, is seeded by the neutrals and is driven by electro-mechanical coupling through the frictional force between the plasma and neutral fluids. The condition for the onset of the instability requires a finite neutral diffusion flow. It is proposed that the NILF mode can provide a physical mechanism to understand the nature and cause of the observed irregularities in the D-region of the Earth’s ionosphere mrane, E. V. and Grandal, B. (1981) J. atmos. rerr. P@. 43, 179; Sinha, H. S. S. (1992) J. armos. terr. Whys. 54, 491. However, its general application in any weakly ionized plasma with an ambient magnetic field is not restricted.
1. INTRODUCTION
In partially ionized plasmas, the number density of the neutral gas exceeds that of the plasma and the dynamics of the neutral fluid cannot be ignored. The dynamical behaviour of such systems cannot be fully described by either the equations of a neutral fluid or those of an ionized fluid alone. Instead, a three-fluid model, consisting of electron, ion and neutral Druids provides a more appropriate and consistent approach to anaiyse the electrodynamic properties of weakly ionized plasmas (Rees, 1989 ; Kelley, 1989). In such plasma systems the collision frequency of electrons and ions with neutral atoms greatly exceeds that of collisions of these particles with one another (Golant ef al., 1980) :
veaI3
Veir
vm;
Via >>
mechanism of neutral and ionized fluids. In the presence of a free energy source, the perturbation can grow, modify the dynamical behaviour of the system and generate anomalous heating, transportation of momentum and energy, etc. The study of such perturbations based on three-fluid dynamics can play ai important role in radio wave propagation (Stenflo, 1985 ; Tsintsadze et al., 1990) in the Earth’s ionosphere, wave processes in laboratory plasmas and in pinch machines (Berezin er al., 1991). Recently Rao et al. (1991) have performed a selfconsistent analysis of three-fluid dynamics to investigate the linear and nonlinear behaviour of acoustictype waves in partially ionized plasmas. In their analysis, the lowest order coupling :
a
II
w.
vii, -v Mi
z
‘.
e’
Such plasmas exist in the Earth’s ionosphere, cometary magnetospheres, planetary ring systems and asteroid zones. They can also be produced in the laboratory with a variable degree of ionization. Linear and nonlinear wave propagation in fully ionized plasma systems has been extensively investigated over the last few decades. However, a consistent analysis of waves and instabilities in partially ionized plasmas based on the three-fluid model has not been fully performed. Consequently, it has become an important area for active research and hence. requires adequate attention to understand the behaviour of neutral gas with partial ionization under the action of external forces. The response of small perturbations to such a medium will provide a tool to investigate the coupling
K
%a,vair
gives vi z v, and ni NNqn, where rt % no/naOis the degree of ionization (ratio of the equilibrium number density of the charged particles to that of the total heavy particles including neutral atoms). The above conclusion holds good if the neutral pressure is ignored or neglected in comparison with the resistive term in the neutral momentum equation. However, the pressure term is retained in higher order corrections wherein the inertial terms for ions and neutrals, together with ion and neutral pressure terms, cause the existence of acoustic-type waves. The role of neutral atoms in destabilizing the drift-dissipative instability of the outer plasma in a tokamak has been highlighted by Sunder et al. (1991) in a shear field. Physically, the instability is explained in terms of unbalanced 1197
1198
C.
B. DWIVEDI and A. C. DAS
ionization of the neutrals coming from either wall or limiter and thus causing the destabilization of driftdissipative instability; otherwise, in the absence of neutrals, it is stable. Although the plasma in the tokamak is fully ionized, the importance of neutrals lies in their ability to affect the fluctuations in the outer plasma of the tokamak by coupling with charged particles through the ionization process. Berezin et al. (1991) and Duch and Griem (1966) have carried out computational modelling of 0 pinches based on the three-fluid model to understand the one-dimensional characteristics of 0 pinch dynamics. They have incorporated the inelastic processes (ionization, recombination, excitation, etc.) and charged particle-atom collisions to describe the behaviour of 0 pinch by solving self-consistent three-fluid equations. From the above introductory description, one can realize the importance of the three-fluid model in describing the plasma phenomena on more realistic grounds. Recently, Thrane and Grandal (1981) and Sinha (1992) have reported the existence of irregularities in the D-region of the Earth’s ionosphere. The origin of these irregularities is attributed to neutral turbulence, through the process of vertical displacement of the air parcel. However, the nature and cause of the observed irregularities have not been firmly established (Thrane and Grandal, 1981). Based on the self-consistently coupled dynamics of the plasma and neutral fluids through collisions, a novel mode of instability is proposed to explain the observed irregularities in the D-region of the Earth’s ionosphere. The instability describes a new mode of response which a weakly ionized plasma in a magnetic field can exhibit. Its nature and underlying physics are different to those of pure plasma and neutral fluid mode instabilities. It produces fluctuations in both neutral and ionized fluids through electro-mechanical coupling on the basic principle of dynamo and motor. To understand the equilibrium first, let us assume that the hydrostatic balancing of the neutral fluid is disturbed. Consequently, neutrals start moving under the action of the pressure gradient and gravitational forces. Due to collisional coupling with ions, a momentum exchange occurs between ions and neutrals and the resulting diffusive friction on the neutrals provides equilibrium stability to the neutrals. Momentum transfer to the ions generates an equilibrium ion current which produces a Lorentz force. This, in collaboration with gravitational and ion pressure gradient forces, governs the equilibrium stability of the ion fluid. Now, if this diffusive dynamic equilibrium is disturbed by a small fluctuation in the neutral fluid, an electro-mechanical mode of oscillation is set up. Its frequency is governed by diffusive ion diamagnetic
and gravitational drifts. Its propagation characteristics are controlled by an electro-mechanical coupling phenomenon, wherein the mechanical energy (associated with the neutral fluid) converts into electrical energy (associated with the plasma fluid) and vice versa, to provide the restoring force for the wave oscillation. The wave propagates in the form of compression and rarefaction with coupled fluctuations in the plasma and neutral fluids. The disturbance in the neutral fluid can be due to either an accidental phenomenon of internal origin or to an external one. We find that this mode of wave oscillation grows due to a free source of energy stored in the form of relative kinetic motion between ions and neutrals. Based on the above arguments, two-dimensional linear analysis of NILF mode instability has been carried out. For simplicity, slab geometry has been considered and inhomogeneity in plasma and neutral fluid density has been allowed. Gravitational force has been introduced in the ion and neutral fluid dynamics, whereas it has been neglected for the electrons. To simplify further, nonelastic processes and the role of negative ions have been ignored. However, their consideration will form a part of future work. The role of negative ions becomes important in the altitude range below 80 km in the Earth’s ionosphere. For the frequency range of interest, I(d/dt)( cc Via, vai ions are assumed to be magnetized and collisional. The dynamics of neutrals is governed by pressure gradient and gravitational and frictional forces. Momentum exchange between neutrals and electrons has been neglected due to their large mass difference. Under these simplifying approximations, a linear dispersion relation and growth rate for neutral induced low frequency (NILF) mode instability have been derived and analysed. It is found that the instability occurs when gravitational and neutral density gradient forces produce a finite neutral diffusion flow. It is concluded that the instability is driven by neutrals due to electromechanical coupling of the free source of energy (available in the form of relative kinetic motion between ions and neutrals) with the fluctuation. Section 2 deals with the introduction of basic equations. In Section 3, the dispersion relation is derived and the growth rate is calculated. Results and discussions are given in Section 4.
2. BASIC EQUATIONS
Consider the two-dimensional propagation of the NILF mode through a weakly ionized plasma in an ambient magnetic field B, along the z-direction. The nlasma ran ~~. and neutral densitvd inhomoeeneities lie in the
Neutral
induced
low frequency
1199
instability
x-direction. The unstable wave is allowed to propagate in the x-y plane, with propagation wave vector k, = k,f + k,$. The required basic equations consist of momentum and continuity equations for the plasma and the neutral gas. Within the fluid model, these equations are given as : dv,
where -Lg + v * (n,v,) = 0.
(2)
Here u should be replaced by (e) for electrons, (i) for ions and (a) for neutral atoms. v,~ is the collision frequency between tl and fi particles. g, is the gravitational acceleration acting on species LX.Under the approximations l(a/at)l CCvia, vai and wCc>>v,,, the momentum equations for plasma and neutral fluids are reduced to, e 0 = -v~-C0&xz-cc,z~, m, 0 = -~v~+m&xi-c~~
Parallel motion for electrons, ions and neutrals has been neglected. The second and third terms in equation (7) describe, respectively, the diffusive and usual ion diamagnetic drifts. Using equations (6)-(g) in the continuity equations for electrons, ions and neutrals, one gets :
Vn n, I
a,(r?,/no)-
(3)
C
--&xv,
lnn,*V,f$
-via(Vi-V,)+gi
a,(&/n,)-
C
--fxVI
lnn,~V,J viaC,Z
V,fi,
h
--zxV,Inn,--
nao
vaiaci
where
-V,i(v,-vi)+ga,
(9)
0
BO
(4) 0 = -c;?
=
BO
(5)
v,ri, +n,*
:
g,,x& p-~VI*nn,ox~ (
w- cl
=(J V,iOCi
>
(10)
eB0 o,=n&c
eB0
ao
Oh=-, llliC
(11)
are the electron cyclotron frequency, ion cyclotron frequency, thermal velocities for electrons, ions and neutrals, respectively. ye, yi and y. are the adiabatic indices for electrons, ions and neutral atoms, respectively. Under the drift approximation, equations (3)-(S) are solved to give :
Equations (9)-(11) form a set of linearly coupled equations to solve for the dispersion relation. Electrons and ions are coupled through the quasi-neutrality condition. In equation (lo), the second and third terms arise from Vlno *vi, and the last one from V,& - vi0 convection parts of the ion continuity equation, The fourth. term in equation (11) arises due to compressibility of the neutral fluid and provides proper phase to cause the fluctuation to grow. It is remarkable to note that the interdependence of neu-
1200
C. B. DWIMDI
tral density and electrostatic potential fluctuations is the consequence of electro-mechanical coupling of plasma and neutral fluids. 3. DISPERSION
RELATION
AND GROWTH
RATE
Under the local approximations klL,, k,L, >> 1, one can do the Fourier analysis of equations (9)-( 11) to eliminate the variables and find the local dispersion relation for the NILF mode as :
1 ‘&,’ k, * V, Inn,, x 2 = 0, +C1---vaiwci
(12)
where : 6 = w-k,
&,,xk *--o&i
k1-c vai -
(c:/wci)i
x
VI In no * kl
and : P= (I-k~~,/k~v~,-V,i~~/vi~g~L~).
and
A. C.
DAS
Evidently the two-dimensional effect modifies the results significantly for ,J,/& % Vi,/Oci- From the mathematical expressions (1.5) for the real and imaginary parts of complex o, one can infer the following ; for the wave propagation in the almost y-direction (1, <
Separating out real and imaginary parts of w gives the frequency (w,) and growth rate (y) of the NILF mode instability :
Wnao L, = z F(o), n”lno
___
B
where :
f- k* *g, +(c~/~~)&xV~Inn,~k, v,i
y= --
(13) This is similar to the relation derived by Thrane and Grandal (1981) with the obvious difference of the wave factor F(w), which is absent in their expression.
k2 c2
’ a V&i
where L, = (V, Inn,,)and L, = (V, Inn,)-’ are, respectively, the equilibrium neutral and plasma density inhomogeneity scale lengths. The third term in equation (13) arises due to wave propagation along the density gradient. The entire term inside parentheses in equation (14), hereafter denoted by A, is a measure of the magnitude of the equilib~um kinetic motion of the neutral fluid and drives the instability. The condition for the growth requires A to be negative but very small in order to satisfy the approximation imposed on the time scale of the instability. It is remarkable to note that the instability is driven by the neutral l-mid. Under the approximations L, x L,, via/V.i>> 1, frequency and growth rate are reduced to :
(15)
4. RESULTS
AND DISCUSSIONS
From the analysis, one concludes that the NILF mode instability arises in a weakly ionized plasma, embedded in an ambient magnetic field, with weak density inhomogeneity in plasma and neutral fluids. This novel mode of instability is driven by a combination of compressibility of the neutral fluid and composite effects of gravitational and diffusive ion diamagnetic (produced through the ion-neutral drag effect) drift convection of the ion plasma density perturbation. Compressibility of the neutral fluid provides the required phase difference among the perturbed quantities to cause the fluctuation to grow. For the existence of the instability, a finite gradient in the neutral and plasma fluids along with the gravitational force is needed. However, the underlying physics and the condition for the onset of the instability
1201
Neutral induced low frequency ins~bi~ity (g,,L, > 0) are different from those of the usual plasma and neutral fluid instabilities. The basic physical m~ha~sm for driving the instability is based on the principle of dynamo and motor, wherein the mutual conversion of electro-mechanical energy takes place through the frictional coupling between ion and neutral fluids. An accidental fluctuation in the neutral fluid seeds the instability. To illustrate the underlying physics let us assume that a small density ripple ii, arises in the neutral fhtid along the y-direction. It produces a perturbed ion current density jX through the dynamo effect. Subsequently, an electrostatic potential fluctuation b; is set up along the y-axis due to 3, x B, force. Now, the fluctuating potential generates a density perturbation n’in the ion and electron fiuids. Furthermore, the dynamic effect of the perturbed potential is transmitted to the neutrals through the motor principle and thereby enhances the original ripple in the neutral fluid. Thus, the instability sets in and grows exponentially, provided proper phase is maintained between the perturbed quantities. Temporal variation of the ion density fluctuation is controlled by its spatial convection. If we consider equation (lo), it appears that under the condition g,,L, > 0, gravitational drift causes the ion density perturbation to convect out of the perturbed region. Contrary to this, the diffusive ion diamagnetic drift tends to convect the ion density fluctuation into the perturbed region, Thus, the relative competition between the two necessitates the latter to exceed the former and hence leads to the condition, c2/g8L, > 1 for the onset of the instability. For typical parameters of D-region plasma ; T, 3 Ti x T. = 200 K, n,, = %Q = no Z IO3 cmS3, q(<10-7) X lo-‘, Yia(> IO4 s- ‘) zs 6 x lo4 s- ‘, B, = 0.35 G and the instability condition requires I& < 6 km. Furthermore, the conditions, k,pi C<1, pi being the ion-Larmor radius, k,&,, << 1, k,L,, klL, >> 1 limit the range (12 km >>-/ZL>> 12 m) of the perturbed scale lengths. For &+co, 2,~ 100 m, La% L,z6 km and A sz IO-“, real and imaginary parts of the complex frequency u result; 0sciIlation frequency w, % 1O-^4 s-‘andthegrowthratey~ 10-4s-‘. Finally, the role of NILF instability in ion-neutral heating (through diffusive damping) and in energy and mass t~ns~~ation is conjectured. To provide a more realistic and full description of the entire range
of the D-region of the Earth’s ionosphere, inclusion of inelastic processes like ionization and recombination and of the dynamics of negative ions is essential. However, the present analysis forms a beginning towards it and highlights the important of neutral dynamics in exciting a new kind ofnormal mode, which a weakly ionized plasma in magnetic and gravitational fields can support. It may be argued that the allowance of neutral dynamics eliminates the classical R-T instability and replaces it by NfLF mode instability. The non-existence of R-T instability in lower parts of the Earth’s ionosphere (below the F-region) favours this view. ~c~~~~fe~~~~e~~~-One of the authors (CBD) acknowledges P. K. Kaw, IPR, Bhat, Gandhinagar, India, P. K. Shukla, Ruhr-Universitat, Bochum, Germany and N. N. Rao and H. S. S. Sinha. PRL. A~edabad. India for fruitful discussions and helpful comments during the completion of the work.
REFERENCES Berezin, Yu A., Fedourk, M. P. and Khudik, V. N. (1991) Modeling of magnetic geld compression of a plasmoid with allowance of ionization. Soo. J. Pfusma Phys. 17, 208.
Duch, D. and Griem, H. R. (1966) Computer study of the dynamic vhase of a small 6 ninth. Phvs. FtUids 9. 1099. Go&t, V. b., Zhihnsky, A. P. and Sakharov, I. I?. (1980) Fundamentals oj Pfaska Physics. Wiley, New York.. Kellev, M. C. (1989) The Earth’s IonosDhere. Plasma Physics and ~~e~rro-~~n*~~~~.Academic Press, San Diego, GA. Rao, N. N., Kaup, D. J. and Shukla, P. K. (1991) Nonlinear acoustic waves in partially ionised collisional plasmas. J. Plasma Phys. 45,285.
Rees, M. H. (1989) Physics and C~rn~try of Upper Atmosphere. Cambridge University Press, Cambridge. Sinha, H. S. S. (1992) Plasma density irregularities in the equatorial B-region produced by neutral turbulence. f. atmos. ferr. Phys. %,49.
Stenflo, L. (1985) Parametric excitation of collisional modes in the high latitude ionosphere. J. geophys. Res. 90, 5355. Sunder. D.. Morozov. D. Kh. and Morozov. N. N. 11991) Rde’of neutral particles in dissipative drift instabihty of the outer plasma in a tokamak. Sov. J. Plasma Pfzys. 17, 166.
Thrane, E. V. and Grandat, B. (1981) Observation of fine scale structure in the mesosphere and lower thermosphere. J. atmos. terr. Phys. 43, 179.
Tsintsadze, N. L., Baudze, T. D., Shukla, P. K. and Stenflo, L. (1990) Stimulated scattering of radio waves off acoustic waves in partially ionised plasmas. J. geophys. Res. 95, 2465.
0
00324633/92 f5.00c0.00 1992 Per$amon Press Ltd
NEUTRAL INDUCED LOW FREQUENCY INSTABILITY IN A WEAKLY IONIZED MAGNETIZED PLASMA C. B. DWNEDI
and A. C. DAS
Physical Research Laboratory, Navrangpura, Ahmedabad-380 009, India (Received
in final form
9
ApFZ?
1992)
Abstract-Using a three-fluid model, a linear local analysis of neutral induced low frequency (NILF) mode instability in an inhomogeneous weakly ionized plasma in a magnetic field has been carried out. It is concluded that this novel instability, electro-mechanical in nature, is seeded by the neutrals and is driven by electro-mechanical coupling through the frictional force between the plasma and neutral fluids. The condition for the onset of the instability requires a finite neutral diffusion flow. It is proposed that the NILF mode can provide a physical mechanism to understand the nature and cause of the observed irregularities in the D-region of the Earth’s ionosphere mrane, E. V. and Grandal, B. (1981) J. atmos. rerr. P@. 43, 179; Sinha, H. S. S. (1992) J. armos. terr. Whys. 54, 491. However, its general application in any weakly ionized plasma with an ambient magnetic field is not restricted.
1. INTRODUCTION
In partially ionized plasmas, the number density of the neutral gas exceeds that of the plasma and the dynamics of the neutral fluid cannot be ignored. The dynamical behaviour of such systems cannot be fully described by either the equations of a neutral fluid or those of an ionized fluid alone. Instead, a three-fluid model, consisting of electron, ion and neutral Druids provides a more appropriate and consistent approach to anaiyse the electrodynamic properties of weakly ionized plasmas (Rees, 1989 ; Kelley, 1989). In such plasma systems the collision frequency of electrons and ions with neutral atoms greatly exceeds that of collisions of these particles with one another (Golant ef al., 1980) :
veaI3
Veir
vm;
Via >>
mechanism of neutral and ionized fluids. In the presence of a free energy source, the perturbation can grow, modify the dynamical behaviour of the system and generate anomalous heating, transportation of momentum and energy, etc. The study of such perturbations based on three-fluid dynamics can play ai important role in radio wave propagation (Stenflo, 1985 ; Tsintsadze et al., 1990) in the Earth’s ionosphere, wave processes in laboratory plasmas and in pinch machines (Berezin er al., 1991). Recently Rao et al. (1991) have performed a selfconsistent analysis of three-fluid dynamics to investigate the linear and nonlinear behaviour of acoustictype waves in partially ionized plasmas. In their analysis, the lowest order coupling :
a
II
w.
vii, -v Mi
z
‘.
e’
Such plasmas exist in the Earth’s ionosphere, cometary magnetospheres, planetary ring systems and asteroid zones. They can also be produced in the laboratory with a variable degree of ionization. Linear and nonlinear wave propagation in fully ionized plasma systems has been extensively investigated over the last few decades. However, a consistent analysis of waves and instabilities in partially ionized plasmas based on the three-fluid model has not been fully performed. Consequently, it has become an important area for active research and hence. requires adequate attention to understand the behaviour of neutral gas with partial ionization under the action of external forces. The response of small perturbations to such a medium will provide a tool to investigate the coupling
K
%a,vair
gives vi z v, and ni NNqn, where rt % no/naOis the degree of ionization (ratio of the equilibrium number density of the charged particles to that of the total heavy particles including neutral atoms). The above conclusion holds good if the neutral pressure is ignored or neglected in comparison with the resistive term in the neutral momentum equation. However, the pressure term is retained in higher order corrections wherein the inertial terms for ions and neutrals, together with ion and neutral pressure terms, cause the existence of acoustic-type waves. The role of neutral atoms in destabilizing the drift-dissipative instability of the outer plasma in a tokamak has been highlighted by Sunder et al. (1991) in a shear field. Physically, the instability is explained in terms of unbalanced 1197
1198
C.
B. DWIVEDI and A. C. DAS
ionization of the neutrals coming from either wall or limiter and thus causing the destabilization of driftdissipative instability; otherwise, in the absence of neutrals, it is stable. Although the plasma in the tokamak is fully ionized, the importance of neutrals lies in their ability to affect the fluctuations in the outer plasma of the tokamak by coupling with charged particles through the ionization process. Berezin et al. (1991) and Duch and Griem (1966) have carried out computational modelling of 0 pinches based on the three-fluid model to understand the one-dimensional characteristics of 0 pinch dynamics. They have incorporated the inelastic processes (ionization, recombination, excitation, etc.) and charged particle-atom collisions to describe the behaviour of 0 pinch by solving self-consistent three-fluid equations. From the above introductory description, one can realize the importance of the three-fluid model in describing the plasma phenomena on more realistic grounds. Recently, Thrane and Grandal (1981) and Sinha (1992) have reported the existence of irregularities in the D-region of the Earth’s ionosphere. The origin of these irregularities is attributed to neutral turbulence, through the process of vertical displacement of the air parcel. However, the nature and cause of the observed irregularities have not been firmly established (Thrane and Grandal, 1981). Based on the self-consistently coupled dynamics of the plasma and neutral fluids through collisions, a novel mode of instability is proposed to explain the observed irregularities in the D-region of the Earth’s ionosphere. The instability describes a new mode of response which a weakly ionized plasma in a magnetic field can exhibit. Its nature and underlying physics are different to those of pure plasma and neutral fluid mode instabilities. It produces fluctuations in both neutral and ionized fluids through electro-mechanical coupling on the basic principle of dynamo and motor. To understand the equilibrium first, let us assume that the hydrostatic balancing of the neutral fluid is disturbed. Consequently, neutrals start moving under the action of the pressure gradient and gravitational forces. Due to collisional coupling with ions, a momentum exchange occurs between ions and neutrals and the resulting diffusive friction on the neutrals provides equilibrium stability to the neutrals. Momentum transfer to the ions generates an equilibrium ion current which produces a Lorentz force. This, in collaboration with gravitational and ion pressure gradient forces, governs the equilibrium stability of the ion fluid. Now, if this diffusive dynamic equilibrium is disturbed by a small fluctuation in the neutral fluid, an electro-mechanical mode of oscillation is set up. Its frequency is governed by diffusive ion diamagnetic
and gravitational drifts. Its propagation characteristics are controlled by an electro-mechanical coupling phenomenon, wherein the mechanical energy (associated with the neutral fluid) converts into electrical energy (associated with the plasma fluid) and vice versa, to provide the restoring force for the wave oscillation. The wave propagates in the form of compression and rarefaction with coupled fluctuations in the plasma and neutral fluids. The disturbance in the neutral fluid can be due to either an accidental phenomenon of internal origin or to an external one. We find that this mode of wave oscillation grows due to a free source of energy stored in the form of relative kinetic motion between ions and neutrals. Based on the above arguments, two-dimensional linear analysis of NILF mode instability has been carried out. For simplicity, slab geometry has been considered and inhomogeneity in plasma and neutral fluid density has been allowed. Gravitational force has been introduced in the ion and neutral fluid dynamics, whereas it has been neglected for the electrons. To simplify further, nonelastic processes and the role of negative ions have been ignored. However, their consideration will form a part of future work. The role of negative ions becomes important in the altitude range below 80 km in the Earth’s ionosphere. For the frequency range of interest, I(d/dt)( cc Via, vai ions are assumed to be magnetized and collisional. The dynamics of neutrals is governed by pressure gradient and gravitational and frictional forces. Momentum exchange between neutrals and electrons has been neglected due to their large mass difference. Under these simplifying approximations, a linear dispersion relation and growth rate for neutral induced low frequency (NILF) mode instability have been derived and analysed. It is found that the instability occurs when gravitational and neutral density gradient forces produce a finite neutral diffusion flow. It is concluded that the instability is driven by neutrals due to electromechanical coupling of the free source of energy (available in the form of relative kinetic motion between ions and neutrals) with the fluctuation. Section 2 deals with the introduction of basic equations. In Section 3, the dispersion relation is derived and the growth rate is calculated. Results and discussions are given in Section 4.
2. BASIC EQUATIONS
Consider the two-dimensional propagation of the NILF mode through a weakly ionized plasma in an ambient magnetic field B, along the z-direction. The nlasma ran ~~. and neutral densitvd inhomoeeneities lie in the
Neutral
induced
low frequency
1199
instability
x-direction. The unstable wave is allowed to propagate in the x-y plane, with propagation wave vector k, = k,f + k,$. The required basic equations consist of momentum and continuity equations for the plasma and the neutral gas. Within the fluid model, these equations are given as : dv,
where -Lg + v * (n,v,) = 0.
(2)
Here u should be replaced by (e) for electrons, (i) for ions and (a) for neutral atoms. v,~ is the collision frequency between tl and fi particles. g, is the gravitational acceleration acting on species LX.Under the approximations l(a/at)l CCvia, vai and wCc>>v,,, the momentum equations for plasma and neutral fluids are reduced to, e 0 = -v~-C0&xz-cc,z~, m, 0 = -~v~+m&xi-c~~
Parallel motion for electrons, ions and neutrals has been neglected. The second and third terms in equation (7) describe, respectively, the diffusive and usual ion diamagnetic drifts. Using equations (6)-(g) in the continuity equations for electrons, ions and neutrals, one gets :
Vn n, I
a,(r?,/no)-
(3)
C
--&xv,
lnn,*V,f$
-via(Vi-V,)+gi
a,(&/n,)-
C
--fxVI
lnn,~V,J viaC,Z
V,fi,
h
--zxV,Inn,--
nao
vaiaci
where
-V,i(v,-vi)+ga,
(9)
0
BO
(4) 0 = -c;?
=
BO
(5)
v,ri, +n,*
:
g,,x& p-~VI*nn,ox~ (
w- cl
=(J V,iOCi
>
(10)
eB0 o,=n&c
eB0
ao
Oh=-, llliC
(11)
are the electron cyclotron frequency, ion cyclotron frequency, thermal velocities for electrons, ions and neutrals, respectively. ye, yi and y. are the adiabatic indices for electrons, ions and neutral atoms, respectively. Under the drift approximation, equations (3)-(S) are solved to give :
Equations (9)-(11) form a set of linearly coupled equations to solve for the dispersion relation. Electrons and ions are coupled through the quasi-neutrality condition. In equation (lo), the second and third terms arise from Vlno *vi, and the last one from V,& - vi0 convection parts of the ion continuity equation, The fourth. term in equation (11) arises due to compressibility of the neutral fluid and provides proper phase to cause the fluctuation to grow. It is remarkable to note that the interdependence of neu-
1200
C. B. DWIMDI
tral density and electrostatic potential fluctuations is the consequence of electro-mechanical coupling of plasma and neutral fluids. 3. DISPERSION
RELATION
AND GROWTH
RATE
Under the local approximations klL,, k,L, >> 1, one can do the Fourier analysis of equations (9)-( 11) to eliminate the variables and find the local dispersion relation for the NILF mode as :
1 ‘&,’ k, * V, Inn,, x 2 = 0, +C1---vaiwci
(12)
where : 6 = w-k,
&,,xk *--o&i
k1-c vai -
(c:/wci)i
x
VI In no * kl
and : P= (I-k~~,/k~v~,-V,i~~/vi~g~L~).
and
A. C.
DAS
Evidently the two-dimensional effect modifies the results significantly for ,J,/& % Vi,/Oci- From the mathematical expressions (1.5) for the real and imaginary parts of complex o, one can infer the following ; for the wave propagation in the almost y-direction (1, <
Separating out real and imaginary parts of w gives the frequency (w,) and growth rate (y) of the NILF mode instability :
Wnao L, = z F(o), n”lno
___
B
where :
f- k* *g, +(c~/~~)&xV~Inn,~k, v,i
y= --
(13) This is similar to the relation derived by Thrane and Grandal (1981) with the obvious difference of the wave factor F(w), which is absent in their expression.
k2 c2
’ a V&i
where L, = (V, Inn,,)and L, = (V, Inn,)-’ are, respectively, the equilibrium neutral and plasma density inhomogeneity scale lengths. The third term in equation (13) arises due to wave propagation along the density gradient. The entire term inside parentheses in equation (14), hereafter denoted by A, is a measure of the magnitude of the equilib~um kinetic motion of the neutral fluid and drives the instability. The condition for the growth requires A to be negative but very small in order to satisfy the approximation imposed on the time scale of the instability. It is remarkable to note that the instability is driven by the neutral l-mid. Under the approximations L, x L,, via/V.i>> 1, frequency and growth rate are reduced to :
(15)
4. RESULTS
AND DISCUSSIONS
From the analysis, one concludes that the NILF mode instability arises in a weakly ionized plasma, embedded in an ambient magnetic field, with weak density inhomogeneity in plasma and neutral fluids. This novel mode of instability is driven by a combination of compressibility of the neutral fluid and composite effects of gravitational and diffusive ion diamagnetic (produced through the ion-neutral drag effect) drift convection of the ion plasma density perturbation. Compressibility of the neutral fluid provides the required phase difference among the perturbed quantities to cause the fluctuation to grow. For the existence of the instability, a finite gradient in the neutral and plasma fluids along with the gravitational force is needed. However, the underlying physics and the condition for the onset of the instability
1201
Neutral induced low frequency ins~bi~ity (g,,L, > 0) are different from those of the usual plasma and neutral fluid instabilities. The basic physical m~ha~sm for driving the instability is based on the principle of dynamo and motor, wherein the mutual conversion of electro-mechanical energy takes place through the frictional coupling between ion and neutral fluids. An accidental fluctuation in the neutral fluid seeds the instability. To illustrate the underlying physics let us assume that a small density ripple ii, arises in the neutral fhtid along the y-direction. It produces a perturbed ion current density jX through the dynamo effect. Subsequently, an electrostatic potential fluctuation b; is set up along the y-axis due to 3, x B, force. Now, the fluctuating potential generates a density perturbation n’in the ion and electron fiuids. Furthermore, the dynamic effect of the perturbed potential is transmitted to the neutrals through the motor principle and thereby enhances the original ripple in the neutral fluid. Thus, the instability sets in and grows exponentially, provided proper phase is maintained between the perturbed quantities. Temporal variation of the ion density fluctuation is controlled by its spatial convection. If we consider equation (lo), it appears that under the condition g,,L, > 0, gravitational drift causes the ion density perturbation to convect out of the perturbed region. Contrary to this, the diffusive ion diamagnetic drift tends to convect the ion density fluctuation into the perturbed region, Thus, the relative competition between the two necessitates the latter to exceed the former and hence leads to the condition, c2/g8L, > 1 for the onset of the instability. For typical parameters of D-region plasma ; T, 3 Ti x T. = 200 K, n,, = %Q = no Z IO3 cmS3, q(<10-7) X lo-‘, Yia(> IO4 s- ‘) zs 6 x lo4 s- ‘, B, = 0.35 G and the instability condition requires I& < 6 km. Furthermore, the conditions, k,pi C<1, pi being the ion-Larmor radius, k,&,, << 1, k,L,, klL, >> 1 limit the range (12 km >>-/ZL>> 12 m) of the perturbed scale lengths. For &+co, 2,~ 100 m, La% L,z6 km and A sz IO-“, real and imaginary parts of the complex frequency u result; 0sciIlation frequency w, % 1O-^4 s-‘andthegrowthratey~ 10-4s-‘. Finally, the role of NILF instability in ion-neutral heating (through diffusive damping) and in energy and mass t~ns~~ation is conjectured. To provide a more realistic and full description of the entire range
of the D-region of the Earth’s ionosphere, inclusion of inelastic processes like ionization and recombination and of the dynamics of negative ions is essential. However, the present analysis forms a beginning towards it and highlights the important of neutral dynamics in exciting a new kind ofnormal mode, which a weakly ionized plasma in magnetic and gravitational fields can support. It may be argued that the allowance of neutral dynamics eliminates the classical R-T instability and replaces it by NfLF mode instability. The non-existence of R-T instability in lower parts of the Earth’s ionosphere (below the F-region) favours this view. ~c~~~~fe~~~~e~~~-One of the authors (CBD) acknowledges P. K. Kaw, IPR, Bhat, Gandhinagar, India, P. K. Shukla, Ruhr-Universitat, Bochum, Germany and N. N. Rao and H. S. S. Sinha. PRL. A~edabad. India for fruitful discussions and helpful comments during the completion of the work.
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