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Current driven ion wave instability in a weakly ionized collisional magnetoplasma
Volume 44A, number 6
PHYSICS LETTERS
16 July 1973
CURRENT DRIVEN ION WAVE INSTABILITY IN A WEAKLY IONIZED COLLISIONAL MAGNETOPLASMA P. KAW Physical Research Laboratory, Ahmedabad 9, India Received 19 April 1973 Ion waves propagating nearly across the magnetic field in a weakly ionized plasma with cross-field electron streaming are shown to be unstable due to dissipative effects.
In this letter we examine the stability of oblique ion-acoustic waves in a weakly ionized collisional plasma with electron streaming across the magnetic field. We find that dissipative effects in the parallel equation of motion for electrons drive the ion waves unstable. Similar instabilities in unmagnetized fully ionized gases arising through other dissipative effects like electron thermal conductivity, electron viscosity etc., have recently been investigated [1—5]. We choose a coordinate system with the magnetic field in the z-direction and an equilibrium electron drift in the x-direction. The parameter range of interest to us is ~e ~ ~e and 1j ~2~v and ~2refer respec~-
tively to the collision and cyclotron frequencies (the dominant collisions are electron—neutral and ion— neutral). Under these conditions, the electron drift in the x-direction could be generated by a d.c. electric field in the y.direction. Such a situation can be produced in the laboratory and also occurs naturally in the equatorial and auroral electrojet regions of the ionosphere. We shall use fluid equations to treat the parallel motion of electrons and a kinetic equation with BGK type collision term for ions. This is justified since Pe > (w kv0) and (w/v~)is arbitrary. Consider an ion wave perturbation propagating obliquely to the magnetic field. Assume k~is large enough so that the parallel phase velocity of the wave is less than electron thermal speed, i.e. (w/kzve) ~ 1. In this case electrons move freely along the lines of force and relationship. try to establish a Boltzmann densitypotential This is modified type by electron— neutral collisions which make the density-potential relationship complex. Neglecting electron inertia, the parallel equation of motion for electrons is given by
e ~ 0
—
—
m az
~ T
~e
mn
—
0
az
—
p
(1)
e
where denotes linear perturbations and ‘~e= 1 or 3 depending on whether the electrons are isothermal or adiabatic (this is a simplistic way of accounting for the electron energy equation; later we point out the modifications introduced by an actual use of the energy equation). Using the usual exp [i(k.r Wi)] dependence for the perturbations and combining (1) with the electron continuity equation, one obtains ‘—‘
—
~
n~/~
.
2
4/[mve(w
—
.
ku
2
= 0ik~e 0) + i/c5 7e~]. (2) In deriving eq. (2) the contribution of the perpendicular motion of electrons to Vv has been ignored since it is down by (ve/cze). This is in contrast to the analysis of field-aligned (k5 = 0) resistive instabilities in weakly ionized plasmas in which the perpendicular motion of electrons is crucial [6—8]. Since i’~~ ~ the ions are essentially unrnagnetized and one can use the linearized kinetic equation for unmagnetized ions viz. e afoj + uVf1 = Lf1 h— ‘0~ (3)
aT
—
—~
—
—
where the right side is a number and momentum conserving BGK type collision term. Assuming exp[i(kr— wi’)] dependence for the perturbation, one can evaluatef~which on integration over velocity space yields the ion density perturbation il~viz. 3u -i—i ~ k• (~f~/~~) IIi f~d = d3v n M (w+iv~_k.u) L ‘ w+iv.—kv —
°
—
—f
.
—
i~.f
.
—
1
(4) Using the quasi-neutrality approximation ~e 427
Volume 44A, number 6
PHYSICS LETTERS
valid for low-frequency, long wavelength modes, one obtains the dispersion relation
I
+
w+iv kv 1
~kv~ [~ iv. ~ /w+iv
‘y~,= I because electron temperature perturbations are immediately smoothed out. Three important points of difference between the
earlier on fully ionizedare unmagnetized and thework present calculation noteworthy: plasmas
+~
--—a z
16 July 1973
(i) The growth rates here are larger by (k2/k~) k•Vo)Pe + ik~~yT/rnj= 0(5) 2/m)l!2 denotes the plasma frequenwhere = (4~ii~>e cy (the subscripts e and i having obvious meanings) and Z(/3) is the plasma dispersion function tabulated by Fried and Conte 19] In the fluid limit for ions when lw+iv~/kv~ one can use the asymptotic form of the Z-function +
iw
(k~/k~)[(w
191 and approximate eq. (5) as —-
- -
where s = [(‘YeTe+ 3T
2 is the sound speed and 1)/M}V ~i + (~/2)~1~ (w4/lk Ik2v?) exp[— w2/2k2v~i 111
cludes ion Landau damping effects. For simplicity, treating the imaginary terms as a perturbation, one obtains w + i’y, where ks 7=
---
--
(m/~ (k2/k~)~e [1
equation of motion for electrons to instability in (ii) Electron -neutral collision lead effects in the parallel contrast to electron—ion collisions for fully ionized gases. This is because normally electron—neutral collisions give a damping of the ion acoustic wave where as electron—ion collisions do not (in the latter, there is only an exchange of momentum between electrons and ions and the wave remains undamped).
w(w + i0~) k2s2 + i(m/1P1~I(k2/k~)v~(w k• u~)= o ~))
=
which could he a sizable i~ctor.
neutrals providing a huge sink of energy. Thus the time-dependence of frequency which plagues the instability in fully ionized gases [3—5] does not bother us here.
(7a)
Finally, we might mention that the results obtained here may have some relevance to the recent observation
(7b)
of non-field-aligned back-scatter echoes in radio aurora
(v 0/s)I
(iii) In our calculations, electrons will acquire a steady state temperature Te in an energy relaxation time (mv~/A~~ after which it will stay constant, the
[10,11]. We note that the second term in the expression for ‘y is destabilizing if v0 > s. Thus electron streaming across the magnetic field leads to instabilities when v~exceeds s. This is a conclusion in the ion-fluid limit which might be modified if the exact dispersion relation (5) is 1111merically solved. If T~ Ti, ion Landau damping can be very large; so the instability will most likely occur only when T~~‘ 7. When finite electron thermal conductivity along the lines of force is taken into account, the result is to modify eq. (7b) to the form
7
=
1 ~ ~i
~fl
~
/c2 ~ + n0T1 ~e k~ ~~i]
( I -- s~-)
(7c)
where x11 is the coefficient of parallel electron thermal conductivity, i.e. electron thermal conductivity adds a
comparable destabilizing term. Furthermore, for large electron thermal conduction along the lines of force.
428
References III B. Coppi and E. Mazzucato, Phys. Fluids 14(1971)134. 121 T.D. Rognlien and S.A. Self. Phys. Rev. Letters 27(1971) 792. 131 P.K. Kaw and A.K. Sundarani. Phys. Lett. 38A (1972) 355.
141 AK. Sundarani and P.K. Kaw, Phys. Fluids, published.
[s1G. Bateman, Phys.
to be
Rev. Lcli. 29(1972)1499.
[61 0. Buneman, Phys. Rev. Lctt. 10(1963)285. 71 D.T. Farley, J. Geophy. Res. 58 (1963) 6083. 181 A. Rogister and ND. Angelo, J. Geophy. Res. 75 (1970)
3879. 191 B.D. Fried and S. Contc, The plasma dispersion function. (New York, Academic Press, 1961). [101 J. Hofstee and PA. Forsyth, J. Atmos. Terres. Phys. 34(1972) 893.
[Ill DR. Mcorcroft, J. Geophy. Res. 77 (1972) 765.
PHYSICS LETTERS
16 July 1973
CURRENT DRIVEN ION WAVE INSTABILITY IN A WEAKLY IONIZED COLLISIONAL MAGNETOPLASMA P. KAW Physical Research Laboratory, Ahmedabad 9, India Received 19 April 1973 Ion waves propagating nearly across the magnetic field in a weakly ionized plasma with cross-field electron streaming are shown to be unstable due to dissipative effects.
In this letter we examine the stability of oblique ion-acoustic waves in a weakly ionized collisional plasma with electron streaming across the magnetic field. We find that dissipative effects in the parallel equation of motion for electrons drive the ion waves unstable. Similar instabilities in unmagnetized fully ionized gases arising through other dissipative effects like electron thermal conductivity, electron viscosity etc., have recently been investigated [1—5]. We choose a coordinate system with the magnetic field in the z-direction and an equilibrium electron drift in the x-direction. The parameter range of interest to us is ~e ~ ~e and 1j ~2~v and ~2refer respec~-
tively to the collision and cyclotron frequencies (the dominant collisions are electron—neutral and ion— neutral). Under these conditions, the electron drift in the x-direction could be generated by a d.c. electric field in the y.direction. Such a situation can be produced in the laboratory and also occurs naturally in the equatorial and auroral electrojet regions of the ionosphere. We shall use fluid equations to treat the parallel motion of electrons and a kinetic equation with BGK type collision term for ions. This is justified since Pe > (w kv0) and (w/v~)is arbitrary. Consider an ion wave perturbation propagating obliquely to the magnetic field. Assume k~is large enough so that the parallel phase velocity of the wave is less than electron thermal speed, i.e. (w/kzve) ~ 1. In this case electrons move freely along the lines of force and relationship. try to establish a Boltzmann densitypotential This is modified type by electron— neutral collisions which make the density-potential relationship complex. Neglecting electron inertia, the parallel equation of motion for electrons is given by
e ~ 0
—
—
m az
~ T
~e
mn
—
0
az
—
p
(1)
e
where denotes linear perturbations and ‘~e= 1 or 3 depending on whether the electrons are isothermal or adiabatic (this is a simplistic way of accounting for the electron energy equation; later we point out the modifications introduced by an actual use of the energy equation). Using the usual exp [i(k.r Wi)] dependence for the perturbations and combining (1) with the electron continuity equation, one obtains ‘—‘
—
~
n~/~
.
2
4/[mve(w
—
.
ku
2
= 0ik~e 0) + i/c5 7e~]. (2) In deriving eq. (2) the contribution of the perpendicular motion of electrons to Vv has been ignored since it is down by (ve/cze). This is in contrast to the analysis of field-aligned (k5 = 0) resistive instabilities in weakly ionized plasmas in which the perpendicular motion of electrons is crucial [6—8]. Since i’~~ ~ the ions are essentially unrnagnetized and one can use the linearized kinetic equation for unmagnetized ions viz. e afoj + uVf1 = Lf1 h— ‘0~ (3)
aT
—
—~
—
—
where the right side is a number and momentum conserving BGK type collision term. Assuming exp[i(kr— wi’)] dependence for the perturbation, one can evaluatef~which on integration over velocity space yields the ion density perturbation il~viz. 3u -i—i ~ k• (~f~/~~) IIi f~d = d3v n M (w+iv~_k.u) L ‘ w+iv.—kv —
°
—
—f
.
—
i~.f
.
—
1
(4) Using the quasi-neutrality approximation ~e 427
Volume 44A, number 6
PHYSICS LETTERS
valid for low-frequency, long wavelength modes, one obtains the dispersion relation
I
+
w+iv kv 1
~kv~ [~ iv. ~ /w+iv
‘y~,= I because electron temperature perturbations are immediately smoothed out. Three important points of difference between the
earlier on fully ionizedare unmagnetized and thework present calculation noteworthy: plasmas
+~
--—a z
16 July 1973
(i) The growth rates here are larger by (k2/k~) k•Vo)Pe + ik~~yT/rnj= 0(5) 2/m)l!2 denotes the plasma frequenwhere = (4~ii~>e cy (the subscripts e and i having obvious meanings) and Z(/3) is the plasma dispersion function tabulated by Fried and Conte 19] In the fluid limit for ions when lw+iv~/kv~ one can use the asymptotic form of the Z-function +
iw
(k~/k~)[(w
191 and approximate eq. (5) as —-
- -
where s = [(‘YeTe+ 3T
2 is the sound speed and 1)/M}V ~i + (~/2)~1~ (w4/lk Ik2v?) exp[— w2/2k2v~i 111
cludes ion Landau damping effects. For simplicity, treating the imaginary terms as a perturbation, one obtains w + i’y, where ks 7=
---
--
(m/~ (k2/k~)~e [1
equation of motion for electrons to instability in (ii) Electron -neutral collision lead effects in the parallel contrast to electron—ion collisions for fully ionized gases. This is because normally electron—neutral collisions give a damping of the ion acoustic wave where as electron—ion collisions do not (in the latter, there is only an exchange of momentum between electrons and ions and the wave remains undamped).
w(w + i0~) k2s2 + i(m/1P1~I(k2/k~)v~(w k• u~)= o ~))
=
which could he a sizable i~ctor.
neutrals providing a huge sink of energy. Thus the time-dependence of frequency which plagues the instability in fully ionized gases [3—5] does not bother us here.
(7a)
Finally, we might mention that the results obtained here may have some relevance to the recent observation
(7b)
of non-field-aligned back-scatter echoes in radio aurora
(v 0/s)I
(iii) In our calculations, electrons will acquire a steady state temperature Te in an energy relaxation time (mv~/A~~ after which it will stay constant, the
[10,11]. We note that the second term in the expression for ‘y is destabilizing if v0 > s. Thus electron streaming across the magnetic field leads to instabilities when v~exceeds s. This is a conclusion in the ion-fluid limit which might be modified if the exact dispersion relation (5) is 1111merically solved. If T~ Ti, ion Landau damping can be very large; so the instability will most likely occur only when T~~‘ 7. When finite electron thermal conductivity along the lines of force is taken into account, the result is to modify eq. (7b) to the form
7
=
1 ~ ~i
~fl
~
/c2 ~ + n0T1 ~e k~ ~~i]
( I -- s~-)
(7c)
where x11 is the coefficient of parallel electron thermal conductivity, i.e. electron thermal conductivity adds a
comparable destabilizing term. Furthermore, for large electron thermal conduction along the lines of force.
428
References III B. Coppi and E. Mazzucato, Phys. Fluids 14(1971)134. 121 T.D. Rognlien and S.A. Self. Phys. Rev. Letters 27(1971) 792. 131 P.K. Kaw and A.K. Sundarani. Phys. Lett. 38A (1972) 355.
141 AK. Sundarani and P.K. Kaw, Phys. Fluids, published.
[s1G. Bateman, Phys.
to be
Rev. Lcli. 29(1972)1499.
[61 0. Buneman, Phys. Rev. Lctt. 10(1963)285. 71 D.T. Farley, J. Geophy. Res. 58 (1963) 6083. 181 A. Rogister and ND. Angelo, J. Geophy. Res. 75 (1970)
3879. 191 B.D. Fried and S. Contc, The plasma dispersion function. (New York, Academic Press, 1961). [101 J. Hofstee and PA. Forsyth, J. Atmos. Terres. Phys. 34(1972) 893.
[Ill DR. Mcorcroft, J. Geophy. Res. 77 (1972) 765.