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Electron-temperature-gradient driven dust-acoustic waves in collisional dusty magnetoplasmas
17 February 1997 PHYSICS LETTERS A
ELSEVIER
Physics Letters A 226 (1997) 196-198
Electron-temperature-gradient driven dust-acoustic waves in collisional dusty magnetoplasmas PK. Shukla a, R. Bingham b, J.T. Mendoqa ‘, D.G. Resendes c a Institutfiir Theoretische Physik II! Ruhr-Universitlit Bochum, D-44780 Bochum, Germany b Rutherford Appleton Laboratory, Chilton, Didcot, Oxfordshire. OX11 OQX, UK c Physics Department, Institute Superior TPcnico, 1096 Lisbon Codex, Portugal
Received 29 November 1996; accepted for publication 10 December 1996 Communicated by V.M. Agranovich
Abstract It is shown that dust-acoustic and dust ion-acoustic waves can become unstable in the presence of an equilibriumelectrontemperature-gradient (ETG) in a collisional dusty magnetoplasma.The growth rate of the ETG-driven dust-acoustic modes is obtained. It is found that nonthermal dust-acoustic fluctuations can cause cross-field anomalous electron diffusion and electron thermal transport in dusty plasmas. The relevance of our investigation to laboratory plasmas is pointed out. PACS: 52.35.F~; 52.35.M~; 52.35.Ra
About five years ago, Rao, Shukla and Yu [ l] demonstrated the existence of extremely low phase velocity (in comparison with the ion thermal velocity) dust-acoustic (DA) waves in an unmagnetized dusty plasma. In DA waves, the tension comes from Boltzmann distributed electron and ion fluids, whereas the dust particle mass provides the inertia. Thus, the DA waves differ significantly from the dust ion-acoustic (DIA) waves [ 21 which involve inertial ion and dust fluids as well as Boltzmann electrons. Both the DA and DIA waves have been observed in laboratory plasmas [3-51 In this paper, we investigate the instability of the DA and DIA waves in a nonuniform collisional dusty plasma which is embedded in an external magnetic field. It is found that free energy stored in an equilibrium electron temperature gradient can be coupled to DA and DIA waves. The electron-temperaturegradient driven DA and DIA wave turbulence can
cause cross-field anomalous particle and heat trans-
ports. Let us consider the propagation of the DA and DIA modes in a nonuniform dusty plasma embedded in an external magnetic field BoZ, where Bo is the strength of the magnetic field and L is a unit vector along the z axis. The equilibrium electron temperature gradient d,Td is along the x axis, whereas equilibrium density profile is assumed to be flat. The dust grain size and the inter-grain spacing are assumed to be much smaller than the characteristic scale lengths (viz. the effective Debye length, the gyroradii etc.). At equilibrium, we have nio = nd( 1 + P), where P = Z~ondo/ns, njo is the unperturbed number density of the particle species j (equals e for electrons, i for ions and d for negatively charged dust grains) and ZdOis the unperturbed number of charges residing on the dust grains. In the electric field E (= -V& where 4 is the electrostatic potential) of low-frequency (in compar-
0375-9601/97/$17.00 Copyright @ 1997 Elsevier Science B.V. All rights reserved. P/I SO375-9601(96)00937-l
PK.
Shukla et al./Physics
ison with the electron collision frequency, Y, which is much smaller than the electron gyrofrequency, uce) DA and DIA modes, the electron number density perturbation for k:uL > WV,, where kz is the component of the wavevector k along the z axis, ute is the electron thermal velocity, and w is the wave frequency, reads
Letters A 226 (1997)
(6) where Ti is the ion temperature. Eqs. (l)-(6) are closed with the help of Poisson’s equation V24 =4re(&l
-hl
E 5?
so
_ *.715!,
(1)
Tea
where Tel < Tea is the perturbation in the equilibrium electron temperature Tea and e is the magnitude of the electron charge. The electron energy equation, which determines Tel, is of the form
where Xell = 3.16u~,v, represents the coefficient of the electron thermal conductivity and c is the speed of light. Here, we have assumed that the electron thermal conduction along the magnetic field lines is much faster than that in a direction transverse to the z axis. From (1) and (2), we have = -(l.l4a,+vr*
x.,,a$
.V)$,
d&r
=
Zoioe
v24,
(4)
where the wave frequency (phase velocity) is assumed to be much larger than the dust gyrofrequency (dust thermal velocity). The ion mass is denoted by md. The ion number density perturbation associated with long-wavelength (in comparison with the ion gyroradius, pi) DIA waves is given by
(7)
a2 + k2c2 1 _ 1.7li we - ~w w,(l+b) > = (;s+b) 2 where
I$
= k . VT*, wx
(8)
= 3.16k;u;/v,,
Wf
=
(nio/n~)k:c~,c~ = T./mi,b = (nio/n~)k~c~/w~i, and ci = Z~ondoTe/n~md. Furthermore, kl is the component of the wavevector k across the z axis. On the other hand, the dispersion relation for the DA waves is found to be 1 - 1.71ia
where VT* = (cT,o/eBo) 2 x V In Td represents the electron diamagnetic drift velocity. The dynamics of the dust grains is governed by
+ &end1 -nit).
We now derive the linear dispersion relation for the DIA and DA waves in the presence of equilibrium electron temperature gradient. For this purpose, we assume that njt ,# and Tel areproportional toexp(ik.riwt) and Fourier transform (3)) (5) and (7) assuming that the wavelength (2r/lkl) is much smaller than the scale length of the equilibrium electron-temperaturegradient. For DIA waves, we have
(3) e0
197
196-198
w;-
$0 wx
=- GM Cl? ’
(9)
where u = nsTi/nioTd, w k~ = k2~C&,,/m,jnio( 1+ (T), and we have assumed that k2hi << 1, Here, ho is the effective Debye length of the dusty plasma [ 61. Letting w = o, + iy and assuming that y << w,, we obtain from (8) the real and imaginary parts of the DIA wave frequency, respectively, wr = OS/( 1 + b) 1’2,
( 10)
and y = 1.71v,(w;
- 1.42w,)/k:u:,.
(11)
On the other hand, (9) gives for the DA waves
where we have assumed that the aligned wave phase velocity is much ion thermal velocity. On the other wavelength (in comparison with pi) have
magnetic fieldlarger than the hand, for short DA waves, we
WI = Zdo~~do/~io>“2~~/~d~“2/~
1 + (7)
112 _ = WDA, (12)
and (13)
198
PK. Shukla et al./Physics
Eqs. ( 11) and ( 13) show that both the DIA and DA waves can become unstable provided that w; is much larger than w,,/( 1 + b) ‘I2 and WDA,respectively. In the following, we demonstrate that nonthermal ETG dust modes can cause cross-field electron diffusion as well as electron energy transport. The electron flux is r, = (ne*ve,) + C.C.,
(14)
where n,t is given by ( 1) and ~~1 = i( c/&)k x 2 is the E x & velocity of the electrons. Substituting for 7”t from (3) into ( 1) and inserting the resulting expression for nei into ( 14), we have
rex
X-
C&O
c ekye14k12
T,oBo
k2v4 z te
(w*
- 1.14wr).
(15)
The electron energy flux is found to be I’,,Td/ 1.7 1~0. We can invoke the mixing length hypothesis to estimate the fluctuation spectrum 1c&k12. Hence, equating the electron excursion length during the E x Bo drift with half of the wavelength, we have
(16) where Wr* = cT~k,d,T~/eBoT~
and LT = Td/
dxT&.
In summary, we have investigated the linear instability of DA and DIA waves in the presence of an equilibrium electron-temperature-gradient in a dissipative dusty plasma embedded in an external magnetic field. It is found that when the dust drift wave frequency Wr+ is larger than the frequencies of the DA and DIA
Letters A 226 (1997) 196-198
waves, the latter are driven at a nonthermal level on account of free energy stored in the equilibrium electron temperature inhomogeneity. Nonthermal DA and DIA waves are found to cause electron diffusion as well as electron thermal conduction across the external magnetic field lines. Thus, our investigation offers a possible mechanism for the excitation of DIA and DA fluctuations and associated cross-field anomalous electron and heat transports in nonuniform dusty magnetoplasmas such as those in radio-frequency discharges as well as in space plasmas. This work was partially supported by the Commission of the European Union (Brussels) through the network “Colloidal Plasmas” of the Human Capital and Mobility program under contract no. CHRXCT94-0602, as well as by the Deutsche Forschungsgemeinschaft (Bonn) through the Sonderforschungsbereich 19 1.
References [ I ] N.N. Rao, PK. Shukla and M.Y. Yu, Planet. Space Sci. 38
(1990) 543. [2] PK. Shukla and V.P Silin, Phys. Ser. 45 (1992) 508. [ 31 J.H. Chu, J.B. Du and Lin I., J. Phys. D 27 ( 1994) 296. A. Barkan, R.L. Merlin0 and N. D’Angelo, Phys. Plasmas 2 (1995) 3563. R.L. Merlino, A. Barkan and N. D’Angelo, in: AIP Conf. Proc. 345: Int. Conf. on Plasma physics, ICPP 1994, eds. PH. Sakanaka and M. Tendler (AIP Press, New York, 1995) pp. 295-302. PK. Shukla, Phys. Plasmas 1 ( 1994) 1362.
ELSEVIER
Physics Letters A 226 (1997) 196-198
Electron-temperature-gradient driven dust-acoustic waves in collisional dusty magnetoplasmas PK. Shukla a, R. Bingham b, J.T. Mendoqa ‘, D.G. Resendes c a Institutfiir Theoretische Physik II! Ruhr-Universitlit Bochum, D-44780 Bochum, Germany b Rutherford Appleton Laboratory, Chilton, Didcot, Oxfordshire. OX11 OQX, UK c Physics Department, Institute Superior TPcnico, 1096 Lisbon Codex, Portugal
Received 29 November 1996; accepted for publication 10 December 1996 Communicated by V.M. Agranovich
Abstract It is shown that dust-acoustic and dust ion-acoustic waves can become unstable in the presence of an equilibriumelectrontemperature-gradient (ETG) in a collisional dusty magnetoplasma.The growth rate of the ETG-driven dust-acoustic modes is obtained. It is found that nonthermal dust-acoustic fluctuations can cause cross-field anomalous electron diffusion and electron thermal transport in dusty plasmas. The relevance of our investigation to laboratory plasmas is pointed out. PACS: 52.35.F~; 52.35.M~; 52.35.Ra
About five years ago, Rao, Shukla and Yu [ l] demonstrated the existence of extremely low phase velocity (in comparison with the ion thermal velocity) dust-acoustic (DA) waves in an unmagnetized dusty plasma. In DA waves, the tension comes from Boltzmann distributed electron and ion fluids, whereas the dust particle mass provides the inertia. Thus, the DA waves differ significantly from the dust ion-acoustic (DIA) waves [ 21 which involve inertial ion and dust fluids as well as Boltzmann electrons. Both the DA and DIA waves have been observed in laboratory plasmas [3-51 In this paper, we investigate the instability of the DA and DIA waves in a nonuniform collisional dusty plasma which is embedded in an external magnetic field. It is found that free energy stored in an equilibrium electron temperature gradient can be coupled to DA and DIA waves. The electron-temperaturegradient driven DA and DIA wave turbulence can
cause cross-field anomalous particle and heat trans-
ports. Let us consider the propagation of the DA and DIA modes in a nonuniform dusty plasma embedded in an external magnetic field BoZ, where Bo is the strength of the magnetic field and L is a unit vector along the z axis. The equilibrium electron temperature gradient d,Td is along the x axis, whereas equilibrium density profile is assumed to be flat. The dust grain size and the inter-grain spacing are assumed to be much smaller than the characteristic scale lengths (viz. the effective Debye length, the gyroradii etc.). At equilibrium, we have nio = nd( 1 + P), where P = Z~ondo/ns, njo is the unperturbed number density of the particle species j (equals e for electrons, i for ions and d for negatively charged dust grains) and ZdOis the unperturbed number of charges residing on the dust grains. In the electric field E (= -V& where 4 is the electrostatic potential) of low-frequency (in compar-
0375-9601/97/$17.00 Copyright @ 1997 Elsevier Science B.V. All rights reserved. P/I SO375-9601(96)00937-l
PK.
Shukla et al./Physics
ison with the electron collision frequency, Y, which is much smaller than the electron gyrofrequency, uce) DA and DIA modes, the electron number density perturbation for k:uL > WV,, where kz is the component of the wavevector k along the z axis, ute is the electron thermal velocity, and w is the wave frequency, reads
Letters A 226 (1997)
(6) where Ti is the ion temperature. Eqs. (l)-(6) are closed with the help of Poisson’s equation V24 =4re(&l
-hl
E 5?
so
_ *.715!,
(1)
Tea
where Tel < Tea is the perturbation in the equilibrium electron temperature Tea and e is the magnitude of the electron charge. The electron energy equation, which determines Tel, is of the form
where Xell = 3.16u~,v, represents the coefficient of the electron thermal conductivity and c is the speed of light. Here, we have assumed that the electron thermal conduction along the magnetic field lines is much faster than that in a direction transverse to the z axis. From (1) and (2), we have = -(l.l4a,+vr*
x.,,a$
.V)$,
d&r
=
Zoioe
v24,
(4)
where the wave frequency (phase velocity) is assumed to be much larger than the dust gyrofrequency (dust thermal velocity). The ion mass is denoted by md. The ion number density perturbation associated with long-wavelength (in comparison with the ion gyroradius, pi) DIA waves is given by
(7)
a2 + k2c2 1 _ 1.7li we - ~w w,(l+b) > = (;s+b) 2 where
I$
= k . VT*, wx
(8)
= 3.16k;u;/v,,
Wf
=
(nio/n~)k:c~,c~ = T./mi,b = (nio/n~)k~c~/w~i, and ci = Z~ondoTe/n~md. Furthermore, kl is the component of the wavevector k across the z axis. On the other hand, the dispersion relation for the DA waves is found to be 1 - 1.71ia
where VT* = (cT,o/eBo) 2 x V In Td represents the electron diamagnetic drift velocity. The dynamics of the dust grains is governed by
+ &end1 -nit).
We now derive the linear dispersion relation for the DIA and DA waves in the presence of equilibrium electron temperature gradient. For this purpose, we assume that njt ,# and Tel areproportional toexp(ik.riwt) and Fourier transform (3)) (5) and (7) assuming that the wavelength (2r/lkl) is much smaller than the scale length of the equilibrium electron-temperaturegradient. For DIA waves, we have
(3) e0
197
196-198
w;-
$0 wx
=- GM Cl? ’
(9)
where u = nsTi/nioTd, w k~ = k2~C&,,/m,jnio( 1+ (T), and we have assumed that k2hi << 1, Here, ho is the effective Debye length of the dusty plasma [ 61. Letting w = o, + iy and assuming that y << w,, we obtain from (8) the real and imaginary parts of the DIA wave frequency, respectively, wr = OS/( 1 + b) 1’2,
( 10)
and y = 1.71v,(w;
- 1.42w,)/k:u:,.
(11)
On the other hand, (9) gives for the DA waves
where we have assumed that the aligned wave phase velocity is much ion thermal velocity. On the other wavelength (in comparison with pi) have
magnetic fieldlarger than the hand, for short DA waves, we
WI = Zdo~~do/~io>“2~~/~d~“2/~
1 + (7)
112 _ = WDA, (12)
and (13)
198
PK. Shukla et al./Physics
Eqs. ( 11) and ( 13) show that both the DIA and DA waves can become unstable provided that w; is much larger than w,,/( 1 + b) ‘I2 and WDA,respectively. In the following, we demonstrate that nonthermal ETG dust modes can cause cross-field electron diffusion as well as electron energy transport. The electron flux is r, = (ne*ve,) + C.C.,
(14)
where n,t is given by ( 1) and ~~1 = i( c/&)k x 2 is the E x & velocity of the electrons. Substituting for 7”t from (3) into ( 1) and inserting the resulting expression for nei into ( 14), we have
rex
X-
C&O
c ekye14k12
T,oBo
k2v4 z te
(w*
- 1.14wr).
(15)
The electron energy flux is found to be I’,,Td/ 1.7 1~0. We can invoke the mixing length hypothesis to estimate the fluctuation spectrum 1c&k12. Hence, equating the electron excursion length during the E x Bo drift with half of the wavelength, we have
(16) where Wr* = cT~k,d,T~/eBoT~
and LT = Td/
dxT&.
In summary, we have investigated the linear instability of DA and DIA waves in the presence of an equilibrium electron-temperature-gradient in a dissipative dusty plasma embedded in an external magnetic field. It is found that when the dust drift wave frequency Wr+ is larger than the frequencies of the DA and DIA
Letters A 226 (1997) 196-198
waves, the latter are driven at a nonthermal level on account of free energy stored in the equilibrium electron temperature inhomogeneity. Nonthermal DA and DIA waves are found to cause electron diffusion as well as electron thermal conduction across the external magnetic field lines. Thus, our investigation offers a possible mechanism for the excitation of DIA and DA fluctuations and associated cross-field anomalous electron and heat transports in nonuniform dusty magnetoplasmas such as those in radio-frequency discharges as well as in space plasmas. This work was partially supported by the Commission of the European Union (Brussels) through the network “Colloidal Plasmas” of the Human Capital and Mobility program under contract no. CHRXCT94-0602, as well as by the Deutsche Forschungsgemeinschaft (Bonn) through the Sonderforschungsbereich 19 1.
References [ I ] N.N. Rao, PK. Shukla and M.Y. Yu, Planet. Space Sci. 38
(1990) 543. [2] PK. Shukla and V.P Silin, Phys. Ser. 45 (1992) 508. [ 31 J.H. Chu, J.B. Du and Lin I., J. Phys. D 27 ( 1994) 296. A. Barkan, R.L. Merlin0 and N. D’Angelo, Phys. Plasmas 2 (1995) 3563. R.L. Merlino, A. Barkan and N. D’Angelo, in: AIP Conf. Proc. 345: Int. Conf. on Plasma physics, ICPP 1994, eds. PH. Sakanaka and M. Tendler (AIP Press, New York, 1995) pp. 295-302. PK. Shukla, Phys. Plasmas 1 ( 1994) 1362.