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Hall instabilities in dusty plasmas
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Planetary and Space Science 51 (2003) 393 – 398 www.elsevier.com/locate/pss
Hall instabilities in dusty plasmas N. D’Angelo∗ Department of Physics and Astronomy, The University of Iowa, Iowa City, IA 52242-1479, USA Received 10 July 2002; accepted 29 January 2003
Abstract A uni*ed treatment is presented of Hall instabilities in collisional dusty plasmas, using -uid equations for all charged components. Two modes are generally possible: one with frequencies in the ion-acoustic range (the Farley–Buneman mode), and the other with frequencies in the dust-acoustic range. The real and imaginary parts of the mode frequencies are obtained as functions of the electric *eld, E0 , perpendicular to the magnetic *eld, B, permeating the plasma. Calculations are performed for laboratory-type plasmas, and for ionospheric-type plasmas. ? 2003 Elsevier Science Ltd. All rights reserved. Keywords: Hall instabilities; Farley–Buneman instability; Dusty plasmas
1. Introduction The Farley–Buneman instability (Farley, 1963; Buneman, 1963) is a Hall current instability *rst observed by 50 MHz radar returns from the equatorial electrojet over Huancayo, Peru (Bowles et al., 1963), and later at stations at high latitudes, around the auroral zone or within the polar cap (see, e.g., Kelley, 1989). When the conditions are satis*ed !ci ¡ i , !ce e , where !ci and !ce are the ion and electron gyrofrequencies and i and e the collision frequencies of ions and electrons with the neutral gas background, an electric *eld E0 perpendicular to the magnetic *eld, B, produces an E0 × B=B2 drift of the electrons, whereas the (unmagnetized) ions are held back by collisions with the neutral gas. Thus, a drift of the electrons relative to the ions, of magnitude ∼ E0 =B exists. If this drift is larger than the local ion-acoustic speed, Cs , an instability is excited, with waves propagating essentially in the direction of the E0 × B=B2 drift of the electrons. The component of the propagation vector, K⊥ , in the E0 × B direction is much 2 larger than the component along B; K ; thus K2 K⊥ . The perpendicular wavelength is on the order of several meters, or larger. In the auroral or polar cap ionosphere, where B 5 × 10−5 T and Cs 400–500 m=s, the condition for ∗
Corresponding author. Fax: +319-335-1753. E-mail address: [email protected] (N. D’Angelo).
excitation of the instability, E0 =B ¿ Cs , requires an electric *eld, E0 & 20–25 mV=m. Since the Farley–Buneman instability can be observed in high-latitude ionograms as the so-called “Slant E Condition (SEC)” signature (Olesen, 1972), routine ionograms can provide information as to how often the ionospheric E0 *eld magnitude is above the 20 –25 mV=m level (see D’Angelo, 1980 for a review of the work at the Danish Space Research Institute and the Danish Meteorological Institute). The Farley–Buneman instability is, thus, utilized as a kind of diagnostic tool of ionospheric electric *elds. Laboratory experiments have veri*ed the predictions of the Farley–Buneman instability theory, which can thus be regarded as being on *rm grounds (Saito et al., 1964; D’Angelo et al., 1974; John and Saxena, 1975). In the last decade much work, both theoretical and experimental, has been done on dusty plasmas and, in particular, on waves and instabilities in dusty plasmas (see, e.g., Nakamura et al., 2000; Shukla and Mamun, 2002). Several of the wave modes are simply modi*cations of standard modes in normal plasmas, whose properties are altered by the presence of (generally negatively) charged dust grains assumed to be so massive that they constitute a *xed and constant background. One dusty plasma mode, however, involves in an essential manner the dust dynamics. It produces extremely low frequency waves, since the dust grains which provide the inertia are so much more massive than normal negative ions. It was discussed by Rao et al. (1990), who named it the “dust-acoustic” mode, and was observed in the
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N. D’Angelo / Planetary and Space Science 51 (2003) 393 – 398
laboratory by Barkan et al. (1995) and by Thompson et al. (1997). The Farley–Buneman instability in a dusty plasma was *rst discussed by Rosenberg and Chow (1998). They considered a situation in which both the ions and the dust grains are unmagnetized, i.e., !ci i and !cd , where !cd is the dust gyrofrequency and the dust collision frequency with the neutral gas background. For the electrons, !ce e . They found that the dust in-uences the instability through its eJect on the equilibrium charge neutrality, and that in certain regimes negative dust can decrease the critical electron drift. In their analysis, the wave velocity is on the order of the ion-acoustic velocity, as in normal plasmas. A Hall current instability in the dust frequency regime in a collisional plasma was investigated by Rosenberg and Shukla (2000), who discussed also possible applications to dusty plasmas in the Earth’s upper mesosphere or E region. Both Rosenberg and Chow (1998) and Rosenberg and Shukla (2000) used linear kinetic theory in their analyses. In the present paper a uni*ed treatment is presented of the Farley–Buneman instability (with frequencies in the ion-acoustic range) and of a much lower frequency Hall instability (with frequencies in the dust-acoustic range). In particular, starting from a situation in which only the electrons are magnetized, the transition is explored to a situation in which both ions and electrons are magnetized and only the dust is unmagnetized. Critical electric *elds are evaluated for excitation of both the Farley–Buneman and the lower frequency Hall instability. The analysis is based on -uid equations for all charged components and propagation of the waves perpendicular to the B *eld (K = 0) is envisaged. The eJect of a small but *nite K is considered in Appendix A. Section 2 of the paper presents the theory of the instabilities. Section 3 analyzes a few situations realizable in the laboratory, while Section 4 deals with ionospheric dusty plasmas. Section 5 contains the conclusions. 2. Theory Consider a dusty plasma in which ni ; ne ; nd and N are the densities of the positive ions, the electrons, the (negatively charged) dust grains and the neutral gas molecules, respectively; Ti ; Te ; Td and TN the corresponding temperatures, and mi ; me ; md and mN the corresponding masses. The neutral gas is taken to be at rest, while vi ; ve and vd indicate the velocities of the three charged components. The dust grains are assumed spherical, all with the same radius a, density %, and a charge −eZ on each. Collisions with the neutral gas occur with frequencies i = Ni Ci ;
e = Ne Ce
and
=
4mN Na2 CN ; md
where i and e are the positive ion-molecule and electron-molecule collision cross-sections, and Ci and Ce the ion and electron thermal speeds. In the expression for
(Baines et al., 1965), CN is the thermal speed of the neutral gas molecules. A uniform and constant magnetic *eld, B, is directed along the positive z-axis of a Cartesian frame of reference. E indicates the electric *eld in the plasma. The equations describing the behavior of the three charged components are the usual continuity and momentum equations. For the negatively charged dust they are: @nd + ∇ · (nd vd ) = 0; @t n d md
(1)
@vd + nd md vd · ∇vd + Td ∇nd + eZnd E + eZnd vd @t
×B = −nd md vd :
(2)
For the positive ions and the electrons we have analogous equations. Eqs. (1) and (2), and the corresponding equations for the ions and the electrons, are linearized around a zero-order state characterized by @ = 0; @t
∇n 0 = 0;
v 0 = const ( = d; i; e)
ˆ with and a uniform and constant electric *eld E0 = −E0 x, E0 ¿ 0 and xˆ the unit vector along the positive x-axis. With !cd = eZB=md , we *nd for the dust grains vd0x =
eZ E0 ; 2 md !cd + 2
(3)
vd0y =
!cd eZ E ; 2 + 2 0 md !cd
(4)
vd0z = 0
(5)
with similar expressions for ion and electron velocities. The linearized equations are Fourier transformed, assuming for the *rst-order quantities a space and time dependence of the type ei(Ky−!t) , with wave propagation along the y-axis. With g = ! − Kvd0y we readily obtain a relation between the relative density -uctuation of the dust #d = nd1 =nd0 and the potential -uctuation ’1 = ki E1 : 2 [(g + i)2 g − K 2 Cd2 (g + i) − !cd g]#d
+
eZ (g + i)K 2 ’1 = 0: md
(6)
With b = ! − Kvi0y and f = ! − Kve0y we also *nd: 2 [(b + ii )2 b − K 2 Ci2 (b + ii ) − !ci b]#i
−
e (b + ii )K 2 ’1 = 0 mi
(7)
N. D’Angelo / Planetary and Space Science 51 (2003) 393 – 398
395
and 2 f]#e [(f + ie )2 f − K 2 Ce2 (f + ie ) − !ce
+
e (f + ie )K 2 ’1 = 0; me
(8)
where #i = ni1 =ni0 and #e = ne1 =ne0 . The condition of charge neutrality in *rst order can be written as (Z#d − #i + (1 − (Z)#e = 0;
(9)
where ( = nd0 =ni0 is the ratio between the zero-order dust density and the zero-order ion density. The dispersion relation is obtained by inserting the expressions of #d ; #i and #e , in terms of ’1 , from Eqs. (6)–(8) into Eq. (9). The dispersion relation was solved numerically, using MAPLE, for several cases of interest to laboratory or ionospheric plasmas.
Fig. 1. Laboratory plasma of Table 1, but without dust (( = 0), N = 5 × 1022 m−3 . Frequency (full line) and growth rate (dotted line) of the Farley–Buneman instabilities as functions of E0 .
3. Laboratory plasmas The theory developed in Section 2 was applied to the case of a laboratory dusty plasma, whose general characteristics are given in Table 1. To begin with, calculations were performed for the case of no dust present. Fig. 1 shows the frequency, !, and the growth rate of the Farley–Buneman instability, in s−1 , as functions of the applied E0 *eld, in V/m. The density of the neutral gas molecules is in this case N = 5 × 1022 m−3 . As expected, the wave frequency, !, is such that the wave phase velocity, !=k, equals the E0 =B speed at all E0 ’s. The threshold electric *eld is ∼ 600 V=m. When dust is added to the plasma, with the characteristics of Table 1 and still with a neutral gas density N =5×1022 m−3 , the results are obtained which are shown in Fig. 2. Now two modes are present, one the higher frequency Farley–Buneman mode, whose features are only slightly diJerent from those of Fig. 1, and the other a much lower frequency mode whose phase velocity Table 1 Parameters of laboratory plasmas
mi = 6:7 × 10−26 Kg mN = 6:7 × 10−26 Kg Ti = 300 K Te = 10000 K TN = 300 K Td = 300 K a = 10−6 m % = 103 Kg= m3 i = 5 × 10−19 m2 e = 10−20 m2 Z = 2397 ( = 2:1 × 10−4 (Z = 0:5 B = 0:4 T K = 100 m−1
Fig. 2. Same as Fig. 1, but with dust present. The dust properties are those of Table 1. !ci =i 0:15.
!=k ∼ 10−2 m=s is on the order of the dust-acoustic speed. The threshold E0 for the former mode is ∼ 700 V=m, whereas that for the latter is ∼ 3000 V=m. For the conditions of Fig. 2, the ratio !ci =i between the positive ion gyrofrequency and the ion-molecule collision frequency is ∼ 0:15, which means that the ions (as well as the dust grains, with !cd = 5 × 10−5 ) are unmagnetized, whereas the electrons, with !ce =e 360, are magnetized. Next, the density of the neutral gas molecules is changed to N = 1 × 1022 m−3 . This still leaves the dust grains unmagnetized and the electrons magnetized, while the ratio !ci =i is ∼ 0:77. Thus, the ion magnetization is now just borderline. Fig. 3 shows the results for this case. There are only small changes in the features of the higher frequency Farley–Buneman mode, but the lower frequency mode has a much lower E0 *eld threshold (∼ 150 V=m) than in Fig. 2. In Fig. 4 the results are shown which were obtained with a further decrease of N to 3 × 1021 m−3 . Here !ci =i = 2:6, !ce =e 6 × 103 and !cd = = 7:7 × 10−4 , with the electrons magnetized, the dust grains unmagnetized, and the ions
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N. D’Angelo / Planetary and Space Science 51 (2003) 393 – 398
Fig. 3. Same as Fig. 2, but N = 1 × 1022 m−3 and !ci =i = 0:77.
Fig. 5. Same as Fig. 2, but N = 1 × 1021 m−3 and !ci =i = 7:7.
appear as a perturbation traveling azimuthally (or as a helix of low pitch), with azimuthal wavelengths on the order of centimeters and azimuthal phase velocity on the order of centimeters per second. 4. Ionospheric plasmas
Fig. 4. Same as Fig. 2, but N = 3 × 1021 m−3 and !ci =i = 2:6.
almost fully magnetized. In the 10 V=m 6 E0 6 104 V=m range, only the lower frequency mode is excited, with a threshold E0 as low as ∼ 10 V=m. Finally, the magnetization of the ions is still increased by further lowering N to 1021 m−3 , which provides an !ci =i of ∼ 7:7. The results for the frequency and the growth rate of the lower frequency mode (the only one excited for 1 V=m 6 E0 6 104 V=m) are shown in Fig. 5. In this case, the threshold electric *eld E0 is as low as ∼ 2 V=m. This eJect due to the variation of !ci =i is observed not only by varying N , at a constant value of the magnetic *eld strength, but also at a constant N by varying the magnitude of B. Thus, the decrease, with increasing !ci =i , of the critical drift for the low frequency wave seems to be largely due to excitation by the ion E × B drift. Excitation of the low frequency (“dust-acoustic”) mode may be looked for in experimental situations similar to those reported, e.g., by Barkan et al. (1995). A “*rerod” at higher potential relative to the surrounding plasma, traps negatively charged dust grains. The “*rerod” is elongated along the magnetic *eld lines, with the static electric *eld at right angles to B. The low frequency (“dust-acoustic”) mode would
The parameters of the ionospheric plasma considered here are given in Table 2. The density of the neutral gas background was chosen as N = 6 × 1016 m−3 . This, with a B *eld of 5 × 10−5 T, provides a ratio !ci =i ≈ 20. The ratio !ce =e is ∼ 1:2 × 105 , whereas !cd = ≈ 6 × 10−2 . Evidently we are dealing here with an ionospheric region above the normal height range, ∼ 100 km to ∼ 120 km, where the usual Farley–Buneman instability is observed. Since the ions are also magnetized in our case, we expect to obtain, from our analysis, excitation of only one mode, namely the low frequency mode. For the plasma parameters of Table 2, we present the results for the frequency and growth rate as functions of the ionospheric electric *eld, Table 2 Parameters of ionospheric plasmas
mi = 3:3 × 10−26 Kg mN = 3:3 × 10−26 Kg Ti = 400 K Te = 1000 K TN = 400 K Td = 400 K a = 10−8 m % = 500 Kg= m3 i = 5 × 10−19 m2 e = 10−20 m2 Z = 2:4 ( = 0:3 (Z = 0:72 B = 5 × 10−5 T N = 6 × 1016 m−3
N. D’Angelo / Planetary and Space Science 51 (2003) 393 – 398
Fig. 6. Ionospheric plasma of Table 2. !ci =i = 20, !ce =e = 1:2 × 105 , !cd = = 6 × 10−2 and K = 1 m−1 . Frequency (full line) and growth rate (dotted line) as functions of E0 .
397
Fig. 7, and K = 0:01 m−1 in Fig. 8. It appears from these results that the low frequency mode can grow in a dusty plasma of the type of Table 2, with ionospheric electric *elds whose magnitude is only a few tenths of a mV/m. The question may be raised whether the dust grains, falling under the in-uence of gravity, can remain near the altitude at which they were injected long enough for the low frequency wave mode to grow. For the conditions envisaged here, the terminal velocity of a dust grain is 9:8= 63 m=s, and with an instability growth rate of ∼ 0:3 s−1 , the dust grains would fall only ∼ 210 m during one growth time. The theory developed in Section 2 allows us, of course, to study also the usual Farley–Buneman instability which occurs in the height range ∼ 100 km to ∼ 120 km, but no numerical results are presented here since they exhibit no special novel features. Rosenberg (2001) has shown that negatively charged dust at these altitudes would increase the critical drift for the Farley–Buneman instability, since the main eJect of negatively charged dust would be to increase the phase speed of the wave. 5. Conclusions
Fig. 7. Same as Fig. 6, but with K = 0:1 m−1 .
Fig. 8. Same as Fig. 6, but with K = 0:01 m−1 .
E0 , perpendicular to the magnetic *eld, B, in the range 0:1 mV=m 6 E0 6 1 V=m. Figs. 6–8 refer to the same ionospheric plasma, but K = 1 m−1 in Fig. 6, K = 0:1 m−1 in
Hall current instabilities in collisional dusty plasmas have been investigated, using a -uid picture for all three charged components, i.e., the positive ions, the electrons and the negatively charged dust grains. In general, two wave modes are possible in such plasmas, a higher frequency Farley– Buneman mode and a lower frequency mode with frequencies in the “dust-acoustic” range. At high neutral gas pressures, where both the dust and the positive ions are unmagnetized (!cd =1, !ci =i 1), the Farley–Buneman mode is the one more easily excited, with threshold electric *elds, E0 , smaller than the threshold *elds for the lower frequency mode. However, as the gas pressure is decreased and the ions become magnetized, the lower frequency mode is the one more easily excited. At !ci =i & 1 − 2, only this mode has a positive growth rate, in the 1 V=m 6 E0 6 104 V=m range, for the laboratory dusty plasma of Section 3. We have also considered the case of an ionospheric dusty plasma at an altitude somewhat above the usual height range (∼ 100 km to ∼ 120 km) of the Farley–Buneman instability. With magnetized ions and electrons, and unmagnetized dust, only the lower frequency mode has a positive growth rate for E0 ¡ 1 V=m. For the ionospheric dusty plasma considered in Section 4, it is found that this mode can be excited by E0 *elds as small as a few tenths of a mV/m. Appendix A. To explore the eJect of a small but *nite Kz , Eqs. (1) and (2) and the analogous ones for the ions and the electrons are linearized, and the space-time dependence of the *rst-order quantities is taken to be of the type ei(Ky y+Kz z−!t) .
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N. D’Angelo / Planetary and Space Science 51 (2003) 393 – 398
and ∼ 10 cm, respectively. Here the threshold E0 is lower (∼ 20 V=m) and !=Ky near threshold is ∼ 3 × 10−2 m=s. An interesting feature is the excitation gap between E0 ≈ 2000 V=m and E0 ≈ 7000 V=m. The eJect of *nite Kz on the low frequency instability was also considered in Rosenberg and Shukla (2002), where a trend to larger growth at small but *nite Kz was found. References
Fig. 9. Laboratory plasma of Table 1, N = 3 × 1021 m−3 and !ci =i = 2:6. Frequency (full line) and growth rate (dotted line) as functions of E0 . Ky = 100 m−1 and Kz = 1 m−1 .
Fig. 10. Same as Fig. 9, but Ky = 600 m−1 and Kz = 60 m−1 .
Thus one obtains a set of 13 homogeneous equations in 13 unknowns, #d ; #i ; #e ; ’1 and the 9 velocity components of the three charged species. The determinant of the coeOcients, D, set equal to zero is the dispersion relation. The equation D = 0 is solved by using MAPLE. Two examples of solutions are shown in Figs. 9 and 10. Both *gures are for a laboratory-type plasma as given by Table 1, with N = 3 × 1021 m−3 and !ci =i = 2:6. For Fig. 9 the choice was made: Ky = 100 m−1 and Kz = 1 m−1 . Excitation of the low frequency mode has a threshold of ∼ 100 V=m and, near threshold, the velocity !=Ky is ∼ 0:1 m=s. In Fig. 10, Ky = 600 m−1 and Kz = 60 m−1 , i.e., the perpendicular and parallel wavelengths are ∼ 1 cm
Baines, M.F., Williams, I.P., Asebiomo, A.S., 1965. Resistance to the motion of a small sphere moving through a gas. Mon. Not. R. Astron. Soc. 130, 63–74. Barkan, A., Merlino, R.L., D’Angelo, N., 1995. Laboratory observation of the dust-acoustic mode. Phys. Plasmas 2, 3563–3565. Bowles, K.L., Balsley, B.B., Cohen, R., 1963. Field-aligned E-region irregularities identi*ed with acoustic plasma waves. J. Geophys. Res. 68, 2485–2501. Buneman, O., 1963. Excitation of *eld aligned sound waves by electron streams. Phys. Rev. Lett. 10, 285–287. D’Angelo, N., 1980. The Farley–Buneman instability in the polar cap ionosphere. Proceedings of the International Conference on Plasma Physics, Nagoya, Japan. Paper 8A3, Vol. II, pp. 97–104. D’Angelo, N., PQecseli, H.L., Petersen, P.I., 1974. The Farley instability: a laboratory test. J. Geophys. Res. 79, 4747–4751. Farley, D.T., 1963. A plasma instability resulting in *eld-aligned irregularities in the ionosphere. J. Geophys. Res. 68, 6083–6097. John, P.I., Saxena, Y.C., 1975. Observation of the Farley–Buneman instability in laboratory plasma. Geophys. Res. Lett. 2, 251–254. Kelley, M.C., 1989. The Earth’s Ionosphere: Plasma Physics and Electrodynamics. Academic Press, San Diego, CA. Nakamura, Y., Yakota, T., Shukla, P.K., 2000. Frontiers in Dusty Plasmas. Elsevier, Amsterdam. Olesen, J.K., 1972. On the polar slant E condition, its identi*cation, morphology and relationship to other electrojet phenomena. In: Frihagen J. (Ed.), AGARD Conference Proceedings, Vol. 97, pp. 27.1–27.19. Rao, N.N., Shukla, P.K., Yu, M.Y., 1990. Dust-acoustic waves in dusty plasmas. Planet. Space Sci. 38, 543–546. Rosenberg, M., 2001. EJect of charged dust on Hall current instabilities in the E region. IEEE Trans. Plasma Sci. 29, 261–266. Rosenberg, M., Chow, V.W., 1998. Farley–Buneman instability in a dusty plasma. Planet. Space Sci. 46, 103–108. Rosenberg, M., Shukla, P.K., 2000. Low-frequency Hall current instability in a dusty plasma. J. Geophys. Res. 105, 23,135–23,139. Rosenberg, M., Shukla, P.K., 2002. Parallel propagation eJects on low-frequency Hall current instability in a dusty plasma. Planet. Space Sci. 50, 261–267. Saito, S., Sato, N., Hatta, Y., 1964. Low frequency oscillations in co-axial cylindrical discharge tubes in a magnetic *eld perpendicular to an electric *eld. Appl. Phys. Lett. 5, 46–47. Shukla, P.K., Mamun, A.A., 2002. Introduction to Dusty Plasma Physics. Institute of Physics Publishing, Bristol and Philadelphia. Thompson, C., Barkan, A., D’Angelo, N., Merlino, R.L., 1997. Dust acoustic waves in a direct current glow discharge. Phys. Plasmas 4, 2331–2335.
Planetary and Space Science 51 (2003) 393 – 398 www.elsevier.com/locate/pss
Hall instabilities in dusty plasmas N. D’Angelo∗ Department of Physics and Astronomy, The University of Iowa, Iowa City, IA 52242-1479, USA Received 10 July 2002; accepted 29 January 2003
Abstract A uni*ed treatment is presented of Hall instabilities in collisional dusty plasmas, using -uid equations for all charged components. Two modes are generally possible: one with frequencies in the ion-acoustic range (the Farley–Buneman mode), and the other with frequencies in the dust-acoustic range. The real and imaginary parts of the mode frequencies are obtained as functions of the electric *eld, E0 , perpendicular to the magnetic *eld, B, permeating the plasma. Calculations are performed for laboratory-type plasmas, and for ionospheric-type plasmas. ? 2003 Elsevier Science Ltd. All rights reserved. Keywords: Hall instabilities; Farley–Buneman instability; Dusty plasmas
1. Introduction The Farley–Buneman instability (Farley, 1963; Buneman, 1963) is a Hall current instability *rst observed by 50 MHz radar returns from the equatorial electrojet over Huancayo, Peru (Bowles et al., 1963), and later at stations at high latitudes, around the auroral zone or within the polar cap (see, e.g., Kelley, 1989). When the conditions are satis*ed !ci ¡ i , !ce e , where !ci and !ce are the ion and electron gyrofrequencies and i and e the collision frequencies of ions and electrons with the neutral gas background, an electric *eld E0 perpendicular to the magnetic *eld, B, produces an E0 × B=B2 drift of the electrons, whereas the (unmagnetized) ions are held back by collisions with the neutral gas. Thus, a drift of the electrons relative to the ions, of magnitude ∼ E0 =B exists. If this drift is larger than the local ion-acoustic speed, Cs , an instability is excited, with waves propagating essentially in the direction of the E0 × B=B2 drift of the electrons. The component of the propagation vector, K⊥ , in the E0 × B direction is much 2 larger than the component along B; K ; thus K2 K⊥ . The perpendicular wavelength is on the order of several meters, or larger. In the auroral or polar cap ionosphere, where B 5 × 10−5 T and Cs 400–500 m=s, the condition for ∗
Corresponding author. Fax: +319-335-1753. E-mail address: [email protected] (N. D’Angelo).
excitation of the instability, E0 =B ¿ Cs , requires an electric *eld, E0 & 20–25 mV=m. Since the Farley–Buneman instability can be observed in high-latitude ionograms as the so-called “Slant E Condition (SEC)” signature (Olesen, 1972), routine ionograms can provide information as to how often the ionospheric E0 *eld magnitude is above the 20 –25 mV=m level (see D’Angelo, 1980 for a review of the work at the Danish Space Research Institute and the Danish Meteorological Institute). The Farley–Buneman instability is, thus, utilized as a kind of diagnostic tool of ionospheric electric *elds. Laboratory experiments have veri*ed the predictions of the Farley–Buneman instability theory, which can thus be regarded as being on *rm grounds (Saito et al., 1964; D’Angelo et al., 1974; John and Saxena, 1975). In the last decade much work, both theoretical and experimental, has been done on dusty plasmas and, in particular, on waves and instabilities in dusty plasmas (see, e.g., Nakamura et al., 2000; Shukla and Mamun, 2002). Several of the wave modes are simply modi*cations of standard modes in normal plasmas, whose properties are altered by the presence of (generally negatively) charged dust grains assumed to be so massive that they constitute a *xed and constant background. One dusty plasma mode, however, involves in an essential manner the dust dynamics. It produces extremely low frequency waves, since the dust grains which provide the inertia are so much more massive than normal negative ions. It was discussed by Rao et al. (1990), who named it the “dust-acoustic” mode, and was observed in the
0032-0633/03/$ - see front matter ? 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0032-0633(03)00025-4
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N. D’Angelo / Planetary and Space Science 51 (2003) 393 – 398
laboratory by Barkan et al. (1995) and by Thompson et al. (1997). The Farley–Buneman instability in a dusty plasma was *rst discussed by Rosenberg and Chow (1998). They considered a situation in which both the ions and the dust grains are unmagnetized, i.e., !ci i and !cd , where !cd is the dust gyrofrequency and the dust collision frequency with the neutral gas background. For the electrons, !ce e . They found that the dust in-uences the instability through its eJect on the equilibrium charge neutrality, and that in certain regimes negative dust can decrease the critical electron drift. In their analysis, the wave velocity is on the order of the ion-acoustic velocity, as in normal plasmas. A Hall current instability in the dust frequency regime in a collisional plasma was investigated by Rosenberg and Shukla (2000), who discussed also possible applications to dusty plasmas in the Earth’s upper mesosphere or E region. Both Rosenberg and Chow (1998) and Rosenberg and Shukla (2000) used linear kinetic theory in their analyses. In the present paper a uni*ed treatment is presented of the Farley–Buneman instability (with frequencies in the ion-acoustic range) and of a much lower frequency Hall instability (with frequencies in the dust-acoustic range). In particular, starting from a situation in which only the electrons are magnetized, the transition is explored to a situation in which both ions and electrons are magnetized and only the dust is unmagnetized. Critical electric *elds are evaluated for excitation of both the Farley–Buneman and the lower frequency Hall instability. The analysis is based on -uid equations for all charged components and propagation of the waves perpendicular to the B *eld (K = 0) is envisaged. The eJect of a small but *nite K is considered in Appendix A. Section 2 of the paper presents the theory of the instabilities. Section 3 analyzes a few situations realizable in the laboratory, while Section 4 deals with ionospheric dusty plasmas. Section 5 contains the conclusions. 2. Theory Consider a dusty plasma in which ni ; ne ; nd and N are the densities of the positive ions, the electrons, the (negatively charged) dust grains and the neutral gas molecules, respectively; Ti ; Te ; Td and TN the corresponding temperatures, and mi ; me ; md and mN the corresponding masses. The neutral gas is taken to be at rest, while vi ; ve and vd indicate the velocities of the three charged components. The dust grains are assumed spherical, all with the same radius a, density %, and a charge −eZ on each. Collisions with the neutral gas occur with frequencies i = Ni Ci ;
e = Ne Ce
and
=
4mN Na2 CN ; md
where i and e are the positive ion-molecule and electron-molecule collision cross-sections, and Ci and Ce the ion and electron thermal speeds. In the expression for
(Baines et al., 1965), CN is the thermal speed of the neutral gas molecules. A uniform and constant magnetic *eld, B, is directed along the positive z-axis of a Cartesian frame of reference. E indicates the electric *eld in the plasma. The equations describing the behavior of the three charged components are the usual continuity and momentum equations. For the negatively charged dust they are: @nd + ∇ · (nd vd ) = 0; @t n d md
(1)
@vd + nd md vd · ∇vd + Td ∇nd + eZnd E + eZnd vd @t
×B = −nd md vd :
(2)
For the positive ions and the electrons we have analogous equations. Eqs. (1) and (2), and the corresponding equations for the ions and the electrons, are linearized around a zero-order state characterized by @ = 0; @t
∇n 0 = 0;
v 0 = const ( = d; i; e)
ˆ with and a uniform and constant electric *eld E0 = −E0 x, E0 ¿ 0 and xˆ the unit vector along the positive x-axis. With !cd = eZB=md , we *nd for the dust grains vd0x =
eZ E0 ; 2 md !cd + 2
(3)
vd0y =
!cd eZ E ; 2 + 2 0 md !cd
(4)
vd0z = 0
(5)
with similar expressions for ion and electron velocities. The linearized equations are Fourier transformed, assuming for the *rst-order quantities a space and time dependence of the type ei(Ky−!t) , with wave propagation along the y-axis. With g = ! − Kvd0y we readily obtain a relation between the relative density -uctuation of the dust #d = nd1 =nd0 and the potential -uctuation ’1 = ki E1 : 2 [(g + i)2 g − K 2 Cd2 (g + i) − !cd g]#d
+
eZ (g + i)K 2 ’1 = 0: md
(6)
With b = ! − Kvi0y and f = ! − Kve0y we also *nd: 2 [(b + ii )2 b − K 2 Ci2 (b + ii ) − !ci b]#i
−
e (b + ii )K 2 ’1 = 0 mi
(7)
N. D’Angelo / Planetary and Space Science 51 (2003) 393 – 398
395
and 2 f]#e [(f + ie )2 f − K 2 Ce2 (f + ie ) − !ce
+
e (f + ie )K 2 ’1 = 0; me
(8)
where #i = ni1 =ni0 and #e = ne1 =ne0 . The condition of charge neutrality in *rst order can be written as (Z#d − #i + (1 − (Z)#e = 0;
(9)
where ( = nd0 =ni0 is the ratio between the zero-order dust density and the zero-order ion density. The dispersion relation is obtained by inserting the expressions of #d ; #i and #e , in terms of ’1 , from Eqs. (6)–(8) into Eq. (9). The dispersion relation was solved numerically, using MAPLE, for several cases of interest to laboratory or ionospheric plasmas.
Fig. 1. Laboratory plasma of Table 1, but without dust (( = 0), N = 5 × 1022 m−3 . Frequency (full line) and growth rate (dotted line) of the Farley–Buneman instabilities as functions of E0 .
3. Laboratory plasmas The theory developed in Section 2 was applied to the case of a laboratory dusty plasma, whose general characteristics are given in Table 1. To begin with, calculations were performed for the case of no dust present. Fig. 1 shows the frequency, !, and the growth rate of the Farley–Buneman instability, in s−1 , as functions of the applied E0 *eld, in V/m. The density of the neutral gas molecules is in this case N = 5 × 1022 m−3 . As expected, the wave frequency, !, is such that the wave phase velocity, !=k, equals the E0 =B speed at all E0 ’s. The threshold electric *eld is ∼ 600 V=m. When dust is added to the plasma, with the characteristics of Table 1 and still with a neutral gas density N =5×1022 m−3 , the results are obtained which are shown in Fig. 2. Now two modes are present, one the higher frequency Farley–Buneman mode, whose features are only slightly diJerent from those of Fig. 1, and the other a much lower frequency mode whose phase velocity Table 1 Parameters of laboratory plasmas
mi = 6:7 × 10−26 Kg mN = 6:7 × 10−26 Kg Ti = 300 K Te = 10000 K TN = 300 K Td = 300 K a = 10−6 m % = 103 Kg= m3 i = 5 × 10−19 m2 e = 10−20 m2 Z = 2397 ( = 2:1 × 10−4 (Z = 0:5 B = 0:4 T K = 100 m−1
Fig. 2. Same as Fig. 1, but with dust present. The dust properties are those of Table 1. !ci =i 0:15.
!=k ∼ 10−2 m=s is on the order of the dust-acoustic speed. The threshold E0 for the former mode is ∼ 700 V=m, whereas that for the latter is ∼ 3000 V=m. For the conditions of Fig. 2, the ratio !ci =i between the positive ion gyrofrequency and the ion-molecule collision frequency is ∼ 0:15, which means that the ions (as well as the dust grains, with !cd = 5 × 10−5 ) are unmagnetized, whereas the electrons, with !ce =e 360, are magnetized. Next, the density of the neutral gas molecules is changed to N = 1 × 1022 m−3 . This still leaves the dust grains unmagnetized and the electrons magnetized, while the ratio !ci =i is ∼ 0:77. Thus, the ion magnetization is now just borderline. Fig. 3 shows the results for this case. There are only small changes in the features of the higher frequency Farley–Buneman mode, but the lower frequency mode has a much lower E0 *eld threshold (∼ 150 V=m) than in Fig. 2. In Fig. 4 the results are shown which were obtained with a further decrease of N to 3 × 1021 m−3 . Here !ci =i = 2:6, !ce =e 6 × 103 and !cd = = 7:7 × 10−4 , with the electrons magnetized, the dust grains unmagnetized, and the ions
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Fig. 3. Same as Fig. 2, but N = 1 × 1022 m−3 and !ci =i = 0:77.
Fig. 5. Same as Fig. 2, but N = 1 × 1021 m−3 and !ci =i = 7:7.
appear as a perturbation traveling azimuthally (or as a helix of low pitch), with azimuthal wavelengths on the order of centimeters and azimuthal phase velocity on the order of centimeters per second. 4. Ionospheric plasmas
Fig. 4. Same as Fig. 2, but N = 3 × 1021 m−3 and !ci =i = 2:6.
almost fully magnetized. In the 10 V=m 6 E0 6 104 V=m range, only the lower frequency mode is excited, with a threshold E0 as low as ∼ 10 V=m. Finally, the magnetization of the ions is still increased by further lowering N to 1021 m−3 , which provides an !ci =i of ∼ 7:7. The results for the frequency and the growth rate of the lower frequency mode (the only one excited for 1 V=m 6 E0 6 104 V=m) are shown in Fig. 5. In this case, the threshold electric *eld E0 is as low as ∼ 2 V=m. This eJect due to the variation of !ci =i is observed not only by varying N , at a constant value of the magnetic *eld strength, but also at a constant N by varying the magnitude of B. Thus, the decrease, with increasing !ci =i , of the critical drift for the low frequency wave seems to be largely due to excitation by the ion E × B drift. Excitation of the low frequency (“dust-acoustic”) mode may be looked for in experimental situations similar to those reported, e.g., by Barkan et al. (1995). A “*rerod” at higher potential relative to the surrounding plasma, traps negatively charged dust grains. The “*rerod” is elongated along the magnetic *eld lines, with the static electric *eld at right angles to B. The low frequency (“dust-acoustic”) mode would
The parameters of the ionospheric plasma considered here are given in Table 2. The density of the neutral gas background was chosen as N = 6 × 1016 m−3 . This, with a B *eld of 5 × 10−5 T, provides a ratio !ci =i ≈ 20. The ratio !ce =e is ∼ 1:2 × 105 , whereas !cd = ≈ 6 × 10−2 . Evidently we are dealing here with an ionospheric region above the normal height range, ∼ 100 km to ∼ 120 km, where the usual Farley–Buneman instability is observed. Since the ions are also magnetized in our case, we expect to obtain, from our analysis, excitation of only one mode, namely the low frequency mode. For the plasma parameters of Table 2, we present the results for the frequency and growth rate as functions of the ionospheric electric *eld, Table 2 Parameters of ionospheric plasmas
mi = 3:3 × 10−26 Kg mN = 3:3 × 10−26 Kg Ti = 400 K Te = 1000 K TN = 400 K Td = 400 K a = 10−8 m % = 500 Kg= m3 i = 5 × 10−19 m2 e = 10−20 m2 Z = 2:4 ( = 0:3 (Z = 0:72 B = 5 × 10−5 T N = 6 × 1016 m−3
N. D’Angelo / Planetary and Space Science 51 (2003) 393 – 398
Fig. 6. Ionospheric plasma of Table 2. !ci =i = 20, !ce =e = 1:2 × 105 , !cd = = 6 × 10−2 and K = 1 m−1 . Frequency (full line) and growth rate (dotted line) as functions of E0 .
397
Fig. 7, and K = 0:01 m−1 in Fig. 8. It appears from these results that the low frequency mode can grow in a dusty plasma of the type of Table 2, with ionospheric electric *elds whose magnitude is only a few tenths of a mV/m. The question may be raised whether the dust grains, falling under the in-uence of gravity, can remain near the altitude at which they were injected long enough for the low frequency wave mode to grow. For the conditions envisaged here, the terminal velocity of a dust grain is 9:8= 63 m=s, and with an instability growth rate of ∼ 0:3 s−1 , the dust grains would fall only ∼ 210 m during one growth time. The theory developed in Section 2 allows us, of course, to study also the usual Farley–Buneman instability which occurs in the height range ∼ 100 km to ∼ 120 km, but no numerical results are presented here since they exhibit no special novel features. Rosenberg (2001) has shown that negatively charged dust at these altitudes would increase the critical drift for the Farley–Buneman instability, since the main eJect of negatively charged dust would be to increase the phase speed of the wave. 5. Conclusions
Fig. 7. Same as Fig. 6, but with K = 0:1 m−1 .
Fig. 8. Same as Fig. 6, but with K = 0:01 m−1 .
E0 , perpendicular to the magnetic *eld, B, in the range 0:1 mV=m 6 E0 6 1 V=m. Figs. 6–8 refer to the same ionospheric plasma, but K = 1 m−1 in Fig. 6, K = 0:1 m−1 in
Hall current instabilities in collisional dusty plasmas have been investigated, using a -uid picture for all three charged components, i.e., the positive ions, the electrons and the negatively charged dust grains. In general, two wave modes are possible in such plasmas, a higher frequency Farley– Buneman mode and a lower frequency mode with frequencies in the “dust-acoustic” range. At high neutral gas pressures, where both the dust and the positive ions are unmagnetized (!cd =1, !ci =i 1), the Farley–Buneman mode is the one more easily excited, with threshold electric *elds, E0 , smaller than the threshold *elds for the lower frequency mode. However, as the gas pressure is decreased and the ions become magnetized, the lower frequency mode is the one more easily excited. At !ci =i & 1 − 2, only this mode has a positive growth rate, in the 1 V=m 6 E0 6 104 V=m range, for the laboratory dusty plasma of Section 3. We have also considered the case of an ionospheric dusty plasma at an altitude somewhat above the usual height range (∼ 100 km to ∼ 120 km) of the Farley–Buneman instability. With magnetized ions and electrons, and unmagnetized dust, only the lower frequency mode has a positive growth rate for E0 ¡ 1 V=m. For the ionospheric dusty plasma considered in Section 4, it is found that this mode can be excited by E0 *elds as small as a few tenths of a mV/m. Appendix A. To explore the eJect of a small but *nite Kz , Eqs. (1) and (2) and the analogous ones for the ions and the electrons are linearized, and the space-time dependence of the *rst-order quantities is taken to be of the type ei(Ky y+Kz z−!t) .
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N. D’Angelo / Planetary and Space Science 51 (2003) 393 – 398
and ∼ 10 cm, respectively. Here the threshold E0 is lower (∼ 20 V=m) and !=Ky near threshold is ∼ 3 × 10−2 m=s. An interesting feature is the excitation gap between E0 ≈ 2000 V=m and E0 ≈ 7000 V=m. The eJect of *nite Kz on the low frequency instability was also considered in Rosenberg and Shukla (2002), where a trend to larger growth at small but *nite Kz was found. References
Fig. 9. Laboratory plasma of Table 1, N = 3 × 1021 m−3 and !ci =i = 2:6. Frequency (full line) and growth rate (dotted line) as functions of E0 . Ky = 100 m−1 and Kz = 1 m−1 .
Fig. 10. Same as Fig. 9, but Ky = 600 m−1 and Kz = 60 m−1 .
Thus one obtains a set of 13 homogeneous equations in 13 unknowns, #d ; #i ; #e ; ’1 and the 9 velocity components of the three charged species. The determinant of the coeOcients, D, set equal to zero is the dispersion relation. The equation D = 0 is solved by using MAPLE. Two examples of solutions are shown in Figs. 9 and 10. Both *gures are for a laboratory-type plasma as given by Table 1, with N = 3 × 1021 m−3 and !ci =i = 2:6. For Fig. 9 the choice was made: Ky = 100 m−1 and Kz = 1 m−1 . Excitation of the low frequency mode has a threshold of ∼ 100 V=m and, near threshold, the velocity !=Ky is ∼ 0:1 m=s. In Fig. 10, Ky = 600 m−1 and Kz = 60 m−1 , i.e., the perpendicular and parallel wavelengths are ∼ 1 cm
Baines, M.F., Williams, I.P., Asebiomo, A.S., 1965. Resistance to the motion of a small sphere moving through a gas. Mon. Not. R. Astron. Soc. 130, 63–74. Barkan, A., Merlino, R.L., D’Angelo, N., 1995. Laboratory observation of the dust-acoustic mode. Phys. Plasmas 2, 3563–3565. Bowles, K.L., Balsley, B.B., Cohen, R., 1963. Field-aligned E-region irregularities identi*ed with acoustic plasma waves. J. Geophys. Res. 68, 2485–2501. Buneman, O., 1963. Excitation of *eld aligned sound waves by electron streams. Phys. Rev. Lett. 10, 285–287. D’Angelo, N., 1980. The Farley–Buneman instability in the polar cap ionosphere. Proceedings of the International Conference on Plasma Physics, Nagoya, Japan. Paper 8A3, Vol. II, pp. 97–104. D’Angelo, N., PQecseli, H.L., Petersen, P.I., 1974. The Farley instability: a laboratory test. J. Geophys. Res. 79, 4747–4751. Farley, D.T., 1963. A plasma instability resulting in *eld-aligned irregularities in the ionosphere. J. Geophys. Res. 68, 6083–6097. John, P.I., Saxena, Y.C., 1975. Observation of the Farley–Buneman instability in laboratory plasma. Geophys. Res. Lett. 2, 251–254. Kelley, M.C., 1989. The Earth’s Ionosphere: Plasma Physics and Electrodynamics. Academic Press, San Diego, CA. Nakamura, Y., Yakota, T., Shukla, P.K., 2000. Frontiers in Dusty Plasmas. Elsevier, Amsterdam. Olesen, J.K., 1972. On the polar slant E condition, its identi*cation, morphology and relationship to other electrojet phenomena. In: Frihagen J. (Ed.), AGARD Conference Proceedings, Vol. 97, pp. 27.1–27.19. Rao, N.N., Shukla, P.K., Yu, M.Y., 1990. Dust-acoustic waves in dusty plasmas. Planet. Space Sci. 38, 543–546. Rosenberg, M., 2001. EJect of charged dust on Hall current instabilities in the E region. IEEE Trans. Plasma Sci. 29, 261–266. Rosenberg, M., Chow, V.W., 1998. Farley–Buneman instability in a dusty plasma. Planet. Space Sci. 46, 103–108. Rosenberg, M., Shukla, P.K., 2000. Low-frequency Hall current instability in a dusty plasma. J. Geophys. Res. 105, 23,135–23,139. Rosenberg, M., Shukla, P.K., 2002. Parallel propagation eJects on low-frequency Hall current instability in a dusty plasma. Planet. Space Sci. 50, 261–267. Saito, S., Sato, N., Hatta, Y., 1964. Low frequency oscillations in co-axial cylindrical discharge tubes in a magnetic *eld perpendicular to an electric *eld. Appl. Phys. Lett. 5, 46–47. Shukla, P.K., Mamun, A.A., 2002. Introduction to Dusty Plasma Physics. Institute of Physics Publishing, Bristol and Philadelphia. Thompson, C., Barkan, A., D’Angelo, N., Merlino, R.L., 1997. Dust acoustic waves in a direct current glow discharge. Phys. Plasmas 4, 2331–2335.