aa r X i v : . [ a s t r o - ph ] M a y Growing drift-cyclotron modes in the hot solaratmosphere
J. Vranjes and S. Poedts
Center for Plasma Astrophysics, Celestijnenlaan 200B, 3001 Leuven, Belgium.
Abstract:
Well-known analytical results dealing with ion cyclotron and drift wavesand which follow from the kinetic theory are used and the dispersion equation, whichdescribes coupled two modes, is solved numerically. The numerical results obtained byusing the values for the plasma density, magnetic field and temperature applicable to thesolar corona clearly show the coupling and the instability (growing) of the two modes. Thecoupling happens at very short wavelengths, that are of the order of the ion gyro radius,and for characteristic scale lengths of the equilibrium density that are altitude dependentand may become of the order of only a few meters. The demonstrated instability of thetwo coupled modes (driven by the equilibrium density gradient) is obtained by using arigorous kinetic theory model and for realistic parameter values. The physical mechanismwhich is behind the coupling is simple and is expected to take place throughout the solaratmosphere and the solar wind which contain a variety of very elongated density structuresof various sizes. The mode grows on account of the density gradient, it is essentially anion mode, and its further dissipation should result in an increased ion heating.1
Introduction
The ion cyclotron resonance and the ion cyclotron mode have been discussed in thecontext of problems related to the heating of the solar corona, e.g., Marsch et al. (1892),Cranmer et al. (1999), Cranmer (2000), Tu & Marsch (2001), Markovskii (2001), Isenberg(2001), Hollweg & Isenberg (2002). This is due to various reasons, such as the evidenceobtained from in situ measurements in the solar wind and coronal holes of resonant ioncyclotron heating (Hollweg & Isenberg 2002), and a preferential heating of coronal ions(with respect to electrons) which is most dominant in the direction perpendicular to themagnetic field lines. The damping of such IC waves is believed to be a good candidate forthe consequent coronal heating and solar wind acceleration Markovskii (2001). However,as in many wave-heating scenarios of the corona proposed in the past, there is usuallythe problem of the source for the required massive generation of such IC waves. In theliterature, various effects have been proposed as sources for the IC mode, like currents(Forslund 1970, Toichi 1971), global resonant MHD modes Markovskii (2001), etc. Yet,we note that these effects themselves need some source, so the problem is not solved butmerely shifted to another problem. The multi-fluid description of the beam-driven IC waveexcitation is presented in Mecheri & Marsch (2007), where a tiny ion-beam populationis assumed to originate from small scale reconnections. A rather large growth rate (ofthe order of 0 . p , where Ω p is the proton gyro-frequency) is obtained for relatively largevalues of the wave number, and for angles of propagation below 60 ◦ .In one of the recent studies dealing with the generation of ion cyclotron waves in thesolar atmosphere Markovskii (2001) it is pointed out that the solar atmosphere containsdensity inhomogeneities of various scales with density gradients in the direction perpen-dicular to the magnetic field. Indeed, detailed studies performed in the past, which includedirect observations, reveal the existence of ray-like structures that span from very largeperpendicular cross sections, like in the case of polar plumes and streamers, down to veryfine filamentary structures with a cross section of the order of a kilometer. More detailson that issue are given in Woo & Habbal (1997), Woo (1996), November & Koutchmy(1996), Karovska & Habbal (1991), and in references cited therein.Fine filaments are visible even from ground-based observations like those during theeclipse in 1991 (November and Koutchmy 1997), showing a slow radial enlargement of thestructures (i.e., in the direction out of the Sun) which is consistent with a low-beta plasma.2n situ measurements by Voyager 1 and Voyager 2 (Woo & Habbal 1997), show that thefinest structures in the slow solar wind at around 9 R ⊙ are about 3 times finer than thosein the fast wind. Assuming a radial expansion, they conclude that the transverse sizesof these highly elongated structures at the Sun are below 1 km. In Karovska & Habbal(1991) an image restoration method is used to study the structures of the quiet Sun, witha spatial resolution of 5” and a temporal resolution of 5.5 minutes. The contour mapspresented in the work reveal the existence of numerous structures of various sizes.The presence of density gradients in the direction perpendicular to the magnetic fieldlines implies the possibility of the existence of a drift wave, a mode with the unique featureof being unstable both in the kinetic and the fluid domain (hence the term universallyunstable mode ). Within kinetic theory, the drift mode is unstable even for a Maxwellianelectron distribution. This is due to the inverse electron Landau damping effect. Withinfluid theory, the mode is also unstable, due to the common effect of the electron collisionsand the ion inertia, in the presence of a background density gradient. A comparisonbetween the kinetic and the fluid instability (Goldston & Rutherford 1995) reveals thatthe resistive one is dominant provided that the electron parallel mean-free path is smallerthan the parallel wavelength. This is essentially the reason for the interest in the driftmodes with very large parallel wavelengths and relatively short perpendicular wavelengths.One particularly interesting feature of the drift wave is related to its intrinsic nonlin-earity (Hasegawa & Sato 1989). The nonlinear terms, which follow from the convectivederivatives in the momentum equations, are comparable to the linear ones provided that k ⊥ ρ s eφ/ ( κT e ) is of the same order as ρ s /L n . Here, φ is the electrostatic wave potential, ρ s = c s / Ω i , Ω i is the ion gyro-frequency, c s = κT e /m i , L n is the scale length of the densitygradient, and ρ s /L n is usually a small quantity. Hence, the mode becomes nonlinear evenfor very small perturbations provided that k ⊥ ρ s is not much larger than unity. Neverthe-less, in the literature dealing with the waves and instabilities in solar plasma, the effectsrelated to the density gradient and the consequent drift waves are usually disregarded.In our recent publications, some aspects of the drift wave instability in the solar plasmahave been discussed. The collisional coupling between the drift and kinetic Alfv´en wavesin the upper solar atmosphere (Vranjes & Poedts 2006) reveals a strongly growing driftmode in the chromospheric plasma. The similar collisional instability of the mode, thoughwith a much smaller increment, has been demonstrated also for the coronal plasma. Inboth cases, the kinetic Alfv´en part of the mode is shown to be collisionally damped. In3he presence of a plasma flow along the magnetic field lines the drift mode is subject toa reactive-type instability, provided that this background velocity has a gradient in theperpendicular direction. Such a problem has been discussed in our recent work (Saleem,Vranjes & Poedts 2007) dealing with solar spicules. The drift mode has been shown tobe unstable for typical spicule characteristic lengths of the density and the shear flowgradients, i.e. in the range of a few hundred meters up to a few kilometers, yielding wavefrequencies of the order of a few Herz.One of the basic properties of the drift mode is the low frequency | ∂/∂t | ≪ Ω i . In thecase when this low-frequency limit is not well satisfied, there appears a coupling betweenthe drift and ion-cyclotron modes. In the limit when | ∂/∂t | ∼ Ω i , this coupling is veryeffective and well known (Ichimaru 1973). In the simplest case, it yields a coupled unstabledrift-cyclotron mode with the instability driven by the plasma density gradient. The firstdemonstration of this instability was performed by Mikhailovskii & Timofeev (1963).To the best of our knowledge, the perpendicular density gradients, and the couplingof the drift and IC modes in the regime ω ∼ Ω i together with the consequent waveinstability, have never been discussed in the context of the heating of the solar corona andthe acceleration of the solar wind. In the present work we describe the basic instabilityof the mode to focuss attention on the existence of such an instability that may bewidespread and may contribute substantially to the problems discussed above. The term’fine structures’, used in the text above for the structures observed in the solar coronastill denotes spatial scales that are much larger than those used in our present study, inSections 2 and 3. In other words, there is no observational support for the existence ofsuch tiny density inhomogeneities, and it is not expected in the near future. Yet, in viewof the variety of density structures at larger and observable spatial scales, and having noobvious physical effect that would prevent inhomogeneities at scales that are even shorterthan those observed so far, such micro scale plasma inhomogeneities cannot be excluded. Following Ichimaru (1973), the plasma dielectric function in the case of a negligible parallelwave vector and for the frequency limit ω ∼ Ω i , and k ⊥ = k y = k within the kinetic theoryfor hot ions and electrons, is given by (in the local approximation) ǫ ( k, , ω ) = 1 + k e k (cid:20) ω ∗ e ω + 1 − Λ ( β e ) (cid:21) k λ D ( − ( ω − ω ∗ ) " Λ ( β ) ω + Λ ( β ) ω − Ω i . (1)Here, β e = k T e T i m e m i ρ L < , β = k ρ L ≫ , (2) ω ∗ = ω ∗ e T i /T e , ω ∗ e = n ′ n T e k y m e Ω e , k e = ω pe v T e , Λ n ( X ) ≡ I n ( X ) exp( − X ) , ρ L = v T i Ω i ,I n denotes the modified Bessel function of the n − th order, and the prime denotes thederivative in the direction perpendicular to both the wave-vector and the magnetic field.Using Λ n ( β ) → (2 πβ ) − / exp( − n / β ) (for β → ∞ ), we have Λ ( β ) ≃ / [(2 π ) / kρ L ] ≡ δ . In view of the first expression in Eq. (2), we have Λ ( β e ) ≃ − β e , while from thesecond one we have δ ≪
1. The dispersion equation then becomes (cid:16) k λ d − δ (cid:17) ω − h Ω i (cid:16) k λ d (cid:17) + ω ∗ (1 − δ ) i ω + ω ∗ Ω i = 0 , (3) λ d = ρ L m e m i + 1 k e T i T e . In the two limits ( ω ≪ Ω i and ω ∼ Ω i ), the two modes are the drift wave and the ICwave, respectively, ω = ω ∗ / (1 + k λ d ) and ω ∼ Ω i [1 + δ/ (1 + k λ d )]. The instabilitymay appear at the point of eventual intersection of the two dispersion curves, and theinstability condition reads:4 ω ∗ Ω i (cid:16) k λ d − δ (cid:17) > h Ω i (cid:16) k λ d (cid:17) + ω ∗ (1 − δ ) i . (4)Below, we apply these expressions to the solar atmosphere in order to see if there is awindow in the relevant parameter domain allowing for the instability. The necessary instability condition (4) can be satisfied for a chosen set of plasma param-eters n , T, B , L n = ( n ′ /n ) − and for a given wavelength. Because of the horizontal andvertical stratification, various values may be considered for the density, the temperatureand the magnetic field. Yet, physically, in order to have an instability, the frequencies ofthe drift and IC modes must become close to each other, and this is most easily controlledby the density inhomogeneity scale-length L n and/or the wave-length. Here, L n is a local,5patially dependent parameter that determines the local properties of the drift-cyclotronmode. Assuming a cylindric elongated density structure with a radius r and with aGaussian radial density distribution n ( r ) = N exp( − r /a ) , where N is the density at the axis of the cylinder, and a determines the radial decrease ofthe density, we have L n ( r ) = n ( r ) /n ′ ( r ) = a / (2 r ). Hence, we have a radially changingscale-length, which goes to infinity at the center and decreases towards the boundary.In the case of an e-folding decrease along r , we have a = r , n ( r ) /N = 37%, and L n /r = 1 /
2. For such a density profile L n is minimum in the outer region of the plasmacolumn. As a result, in the eigen-mode analysis of the drift wave in the cylindric geometrythe amplitude and the increment of the drift mode are maximum in the same region(Bellan 2006, Vranjes & Poedts 2005).Hence, Eq. (3) is solved numerically in terms of the wavelength and the density scalelength, for parameter values applicable to the solar corona. As an example we take B = 10 − T, n = 10 m − , T e = T i = 10 K, that may be used to describe the physicalproperties of the plasma at the altitude of around one solar radius, and we have chosena very short density inhomogeneity scale-length, L n = 5 m. In the case of the Gaussianprofile discussed above and for the e-folding decrease, this yields the characteristic radiusof the structure r ≃
10 m. Such small values for L n are necessary to obtain high valuesfor the drift wave frequency because it is proportional to 1 /L n . For these parameters, theplasma beta is 0 . ρ L = 0 .
94 m. For larger wavelengths,the frequencies of the two modes become well separated and the instability vanishes.The maximum increment ω i is ≈ ω r = 102830 Hz. Thisvalue of the increment is lower than the approximate theoretical value (Mikhailovskii &Timofeev 1963) given by Ω i ( m e /m i ) / . The small values for L n imply a relatively shorttime for the existence of such structures, making them difficult to detect. As seen fromFig. 1, the frequencies of the corresponding modes are high, of the order of 10 Hz. Onthe other hand the ion collision frequency ν ii = 4 n i ( π/m i ) / [ e i / (4 πε )] L ii / [3( κT i ) / ]for the given parameters is about 10 − Hz. The perpendicular ion diffusion coefficient(Chen 1988) is D ⊥ ≈ κT i ν i / ( m i Ω i ) = 0 .
01 m /s. The diffusion velocity in the direction6f the given density gradient is D ⊥ ∇ n/n = 2 mm/s only. So we have about 7 ordersof magnitude difference for the two characteristic times, and this is enough time for theinstability to develop before the equilibrium density structure disappears.In Fig. 2, we fix the wave-length at λ = 0 .
95 m, for the same parameter values as inFig. 1, and calculate the frequency in terms of the density length scale L n . The maximumincrement ω i = 5361 Hz is obtained at L n = 5 . ω r = 102504 Hz. Compared to the recent results of Mecheri& Marsch (2007), the present growth-rate driven by the density gradient appears to besmaller.We note that in some studies, e.g., Coles and Harmon (1989), the existence of a wave-number cutoff has been found at the ion inertial length λ i = c/ω pi , preventing densityperturbations for wavelengths shorter than λ i . However, this should not be confusedwith the density structures in our work. We are dealing with electrostatic perturbationsat considerably shorter time and space scales that are presently difficult to detect. Moreimportantly, there seems to be no definite consensus about the nature and the origin of thementioned wave-length cutoff. Although it is a separate issue, in our view it seems verylikely that the mentioned cutoff should be attributed to the problem of electromagneticperturbations at spatial/time scales satisfying ( ω/k ) ≪ c , and for wavelengths belowthe ion inertial length. As it is known from plasma theory (Shukla et al. 2001; Stamper& Tidman 1973; Vranjes et al. 2007; Yu & Stenflo 1985) in this case, the displacementcurrent in the Amp`ere law ∇ × B = µ ~j + µ ε ∂ ~E/∂t can be omitted, and the ionperturbations become negligible. Further, setting Amp`ere’s law (without the last term)in the electron continuity equation reveals that the electron density perturbations vanish. The instability discussed here implies very short scale lengths for the inhomogeneity ofthe equilibrium density and/or a very weak magnetic field. Only in such circumstancescan the frequencies of the drift and IC modes become close to each other so that thetwo modes can effectively couple. Short density scale lengths are presently not directlyobservable in the solar atmosphere. However, we have learned that improvement of theresolution in observations of phenomena in the solar atmosphere tends to result in anincreased variety and complexity of the density and/or magnetic field structures at short7cales, showing very tiny filaments at scales below 1 km. Therefore, one may expecta larger variety at even shorter scales. The present analysis clearly demonstrates theinstability of perpendicularly propagating modes at frequencies in the range of the ioncyclotron frequency and at wavelengths of the order of the ion gyro radius. The tinyfilaments are very elongated, they may extend to many solar radii, and the growth ofthe mode and the consequent dissipation and heating of ions may take place over largedistances. The numbers used here are for the solar corona however, the large radial (fromthe Sun) length of the structures and the consequent decreasing of the magnetic fieldintensity implies larger density scale lengths at which the instability takes place. Thiscan be easily shown by reducing the magnetic field to 10 − T and the number density byone order of magnitude. As a result, the necessary density scale length L n for the unstablemodes becomes of the order of 60 meters. Therefore the development of unstable growingmodes may take place at large distances along the same density filaments that pervadethe corona and spread within the solar wind. The presence of hotter ions in and aroundsuch filamentary structures should be interpreted as an indication and a signature of theinstability.There have been intensive searches for possible mechanisms for the excitation of suchmodes. Our analysis is based on one such mechanism that is well known, but not usedin the context of the solar plasma, and it requires a step beyond the widely used MHDmodel. At higher densities or temperatures the plasma beta becomes higher so thatelectromagnetic effects should be included. However, it is known that eventual bendingof the magnetic field lines implies an additional obstacle for electron motion along themagnetic lines, making the drift-type modes even more unstable (Vranjes et al. 2007b).The parameter values that we use here are realistic and the instability that has beendemonstrated for the given cases is thus physically very likely.Acknowledgements:These results are obtained in the framework of the projects G.0304.07 (FWO-Vlaanderen),C 90205 (Prodex), GOA/2004/01 (K.U.Leuven), and the Interuniversity Attraction PolesProgramme - Belgian State - Belgian Science Policy.8 eferences [1] Bellan, P. M. 2006, Fundamentals of Plasma Physics (Cambridge Univ. 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1. The frequencies and the increment (multiplied by 10) of the coupled drift-cyclotronmode in terms of the wave-length.2. The frequencies and the increment of the coupled drift-cyclotron mode in terms ofthe characteristic density gradient length. Here, the increment is multiplied by 10.11 .8 1.0 1.2 1.4 1.6 1.8 2.00.04.0x10 growth rate (x 10) drift modeIC mode [m] f r equen cy [ H z ] .5 5.0 5.5 6.0 6.5 7.00.04.0x10 growth rate (x 10)drift modeIC mode f r equen cy [ H z ] L nn