Interstellar Turbulent Magnetic Field Generation by Plasma Instabilities
aa r X i v : . [ a s t r o - ph . H E ] J un Interstellar Turbulent Magnetic Field Generation byPlasma Instabilities
R. C. Tautz • J. TriptowAbstract
The maximum magnetic field strength gen-erated by Weibel-type plasma instabilities is estimatedfor typical conditions in the interstellar medium. Therelevant kinetic dispersion relations are evaluated byconducting a parameter study both for Maxwellian andfor suprathermal particle distributions showing thatmicro Gauss magnetic fields can be generated. It isshown that, depending on the streaming velocity andthe plasma temperatures, either the longitudinal or atransverse instability will be dominant. In the presenceof an ambient magnetic field, the filamentation insta-bility is typically suppressed while the two-stream andthe classic Weibel instability are retained.
Keywords plasmas — magnetic field — interstellarmedium — instabilities — counterstream
Galactic magnetic fields are ubiquitous (see Beck et al.1996, for an overview). Even in galaxies at high red-shifts, magnetic fields have been found (Bernet et al.2008). The correlation between far-infrared radiationof massive stars and radio emission produced by syn-chrotron radiation of energetic particles in the sur-rounding magnetic fields (Murphy 2009). Such impliesa connection between the formation of massive starsand galactic magnetic fields.The generally accepted model for the generation ofgalactic magnetic fields is found in the dynamo pro-cess (Beck et al. 1996; Brandenburg and Subramanian2005; Kulsrud 2010), which, however, requires a seed
R. C. TautzJ. TriptowZentrum f¨ur Astronomie und Astrophysik, Technische Univer-sit¨at Berlin, Hardenbergstraße 36, D-10623 Berlin, Germany magnetic field (Schlickeiser 2005; Schober et al. 2012).Among other processes (e. g., Ryu et al. 2012; Durrer and Neronov2013), seed fields can be generated by plasma insta-bilities for example in the neighborhood of massivestars that ionize the surrounding interstellar medium(Schlickeiser 2012). A special class of such instabilitiesgenerates modes that purely grow in time and do notpropagate—the so-called “aperiodic” modes (Weibel1959; Tautz and Lerche 2012a). The fact that suchmodes can be emitted spontaneously even in unmag-netized plasmas (Yoon 2007; Tautz and Schlickeiser2007; Yoon and Schlickeiser 2012; Lazar et al. 2012)again underscores the validity of the process. Onsmaller scales, magnetic fields play an important rˆole inthe formation of molecular clouds (Inoue and Inutsuka2012), star formation, and thermally unstable interstel-lar flows (Mantare and Cole 2012). Furthermore, ape-riodic modes are essential for particle acceleration atcosmic shocks (e. g., Reville et al. 2008; Niemiec et al.2010). In general, the coupling of matter and magneticfields is confirmed by the typical scaling B ∝ √ n forrelatively high particle densities (Heiles and Crutcher2005).Because of the typically low plasma densities, therelevant processes have to be described using kineticplasma theory (see, e. g., Davidson 1983; Schlickeiser2002; Tautz 2012, for an introduction), which hasa long tradition. Much of the progress in cata-loging waves in plasmas, both non-relativistically aswell as relativistically, has ably been summarized byClemmow and Dougherty (1969) and, with astrophysi-cal applications much to the fore, by Schlickeiser (2002),where copious references to the many advances in un-derstanding such waves are to be found. Typically, con-centration is focused on simplified geometries, for exam-ple modes propagating parallel or perpendicular withrespect to a given symmetry axis such as a streamingdirection or an ambient magnetic field. A considerable k Γ = ℑ ( ω ) Γ max k max k end Fig. 1
Schematic plot of the imaginary frequency part—the growth rate, Γ —as a function of the wavenumber inarbitrary units. The maximum growth rate, Γ max , lends itsname to the associated wavenumber, k max . The wavenum-ber at which the growth rate vanishes, in contrast, is labeled k end . The shaded area denotes the region of instability. amount of work has been done on oblique propagat-ing wave modes (e. g., Bret et al. 2006; Gremillet et al.2007) and general coupling effects between the variousmodes (e. g., Tautz et al. 2006, 2007; Tautz and Lerche2012b).Here, the maximum magnetic field strength gen-erated by a special class of plasma instabilities—so-called “Weibel-type” instabilities (Weibel 1959; Fried1959; Achterberg and Wiersma 2007; Tautz and Lerche2012a)—will be investigated by means of a parameterstudy. Such instabilities have been the focus of in-tense research for some time regarding both their lin-ear and non-linear stages. Based on a recently devel-oped method (Tautz 2011), the maximum growth rateand the associated wavenumber can be efficiently deter-mined. In contrast to the work mentioned above, whichexplained the generation of seed magnetic fields in theearly universe, we aim at small-scale magnetic fieldgeneration using present-day conditions. Instead of afully non-linear calculation (see, e. g., Achterberg et al.2007), a simple estimation will be used that has beenconfirmed by numerical particle-in-cell simulations.Throughout, Gaussian cgs units will be used.This article is organized as follows: In Sec. 2, the dis-persion relation are introduced together with the initialdistribution functions that describe the streaming andtemperature anisotropies. The relations used to esti-mate the maximum magnetic field strength and spatialscales on which it varies are introduced in Sec. 3. InSec. 4, a parameter study is conducted to illustrate themagnetic field growth for various combinations of theparameters for particle density, electron temperature, v || f ( v || ) M.B. κ = 1 κ = 2 κ = 4 Fig. 2 (Color online) The Maxwell-Boltzmann (black dot-ted line) and kappa-type distribution functions in arbitraryunits. The latter distribution is shown for different values ofthe index κ . The shaded area illustrates the suprathermaltail of the kappa-type distribution function. and streaming velocities, which all form the anisotropypattern responsible for the instability behavior. Addi-tionally, the effect due to a suprathermal particle pop-ulation will be demonstrated. Sec. 5 provides a shortsummary and a discussion of the results. According to the linearized Vlasov theory (see, e. g.,Davidson 1983; Schlickeiser 2002; Tautz 2012, for anintroduction), magnetic field growth can be describedin terms of plasma instabilities. By limiting ourselves tothe simplified cases of parallel and perpendicular wavevectors with respect to a given symmetry axis, threedispersion functions (see Tautz and Schlickeiser 2005a,2006) can be derived, which describe: (i) the paral-lel longitudinal mode, D ℓ ; (ii) the parallel transversemode, D t ; and (iii) the perpendicular ordinary wavemode, D ⊥ . The solution of the dispersion relations, D i = 0, then yields the (complex) frequency as a func-tion of the wave vector, where the imaginary frequencypart describes growth (if positive) or damping (if nega-tive). For the velocity distribution functions introducedin subsection 2.2, the dispersion relations are summa-rized in Appendix A.2.1 Maximum growth rateHere, however, we are more interested in the maximum growth rate, which, according to τ ∼ Γ − , has theshortest characteristic growth time. By using the im-plicit function theorem, the maximum growth rate can ( k ⊥ w ⊥ / Ω) / Γ / Ω Fig. 3
Growth rate as obtained from the analytical so-lution of the dispersion relation for perpendicular wavepropagation (lines) in comparison to the maximum growthrate resulting from a PIC simulation (dots). The solidand dashed lines correspond to the cases of T k = T ⊥ and T k = 2 T ⊥ , respectively, with ( v /w k ) = 2 in both cases (cf.Tautz and Schlickeiser 2006; Tautz and Sakai 2007). be directly obtained (Tautz 2011) together with the as-sociated “maximum” wavenumber fromd ω d k = − ∂D ( ω, k ) /∂k∂D ( ω, k ) /∂ω , (1)which, by requiring d ω/ d k = 0 in order to have anextremum of the growth rate as a function of thewavenumber, can be expressed as ∂D ( ω, k ) /∂k = 0 to-gether with D ( ω, k ) = 0. Thus, the maximum growthrate and the associated wavenumber are obtained as D ( ω, k ) = 0 ∧ ∂D ( ω, k ) /∂k = 0 ⇒ { k max , Γ max } . (2)The typical shape of the growth rate (see Fig. 1) ensuresthat the result indeed corresponds to the maximumgrowth rate. While the largest unstable wavenumber, k end , can often be determined analytically, the unstablewavenumber associated with the maximum growth rate, k max , is available only by numerically solving Eqs. (2).2.2 Distribution functionThe required free energy is provided through aniso-tropies in the form of: (i) two interpenetrating (“coun-terstreaming”) components with (ii) different tempera-tures in the directions parallel and perpendicular to thestreaming direction. For the initial distribution func-tion, two different forms are assumed. The first one isa Maxwellian (Tautz and Schlickeiser 2005a, 2006), f ( v ) = C − exp (cid:18) − v ⊥ w ⊥ (cid:19) X j = ± exp " − (cid:0) v k + jv (cid:1) w k , (3) where w k , ⊥ = r k B T k , ⊥ m (4)are the thermal velocities and where C = 2 π / w ⊥ w k is a constant factor to ensure the normalization of f .The second one is a kappa-type distribution to in-clude suprathermal particles (Lazar et al. 2008, 2010) f ( v ) = ˜ C − X j = ± " v ⊥ κθ ⊥ + (cid:0) v k + jv (cid:1) κθ k − ( κ +1) , (5)where the index κ characterizes the fraction of supra-thermal particles. The normalization factor is nowgiven as˜ C = 2 π / θ ⊥ θ k κ / Γ ( κ − / Γ ( κ + 1)and where the (modified) thermal velocities are θ k , ⊥ = r κ − κ w k , ⊥ . (6)Note that both distribution functions are limitedto non-relativistic temperatures and streaming veloc-ities. Relativistic (Tautz and Schlickeiser 2005b; Tautz2010) and semi-relativistic (Zaheer and Murtaza 2007;Tautz and Shalchi 2008) generalizations would requirethe use of relativistic dispersion relations and are, there-fore, considerably more difficult. Relativistic effectsmay be extremely important (Tautz et al. 2006) butonly if the relevant parameters are truly relativistic(Schaefer-Rolffs and Schlickeiser 2005). The maximum growth rate and the correspondingwavenumber in Eq. (2) can be used to estimate themaximum turbulent magnetic field strength generatedby the instability. Additionally, the associated spatialscales can be determined. Consider both in turn.3.1 Maximum magnetic field strengthAccording to Schlickeiser (2005), the maximum fieldstrength can be estimated from the condition that theLarmor radius in the generated magnetic field strengthbe comparable to the characteristic length scale. Thelatter is given through the wavenumber for which thegrowth is maximal, which leads to R L ∼ k − . (7) Table 1
Parameter values for the streaming velocity, v ,the temperature, T , the particle number density, n , and theambient magnetic field strength, B . Symbol Values Reference v
10 – 20 km/s Nehm´e et al. (2008) v
60 – 150 km/s Aalto et al. (1999) v . c – 0 . c Zweibel and Shull (1982) T
10 – 10 K Karttunen et al. (2007) n − – 10 cm − — B × − G Beck et al. (1996)With R L = v/Ω , where v = v is the streaming veloc-ity and where Ω = qB max / ( mc ) is assumed to be thegyrofrequency that results from the emergent field, themaximum magnetic field strength can be estimated to B max ∼ η mcq v k max . (8)Note that the presence of a background magnetic fieldstrength requires a considerably more complex calcula-tion (e. g., Kato 2005) involving the currents inducedby the background magnetic field. However, for the pa-rameters typically found in the interstellar medium, thestrongest magnetic fields usually stem from the longi-tudinal mode as shown below, which is unaffected bythe presence of a background magnetic field.The additional factor, η ∼ .
01, is due to the factthat numerical simulations typically show a somewhatreduced maximum magnetic field strength (Schlickeiser2005). The maximum growth rate and the correspond-ing wavenumber, in contrast, have been reproducedwith fairly good accuracy, as confirmed by Fig. 3.Both the positions and the magnitudes of the maxi-mum growth rate are in agreement with each other (seeTautz and Schlickeiser 2006; Tautz and Sakai 2007).For electrons, the maximum magnetic field resultingfrom Eq. (8) can be expressed as (cid:18) B max µ G (cid:19) ≈ . (cid:18) v km/s (cid:19) (cid:18) k max cm − (cid:19) , (9a)for electrons or, alternatively, as (cid:18) B max µ G (cid:19) ≈ . × − (cid:18) v km/s (cid:19)(cid:16) n e cm − (cid:17) / ˜ k max , (9b)where the usual normalization ˜ k = ck/ω p ,e is employedwith ω p ,e = p πn e q /m e the electron plasma fre-quency. The fact that the largest unstable wavenumber, k end , is typically of the same order of magnitude (cf.Fig. 1) as the maximum unstable wavenumber, k max ,allows one to use the first as a rough estimate, whereasthe second gives the more precise result.
50 100 150 200 25000.0020.0040.0060.0080.01 v [km/s] B m a x [ µ G ] T [K] B m a x [ µ G ] −5 −4 −3 −2 n [cm − ] B m a x [ G ] Fig. 4
Maximum magnetic field strength generated by thelongitudinal dispersion relation as a function of different pa-rameters. In the upper panel , the streaming velocity, v , isvaried, while other parameters are chosen as T = 10 Kand n = 10 cm − . In the middle panel , v = 100 km/sand n = 10 cm − . In the lower panel , v = 100 km/sand T = 10 K (solid line) as well as v = 150 km/s and T = 10 K (dashed line). L min = 2 π/k end L L max =2 π/k min . Even though usually k min →
0, the maxi-mum spatial scale, L max will be finite, being set by thesize of the localized intense gaseous streaming region(Schlickeiser 2005). A mean scale is given by h L i = 2 πk max = 3 . × (cid:16) n e cm − (cid:17) − / ˜ k − . (10) −2 −1 T ⊥ /T || B m a x [ µ G ] −2 −1 T || [K] B m a x [ µ G ] Fig. 5
Maximum magnetic field strength for the paralleltransverse dispersion relation. The upper panel illustratesinfluence of the temperature anisotropy on the maximummagnetic field strength for T k = 10 K (solid line) and T k =10 K (dashed line). In the lower panel , the huge influence ofthe thermal spread in the particle ensemble is demonstratedfor T ⊥ /T k = 2. In both panels, the particle density is chosenas n = 10 cm − . With ˜ k max being of the order unity, this corresponds to1000 km . h L i .
10 m for 10 − cm − . n e . cm − . In this section, the resulting maximum magnetic fieldstrength is presented that can be expected from thesaturation condition of the linear instability phase. Thecase of a Maxwellian distribution and the modificationsintroduced by a supra-thermal particle population willbe discussed in turn.4.1 Maxwellian distributionIn the upper and middle panels of Fig. 4, the maxi-mum magnetic field strength as obtained from Eqs. (9)is shown as a function of the counterstreaming veloc-ity, v , and as a function of the temperature enteringthe thermal velocity, respectively. For large tempera-tures and a moderate streaming velocities, the resultingmaximum magnetic field strength is relatively low, asconfirmed by the upper panel of Fig. 4, where v ≪ w with w the thermal velocity as defined in Eq. (4). Incontrast, the influence of the (in this case: isotropic)temperature on the resulting field strength for the case v [km/s] B m a x [ µ G ]
3 6 9 12 15 18 21 242.22.42.62.83 B [fG] τ [ m s ] Fig. 6
Maximum magnetic field strength and growth timefor the transverse, ordinary-mode dispersion relation. Inthe upper panel , an ambient magnetic field is shown to haveonly a moderate influence by the comparison of the two caseswith B = 3 fG (solid line) and B = 25 . lower panel , the growth time—i. e., the inverse growthrate—is illustrated for a variable background magnetic field,with v = 250 km/s fixed. In both panels, other parametersare chosen as T = 20 K and n = 10 cm − . w . v is shown in the middle panel. This compari-son confirms that, for thermal velocities exceeding thestreaming velocity, the instability rate is drastically re-duced as the thermal spread is no longer the dominantfeature of the particle distribution.In addition, the lower panel in Fig. 4 depicts themaximum field strength for two different temperaturesas the particle density (which enters both the growthand the normalized wavenumber via the plasma fre-quency) is varied. Note that, for normalized variablesas used in the dispersion relations (cf. Appendix A),Eq. (9) is the only density dependence—at least if col-lisions are completely neglected as done throughout thederivation of the dispersion relations.In the upper panel of Fig. 5, the effect of the temper-ature anisotropy on the transverse mode is shown fortwo different parallel temperature values. Note that,in contrast to the longitudinal two-stream mode, herea higher temperature increases the resulting magneticfield so that, in the limit of a cold plasma, the transverseWeibel instability is suppressed. In addition, while themaximum field strength is saturated as the tempera-ture ratio is increased, the parallel thermal velocity hasa steady influence with B max ∝ T / k as confirmed inthe lower panel. While the longitudinal mode is in- sensitive to the presence of a homogeneous backgroundmagnetic field, the transverse Weibel instability is mod-ified in that the resulting modes are no longer aperiodic.Instead, they have a real frequency part, ω r ≈ Ω with Ω = qB / ( mc ) the gyro-frequency (Tautz and Shalchi2008); these have been named mirror modes.In contrast, the ordinary-mode wave—also known asthe filamentation instability—is generated only in thepresence of an ambient magnetic field. However, de-pending on the plasma parameters, this backgroundfield must not be too strong because otherwise theinstability will be suppressed (Tautz and Sakai 2007;Stockem et al. 2007, 2008). Fig. 6 illustrates that astronger background magnetic field results in a strongerinstability; however, there is a critical backgroundmagnetic field strength above which the instability isquickly suppressed (Fig. 5 in Tautz 2011). In the limitof a cold plasma, the critical magnetic field strength isgiven by (Stockem et al. 2008) B crit = ω p mqc v √ γ (11a) ≈ . × − (cid:18) v km/s (cid:19) (cid:16) n cm − (cid:17) / . (11b)In contrast to the Weibel instability, both the growthrate and the maximum unstable wavenumber are de-creased for a warm plasma; accordingly, B crit as givenby Eq. (11) represents an upper level to the critical fieldstrength. Thus, for non-relativistic streaming velocitiesand moderate densities, the instability will almost al-ways be suppressed. Apart from the suppression, thebackground magnetic field has only a moderate influ-ence both on the resulting maximum turbulent fieldstrength and on the instability growth time, as con-firmed in both panels of Fig. 6.4.2 Suprathermal distributionThe effect of a suprathermal particle population is il-lustrated in Fig. 7. For the longitudinal mode, the up-per panel shows the ratio of the respective maximummagnetic field strengths as the counterstreaming veloc-ity is varied. It is confirmed that, for larger values ofthe spectral index, κ , the difference becomes less pro-nounced.In the lower panel of Fig. 7, the characteristic in-stability growth times are compared, which are simplygiven by τ = Γ − with Γ the growth rate. The ob-servation is that, depending on the precise choice ofthe instability parameters and the spectral index of thesupra-thermal tail, the instability can growth faster orslower compared to the Maxwellian case.From the detailed investigation of the kappa-typedistribution (Lazar et al. 2008, 2010), it is known that, v /c τ M W / τ κ v /c B m a x , M W / B m a x , κ Fig. 7
Modifications of the longitudinal mode introducedby the use of a kappa distribution function with κ = 2(solid lines), κ = 3 (dashed lines), and κ = 4 (dotdashedlines). Shown are the maximum magnetic field strengths (upper panel) and the instability growth times (lower panel) in relation to the Maxwellian case. for parallel wave propagation, the Maxwellian distri-bution provides an upper limit to the growth rate—and, accordingly, a lower limit to the instability growthtime; for perpendicular wave propagation, the situationis reversed so that the growth rate exceeds that for theMaxwellian. However, as shown in Fig. 7, the correc-tion factors are usually close to unity if the streamingvelocity is not too small. In this paper, three linear plasma instabilities have beeninvestigated, which are: (i) the longitudinal two-streaminstability, (ii) the classic Weibel instability, and (iii)the perpendicular filamentation instability with the lat-ter two being transverse modes. In contrast to previ-ous investigations that relied on normalized parametersand determined the instability growth rate (typicallyin relation to the plasma frequency) as a function ofthe wavenumber, here the maximum turbulent mag-netic field strength, B max has been investigated thatcan be generated by these unstable modes. Whereasan exact determination of B max requires knowledge ofthe non-linear behavior of the instability, a simple esti-mate has been used that involves only the wavenumberassociated with the maximum growth rate. For parameter values that are typical for environ-ments such as molecular clouds or the diffusive inter-stellar medium, the following main results have beenfound: • for the longitudinal mode (two-stream instability),the temperature has an overarching influence, whichcan be understood when bearing in mind that the ratio of oriented streaming and random thermal mo-tion dictates the resulting instability rate. For lowtemperatures, therefore, magnetic field generation isconsiderably more efficient; • the transverse electromagnetic mode (classic Weibelinstability), in contrast, is more efficient for highplasma temperatures with the perpendicular temper-ature being significantly higher than the parallel tem-perature; • the perpendicular ordinary-wave mode (filamenta-tion instability) is most efficient for cold plasmas.Finite temperatures decrease the resulting instabil-ity rate but tend to stabilize the mechanism providedthat the parallel temperature exceeds the perperpen-dicular temperature; • for all modes, the maximum field strength scales withthe square root of the particle density; however, inthat regard it has to be noted that collisional effectshave been neglected throughout. • while an ambient magnetic field leaves the longitudi-nal mode unaffected, the other modes are modified inthat: (i) the transverse mode has now an oscillationfrequency of the order of the gyrofrequency; and (ii)the perpendicular mode is, for the parameters con-sidered here, suppressed even for a magnetic field aslow as . . β ).An additional uncertainty is owed to the fact thatonly the cases have been investigated with unstablemodes oriented parallel and perpendicular to a givensymmetry axis. It has been known that the fastestgrowing mode usually has an oblique axis of wave prop-agation (Dieckmann et al. 2006). In that case, however,mode-coupling effects have to be taken into account (Tautz et al. 2007; Tautz and Lerche 2012b). Further-more, the presence of multiple particle species suchas electrons, positrons, and ions introduce additionaleffects in the resulting growth rate (Tautz and Sakai2008). These effects, which will modify the maximummagnetic field strengths presented here, will thereforebe incorporated in future work. Acknowledgements
J. T. thanks D. Breitschwerdtfor the supervision of her thesis.
A Dispersion Relations
For a any given distribution function with a specified anisotropy pattern, the dispersion relations D ℓ , D t , and D ⊥ can be evaluated. The resulting equations relating ω ∈ C and k k or k ⊥ ∈ R are usually non-linear and oftentranscendental. In most cases, a numerical solution is required, even though a series expansions can often lead toreasonable approximative solutions.It should be mentioned that there are investigations without specifying a distribution function (e. g.,Schaefer-Rolffs and Lerche 2006; Tautz and Lerche 2012b), which has shed light on the general behavior of theinstability. A comparison of the instability for various distribution functions (Schaefer-Rolffs and Tautz 2008) hasshown that the mechanism is indeed robust and does not strongly depend on the precise form of the distribution,as long as the anisotropy clearly dominates over the thermal spread of the particle ensemble.Furthermore, note that all dispersion relations are valid in the non-relativistic regime only, i. e., for counter-streaming and thermal velocities small compared to the speed of light. For a discussion of relativistic effects see,e. g., Schaefer-Rolffs and Schlickeiser (2005); Tautz and Schlickeiser (2005b); Tautz and Lerche (2012b).A.1 Maxwellian distributionFor the Maxwellian distribution from Eq. (3), the temperature- and κ -dependent parameters θ k and θ ⊥ nowplay the role of the thermal velocities in the sense that, by calculating the first moment of the distribution, theappropriate θ is obtained.For a plasma consisting of multiple particle species (denoted with the index a ), the longitudinal dispersionrelation, D ℓ , reads (Tautz and Schlickeiser 2005a) D ℓ = k − X a (cid:18) ω p ,a w k (cid:19) (cid:20) Z ′ (cid:18) ω − v kkw k (cid:19) + Z ′ (cid:18) ω + v kkw k (cid:19)(cid:21) = 0 , (A1)where Z ′ is the derivative of the plasma dispersion function, Z ( x ) = 1 √ π Z ∞−∞ d t e − t t − x = i √ π e − x [1 + erf( ix )] , (A2)where the first form is valid for ℑ ( x ) > B , is present—theleft-handed and right-handed modes—which can be expressed as (Tautz and Schlickeiser 2005a) D ± t = ω − c k − X a ω ,a ∓ X a ω ,a w k Ωk (cid:20) Z (cid:18) ω − kv ± Ωkw k (cid:19) + Z (cid:18) ω + kv ± Ωkw k (cid:19)(cid:21) − X a ω ,a (cid:18) w ⊥ w k (cid:19) (cid:20) Z ′ (cid:18) ω − kv ± Ωkw k (cid:19) + Z ′ (cid:18) ω + kv ± Ωkw k (cid:19)(cid:21) . (A3)Note that, due to the linear factor Ω in the terms containing Z ( . . . ), the dispersion relation is greatly simplifiedfor an unmagnetized plasma, i. e., where B = 0.For perpendicular wave propagation, the dispersion relation for the ordinary-wave mode reads (Tautz and Schlickeiser2006) D ⊥ = ω − c k + X a ω ,a + X a ω ,a w k + 2 v w ⊥ (cid:20) − F (cid:18) ,
1; 1 + ωΩ , − ωΩ ; − k w ⊥ Ω (cid:19)(cid:21) , (A4)where F ( a, b ; c, d ; z ) is the generalized hypergeometric function.A.2 Suprathermal distributionFor the kappa-type distribution function that includes particles forming a so-called supra-thermal tail, the longi-tudinal dispersion relation reads (Lazar et al. 2008) D ℓ = k + X a ω ,a θ k (cid:20) − κ + ω − kv kθ k Z κ (cid:18) ω − kv kθ k (cid:19) + ω + kv kθ k Z κ (cid:18) ω + kv kθ k (cid:19)(cid:21) , (A5) where one must take care not to confuse κ (the power-law index in the distribution function) with k (the wavenum-ber). The modified plasma dispersion function (Summers and Thorne 1991) is given through Z κ ( x ) = 1 √ πκ Γ ( κ ) Γ ( κ − / Z ∞−∞ d t (cid:0) x /κ (cid:1) − ( κ +1) t − x , (A6)where Γ ( z ) is the Gamma function. Again, Eq. (A6) is valid for ℑ ( x ) > D ± t = ω − c k − X a ω ,a ∓ X a ω ,a θ k Ωk (cid:20) ˜ Z κ (cid:18) ω − kv ± Ωkθ k (cid:19) + ˜ Z κ (cid:18) ω + kv ± Ωkθ k (cid:19)(cid:21) + 12 X a ω ,a (cid:18) θ ⊥ θ k (cid:19) (cid:20) ω − kv ± Ωkθ k ˜ Z κ (cid:18) ω − kv ± Ωkθ k (cid:19) + ω + kv ± Ωkθ k ˜ Z κ (cid:18) ω + kv ± Ωkθ k (cid:19)(cid:21) , (A7)where, for the plasma dispersion function, a new form has been introduced as˜ Z κ ( x ) = 1 √ πκ Γ ( κ ) Γ ( κ − / Z ∞−∞ d t (cid:0) x /κ (cid:1) − κ t − x , (A8a)which is related to Z κ ( x ) in Eq. (A6) as˜ Z κ ( x ) = (cid:18) x κ (cid:19) Z κ ( x ) + xκ (cid:18) − κ (cid:19) . (A8b)In general, the ordinary-wave mode for perpendicular wave propagation would involve a rather tedious integral.Therefore, Lazar et al. (2010) used the large-wavelength limit, in which case the dispersion relation can be writtenin simplified form as D ⊥ ( kR L ≪ ≈ ω − c k − X a ω ,a − k X a ω ,a v ω − Ω " (cid:18) w ⊥ v (cid:19) , (A9)which agrees with the corresponding expansion of the dispersion relation for the case of a Maxwellian distributionfunction, Eq. (A4). In the opposite limit of small wavelengths, the dispersion relation reads D ⊥ ( kR L ≫ ≈ ω − c k − X a ω ,a + X a ω ,c (cid:18) θ k θ ⊥ (cid:19) " (cid:18) − κ (cid:19) v θ k . (A10)A discussion of the applicability and numerical solutions connecting the two limiting cases has been given byLazar et al. (2010). References