Wave instabilities in an anisotropic magnetized space plasma
aa r X i v : . [ a s t r o - ph ] S e p Astronomy & Astrophysics manuscript no. DzhaKuSta-R c (cid:13)
ESO 2018November 21, 2018
Wave instabilities in an anisotropic magnetized space plasma(Research Note)
N.S. Dzhalilov , , , V.D. Kuznetsov and J. Staude Astrophysikalisches Institut Potsdam (AIP), An der Sternwarte 16, D-14482 Potsdam, Germany, Institute of Terrestrial Magnetism, Ionosphere and Radio Wave Propagation of the Russian Academy of Sciences(IZMIRAN), Troitsk City, Moscow Region, 142190 Russia, Shamakhy Astrophysical Observatory of the Azerbaijan Academy of Sciences (ShAO), Baku Az-1000, Azerbaijane-mail:
[email protected]; [email protected]; [email protected]
Received ... ; accepted 1 July 2008
ABSTRACT
Aims.
We study wave instability in an collisionless, rarefied hot plasma (e.g. solar wind or corona). We consider theanisotropy produced by the magnetic field, when the thermal gas pressures across and along the field become unequal.
Methods.
We apply the 16–moment transport equations (obtained from the Boltzmann-Vlasov kinetic equation) includ-ing the anisotropic thermal fluxes. The general dispersion relation for the incompressible wave modes is derived.
Results.
It is shown that a new, more complex wave spectrum with stable and unstable behavior is possible, in contrastto the classic fire-hose modes obtained in terms of the 13–moment integrated equations.
Key words.
MHD – Instabilities – Plasmas – Waves – Turbulence – Sun: corona – Solar wind
1. Introduction
An almost collisionless, rarefied, hot, magnetized spaceplasma such as that of the solar corona is anisotropic andinhomogeneous, in particular in the cross-field direction(see Aschwanden (2005)). There have been observations ofa thermal anisotropy of T ⊥ / T k ∼ − T ⊥ / T k > T k > T ⊥ is alsofound in solar wind observations (see Marsch (2006)). Dueto the anisotropy in the kinetic temperatures of protonsand heavy ions, the corresponding partial pressures be-come anisotropic, in addition to the total thermal pres-sure such that p ⊥ = p k . It is now generally accepted thatthe observed large ion temperature anisotropies are relatedto the physical mechanism by which the solar corona andsolar wind are heated (see Hollweg & Isenberg (2002) andMarsch (2006)).In these circumstances, it is difficult to develop a tradi-tional hydrodynamical description of the plasma. We there-fore attempt to extend the MHD approximation by consid-ering the anisotropy of the magnetized plasma. We considerthe large-scale wave peculiarities that can appear in a col-lisionless plasma. The large-scale plasma motions are usu-ally described by a fluid approximation, and the integratedmoment equations derived from the Boltzmann-Vlasov ki-netic equation are used. If collisions between particles arerare and a strong magnetic field approximation is valid, Send offprint requests to : J. Staude the usual MHD equations have to be replaced by otherequations for which the fluid approximation is valid too.It was shown, e.g. by Grad (1949), Chew et al. (1956), andRudakov & Sagdeev (1958), that for a collisionless plasma– mainly across a magnetic field – the fluid approach can beused. However, these so-called 13–moment equations can-not be used to describe a plasma of arbitrary anisotropicpressure. We therefore use the 16–moment equations.
2. Basic equations and wave equations
The 16–moment set of equations was used bymany authors in different theoretical approaches,especially for modeling the solar wind (seeDemars & Schunk (1979), Olsen & Leer (1999), Li (1999),and Lie-Svendsen et al. (2001)). The 16–moment set oftransport equations for the collisionless plasma in thepresence of gravity g but without magnetic diffusivity isgiven as follows (see e.g. Oraevskii et al. (1985)): dρdt + ρ div v = 0 , (1) ρ d v dt + ∇ ( p ⊥ + B π ) − π ( B · ∇ ) B = ρ g ++( p ⊥ − p k )[ h div h + ( h · ∇ ) h ] + h ( h · ∇ )( p ⊥ − p k ) , (2) ddt p k B ρ = − B ρ (cid:20) B ( h · ∇ ) (cid:18) S k B (cid:19) + 2 S ⊥ B ( h · ∇ ) B (cid:21) , (3) ddt p ⊥ ρB = − Bρ ( h · ∇ ) (cid:18) S ⊥ B (cid:19) , (4) ddt S k B ρ = − p k B ρ ( h · ∇ ) (cid:18) p k ρ (cid:19) , (5) Dzhalilov, Kuznetsov & Staude: Wave instabilities in an anisotropic plasma (RN) ddt S ⊥ ρ = − p k ρ (cid:20) ( h ·∇ ) (cid:18) p ⊥ ρ (cid:19) + p ⊥ ρ p ⊥ − p k p k B ( h ·∇ ) B (cid:21) , (6) d B dt + B div v − ( B · ∇ ) v = 0 , div B = 0 , (7)where ∇ = ∇ k + ∇ ⊥ , ∇ k = h ( h · ∇ ) ,ddt = ∂∂t + ( v · ∇ ) , v = v k + v ⊥ , h = B B , (8)and S k and S ⊥ are the heat fluxes along the magnetic fieldof parallel and perpendicular thermal motions. If the ther-mal fluxes are neglected, S ⊥ = 0 and S k = 0, we obtain theequations describing the laws of the change in longitudinaland transverse thermal energy along the trajectories of theplasma (the left-hand parts of Eqs. (3) and (4)). These so-called “double-adiabatic” invariants and Eqs. (1), (2), and(7) also form a closed system of equations, the CGL (Chew-Goldberger-Low) equations (see Chew et al. (1956)). Byusing the CGL-equations, we would however obtain incom-plete equations instead of Eqs. (5, 6). This is because, byderiving the CGL equations, authors so far omitted with-out proof the third moments of the distribution functionand therefore the thermal fluxes (see Chew et al. (1956)and Baranov & Krasnobayev (1977)). The equations de-rived for the 16–moment set, in our case Eqs. (1–7), in-clude the thermal fluxes; they are more complete, and theCGL equations cannot be derived from these equationsas a special case. For simplicity, we assume that the ba-sic initial equilibrium state of the plasma is homogeneous( g = 0, and the quantities v , ρ , p ⊥ , p k , B , S ⊥ , and S k are constant). Equations (1) to (7) automatically satisfythis equilibrium state with non-zero thermal fluxes. We con-sider small linear perturbations of all physical variables,for example pressure in the form p = p + p ′ ( r, t ), where p ′ ( r, t ) ∼ exp i ( k · r − ωt ), ω is the wave frequency, and k is the wave number. For the perturbations, we obtainlinear wave equations. Even if we insert zero initial heatfluxes S k = S ⊥ = 0, the perturbations of these functionswill never become zero: S ′k = 0, S ′⊥ = 0. Using the 16-moment equations, we should derive more reliable resultsabout the wave properties in an anisotropic plasma thanwith the CGL equations based on the 13-moment equa-tions.In the presence of an external magnetic field, the initialcollisionless heat fluxes should be defined by solutions of thekinetic equations. We should, however, use some appropri-ate estimate as a parameter. The heat flux functions shouldbe estimated by taking the thermal energy density of theelectrons multiplied by the particle stream speed along themagnetic field u : S k ≈ n e k B T k u δ = δu p k . Hollweg(1974, 1976) provided some estimates of the correction pa-rameter δ ( α in these papers) by assuming realistic shapesof electron distribution functions and comparing the resultswith space observations. We note that δ depends on themagnetic field. In the range of B = 0 . −
100 G, the esti-mates give δ ≈ − .
1. In the same way, S ⊥ ≈ δu p ⊥ ,and we define the parameter γ = (3 / δv /c k . We notethat Marsch & Richter (1987) quoted values of γ measuredin the solar wind. We introduce dimensionless parameters and note thatthe indices ”0” of physical parameters are omitted for sim-plicity: α = p ⊥ p k , c k = p k ρ , β = B πp k = v c k , (9)¯ S k = S k p k c k , ¯ S ⊥ = S ⊥ p ⊥ c k , l = cos φ, (10)where φ is the angle between wave vector and magneticfield, and the indices k and ⊥ correspond to the valuesof the parameters along and across the magnetic field, re-spectively. We note that our β is defined to be inverselyproportional to the more commonly used plasma beta, β = 2 /β plasma . With the parameter γ defines above, wehave ¯ S k = ¯ S ⊥ = γ .In analogy with the usual MHD equations used e.g.by Somov et al. (2007), there are two independent wavebranches in the plasma: waves that do not compress theplasma (div v = 0) and waves that compress the plasma(div v = 0). We restrict ourselves to the incompressiblewave modes.After inserting into the wave equations the condition ofincompressibility ( k · v ) = 0 and ρ ′ = 0, we obtain as usualthe parametric dispersion equation. This is a polynomialequation of 6 th order in the frequency of oscillations. Forthe parameter Z = ω/ ( c k k k ), this equation can be writtenin the form c Z + c Z + c Z + c Z + c Z + c Z + c = 0 , (11) c = 3 (1 − α ) (cid:2) α ( l −
1) + α (1 + l ) + 2 ( β − l ) (cid:3) ,c = 2 γ (cid:2) β ( α −
2) + 3 α (1 − l ) + α (4 l −
3) + 2 l (cid:3) ,c = 4 l + 2 β (4 α −
5) + 2 α (1 + 3 l ) + α ( l − −− α (1 + 11 l ) , c = − γ (cid:2) β ( α −
2) + α (2 l − − α (cid:3) ,c = 6 l + α ( l − − lα + 2 β (2 − α ) ,c = 2 γ (2 lα − l − α ) , c = α + lα − l. All coefficients are real and, consequently, all solutions arereal or conjugate complex. In the usual isotropic MHD case,only Alfv´en waves with ω = k k v A are present, the phasevelocities of which are equal to each other in both directionswith respect to the magnetic field. Instead of assuming that Z = β in the isotropic MHD, we determined now the 6 thorder Eq. (11) in the anisotropic case. With the heat fluxesfor which γ = 0, odd nonzero coefficients c , c , and c generate wave propagation velocities that depend on thedirection of the magnetic field.
3. Limiting and special cases
In this section we investigate the most important cases ofEq. (11) which can be solved analytically. zhalilov, Kuznetsov & Staude: Wave instabilities in an anisotropic plasma (RN) In the case l = 1 or k = k k , the six roots of Eq. (11) aresimple. The first pair of roots corresponds to a pair of stablemodes ω = ± k k c k . (12)The properties of these modes do not depend on the mag-netic field. They are isotropic in terms of the direction ofmagnetic field. This implies that stable waves propagatealong and against the magnetic field with the same phasevelocity. In these modes, ρ ′ = 0, v ′ = 0, B ′ = 0, p ′k = 0,and S ′k = 0. However, p ′⊥ ∼ S ′⊥ = 0. The restoring force forthese modes is therefore ∇ p ⊥ . Motions of particles acrossthe magnetic field cause a perturbation of the plasma pres-sure p ⊥ , but this is compensated by the generation of a heatflux. At the same time no hydrodynamic motion is producedby these modes, v = 0 as p ′k = 0. We therefore have unusualthermal waves: they are stable waves (Im( ω ) = 0) thatpropagate (Re( ω ) = 0) with the parallel sound speed c k .Analogous to sound waves, they are not dispersive modes.The modes of this branch are named “isotropic thermal”waves.The second pair of roots corresponds to ω/k k c k = ± p α + β − , (13)which are conventional isotropic fire-hose waves. The wavesbecome unstable if α + β < p k > p ⊥ + 2 p mag . The fire-hose instability therefore disappears for β ≥ α ≥ ω = k k v if α = 1.The third pair of solutions corresponds to2 ωk k c k = − γ ± r γ + 12 1 − α − α . (14)The instability condition ( α − / (2 − α ) > γ /
12 is obeyedif 1 < α <
2. For γ = 0, the unstable modes begin topropagate, Re( ω ) = 0. For γ = 0, stable waves outside theregion 1 < α < γ = 0, the stable waves for which ( α − / (2 − α ) < γ / γ , the prograde wavespropagate more slowly, but retrograde modes travel faster.There is a crossing of these branches on the axes of α for γ = 0 at V ph = 0, and for γ > V ph <
0. Between thecrossing points of the branches and α = 2, an instabilityarises. In contrast to the fire-hose and the thermal modes,these modes in the instability region are traveling, Re( ω ) =0. They do not depend on β . For these waves p ′k p k = 2(1 − α ) B ′ B , p ′⊥ p ⊥ = 1 − α − α B ′ B ,S ′k ∼ p ′k , S ′⊥ ∼ α − α − α B ′ B .
These modes are generated by pressure anisotropy: if α = 1,they disappear. The limit β ≫ Z ∼ O ( β ). Other isotropic thermal andthermally anisotropic waves remain unchanged. In the limit β ≪
1, the first pair of solutions of Eq.(11) sim-plifies to Z ≈ α −
1, which is the same solution as that forthe parallel fire-hose waves. The other thermal modes aredescribed by a polynomial equation of 4 th order. However,for the limit γ →
0, we obtain Z ≈ ( − b ∗ ± √ D ) / (2 a ∗ ) , D = b ∗ − a ∗ c ∗ , (15)where c ∗ = 3(1 − α )[ α (1 − l ) − l ] , a ∗ = α + l ( α − , and b ∗ = α (1 − l ) − α (1 + l ) + 10 l. These solutions caneasily be investigated. In the region 0 < α < α, l ), we have
D < Z ) = 0 andIm( Z ) = 0. This implies that both thermal waves becomeunstable, and D > α ≥
1. In this case for unstablemodes, Re( Z ) = 0 and instability disappears if α ≥
2. Thethermal wave branches have crossing points, where theirphase velocities coincide. For example, this occurs if l = 1and α = 0 . We consider the important special case of arbitrary β and γ = 0. Even though the absence of fluxes, S ⊥ =0 and S k = 0, is far from reality, this simplifiedcase was investigated using the 13–moment equations(e.g. Kato et al. (1966), Baranov & Krasnobayev (1977),and Kuznetsov & Oraevskii (1992)). Substituting γ = 0into Eq. (11), we obtain a cubic equation for ζ = Z : c ζ + c ζ + c ζ + c = 0 . (16)We have symmetrically only three pairs of solutions, Z = ±√ ζ . The analytical solutions ζ , ζ , and ζ allow us toinvestigate in significant detail the dependence of the so-lutions on the parameters α, β , and φ . For a small devia-tion of the propagation angle ( φ = 0), the situation dif-fers significantly from that of the parallel propagation. Allthree modes interact, and this interaction occurs in tworanges of α : α > α c and α < α c . The critical value of α = α c = 4 l/ (1 + l ) is the singular point for c = 0.With increasing propagation angle, the interaction domainsexpand. In these domains, the growing rates also becomelarger and we obtain a mixture of modes — a turbulentwave motion. To investigate the role of the thermal parameter γ , wesolved the 6 th order polynomial equation, choosing morerealistic values of γ <
1. In Fig.(1), the phase velocities andinstability rates are shown. We observe thata) waves with positive and negative velocities are different
Dzhalilov, Kuznetsov & Staude: Wave instabilities in an anisotropic plasma (RN) b = 1l = 0.5g = 0.1 a faai i a a R Z
K3K2K10123 i afi fa b = 1l = 0.5g = 0.1 a I Z
Fig. 1.
Phase velocities V ph = Re( Z ) of the three modes as a function of α for fixed parameters β = 1, φ = π/
4, and γ = 0 . Z ) of the three modes (right picture). The labels at the curves correspond tomodified fire-hose waves ( f ), to isotropic thermal waves ( i ), and to thermally anisotropic waves ( a ).and all three wave branches become coupled;b) maxima of the instability rate strongly depend on γ .For some parameters, all three modes become unstable. To compare our results with the classic fire-hose modesbased on the CGL equations, we use the initial set of equa-tions, omit Eqs. (5) and (6), and substitute into the others S k = S ⊥ = 0. These CGL equations for incompressiblewaves provide two pairs of solutions: Z = ± i √ A, A = (1 − α )[ α (1 + l ) − l ] + 2 β (2 − α ) α (1 + l ) − l . (17)The fire-hose instability condition is that A >
0. In theparallel propagation case ( l = 1), this condition is the mostfamiliar case of α + β <
1. We obtain the same result for l = 1, if β = 0. The instability condition is more completefor the oblique propagation case if β >
0, but this stronglydiffers from our results based on the 16–moment equations.For the classic fire-house instability, we always have thatRe( ω ) = 0. In our case, Re( ω ) = 0 in most cases due tothe coupling of these modes with other thermal modes. Forthe classic modes in the relations, for example, p ′k /p k = n B ′ /B and p ′⊥ /p ⊥ = n B ′ /B , the coefficients are n = − n = 1. In our case, these coefficients are completefunctions of all parameters. The main difference in our casecompared to that for the CGL equations is the appearanceof two additional thermal branches, even if γ = 0.
4. Conclusion
To investigate the peculiarities of large-scale wave motionsin a collisionless magnetized plasma, we have applied the16–moment transport equations, derived as integrated mo-ments of the kinetic equations. In earlier similar attempts, the 13–moment equations were used. However, these equa-tions exclude without any reason the thermal fluxes andare therefore incomplete.Anisotropy is the main feature of a collisionless plasmawith a strong magnetic field. In the present study, the pres-sure anisotropy was described by the parameter α and theheat fluxes by γ . By assuming that γ = 0 we were unable toderive the 13–moment equations, or by assuming that both α = 1 and γ = 0 we did not obtain the isotropic MHDcase. The 16–moment equations were in principle differ-ent equations. Using these equations, we illustrated that awide unstable and stable wave spectrum in the collisionlessanisotropic plasma was possible, even in the incompress-ible approximation. If γ = 0 (heat fluxes are present), thewaves propagated along and against the magnetic field atdifferent speeds. This behavior differed from that of theusual isotropic MHD case. The coupled wave spectrum, in-cluding modified fire-hose modes, strongly depends on themagnetic field value (parameter β ), pressure anisotropy pa-rameter α , heat flux parameter γ , and wave propagationangle φ with respect to the magnetic field. The deduced in-stability increments are rather large. We have derived thegeneral instability condition for incompressible waves. Acknowledgements.
We are vary thankful to Bernhard Kliem,Gottfried Mann, and the referee whose critical comments helped toimprove an earlier version of this paper. The present work has beensupported by the German Science Foundation (DFG) under grant No.436 RUS 113/931/0-1 (R) which is gratefully acknowledged.
References
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