mhd waves and instabilities of a temperature-anisotropic plasma in the solar corona as a source of...

ISSN 1063-7737, Astronomy Letters, 2011, Vol. 37, No. 9, pp. 649–655. c© Pleiades Publishing, Inc., 2011.Original Russian Text c© N.S. Dzhalilov, V.D. Kuznetsov, 2011, published in Pis’ma v Astronomicheskiı Zhurnal, 2011, Vol. 37, No. 9, pp. 706–712.

MHD Waves and Instabilities of a Temperature-Anisotropic Plasmain the Solar Corona As a Source of Its Heating

N. S. Dzhalilov1, 2* and V. D. Kuznetsov1**

1Pushkov Institute of Terrestrial Magnetism, Ionosphere, and Radio Wave Propagation, Russian Academy ofSciences, Troitsk, Moscow oblast, 142190 Russia

2Shamakha Astrophysical Observatory, National Academy of Sciences of Azerbaijan, Baku, Az-1000Azerbaijan

Received April 18, 2011

Abstract—The MHD instabilities of a temperature-anisotropic coronal plasma are considered. We showthat aperiodic mirror instabilities of slow MHD waves can develop under solar coronal conditions for weakmagnetic fields (B < 1 G) and periodic ion-acoustic instabilities can develop for strong magnetic fields(B > 10 G). We have found the instability growth rates and estimated the temporal and spatial scales ofdevelopment and decay of the periodic instability. We show that the instabilities under consideration canplay a prominent role in the energy balance of the corona and may be considered as a large-scale energysource of the wave coronal heating mechanism.

DOI: 10.1134/S0320010811090038

Keywords: plasma astrophysics, solar corona, solar wind, anisotropic magnetohydrodynamics,MHD instabilities, plasma heating.

INTRODUCTION

The solar coronal heating problem still remainsone of the unsolved problems in astrophysics. It isgenerally believed that two mechanisms are mainlyresponsible for plasma heating and they both arerelated to a magnetic field: magnetic reconnections(and the currents and microflares produced by them)and Alfven wave dissipation (Longcope 2004; Pe-ter and Gudiksen 2005; Aschwanden 2006; Klim-chuk 2006). In principle, Alfven waves can deliver asufficient energy flux from the lower atmosphere intothe corona to provide a balance between the energylosses through radiation and heat transport by elec-tron heat conduction and the solar wind. However,under coronal conditions, an efficient Alfven wavedissipation mechanism that would operate globally(irrespective of the magnetic field configuration) isunknown. For example, the resonant Alfven waveabsorption mechanism (Ionson 1978; Hollweg 1987)can be realized only in a very narrow layer at the topof magnetic loops, where the direction of the densitygradient is perpendicular to that of the magneticfield. All of the attempts to model coronal plasmaheating made so far are based exclusively on ordinary

*E-mail: [email protected]**E-mail: [email protected]

MHD equations for an isotropic plasma. How-ever, present-day observations and measurements,along with kinetic modeling results, suggest thatthe distribution functions of particles (particularlyions) are not described by a Maxwellian distribu-tion. Starting from the base of the corona, thehot plasma density falls sharply with height andthe particle collision frequency decreases sharply.The plasma becomes strongly magnetized, i.e., theLarmor radius is much less than the particle meanfree path, rBe,i � λe,i, even at low magnetic fieldstrengths (B ∼ 10−3 G). As a result, the plasmabecomes temperature-anisotropic relative to themagnetic field direction, i.e., the transverse andlongitudinal plasma pressures and temperatures areunequal, p⊥/p|| = T⊥/T|| �= 1. In the upper coronaand coronal holes, the ion component of the plasma isessentially collisionless and temperature-anisotropic.Under such conditions, the approximation of idealisotropic MHD is inapplicable. This can be easilyverified by considering basic average coronal plasmacharacteristics. In the universally accepted notation,the physical parameters of the solar coronal plasmaare:

Te∼= Ti = 106 K, ne

∼= np = 109 cm−3, vT e ≈ 4 ×103 km s−1, vT i ≈ 100 km s−1, τe ≈ 10−2 s, τi ≈0.8 s, λe ≈ 40 km, λi ≈ 80 km, λD ≈ 0.2 cm, λT =

649

650 DZHALILOV, KUZNETSOV

kTe

μmpg≈ 2300 km, cs ≈ 100 km s−1, and for the range

of magnetic field strengths B = 0.1–100 G: rBe ≈200–0.2 cm, rBi ≈ 9000–9 cm, τBe ≈ 10−6–10−9 s,τBi ≈ 10−2–10−5 s, vA ≈ 10–104 km s−1.

As we see, the cyclotron radii and periods are verysmall compared to all of the other scales. For suchconditions, we can use the most general MHD equa-tions for an anisotropic plasma that were derivedby many authors as equations for the moments ofthe kinetic particle distribution function (see, e.g.,Oraevsky et al. 1985; Ramos 2003). These equationsmay be considered as a generalization of the Chew–Goldberger–Low approximation (Chew et al. 1956)that allows for the heat fluxes of a collisionlessplasma.

To simplify the problem, we will consider a colli-sionless plasma composed of ions alone. The propa-gation of MHD waves in an anisotropic homogeneousplasma with allowance made for the heat fluxes alongthe magnetic field and possible types of instabilitieswere considered previously (Dzhalilov et al. 2008;Kuznetsov and Dzhalilov 2010; Dzhalilov et al. 2010).Here, we will examine how the instabilities foundcan be realized under solar coronal conditions. Thebasic idea put forward here is that the Alfven wavesor fast MHD waves being generated in the lowersolar atmosphere (where the plasma is predominantlycollisional and isotropic) gradually enter a weakly col-lisional temperature-anisotropic plasma, where fire-hose, mirror, and other instabilities related to theheat flux can develop. We suggest considering theseinstabilities as a large-scale energy source of the wavecoronal plasma heating mechanism.

WAVES IN AN ANISOTROPIC PLASMA

For a single-fluid model, the MHD transportequations in a temperature-anisotropic plasma with-out any dissipative effects can be represented as(Oraevsky et al. 1985; Ramos 2003)

dρ

dt+ ρdivv = 0, (1)

ρdvdt

+ ∇(

p⊥ +B2

8π

)− 1

4π(B · ∇)B (2)

= ρg + (p⊥ − p‖)[hdivh + (h · ∇)h]

+ h(h · ∇)(p⊥ − p‖),

d

dt

p‖B2

ρ3+

B2

ρ3

[B(h · ∇)

(S‖B

)(3)

+2S⊥B

(h · ∇)B

]= 0,

d

dt

p⊥ρB

+B

ρ(h · ∇)

(S⊥B2

)= 0, (4)

d

dt

S‖B3

ρ4+

3p‖B3

ρ4(h · ∇)

(p‖ρ

)= 0, (5)

d

dt

S⊥ρ2

+p‖ρ2

[(h · ∇)

(p⊥ρ

)(6)

+p⊥ρ

p⊥ − p‖p‖B

(h · ∇)B

]= 0,

dBdt

+ Bdivv − (B · ∇)v = 0, divB = 0, (7)

where ddt = ∂

∂t + (v · ∇) is the total derivative, vis the plasma velocity, ρ is the density, p⊥ and p||are the transverse and longitudinal gas pressures,S|| and S⊥ are the heat fluxes along the magneticfield produced by the longitudinal and transversethermal motions of particles, B is the magnetic fieldstrength, h = B

B is a unit vector along the magneticfield, and g is the gravity. Equations (1)–(7) differfrom the Chew–Goldberger–Low equations by theadditional two equations (5) and (6) that describethe evolution of the heat fluxes. Let the plasma inthe basic equilibrium state be homogeneous, g = 0,and {v0, ρ0, p⊥0, p||0, S⊥0, S||0, B0} = const �= 0. Forlinear waves with an amplitude ∼ exp i(k · r− ωt)(where ω is the oscillation frequency in the frameof reference comoving with the fluid, k is the wavenumber), apart from incompressible Alfvenic modes(V 2 = α + β − 1), the system of equations (1)–(7)admits the following dispersion equation for com-pressible modes (Kuznetsov and Dzhalilov 2010;Dzhalilov et al. 2010):

8∑n=0

Cn(α, β, γ, l)V n = 0, (8)

V =ω

k||c||, l = cos2 θ, α =

p⊥p||

,

β =V 2

A

c2||

, γ =S||

p||c||=

S⊥p⊥c||

,

C0 = 3(2s + r), C1 = −4γ(3s + r),

C2 = 2s(2γ2 − 5) + 3(l − 3r),C3 = 4γ(s + r − l), C4 = 2s + 7r − 9l,

C5 = 4γl, C6 = 7l − r,

C7 = 0, C8 = −l, s = α2(1 − l),r = l − β − α(2 − l).

Here, c2|| = p||/ρ is the square of the speed of sound

along the magnetic field, k|| = k cos θ, θ is the wave

ASTRONOMY LETTERS Vol. 37 No. 9 2011

MHD WAVES AND INSTABILITIES 651

propagation angle relative to the magnetic field,and V is the normalized complex phase velocity. Inthe above notation, the indices “0” in the basic-stateparameters are omitted for simplicity. Analysis of thedispersion equation and some of its solutions werepresented by Ramos (2003), Dzhalilov et al. (2008),and Kuznetsov and Dzhalilov (2010). Here, the so-lutions obtained by Kuznetsov and Dzhalilov (2010)and Dzhalilov et al. (2010) are important to us. Inthese solutions, the properties of the waves are modi-fied by the heat fluxes (the waves become asymmetricwith respect to the propagation direction — alongor opposite to the heat flux) and instabilities emergeas a result of the intersection between the dispersioncurves of various waves.

Let us estimate the ranges of variation in problemparameters for the coronal plasma. Suppose thatc|| = cs under coronal conditions. Then, the magneticfield parameter β ≈ 0.5B2

0 [G] ≈ 10−2−104 and theheat flux parameter γ ≈ 3

4δv0/cs ∼ εβ1/2, where 1 �ε ∼ 0.1. The case with α > 1, i.e., when the particlesare more heated in the transverse direction, T⊥ > T||,is encountered most frequently in observations. Thiscan be explained by the fact that the particles locallyheated in the longitudinal direction (α < 1) have anopportunity to leave the heating place rapidly alongthe magnetic field. However, the state with transverseheating can exist for a fairly long time. Clearly, theparameter α for the corona (not the outer corona) willbe close to unity. Consider a fixed value of α = 1.5.

We numerically found all solutions of the eighth-order dispersion equation (8) for complex values ofthe phase velocity V as a function of the magneticfield B0. For the case of l = 0.5, Fig. 1 shows eightcurves corresponding to the real parts of these solu-tions, Vph = Re(V )cs. These are forward waves withVph > 0 and backward waves with Vph < 0 above andbelow the zero level, respectively. The extreme fastestmodes with number 4 correspond to the prototype offast magnetoacoustic waves in an isotropic plasma.The slowest waves with number 1 are the proto-types of slow magnetoacoustic waves in an isotropicplasma. Slow (No. 2, Vph < cs) and fast (No. 3,Vph > cs) modes related to the presence of a heatflux along the magnetic field emerge between thesemagnetoacoustic modes. These modes are the hy-drodynamic prototypes of kinetic ion-acoustic waves.The dashed straight lines and dotted curves in thesame figure indicate the locations of the speed ofsound and the Alfven velocity, respectively. For amagnetic field B0 > 2 G, the fast MHD waves (No. 4)propagate almost with the Alfven velocity. Sincethe influence of the heat flux on these modes is veryweak, the forward and backward modes are almostsymmetric and stable. The situation with the slow

−40010.01 0.1 10 100

−300

−200

−100

0

100

200

300

400

B0, G

Vp

h

α = 1.5l = 0.5

1

2

34

43

1

2

Fig. 1. Phase velocities (km s−1) of all possible eightmodes of the dispersion equation (8) versus magneticfield strength B0 [G]; the curve numbers point to thetypes of waves: 1—the prototypes of slow MHD waves,4—the prototypes of fast MHD waves, and 2 and 3—slow and fast ion-acoustic waves; Vph > 0 for forwardwaves and Vph < 0 for backward waves; the two straightdashed lines point to the position of the speed of soundalong the magnetic field, cs = 100 km s−1, and the dottedcurves correspond to the position of the Alfven velocity±VA(B0).

modes (No. 1) is the reverse one: because of theheat flux, the forward and backward modes becomeasymmetric and these branches merge together andvanish at B0 < 1.5 G. This implies that an aperiodic(Reω = 0) mirror instability emerges (Vedenov andSagdeev 1958; Chandrasekhar et al. 1958). The ion-acoustic modes also become asymmetric; the forwardmodes are stable, while the backward slow (No. 2)and fast (No. 3) waves become unstable while merg-ing together at B0 > 6 G. This instability is periodic,Reω �= 0. Thus, out of the eight modes found, onlythe slow MHD waves and backward ion-acousticmodes can become unstable. These waves and relatedinstabilities are of considerable interest as a possiblelarge-scale energy source of the wave coronal heatingmechanism and should be investigated in more detail.To ascertain how a change in the wave propagationangle affects the properties of unstable waves, Fig. 2presents the phase velocities of these modes in threecases: longitudinal propagation l = 1, oblique propa-gation l = 0.5, and quasi-transverse propagation l =0.1. As we see, the periodic ion-acoustic instability(solid curves) emerges in all cases, but the wavevelocity falls sharply with increasing propagation an-gle. The range of velocities for these waves is 25–



10.01 0.1 10 100

100V

ph

α = 1.5

0

−100

−200

B0, G

Fig. 2. Dependence Vph(B0) for two unstable modes(the dashed lines are forward and backward slowMHD waves; the solid lines are backward slow and fastion-acoustic waves) for various wave propagation angles;l = 1, 0.5, 0.1 correspond to these propagation angles:from top to bottom for the dashed lines and from bottomto top for the solid ones.

200 km s−1. The point at which the fast and slowmodes merge together is virtually independent of thepropagation angle. This implies that the instabilitythreshold in all cases lies at B0 = 6 G. For slowwaves (dashed lines), the wave propagation veloc-ity also decreases with increasing propagation an-gle. The mirror instability disappears for longitudinalwaves. The threshold for this instability is shifted asthe propagation angle changes. Figure 3 presentsthe growth rates of the instabilities Γ1 = Imω/(kc||)(solid curves) and the ion-acoustic instability growthrate Γ2 = Imω/Reω (dashed curves) for various l.The regions B0 < 1.5 G and B0 > 6 G correspond tothe aperiodic mirror instability and the periodic ion-acoustic instability, respectively. The mirror instabil-ity growth rate has a maximum near l ∼ 0.3. Theion-acoustic instability growth rate decreases withincreasing propagation angle.

The longitudinal backward ion-acoustic modesturn out to be most unstable. Here, it is important tonote that these two instabilities emerge independentlyin different magnetic field regions: the mirror and ion-acoustic instabilities emerge, respectively, in weakfields with B0 < 1 G (a quiet corona, coronal holes,the solar wind, etc.) and strong fields with B0 > 10 G(loops above active structures).

Let us now compare our results with observa-tions. In present-day measurements, the waves inthe corona are observed both in the Doppler shifts

10 100

Γ

α = 1.5

2.0

B0, G

1.5

1.0

0.5

010.10.01

Fig. 3. Dependence of the instability growth rates Γ(B0)for slow MHD waves (the aperiodic mirror instabilitythat emerges at B0 < 1.5 G) and backward ion-acousticwaves (the periodic ion-acoustic instability that emergesat B0 > 6 G). The curves from top to bottom correspondto l = 0.3, 0.5, 0.1, 0.01, 0.7 for the mirror instability andl = 1, 0.7, 0.5, 0.3, 0.1, 0.01 for the ion-acoustic instabil-ity. The dashed curve indicates the instability growth rateof ion-acoustic modes that does not depend on l.

of emission line profiles and in the images of coronalstructures (Aschwanden 2006; Prasad et al. 2011).The observations of waves in images suggest thatthe observed wave is a compression one and it isunrelated to the Alfven oscillations even if the ve-locity of the wave train almost coincides with theAlfven velocity. Upward traveling wave modes witha phase velocity Vph = 65−200 km s−1 and a periodP = 3−5 min and Vph ∼ 2100 km s−1 and P ≈ 6 s areobserved in closed structures (magnetic loops). Out-going compression waves with Vph = 75−150 km s−1

and P = 10−30 min are also observed in open struc-tures (in particular, coronal holes). Outside the Sun,P ≈ 6 min at a distance of about two solar radii andP ≈ 20−50 min at a distance of about 20 Mm. Allthese velocities and periods fall into the spectrum ofwaves in an anisotropic coronal plasma found. Inparticular, it follows from Fig. 1 that the waves witha velocity of 65–200 km s−1 correspond to unstableion-acoustic modes, while the high phase velocities,2000 km s−1, correspond to fast magnetoacousticmodes propagating almost with the Alfven velocity.

THE ION-ACOUSTIC INSTABILITYAND THE CORONAL HEATING PROBLEM

The main condition for ion-acoustic instability de-velopment is the propagation of modes opposite to the



heat flux direction. At the base of the corona wherea steep positive temperature gradient exists, the heatflux is always directed downward. This agrees wellwith Spitzer’s law that is used to construct the coro-nal model. It can be assumed that because of theheat flux continuity condition, this situation will beretained in the upper corona until the temperaturegradient changes its sign. In the outer corona, theelectron heat conduction is insignificant and the solarwind is the dominant energy loss component in theenergy balance. Thus, the waves propagating frombelow along a magnetic flux tube encounter the heatflux that produces an instability. Consider the energyaspect of this instability. The energy flux densitydelivered by the waves is

F =12ρ0v2Vph, (9)

where v is the wave velocity amplitude. If we specifythe function F [erg cm−2 s−1] based on observationaldata, then Eq. (9) allows the wave amplitude v =√

2F/(ρ0Vph) to be estimated. We have the follow-ing estimates from currently available observations(Aschwanden 2006): B0 ≈ 0.1 G, F ≈ 3 × 105 forthe quiet Sun; B0 ≈ 0.5 G, F ≈ 8 × 105 in a coronalhole; B0 ≈ 10−100 G, F ≈ 107 above active regions.Clearly, the equilibrium plasma parameters (densityand temperature) change with magnetic field insidethe magnetic flux tube, which, in turn, affects theenergy balance conditions. The above values canbe approximated by a simple dependence, F ≈ (3 +0.97B0) × 105 [erg cm−2 s−1]. Using the observa-tional data for the periods and phase velocities in var-ious magnetic structures from the preceding section,we will then obtain the following estimates. In loopstructures for periods P = 3−5 min, the wave velocityamplitude must be vc = 200−400 km s−1, while forwaves with a period P = 6 s we will obtain vc =70 km s−1. In open structures for waves with a periodP = 10−15 min, the velocity vc = 25−35 km s−1.Here, the wave energy flux is assumed to be com-pletely converted into plasma heating. Let us nowrepresent the wave amplitude for our unstable solu-tion as

vw =

√2F

ρ0Vphet/τ , (10)

τ =1

Im(ω)=

P

2π

∣∣∣∣Re(V )Im(V )

∣∣∣∣ .Here, τ is the characteristic instability growth timeand P is the oscillation period.

Specifying the function F ≈ (3 + 0.97B0) ××105 [erg cm−2 s−1] that approximates the obser-vational data, we can calculate the model amplitude

10

V,

km/s

α = 1.5

B0, G

300

10.1 1000.01

l = 0.5200

100

0

−100

−200

−300

2

3

Fig. 4. Amplitude of backward ion-acoustic modes at theonset of instability development (t = 0) versus magneticfield (above the zero level); below the zero level—thecorresponding phase velocities for l = 0.5. The dottedline indicates the position of the speed of sound along themagnetic field.

vw(t). Suppose that the backward ion-acousticmodes were generated at time t = 0 (at the onsetof instability). Their amplitudes at this time arepresented in Fig. 4. Also shown here are the cor-responding phase velocities for l = 0.5. We see thatthe phase velocities are greater than the velocityamplitude. The velocity amplitude found in the in-stability region (B0 = 10−100 G) is smaller than thecritical one, vw(0) < vc = 400 km s−1. Outside theinstability region, the velocity amplitude for backwardslow and fast ion-acoustic waves is always less thanthe speed of sound. These velocities increase withmagnetic field. At the instability threshold, theybecome identical and equal to the speed of sound,v = cs. The velocity amplitude for an unstable wave isgreater than the speed of sound. This implies that forthe properties of the instability under consideration tobe studied in detail, nonlinear effects must be takeninto account. In Fig. 5, the solid curve indicatesthe dependence τ(B0) and the dashed lines indicatethe time after which the equality vw(t = τc) = vc =400 km s−1 is established. As we see, this timeis mainly less than the oscillation period, implyingthat the dissipation of waves is very strong. Undercoronal conditions, rapid instability development andstrong growth in wave amplitude may be consideredas one of the energy supply mechanisms and asa large-scale energy source of the wave coronalheating mechanism, bearing in mind that the heating



τ,

min

α = 1.5

B0, G

5

100

4

3

2

1

5010510

Fig. 5. Characteristic growth time of the backward ion-acoustic instability τ (solid curve) calculated for P =3 min and the time τc when the critical velocity amplitudeat which an energy balance is achieved in an elementaryvolume is reached (dashed curves). The dashed curvesfrom top to bottom correspond to the values of τc(l) at l =1, 0.7, 0.5, 0.3, 0.1, 0.01, with τ being independent of l.

mechanism itself, i.e., the transfer of wave energyto plasma ions, is related to the nonlinear stageof instability development and is realized on muchsmaller scales.

Let us now consider the spatial instability scales.The dissipation length scale for waves leading to adensity perturbation is estimated as (Priest 1982)

d ≈∣∣∣∣ρ0

ρ′1k

∣∣∣∣ =P

2π

∣∣∣∣ρ0

ρ′Vph

∣∣∣∣ . (11)

Dzhalilov et al. (2010) provided the formulas

v||cs

= w1ρ′

ρ0,

v⊥cs

= w2ρ′

ρ0, (12)

where the coefficients w1 and w2 are determined bythe solutions of the dispersion equation (8). If wedenote w =

√w2

1 + w22, then we find ρ0/ρ

′ = wcs/vw.Taking this into account, from (10) we will obtaind = d0e

−t/τ . Figure 6 presents the dependence d(B0)at the initial density perturbation growth stage. Inthe instability region, this scale lies within the range1–10 Mm. For waves with a period of 3 min, asfollows from Fig. 4, the characteristic wavelength isestimated to be λ = PVph ≈ 40 Mm. Thus, the wavedissipation length scales are much smaller than thewavelength itself. This again suggests that the dis-sipation of waves through instability development is

10

d, k

m

α = 1.5

B0, G10.10.01

10000

100

9000

8000

7000

6000

5000

4000

3000

2000

1000

Fig. 6. Spatial density perturbation scale at the onset ofion-acoustic instability versus magnetic field for variouspropagation angles: the curves from top to bottom corre-spond to l = 1, 0.7, 0.5, 0.3, 0.1.

highly efficient. This length scale decreases exponen-tially with instability development. Total dissipationwill occur when this length scale decreases to the

cyclotron radius, d ≈ rB ≈ 1100B0[G]

[km]. This will

occur in a time tB ≈ −τ ln(rB/d0). This time isshown in Fig. 7. It is several periods.

DISCUSSION

We considered all possible types of MHD wavesunder conditions of a temperature-anisotropic solarcoronal plasma. We showed that the heat fluxes alongthe magnetic field cause the waves to become asym-metric relative to the heat flux direction. Because ofthe anisotropy in pressure and because of the heatflux, an instability is excited at some plasma parame-ters. All these properties of anisotropic MHD wavesare absent in ordinary isotropic MHD. Apart fromthe classical fire-hose instability related to incom-pressible Alfvenic modes, the instabilities related tocompressible wave modes can develop. An aperiodicmirror instability of slow MHD waves emerges in re-gions with a weak magnetic field and the ion-acousticwaves that propagate opposite to the heat flux di-rection become unstable in regions with a strongmagnetic field. This instability is periodic. We con-sidered the described instabilities only qualitativelyand showed that the development of an ion-acousticinstability could become an efficient solar coronalplasma heating mechanism. The energies of the



t B,

min

α = 1.5

B0, G

40

30

20

10

5010510

50

100

Fig. 7. Total ion-acoustic instability decay time as afunction of the magnetic field and propagation angle: fromtop to bottom l = 1−0.01.

Alfven and fast magnetoacoustic waves propagatingfrom the lower atmosphere into the corona can bean energy source of the instability development. Themain requirement for the emergence of a periodic ion-acoustic instability (the propagation of waves frombelow opposite to the heat flux along a magneticflux tube) holds good in the solar corona. Based onobservations, we showed that it takes a time less thanone period between the initial instability stage and theattainment of the critical amplitude needed to providean energy balance. This explains why traveling waveswith periods of 3–5 min rather than standing onesare observed in closed loop structures. The reasonis that the instability develops rapidly and the waveshave no time to propagate for the total coronal looplength; they decay. We estimated the decay length for

unstable waves. A numerical analysis of the nonlinearinstability development is needed for a more rigorousconsideration of the qualitative results obtained.

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Translated by V. Astakhov


mhd waves and instabilities of a temperature-anisotropic plasma in the solar corona as a source of...

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