Nonlinear excitation of small-scale Alfvén waves by fast waves and plasma heating in the solar atmosphere

NONLINEAR EXCITATION OF SMALL-SCALE ALFVÉN WAVES BYFAST WAVES AND PLASMA HEATING IN THE SOLAR ATMOSPHERE

YURIY VOITENKO1,2 and MARCEL GOOSSENS1

1Centre for Plasma Astrophysics, K.U. Leuven, Celestijnenlaan 200B, 3001 Heverlee, Belgium(e-mail: [email protected])

2Main Astronomical Observatory of the National Academy of Sciences of Ukraine,27 Zabolotnoho St., 03680 Kyiv, Ukraine

(Received 15 November 2001; accepted 29 May 2002)

Abstract. We study a nonlinear mechanism for the excitation of kinetic Alfvén waves (KAWs)by fast magneto-acoustic waves (FWs) in the solar atmosphere. Our focus is on the excitation ofKAWs that have very small wavelengths in the direction perpendicular to the background magneticfield. Because of their small perpendicular length scales, these waves are very efficient in the energyexchange with plasmas and other waves. We show that the nonlinear coupling of the energy of thefinite-amplitude FWs to the small-scale KAWs can be much faster than other dissipation mechanismsfor fast wave, such as electron viscous damping, Landau damping, and modulational instability. Thenonlinear damping of the FWs due to decay FW = KAW + KAW places a limit on the amplitude ofthe magnetic field in the fast waves in the solar corona and solar-wind at the level B/B0 ∼ 10−2. Inturn, the nonlinearly excited small-scale KAWs undergo strong dissipation due to resistive or Landaudamping and can provide coronal and solar-wind heating. The transient coronal heating observedby Yohkoh and SOHO may be produced by the kinetic Alfvén waves that are excited by parametricdecay of fast waves propagating from the reconnection sites.

1. Introduction

In-situ and remote observations show the ubiquitous presence of MHD waves inthe solar corona and solar wind. These waves can be excited by different drivers,like MHD plasma instabilities and mechanical plasma motions (see, e.g., Verheest,1977; Heyvaerts and Priest, 1983; Fushiki and Sakai, 1994; Ruderman et al., 1997).The flux of MHD waves can provide an efficient plasma energization if the wavelengths and periods are sufficiently short. However, the MHD waves are excitedat length scales and time scales that are far longer than those required for efficientdissipation. It is therefore evident that the energy carried by MHD waves can onlybe available for the energization of the plasma if there exists a mechanism that cantransport the energy from the long wavelengths at excitation to the required shortwavelengths for dissipation.

The processes that are able to transport energy to dissipative length-scales at-tract great interest in the space plasma physics community. Resonant absorption(Ionson, 1978; Goossens, 1991) and phase mixing (Heyvaerts and Priest, 1983) aretwo popular linear mechanisms for transferring energy towards small length-scales.

Solar Physics 209: 37–60, 2002.© 2002 Kluwer Academic Publishers. Printed in the Netherlands.

38 YURIY VOITENKO AND MARCEL GOOSSENS

Another class of theories of energy transport from long length-scales to short,dissipative length-scales, is based on the nonlinear interactions of waves. Nonlin-ear interaction and turbulent cascade of wave energy have been widely discussedrecently in the context of space plasma energization (Marsch, 1997; Nakariakov,Roberts, and Murawski, 1997; Matthaeus et al., 1999; Li et al., 1999).

Fast MHD waves (FWs) can be excited in the corona by perturbations of themagnetic field lines which are anchored into the dense convective zone and dis-placed by the plasma motions there (Tirry and Berghmans, 1997; De Groof, Tirry,and Goossens, 1998; De Groof and Goossens, 2000). The consequent dissipationof these waves in resonant layers can contribute to coronal heating (De Groof andGoossens, 2000, and references therein).

The time required for creating of sufficiently short length-scales for efficientdissipation, and the nature of this dissipation, are not very well known. The setuptime of the linear resonance has been first calculated by Kappraff and Tataro-nis (1977), and the theory has been further improved and developed using dif-ferent models by Wright and Rickard (1995), Tirry, Berghmans, and Goossens(1997), Tirry and Berghmans (1997), Ruderman and Wright (2000), and De Groofand Goossens (2000). Wright and Rickard (1995) considered a nonuniform one-dimensional magnetohydrodynamic cavity driven by a prescribed random motion.Tirry, Berghmans, and Goossens (1997) and Tirry and Berghmans (1997) consid-ered periodic driving at footpoints polarized normal to the magnetic surfaces (Tirry,Berghmans, and Goossens, 1997), and in the magnetic surfaces (Tirry and Bergh-mans, 1997). De Groof and Goossens (2000) studied stochastic driving polarizednormal to the magnetic surfaces and showed that the fast wave builds up in thecorona and couples energy to the resonant Alfvén waves in resonant layers. Allthe above studies used linear MHD and found that coupling of fast MHD wavesto Alfvén waves, and vice versa, is an ubiquitous phenomenon. In agreement withMann, Wright, and Cally (1995) they found that the length scale of the resonantAlfvén wave, that is the width at half maximum of the amplitude, decreases as1/t . Length scales of 100 m can be obtained in 3 hours. This value for the setuptime is definitely shorter than the lifetime of a loop but is longer than the observedtime scales of minute or less in the coronal heating process, as seen by Yohkohand SOHO (Shimizu et al., 1992; Berger et al., 1999; Berghmans, McKenzie,and Clette, 2001). Therefore, the calculation of the resonant absorption in thecorona require consideration of a quickly varying background with mobile resonantsurface(s) that can destroy the resonance. To avoid this difficulty, anomalous trans-port coefficients (resistivity and viscosity) produced by current-driven and velocityshear-driven instabilities in the resonant layers can be invoked. It might speed upthe creation of small length scales, so that this is definitely not yet a closed matter.

On the other hand, it is widely believed that impulsive heating events of timescales ranging from seconds to tens of minutes are produced by magnetic reconnec-tion events at the base of solar corona (Harrison et al., 1999). The fast waves canbe driven by a pinching current sheet in reconnection events (Fushiki and Sakai,

NON-LINEAR EXCITATION OF KINETIC ALFVEN WAVES BY FAST WAVES 39

1994). These waves propagate away from the reconnection sites and can producefast plasma heating provided they are sufficiently quickly dissipated.

Compressional oblique fast waves can be excited by phase mixing of parallel-propagating waves whose magnetic perturbation has a component parallel to theperpendicular gradient of the Alfvén velocity in the pressure-balanced structuresin the solar corona and solar wind. However, observations show that magnetic per-turbations are almost incompressible in the solar wind (Marsch, 1991). There arealso coronal observations suggesting Alfvén or kink modes (Koutchmy, Zhugzda,and Locans, 1983; Doyle, Banerjee, and Perez, 1998), even under the presence ofa strong driver, e.g., flare (Aschwanden et al., 1999).

In view of the theoretical prediction that fast waves can be easily excited, and theobservational evidence of a low compressional component of the wave magneticfield in the solar atmosphere and solar wind, we suggest that the fast waves can bedissipated more efficiently than by the Landau/collisional damping or linear modetransformation. In the present paper we study a nonlinear process that can stronglyaccelerate the evolution of wave energy towards small length-scales: the parametricdecay of a large-scale pump fast wave into Alfvén waves. In the framework oftwo-fluid MHD we show that fast waves are nonlinearly coupled to the kineticAlfvén waves. The kinetic Alfvén waves we refer to are Alfvén waves with shortwavelengths across the background magnetic field B0 (large perpendicular wavenumbers k⊥). When the perpendicular wave number steadily increases, the Alfvénwave acquires a significant electric field parallel to the equilibrium magnetic field,E‖ ‖ B0. Because of this parallel electric field the AW begins to interact with theplasma species and accelerates plasma particles. An oblique Alfvén wave acquireskinetic properties and is thus called a kinetic Alfvén wave. Because of their abil-ity to interact strongly with space plasmas, kinetic AWs (KAWs) are now underintensive investigation (Hollweg, 1999; Voitenko and Goossens, 2000b, 2002, andreferences therein).

The non-local three-wave resonance of a large-scale pump fast wave with small-scale daughter KAWs is made possible by the highly anisotropic nature of KAWswhich are still low-frequency despite of the high oblique wave numbers. Thisprocess introduces a non-local transfer of wave energy from large scales (in the fastwave mode) towards small scales (in the kinetic Alfvén mode) where the waves arestrongly damped heating the plasma. On the other hand, the nonlinear damping ofFWs places a limit on the FW amplitude in the solar corona and solar wind.

https://www.researchgate.net/publication/234219304_Kinetic_Physics_of_the_Solar_Wind_Plasma?el=1_x_8&enrichId=rgreq-2898039fbf687c273eae99172ab77f97-XXX&enrichSource=Y292ZXJQYWdlOzIyNjEzMjY1OTtBUzoxNDQ1ODc1OTkzODg2NzJAMTQxMTQ4Mzc3ODAzNg==


2. Kinetic Alfvén Waves in 2-Fluid MHD

2.1. PLASMA MODEL AND BASIC EQUATIONS

The current density, j = ∑s qsnsvs , and the charge density,Q = ∑

s qsns , inducedby AWs, have to be calculated using a suitable mathematical plasma model. Themost popular models are based on the single-fluid MHD equations, the two- ormulti-fluid MHD equations, and kinetic Vlasov equations. We use the mathemati-cal model of resistive two-fluid MHD that permits us to take into account the mostimportant linear and nonlinear effects in the short-scale AWs. In two-fluid MHDthe electron and ion fluids are allowed to move in separate ways, but are coupledby the collective electromagnetic fields and by the electron–ion friction force.

The parallel friction force, which is responsible for the (parallel) resistivityalong B0, allows us to estimate the threshold amplitude of the large-scale pumpFW. We drop the perpendicular friction force in AWs responsible for the perpen-dicular resistivity, because its contribution to the wave dissipation is

(λ‖/λ⊥

)2

times smaller than the contribution due to the parallel resistivity, λ‖ and λ⊥ arethe parallel and perpendicular wavelengths. The effect of kinetic Landau dampingwill be included semi-empirically.

We find it convenient to introduce Boltzmann-like potentials instead of thenumber density ns:

φs = −T sqs

lnns

n0, (1)

where qe = e for ions and qi = −e for electrons, T s is temperature, and n0 isthe equilibrium number density. At this point it is instructive to note that classicAlfvén waves in a uniform plasma of infinite extent do not have any variation indensity. Kinetic AWs do have variations in density. Since it is these waves that weare interested in, φ is a convenient variable. Then, by separating the magnetic fieldinto its background part and its part due to the waves, B → B0+ B, we write theequation of motion and the continuity equation as

∂vs∂t

= qs

ms

(1

cvs × B0 + fs

), (2)

∂

∂tφs + vs · ∇φs − ms

qsV 2T s∇ · vs = 0, (3)

where V 2T s = Ts/ms . As usually, temperature variations are neglected in the Alfvén

waves for which the plasma pressure perturbations are determined by the numberdensity perturbations.

The function fs is the charge force per unit mass and unit charge. It includes thelinear electric force, the collisional friction force, the pressure gradient force, andthe nonlinear force:


fs = E − νme

ne2j‖ + ∇φs + Fs, (4)

where ν = 0.51νe, νe is the electron collision frequency. The velocity difference isexpressed through the current (the quasi-neutrality condition ne = ni ≡ n is used):

(vi − ve) = 1

nej, (5)

and the nonlinear force is

Fs = 1

cvs × B − ms

qs(vs · ∇) vs . (6)

The wave electromagnetic fields E and B obey the standard Maxwell equations:

∇ × B = 4π

cj + 1

c

∂E∂t, (7)

∇ × E = −1

c

∂B∂t, (8)

∇ · E = 4πQ. (9)

2.2. ALFVÉN WAVE EIGENMODE EQUATION

We use for the AWs the quasi-neutrality condition,

− (Te/Ti) φi = φe ≡ φ,

instead of Gauss’ law (9), and we eliminate all the quantities in Equations (2)–(9)in terms of φ. After calculations similar to those given by Voitenko and Goossens(2000) (see Appendix), we arrive at the following nonlinear equation for AWs interms of the effective density potential φ:(

∂2

∂t2− V 2

AK2∇2

‖ − 2γL∂

∂t

)φ = Ntot. (10)

γL is the linear damping rate due to resistive dissipation (dominant at lower fre-quencies),

γL = −0.5νδ2e k

2⊥

1 + δ2e k

2⊥, (11)

where δe = √c2me/4πne2 is the electron inertial length. Different expressions

for γL can be introduced ad hoc when other dissipation/excitation mechanismsare more efficient. For example, Landau damping is stronger at higher frequencies(see Landau damping rate (49) in Section 4.1). K is the dispersion function in theperpendicular wave number space,


K2 =1 + me

mi+ µT

1 + me

mi+memiβ−1µT

1 +

(β − me

mi

)µT(

1 + me

mi+µT

)2

, (12)

where µT = ρ2T k

2⊥, ρ2T = V 2

T / 2i , V

2T = T /mi , T = Ti + Te, i = eB0/mic

is the ion-cyclotron frequency, VA = B0/√

4πnmi is the Alfvén velocity. Theterm proportional to β = V 2

T /V2

A in brackets is due to the gas compressibility,the subscript ‖ denotes the direction of the equilibrium magnetic field B0 (thatis E‖ ‖ B0, etc.). The terms that are due to the magnetic compressibility B‖ areproportional to β = V 2

T /V2

A; they are hidden in the normalized temperatures Ts =Ts/ (1 + β) (e.g., the normalized thermal velocity V 2

T = V 2T / (1 + β), and so on).

In what follows, (1 + β)−1 always appears as a factor multiplying temperatures.Hence, we shall drop the superfluous bar in what follows.

Expression (12) is an improved and corrected version of the AW dispersiongiven by Voitenko and Goossens (2000). In a plasma with me/mi � β � 1, itsuffices to use the following expression for K for our present purposes:

K2 = 1 + µT . (13)

The nonlinear part of (10) is

Ntot = V 2T e(

1 + δ2e k

2⊥) [Ne + ∇‖Nei − me

mi

V 2A

2∇2

⊥Ne + me

mi

(1 −K−2

k

)Ne +

+memiK−2k Ni −

me

mi

1

i

∂

∂t

(1 −K−2

k

) (V 2A∇2

⊥)−1Nf ‖

],

(14)

where the nonlinear second-order terms Ne, Ni , Nei and Nf ‖ are given in theAppendix.

3. Linear Properties of Kinetic Alfvén Waves

The linear approximation of (10) in Fourier space gives the linear dispersion ofsmall-scale AWs:

ω = kzVAK,

where K is given by (12).For β = me/mi the Alfvén wave becomes dispersiveless. With β = me/mi

(which means ρ2T = δ2

e ) in (12), the wave frequency no longer depends on the per-pendicular wave number. This fact has the following physical interpretation. Thethermal effects are characterized by the perpendicular length-scale ρT , and tend toaccelerate the wave. The parallel electron inertia effects, which are characterized


by the length-scale δe, tend to decelerate the wave. When the characteristic length-scale of the thermal effects is equal to the characteristic length-scale of electroninertia effects, ρT = δe, these effects compensate each other, and the wave becomesdispersiveless.

3.1. ION AND ELECTRON VELOCITIES

Calculating the linear force fs in terms of φ we obtain

fe = µ−1s

[(1 + µs V

2A

V 2T e

+ me

mi

)K2 − β

]∇⊥φ +

+ V2

A

V 2T e

K2∇‖φ + iω i

V 2Ak

2

(1 + Ti

Te

)b0 × ∇φ

(15)

for electrons, and

fi = fe −(

1 + Ti

Te

)∇φ (16)

for ions. Here b0 = B0/B0.Then from

vs⊥ = c

B0

[1

s

∂

∂tfs⊥ − b0 × fs⊥

](17)

and from

∂

∂tvs‖ = qs

msfs‖ (18)

we find the ion and electron velocities in AWs as

vi⊥ = c

B0µ−1s

[1

i

∂

∂t

((1 + µs V

2A

V 2T e

+ me

mi

)K2 − µT − 2β

)∇⊥φ −

−((

1 + µs V2

A

V 2T e

+ me

mi

)K2 − µT − β

)b0 × ∇⊥φ

],

(19)

ve⊥ = c

B0µ−1s

[−((

1 + µs V2

A

V 2T e

+ me

mi

)K2 − β

)b0 × ∇⊥φ −

−β 1

i

∂

∂t∇⊥φ

],

(20)

∂

∂tvi‖ = e

me

V 2A

V 2T e

[me

miK2 − β

]∇‖φ, (21)


∂

∂tve‖ = − e

me

V 2A

V 2T e

K2∇‖φ. (22)

From these expressions we notice the following properties for dispersive AWswith µT > ω/ i . Firstly, the perpendicular current is not dominated by the ion po-larization drift, jP⊥‖∇⊥φ, but by the B0 ×∇⊥φ drift, jE⊥⊥∇⊥φ, analog of the elec-tric E × B drift. Secondly, this perpendicular current, jE⊥, arises because the driftvelocity of the electrons is faster than that of the ions. And thirdly, there is a parallelcurrent, carried mainly by the electrons. Note, however, that the current closure issupplied by the parallel electron motion and perpendicular ion polarization drift.

3.2. PLASMA COMPRESSIBILITY

The electron density fluctuations and compressional perturbations of the magneticfield are essential for small-scale AWs. Eliminating all the quantities in the lin-earized version of Equations (2)–(9) in terms of φ and the perpendicular wavemagnetic field, B⊥, we obtain

∂

∂tB⊥ = −cµ−1

s ∇‖[((

1 + me

mi

)K2 − β

)b0 × ∇⊥φ + β 1

i

∂

∂t∇⊥φ

]. (23)

Then the relative importance of the number density fluctuations with respect to themagnetic fluctuations in KAWs is

n

n0= −ik⊥δi K

K2 − βB⊥B0, (24)

where δi = √c2mi/4πne2 is the ion inertial length. In the perpendicular wave

number range k⊥ρT < 1 < k⊥δi < 1/√β the perturbation of the number density is

even more pronounced than the magnetic field perturbation. This condition followsfrom (24) when we put there n/n0 > B⊥/B0 and account for ρT = δi

√β.

3.3. MAGNETIC COMPRESSIBILITY

From the perpendicular component of Ampère’s law (7) we get the magnetic com-pressibility in AWs (φ − B‖ relation):

B‖ = −c iV 2

A

(1 + Ti

Te

)φ. (25)

This relation may be written in the form

B‖B0

= −β nn0, (26)

showing that the gas compression is more important for AWs than the magneticcompression in a low-β plasma. The gas and magnetic compressions are anti-correlated, B‖n < 0, and in this sense the kinetic AW is similar to slow magneto-acoustic waves.


Comparison of (24) and (26) shows that the shear component of the magneticfield B⊥ is always dominant in AWs.

The large-scale AWs cannot produce any density and parallel velocity perturba-tions, and cannot produce compressional magnetic field perturbations. However,an accumulation of Alfvénic fluctuations at perpendicular wave-lengths at ion-gyroradius length-scales gives rise to both density and magnetic pressure perturba-tions. Due to these perturbations, short-scale high-frequency AWs are effectivelycoupled to large-scale low-frequency FWs and extract energy from them.

4. Nonlinear Decay of Fast Waves into Short-Scale Alfvén Waves

In this section we use the nonlinear eigenmode equation (10) to study the paramet-ric excitation of small-scale AWs by a pump fast wave.

4.1. RESONANT CONDITIONS

Let us consider a pump FW with a frequency ωP propagating at an angle to thebackground magnetic field, kP= (kP⊥; 0; kPz), axis z is parallel to B0. Any pair ofAWs can be nonlinearly coupled to this pump wave provided that the followingresonant conditions are satisfied:

k1 + k2 − kP = 0, (27)

ω1 + ω2 − ωP = 0. (28)

In principle, a frequency mismatch ( = ω1 + ω2 − ωP is allowed for as long as itis smaller than the rate of nonlinear interaction γNL, ( � γNL. In this paper weconcentrate on exact resonance, ( = 0.

There are two possible channels for the decay of fast waves into KAWs - de-cay into counter-propagating waves, sign (k1z) = −sign (k2z), and decay intoco-propagating waves, sign (k1z) = sign (k2z).

For non-perpendicular propagation of the fast wave we have to use the fullmatching condition for the parallel wave numbers:

k1z + s2 |k2z| = kPz, (29)

where the s2 = 1 stands for KAWs propagating in the same direction (that is in thedirection kPz), and the s2 = −1 stands for KAWs propagating in opposite directions(we assumed the wave 1 propagating in the same direction as the pump wave, thenthe wave 2 propagates in the opposite direction). In the limit of quasi-perpendicularpropagation of the fast wave (small kPz), this last case can be reduced to the dipoleapproximation in the parallel direction. But in the case of finite kPz, and if the wavespropagate in the same direction, the decay becomes selective.


As we are interested in a non-local process of energy transport into a region ofhigh perpendicular wave numbers, k2⊥,k1⊥ � kP⊥, we have −k1⊥ ≈ k2⊥ ≡ k⊥and K (k2⊥) ≈ K (k1⊥) ≡ K. The frequency matching condition (28) and theparallel wave number matching condition (29) can be solved, giving frequencies ofdecay waves for given K = K (k⊥) and S2 = −1 as

ω1 = 1

2

(1 + kPz

kP

K

KP

)ωP, (30)

ω2 = 1

2

(1 − kPz

kP

K

KP

)ωP. (31)

Then we have the following restrictions on the possible perpendicular wavenumbers of KAWs. For s2 = 1 (parallel propagating decay waves), we find from(28) that the KAWs’ wave numbers should be close to a fixed value, µT ≈ µ∗

T ≡(1 + β) k2

P⊥/k2Pz . The decay into parallel propagating KAWs is thus selective (re-

stricted by the matching conditions).For s2 = −1 (anti-parallel propagating decay waves), the matching conditions

reduce to requirement ω2 > 0, which can be satisfied in a wide range of perpen-dicular wave numbers µT < µ∗

T . This last restriction is not so severe and we willconsider here only the decay into counter-propagating KAWs.

4.2. NONLINEAR GROWTH RATE

To derive the nonlinear second-order parts of the equations for the KAWs 1 and 2,we use the linear relation for KAWs given in the previous section and the standardlinear relation for the pump FW:

φP = meV2T e

eKPBP‖B0

≡ meV2T e

eKPbP‖ (32)

(in terms of number density this means nP/n0 = KPBP‖/B0). The perpendicularelectron and ion velocity in the pump FW are equal, VPe⊥ = VPi⊥ ≡ VP⊥,

VP⊥ = eP⊥ωP

kP⊥bP‖, (33)

and the magnetic field components are related through ∇ · B = 0 as

BP⊥ = −eP⊥kPz

kP⊥BP‖. (34)

The function KP accounts for the thermal correction in the oblique fast waves:

KP = ωP

kPVA=√

1 + β k2P⊥k2

P

. (35)


We assume kPz > 0 and consider excited waves with high perpendicular wavenumbers, |k1⊥| ≈ |k2⊥| � kP⊥ (dipole approximation in the perpendicular direc-tion). It is useful to factorize the wave functions φ1 and φ2 into an exponentialphase dependence and a slowly varying amplitude, +1,2 = +1,2 (t):

φ1,2 = +1,2 exp(−iω1,2t + ik1,2 · r

).

Then the equations for resonant short-scale Alfvén waves 1 and 2, coupled to thelarge-scale pump fast wave P, are obtained from the nonlinear eigenmode equation(10) as[

∂

∂t− γL1

]+1 = i

ωP

2N2bP‖+∗

2, (36)

[∂

∂t− γL2

]+∗

2 = −i ωP

2N1b

∗P‖+1. (37)

γL1 and γL2 are the rates of the linear interaction for waves 1 and 2 (decrementor increment), bP‖ = BP‖/B0 is the normalized (parallel) component of the pumpmagnetic field.

The nonlinear coupling coefficients N1,2 for the waves 1 and 2,

N1,2 = −KPω1,2

ωP− 1

K2

2ω2,1

ωP+ s1,2 K

KP

kPz

kP+ kP⊥ · k1,2⊥

k2P⊥

×

×((1 −KP)

(1 − s1,2 K

KP

kPz

kP

)+ β

(1 − 1

K2

)2

+ β

K2

),

(38)

follow from (14) and (61)–(64) in the Appendix, where all the KAW perturbationsare eliminated in terms of φ1,2 by means of (19)–(26), and all fast wave pertur-bations are eliminated in terms of bP‖ by means of (32)–(35). In (36)–(38) weused the low-β approximation (13) for the dispersion functions K1 ≈ K2 ≡ K,K = 1 + µT and kept only the dominant terms.

Looking for exponentially growing (or decaying) solutions, +1,2 ∼ exp (γtott),we find the total growth (or damping) rate

γtot = γL1 + γL2

2±√(

γL1 − γL2

2

)2

+ γ 2NL, (39)

where γNL is the rate of the nonlinear pumping of FW energy into daughter KAWs.In terms of total FW magnetic field bP = BP/B0, bP‖ = bPkP⊥/kP, the nonlineargrowth rate

γNL = ωP

2

√N1N2

kP⊥kP

|bP| . (40)


Looking for the high perpendicular wave numbers of the excited KAWs and keep-ing the terms that can be large for large k2

1⊥ ≈ k22⊥ ≡ k2

⊥ � k2P⊥, the coupling

coefficients (38) can be written as

N1 = s1K

KPcos θP − cos θ2

KP√1 − q2

i

ωP

√µT

β×

×((1 −KP)

(1 − s1 K

KPcos θP

)+ β

(1 − 1

K2

)2

+ β

K2

),

(41)

N2 = s2K

KPcos θP + cos θ2

KP√1 − q2

i

ωP

√µT

β×

×((1 −KP)

(1 − s2 K

KPcos θP

)+ β

(1 − 1

K2

)2

+ β

K2

).

(42)

Here θP is the propagation angle of the fast wave, cos θP = kPz/kP, and θ2 isthe angle between the perpendicular wave vectors of the pump fast wave and theKAW 2, cos θ2 = kP⊥ · k2⊥/ (|kP⊥| |k2⊥|).

The nonlinear pumping rate γNL is a function of cos θP and cos θ2. γNL attainsits maximum at some value of (ωP/ i) cos θ2 that depends on µT . The depen-dence of this maximal value of γNL on µT is shown in Figure 1 for cos θP = 0.2.Corresponding values of (ωP/ i) cos θ2 vary in the range 5–10. As γNL increaseswith k⊥, the maximal growth rate occur at maximally possible µT � µ∗

T ≈ 26.4determined by the resonant conditions (for cos θP = 0.3; 0.4; 0.5 the maximalµ∗T = 11.1; 5.8; 3.3, respectively). Once the wave vector of FW deviates from

the purely parallel or perpendicular direction (kPz, and kP⊥ �= 0), the FW beginsto decay nonlinearly into KAWs. This process can become already pronounced forrelatively small deviations from parallel or perpendicular propagation.

The physical picture of the decay is as follows. The nonlinear electron velocityoscillations, vNL2,1e, are nonlinearly driven at spatio-temporal scales of KAW 2 andKAW 1 by, respectively, beatings of KAW 1 and KAW 2 with the pump magneticfield: vNL2,1e ∼ vKAW1,2e ×BP. The destabilizing terms in (40), which appear with coeffi-cients s1,2, follow from the divergence of this nonlinear electron velocity, ∼ ∇·vNLe ,pumping the electron density oscillations in accordance with Equation (3) for theelectrons. These nonlinear electron density oscillations interact constructively withnonlinear ion density and parallel current oscillations (remaining terms in (40)),resulting in the KAW’s reinforcement.

Note that if the nonlinear growth rate overcomes the frequency of one of decaywaves, then we have to deal with a modified parametric decay for which the growthrate is (Yukhimuk et al., 1999)

γMD =√

3

22/3

(ω

γNL

)1/3

γNL. (43)


Figure 1. Growth rate of the three-wave resonant decay FW = KAW + KAW for fast-wave amplitudebP = 0.1 as function of µT . The dashed curve shows the relative nonlinear pumping rate γNL/ωP.Curve 1: total growth rate γtot/ωP = γNL/ωP + γC/ωP in the collisional regime (for the case ofpump frequency ωP � νe). Curve 2: total growth rate

γtot/ωP = (γL1/ωP + γL2/ωP) /2 ±√(γL1/ωP − γL2/ωP)

2 /4 + γ 2NL/ω

2P

in the collisionless regime (ωP > νe), when Landau damping dominates.

For a fast wave propagating at an angle close to π/2 (quasi-perpendicular prop-agation), we can use the dipole approximation in the parallel direction also:

k1‖ ≈ −k2‖ ≡ k‖.

The excited KAWs can then only propagate in opposite directions along the back-ground magnetic field B0, and

ω1 = ω2 = ωP

2.

In this case γL1 = γL2 and the total growth (or damping) rate is reduced to

γtot = γNL + γL.

4.3. GROWTH RATE AND THRESHOLD AMPLITUDE OF THE DECAY:COLLISIONAL REGIME

We can find the dissipative threshold of the instability from the marginal conditionγNL + γL = 0, where γd is the linear damping rate of the excited Alfvén waves.For the low-frequency fast waves in the frequency range ωP = 0.01–10 s−1 (e.g.,excited in the corona by photospheric motions), the collisional damping of thedaughter KAWs dominates:


γL1 = γL2 = γC ≈ −memi

1

2βνe(ρ2T k

2⊥). (44)

The (dimensionless) total growth rate of small-scale KAWs in the collisionalregime,

γtot

ωP= γNL

ωP+ γC

ωP, (45)

is shown in Figure 1 (curve 1) for the pump amplitude bP = 0.05 and for the pumpfrequency ωP = νe.

The threshold amplitude of the fast wave in the corona in the collisional regimeis

bCP = me

mi

1

β

µ∗T√

N1(µ∗T

)N2(µ∗T

)√1 − cos2 θP

νe

ωP. (46)

bCP only weakly depends on β and cos θP for typical values of β = 0.05–0.3 inthe corona, and for oblique propagation of the fast wave, cos θP = 0.2–0.5. In thisrange

bCP ≈ 10−2 νe

ωP. (47)

This expression shows that the collisional threshold for a pump frequency as low asωP ≈ 0.1νe is

∣∣bCP ∣∣ ≈ 0.1. For higher frequencies the threshold decreases to values∣∣bCP ∣∣ � 10−2 at ωP ≈ νe. At higher pump frequencies, ωP > νe, the instabilityswitches into the collisionless regime where Landau damping of KAWs dominates.

4.4. GROWTH RATE AND THRESHOLD AMPLITUDE OF THE DECAY:COLLISIONLESS REGIME

There is a possibility that high-frequency fast waves can be launched into thecorona by magnetic reconnection events. In this case the high-frequency,

ωk

νe� 1√

2π

VA

VT eK (48)

(approximately ωk/νe > 1), part of the spectrum of the excited KAWs undergoescollisionless Landau damping (Voitenko and Goossens, 2000c). Also, in a morerarefied solar wind the Landau damping is stronger than the other linear dissipationmechanisms for KAWs. In these cases the collisional decrement, γC , has to bereplaced in (10), (36), and (37) by the Landau damping rate, which is different forthe waves 1 and 2:

γ1,2Ld

ωP= −

√π

8

Te

Ti

VA

VT e

µT

K

ω1,2

ωP. (49)

Then from (39) we find the net growth rate


γtot

ωP= −

√π

32

Te

Ti

VA

VT e

µT

K+

+√(√

π

32

Te

Ti

VA

VT e

µT

K

)2 (qK

KP

)2

+N1N2(1 − cos2 θP

) |bP|24,

(50)

where the resonant conditions are used. The wave number dependence of the growthrate in the collisionless (Landau) regime is shown in Figure 1 (curve 2). Note thatin the collisionless regime the decay switches on when µT overcomes some criticalvalue.

The collisionless (Landau) dissipative threshold is

bLP = µT

K

√π

4

me

βmi

1

N1N2(1 − cos2 θP

) (1 − cos2 θPK2

K2P

). (51)

bLP is a decreasing function of µT and attains its minimum bLthr at maximal possibleµT � µ∗

T . The values of this minimum determine the instability threshold as afunction of cos2 θP. Note that in the limit µT → µ∗

T we obtain bLP → 0, i.e.,there is no longer a threshold for the instability. Noting from (30) that the limitµT → µ∗

T implies ω1 → 0, that is k1z → 0 (λ1z → ∞), the real collisionlessthreshold has to be calculated in a different limit, µT → µm < µ∗

T , where µmcorresponds to a maximal parallel wavelength allowed by the parallel system sizeLz: λ1z (µm) = 2πVAK (µm) = Lz.

For the fast wave amplitudes well above the thresholds, bP � bC,LP , the growth

rate of the instability is determined by γNL.

5. Fast Wave Decay and Transient Heating Events in the Solar Atmosphere

In many cases, the decay of fast wave into counter-propagating KAWs can be adominant dissipation mechanism for coronal fast waves. As a particular examplewe examine the possible role of this process in the heating events in the solaratmosphere.

Transient brightenings on time scales of about one minute or less have beenfound from soft X-ray Yohkoh observations of active regions and from recent ultra-violet SOHO observations of the quiet Sun (Shimizu et al., 1992; Innes et al., 1997;Berger et al., 1999; Berghmans, McKenzie, and Clette, 2001). It is believed thatmost of these explosive events are closely related with the interaction of magneticfluxes, separated by current sheets. Magnetic reconnection in current sheets mayproduce reconnection outflows and consequent plasma heating and line broadeningdue to plasma turbulence excited by the outflows, like in solar flares (Voitenko andGoossens, 2000a, 2002).


Figure 2. Schematic representation of the nonlinear excitation of kinetic Alfven waves by fast wavesemitted from a collapsing current sheet. The background magnetic field is perpendicular to the planeof the paper.

On the other hand, a considerable fraction of the energy can be released by thedynamical evolution of the current sheets themselves, especially when fast mag-netic reconnection cannot develop. So, Fushiki and Sakai (1994) have shown thatthe fast waves can be produced in the solar atmosphere by a pinching current sheet.In turn, the fast waves undergo a modulational instability if the wave amplitudesovercome a critical value, bP‖ > Vs/VA (Sakai, 1983; Fushiki and Sakai, 1994).The slow mode waves excited by the modulational instability of the fast waves aredamped giving rise to plasma heating through Landau damping.

We consider the possibility that the fast waves with ωP ∼ 10 − 100 s−1 emittedfrom the magnetic reconnection events heat the surrounding plasma by the heav-ily damped KAWs that are excited by the parametric decay instability of the fastwaves. For the coronal parameters the threshold amplitude (47) of the parametricdecay is small, bCP‖ ≈ 10−2 for ωP ≈ νe ≈ 10 s−1 in the corona. The process ofnonlinear KAWs’ excitation in the plasma around current sheets is schematicallyshown in Figure 2.

The growth rate of the modulation instability is (Sakai, 1982)

γMI

ωP= cos θP

KP

√b2

P‖ − β

2, (52)

and its threshold amplitude

bMIP‖ =√β

2. (53)


Figure 3. Maximal relative growth rates of the decay FW → KAW + KAW instability γtot/ωP (curves1 and 2) and modulational instability γMI /ωP (dashed curve) as functions of FW propagation angle,cos θP. Curve 1 shows the relative growth rate γtot/ωP of the decay instability in the collisionalregime (ωP � νe), curve 2 - in the collisionless Landau (ωP > νe) regime. The FW amplitude isbP = 0.2, β = 0.05.

bMIP‖ is rather high in the corona, bMIP‖ � 0.2. Comparing (47) with (53), we seethat the parametric decay has a lower collisional threshold than the modulationinstability, bPDP‖ < bMIP‖ , if the wave frequency ωP > 10−2√2/βνe. This conditiongives ωP > 0.3 (wave periods τP < 20 s) for the solar corona where νe ≈ 10 s−1,β ≈ 0.05. As the frequencies of FWs excited by collapsing current sheets are byfar larger, ωP ≈ 24VA/d, d is the current sheet width (Fushiki and Sakai, 1994),the decay threshold is much lower than modulational.

In the collisionless regime, the parametric decay (50) is stronger than the modu-lation instability (52) for all propagation angles of FW in a plasma with β > 0.03.This condition is satisfied almost everywhere in the corona. Only for quasi-parallelFW, cos θP > 0.6, the increment of modulational instability overcomes the decayincrement in a plasma with β < 0.03.

The numerical comparison of the decay and modulational increments shows(see Figure 3) that the decay instability dominates under coronal conditions. Notethat the plasma and wave parameters in Figure 3 are chosen such as to make themodulational instability over-threshold: for higher β > 0.05, or for lower bP < 0.2,the modulational instability disappears, while the decay increment only slightlyreduces.

For the FW with amplitude bP = 0.1 (modulational instability is below thresh-old) and frequency ωP = νe = 30 s−1 (collisional regime) pumped from the currentsheet, we can estimate the typical time for the decay instability to develop and theplasma to heat:


τtot ∼ τNL ∼ τc ∼ 1 s

in a wide range of propagation angles 0.1 < cos θP < 0.6. Such waves damp inthe vicinity of the source (current sheet), and damp within the distance LFW =VAτtot ≈ 103 km. Therefore the heated volume is 103 × 103 × Lz km3 (Lz is thelength of the current sheet, Lz ≈ 103 km). Even if the initial FWs are emittedextremely obliquely, cos θP < 0.1 (which is unlikely for reasonable length scalesof current sheets), their refraction during propagation in the region of strongermagnetic field (away from reconnection region) quickly increases the cos θP tocos θP > 0.1.

For the high-frequency fast waves with ωP = 300 s−1 > νe (collisionlessregime)

τtot ∼ τNL ∼ τL ∼ 0.1s

in a wide range of propagation angles 0 < cos θP < 0.6. Such waves damp at shortdistances from the source, LFW = VAτtot ≈ 100 km.

The real heating time in this model is determined by the magnetic reconnectiontime scale, which is in the range 10–100 s, and the volume heated by the waves isrestricted in size to ∼ 102 –103 km, equal to the dissipation distance of fast waves.A significant heating beyond this volume may be due to the thermal conductionfrom it.

6. Discussion and Conclusions

We have studied the parametric excitation of high-wave number AWs by large-scale pump FWs. This is a new channel for the dissipation of fast waves in thesolar atmosphere where fast waves nonlinearly couple their energy to Alfvén waveswith short wavelengths across B0. In the framework of two-fluid MHD we haveinvestigated a resonance parametric decay of the pump FW into pairs of short-scale KAWs: FW→KAW+ AW. The process is caused by the scalar nonlinearinteraction that is proportional to the scalar product of the wave vectors of theinteracting waves. It can strongly accelerate the evolution of wave energy towardssmall length scales in astrophysical plasmas. The nonlinear coupling is strong forfast waves launched with amplitudes of the order of 0.01–0.1 for BP/B0. Thenonlinear damping of such finite-amplitude fast waves is much stronger than thelinear damping mechanisms based on classical transport coefficients and Landaudamping. For oblique fast waves, the linear dissipation is due to electron viscous(γ vis) and Landau (γ L

Pe) damping (Hung and Barnes, 1973), and the electron viscousdamping passes over into electron Landau damping as ωP/νe passes through unity,and proton Landau damping is restricted to cos θP > 0.6 and is weak for coronalvalues of β, β < 0.4. Then we can estimate linear FW damping in the range0 < cos θP < 0.5 as


γP

ωP∼ γ vis

ωP≈ − (0.05 − 0.5)

ωP

νeβ, (54)

for ωP < νe (collisional regime) and (e.g., Perkins, 1973)

γP

ωP∼ γ L

Pe

ωP= −

√π

8

√me

mi

VS

VA

1 − cos2 θP

KP cos θPexp

(− ω2

P

2V 2T ek

2 cos2 θP

)(55)

for ωP > νe (collisionless regime).Using these relations, we estimate that, for typical νe ≈ 10 s−1 and β ≈ 0.1 in

the corona, the amplitudes bP > 10−2 provide the dominant damping of fast wavesdue to decay into KAWs.

For the heating problem in general, the scalar parametric decay gives an ex-cellent prediction of the thermal explosion that follows the injection of finite-amplitude FWs. Of course, the enhanced energy release is restricted by the dy-namical back-reaction of the heating on the waves. As a particular example weconsidered the decay of fast waves emitted by collapsing current sheets in themagnetic reconnection events at the base of solar corona. This process providesfast local heating of coronal plasma in the volume of about 103 × 103 × 103 km3.

An important feature of this process is that it is non-local in wave numberspace: the parametric decay of large-scale FWs into a spectrum of short-scaleKAWs introduces a jump-like transport of wave energy directly into the dissipativewave number domain. The excited KAWs have very short wavelengths in the planeperpendicular to B0, and thus are damped almost immediately by the linear kineticor collisional dissipation (Voitenko and Goossens, 2000c). As a consequence, theoverall time scale of the heating process is determined mainly by the characteristictime of the nonlinear mode conversion or by the relaxation time of the currentsheets. Our estimations show that the overall time scale of the heating can easilybe in the (sub-)minute range, as observed by SOHO at the base of solar corona.

The nonlinear excitation of KAWs by fast waves can be important for the heat-ing process of coronal magnetic loops. The compressional fast eigenmodes can beaccumulated in a loop under the action of a stochastic driver at loop’s footpoints, asis discussed in Section 1. Once the FW amplitude overcomes the threshold value,which is in the range bCP‖ = 0.01–0.1 for coronal loops, the pairs of resonantKAWs are nonlinearly excited. This process recalls linear resonant absorption insense that it occurs on resonant magnetic surfaces. But nonlinear resonant con-ditions require the pump wave frequency to be matched with the local Alfvénfrequency doubled, because two Alfvén waves are involved. And, of cause, thephysics of nonlinear mode conversion is quite different. The nonlinear resonantabsorption has an advantage in comparison with linear resonant absorption since itdoes not require a preliminary setup stage for the creation of small length scales.The nonlinearly excited KAWs possess already small perpendicular wavelengthsand therefore strongly damp heating plasma. In view of potential importance ofthis process for the problem of coronal loop’s heating, it will be discussed in moredetails elsewhere (Voitenko and Goossens, in preparation).


The compressional (fast mode) waves may be gradually excited in the solarwind by the phase mixing (refraction) of the slab Alfvénic waves that propagateinitially along B0 and enter the region with transversal inhomogeneity. The wavesthat are polarized in the direction of plasma inhomogeneity correspond to the fastwave mode. Due to refraction the component of the wave vector in the direction ofthe inhomogeneity increases steadily. Once the growing compressional componentof the fast wave exceeds the threshold value, the parametric excitation of KAWsand consequent plasma heating switches on, providing a gradual heating of thesolar wind with distance from the Sun. This mechanism also provides the sourcefor the sunward propagating waves in the solar wind.

There are many other objects that may be influenced by the process of para-metric decay of fast waves. Plasma energization and anisotropic fluctuations ofdensity and magnetic field in the interstellar plasma and in radio jets are examplesof possible applications. In the previous section we have shown that the nonlineardecay of large-scale fast waves into small-scale AWs is a very feasible energizationmechanism of the coronal plasma. This process can also explain the replenishmentof small-scale high-frequency KAWs, observed in the solar wind (Matthaeus et al.,1999). Finally, the astrophysical implications for anisotropic heating of stellar plas-mas are straightforward. Segregation from MHD compressional motions not onlyallows strong heating of the coronal plasma but also permits significant temperatureanisotropy of plasma species to be maintained by the decay action of large-scalefast waves in stars like the Sun.

Generically, the coupled-mode equation for small-scale KAWs does not dependupon a particular configuration of the magnetoplasma. Qualitative ideas about theeventual evolution of mode energies lead to scenarios that depend mainly uponnonlinear coupling coefficients and linear growth/damping rates. In our case, thenonlinearly unstable small-scale KAWs are excited by a large-scale pump fastwave. This scenario models a jump-like transport of wave energy directly into thedissipative wave number domain (there is no need for the energy cascade involvingwave-wave interactions that are local in frequency and wave number).

The influence of the driving on the primary modes is such that they are reducedto threshold amplitudes ultimately. As the threshold amplitudes are small (muchless than unity in terms of background magnetic field), we do not need high am-plitudes of fast waves for this nonlinear process. In turn, the threshold amplitudeplaces a limit on the amplitudes of fast waves that can persist in a plasma. Thetypical value for this limiting amplitude is B/B0 ∼ 10−2 in the solar corona andsolar wind.

The mechanism investigated in the present paper may play an important role inthe impulsive heating events observed by Yohkoh and SOHO, and in the extendedquasisteady heating of the solar corona and solar wind. To this end we would liketo stress the need for further theoretical and experimental study of the role of short-scale KAWs in space plasmas and a cautious discussion thereof.


Acknowledgements

This research is supported by the FWO-Vlaanderen grants G.0335.98 and G.0344.98.We are grateful to an anonymous referee for his/her helpful comments and sugges-tions.

Appendix

As in Voitenko and Goossens (2000), we introduce the potentials ϕ and ψ as

E⊥ = −∇⊥ϕ − 1

c

∂

∂tA⊥,

E‖ = −∇‖ (ϕ + ψ) .The Poisson equation gives the following relation between φi and φe ≡ φ:

φi = V 2T i

ω2pi

∇2ϕ − Ti

Teφe ≡ −ηϕ − Ti

Teφ. (56)

Expressing E, B and j‖ through potentials, we get

V −2T e

∂2

∂t2φ = −∇2

‖ (ϕ + ψ − φ)+ νδ2e∇2∇2

‖ψ +

+ 1

2e

∂2

∂t2

(∇2‖ψ − ∇2

⊥ϕ)+ me

mi

1

c

1

i

∂2

∂t2B‖ +Ne

(57)

from the electron continuity equation, and

−mimeV −2T e

((1 − V 2

T i

1

2∇2

⊥

)∂2

∂t2− V 2

T i∇2‖

)φ =

= V −2T i

((1 − V 2

T i

1

2∇2

⊥

)∂2

∂t2− V 2

T i∇2‖

)ηϕ − 1

c

1

i

∂2

∂t2B‖+

+ ∇2‖(−ϕ − ψ + νδ2

e∇2ψ)+ 1

2

∂2

∂t2

(∇2‖ψ − ∇2

⊥ϕ)+Ni

(58)

from the ion continuity equation, where ν = ν∂−1t .

Two other equations are obtained from the Ampére’s law. The parallel compo-nent gives

δ2e∇2ψ =

(1 + me

mi

)(ϕ + ψ − φ − νδ2

e∇2ψ) +

+memi

(1 + Ti

Te

)φ + me

miηϕ + ∇−1

‖ Nei.

(59)


Taking ∇⊥× of the perpendicular part of (7) we get the equation for the parallelcompressional component of wave magnetic field:[

∂2

∂t2− V 2

A∇2

]1

c

∂

∂tB‖ = i

(1 + Ti

Te

)∂

∂t∇2

⊥φ +Nf ‖. (60)

Up to second order, the nonlinear terms in (57)–(60) are

Ne = −V −2T e

∂

∂t(ve · ∇φ)+

+∇ ·(

Fe‖ + 1

2e

∂2

∂t2Fe⊥ − 1

e

∂

∂t[b0 × Fe⊥]

) (61)

and

Ni = Ti

TeV −2T i

∂

∂t(vi · ∇φ)+

+∇ ·(

Fi‖ + 1

2

∂2

∂t2Fi⊥ − 1

∂

∂t[b0 × Fi⊥]

),

(62)

Nei = −[me

miFi‖ + Fe‖ −me ∂

∂t(φi

Tivi‖ + φe

Teve‖)

], (63)

and

Nf ‖ = −∇⊥ × V 2A∂

c∂t

(eφ

Te∇2A⊥

)−

− i ∂∂t

∇⊥ ×[− 1

i

∂

∂t

[Fi + me

miFe

]+ b0 × (Fi − Fe)

].

(64)

The last term, Nf ‖, originates from the nonlinear magnetic compression. Wehave ignored its contribution in our study of the vectorial nonlinear interactionwhich is local in k-space. Here we take Nf ‖ into account as it may become signifi-cant in the non-local scalar interactions.

Neglecting in (57)–(60) deviations from quasi-neutrality that are important atDebye wavelengths, η ≈ 0, and eliminating all wave potentials in favor of φ, weobtain the following equation for the (coupled) quasi-perpendicular Alfvén andslow magneto-acoustic waves:


V −2T e

[(1 + me

mi

) (1 − νδ2

e∇2⊥)− δ2

e∇2⊥

]∂2

∂t2

∂2

∂t2φ −

−V −2T e V

2A∇2

‖

[∂2

∂t2

(1 + me

mi+ β (1 − νδ2

e∇2⊥)− V 2

T

1

2∇2

⊥

)− V 2

T∇2‖

]φ

= + ∂2

∂t2

[Ne + ∇‖Nei

]− me

mi

V 2A

2

∂2

∂t2∇2

⊥Ne+

+memi

[∂2

∂t2− V 2

A∇2‖

](Ne − 1

iv2k∇2

‖

(∇2

⊥∂

∂t

)−1

Nf ‖

)+ me

miNi.

(65)

All thermal quantities with bar are defined with the normalized temperatures T ≡T / (1 + β).

After Fourier-analyzing (65) in configuration space, we arrive to (10–14) for(kinetic) Alfvén wave branch.

References

Aschwanden, M. J., Fletcher, L., Schrijver, C. J., and Alexander, D.: 1999, Astrophys J. 520, 880.Berger, T. E., De Pontieu, B., Schrijver, C. J., and Title, A. M.: 1999, Astrophys J. 519, L97.Berghmans, D., McKenzie, D., and Clette, F.: 2001, Astron. Astrophys. 369, 291.Cadez, V. M., Csik, A., Erdelyi, R., and Goossens, M.: 1977, Astron. Astrophys. 326, 1241.Cranmer, S. R., Field, G. B., and Kohl, J. L.: 1999, Astrophys. J. 518, 937.De Groof, A., Tirry, W. J., and Goossens, M.: 1998, Astron. Astrophys. 335, 329.De Groof, A. and Goossens, M.: 2000, Astron. Astrophys. 356, 724.Doyle, J. G., Banerjee, D., and Perez, M. E.: 1998, Solar Phys. 181, 91.Erdelyi, R., Doyle, J. G., Perez, M. E., and Wilhelm, K.: 1998, Astron. Astrophys. 337, 287.Fushiki, T. and Sakai, J.-I.: 1994, Solar Phys. 150, 117.Goossens, M.: 1991, in Eric R. Priest, Alan W. Wood (eds.) Advances in Solar System Magnetohy-

drodynamics, Cambridge University Press, Cambridge, p. 137.Harrison, R. A., Lang, J., Brooks, D. H., and Innes, D. E.: 1999, Astron. Astrophys. 351, 1115.Heyvaerts, J. and Priest, E. R.: 1983, Astron. Astrophys. 117, 220.Hollweg, J. V.: 1999, J. Geophys. Res. 104, 14811.Hung, R. J. and Barnes, A.: 1973, Astrophys. J. 180, 271.Innes, D. E., Inhester, B., Axford, W. I., and Wilhelm, K.: 1997, Nature 386, 811.Ionson, J. A.: 1978, Astrophys. J. 226, 650.Kappraff, J. M. and Tataronis, J. A.: 1977, J. Plasma Phys. 18, 209.Koutchmy, S., Zhugzda, Y. D., and Locans, V.: 1983, Astron. Astrophys. 120, 185.Laing, G. B. and Edwin, P. M.: 1995, Solar Phys. 157, 103.Li, X., Habbal, S. R., Hollweg, J. V., and Esser, R.: 1999, J. Geophys. Res. 104, 2521.Mann, I. R., Wright, A. N., and Cally, P. S.: 1995, J. Geophys. Res. 100, 19441.Marsch, E.: 1991, ‘Kinetic Physics of the Solar Wind’, in R. Schwenn and E. Marsch (eds.), Physics

of the Inner Heliosphere, Vol. II, Springer-Verlag, Heidelberg, p. 45.Marsch, E. and Tu, C. Y.: 1997, Astron. Astrophys. 319, L17.Matthaeus, W. H., Zank, G. P., Leamon, R. J., Smith, C. W., and Mullan, D. J.: 1999, Space Sci. Rev.

87, 269.




Nakariakov, V. M., Roberts, B., and Muravski, K.: 1997, Solar Phys. 175, 93.Perkins, F.: 1973, Astrophys. J. 179, 637.Ruderman, M. S. and Wright, A. N.: 2000, Phys. Plasmas 7, 3515.Ruderman, M. S., Berghmans, D., Goossens, M., and Poedts, S.: 1997, Astron. Astrophys. 320, 305.Sakai, J.-I.: 1983, Solar Phys. 84, 109.Shimizu, T., Tsuneta, S., Acton, L. W., Lemen, J. R., and Uchida, Y.: 1992, Publ. Astron. Soc. Japan

44, L147.Tirry, W. J., Berghmans, D., and Goossens, M.: 1997, Astron. Astrophys. 322, 329.Verheest, F.: 1977, Astrophys. Space Sci. 46, 165.Voitenko, Yu. M. and Goossens, M.: 2000a, Physica Scripta T84, 194.Voitenko, Yu. M. and Goossens, M.: 2000b, Astron. Astrophys. 357, 1073.Voitenko, Yu. M. and Goossens, M.: 2000c, Astron. Astrophys. 357, 1086.Voitenko, Yu. M. and Goossens, M.: 2002, Solar Phys. 206, 285.Wilhelm, K., Marsch, E., Dwivedi, B. N. et al.: 1998, Astrophys. J. 500, 1023.Wright, A. N. and Rickard, G. J.: 1995, Astrophys. J. 444, 458.Yukhimuk, A. K., Yukhimuk, V. A., Sirenko, O. K., and Voitenko, Yu. M.: 1999, J. Plasma Phys. 62,

53.

Documents

Nonlinear excitation of small-scale Alfvén waves by fast waves and plasma heating in the solar atmosphere