astron. astrophys. 357, 1073–1085 (2000) astronomy and...

Astron. Astrophys. 357, 1073–1085 (2000) ASTRONOMYAND

ASTROPHYSICS

Nonlinear decay of phase-mixed Alfven waves in the solar corona

Yu. Voitenko1,2 and M. Goossens1

1 Centre for Plasma Astrophysics, K.U.Leuven, Belgium ([email protected])2 Main Astronomical Observatory, Kyiv, Ukraine

Received 22 November 1999 / Accepted 21 March 2000

Abstract. This paper presents an analytical investigation of thenonlinear interaction of phase-mixed Alfven waves in the frame-work of two-fluid magnetohydrodynamics. It focuses on theparametric decay of the phase-mixed pump Alfven wave intotwo daughter Alfven waves. This parametric decay is a non-linear phenomenon which does not occur in ideal MHD sinceit is induced by the combined action of finite wave amplitudeand the non-zero gyroradius of the ions. In contrast to intu-itive expectation, the effects of the non-zero gyroradius of theions are already important at length scales that are substantiallylonger than the ion gyroradius. The parametric decay occursfor relatively small wave amplitude and is more efficient thancollisional damping.

Key words: instabilities – Magnetohydrodynamics (MHD) –plasmas – waves – Sun: corona

1. Introduction

The physical mechanisms responsible for the heating of thesolar corona still lack unambiguous identification. Resonantabsorption and phase mixing (see Goossens 1994 and refer-ences therein) are two popular theories of Alfven wave heatingwhich involve spectral energy transfer towards small length-scales. Both are due to the transverse plasma inhomogeneity,typical for the corona. Phase mixing of Alfven waves whichleads to enhanced wave dissipation and consequent plasma heat-ing, has been first proposed as a coronal heating mechanism byHeyvaerts & Priest (1983). Since then, phase mixing has beenstudied both for open and closed magnetic configurations (seerecent papers by Ofman & Davila 1995; Hood et al. 1997a,b;Ruderman et al. 1998; De Moortel et al. 1999). The first con-clusions on the asymptotic behaviour of the wave amplitudeswith height,∼ exp

(−z3/z3c

), or time,∼ exp

(−t3/t3d), havebeen recovered for the typical coronal conditions, when thelength scale of the plasma inhomogeneity along the magneticfield lines,L‖, is very large compared to that of the plasma inho-mogeneity transverse to the magnetic field lines,L⊥,L‖ L⊥(e.g., Hood et al. 1997a). The only notable deviation from this

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behaviour was found by Ruderman et al. (1998), who presumedad hoc that the scale height of wave damping is comparable tothe vertical inhomogeneity scale.

Phase mixing of AWs induced by plasma flows has beenstudied by Ryutova & Habbal (1995). However, their resultsonly apply to sub-Alfvenic flows,v0 VA, as has been notedby Ruderman et al. (1999). Ruderman et al. (1999) relaxed therestriction of sub-Alfvenic flows and their investigation showsa good agreement between the linear analytical solutions andthe nonlinear numerical simulations in the model of one-fluidnonlinear resistive MHD.

These results were obtained in the one-fluid MHD approx-imation. However, with the creation of short transverse length-scales in Alfven waves, the Alfven waves become essentiallytwo-dimensional in the sense that they have long-wavelengthsalong the magnetic field and short-wavelengths across it. In thissituation the ion polarization drift in the perpendicular direc-tion creates a charge separation acrossB0, while field-alignedelectron flows tend to cancel this charge separation, and thusthe motions of the ions and electrons decouple from other. Thefinite temperature and/or electron inertia effects prevent com-plete charge cancellation, and a longitudinal wave electric fieldEz‖B0 arises. The presence of the longitudinal wave electricfield, Ez‖B0, and current,jz ‖ B0, bring about many newproperties for Alfven waves, including the effective interac-tion of waves with plasma particles (either via kinetic effectsat Cherenkov resonance, or via collisions), as well as the non-linear interaction with other modes and among Alfven wavesthemselves. Since an adequate description of these effects re-quires kinetic plasma theory, these short-scale Alfven waves arecalled kinetic Alfven waves (KAWs).

Linear and nonlinear properties of KAWs have been ex-tensively studied in the astrophysical and geophysical con-text. Examples are plasma heating (de Azevedo et al. 1994;Voitenko 1994), and current drive (Elfimov et al. 1996) in coro-nal loops in the solar atmosphere, instability of the inter-stellar plasma (Shukla et al. 1989), particles acceleration ingalactic radio jets (Bodo & Ferrari 1982), impulsive energyrelease in solar flares (Voitenko 1996a,b, 1998c), wave in-stability (Voitenko et al. 1990 ; Hasegawa & Chen 1992) andplasma turbulence (Voitenko 1996c) in the Earth’s magneto-sphere, anomalous magnetic diffusion in coronal current layers

1074 Y. Voitenko & M. Goossens: Nonlinear decay of phase-mixed Alfven waves

(Voitenko 1995). The nonlinear properties of KAWs are of greatimportance (Shukla & Stenflo 1995). In fusion devices, the effi-ciency of the energy exchange between waves and plasma par-ticles caused by kinetic properties of KAWs has been provenrecently both by theory and by experiment (Jaun et al. 1997).

It is thus surprising, that almost no attention has been paidto these properties of the KAWs created by the phase-mixingprocess in the non-uniform corona. Indeed, there are severalpapers studying the creation of short transverse length-scalesin Alfv en waves and the eventual wave dissipation. There arealso a few papers which concentrate on nonlinear effects inphase mixing (Nakariakov et al. 1997), and resonant absorption(Poedts & Goedbloed 1997). Ofman & Davila (1997) studiedboth the nonlinear excitation of sound waves and plasma heat-ing by AWs in a nonuniform plasma in coronal holes. But theseinvestigations have been carried out in the one-fluid MHD ap-proximation, missing important properties of Alfven waves withlarge transverse gradients developed by phase mixing.

The lack of interest in KAW in solar physics is due to anargument based on two characteristic transverse length scales,namely the Alfven wave dissipation length scaleld and the pro-ton gyroradiusρi, with ld being much longer thanρi. The factthatρi ld so that Alfven waves are damped long before lengthscales of the order ofρi are set up by the phase mixed Alfvenwaves leads to the intuitive conclusion that non-zero proton gy-roradius effects are unimportant for Alfven waves. However,this intuitive conclusion can be erroneous.

In the present article we shall show that the finite (ion) Lar-mor radius (FLR) effects in Alfven waves can come into playat length scales which are much longer than bothρi and ld.In this paper we study the parametric decay of a phase-mixedAW into two daughter AWs, induced by the combined action ofthe finite wave amplitude and FLR effects. This is an importantnonlinear process which significantly modifies the wave spec-tral dynamics of phase mixing. So we are not concerned withthe mechanisms that create phase mixed Alfven waves. Instead,we try to find out how the nonlinearity induced by phase mixingaffects the wave spectral dynamics.

A significant input of energy into a plasma, as observed inthe solar corona, can only be achieved if the launched waveshave sufficiently large amplitudes. E. g., AWs can balance theenergy loss from the loop structures of active regions and fromcoronal holes, if the wave magnetic field amounts to 1-5% ofthe background magnetic fieldB0. Nonlinear effects becomeimportant for waves of such amplitudes, and the ability of thewaves to participate in different kinds of nonlinear interactionhave to be examined.

It is well known that ideal MHD Alfven waves cannotinteract among themselves. Nonlinear coupling of AWswith other waves has been studied for a parallel-propagatingpump wave (Galeev & Sagdeev 1979; Goldstein 1978;Vi nas & Goldstein 1991; Jayanti & Hollweg 1993; Ghosh etal. 1993). In particular, the parametric decay into oblique waveswas studied analytically (Vinas & Goldstein 1991) and numer-ically (Ghosh et al. 1993), and the consequent development ofanisotropic turbulent cascade has been observed in numerical

simulations (Ghosh & Goldstein 1994). In the Discussion weshall go into the details of how our approach differs from thatused in the papers cited above. Here it suffices to note thatwe start with spectral anisotropy (due to phase mixing) whichthen initiates the nonlinear spectral dynamics. The papers citedabove consider the reverse sequence.

We show that the influence of the three-wave nonlinear inter-action becomes important with the creation of small transverselength-scales in the phase-mixed Alfven waves, and can signif-icantly change the picture of the phase mixing in the corona. Inorder to describe this process, we have to take into account thatthe motions of the electrons and the ions in AWs become decou-pled from each another at small length-scales, and that the wavedynamics can be adequately described by kinetic theory. Thenonlinear kinetic theory of KAWs, including nonlinear three-wave resonant interaction and linear Cherenkov wave-particleresonant interactions, has been developed by Voitenko (1998a).The collisional effects can be self-consistently taken into ac-count in this theory by including collisional integrals into theVlasov equations.

However, when we consider the low-frequency part of theAlfv en wave spectrum in the solar corona,ω . 1 − 100 s−1,we note that irrespectively of the KAW wavelengths, the lin-ear kinetic damping is weaker than the collisional dissipationin this frequency range (Voitenko & Goossens 2000). Neglect-ing kinetic wave-particle interaction, some properties of thesewaves, such as thermal wave dispersion, collisional dissipation,and nonlinear three-wave interaction, can be described in theframework of the much simpler two-fluid MHD model. As ispointed out in the Discussion, the high-frequency AWs withω & 1 − 100 s−1, which can be excited by the chromosphericmagnetic activity (Axford & McKenzie 1992), can be also ap-proximately described by two-fluid MHD with kinetic effectstaken from kinetic theory.

The paper is arranged as follows. The basic equations arepresented in Sect. 2. The second-order two-fluid MHD theoryof the nonlinear plasma motions in short-scale Alfven waves isdeveloped, and the Alfven wave eigenmode equation is derivedin Sect. 3. The collisional (linear) damping of short-scale Alfvenwaves results in the same asymptotic behaviour of the waveamplitude as was found by Heyvaerts & Priest (1983) (Sect. 4).The parametric decay of a pump oblique AW into two daughteroblique AWs is considered in Sect. 5, and the results are appliedto a phase-mixed AW propagating upward in the solar corona inSects. 6 and 7. We discuss possible consequences of this processin the solar corona in Sect. 8 and present our conclusions in thelast section, Sect. 9.

2. Basic equations

The wave electromagnetic fields obey Maxwell’s equations

∇ × B =4πc

j +1c

∂E∂t, (1)

∇ × E = −1c

∂B∂t, (2)

Y. Voitenko & M. Goossens: Nonlinear decay of phase-mixed Alfven waves 1075

∇ · E = 4πQ, (3)

where the current density,j=∑

s qsnsvs, and the charge den-sity,Q =

∑s qsns, have to be calculated using a suitable math-

ematical model of the plasma. The most popular models arebased on the ideal MHD equations, the two-fluid MHD equa-tions, and kinetic Vlasov equations.

As discussed in the Introduction, we use the mathematicalmodel of two-fluid MHD in order to take into account some im-portant linear and nonlinear effects in the low-frequency short-scale AWs. In two-fluid MHD the electron and ion fluids areallowed to move in separate ways, but are coupled by collectiveelectromagnetic fields and by the electron-ion friction force.The equations of motion for the electrons and ions are:

∂ve

∂t+ (ve∇)ve =

− e

me

(E +

1cve × B

)− Te

mene∇ne + R‖, (4)

∂vi

∂t+ (vi∇)vi =

e

mi

(E +

1cvi × B

)− Ti

mini∇ni − me

miR‖, (5)

wherepe,i = ne,iTe,i is the ion/electron pressure,Te,i = const(but we do not requireTe = Ti). The parallel friction force,R‖ ‖B0, responsible for the (parallel) resistivity alongB0, is

R‖ = −ν (ve‖ − vi‖),

whereν = 0.51νe, νe is the electron collision frequency.Note that we have dropped the perpendicular friction force,

responsible for the perpendicular resistivity, because its contri-bution in the wave dissipation is

(λ‖/λ⊥

)2times smaller than

contribution of the parallel resistivity (λ‖ andλ⊥ are the paralleland perpendicular wavelengths).

The particles number densities obey the continuity equations

∂

∂tns + ∇(nsvs) = 0, (6)

where the subscripts denotes particles species,s = e, i.

3. Alfven wave eigenmode equation

Instead of the number densityns, it is convenient to introducethe Boltzman-like potentials

φs = −T s

qslnns

nn, (7)

whereqs = e for ions and−e for electrons, andnn is a constant(in a uniform plasma, one can take the equilibrium number den-sity n0 for nn). Then, by separating background and perturbedmagnetic fieldB → B0+ B, we can rewrite the Eqs. (4)-(6) as

∂vs

∂t=

qsms

(1cvs × B0 + Es

), (8)

∂

∂tφs + vs · ∇φs − ms

qsV 2

Tα∇ · vs = 0. (9)

The functionEs is the charge force per unit mass and unit charge,which includes the linear electric force, the collisional frictionforce, the pressure gradient force, and the nonlinear force:

Es = E−νme

ne2j‖ + ∇φs + fs. (10)

The velocity difference is expressed through the current (thequasineutrality conditionne = ni ≡ n is used):

(vi − ve) =1ne

j, (11)

and the nonlinear force is

fs =1cvs × B − ms

qs(vs · ∇)vs. (12)

Note that the functionsφs in (7)-(10) contain both linear andnonlinear parts.

Multiplying B0 × (8) we find

∂

∂tvs × B0 =

qsms

[−B0 × Es − 1

cB2

0vs +1c

(B0 · vs)B0

]. (13)

Inserting this into∂(8)/∂t, we get an equation for the velocityin the plane⊥ B0 as(

1 +1

Ω2s

∂2

∂t2

)∂

∂tvs⊥ =

qsms

1Ωs

∂

∂t

[1Ωs

∂

∂tEs⊥ − b0 × Es⊥

], (14)

whereb0 = B0/B0 is the unit vector in the direction of equi-librium magnetic field (that isE‖ = b0E‖, etc.),Ωs is thecyclotron frequency.

For the motions that are slower than the cyclotron gyrationof the particles, as the low-frequency MHD waves, the electronand ion velocities in the plane, normal toB0, may be found ina drift approximation:

∂

∂tvs⊥ =

qsms

1Ωs

∂

∂t

[1Ωs

∂

∂tEs⊥ − b0 × Es⊥

]. (15)

The velocity of the particles along the magnetic field followsthe equation

∂

∂tvs‖ =

qsms

Es‖. (16)

Using (14) and (13), we eliminate the particle velocity fromthe continuity Eq. (9):

∂2

∂t2φs +

∂

∂t(vs · ∇φs) − V 2

Tα∇ ·(Es‖ +

1Ωs

∂

∂t

[1Ωs

∂

∂tEs⊥ − b0 × Es⊥

])= 0, (17)


or, inserting the explicit expression forEs,

∂2

∂t2

(1 − V 2

Tα∇ · 1Ω2

s

∇⊥

)φs − V 2

Tα∇ · ∇‖φs +

V 2Tα∇ · 1

Ωs

∂

∂t[b0 × ∇⊥φs] = V 2

Tα∇ ·((E−meν

ne2j)

‖+

1Ω2

s

∂2

∂t2E⊥ − 1

Ωs

∂

∂t[b0 × E⊥]

)−

∂

∂t(vs · ∇φs) +

V 2Tα∇ ·

(fs‖ +

1Ω2

s

∂2

∂t2fs⊥ − 1

Ωs

∂

∂t[b0 × fs⊥]

); (18)

hereV 2Tα = Tα/mα. Taking into account the quasineutrality

condition,− (Te/Ti)φi = φe ≡ φ, and neglecting couplingto the fast magnetosonic waves (via terms with vector prod-ucts), we rewrite the continuity Eq. (18) for electrons and ionsseparately:

V −2Te

(∂2

∂t2φ− V 2

Te∇ · ∇‖φ)

=∇·(E−ν me

ne2j)

‖+Ne; (19)

mi

meV −2

Te

(∂2

∂t2

(1 − V 2

Ti∇ · 1Ω2 ∇⊥

)φ− V 2

Ti∇ · ∇‖φ)

=

−∇ ·((

E−ν me

ne2j)

‖+

1Ω2

∂2

∂t2

(E−ν me

ne2j)

⊥

)−Ni, (20)

whereNe andNi denote the following nonlinear terms:

Ne = −V −2Te

∂

∂t(ve · ∇φe) +

∇ ·(fe‖ +

1Ω2

e

∂2

∂t2fe⊥ − 1

Ωe

∂

∂t[b0 × fe⊥]

); (21)

and

Ni =Ti

TeV −2

Ti

∂

∂t(vi · ∇φ) +

∇ ·(fi‖ +

1Ω2

∂2

∂t2fi⊥ − 1

Ω∂

∂t[b0 × fi⊥]

), (22)

One more equation is obtained from the parallel component ofAmpere’s law:

∂

∂t

c

4πen[∇ × B]‖ =

∂

∂t

1enj‖ =

(∂

∂tvi‖ − ∂

∂tve‖) =

e

me

(me

miEi‖ + Ee‖

), (23)

or, explicitly,

∇‖φ = −(E − ν

me

ne2j)

‖− δ2e [∇ × ∇ × E]‖ +Nei, (24)

where the electron skin lengthδ2e = c2/ω2pe and the nonlinear

term

Nei = −δ2ee

cTe[∇ × B]‖

∂

∂tφ−[

me

mifi‖ + fe‖ +

n

n0

(E− ν

ne2j + ∇φ

)‖

], (25)

The set of Eqs. (19), (20) and (22) constitute a basis forthe investigation of linear and nonlinear properties of short-scale Alfven waves. Further simplification may be achieved byexpressing the electromagnetic fields through electromagneticpotentials. A usual choice isϕ andA, such that

E = − ∇ϕ− 1c

∂

∂tA

and

B = ∇ × A.

However, here it is convenient to introduce an effective po-tentialψ for the field-aligned inductive (solenoidal) part of elec-tric field,

EA‖ = −∇‖ψ,

so thatψ is related to the parallel component of the potentialAthrough the equation

−1c

∂

∂tA‖ = −∇‖ψ.

The potential part of the electric field is described by the usualpotentialϕ:

Eϕ = −∇ϕ.This choice enables us to express bothϕ andφ in terms ofψ.Namely, insertion∇‖φ from (22) into (19) givesφ in terms ofψ (note that only the inductive part of the electric field entersAmpere’s law):

V −2Te

∂2

∂t2φ = −∇2

‖δ2e∇2ψ + ∇‖ ·Nei +Ne. (26)

Furthermore, when we apply the inverse operator∇−2‖ to (19)

we get a formal solution with respect toϕ:

ϕ=−ψ+φ−V −2Te ∇−2

‖∂2

∂t2φ−∇−2

‖ ∇·ν me

ne2j‖ +∇−2

‖ Ne.(27)

Inserting this into (20) we get

−mi

meV −2

Te

[∂2

∂t2

(1 − V 2

T ∇ · 1Ω2 ∇⊥

)− V 2

T ∇ · ∇‖+

me

mi

∂2

∂t2+me

mi

1Ω2

∂2

∂t2∇2

⊥∇−2‖

∂2

∂t2

]φ = Ni −Ne −

1Ω2

∂2

∂t2∇2

⊥[−ψ − ∇−2

‖ ∇ · ν me

ne2j‖ + ∇−2

‖ Ne

]+

∇⊥1

Ω2

∂2

∂t2EA⊥, (28)

whereV 2T = (1 + Te/Ti)V 2

Ti. We can now removeφ fromEq. (28) by means of (26):

mi

me

[∂2

∂t2

(1+

me

mi− V 2

T ∇ · 1Ω2 ∇⊥

)− V 2

T ∇ · ∇‖+

me

mi

1Ω2

∂2

∂t2∇2

⊥∇−2‖

∂2

∂t2

]∇2

‖δ2e∇2ψ − 1

Ω2

∂2

∂t2∂2

∂t2∇2ψ =


mi

me

[∂2

∂t2

(1+

me

mi− V 2

T ∇ · 1Ω2 ∇⊥

)− V 2

T ∇ · ∇‖+

me

mi

1Ω2

∂2

∂t2∇2

⊥∇−2‖

∂2

∂t2

] (∇‖ ·Nei +Ne

)+

∂2

∂t2[Ni −Ne] −

1Ω2

∂2

∂t2∇2 ∂

2

∂t2

(∇−2

‖ ∇ · ν me

ne2j‖ + ∇−2

‖ Ne

). (29)

When we multiply this equation byΩ2∇2‖δ

2e and treating the

right-hand part as a perturbation, we get the eigenmode equation∇2

‖V2A

[∂2

∂t2

(1+

me

mi− V 2

T ∇ · 1Ω2 ∇⊥

)− V 2

T ∇2‖

]

− ∂4

∂t4(1 − δ2e∇2

⊥)− ν

∂

∂t

∂2

∂t2δ2e∇2

∇2

‖δ2e∇2ψ =

∂2

∂t2

∂2

∂t2(∇‖Nei +Ne

)+

me

mi

[∇2

‖V2A (Ni −Ne) − δ2i ∇2 ∂

2

∂t2Ne

]. (30)

where the collisional term is expressed in terms ofψ,

∂

∂t∇ · me

ne2j‖ = ∇ · c

2me

4πne2

[∇ × 1

c

∂

∂tB]

‖= ∇2

‖δ2e∇2ψ.

The differential order of this equation may be reduced by drop-ping the term∼ V 2

T ∇2‖, which has a small effect on the Alfven

wave branch in a low-β plasma:V 2

A∇2‖

(1 − V 2

T ∇ · 1Ω2 ∇⊥

)− ∂2

∂t2(1 − δ2e∇2

⊥) −

ν∂

∂tδ2e∇2

∇2

‖δ2e∇2ψ =

∂2

∂t2(∇‖Nei +Ne

)+

me

mi

[∇2

‖V2A (Ni −Ne) − δ2i ∇2 ∂

2

∂t2Ne

]. (31)

In the above expressionsρ2i =V

2T /Ω

2i = (1 + Te/Ti)V 2

Ti/Ω2i

and the ion skin-lengthδ2i = (mi/me)δ2e .Eq. (31) is the key equation of the present paper. It is the

(second-order in∂/∂t) nonlinear Alfven wave eigenmode equa-tion in a non-uniform plasma. It contains the nonzero ion-gyroradius correction (∼ V 2

T ∇ · Ω−2∇⊥), the electron iner-tia correction (∼ δ2e∇2

⊥), and the collisional dissipative term(∼ ν) in the linear (left-hand side) part. The nonlinear (right-hand side) part contains second-order terms coming from theelectron and ion continuity equations (∼ Ne,Ni) and from theparallel component of Ampere’s law (∼ Nei).

The Alfven mode Eq. (31) is linearly decoupled from the fast(slow) mode equation because the fast (slow) wave has muchhigher (lower) frequencies than the Alfven wave for the samewavenumber and for large propagation angles. The nonlinearpart of (31) contains the beatings of the pump AW with thefluctuations corresponding to different modes, including fast,Alfv en and slow modes. However, the secondary waves areeffectively enhanced from the background noise by the beat-ings which are in resonance with the excited waves. Due to the

separation of the oblique Alfven mode from the other modesin the (ω − k) space, the resonance can occur with the low-wavenumber fast wave or with the low-frequency slow wave.These resonances are less efficient than the resonance amongoblique Alfven waves, which are close in the (ω − k) space.There are two other important properties of the fast and slowwaves which make it difficult for them to be excited by thethree-wave resonant interaction with Alfven waves in the solarcorona. The growth of the fast wave is suppressed by: (1) strongrefraction of the isotropic (ω =

√k2

z + k2⊥VA) fast wave in a

nonuniform plasma, quickly bringing the wave out of the res-onance; (2) by ion temperature anisotropyTi⊥/Ti‖ > 1, veryprobable in the corona in view of recent SOHO observations.The growth of the slow wave is suppressed by its strong Landaudamping unlessTe/Ti 1, which is unlikely to be the case inthe corona. Thereto, direct comparison shows that the increment(56) is much larger than the increment of decay into slow andfast waves (Yukhimuk et al. 1999).

On the contrary, the Alfven mode is less sensitive to theabove mentioned stabilizing factors and has a highly anisotropicdispersion making it possible for the wave triads to remain inresonance even in a nonuniform plasma. That is why our focusis on the excitation of highly oblique Alfven waves by a highlyoblique pump Alfven wave set up by phase mixing due to plasmanonuniformity. The situation can be different with the parallel-propagating pump AW.

4. Alfven wave phase mixing and collisional dissipationin corona

Consider the evolution of an AW, excited at the magnetic fieldlines, where the length scale of the transverse inhomogeneity ofthe equilibrium isL⊥ and the length scale of the field-aligned in-homogeneity of the equilibrium isL‖. BothL⊥ andL‖ can varyover a wide range in the solar corona, but usuallyL‖ L⊥.Take a wave frequency for which the parallel wavelengthλ‖ ismuch shorter than the length-scale of the field-aligned equilib-rium inhomogeneity,λ‖ < L‖. Then, as the wave propagates,the perpendicular length-scale of the wave decreases in time asλ⊥/L⊥ = τA/t (τA is the wave period). Therefore, even if theinitially excited AW has a smooth distribution in the directionof the plasma inhomogeneity, phase mixing creates short wavelength-scalesλ⊥ < L⊥ in a few wave periods. In open mag-netic configurations, where waves propagate upward from thefootpoints, this corresponds to heights of the order of a few field-aligned wavelengths. From now on we shall study the dynamicsof Alfv en waves, in which short perpendicular wavelengths,λ⊥ < L⊥, are either initially determined by a (unspecified)generator, or are created by phase-mixing.

In this situation we consider the evolution of theω-th har-monic of the wave field using a WKB-ansatz:

A = A0 × exp(

−iωt+ i

∫k · dr +

∫γdt

). (32)

The wave vectork and the growth/damping rateγ have to becalculated from the local dispersion relation, and the integration


is along the path of the wave propagation. The effects of plasmainhomogeneity for these waves come about through the spatialdependence of the plasma parameters along the path of the wavepropagation. This path is determined by the ray equations forthe wave vectork and positionr:

dkdt

= −∂ω

∂r= −ω

(1VA

∂VA

∂r

); (33)

drdt

=∂ω

∂k. (34)

In accordance with the above, the wave frequency here isdetermined by the local dispersion relation found from the dy-namic Eq. (31) for Alfven waves. When we neglect the nonlinearterms in (30), we get the linear Alfven wave frequency,ω, andthe linear damping rate,γc, which accounts for the thermal andelectron inertia effects, and for electron-ion collisions:

ω2 = k2‖V

2A

[1 + k2

⊥ρ2i

1 + k2⊥δ2e

− β1

1 + k2⊥ρ

2i

]; (35)

γc = −0.5νk2

⊥δ2e

1 + k2⊥δ2e

[1 + β

(1 + k2

⊥δ2e

)5(1 + k2

⊥ρ2i )

2

], (36)

whereβ = V 2T /V

2A. Since we ignore the magnetic compress-

ibility, β < 1. For a low-β plasma, a good approximation to(35) and (36) in a wide range of parameters is

ω2 = k2‖V

2A

1 + k2⊥ρ

2i

1 + k2⊥δ2e

; (37)

γc = −0.5νk2

⊥δ2e

1 + k2⊥δ2e

. (38)

When we neglect the electron inertia term, we find the AW dis-persion to be close to the KAW dispersion, known from thekinetic theory (Hasegawa & Chen 1976), for all values of thedispersion parameterk2

⊥ρ2i . In the limitk2

⊥ρ2i 1 they are iden-

tical.It is convenient to introduce the dispersion functionK (k⊥):

K2 (k⊥) =1 + k2

⊥ρ2i

1 + k2⊥δ2e

, (39)

and to write the AW dispersion relation in the form

ω = kzVAK (k⊥) .

The functionK (k⊥) is the factor by which the parallel phasevelocity of the AWs with perpendicular wavelength2π/k⊥ ex-ceedsVA.

In the case of weak AW dispersion (k2⊥δ

2e , k

2⊥ρ

2i 1),

K (k⊥) ' 1, we recover by means of Eq. (36) the asymptoticresult of Heyvaerts & Priest (1983) for the wave damping withheightz:

A = A0 × exp

(∫γc

(∂ω

∂k z

)−1

dz

)=

A0 × exp(

−z3

z3c

), (40)

wherezc is the characteristic height of the resistive (collisional)wave dissipation,

zc =[

112me

mi

νe

VAL2A

]− 13(ω

Ωi

)− 23

. (41)

However, in addition to collisional dissipation, the small-scale AWs undergo also nonlinear interaction, described by theright-hand side of (31). Therefore, the amplitude of the pumpAW changes not only because of collisional dissipation, but alsodue to nonlinear interaction. Nonlinear AWs interaction and thequestion which process dominates for given plasma and waveparameters is investigated in the following sections.

5. Parametric decay and damping of the phase-mixed AWin the corona

5.1. Dynamic equation and coupling coefficientsof the three-wave AW interaction

We shall proceed further in the local approximation and ex-pand all perturbations that appear in the eigenmode Eq. (31)into Fourier series. We use the second-order approximation forall quantities, which are present in the nonlinear right-hand side,expressing all perturbations through the parallel component ofthe electromagnetic potentialAz. Then, we average over timeand space intervals which are short compared to the intervalsover which the amplitudes of the waves vary, butω 4 t 1,k· 4 r 1, and obtain the dynamic equations describing theslow variations of the wave amplitude:(

∂

∂t+ vk · ∇−γk

)Aσ

k =∑1,2

δ(σk−σ1k1−σ2k2)δ (σωk − σ1ω1 − σ2ω2)

Us1,s2(σk, σ1k1, σ2k2)Aσ11 Aσ2

2 , (42)

wheresi = ∓1 for kiz ≶ 0, i = k, 1, 2 (kz is chosen to be> 0).The coefficientss1,2 = ±1 in the Eq. (42) take into account

that the phase velocity ofµ-th wave,Vµ = ωµ/kµz, can bepositive (Vµ > 0 with kµz > 0) or negative (Vµ < 0 with kµz <0), and the indicesσ,σ1 andσ2 denote complex conjugate termsA+1

µ ≡ Aµ andA−1µ ≡ A∗

µ (note that only real parts, ReAµ =0.5(Aµ +A∗

µ

), should be taken of the complex amplitudes in

binary combinations).The coupling coefficient of the three-wave resonant interac-

tion is

Us,s1,s2(σk,σ1k1, σ2k2) =VA

4B0(s1K1 − s2K2)

Kk

κ2(

s1κ

21

K1+ s2

κ22

K2+ s

κ2

Kk

)σ1σ2 (k1 × k2)z , (43)

whereκ = k⊥ρi is the kinetic factor, accounting for temper-ature (FLR and electron pressure) effects. From (43) we seethat three-wave resonant interaction among AWs is local ink-space, i.e. the wavenumbers of the effectively interacting AWsare of the same order,k⊥ ∼ k1⊥ ∼ k2⊥. Note that the non-zero


value ofU(k,k1,k2) and the non-linear (three-wave) couplingamong AW triads becomes possible due to the finite value ofthe kinetic factorκ. In MHD Alfv en wavesκµ = 0, makingK = K1 = K2 = 1 so thatU(k,k1,k2) = 0 for resonanttriads in agreement with the well-known fact that three-waveresonant interaction is impossible among ideal MHD Alfvenwaves.

Since we consider AWs with frequenciesω2 Ω2 in alow-β coronal plasma, we have dropped the terms of the order∼ ω2/Ω2 and∼ β in comparison to those retained in the ex-pression for the matrix element (43). In this situation, the mostimportant effect for parametric decay is the vector nonlinearity,∼ (k1 × k2)z, retained in (43). The converse situation, wherethe scalar nonlinearity,∼ k1 · k2, becomes more important, isconsidered by Voitenko & Goossens (in preparation).

The matrix elementU(k,k1,k2) is the key mathematicalobject in the non-linear wave theory. Once this matrix elementis found, we can follow theoretical methods developed earlierin order to study nonlinear properties of a given mode.

Using expressions (42)-(43), we study the parametric decayof AWs in the next section. This nonlinear process is importantfor the AWs and often plays a crucial role in wave propaga-tion, nonlinear saturation of instabilities, and spectral dynamics(Voitenko 1998a,b).

5.2. Parametric decay instabilityof the short-scale Alfven waves

The parametric decay of a pump oblique AW into twodaughter oblique AWs in a hot plasma has been first ex-amined by Erokhin et al. (1978), followed by other au-thors (see e.g. Volokitin & Dubinin 1989; Voitenko 1996c;Yukhimuk & Kucherenko 1993). The common approach so farwas to consider weak wave dispersion,k2

⊥ρ2i 1, and to as-

sume that all three interacting AWs propagate in the same di-rection along the background magnetic field. However, it wasshown in the framework of kinetic theory by Voitenko (1998a)that the strongest decay of the weakly dispersing AW was missedin the previous investigations. Indeed, the strongest decay oc-curs when the daughter waves propagate in opposite directionsalong the background magnetic field. Also, in contrast with pre-vious studies, we consider the influence of resistive dissipationof AWs and the detuning of the perpendicular wavenumbers,caused by the transverse inhomogeneity. The inclusion of theseeffects is important when investigating the phase mixing of thelow-frequency part of the AWs in the corona. Hence, we shallstudy the KAW decay instability for an arbitrary value of thekinetic factorκ, based on the dynamic Eq. (42) found from thetwo-fluid resistive MHD plasma model.

Let us consider a pump KAW with frequencyωk, wave vec-tork, and amplitudeAk, propagating in the positivez-direction,i.e. kz > 0. From (42) it follows that the waves(ω1,k1, A1)and(ω2,k2, A2) can effectively interact with the pump wave,provided the following resonant conditions:

k1 + k2 = k;

ω1 + ω2 = ωk (44)

are satisfied. The dynamical equations for waves1 and2, cou-pled via the parametric pump wave, are:(∂

∂t− γ1

)A1 = −U(k1,k,−k2)AkA

∗2; (45)

(∂

∂t− γ2

)A∗

2 = −U(−k2,−k,k1)A∗kA1. (46)

Acting by the operator(∂/∂t − γ2) on the Eq. (45) and elim-inatingA∗

2 by use of Eq. (46), we obtain the equation for theamplitudeA1 (the corresponding equation for the amplitudeA2 is obtained in the same way):(

∂2

∂t2− (γ1 + γ2)

∂

∂t+ γ1γ2

)A1,2 =

U(k1,k,−k2)U(−k2,−k,k1) |Ak|2A1,2. (47)

An exponential solution to (47),∼ exp(δt), has indices

δ =γ1 + γ2

2±√(

γ1 − γ2

2

)2

+ γ2NL,

where the rate of non-linear interaction is

γ2NL = Ω2

i

(VA

4VTi

)2K1K2(K2 − s2Kk)(s1Kk −K1)

κ2κ

21κ

22(

s1κ

21

K1+ s2

κ22

K2+

κ2

Kk

)2

|κ1 × κ2|2z∣∣∣∣Bk

B0

∣∣∣∣2

. (48)

The waves1 and2 grow if

δ > 0. (49)

For the rate of the non-linear interactionγNL to be real, i.e.γ2

NL > 0, the dispersion functionsKµ have to satisfy the decaycondition that

(K2 − s2Kk)(s1Kk −K1) > 0. (50)

In the case of weak wave-particles interaction,γ1, γ2 γNL,the growth rate of the waves1 and2 is determined by the rateof the non-linear interaction:δ ∼ γNL.

The decay condition (50) implies that at least one dispersionfunction of the decay product should be smaller than the disper-sion function of the pump wave (γ2

NL < 0 for arbitrarys1 ands2if bothK1 andK2> Kk). In what follows we chooseK1 < Kk,so that decay is possible: into (1) parallel-propagating daugh-ter KAWs, k1z > 0, k2z > 0 (s1 = s2 = 1), and into (2)counterstreaming daughter KAWs,k1z > 0, k2z < 0 (s1 = 1,s2 = −1) (herekz > 0 is chosen for the pump wave).Thesedecay channels are shown on Fig. 1 as parallelograms in the(ω, kz)-plane, reflecting matching conditions for correspondingAW triads.

Let us consider these two cases separately for the weaklydispersed KAWs,κ2,κ2

1 ,κ22 1 (we shall not consider here

strongly dispersed AWs,κ2,κ21 ,κ

22 1, which are unlikely

for the AW phase-mixing in corona).


Fig. 1.Two possible channels for decay of the pump KAWk: 1) decayinto two parallel-propagating KAWsk → 1+2 (k1z > 0,k2z > 0); 2)decay into two counterstreaming KAWsk → 1′ +2′ (k′

1z > 0, k′2z <

0), where we have takenk′1⊥ = k1⊥ for simplicity. The dispersion of

the MHD Alfven waveω = |kz| VA is shown for reference.

5.3. Excitation of the parallel-propagating decay AWs(k1z > 0, k2z > 0)

As follows from (50) withs1 > 0 and s2 > 0, the decayprocess including parallel-propagating KAWs(ωk,k, Ak) →(ω1,k1, A1) + (ω2,k2, A2) can take place if the dispersionfunctions of the decay productsK1 andK2 are smaller andlarger respectively than the dispersion functionKk of the pumpKAW:

K1 < Kk < K2. (51)

Since the functionK is a monotonous function of its argumentκ, we conclude that the wavenumbers of the interacting waveshave to satisfy the following sequence of inequalities:

0 < k1⊥ < k⊥ < k2⊥ < 2k⊥. (52)

The last inequality here follows from a combination of the res-onant condition for the wave vectors and the second inequality.

For κ2,κ2

1 ,κ22 1, we use the asymptotic expression

K (kµ⊥) = 1 + 0.5k2µ⊥ρ

2i in (48) so that

γNL = ΩiVA

8VTi

(1 +

Te

Ti

)(κ2 + κ

21 + κ

22)√

(κ22 − κ

2k)(κ2

k − κ21)

κκ1κ2

|κ1 × κ2|z∣∣∣∣Bk

B0

∣∣∣∣ . (53)

The dependence of the normalized decay rate,

γNL = γNL

[Ωi

VA

8VTi

(1 +

Te

Ti

) ∣∣∣∣Bk

B0

∣∣∣∣κ2]−1

, (54)

on the wavenumbers of the decay waves is shown on the Fig. 2.The spectrum of the generated daughter waves has a sharp

maximum. To find the wavenumbers of the daughter waves

Fig. 2. The dependence of the normalized rate of parametric decay onthe wavenumbers of the parallel-propagating decay waves,k1⊥/k⊥andk2⊥/k⊥.

which are most effectively excited, we determine the maximumof γNL given by (53) with respect to the wavenumbers of thedecay products. Thus we obtain the rate of resonant decay in-stability into parallel-propagating AWs as

γNL = 0.4ΩiVA

VTiκ

3∣∣∣∣Bk

B0

∣∣∣∣ , (55)

which is achieved forκ1 = 0.717κ,κ2 = 1.49κ (here we takeTe = Ti for simplicity).

Expression (55) shows a strong dependence of the interac-tion rate on the perpendicular wavenumber of the pump wave,γNL ∼ k3

⊥, and this results in a rapid decrease of the interactionrate with decreasingk⊥.

5.4. Counterstreaming decay KAWs (k1z > 0, k2z < 0)

Eq. (53) is now replaced with

γNL = ΩiVA

4VTi

√1 +

Te

Ti

(κ2 + κ21 − κ

22)√

(κ2 − κ21)

κκ1κ2

|κ1 × κ2|z∣∣∣∣Bk

B0

∣∣∣∣ . (56)

The dependence of the normalized decay rateγNL on thewavenumbers,

γNL = γNL

(Ωi

VA

4VTi

√1 +

Te

Tiκ

2∣∣∣∣Bk

B0

∣∣∣∣)−1

=

(κ2 + κ21 − κ

22)√

(κ2 − κ21)

κ3κ1κ2

,

is shown on the Fig. 3.When we compare the counterstreaming decay waves with

the parallel-propagating decay AWs, we find that the spectrumof the counterstreaming decay waves is concentrated aroundanother pair of wavenumbers:κ1 = 0.776κ, κ2 = 0.442κ.With these wavenumbers the maximum growth rate is

γNLk = 0.3Ωi

VA

VTiκ

2∣∣∣∣Bk

B0

∣∣∣∣ . (57)


Fig. 3. The normalised rate of parametric decay depending on thewavenumbers of counterstreaming decay waves,k1⊥/k⊥ andk2⊥/k⊥.

With decreasingk⊥, the decrease of the rate of decay intocounterstreaming decay waves is not as strong as for parallel-propagating decay waves. This means that the decay into coun-terstreaming AWs is more effective for the weakly dispersingwaves, created by phase mixing. Consequently, (57) gives agood approximation for the damping rate of the pump AW, andfor the rate of the three-wave AWs interaction ink-space aswell, if the decay into counterstreaming waves is not forbidden.

Summarizing, there are significant differences between thedecay into parallel-propagating waves, and decay into counter-streaming (antiparallel-propagating) waves. Namely, the decayof a weakly-dispersing AW into counter-streaming waves:

1. is much stronger for a weakly dispersed AW in a homoge-neous plasma, but

2. is more sensitive to the field-aligned inhomogeneity;3. transfers the wave energy into lower wavenumbers only,ks⊥ < k0⊥, thus reducing collisional dissipation, whereasthe decay into parallel-propagating waves excite bothshorter- and longer-wavelength waves,ks⊥ ≷ k0⊥;

4. in part, reverses the direction of wave energy propagation.

Expressions (55) and (57) will be used to estimate the non-linear damping of the phase-mixed AWs with height in solarcorona.

5.5. Collisional threshold of decay instability

Let us estimate the threshold amplitude for the decay instabil-ity of the phase-mixed pump in the corona, where the growthof the decay products due to nonlinear coupling via the pumpwave may be balanced by their collisional dissipation. In a localapproximation, the threshold amplitude for the pump wave toexcite waves 1 and 2 is obtained from the marginal condition fordecay (49),γ1γ2 = γ2

NL, whereγ1 andγ2 are to be determinedfrom (38).

In the coronal plasma, the collisional damping is

γcL(1,2) ≈ −0.5νk2(1,2)⊥δ

2e , (58)

and the marginal decay condition is

γ2NL = 0.25ν2k2

1⊥k22⊥δ

4e . (59)

For the strongest counterstreaming decay (57), the thresholdappears to be independent of the perpendicular wavenumbersor other wave characteristics, and the threshold value dependsonly on plasma parameters:∣∣∣∣Bk

B0

∣∣∣∣ = 0.57me

mi

ν

Ωi

VA

VTi. (60)

The actual value of the threshold with the typical coronal pa-rameters is extremely low:∣∣∣∣Bk

B0

∣∣∣∣ = 10−6 − 10−7. (61)

The decay condition (49) is certainly satisfied for the AWs withBk/B0 ∼ 0.01, able to heat the corona.

Eqs. (55) - (61) clearly demonstrate that the phase-mixedAWs withκ /= 0 can become nonlinearly unstable at very smallamplitudes,Bk/B0 & 10−6. With these amplitudes, the para-metric decay becomes stronger than the collisional dissipationof the phase-mixed AW,γNL > γcL.

5.6. Nonuniformity thresholds

We consider three effects of nonuniformity on the parametricdecay of phase-mixed AW.

1. It should be noted that although the decay of theκ2 1

pump wave into counterstreaming KAWs is local ink⊥-space,it is non-local inkz-space because|k2z| /kz . κ

2 1. Thismeans that our results on this decay are applicable as long as thelarge parallel wavelength of the counterstreaming (decay) KAWis shorter than the inhomogeneity length scale,|k2z|−1 . L‖.Although our results can be used for waves withλ2z ∼ L‖ asorder of magnitude estimates, for larger wavelengths they can besignificantly affected by the non-uniformity. The correspondingmarginal perpendicular wavenumber of the pump wave and itsparallel wavelength are related throughκ

2 ∼ λz/L‖.2. Also, the efficiency of the decay including counterstream-

ing AWs in a nonuniform plasma may be reduced by the finitecorrelation length along the field lines,Lc‖. This correlationlength may be estimated asLc‖ ≈ VA/ts, wherets is the dura-tion of the quasi-periodic pulse that excites the pump wave. Wecan estimate the corresponding threshold for decay into coun-terstreaming KAWs from the condition that the waves shouldoverlap during the growth time (γNL

k > VA/Lc‖):∣∣∣∣Bk

B0

∣∣∣∣thr

≈ 2.5√β(kzκ

2Lc

)−1 ωk

Ωi, (62)

which can be high for the waves with short correlation lengths.3. The progressive detuning of the perpendicular wavenum-

bers of the interacting AWs triplet can limit the decay if thetime of the nonlinear interaction is shorter than the time re-quired to change the wavenumbers. The physics of this processis as follows. In the course of time, the initially most effectively


interacting triplet of resonant waves is removed from its topposition in the wavenumber space, because the different tempo-ral behaviour of the constant and evolving components of theperpendicular wave vectors changes the wavenumber’s ratiosin the triplet. The corresponding limitation on the pump waveamplitude may be found from the conditionγNL > ω/k⊥LA.

6. Nonlinear damping of the phase-mixed AWs in corona

As in the Discussion, we take the length-scale of the field-aligned inhomogeneity in a coronal holeL‖ ∼ 105 km, thetransverse inhomogeneity length-scale ranging as1 km. L⊥ .103 km, and wave amplitudes ranging from0.01 to 0.03 forB/B0.

For wave frequenciesω > 1 s−1 (see Discussion), thenonuniformity restrictions for the decay into counterstreamingwaves may be satisfied. Then, using the damping rate (57) in(32), we find that the pump wave amplitude decays with heightas

A = A0 × exp(

− z3

z3N

), (63)

where the distance of nonlinear damping

zN =[(

0.1ω2V2A

Ω2i

V −3A L−2

A

)ΩiVTi

VA

∣∣∣∣Bk

B0

∣∣∣∣]−1/3

. (64)

To compare this height with the field-aligned wavelength werewritezN as

zN = λz

[(2π)2

10ρiλz

L2A

∣∣∣∣Bk

B0

∣∣∣∣]−1/3

, (65)

where the coefficient beforeλz strongly depends on the perpen-dicular nonuniformity length-scaleLA⊥.

Let us apply the above results to phase-mixed Alfven wavesin a coronal hole. Takingρi ∼ 1 m, Bk/B0 ∼ 10−2, andλz ∼ 107 m, we obtain a low damping height in the coronalholes, varying in the range fromzN & λz with LA⊥ = 1 km,up to zN = 102λz with LA⊥ = 103 km. Note that the iongyroradius is nevertheless very small compared toλ⊥ at theseheights.

The corresponding transverse wavelengths,λ⊥N = 0.7 kmwith LA⊥ = 1 km andλ⊥N = 7 km withLA⊥ = 103 km, aremuch longer than the dissipative length-scale,ld ∼ 102 m. Thisis exactly the point we made in the Introduction that the effectof FLR in Alfv en waves comes into play at length scales whichare much longer thanld.

7. Collisional dissipation versus parametric decayinto counterstreaming waves

To find the linear/nonlinear damping of the AW with height, weproceed further using a WKB-ansatz (32) for the AW ampli-tude. The damping rate includes linear collisional damping andnonlinear damping due to parametric decay:

γ = γν − γNLk = −0.5νk2

⊥δ2e − 0.3Ωi

VA

VTiκ

2∣∣∣∣Bk

B0

∣∣∣∣ .

Taking integrals in the exponent we find the same behaviourof the amplitude with height as in the case of collisional dampingof the phase-mixed wave, but now with the cumulative dissipa-tion distancezcN :

A = A0 × exp(

− z3

z3cN

), (66)

wherezcN is determined by the relation

1z3cN

=1z3c

(1 +

z3c

z3N

), (67)

and the nonlinear damping height is given by (64). The col-lisional damping heightzc is determined by (41), which werewrite in the form

zc =[(

0.1ω2V2A

Ω2i

V −3A L−2

A

)νeme

mi

]−1/3

. (68)

In a coronal hole it can be estimated aszc =(102 − 103

)×λz,which corresponds to the transverse wavelengthsλ⊥ ∼ 102 m.

We see that the characteristic height of the wave damping,zcN , can be very different from that predicted by resistive MHD,zc, if zc > zN , which gives the condition for the nonlinearly-dominated phase-mixing:∣∣∣∣Bk

B0

∣∣∣∣ > BN =√me

mi

νe

Ωi

VA

VTe. (69)

The corresponding threshold amplitudeBN does not depend onthe wave parameters and is almost the same as the collisionalthreshold for the parametric decay itself (60), which is a con-sequence of the local character of the AW parametric decay ink⊥-space.

If condition (69) is satisfied, the initial Alfven waves, excitedat the base of the solar corona, are damped at heights∼ zN . Fortypical coronal hole conditions discussed above, we obtain am-plitudes of AWs, leading to the nonlinearly-dominated regimeof phase-mixing:∣∣∣∣Bk

B0

∣∣∣∣ > 10−6. (70)

If the nonthermal broadening known from the spectral ob-servations is due to AWs, then the wave amplitudesBk/B0 =0.01−0.03, and condition (70) is well satisfied. Then the damp-ing height in this nonlinearly-dominated regimezcN ' zN .

The competition of the linear and nonlinear damping mech-anisms for the phase-mixed AWs, including parametric decayinto both parallel-propagating and counterstreaming waves, col-lisional dissipation and Landau damping, is studied in more de-tail in the accompanying paper (Voitenko & Goossens 2000).

8. Discussion

The results obtained in the present paper show that the processof phase-mixing, which is inevitable in the non-uniform solarcorona, can switch on the parametric decay of AWs into a spec-trum of secondary AWs. As a result, the waves generated at the


base of a coronal magnetic structure undergo a transition fromlaminar to nonlinear (or even turbulent) propagation at a heightz = zN , determined by the plasma and wave parameters. Con-sequently, the whole picture of the AW dynamics in a coronalplasma may be considerably modified, and not only at heightsz > hc, but also atz < zN , because a part of the wave fluxcan reverse its direction of propagation if the time of parametricdecay into counterstreaming AWs is shorter than the time ofwave dissipation. The energy transferred to the inward propa-gating AWs will not directly contribute to accelerating the solarwind; but it can again couple nonlinearly to upward propagatingwaves, increasing the transverse wavenumbers.

We found that the collisional wave dissipation of the phase-mixed AW, represented by the Ohmic dissipation of the parallelwave current, is much weaker than the nonlinear damping dueto parametric decay. The linear damping rate due to the ion-ioncollisions (shear ion viscosity) can in some cases be comparableto Ohmic dissipation (Hasegawa & Chen 1976), but can hardlyexceed it. Under coronal hole conditionsγei/γii ∼ 10.

In Sect. 4 our results have been applied to the phase-mixedAlfv en waves in a coronal hole. The influence of vertical in-homogeneity in coronal holes on the AW phase mixing hasbeen investigated in great detail in the linear approximation(Ruderman et al. 1998; De Moortel et al. 1999). The nonlineargeneration of fast MHD waves by phase-mixed AWs has beenconsidered by Nakariakov et al. (1997), but this process seemsto be less effective for waves with amplitudes∼ 0.01, as sug-gested by observations. It is therefore important to check if anonlinear process can influence phase mixed AWs in the corona,and in coronal holes in particular.

The length-scale of the field-aligned inhomogeneity in acoronal hole is primarily determined by the characteristic scaleof the density variation with height,L‖ ∼ hg ∼ 105 km. Thetransverse inhomogeneity length-scale is uncertain. IfL⊥ isdetermined by the visible plume sizes, then one can takeL⊥ ∼103 km. However, it is quite possible that the corona is muchmore structured in the horizontal direction, up toL⊥ ∼ 1 km(Woo 1996).

The wave frequency is also not an accurately known pa-rameter in the problem. From the point of view of the powerpresent in the photospheric motions, and in the magnetic fieldsof the chromosphere and corona, it can range fromω ∼ 10−3

s−1, for the waves excited by granular motions, up toω ∼ 106

s−1, for waves excited by the small-scale magnetic activity inthe chromospheric network (Axford & McKenzie 1992). Per-sistent nonthermal plasma motions in the corona have beenconfidently detected by means of spectroscopic observations,(Saba & Strong 1991; Tu et al. 1998; Hara & Ichimoto 1999,and references therein). Wave amplitudes estimated from thesedata range from0.01 to0.03 forB/B0, and energy requirementscan be satisfied.

However, because of the lack of observational evidencethat the low-frequency global modes supply the energy forthe high-temperature corona, one can suppose that either thewave periods fall below the available time resolution,τ < 1 s(ω & 1 s−1

), or the wave coherence lengths can not yet be spa-

tially resolved, i.e., at least, wavelengthλ . 106 m. This lastcondition in turn maps onto the frequency scale asω & 1 s−1

for Alfv en waves ifλ is the parallel wavelength. In principle,Alfv en waves withλ⊥ 106 m can be highly localized inplanes perpendicular toB0, which makes them unresolved. Butthe direct footpoint excitation of short-scale waves requires toomuch power in the photosphere concentrated at small scales, andany alternative process involving an evolutional structurizationin the corona should be observed at the initial large-scale stageif ω < 1 s−1.

In the closed magnetic configurations, like coronal loops,the random global photospheric motions can excite short-scalestanding AWs localized around resonant field lines, wherethe Alfven travel time is equal to the photospheric time-scaleat which the photospheric motions contain enough energy(De Groof et al. 1998). The problem here is that the setup timeseems to be long compared to≤ 1-minute variations of thefine-scale chromospheric activity, repulsing the dynamics of thecoronal heating process (Berger et al. 1999).

The above consideration suggests that the waves responsiblefor coronal heating have frequenciesω & 1 s−1, or even higher(Tu et al. 1998; Wilhelm et al. 1998). Moreover, this suggestioncan help to circumvent some difficulties with in situ spacecraftobservations of waves in the solar wind. Using thek−1 spec-tra derived from the Helios (and later Ulysses) observations,Roberts (1989) has shown that the wave energy contained inthose spectra, is not sufficient to accelerate the solar wind whenextrapolated to the corona. This difficulty can be circumventedby a spectrum of AWs with higher frequencies, which can beeffectively dissipated by cyclotron damping when the wavesreach the height where the cyclotron frequency equals the wavefrequency (Axford & McKenzie 1992; Tu et al. 1998).

At the same time, small perpendicular length scales dooften develop; they are important for AWs and open var-ious alternative dissipation channels (Matthaeus et al. 1999;Hollweg 1999). The structurization can be very efficient for thehigh-frequency AWs in a nonuniform coronal plasma becausethey undergo strong phase mixing and quickly develop strongtransverse gradients that accelerate the nonlinear interaction.Nonlinear decay initially spreads the input wave spectrum outover short wavelengths across the magnetic field as shown onFigs. 2 and 3, and over long wavelength in parallel direction(not shown). With sufficient driver power, a further growth ofthe secondary waves gives rise to the nonlinear coupling amongthem resulting in the anisotropic turbulent cascade.

Another possible scenario is that anisotropic turbulentcascade is excited by the parametric instability of parallel-propagating AW (Ghosh & Goldstein 1994). Note that our ap-proach differs from that in the papers (Vinas & Goldstein 1991;Ghosh et al. 1993; Ghosh & Goldstein 1994) in both the physi-cal model and the origin of nonlinearity involved. While it isquite natural to take the parallel dispersion for the parallel-propagating waves into account (Vinas & Goldstein 1991), itis equally natural that the perpendicular dispersion is importantfor the highly oblique Alfven wave. The planar model used inthe above papers restricts all waves to propagate in the(x− z)


plane. Since our leading nonlinearity contains the vector productof perpendicular wave vectors, the2 1

2 D simulations reduce it tozero and hence can not account for it. Also, the use of one-fluidor reduced two-fluid equations rules out important nonlinearcouplings of oblique AWs. This happens, in particular, whenthe nonlinear terms due to electron and ion electric drifts areassumed to be cancelled by each other.

It is not certain presently if the nonlinear interaction of AWsalone can reproduce the one-dimensional powerk−1 spectrawith most (80%) fluctuation energy hidden in the small-scaletransverse spectra as suggested by observations of the solar wind(Matthaeus et al. 1999 and references therein). The analysis ofpossible causes of the appearance of anisotropic spectra sug-gests (Ghosh et al. 1998) that their most probable source is in thesolar corona, where the solar wind originates. The anisotropicspectra can indeed be developed in the corona by the wave-wave interactions that we study, but with an even flatter parallelpower spectral index−1/2 (the perpendicular power index is−2) (Voitenko 1998b). Such a flat spectrum can steepen towardsan index−1 by the Landau damping which has a∼ kz depen-dence on the parallel wave number, but exact final spectra canbe obtained via numerical simulations.

The dynamics of the spectrum of parallel-propagating AWshas been discussed by Agim et al. (1995) for waves gener-ated by a proton beam. On the contrary, our input phase mixedspectra are essentially anisotropic. For example, if the initialone-dimensional spectrum of AWs is∼ k

−1/2z at the base of

corona, the two-dimensional spectrum created by phase mix-ing is ∼ k

−1/2z k

−1/2x , x is in the direction of inhomogeneity.

This spectrum is restricted to the (z − x) plane and is thusparametrically unstable in the sense we are discussing, excitingwaves withky /= 0. This is a feature of the vector nonlinearity(∼ (k1 × k2)z), involving waves propagating in the differentdirections acrossB0.

As we restricted our analysis to the wavelengthsλz shorterthan the vertical inhomogeneity length-scale,L‖ ∼ 105 km,ω & 10−2 s−1. Also, the competition of the parametric decayinto counterstreaming waves with collisional dissipation maybe reduced in the low-frequency range,10−2 s−1 < ω < 10s−1, where this decay channel is prohibited by the field-alignedplasma inhomogeneity and decay into parallel-propagatingwaves comes into play (see Voitenko & Goossens 2000). Al-though, as we discussed above, this low-frequency range tendsto be excluded from the observational point of view, the corre-sponding linear and nonlinear wave damping is considered byVoitenko & Goossens (2000).

Here we neglected kinetic damping effects in comparisonto collisional dissipation. To describe the dissipation of highfrequency AWs, one has, in principle, to use the kinetic plasmamodel, accounting for the Landau damping. However, notingthat the two-fluid MHD dispersion of AWs is close to theexact kinetic dispersion in the whole range of perpendicularwavenumbers and frequencies, one can use the two-fluid MHDmodel with the semi-empirically included Landau damping forthe high-frequency AWs also. Moreover, the Landau damping it-

self may be reduced by the quasilinear relaxation of the velocity-space distribution, in which case the competition of the dampingdue to collisions and decay into counterstreaming waves can ex-tend into high-frequency region, up toω . 106 s−1.

The complicated interplay of the different damping mecha-nisms in the different frequency ranges is studied by Voitenko& Goossens (2000).

9. Conclusions

The main conclusions of the present paper are:

1. Phase mixing of finite-amplitude Alfven waves in the coronagives rise to nonlinear coupling of AWs and to spectral en-ergy redistribution. The strongest coupling in the plasma,which is uniform alongB0, includes counter-propagatingwaves.

2. The nonlinear effects of a non-zero ion Larmor radius onthe finite-amplitude Alfven waves comes into play at lengthscales which are much longer than the dissipation length-scaleld. In the coronal holes, the phase-mixing switches tothe nonlinear phase at the heights equal to10−100 parallelwavelengthsλz, where the transverse wavelengthλ⊥ ∼ 104

m while the dissipation length-scale isld ∼ 102 m.3. Nonlinear coupling introduces a spectral redistribution of

the wave energy, thus changing the rate of the dissipation ofwave flux. The influence of the nonlinear interaction on theactual dissipation of the wave flux and consequent plasmaheating depends on the frequency of the waves and is asubject of further investigations. The part of wave energywhich is transferred to smaller scales can be damped rapidlyby collisional dissipation.

4. The distance at which the phase-mixing undergoes a tran-sition from a laminar to a nonlinear stage depends on thewave frequency and amplitude. In the case of the decayinto counterstreaming waves this distance is proportional toω−2/3 (Bk/B0)

−1/3 and is given by (64).

Acknowledgements.This work was supported by the FWO-Vlaanderen grant G.0335.98 and by Intas grant 96-530. Yu.V. acknowl-edges the financial support by the KUL and by Ukrainian FFR grant2.4/1003.

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