Nonlinear coupling of Alfven waves with widely different cross-field

wavelengths in space plasmas

Yuriy M. Voitenko1 and Marcel GoossensCentre for Plasma Astrophysics, K.U. Leuven, Heverlee, Belgium

Received 30 October 2004; revised 13 March 2005; accepted 29 March 2005; published 9 July 2005.

[1] Multiscale activity and dissipation of Alfven waves play an important role in anumber of space and astrophysical plasmas. A popular approach to study the evolutionand damping of MHD Alfven waves assumes a gradual evolution of the wave energyto small dissipative length scales. This can be done by local nonlinear interactions amongMHD waves with comparable wavelengths resulting in turbulent cascades or by phasemixing and resonant absorption. We investigate an alternative nonlocal transport of waveenergy from large MHD length scales directly into the dissipation range formed by thekinetic Alfven waves (KAWs). KAWs have very short wavelengths across the magneticfield irrespectively of their frequency. We focus on the nonlinear mechanism for theexcitation of KAWs by MHD Alfven waves via resonant decay AW ! KAW1 + KAW2.The resonant decay conditions can be satisfied in a rarified plasmas, where the gas/magnetic pressure ratio is less than the electron/ion mass ratio. The decay is efficient atlow amplitudes of the magnetic field in the MHD waves, B/B0 � 10�2. In turn, thenonlinearly driven KAWs have sufficiently short wavelengths for the dissipative effects tobecome significant. Therefore the cross-scale nonlinear coupling of Alfven waves canprovide a mechanism for the replenishment of the dissipation range and the consequentenergization in space plasmas. Two relevant examples of this scenario in the solar coronaand auroral zones are discussed.

Citation: Voitenko, Y. M., and M. Goossens (2005), Nonlinear coupling of Alfven waves with widely different cross-field

wavelengths in space plasmas, J. Geophys. Res., 110, A10S01, doi:10.1029/2004JA010874.

1. Introduction

[2] MHD Alfven waves and turbulence constitute anessential ingredient of magnetized laboratory and spaceplasmas. In situ and remote observations show that MHDwaves are present in the solar corona, solar wind, andplanetary magnetospheres. It is believed that the plasmaenergization that accompanies an enhanced activity ofMHD waves has its source in these waves. However, theenergy of MHD waves resides in length scales and time-scales that are far longer than those required for efficientdissipation. Hence the energy carried by MHD waves canonly contribute to the energization of the plasma if theenergy can be transported from the long wavelengths atexcitation to the required short wavelengths for dissipa-tion. Most attention so far has been given to processes thatgradually decrease the wavelengths across the backgroundmagnetic field. These are so-called perpendicular turbulentcascade [Matthaeus et al., 1998] and phase mixing/resonant absorption [Goossens, 1994]. The disadvantagesof these mechanisms are well known. In particular, they

can work only under restricting special conditions, like thepresence of a fraction of counterpropagating waves for theturbulent cascade [Leamon et al., 2000; Dmitruk et al.,2001] or strong plasma inhomogeneity for the phasemixing/resonant absorption [Goossens, 1994]. In someimportant cases, like heating of the solar corona andacceleration of the solar wind, these restrictions set boundsto wave frequency, specific for each mechanism.[3] The problems with plasma energization by waves

particularly shows up in recent SOHO observations of thesolar corona. Spectroscopic SOHO observations of thewidths of extreme ultraviolet emission lines indicate thations are very hot in the high corona at 1.5–5 solar radii[Kohl et al., 1997]. Moreover, the ion kinetic temperaturesseem to be highly anisotropic with the perpendicular (withrespect to the background magnetic field) temperaturebeing much higher than the parallel temperature [Kohl etal., 1997; Dodero et al., 1998]. These observations suggestthat the ions are energized (heated and/or accelerated)anisotropically, mainly across the magnetic field. Anattempt to attribute the perpendicular energization of theions in the solar corona to the ion-cyclotron heating byhigh-frequency waves (see recent papers by Marsch andTu [2001], Cranmer [2002], and Hollweg and Isenberg[2002], and references therein) leads to the question aboutthe origin of such high-frequency (102 � 104 Hz) waves at

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 110, A10S01, doi:10.1029/2004JA010874, 2005

1On leave from Main Astronomical Observatory, Kyiv, Ukraine.

Copyright 2005 by the American Geophysical Union.0148-0227/05/2004JA010874$09.00

A10S01 1 of 11

those large heights in the corona. If these waves areexcited at the bottom of the solar atmosphere, it is difficultto explain how they can cross the chromosphere [DePontieu et al., 2001] and propagate so far [Voitenko andGoossens, 2002a].[4] Various observations suggest a significant antisun-

ward flux of MHD waves propagating at the photosphericlevels [Ulrich, 1996], and throughout the whole solarcorona [Cranmer, 2004]. The source for this flux can bethe convective motions at photosphere, which can generatevery low-frequency waves Alfven waves with f = 10�4 �10�2 Hz. However, even if a significant fraction of Alfvenwaves observed in the photosphere has periods of 5 min[Ulrich, 1996], the frequencies of coronal waves remainuncertain. First, strong reflection of long-period wavesshould considerably reduce their flux reaching corona[Leer et al., 1982], and, second, the waves with shorterperiods can be generated by high magnetic activityobserved at all levels from the photosphere to the coronalbase. In particular, the magnetic activity (magnetic recon-nection and restructuring) in the magnetic network cangenerate waves in an intermediate-frequency band f �1 Hz [Ryutova et al., 2001], which are less sensitive tothe reflection and can easily reach the corona. The wavesin the same frequency band can be generated also byinteracting magnetic structures at the chromosphere-transi-onion region levels (given involved length scales �108 cmand Alfven velocity �108 cm/s). Having in mind thestrong reflection of the low-frequency waves ( f < 10�2

Hz) on the way from photosphere to corona [Leer et al.,1982] and strong damping of the high-frequency waves( f > 1 Hz) there [De Pontieu et al., 2001], we suggest thatjust the intermediate-frequency waves ( f � 1 Hz) providethe main energy source for the coronal heating andacceleration of the solar wind at 1.5 � 5 R�. Unfortu-nately, there is not enough observational evidences toconclude which waves from which frequency band producethe nonthermal broadening observed in the corona. Untilmore observational support is obtained, both models basedon low or intermediate frequency waves are possible, butwe will only analyze a possibility based on the intermediatefrequency range.[5] One can envisage that the upward flux of MHD

waves generates the presupposed ion-cyclotron waves insitu. The direct turbulent cascade toward higher frequen-cies [Hollweg, 1986; Tu and Marsch, 1997], and thegeneration by currents carried by MHD waves [Markovskii,2001; Markovskii and Hollweg, 2002] have been proposedas local and nonlocal transport mechanisms of waveenergy to the ion-cyclotron dissipation range. However,under the solar corona conditions the turbulent cascade isanisotropic and proceeds toward high cross-field wavenumbers rather than to high field-aligned wave numbersrequired for ion-cyclotron resonance [Matthaeus et al.,2003]. Phase mixing in transversally nonuniform plasmaworks in the same direction, but it seems to be too slowin the upper corona for providing the necessary rate oftransfer of wave energy to the dissipative length scales,which are of the order of ion (proton) gyroradius rp. Thenecessary fraction of counterpropagating waves, requiredfor the perpendicular turbulent cascade, can be easilygenerated by partial reflection in the corona for the low-

frequency (<10�2 Hz) Alfven waves [Leamon et al.,2000; Matthaeus et al., 2003]. Therefore there are nodifficulties with cascade models in this low-frequencyregime. However, in the case of higher wave frequencies,f � 1 Hz, considered in our model, the reduced reflectionmakes the turbulence dissipation negligible, as is illustratedby Dmitruk and Matthaeus [2003, Figure 7]. Nonlocaltransport due to current instabilities [Markovskii, 2001;Markovskii and Hollweg, 2002] can only be efficient whenthe MHD waves have strong shear flows or currents.Hence this type of transport requires prestructured Alfvenwaves with sufficiently small scales either across thebackground magnetic field or in the field-aligned direction(which in turn implies already high-frequency Alfvenwaves).[6] In the auroral regions of the terrestrial magnetosphere,

the multiscale activity of Alfven waves in both MHD andin kinetic modes is a well-documented fact. Moreover,recent satellite observations leave little room for doubtabout the KAWs being responsible for many intense eventsin the aurora [Wygant et al., 2002; Chaston et al., 2003].The kinetic Alfven waves (KAWs) we refer to are Alfvenwaves with short wavelengths across the background mag-netic field B0 (large perpendicular wave numbers k? � kk).The linear and nonlinear effects in KAWs due to shortcross-field wavelengths become especially pronouncedwhen the wavelength becomes comparable to the kinematicscales of the plasma species. These are the ion gyroradiusrp and the electron inertial length de (k?rp � 1 and/ork?de � 1). At the same time, in many cases these effectsare already efficient with k?rp(de) � 0.01 � 0.1. Althoughthe generation mechanism for auroral KAWs is not yetidentified, simultaneous observations of the strong down-ward flux of large-scale MHD Alfven waves suggest thatthe source for KAWs is provided by these long-scaleAlfven waves [Wygant et al., 2002]. In other cases, whenthe large-scale currents are energetically more important,the KAWs can be excited by current-driven instabilities[Voitenko et al., 1990; Seyler and Wu, 2001].[7] The nonlinear coupling of waves with widely dif-

ferent wave lengths, such as resonant decay, provides anonlocal cross-scale spectral transfer and can facilitate thedissipation of MHD waves and consequent energy release.One of the well-known properties of the resonant decay isto produce nonlocal energy transport in the wave numberspace in the so-called dipole or adiabatic approximation[Maslennikov et al., 1995]. However, previous work onnonlocal transport was focused mostly on the decay ofhigh-frequency (ion-cyclotron) waves. In the present paperwe show that the low-frequency finite-amplitude MHDAlfven waves are nonlinearly coupled to kinetic Alfvenwaves. We focus on the nonlinear processes that canstrongly accelerate the evolution of Alfven waves towardsmall length-scales: the three-wave resonant decay oflarge-scale MHD Alfven waves into kinetic Alfven waves,AW ! KAW1 + KAW2. The nonlocal three-wave resonanceof a large-scale MHD wave with small-scale daughterKAWs is possible because of the highly anisotropic natureof KAWs which are still low-frequency despite of the highoblique wave numbers.[8] KAWs are able to interact strongly with plasma

species and with other wave modes, and hence they attract

A10S01 VOITENKO AND GOOSSENS: NONLINEAR EXCITATION OF KAWS

2 of 11

A10S01

great interest [Hollweg, 1999; Stasiewicz et al., 2000;Voitenko and Goossens, 2003; Onishchenko et al., 2004].The kinetic and inertial regimes of KAWs have beenobserved in the laboratory [Leneman et al., 1999; Kletzinget al., 2003] and in space [Leamon et al., 2000; Wygant etal., 2002; Chaston et al., 2003; Stasiewicz et al., 2004]plasmas. However, some aspects of the KAWs’ theory,especially their interaction with beams and currents, andnonlinear interactions, are still insufficiently explored. Asfar as the nonlinear interaction is concerned, this can beunderstood in view of a variety of nonlinear factors that cancome into play. Many nonlinear terms can drive nonlinearperturbations of the plasma variables which are typical forKAWs, such as number density of plasma species, as well asfield-aligned and cross-field currents. In this paper wefollow the nonlinear second-order theory of KAWs devel-oped in the framework of two-fluid MHD [Voitenko andGoossens, 2002b].

2. Nonlinear Interaction of KAWs With MHDAlfven Waves

2.1. Nonlinear Eigenmode Equation for KAWs

[9] We consider a uniform hydrogen plasma immersed ina background magnetic field B0. Since the density pertur-bations are typical for KAWs, we use the effective densitypotential f,

f ¼ Te

elnne

n0; ð1Þ

as independent variable. Here ne is the electron numberdensity perturbation, n0 is the unperturbed number density,Te is the electron temperature, and �e is the electron charge.The nonlinear (up to second order in perturbationamplitudes) eigenmode equation for KAWs is [Voitenkoand Goossens, 2002b]:

@2

@t2� V 2

AK2r2

k � 2gd@

@t

� �f ¼ Ntot: ð2Þ

Here gd is the linear damping/growth rate due to wave-particle interaction, VA = B0/

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi4pnmp

pis the Alfven velocity,

rk is the spatial derivative along B0, and K is the dispersionfunction such that the wave dispersion in wave numberspace is w = kkVAK. In low-b plasmas K can beapproximated as

w2

k2kV2A

¼ K2 1þ mT1þ ce

; ð3Þ

where mT = rT2 k?

2 , rT2 = VT

2 /Wp2, VT

2 = T/mp, T = Tp + Te,Wp = eB0/mpc is the proton cyclotron frequency, ce = de

2k?2 =

bm�1mT, de is the electron inertial length, bm = b(mp /me),and b = VT

2 /VA2 is the gas/magnetic pressure ratio. The

KAW dispersion function K has the following properties:for me/mp < b < 1, K > 1, and for b < me /mp, K < 1.

[10] The nonlinear part Ntot of (2) is quite complicated:

1þ d2ek2?

V 2Te

Ntot ¼� V�2Te

me

mp

�4p þ 1� �v�2k

� �� �@

@tve

�� r

þ @

@trk vek � vpk� �

þ �v�2k

@

@tvp � ve� �

� r�f

þ me

mp

�4p

� �rk � Fek þ

me

mp

me

mp

�4p

�

þ me

mp

1� �v�2k

� 1

Wp

@

@tr? � b0 � Fe?½ �

þ me

mp

1� �v�2k

� �rk � Fek � Fpk

� �þ me

mp

�v�2k

1

Wp

@

@tr? � b0 � Fe? � Fp?

� � þ me

mp

�v�2k

1

W2p

@2

@t2r? � Fp?: ð4Þ

[11] The eigenmode KAW equation (2) is derived fromthe electron and ion continuity equations and from theparallel and perpendicular components of the Ampere law.This set of four equations contains five variables. Three ofthem describe the electromagnetic wave field (j, Az, Bz),and the remaining variables are the plasma density responsedue to the electrons (ne) and protons (np). We close thesystem by the quasi-neutrality constraint, n � ne = np, andeliminate all other variables in favor of f (i.e., n).[12] The nonlinear second-order terms in (4) come from

the electron and ion continuity equations and from theparallel and perpendicular components of the Ampere law.In some papers, a linear Boltzmann-like distribution of theion density perturbation is used for obtaining closure[Onishchenko et al., 2004]:

�fp ¼Te

e

np

n0¼ exp �r2pk

2?

� �I0 r2pk

2?

� �� 1

h ij;

where I0(rp2k?

2 ) is the zero-order modified Bessel function.This approach misses the nonlinear part of the ion densityperturbation and can only be justified when the KAWsinteract with ‘‘electronic’’ waves, which do not perturb theions. Instead, we use the ion continuity equation

@

@tfp þ vp � rfp �

mp

eV 2Tpr � vp ¼ 0; ð5Þ

which keeps the nonlinearities due to the convectivederivative and the divergence of the nonlinear perturbationsof the ion velocity.[13] The nonlinear force is

Fs ¼1

cvs � B� ms

qsvs � rð Þvs: ð6Þ

The relative number density fluctuations with respect tomagnetic fluctuations in KAWs is

n

n0¼ �ik?dp

K

K2 � bB?

B0

:

A10S01 VOITENKO AND GOOSSENS: NONLINEAR EXCITATION OF KAWS

3 of 11

A10S01

The magnetic compressibility is

Bk

B0

¼ �bn

n0: ð7Þ

This shows that in a low-b plasma, similarly to the slowmode, the gas compression is more important for KAWsthan the magnetic compression. In terms of fk, we have thefollowing expressions for the perturbed velocity andmagnetic field due to KAWs:

vp? ¼ c

B0

m�1Te

1

Wp

@

@t1� bð Þr? � b0 �r?

� �fk ; ð8Þ

ve? ¼ c

B0

m�1Te �b

1

Wp

@

@tr? � 1þ mTð Þb0 �r?

� �fk ; ð9Þ

vez ¼ se

me

VA

V 2Te

Kfk ; ð10Þ

vpz ¼ �se

me

VA

V 2Te

me=mp � b1þ mT

Kfk ; ð11Þ

B? ¼ sc

VA

K

mTeb0 �r?fk½ �; ð12Þ

where mTe = mTTe/(Te + Tp) and s � sign(kz).

2.2. Resonant Decay

[14] We concentrate here on the processes induced by thescalar nonlinearities [Voitenko and Goossens, 2002b]: res-onant decays of the large-scale pump Alfven and fastmagnetoacoustic waves into KAWs. These processes resultin a jump-like spectral transport of MHD wave energydirectly into the dissipation range (i.e., dissipative lengthscales). We consider the most efficient three-wave resonantinteractions [Sagdeev and Galeev, 1969]. The beatingsbetween the ‘‘pump’’ wave with wave vector kP and thetrial KAW k1 can efficiently drive the complimentary trialKAW k2 with spatiotemporal scales that coincide with thescales of the beatings: k2 = kP ± k1; w(k2) = wP(kP) ± w(k1).If there is initially a low (e.g., thermal) level of KAWs, theefficient amplification of KAWs occurs if there is a positiveback reaction on the wave k1 from the wave k2, i.e., whenthe beatings between the KAW k2 and the pump kP in turndrive k1: k1 = kP ± k2; w (k1) = wP(kP) ± w(k2). In this casethe process takes the form of a (parametric) decay, wherethe pump wave decays into a spectrum of wave pairs whichsatisfy the resonant conditions for wave vectors

k1 þ k2 ¼ kP; ð13Þ

and wave frequencies

w k1ð Þ þ w k2ð Þ ¼ wP kPð Þ: ð14Þ

[15] Vectorial nonlinear interaction of KAWs has beenstudied by Voitenko [1998b] (kinetic theory) and byVoitenko and Goossens [2000, 2002b] (two-fluid resistiveMHD). This interaction is proportional to k1? � k2? andis therefore local in k space, jk1?j � jk2?j � jkP?j. Scalarinteraction is formally of order b, but since it is proportionalto k1? � k2?, the resonant nonlocal interaction is possiblewhere jk1?j � jk2?j � jkP?j. Indeed, from the resonantcondition kP? = k1? + k2?, we have k1? � �k2? and thenk1? � k2? � jk1? � k2?j for the nonlocal interaction.[16] It is useful to factorize the KAW wave functions f1

and f2 into an exponential phase dependence and a slowlyvarying amplitude, F1,2 = F1,2(t):

f1;2 ¼ F1;2 exp �iw1;2t þ ik1;2 � r� �

:

The equations for the amplitudes of the resonant short-scaleAlfven waves 1 and 2, coupled to the large-scale pumpwave P, are then obtained from the nonlinear eigenmodeequation (2) as

@

@t� gd1

� �F1 ¼ U1;�2;PF2

*bP; ð15Þ

@

@t� gd2

� �F2* ¼ U�2;1;�PF1bP*: ð16Þ

Here bP = jbPj, bP = BP/B0 is the normalized magnetic fieldof the pump wave. The coupling coefficients are

U1;�2;P ¼ �iw1F2*bP

� ��1Ntot 1;�2; Pð Þ;

U�2;1;�P ¼ iw2F1bP*� ��1

Ntot �2; 1;�Pð Þ:

Ntot has to be calculated from (4) by eliminating all thepump variables in favor of bP and by eliminating all theKAW variables in favor of F. The total growth (or damping)rate of exponentially growing (or decaying) solutions of thesystem (15–16), F1,2 � exp (gtott), is then

gtot ¼gd1 þ gd2

2�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffigd1 � gd2

2

� �2þ g2NL

r; ð17Þ

where gNL is the nonlinear growth rate, the rate of thenonlinear pumping of MHD wave energy into daughterKAWs:

gNL ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiU1;�2;PU�2;1;�P

pbPj j: ð18Þ

2.3. Nonlinear Decay of MHD Alfven Waves Intokinetic Alfven Waves

[17] We consider a pump AW with a frequency wP � Wp

and wave vector kP = (kPx; 0; kPz), kPx � kPz that decays intopairs of resonant KAWs with k1? �k2? � kP? and withapproximately equal dispersions, K(k1?

2 ) K(k2?2 ). In that

case Ntot simplifies considerably because only the cross-

A10S01 VOITENKO AND GOOSSENS: NONLINEAR EXCITATION OF KAWS

4 of 11

A10S01

field electromagnetic perturbations and the velocity pertur-bations are induced by MHD Alfven waves (bP ? B0):

vPp ¼ �sPVA bP � iwP

Wp

b0 � bP

� �; ð19Þ

vPe ¼ �sPVAbP; ð20Þ

BPz ¼ vPpz ¼ vPez ¼ 0: ð21Þ

[18] The frequency matching condition, w1 + w2 = wP,and the parallel wave number matching condition, s1w1 +s2w2 = wPK, can be satisfied only for antiparallel propaga-tion of the daughter KAWs (we take sP = kPz/jkPzj = 1). So,

assuming s1 = 1, we must have s2 = �1. This determinesthe KAW frequencies as

w1 ¼1þ K

2wP ð22Þ

and

w2 ¼1� K

2wP: ð23Þ

Hence this process is only possible if K < 1, i.e., in a verylow-b plasma with b < me/mp. The corresponding resonanttriplet is shown in Figure 1. Contrary to the decay KAW !KAW1 + KAW2 (see Figure 2), where both the costreamingand the counterstreaming propagation of KAW products areallowed, the MHD Alfven waves can only decay intocounterstreaming KAW products. This is due to the fact thatthe MHD AW dispersion has a steeper slope than that of anyKAW dispersion when bm � b(mp/me) < 1.[19] The KAWs in a bm < 1 plasma are sometimes called

inertial Alfven waves because their dispersion is deter-mined mainly by the parallel electron inertia. In thesecircumstances we obtain the following expressions for theelectron and ion ponderomotive forces induced by thebeatings of the pump Alfven wave and the KAW withbP � r?fk � bP � r?fk:

mTeFe ¼ K2 1� sPs1

K

� �bP � r?ð Þfkb0 þ sK

k2?VA

Wp

fk b0 � bP½ �;

ð24Þ

mTeFp ¼ 1� sPs1

K

� �bP � r?ð Þfkb0

� VA

Wp

sP bP � r?ð Þ b0 �r?½ �fk

þ VA

Wp

sP bP � r?ð Þ 1

Wp

@

@tr?fk : ð25Þ

Figure 1. A parallelogram in the (w, kz) plane reflectingthe resonant conditions for the nonlinear coupling amongthree waves: pump MHD Alfven wave and two counter-streaming kinetic Alfven waves (KAWs) (w1 + w2 = wP;k1z + k2z = kPz; k1? + k2? = kP? = 0).

Figure 3. The normalized nonlinear pumping rate �gNL =gNL/[Wp

ffiffiffiffiffiffiffiffiffiffiffiffiffimp=me

pjbPj] as function of bm and the normalized

perpendicular wave vector dek?. For decreasing bm the peakof the KAW spectrum shifts toward higher wave numbers.

Figure 2. A resonant parallelogram in the (w, kz) plane forthe nonlinear coupling of the pump KAW with two productKAWs (primes indicate counterstreaming KAWs). Contraryto the AW ! KAW1 + KAW2 decay, the resonant decay ispossible here into two costreaming KAWs (w1 + w2 = wP;k1z + k2z = kPz; k1? + k2? = kP? = 0) since both the steeperand the flatter dispersion slopes are allowed for the productKAWs in comparison to that of the pump KAW.

A10S01 VOITENKO AND GOOSSENS: NONLINEAR EXCITATION OF KAWS

5 of 11

A10S01

[20] The coupling coefficients of KAWs with a pumpMHD Alfven wave are found as

U1;�2;P ¼ is11

Kbm � K2� � d2ek

2?

1þ d2ek2?VA eP � k2?ð Þ; ð26Þ

U�2;1;�P ¼ �is21

Kbm � K2� � d2ek

2?

1þ d2ek2?VA eP � k1?ð Þ: ð27Þ

Here eP is the magnetic polarization vector of the pumpwave, eP = BP/BP. The total growth rate of KAWs is givenby (17), where the nonlinear pumping rate is found from(18) as

gNL ¼ffiffiffiffiffiffimp

me

rbm � 1j j d3ek

3?ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ bmd2ek

2?

q1þ d2ek

2?

� �3=2 bPj jWp: ð28Þ

Similarly to the case of decay KAW ! KAW1 + KAW2,the rate of the nonlocal resonant decay AW ! KAW1 +KAW2 is independent of the pump wave frequency. Onlyin the regime of modified decay (see below) such adependency occurs and scales as wP

1/3. A three-dimensional(3-D) plot illustrating how the nonlinear pumping ratedepends on bm and the perpendicular wave numbers of theexcited KAWs is shown in Figure 3. As long as bm issmaller than 1, the MHD Alfven waves decay verystrongly, exciting KAWs, but for bm ! 1, gNL ! 0. Thedecay is impossible for bm > 1, where the resonantconditions cannot be satisfied.[21] For any bm < 1, a broadband KAW spectrum is

excited, but the strongest nonlinear pumping is in the KAWs

with kS?2 = (1 +

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 3=bm

p)de�2, with the (maximum)

pumping rate

gNLmax ¼ Wp

ffiffiffiffiffiffiffiffi3mp

me

r1� bmj j

ffiffiffiffiffiffibm

pþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi3þ bm

p� �2ffiffiffiffiffiffibm

pþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi3þ bm

p� �2 bPj j: ð29Þ

[22] These properties of the nonlinear growth rate, to-gether with the dissipative properties of KAWs, create aninteresting feedback loop for the plasma/MHD wave fluxsystem. When the flux of MHD Alfven waves enters theregion where bm < 1, nonlinearly driven KAWs are gener-ated and tend to heat the plasma up to the level where the bmis kept slightly below 1, bm = b*m ] 1. When used in (28),this particular value b*m provides a dynamical equilibriumwhen the nonlinear growth rate gNL, at which the energy ispumped in the plasma via intermediate KAWs, balances theenergy lost from the plasma due to emission, thermalconduction, and plasma acceleration. With different powersof the launched MHD Alfven flux, the interplay of theseprocesses can set up different regimes, from weak additionalheating in the vicinity of the transition point bm = 1 to thecreation of an extended region where bm ] 1, which ends upwith accelerated flows of heated plasma, where bm > 1.

2.4. Modified Decay

[23] The decay AW ! KAW1 + KAW2 is characterized byquite different frequencies of excited KAWs. For bm ] 1

w2

w1

¼ 1� K

1þ K� 1:

Owing to the frequency separation of the product KAWs,the growth rate of the decay, calculated by use of (29), oftenappears to be larger than the lower frequency w2. In this case(29) cannot be considered as a true growth rate and thetheory should be modified taking into account the nonlinearmodification of the lower KAW frequency. In the regimewhere the frequency of the low-frequency KAW satellite isdetermined by the nonlinear effects,

w2 � w2k ¼1� K

2wP;

the resonant decay takes the form of a modified decay. Inthis regime of strong decay we ignore gd1 gd2, and insteadof (15–16) we obtain from (2)

2w1kw2f1* ¼ Ntot �1; 2;�Pð Þ;

�w22f2 ¼ Ntot 2;�1; Pð Þ:

The resulting dispersion equation is

w32 ¼ �2w2kg

2NL; ð30Þ

where gNL is given by (28). This equation has the unstablesolution

w2 ¼ 2�2=3 1þ iffiffiffi3

p� � w2k

gNL

� �1=3

gNL; ð31Þ

with the growth rate

gMD 1:1w2k

gNL

� �1=3

gNL: ð32Þ

This growth rate has a �jbPj2/3 dependence on the waveamplitude.

3. Saturation of the Decay and KineticAlfven Turbulence

[24] Let us consider the excitation of KAW turbulenceby a MHD Alfven wave, pumping energy into a modekS? � de

�1 (wave vector) with rate 2gNLmax. When theamplitude of the growing mode reaches a threshold(saturation) value, the parametric decay of the nonlinearlyexcited KAWs with kS stops growing and spreads out thewave spectrum. The decay into parallel-propagating KAWsinitially spreads the spectrum into both higher and lowerwave number domains, k1? < kS? < k2?, whereas the decayonly into counterstreaming KAWs spreads the spectrum intolower wave number domain, k1? < k2? < kS? [Voitenko,1998a].[25] The relevance of the decay into counterstreaming

waves for the problem of the nonlinear saturation can beunderstood as follows. The time needed for an efficientexchange of energy among the counterstreaming KAWs,tNL � gNL

�1, has to be shorter than the overlapping time,Lcor/2VA, during which the counterstreaming wave travels inthe field of the pump wave:

Lcor

2VA

^ g�1NL : ð33Þ

A10S01 VOITENKO AND GOOSSENS: NONLINEAR EXCITATION OF KAWS

6 of 11

A10S01

Since the condition for saturation is gNLmax�1 = gk

�1, we getfrom (33) a quite reasonable estimate for the possiblecorrelation length: Lcor ^ 3lz, where lz is the wavelengthalong B0 (lz � 109 cm for waves with frequencies around1 Hz in the solar corona).[26] The dynamics and direction of the nonlinear KAW

energy transfer in k-space has been investigated in a modelof the triads interacting most effectively in the cascadingprocess [Voitenko, 1998b]. In this model the spectral energycascade is considered as a series of consequent decaysbeginning with the first step of the source wave decay intotwo secondary waves (decay products), having the largestgrowth rate. Any decay product of the nth step in turnexcites two decay products of the (n + 1)th step and so on,eventually resulting in an energy transfer in the k space, ofwhich the dynamical properties are determined by the formof the matrix element of three-wave resonant interactionamong KAWs U(k, k1, k2). The wave numbers of thedominant decay products at any step are determined bymaximizing jU(k1, k, �k2)U(�k2, �k, k1)j. In the case ofthe most effective decay into counterstreaming secondarywaves, these wave numbers are k1? 0.776ks? and k2? 0.442ks? in the limit of weak wave dispersion (we put Te =Ti for simplicity). Further pumping of energy into KAWswith kS? waves by MHD waves causes further spreading ofthe wave spectrum and formation of the turbulent cascade.[27] In the KAW turbulence formed by dominant triads

the energy tends to flow toward smaller k, forming a so-called inverse cascade. As the electron Landau damping andthe collisional dissipation of KAWs are highly reduced atsmaller k?, this region can be considered as an inertial one(no sink no source), and an inertial-range power-law spectracan be formed at k? < ks?. The strongest turbulent cascadeincluding interaction among counterstreaming waves formsthe inertial-range energy spectrum [Voitenko, 1998b]:

Wk � k�1=2z k�2

? : ð34Þ

This spectrum can exist if there is a sink at some wavenumber k*?, 0 < k*? < kS?, otherwise we encounter an infiniteenergy growth while k? ! 0. If such a sink exists, theenergy of the turbulence is concentrated at k? ^ k*?. It ispossible that the nonadiabatic ion acceleration providessuch sink, but in this case the inertial range can be quiteshort.

4. Two Examples: Solar Corona and AuroralZones of the Terrestrial Magnetosphere

4.1. Nonlinear Excitation of KAWs by MHD AlfvenWaves in the Solar Coronal Holes

[28] The nonlinear properties of KAWs provide verypromising plasma energization mechanisms that can heatplasma and accelerate the solar wind in coronal holes. Letus start from an initially static equilibrium where the plasmais heated locally, only at low heights (e.g., by acousticshock waves), and the plasma temperature and densitydecrease with heliocentric distance along open magneticfield lines. The gas/magnetic pressure ratio b should alsodecrease, practically to zero, at a distance of the order ofthe hydrostatic scale height. Now, assume a flux of MHD

Alfven waves, launched in the corona by photosphericmotions (or by magnetic reconnection events at the coronalbase). These waves propagate upward unattenuated up tothe height zh where bm(zh) becomes equal to 1. However, assoon as bm drops below 1, MHD Alfven waves undergo astrong parametric decay into KAWs. Owing to their shortperpendicular wavelengths, the nonlinearly excited KAWsdissipate via collisional or collisionless wave-particle inter-action. This, in turn, gives rise to plasma heating andparticles acceleration. Thus the flux of KAWs that prop-agates further upward can easily increase bm again wellabove 1 in the high corona and provide the energy sourcefor the solar wind acceleration. This should eventually resultin a new, dynamic equilibrium, in which (1) the firsttransition point from bm > 1 to bm < 1 is shifted to largerheights z1 > zh (result of the dissipation of downward KAWflux); (2) there exists a region z1 < z < z2 where bm(z) is kept(by the dynamical back reaction on the KAWs damping) at alevel below 1, so as to provide an efficient conversion of theupward MHD Alfven flux into two counterstreaming KAWfluxes; (3) there is a second transition point from bm < 1 tobm > 1, z = z2, beyond which the plasma is strongly heatedby the upward KAW flux and expands forming solar wind.[29] This behavior of bm with height is supported by

models of the corona based on the observations. Fornumerical example, we take the model magnetic field[Banaszkiewicz et al., 1998] with B0 = 8 G at the coronalbase, r = 1.1,

B0 ¼ 1:789� 2

r3þ 4:5

r5þ 1

1:538 r þ 1:538ð Þ2

!Gð Þ; ð35Þ

and the model coronal density [Esser et al., 1999] with n0 =4 � 107 cm �3 at r = 1.1,

n0 ¼ 0:5� 106 � 2:494

r3:76þ 1:034

r9:6410þ 3:711

r16:86102

� �cm�3� �

;

ð36Þ

where the heliocentric distance r is measured in solar radii.The expression for bm is

bm ¼ mp

me

4pn Te þ Tp� �B2

¼ 3:24� 10�12 n cm�3ð Þ Te �Kð Þ þ Ti�Kð Þ½ �

B2 Gð Þ : ð37Þ

[30] Then, with Tp + Te = 2 � 106 K, from (35)–(37) weget the radial dependence of plasma bm:

bm ¼ 2:494

r3:76þ 1:034

r9:64� 10þ 3:711

r16:86� 102

� �

� 2

r3þ 4:5

r5þ 1

1:538 r þ 1:538ð Þ2

!�2

; ð38Þ

which is shown in Figure 4. It is remarkably that bm dropsbelow 1 around r = 1.5, where a strong cross-fieldenergization of heavy ions is observed by SOHO. Inaccordance with our results, this is a region where the

A10S01 VOITENKO AND GOOSSENS: NONLINEAR EXCITATION OF KAWS

7 of 11

A10S01

upward propagating MHD Alfven waves can decay in aspectrum of KAWs.[31] To proceed with our numerical estimate of the decay

rate, we take the following plasma parameters in coronalholes at r = 1.37: Te = 0.8 � 106 K, Tp /Te = 1.5, ne = 1.6 �106 cm�3, B0 = 3.2 G. Then we estimate bm = 0.97, which isstill very close to 1, Alfven velocity VA = 5.5 � 108 cm s�1,proton cyclotron frequency Wp = 3.1 � 104 s�1, and thecollisional frequency

ne ¼4ffiffiffiffiffiffi2p

pLe4ne

3ffiffiffiffiffiffime

pT3=2e

¼ 0:2 s�1; ð39Þ

where the Coulomb logarithm L = 23.[32] Let us first consider the very low frequency regime

of the decay, wk ] 2 � 10�2 s�1 (3-min wave periods). Asthe collisional (resistive) dissipation of KAWs is strongerthan the Landau damping at these wave frequencies[Voitenko and Goossens, 2000], we use the collisionaldamping as an estimate of the dissipation rate:

gd1 ¼ gd2 ¼ gd ¼ �0:25nek2?d

2e

1þ k2?d2e

; ð40Þ

and

gtot ¼ gd þ gNL:

Then, from the threshold condition gd + gNL = 0, wecalculate the threshold amplitude bP = 2.6 � 10�6. This is avery small value, which amounts to only a tiny fraction,<10�4, of the wave power deduced from the nonthermalbroadening observations, dv/VA = B/B0 = 3 � 10�2

[Banerjee et al., 1998].[33] Let us assume that the wave energy participating in

the decay is jbPjeff = jBPjeff /B0 = 10�4 (that is less than 1%of the pump Alfven wave spectrum keeps coherence duringthe decay time), and the waves propagate in the regionwhere bm = 0.99. Then we get the growth rate of themodified decay gMD = 1.98 � 10�2 s�1, which is an upper

bound of our approximation that gMD < wP. Formally, wecan have a much higher growth rate for smaller bm and/orwith larger jbPjeff, but we are restricted by the condition ofweak growth/damping, i.e., gtot should be smaller than w1,gtot < w1 ] wP, which can be violated for the low-frequencyMHD waves. On the one hand, that means that the pumpAlfven waves, which propagate in the region of decreasingdensity, will efficiently loose their energy at the verybeginning once they cross the point where bm = 1. However,as bm further decreases, the decay regime becomes toostrong for our perturbative analysis to be applicable. Onlyfor values of bm sufficiently close to 1, bm = 0.997, weobtain gMD = 6 � 10�3 s�1 which reasonably satisfies theconditions of resonant decay.[34] Therefore the problem with very low frequency AWs

(and hence KAWs) is that their frequency drops below thedecay rate in a wide range of the parameter space, where ourtheory is inapplicable. To be tractable in the framework ofour theory, all the process of the MHD AWs conversion intoKAWs has to fulfill before the waves escape in the regionwith different plasma parameters. For example, the mini-mum decay time tractable for our theory, tMD � gMD

�1 =167 s, implies that the waves propagate in the extendedregion of about 1 solar radius where bm is kept very close to1, bm 0.997.[35] However, in view of strong reflection of very low

frequency waves ( f < 10�1 Hz) between the photosphereand the corona, the waves of the intermediate-frequencyband f � 1 Hz seem more preferable. The source for thesewaves can be magnetic activity in the magnetic network[Ryutova et al., 2001]. With the intermediate frequencies2pfP = wP � 2, we can easily have much faster decay in asmall region around r = 1.5, where bm 0.73. For example,for very small wave amplitudes, jbPjeff = 10�4, we obtaintMD � gMD

�1 � 1 s so that the extension of such region isreasonable small, �10�2 solar radii. For higher wave ampli-tudes jbPjeff > 10�4 the decay occurs at lower heights,rmin < r < 1.5, where bm(rmin) = 1 > bm(r) > 0.73. Theflux of MHD waves, coming from the region r < rmin wherebm > 1, meets initially very small 1 � bm, satisfying thecondition gtot < w, but providing sufficiently fast conversionof wave energy. The mechanism is working in such a way asto keep b close to me/mp so that the values bm = 0.8–0.9 arereasonable. The critical height for the decay rmin is situatedsomewhere between 1.2 and 1.3 solar radii.[36] In this respect it is interesting to mention the spectral

line observations of heavy ions in a solar coronal hole. Linewidths were found to increase with height to 1.15 R�, as isexpected from the undamped propagation of the MHDAlfven waves. However, starting from some height around1.2 R�, the lines were constant. These observations wereinterpreted in terms of a wave flux that undergoessignificant dissipation above 1.2 R� [Banerjee et al.,1998; Moran, 2003]. These observations can be interpretedas an indication of the nonlinear conversion of the MHDAlfven waves into dissipative KAWs that should occur atthese heights in our model. Actually, even withoutimmediate damping of the product KAWs, one shouldexpect the observed behavior of line widths because theshort cross-field wave lengths of KAWs make theminvisible in heavy ion observations. Indeed, from therelations de

2 � rp2 � ri

2 for the heavy ions and k?2 de

2 � 1

Figure 4. The plasma bm dependence on the heliocentricdistance z (in solar radii). The resonant decay occurs at thedistances where bm is less than 1.

A10S01 VOITENKO AND GOOSSENS: NONLINEAR EXCITATION OF KAWS

8 of 11

A10S01

for the nonlinearly driven KAWs, we find that k?2 ri

2 � 1. Inthis case the wave electric field, experienced by heavy ions,is smoothed out by the averaging over large ion cyclotronorbits crossing many wave lengths:

Eeff? ¼ I0 r2i k

2?

� �exp �r2i k

2?

� �E? 1

r2i k2?E? � E?:

The related E � B0 drift is thus highly reduced.[37] The fastest-growing KAWs have high perpendicular

wave numbers, k?2 de

2 ^ 1, and can accelerate the ions acrossthe background magnetic field nonadiabatically [Voitenkoand Goossens, 2004]. The condition for the nonadiabaticcross-field acceleration of oxygen ions O+5 is

mi

5mp

kxVA

Wp

1þ mpK

� Viz

VA

� �By

B0

> 1;

where Viz is the bulk velocity of the oxygen ions withrespect to protons. Initially, Viz = 0, and for the nonlinearlyexcited KAWs with kxde ^ 1 we obtain the followingcondition for relative KAW amplitudes:

BKAWy

B0

>5mp

mi

ffiffiffiffiffiffime

mp

rK

kxde 1þ mp� � ¼ 1� 3ð Þ � 10�3: ð41Þ

Since the relative amplitudes of coronal MHD waves arejbPj 0.05, condition (41) becomes realistic even for asmall fraction (]0.1) of MHD wave energy transferred toKAWs, and O+5 ions can be strongly heated in theperpendicular direction. However, owing to the highefficiency of the nonlinear excitation of KAWs, we couldexpect much more energy to be transferred to KAWs, of theorder of the initial MHD wave energy. In this case theprotons can also be heated by KAWs. The condition fornonadiabatic proton acceleration is By

KAW/B0 ^ 0.01.[38] The IPS observations [Coles et al., 1995; Grall et al.,

1997] put limitations on the properties of the turbulentfluctuations in the solar corona. These observations showthat the density spectrum is very anisotropic at heights <6solar radii and the anisotropy is confined to the smallerscales. For lower heights the cutoff occurs at shorter scales,and the near-cutoff spectrum is substantially enhanced overthe background Kolmogorov power-law spectrum, typicalfor larger scales. These IPS observations suggest that thesmall-scale anisotropic turbulence is due to a differentphysical mechanism than the large-scale Kolmogorov tur-bulence [Grall et al., 1997]. The model of KAW turbulenceexcited below 3 solar radii is consistent with these features.The inconsistency between the cutoff length scale of a few100 m observed at 5 solar radii, and the MHD-driven KAWlengths of a few 10 m at 1.5 solar radii, can be onlyapparent. Firstly, having in mind observed tendencies, thecutoff length scale at 1.5 R� can be considerably shorterthan a few 100 m observed at 5 R�. Second, the MHD-driven spectrum of KAWs, shown on Figure 3, is not final,and the mutual nonlinear interactions among excited KAWsintroduce a spectral shift of the energy-containing wavelengths toward longer scales.[39] An essential feature of this heating mechanism is its

relation to the region of bm ] 1 at some distance from theSun. From the models for magnetic field (35) and plasma

density (36) it follows that bm can drop below 1 at 1.2 � 1.5solar radii. An efficient excitation of the KAWs is possiblethere if the nonlinear growth rate is considerably higher thantheir damping rate due to wave-particle interactions. Thenthe damping distance for KAWs can be much longerthan the excitation distance, and the flux of KAWs excitedat 1.2 � 1.5 solar radii can propagate further upward andheat plasma up beyond 2 solar radii. Under the coronalhole conditions, the dissipation of KAWs is mainly deter-mined by two quite different processes: resonant electronLandau damping and nonresonant nonadiabatic accelera-tion of the ions. The first one results in an increase of thefield-aligned (parallel) energy in the electron component,and the second one results in an increase of the cross-field(perpendicular) energy of the ions. Because of the weakcollisions, the efficiency of the electron heating is reducedby the quasi-linear modification of the electron distributionat resonant velocities. At the same time, the cross-fieldheating of protons and heavy ions is nonresonant and canbe strong [Voitenko and Goossens, 2004]. This can ac-count for the weaker energization of the electrons incomparison to the ion energization in holes [Wilhelm etal., 1998; Esser et al., 1999]. Unfortunately, the KAWdamping rate due to nonadiabatic ion acceleration is stillnot estimated analytically.

4.2. Nonlinear Excitation of KAWs by MHD AlfvenWaves in the Auroral Zones

[40] A high level of KAWs is continuously observed bysatellites in the auroral zones of the terrestrial magneto-sphere at heights 1.5 � 6 RE (RE is Earth’s radius), wherethese waves provide plasma heating and particle accelera-tion. Moreover, recent satellite observations indicate that theKAW flux is responsible for the most intense eventsobserved in the aurora [Wygant et al., 2002; Chaston etal., 2003]. However, the origin of these KAWs is stilluncertain. At the same time satellites measure a large energyflux of large-scale MHD Alfven waves that are excited bythe solar wind interacting with the magnetosphere, and/orby the magnetic reconnection in the geomagnetic tail. TheseMHD waves propagate downward along the geomagneticfield lines, and their flux is sufficient to power the measuredflux of KAWs. This observational fact suggests [Wygant etal., 2002] that the KAWs can be generated by MHD Alfvenwaves. If we take the amplitude of MHD Alfven wave bP =0.1 and wP/Wp = 0.01–0.1 [Wygant et al., 2002], then wefind from (29) a strong nonlinear pumping of energy intoKAWs gNLmax � w1 ] wP in the region where bm is still veryclose to unity, bm ’ 0.99. For bm ] 1, the fastest-growingKAWs have perpendicular wave numbers k?de ^ 1. TheLandau damping rate of these KAWs [Lysak and Lotko,1996] is weaker than the nonlinear pumping rate,gL � w1 � gNLmax. Hence the nonlinear excitation ofKAWs by the decay of MHD Alfven waves can be veryefficient in the auroral regions. The KAWs excited myMHD waves, in turn, interact among themselves and formturbulent spectra observed in the aurora [Voitenko, 1998b].[41] For O+ ions, the condition of nonadiabatic accelera-

tion can be written as

kxVA

Wp

1þ mpK

� Viz

VA

� �By0

B0

>Wi

Wp

¼ 1

16;

A10S01 VOITENKO AND GOOSSENS: NONLINEAR EXCITATION OF KAWS

9 of 11

A10S01

which gives very low critical amplitudes By0 /B0 > 1.5 �10�3. KAWs with supercritical amplitudes are alwaysobserved in the regions of strong cross-field heating of O+

ions. These KAWs are accessible for the nonadiabaticacceleration of O+ across the background magnetic field andare thus responsible for the observed heating and formationof ion conics.

5. Summary

[42] We have investigated a new channel for nonlocalspectral transport of energy induced by three-wave resonantcoupling among low-frequency large-scale MHD Alfvenwaves and small-scale kinetic Alfven waves. The nonlinearcoupling coefficients and the growth rate are calculated forthe resonant decay of a large-scale MHD Alfven wave intopairs of coupled KAWs, AW ! KAW1 + KAW2. Bothelectron and ion nonlinear dynamics turn out to be essentialfor the nonlinear coupling, resulting in a jump-like transportof MHD wave energy directly in the dissipation rangeformed by KAWs.[43] Several important advantages of this nonlocal trans-

port are revealed: (1) it is not restricted to any specificwavelengths or frequencies of initial MHD Alfven waves,(2) it does not require the preexistence of counterpropagat-ing waves or plasma nonuniformities, (3) it is efficient forvery small amplitudes of MHD waves. In uniform plasmasthere is the restriction that the gas/magnetic pressure ratio bshould be below the electron/ion mass ratio, bm � bmp /me <1 (we assumed the protons to be the dominant ion species,mi = mp). However, in a plasma that is nonuniform acrossthe magnetic field, the Alfven wave can propagate in formof surface waves, and the restriction bm < 1 is relaxed in theregions where the frequency of the surface wave overcomesthe local Alfven frequency and the resonant KAWs can benonlinearly excited.[44] The main conclusion is that the resonant decay can

strongly accelerate the spectral evolution of wave energyfrom MHD length scales to dissipative short length scales ina variety of space plasmas. In particular, our results indicatethat the flux of MHD Alfven waves can energize plasma bymeans of nonlinearly driven KAWs in the solar corona atheights 1.3 � 5 solar radii and in the auroral zones of theEarth magnetosphere at heights 1.3 � 5 Earth radii. Thenonlinear excitation of KAWs in the extended solar coronacan be caused by upward propagating Alfven MHD waves,launched from the convection zone or generated by mag-netic reconnection events at the coronal base. Short trans-versal wavelengths of the order 10 m make KAWsaccessible for the nonadiabatic acceleration of oxygen ionsif the KAW/background magnetic field ratio exceeds 0.005.In the terrestrial magnetosphere, a large flux of MHDAlfven waves is generated by magnetic reconnection inthe magnetotail and propagates downward along auroralfield lines. The wave amplitudes are sufficient to power theenergy release in the auroral zones by means of nonlinearlyexcited KAWs.[45] The eventual energy release is produced by KAWs

and is distributed between the field-aligned degree offreedom for the electrons and the cross-field degree offreedom for the ions [Voitenko and Goossens, 2004]. Withhigh oblique wave numbers, the electric field of nonlinearly

driven KAWs has important components parallel to theequilibrium magnetic field, Ek k B0, and parallel to thecross-field wave vector, E? k k?. Because of the parallelelectric field the KAWs undergo the Cherenkov resonancewith the electrons and heat and accelerate them along B0.This process can explain the high electron energy in theparallel (field-aligned) degree of freedom observed in aurora[Wygant et al., 2002]. On the other hand, the perpendicularelectric field E? k k? leads to a strong cross-fieldacceleration of the ions which occurs in the vicinity ofdemagnetizing KAW phases [Voitenko and Goossens,2004]. This process provides an alternative to the ion-cyclotron explanation for the intense cross-field energiza-tion of ions observed in the solar corona [Dodero et al.,1998] and in the aurora [Norqvist et al., 1998].

[46] Acknowledgments. This work was supported by the FWOVlaanderen (research project G.0178.03) and by Onderzoeksfonds K. U.Leuven (research project OT/02/57).[47] Shadia Rifai Habbal thanks both referees for their assistance in

evaluating this paper.

ReferencesBanaszkiewicz, M., W. I. Axford, and J. F. McKenzie (1998), An analyticsolar magnetic field model, Astron. Astrophys., 337, 940.

Banerjee, D., L. Teriaca, J. G. Doyle, and K. Wilhelm (1998), Broadeningof SI VIII lines observed in the solar polar coronal holes, Astron. Astro-phys., 339, 208.

Chaston, C. C., J. W. Bonnell, C. W. Carlson, J. P. McFadden, R. E. Ergun,and R. J. Strangeway (2003), Properties of small-scale Alfven waves andaccelerated electrons from FAST, J. Geophys. Res., 108(A4), 8003,doi:10.1029/2002JA009420.

Coles, W. A., R. R. Grall, M. T. Klinglesmith, and G. Bourgois (1995),Solar cycle changes in the level of compressive microturbulence near theSun, J. Geophys. Res., 100, 17,069–17,080.

Cranmer, S. R. (2002), Coronal holes and the high-speed solar wind, SpaceSci. Rev., 101, 229.

Cranmer, S. R. (2004), Observational aspects of wave acceleration in openmagnetic regions, in SOHO 13: Waves, Oscillations and Small-ScaleTransients Events in the Solar Atmosphere: A Joint View from SOHOand TRACE, ESA SP-547, edited by H. Lacoste, pp. 353–362, Eur. SpaceAgency, Noordwijk, Netherlands.

De Pontieu, B., P. C. H. Martens, and H. S. Hudson (2001), Chromosphericdamping of Alfven waves, Astrophys. J., 558, 859–871.

Dmitruk, P., and W. Matthaeus (2003), Low-frequency waves and turbu-lence in an open magnetic region: Timescales and heating efficiency,Astrophys. J., 597, 1097.

Dmitruk, P., W. H. Matthaeus, L. J. Milano, and S. Oughton (2001), Con-ditions for sustainment of magnetohydrodynamic turbulence driven byAlfven waves, Phys. Plasmas, 8, 2377.

Dodero, M. A., E. Antonucci, S. Giordano, and R. Martin (1998), Solarwind velocity and anisotropic coronal kinetic temperature measured withthe O VI doublet ratio, Solar Phys., 183, 77.

Esser, R., et al. (1999), Plasma properties in coronal holes derived frommeasurements of minor ion spectral lines and polarized white light in-tensity, Astrophys. J., 510, L63.

Goossens, M. (1994), Alfven wave heating, Space Sci. Rev., 68, 51.Grall, R. R., W. A. Coles, S. R. Spangler, T. Sakurai, and J. K. Harmon(1997), Observations of field-aligned density microstructure near the Sun,J. Geophys. Res., 102, 263.

Hollweg, J. V. (1986), Transition region, corona, and solar wind in coronalholes, J. Geophys. Res., 91, 4111.

Hollweg, J. V. (1999), Kinetic Alfven waves revisted, J. Geophys. Res.,104, 14,811.

Hollweg, J. V., and P. A. Isenberg (2002), Generation of the fast solar wind:A review with emphasis on the resonant cyclotron interaction, J. Geo-phys. Res., 107(A7), 1147, doi:10.1029/2001JA000270.

Kletzing, C. A., S. R. Bounds, J. Martin-Hiner, W. Gekelman, andC. Mitchell (2003) , Measurements of the shear Alfven wave dispersionfor finite perpendicular wave number, Phys. Rev. Lett., 90, 035004.

Kohl, J. L., et al. (1997), First results from the SOHO Ultraviolet Corona-graph Spectrometer, Solar Phys., 175, 613.

Leamon, R. J., W. H. Matthaeus, C. W. Smith, G. P. Zank, and D. Mullan(2000), MHD-driven kinetic dissipation in the solar wind and corona,Astrophys. J., 537, 1054.

A10S01 VOITENKO AND GOOSSENS: NONLINEAR EXCITATION OF KAWS

10 of 11

A10S01

Leer, E., T. E. Holzer, and T. Fla (1982), Acceleration of the solar wind,Space Sci. Rev., 33, 161.

Leneman, D., W. Gekelman, and J. Maggs (1999), Laboratory observationsof shear Alfven waves launched from a small source, Phys. Rev. Lett., 82,2673.

Lysak, R. L., and W. Lotko (1996), On the kinetic dispersion relation forshear Alfven waves, J. Geophys. Res., 101, 5085.

Markovskii, S. A. (2001), Generation of ion cyclotron waves in coronalholes by a global resonant magnetohydrodynamic mode, Astrophys. J.,557, 337.

Markovskii, S. A., and J. V. Hollweg (2002), Parametric cross-field currentinstability in solar coronal holes, J. Geophys. Res., 107(A10), 1329,doi:10.1029/2001JA009140.

Marsch, E., and C. Y. Tu (2001), Evidence for pitch angle diffusion of solarwind protons in resonance with cyclotron waves, J. Geophys. Res., 106,8357.

Maslennikov, D. I., V. S. Mikhailenko, and K. N. Stepanov (1995), Non-linear theory of ion-cyclotron parametric decay instability of a plasmawith ions of one species, Plasma Phys. Rep., 21, 749.

Matthaeus, W. H., S. Oughton, S. Ghosh, and M. Hossain (1998),Scaling of anisotropy in hydromagnetic turbulence, Phys. Rev. Lett.,81, 2056.

Matthaeus, W. H., D. J. Mullan, P. Dmitruk, L. J. Milano, and S. Oughton(2003), MHD turbulence and heating of the open field-line solar corona,Nonlin. Proc. Geophys., 10, 93.

Moran, T. G. (2003), Test for Alfven wave signatures in a solar coronalhole, Astrophy. J., 598, 657.

Norqvist, P., M. Andre, and M. Tyrland (1998), A statistical study of ionenergization mechanisms in the auroral region, J. Geophys. Res., 103,23,459.

Onishchenko, O. G., O. A. Pokhotelov, R. Z. Sagdeev, L. Stenflo, R. A.Treumann, and M. A. Balikhin (2004), Generation of convective cells bykinetic Alfven waves in the upper ionosphere, J. Geophys. Res., 109,A03306, doi:10.1029/2003JA010248.

Ryutova, M., S. Habbal, R. Woo, and T. Tarbell (2001), Photospheric net-work as the energy source for the quiet-Sun corona, Solar Phys., 200,213.

Sagdeev, R. Z., and A. A. Galeev (1969), Nonlinear Plasma Theory, W. A.Benjamin, New York.

Seyler, C. E., and K. Wu (2001), Instability at the electron inertial scale,J. Geophys. Res., 106, 21,623.

Stasiewicz, K., et al. (2000), Small scale Alfvenic structure in the aurora,Space Sci. Rev., 92, 423.

Stasiewicz, K., Y. Khotyaintsev, and M. Grzesiak (2004), DispersiveAlfven waves observed by Cluster at the magnetopause, Phys. Scr. T,107, 171.

Tu, C.-Y., and E. Marsch (1997), Two-fluid model for heating of the solarcorona and acceleration of the solar wind by high-frequency Alfvenwaves, Solar Phys., 171, 363.

Ulrich, R. K. (1996), Observations of magnetohydrodynamic oscillations inthe solar atmosphere with properties of Alfven waves, Astrophys. J., 465,436.

Voitenko, Y. M. (1998a), Three-wave coupling and parametric decay ofkinetic Alfven waves, J. Plasma Phys., 60, 497.

Voitenko, Y. M. (1998b), Three-wave coupling and weak turbulence ofkinetic Alfven waves, J. Plasma Phys., 60, 515.

Voitenko, Y., and M. Goossens (2000), Nonlinear decay of phase-mixedAlfven waves in the solar corona, Astron. Astrophys., 357, 1073.

Voitenko, Y. M., and M. Goossens (2002a), Excitation of high-frequencyAlfven waves by plasma outflows from coronal reconnection events,Solar Phys., 206, 285.

Voitenko, Y. M., and M. Goossens (2002b), Nonlinear excitation of small-scale Alfven waves by fast waves and plasma heating in the solar atmo-sphere, Solar Phys., 209, 37.

Voitenko, Y. M., and M. Goossens (2003), Kinetic excitation mechanismsfor ion-cyclotron kinetic Alfven waves in Sun-Earth connection, SpaceSci. Rev., 107, 387.

Voitenko, Y. M., and M. Goossens (2004), Cross-field heating of coronalions by low-frequency kinetic Alfven waves, Astrophys. J., 605, L149.

Voitenko, Y. M., and M. Goossens (2005), Cross-scale nonlinear couplingand plasma energization by Alfven waves, Phys. Rev. Lett., 94, 135003.

Voitenko, Y. M., A. N. Kryshtal, P. P. Malovichko, and A. K. Yukhimuk(1990), Current instability and generation of kinetic Alfven wavesin the magnetosphere of the Earth (in Russian), Geomagn. Aeron.,30, 402.

Wilhelm, K., et al. (1998), The solar corona above polar coronal holes asseen by SUMER on SOHO, Astrophy. J., 500, 1023.

Wygant, J. R., et al. (2002), Evidence for kinetic Alfven waves and parallelelectron energization at 4–6 RE altitudes in the plasma sheet boundarylayer, J. Geophys. Res., 107(A8), 1201, doi:10.1029/2001JA900113.

�����������������������M. Goossens and Y. M. Voitenko, Centre for Plasma Astrophysics,

K.U. Leuven, Celestijnenlaan 200B, B-3001 Heverlee, Belgium. (yuriy.voitenko@wis.kuleuven.ac.be)

A10S01 VOITENKO AND GOOSSENS: NONLINEAR EXCITATION OF KAWS

11 of 11

A10S01