ION HEATING BY A SPECTRUM OF OBLIQUELY PROPAGATING …space.ustc.edu.cn/users/1166164509JDEkSVp1eWMzZ24... · 2 0 = 0.02, (b) B2 k /B 2 0 = 0.067, and (c) B 2 k /B 0 = 0.08. (a) (b)

The Astrophysical Journal, 704:743–749, 2009 October 10 doi:10.1088/0004-637X/704/1/743C© 2009. The American Astronomical Society. All rights reserved. Printed in the U.S.A.

ION HEATING BY A SPECTRUM OF OBLIQUELY PROPAGATING LOW-FREQUENCY ALFVEN WAVES

Quanming Lu1

and Liu Chen2,3

1 CAS Laboratory of Basic Plasma Physics, School of Earth and Space Sciences, University of Science and Technology of China, Hefei 230026, China2 Institute for Fusion and Simulation, Zhejiang University, Hangzhou 310027, China

3 Department of Physics and Astronomy, University of California, Irvine, CA 92697, USAReceived 2009 May 5; accepted 2009 August 31; published 2009 September 24

ABSTRACT

Ion stochastic heating by a monochromatic Alfven wave, which propagates obliquely to the background magneticfield, has been studied by Chen et al. It is shown that ions can be resonantly heated at frequencies a fraction ofthe ion cyclotron frequency when the wave amplitude is sufficiently large. In this paper, the monochromaticwave is extended to a spectrum of left-hand polarized Alfven waves. When the amplitude of the waves issmall, the components of the ion velocity have several distinct frequencies, and their motions are quasi-periodic.However, when the amplitude of the waves is sufficiently large, the components of the ion velocity have aspectrum of continuous frequencies near the ion cyclotron frequency due to the nonlinear coupling betweenthe Alfven waves and the ion gyromotion, and the ion motions are stochastic. Compared with the case of amonochromatic Alfven wave, the threshold of the ion stochastic heating by a spectrum of Alfven waves ismuch lower. Even when their frequencies are only several percent of the ion cyclotron frequency, the ionscan also be stochastically heated. The relevance of this heating mechanism to solar corona is also discussed.

Key words: solar wind – Sun: corona – waves

1. INTRODUCTION

Alfven waves are considered to play a crucial role in heatingof plasmas, such as the solar corona and magnetic fusion devices(Nekrasov 1970; Hollweg 1978; Lieberman & Lichtenberg1973; Karney 1979; Abe et al. 1984). Numerous theoretical andexperimental papers have been published to investigate resonantheating of ions by Alfven waves (Isenberg & Hollweg 1983;Cranmer et al 1999; Li et al. 1999; Tu & Marsch 2001; Lu et al.2006a, 2006b). In these works, the cyclotron resonant condition(ω − k||v|| = nΩ0, where n is an integer, ω and k are thefrequencies and wavevectors of the Alfven waves, respectively.v is the particle velocity in the laboratory frame. The subscript“‖” denotes the component parallel to the background magneticfield and Ω0 is the cyclotron frequency of the particle) isnecessary for ion heating by the Alfven waves, and in generalthe frequencies of the applied Alfven waves are comparableto the cyclotron frequency. Recently, Chen et al. (2001) foundthat an obliquely propagating Alfven wave with sufficientlylarge amplitude can break the magnetic moment invariant atfrequencies a fraction of the ion cyclotron frequency, and thusion stochastic heating by such sub-cyclotron resonance at lowfrequencies is possible. Guo et al. (2008) further pointed out thatthe ion heating occurs when the cyclotron resonances at sub-cyclotron frequencies start to overlap with their correspondingneighboring resonances and then leads to global stochasticity.When a spectrum of Alfven waves is considered, White et al.(2002) demonstrated that the amplitude threshold of the Alfvenwaves for ion stochastic heating can be significantly decreased.Evidences of ion heating by low-frequency Alfven waves havealso been found in laboratory experiments (Gates et al. 2001;Fredrickson et al. 2002; Zhang et al. 2008).

In the present paper, with test particle calculations we inves-tigate ion stochastic heating by a spectrum of obliquely propa-gating Alfven waves with left-hand polarization. By analyzingthe spectrum of the components of the ion velocity as well asthe evolution of the ion temperature, a more precise determina-

tion of the threshold for ion stochastic heating for a spectrum ofoblique Alfven waves is investigated.

The paper is organized as follows. In Section 2, the simulationmodel is presented, and the simulation results are described inSection 3. In Section 4, we discuss and summarize our results.

2. SIMULATION MODEL

A spectrum of left-hand circularly polarized Alfven waves isconsidered in this paper, and the waves propagate obliquely tothe background magnetic field. The dispersion relation of theAlfven waves is ω = kzvA, where vA = B0

/(4πn0mi)1/2 is the

Alfven speed and the background magnetic field is B0 = B0iz.Thus, in the wave frame we have the wave magnetic field (Chenet al. 2001)

Bw =N∑

k=1

Bk[−cos(α) sin(ψk)ix +cos(ψk)iy +sin(α) sin(ψk)iz],

(1)where ψk = kxx + kzz + ϕk , tan(α) = kx

/kz, and ϕk is the

random phase for mode k. N is the number of wave modes.The particles move in the magnetic field as described by thefollowing equations:

mi

dv

dt= qiv × (B0 + Bw), (2)

dr

dt= v, (3)

where the subscript i indicates physical quantities associatedwith ion species i. In this paper, we consider particle motionsin the wave frame and the wave electric field is eliminated. Theequations are solved with Boris algorithm (Birdsall & Langdon,2005), where the kinetic energy of the particle is conserved in thecalculation of cyclotron motion. The time step is Ωpt = 0.025,where Ωp is the proton cyclotron frequency.

743

744 LU & CHEN Vol. 704

Figure 1. Poincare plot for a monochromatic Alfven wave, and the parameters are ω = 0.25, α = 45◦, and (a) B2k /B2

0 = 0.02, (b) B2k /B2

0 = 0.067, and (c)B2

k /B20 = 0.08.

(a) (b) (c)

Figure 2. Power spectrum of the x component of the ion velocity vx (t), which is obtained by FFT the time series of vx (t) from Ωpt = 0 to Ωpt = 26214.4. Theparameters are the same as in Figure 1.

3. SIMULATION RESULTS

In the laboratory frame the waves propagate in the positive zdirection, whose phase velocity is equal to the Alfven speed vA.Therefore, in the wave frame an initially cold ion distributionhas velocity vx = 0, vy = 0, and vz = −vA. In this paper,we investigate the effects of the number of wave modes on ionheating by the oblique Alfven waves. At first, we show theresults for the monochromatic wave, and then the effects of thenumber of wave modes on ion heating are investigated

3.1. The Monochromatic Alfven Wave

To study ion heating, a Poincare plot of λ = vz/vA, ψB =cos(kxx + kzz + ϕk), formed by taking points when vy = 0and vy > 0, is constructed in the wave frame. Poincare plot isthe intersection of an orbit in the state space of a continuousdynamical system with a certain lower dimensional subspace,called the Poincare section. Poincare plot preserves manyproperties of orbits of the original dynamical system and hasa lower dimensional state space. Therefore, it is a useful toolto analyze the properties of a dynamical system (Lieberman& Lichtenberg, 1983). Figure 1 describes the Poincare plot fora monochromatic Alfven wave, and the parameters are ω =0.25Ωp, α = 45◦, and (a) B2

k

/B2

0 = 0.02, (b) B2k

/B2

0 = 0.067,and (c) B2

k

/B2

0 = 0.08. With the increase of the wave amplitude,the motions of the ion become stochastic due to the resonancewith the wave at sub-cyclotron frequencies, and the thresholdis about B2

k

/B2

0 = 0.067 (correspondingly, Bk

/B0 ≈ 0.259,

which is consistent with the results of Chen et al. 2001). Whenthe amplitude is B2

k

/B2

0 = 0.08, the ion can readily diffuse fromvz = −vA to about vz = 0.7vA, and its motions are stochastic.

The ion stochastic motions can also be demonstrated inFigure 2, which shows the power spectrum of the x componentof the ion velocity vx(t). We first obtain a time series of vx(t)from Ωpt = 0 to Ωpt = 26214.4 by solving Equations (2)and (3), and then calculate its power spectrum by Fast FourierTransform (FFT) the time series of vx(t). The parameters arethe same as in Figure 1. When the wave amplitude is small(B2

k

/B2

0 = 0.02), the ion motions have two main frequencies:one is near 0.25Ωp and the other is near 1.0Ωp. These twofrequencies correspond to the frequencies of the Alfven waveand ion gyromotion, respectively, and the ion motions are quasi-periodic. With the increase of the wave amplitude, the ionmotions have more and more distinct frequencies. When thewave amplitude is sufficiently large, the ion motions have acontinuous frequency spectrum, which is stochastic. For the caseB2

k

/B2

0 = 0.08, the main frequencies are concentrated between0.8Ωp and 1.3Ωp with a continuous spectrum. Consistent withthe results of Poincare plot and the plot of x–vx , the thresholdof the ion stochastic motions is around B2

k

/B2

0 = 0.067.Figure 3 shows the time evolution of the parallel and per-

pendicular temperatures for the wave amplitude B2k

/B2

0 = 0.02and B2

k

/B2

0 = 0.08. In the figure, A and B denote T||/T||0

and T⊥/T⊥0 for amplitude B2

k

/B2

0 = 0.02, while C and Ddenote T||

/T||0 and T⊥

/T⊥0 for amplitude B2

k

/B2

0 = 0.08.Here, the subscript “0” stands for the initial values of phys-ical quantities. Initially, particles are evenly distributed in aregion with size 24π × 24π in the z − x plane, and the re-gion is divided into 48 × 48 grids. The thermal velocity of theparticles is 0.1vA, and there is no drift velocity in the labora-tory frame. The total number of particles is 230,400. Doubleperiodic boundary conditions are used for the particles: if one

No. 1, 2009 ION HEATING BY LOW-FREQUENCY ALFVEN WAVES 745

Figure 3. Time evolution of the parallel and perpendicular temperatures for thewave amplitude B2

k /B20 = 0.02 and B2

k /B20 = 0.08. A monochromatic Alfven

wave with ω = 0.25, α = 45◦ is used. In the figure, A and B denote T||/T||0and T⊥/T⊥0 for amplitude B2

k /B20 = 0.02, while C and D denote T||/T||0 and

T⊥/T⊥0 for amplitude B2kB

20 = 0.08.

moves out of one boundary, it will enter from the oppositeboundary. In this and the following subsections, we calculatethe parallel and perpendicular temperatures using the follow-ing procedure: we first calculate T|| = mi/kB〈(vz − 〈vz〉)2〉,T⊥ = mi/2kB〈(vx − 〈vx〉)2 + (vy −〈vy〉)2〉 in every grid (wherethe bracket 〈•〉 denotes an average over a grid cell), and thenthe temperatures are averaged over all grids. In this way, wecan eliminate the effects of the average velocity at each loca-tion on the thermal temperature. From the figure, we can findthat ions can be rapidly heated before Ωpt ≈ 100, and suchmechanism to heat the ions is due to the phase mixing betweenions, which has been discussed in Lu & Li (2007) and Li et al.(2007). In the process of phase mixing, ions are first picked upin the transverse direction by the Alfven wave and obtain anaverage transverse velocity, then the parallel thermal motions ofions produce phase mixing (randomization) among ions leadingto ion heating. After the phase mixing, the ions are stochasti-cally heated by the mechanism discussed in this paper for theamplitude B2

k

/B2

0 = 0.08, and the perpendicular temperature ismuch larger than the parallel temperature. The ions have largetemperature anisotropy, which may excite ion cyclotron waves(Gary et al. 2003; Lu et al. 2006a, 2006b; Lu & Wang 2006).For the amplitude B2

k

/B2

0 = 0.02, which is smaller than thethreshold of the stochastic heating, there is no further heating.

3.2. The Alfven Waves with Two Wave Modes

Here we consider the Alfven waves with two wave modes,and their frequencies are 0.25Ωp and 0.33Ωp, respectively. Theypropagate along the same direction with α = 45◦. The randomphases of these two modes are 0◦ and 30◦, respectively, and theratio of their amplitudes is B2

k2

/B2

k1 = 0.6295. With a spectrumconsisting of waves with different frequencies, a Poincareplot cannot be used to investigate the ion stochastic heating.However, similar to the Poincare plot for the monochromaticAlfven wave, we still can construct a plot of λ = vz/vA,ψB = (

∑k Bk cos ψk)/

∑k Bk , by taking points when vy = 0

and vy > 0 in the wave frame. Figure 4 shows such plotfor (a)

∑k B2

k

/B2

0 = 0.01, (b)∑

k B2k

/B2

0 = 0.018, and (c)∑k B2

k

/B2

0 = 0.04. Similar to Figure 1, which describes theresults of the monochromatic wave case, with the increaseof the wave amplitude, the z component of the ion velocitycan be diffused to a large value. For example, the ion can bediffused from vz = −vA to about vz = −0.95vA, −0.6vA,and 0.7vA for the wave amplitude

∑k B2

k

/B2

0 = 0.01, 0.018,and 0.04, respectively. The maximum value of vz which theion can be diffused to by the Alfven waves increases abruptlyaround the amplitude

∑k B2

k

/B2

0 = 0.018. Therefore, we cansuppose that the threshold for the ion stochastic heating is about∑

k B2k

/B2

0 = 0.018.Figure 5 shows the power spectrum of the x component of

the ion velocity vx(t), which is obtained by FFT the time seriesof vx(t) from Ωpt = 0 to Ωpt = 26214.4, as in Figure 2. Theparameters are the same as in Figure 4. When the wave amplitudeis small (

∑k B2

k

/B2

0 = 0.01), the ion motions have three mainfrequencies, and they are near 0.25Ωp, 0.33Ωp, and 1.0Ωp,respectively. The former two frequencies correspond to thefrequencies of the Alfven waves, and the last one corresponds tothat of the ion gyromotion. The ion motions are quasi-periodic.When the wave amplitude is sufficiently large, the ion motionshave a continuous frequency spectrum, and they are stochastic.Around

∑kB

2k

/B2

0 = 0.018, the spectrum becomes continuous,and the ion motions are stochastic. When

∑kB

2k

/B2

0 = 0.04,the continuous spectrum of the ion motions extends from 0.75Ωpto 1.3Ωp.

Figure 6 shows the time evolution of the parallel and perpen-dicular temperatures for the wave amplitude

∑k B2

k

/B2

0 = 0.01and

∑k B2

k

/B2

0 = 0.04. In the figure, A and B denote T||/T||0 andT⊥/T⊥0 for amplitude

∑k B2

k

/B2

0 = 0.01, while C and D denoteT||/T||0 and T⊥/T⊥0 for amplitude

∑k B2

k

/B2

0 = 0.04. The initial

Figure 4. Plot of λ = vz/vA, ψB = (∑

k Bk cos ψk)/∑

k Bk for (a)∑

k B2k

/B2

0 = 0.01, (b)∑

k B2k

/B2

0 = 0.018, and (c)∑

k B2k

/B2

0 = 0.04, by taking points whenvy = 0 and vy > 0 in the wave frame. The Alfven waves have two wave modes, and their frequencies are 0.25Ω0 and 0.33Ω0, respectively. They propagate along thesame direction with α = 45◦. The random phases of these two modes are 0◦ and 30◦, respectively.

746 LU & CHEN Vol. 704

(a) (b) (c)

Figure 5. Power spectrum of the x component of the ion velocity vx (t), which is obtained by FFT the time series of vx (t) from Ωpt = 0 to Ωpt = 26214.4. Theparameters are the same as in Figure 4.

Figure 6. Time evolution of the parallel and perpendicular temperatures forthe wave amplitude

∑k B2

k /B20 = 0.01 and

∑k B2

k /B20 = 0.04. The Alfven

waves have two wave modes, and their frequencies are 0.25Ω0 and 0.33Ω0,respectively. They propagate along the same direction with α = 45◦. Therandom phases of these two modes are 0◦ and 30◦, respectively. In the figure,A and B denote T||/T||0 and T⊥/T⊥0 for amplitude

∑k B2

k /B20 = 0.01, while C

and D denote T||/T||0 and T⊥/T⊥0 for amplitude∑

k B2k /B2

0 = 0.04.

and boundary conditions are the same as discussed in the abovesubsection. Similar to the case with the monochromatic Alfvenwave, we can find that ions can be rapidly heated by phase mix-ing before Ωpt ≈ 100. Then the ions are stochastically heatedby the mechanism for the amplitude

∑k B2

k

/B2

0 = 0.04, andthey have temperature anisotropy with the perpendicular tem-perature much larger than the parallel temperature. However,for the amplitude

∑k B2

k

/B2

0 = 0.01, there is no further heat-ing after Ωpt ≈ 100.

We also consider the effects of the amplitude ratio B2k2

/B2

k1

on ion motions by keeping∑

k B2k

/B2

0 as a constant. Figure 7shows a plot of λ = vz

/vA, ψB = (

∑k Bk cos ψk)/

∑k Bk , by

taking points when vy = 0 and vy > 0 in the wave frame for (a)B2

k2

/B2

k1 = 0.1, (b) B2k2

/B2

k1 = 0.17, and (c) B2k2

/B2

k1 = 0.4,while

∑k B2

k

/B2

0 is kept as 0.04. With the increase of theamplitude B2

k2, the ion motions tend to be stochastic. AroundB2

k2

/B2

k1 = 0.17, the ion motions become stochastic, and the zcomponent of the ion velocity can be diffused from vz = −vA toabout vz = 0.6vA. When B2

k2 approaches to B2k1, the stochasticity

of the ion motions increases. For B2k2

/B2

k1 = 0.4, vz can bediffused from vz = −vA to about vz = 0.7vA. This can also bedemonstrated in Figure 8, which describes the power spectrumof the x component of the ion velocity vx(t), which is obtained by

FFT the time series of vx(t) from Ωpt = 0 to Ωpt = 26214.4, asin Figure 2. The parameters are the same as in Figure 7. When theamplitude ratio is small (B2

k2

/B2

k1 = 0.1), the ion motions haveseveral distinct frequencies. The main frequencies concentrateon about 0.25Ωp and 1.0Ωp, respectively, which correspond tothe main frequency of the Alfven waves and the frequency ofthe ion gyromotion. The ion motions are quasi-periodic. Theion motions begin to have a continuous frequency spectrum forabout B2

k2

/B2

k1 = 0.17, and become stochastic.

3.3. The Alfven Waves with a Spectrum

In order to investigate the effects of the number of wave modeson the ion motions, we keep α = 45◦ and

∑k B2

k

/B2

0 = 0.013.The frequencies of the waves extend from ω1 = 0.25Ωp toωN = 0.33Ωp, and N is the number of wave modes used inour calculations. The frequencies of the wave modes can becalculated as follows: ωj = ω1 + (j − 1)Δω (j = 1, 2, . . . , N ),where Δω = (ωN − ω1)/(N − 1). The amplitude of individualwave modes satisfies the relation (Bj/B1)2 = (ωj/ω1)−q , andq is chosen as 1.667. This means that the power spectrumof the Alfven waves has an index of −1.667, which is agenerally accepted value for the power spectrum of magneticfluctuations found in the solar wind (Villante 1980; Bavassano& Smith 1986). Figure 9 constructs a plot of λ = vz/vA,ψB = (

∑k Bk cos ψk)/

∑k Bk , by taking points when vy = 0

and vy > 0 in the wave frame for (a) N = 1, (b) N = 2, (c)N = 5, and (d) N = 21. With the increase of wave modes,the ion motions become stochastic. From the figure, we canfind that the ion motions for (c) N = 5 and (d) N = 21 arestochastic. This can also be verified in Figure 10, which showsthe power spectrum of the x component of the ion velocity vx(t)for (a) N = 1, (b) N = 2, (c) N = 5, and (d) N = 21. Thepower spectrum is obtained by FFT the time series of vx(t) fromΩpt = 0 to Ωpt = 26214.4, as in Figure 2. For (a) N = 1 and (b)N = 2, the ion motions are quasi-periodic. It has several distinctfrequencies, which correspond to that of the Alfven wave modesand the ion gyromotion. For (c) N = 5 and (d) N = 21, theion motions which concentrate near Alfven wave frequencies arevery weak, while the motions near its gyromotion are very strongand its frequencies have a continuous spectrum. Therefore, itsmotions are stochastic. We also find that if we further increasethe number of the wave modes, there is no obvious difference.

Figure 11 shows the time evolution of the parallel and per-pendicular temperatures for different numbers of wave modes.In the figure, A and B denote T||/T||0 and T⊥/T⊥0 for the num-ber of wave modes N = 2, while C and D denote T||/T||0 andT⊥/T⊥0 for the number of the wave modes N = 5. The other

No. 1, 2009 ION HEATING BY LOW-FREQUENCY ALFVEN WAVES 747

Figure 7. Plot of λ = vz

/vA, ψB = (

∑k Bk cos ψk)/

∑k Bk , by taking points when vy = 0 and vy > 0 in the wave frame for (a) B2

k2

/B2

k1 = 0.1 (b) B2k2

/B2

k1 = 0.17,and (c) B2

k2

/B2

k1 = 0.4, while∑

k B2k

/B2

0 is kept as 0.04. The Alfven waves have two wave modes, and their frequencies are 0.25Ω0 and 0.33Ω0, respectively. Theypropagate along the same direction with α = 45◦. The random phases of these two modes are 0◦ and 30◦, respectively.

(a) (b) (c)

Figure 8. Power spectrum of the x component of the ion velocity vx (t), which is obtained by FFT the time series of vx (t) from Ωpt = 0 to Ωpt = 26214.4. Theparameters are the same as in Figure 7.

Figure 9. Plot of λ = vz

/vA, ψB = (

∑k Bk cos ψk)/

∑k Bk , by taking points when vy = 0 and vy > 0 in the wave frame for (a) N = 1, (b) N = 2, (c) N = 5, and

(d) N = 21. We keep α = 45◦, and∑

k B2k

/B2

0 = 0.013. The frequencies of the waves extend from ω1 = 0.25Ω0 to ωN = 0.33Ω0.

(a) (b) (c) (d)

Figure 10. Power spectrum of the x component of the ion velocity vx (t) for (a) N = 1, (b) N = 2, (c) N = 5, and (d) N = 21. The power spectrum is obtained byFFT the time series of vx (t) from Ωpt = 0 to Ωpt = 26214.4. The parameters are the same as in Figure 9.

748 LU & CHEN Vol. 704

Figure 11. Time evolution of the parallel and perpendicular temperatures fordifferent numbers of wave modes. In the figure, A and B denote T||/T||0 andT⊥/T⊥0 for the number of wave modes N = 2, while C and D denote T||/T||0and T⊥/T⊥0 for the number of the wave modes N = 5. The other parametersare the same as in Figure 9.

parameters are the same as in Figure 9. The initial and boundaryconditions are the same as discussed in the above subsections.Similar to the cases in the above subsections, we can find thations can be rapidly heated by phase mixing before Ωpt ≈ 100.Then, for N = 5, the ions are stochastically heated with largetemperature anisotropy. However, for N = 2, no further heatingcan be found after Ωpt ≈ 100.

Figure 12 shows that the threshold of ion stochastic motionsfor the different number of wave modes and frequencies, (a)is for α = 45◦ and (b) for α = 60◦. The range for thefrequencies is kept as ωN − ω1 = 0.08Ω0. In general, withthe increase of the number of wave modes, the thresholddecreases, especially for the low-frequency waves. Therefore,if a spectrum of waves is considered, the threshold can bemuch lower than that of a monochromatic wave. When thenumber of wave modes is sufficiently large, there is no obviousdifference. With the increase of the wave frequencies and thepropagation angle α, the threshold of ion stochastic motionsdecreases. In summary, if a spectrum of Alfven waves isconsidered, the threshold of ion stochastic motions can be muchlower than that of the monochromatic wave. For example, forω1= 0.05Ω0, N = 21, α = 45◦, the threshold of amplitudecan be as low as

∑k B2

k

/B2

0 = 0.052. We can also find thatwith the increase of the propagation angle α, the ions can bestochastically heated with more efficiency. Figure 13 shows thetime evolution of the parallel and perpendicular temperatures

Figure 13. Time evolution of the parallel and perpendicular temperatures fordifferent propagation angle α, the amplitude of Alfven waves is

∑k B2

k /B20 =

0.028 with the number of the waves modes N = 2, and the other parametersare the same as in Figure 9. In the figure, A and B denote T||/T||0 and T⊥/T⊥0for the propagation angle α = 45◦, while C and D denote T||/T||0 and T⊥/T⊥0for the propagation angle α = 60◦.

for different propagation angle α, the amplitude of Alfvenwaves is

∑k B2

k

/B2

0 = 0.028 with the number of the wavesmodes N = 2, and the other parameters are the same as inFigure 9. In the figure, A and B denote T||/T||0 and T⊥/T⊥0 forthe propagation angle α = 45◦, while C and D denote T||/T||0and T⊥/T⊥0 for the propagation angle α = 60◦. Obviously, theheating of the ions is more efficient for the propagation angleα = 60◦.

4. SUMMARY AND DISCUSSION

Extended from the paper of Chen et al. (2001), we investigatethe ion stochastic heating by a spectrum of low-frequencyAlfven waves, which are left-hand polarized and propagateobliquely to the background magnetic field. The results showthat when a spectrum of Alfven waves is considered, thethreshold of ion stochastic heating is much lower than that ofthe monochromatic wave. Even when the frequencies of Alfvenwaves are several percent of the ion cyclotron frequency, ionsmay also be stochastically heated. When the amplitude of thewaves is sufficiently large, the velocity of the ion stochasticmotions has a continuous spectrum of frequencies near theion cyclotron frequency due to the nonlinear coupling betweenthe ion gyromotion and the Alfven waves, which leads to ionstochastic motions.

(a) (b)

Figure 12. Threshold of ion stochastic motions for the different number of wave modes and frequencies, (a) is for α = 45◦ and (b) for α = 60◦. The range for thefrequencies is kept as ωN − ω1 = 0.08Ω0.

No. 1, 2009 ION HEATING BY LOW-FREQUENCY ALFVEN WAVES 749

Low-frequency Alfven waves are thought to be pervasive inthe solar corona (Belcher et al. 1969; Tu & Marsch 2001). Theenergy of these low-frequency Alfven waves can be transferredto that of higher frequencies by the perpendicular cascade inincompressible MHD turbulence (see, e.g., Cranmer & van Bal-legooijen 2003 and references therein), or by three-wave inter-actions in compressible MHD turbulence with the existence offast magnetosonic waves (Chandran 2005). The perpendicularcascade can lead to Alfven waves to have wavevectors perpen-dicular to the background magnetic field. Then, the mechanismsof ion stochastic heating discussed in this paper may have rele-vance with solar corona heating.

The authors acknowledge useful discussions with Zehua Guo.Q.M.L. acknowledges support of the National Science Founda-tion of China (40725013, 40674093), and Chinese Academy ofScience (KJCX2-YW-N28). L.C. acknowledges support of USDoE and NSF grants, and National Basic Research of Chinaunder Grant No. 2008CB717806.

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