ION-CYCLOTRON INSTABILITY DUE TO THE THERMAL ANISOTROPY OF DRIFTING ION SPECIES.pdf

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JOURNAL OF GEOPHYSICAL RESEARCH, VOL. , NO. , PAGES 1–5,

ION-CYCLOTRON INSTABILITY DUE TO THE

THERMAL ANISOTROPY OF DRIFTING ION

SPECIES

L. Gomberoff and J. A. ValdiviaDepto. de Fısica, Facultad de Ciencias, Universidad de Chile

Abstract. In a recent study of ion-cyclotron waves generated by the thermal anisotropyof oxygen ions, it was shown that the heavy ion drift velocity and a large thermal anisotropyof the heavy ions can destabilize proton-cyclotron waves [Gomberoff and Valdivia , 2002].Here this study is extended to alpha particles in order to show that a much smaller ther-mal anisotropy is required to trigger strong proton-cyclotron waves. It is also shown thatunder some conditions, the alpha particle branch of the dispersion relation becomes un-stable for frequency values beyond the proton gyrofrequency. This instability occurs forvery large alpha particle thermal anisotropy and very low β α = v2

th/v2

A, where vth and

vA are the thermal and Alfven velocity, respectively. The maximum growth rate of thisbranch of the dispersion relation occurs for drift velocities of the alpha particles largerthan those which drive the maximum growth rate of the proton cyclotron instability. Fi-nally, it is demonstrated that the combined effect of oxygen ions and alpha particles leadto a complex unstable spectrum, and to an enhancement of the proton-cyclotron insta-bility. This mechanism is like a cascade effect in which low frequency ion-cyclotron wavescan drive unstable high frequency ion-cyclotron waves, through anisotropic heating andacceleration of heavy ions. These results may be relevant to the understanding of theheating process of the fast solar wind in coronal holes.

1. Introduction

It is generally believed that heating and acceleration of heavy ions in the lower corona is due to resonant absorptionof Alfven waves [see, e.g., Gomberoff et al., 1996; Marsch ,1998; Tu and Marsch , 1999; Hu and Habbal , 1999; Cran-

mer et al ., 1999a, 1999b, 2000; Isenberg and Hollweg , 1983]. However, low frequency Alfven waves (0.001-0.1Hz) whichdominate the power spectrum of magnetic fluctuations inthe solar wind for distances larger than 0.3 AU, are notlikely to heat and accelerate the heavy ions to the observedvalues. On the other hand, high frequency Alfven waves (10-104 Hz) that might be responsible for ion-cyclotron heatingand acceleration of these ions, have not yet been observedeither in the solar wind or in the corona [Cranmer et al .,1999a, 1999b]. However, there are several proposals as tohow these waves might be generated. Thus, it has been sug-gested that high frequency Alfven waves can be generatedby a turbulent cascade [see, e. g., Tu and Marsch , 1995]. Ithas also been suggested that high frequency Alfven wavescan be generated by microflares that are then converted to

ion-cyclotron waves at higher coronal altitudes [Axford and Mckenzie , 1992, 1996].

Not only high frequency waves have not been observed,but there are other problems with ion-cyclotron heating,such as, the observations on the preferential heating of O+5

[Kohl et al ., 1998, 1999a, 1999b] which are not satisfactorily

Copyright by the American Geophysical Union.

Paper number .0148-0227/02/$9.00

explained by this mechanism [Tu and Marsch , 2001]. Thereare also other approaches to this problem. These includefast and oblique propagation of ion-cyclotron waves [Leamon et al., 2000; Li and Habbal , 2001; Markowskii , 2001; Holl-weg and Markowskii , 2002], the need to develop a kinetictreatment [Isenberg et al ., 2001; Isenberg , 2001], fast shock

heating [Lee and Wu , 2000; Lee , 2001], turbulence-drivenion-cyclotron waves [Li et al ., 1999], and others. Never-theless, there seems to be some observational evidence thatparallel propagating ion-cyclotron waves are at the origin of ion heating and acceleration [Tu and Marsch , 2001; Isenberg

et al ., 2001, Isenberg , 2001].Observations of heating and acceleration of heavy ions

in the solar corona by SOHO, have shown very large ther-mal anisotropy for O+5 ions with values ranging between10 < T ⊥/T < 100 [Kohl et al ., 1998, 1999a, 1999b; Li et al ., 1998; Cranmer et al ., 1999a, 1999b]. In a previ-ous paper [Gomberoff and Valdivia , 2002], it was suggestedthat the heavy ion drift velocity relative to the protons,and large thermal anisotropy can trigger a cascade of ion-cyclotron waves to much higher frequencies. These waves

would then be absorbed by ion species having larger chargeto mass ratio. Ion-cyclotron waves due to heavy ion thermalanisotropy have also been considered by Gary et al., [2001],and Ofman et al., [2001]. However, these authors did notinclude the important effect of heavy ion drift velocity [see,Gomberoff and Valdivia , 2002].

Here we continue to develop this idea by includinganisotropic alpha particles into the problem. It is shownthat due to the much larger density of the alpha particlesrelative to oxygen ions in the solar wind, a much smallerthermal anisotropy is required to trigger large growth ratesof the proton-cyclotron instability [see Gomberoff and Val-divia, 2002]. The observation that a species drift velocity

1



2 GOMBEROFF ET AL.: ION CYCLOTRON WAVES

and thermal anisotropy can lead to an instability beyond the

species gyrofrequency, provides a complementary ion heat-

ing and acceleration mechanism in the solar corona.

We also analyze the very complex situation in which the

unstable spectrum is produced by two thermally anisotropic

heavy ion components. The combined effect of two heavy

ion components can trigger very strong ion-cyclotron waves

above their corresponding gyrofrequency, and the minor ionscan also contribute to enhance the proton cyclotron insta-

bility produced by just one species [Gomberoff and Valdivia ,

2002].

The paper is organized as follows. In Section 2. we derive

the general ion-cyclotron dispersion relation for any number

of drifting heavy ion species using the semi-cold approxima-

tion. In Section 3 we derive and discuss the particular case

when the heavy ion species are drifting alpha particles. In

Section 4, we discuss the case of two drifting ion compo-

nents: O+6 and alpha particles. In Section 5, the results are

summarized.

2. Ion Cyclotron Dispersion RelationWe consider a plasma in an external magnetic field B0,

composed of electrons, protons, and a number of heavy ion

species drifting relative to the protons, along the external

magnetic field. The jth minor heavy ion species has normal-

ized velocities, U j = V j/V A, where V j is the heavy ion veloc-

ity normalized to the Alfven velocity, V A = B0/

(4πnpmp),

with np as the proton density and mp as the proton mass.

The dispersion relation for ion-cyclotron waves moving in

the direction of the external magnetic field, assuming bi-

Maxwellian distribution functions, is (see, e. g., Gomberoff ,

[1992]),

y2 =

j

z jηjAj

M j− z jηj(x − yU j) −

z jηj

M 2j yβ 1/2j

Z (ξj)Gj,(1)

Gj = [Aj(1 − M j(x − yU j)) − M j(x − yU j)], (2)

where the sum is over all ion species, including the pro-

ton background. In Eq. (1), x = ω /Ωp, y = k V A/Ωp, ηj =

nj/np, Aj = T ⊥j/T j−1 is the ion thermal anisotropy, M j =

mj/z jmp (with mj the ion mass and z j the degree of ioniza-

tion of the heavy ion), Z is the plasma dispersion function

[Fried and Conte , 1961], ξj = (M j(x − yU j) − 1)/M jyβ 1/2

j ,

β j = v2th,j/V 2A where v2th,j = 2KT j/mj is the thermal ve-

locity of ion components, and K is the Boltzmann constant.In this equation we have assumed a current-free system,

and the bulk protons has been chosen to be the rest frame

[Gomberoff , 1991]. We shall assume all betas to be much

smaller than 1, so that we can use the semi-cold approxi-

mation [see, e.g., Gomberoff , 1992]. Thus, using the large

argument expansion of the Z-functions, and assuming ω to

be complex, we obtain from the real part of Eq. (1) the cold

plasma dispersion relation,

y2 =

j

z jM jηj(x − yU )2

1 − M j(x − yU j) , (3)

On the other hand, assuming |Re(ξj)| |Im(ξj)|, from theimaginary part of Eq. (1) we obtain the growth/dampingrate,

γ =

j

(π)z jηj

M 2j yβ 1/2j

Gj

F (x, y)exp[−(

1 − M j(x − yU j)

M iyβ 1/2i

)2] (4)

where

F (x, y) =

j

z jM jηj(x − yU j)(2 − M j(x − yU j)

(1 − M j(x − yU j))2 . (5)

The sign of the growth (or dissipation) rate γ is controlledby G, Eq. (2), and for the case of the proton background ityields, Gp = Ap(1 − x) − x. If the proton background has nothermal anisotropy, then the protons contribute with pureabsorption. On the other hand, if Ap > 0 it will lead to in-stability for x < Ap/(Ap+1) [see, e.g., Gomberoff and Neira ,1983; Gomberoff and Vega , 1987; Gomberoff and Elgueta ,1991]. Even though these considerations may affect the to-tal growth rate in situations of interest, we shall take Ap = 0for the rest of the paper in order to simplify the problem.Similarly, if the ion drift velocity is U j = 0, then the instabil-ity occurs always below the gyrofrequency of the anisotropicspecie.

3. Alpha Particles

As pointed out in the introduction, in a previous paper[Gomberoff and Valdivia , 2002], we studied the effect of drift-ing O+6 ions on the ion cyclotron instability triggered by thethermal anisotropy of oxygen ions. It was shown that dueto the oxygen drift velocity and large thermal anisotropy,proton cyclotron waves can be generated. Since this effectdepends on the species thermal anisotropy and branching

ratio, the maximum growth rate for O

+6

was less than 10

−3

for a thermal anisotropy of AO+6 = 100. Here we shallstudy the same effect, but for alpha particles that are muchmore abundant than oxygen ions. Therefore, we expect thatmuch smaller alpha particle thermal anisotropy is requiredto trigger proton cyclotron waves.

Thus, for the particular case of stationary protons anddrifting alpha particles Eqs. (2) and (3) yield,

y2 = x2

1 − x +

4ηα(x − yU α)2

1 − 2(x − yU α), (6)

and

γ p,α =

(π)ηα

2yβ 1/2α

Gα

F (x, y)exp[−( 1 − 2(x − yU α)

2yβ 1/2α

)2] (7)

−

(π)x

yβ 1/2p

1

F (x, y)exp[−(

1 − x

yβ 1/2p

)2],

with

Gα = Aα(1 − 2(x − yU α)) − 2(x − yU α),

and

F (x, y) = (2 − x)x

(1 − x)2 +

8ηα(1 − (x − yU α))(x − yU α)

(1 − 2(x − yU α))2 .



GOMBEROFF ET AL.: ION CYCLOTRON WAVES 3

As it is well known, when there is no drift velocitybetween the ion species the dispersion relation has twobranches each going asymptotically to the corresponding gy-rofrequency [see, e.g., Gomberoff and Neira , 1983; Gomberoff

and Elgueta , 1991]. In this case, the instability occurs forfrequencies below the thermally anisotropic species gyrofre-quency.

On the other hand, when the ion species are drifting rel-

ative to each other, the situation is different. In Fig. 1awe show the dispersion relation Eq. (6) for ηα = 0.05 andU α = 0.2. The upper curve of the dispersion relation cor-responds to the proton branch (R2), and the other to thedrifting alpha particles (R1). Note that the latter satisfiesx − yU α 0.5. In Fig. 1b we display the correspond-ing growth rate, Eq. (7), for Aα = 6, β α = 0.004 andβ p = 10−6. As it follows from Fig. 1b, the maximumproton cyclotron growth rate is now larger than 10−3 foran alpha particle thermal anisotropy of only Aα = 6 [seeGomberoff and Valdivia, 2002]. The value β p = 10−6 usedin Fig. 1b is unrealistic at the base of the solar corona,because it would imply a relativistic Alfven velocity. Forthis reason in Fig. 1c we have increased β p by two ordersof magnitude to β p = 10−4, which is a plausible value of

β p in this region. Due to the larger proton absorption rate,there is a decrease of the high frequency instability range. Itis important to emphasize that both instability regions aresolutions of the branch of the dispersion relation which goesasymptotically to the proton gyrofrequency (R2).

In order to study the dependence of γ max for the branchof the growth rate closer to the proton gyrofrequency (seeFig. 1b and 1c), we have constructed a 3-dimensional plotof γ max vs. U α and β α (Fig. 2a) and γ max vs. U α andβ p (Fig. 2b) for Aα = 6. As expected, as we increaseβ p the ion-cyclotron waves are strongly affected by theproton absorption. The maximum growth rate of the ion-cyclotron waves depends strongly on the alpha’s drift veloc-ity, as shown in Fig. 3 at multiple cross sections of Fig. 2afor fixed β α = 0.001, 0.005, 0.01, as indicated in the corre-

sponding figures.The β p of the protons will be chosen very small to il-lustrate that the high frequency waves, close the the pro-ton gyrofrequency, can indeed be generated by the thermalanisotropy of the drifting minor ions. For Oxygen ions ithas been shown in Gomberoff and Valdivia [2002], that aswe increase β p the growth rate of the waves is partially re-duced suggesting that the protons are capable to absorbingthese waves. In this paper we are concerned with character-izing the parameters controlling the generation of the highfrequency proton cyclotron waves in the initial stages, andfor that β p = 10−6 has been chosen very small in orderto emphasize this effect. Moreover, an exact calculation of Eq. (1) (to be published elsewhere) shows that the semi-cold approximation gives much too large proton absorption

rates close to ω Ωp. This is to be expected because asβ p increases, the semi-cold approximation breaks down forω close to Ωp. However, for the purpose of the present pa-per, the semi-cold approximation is perfectly appropriate.After this linear regime, the evolution of the energy trans-fer between ions must be described in a nonlinear fashion,which requires a different approach that will be publishedelsewhere.

We shall now show that the lower branch of the disper-sion relation - the one satisfying x − yU α 1/2 (see R1in Fig. 1a) - can also become unstable, but for rather largevalues of the alpha thermal anisotropy, Aα. In order to illus-trate this result, in Fig. 4a we have plotted γ vs. frequency

for Aα = 500, β α = β p = 10−6, and U α = 0.6. Fromthis figure it follows that, for these conditions, the spectrumhas a double hump and is very broad. As the drift velocitygoes down, the double humped spectrum shows now a singlepeak. This effect is illustrated in Fig. 4b for U α = 0.45, andin Fig. 4c for U α = 0.24. In Fig. 5a we have plotted γ max

for x > 1, as a function of Aα and β α for fixed U α = 0.5and β p = 10−6. As expected, there is a sharp increase of

γ max with Aα 200. In Fig. 5b we have done the same asin Fig. 5a, but for γ max as a function of Aα and U α, for fixedβ p = β α = 10−6. Here we see an interesting interplay be-tween the alpha’s drift velocity and its thermal anisotropy,as it is illustrated in the cross sections of Fig. 6, for severalvalues of U α = 0.2, 0.3, 0.4, 0.6, 0.8, which are indicated inthe corresponding figure.

4. Alpha Particles and Oxygen Ions

In Gomberoff and Valdivia [2001], it was contended thatsince we are dealing with a linear theory, the combined effectof many heavy ion species could lead to an important growth

rate of the proton-cyclotron instability. To study this situ-ation, we add to the model an additional highly anisotropiccomponent, O+6. We note that observations show a largethermal anisotropy for O+5 ions [Kohl et al ., 1998; Li et al .,1998; Kohl et al ., 1999a, 1999b]. However, we assume a largethermal anisotropy for O+6 as well, because its branchingratio is much larger than that of O+5 ions [see, Gomberoff

and Valdivia , 2002].The corresponding dispersion relation and growth rate

are given by,

y2 = x2

1 − x +

4ηα(x − yU α)2

1 − 2(x − yU α) +

16ηO+6(x − yU O+6)2

1 − 16

6 (x − yU O+6)

, (8)

and

γ = γ p,αF (x, y)

F (x, y) (9)

+6

(π)ηO+6

(166

)2yβ 1/2

O+6

GO+6

F (x, y)exp[−(

1 − ( 166

)(x − yU O+6)

(166

)yβ 1/2

O+6

)2],

with

GO+6 = AO+6(1 − (16

6 )(x − yU O+6)) − (

16

6 )(x − yU O+6)

and

F (x, y) = F (x, y)

+ 16ηO+6(x − yU O+6)(2 − (16

6 )(x − yU O+6))

(1 − ( 166

)(x − yU O+6))2) .

In the case when there is more than one heavy ion specie,the situation is, of course, more complicated. Since there aremany parameters involved, a systematic analysis is difficult.In the following we shall illustrate the main features.

To this end, in Fig. 7a we begin by plotting the dispersionrelation, Eq. (8), when the drift velocity of the heavy ionsis zero relative to the protons. In Fig. 7b we show the cor-responding growth rates, Eq. (9), for AO+6 = 100, Aα = 6,




β α = 0.005, β O+6 = 0.0004, and a very low β p = 10−6. Avery small β p has been used in order to illustrate the resultsby diminishing proton absorption effects. The first p eak isdue to the O+6 ions, i.e., the dispersion curve denoted bythe dotted line (R3) in fig 7a. The second peak correspondsto the alpha particles, and has been denoted by the solidline (R2) in Fig. 7a.

In the following figures, we vary the drift velocities of theheavy species without changing the other parameters.

Thus in Fig. 8a we show the instability regions forU α = 0.2, and U O+6 = 0. The roots are denoted by (R1),(R2), (R3), and (R4), but only the first three belong tothe region of interest. The unstable roots of Fig. 8a corre-spond to (R2), solid line, and (R3), dotted line, as indicatedthere. The (R2) root is clearly double humped, while the(R3) root is single humped. Figure 8a is similar to Fig.1b, but with another peak due to O+6. As U O+6 increasesthere is a large increase of the high frequency band of the(R3) root as shown in Figs. 8b and 8c for U O+6 = 0.1and U O+6 = 0.15, respectively. From Figs. 8b and 8cone can clearly see how the O+6 ions start generating theion-cyclotron waves. The effect of this branch (R3) is due

to oxygen thermal anisotropy and corresponds to Fig. 2bof Gomberoff and Valdivia [2002]. The dispersion relationshown in Fig. 8d, for U α = 0.2 and U O+6 = 0.1, suggests acomplicated interplay among its different branches.

In Fig. 9a we have taken U O+6 = U α = 0.2, and in Fig.9b we have increased U O+6 = 0.25. The effect of increas-ing the drift velocity of O+6 beyond U α is to stabilize thewaves. The waves generated by the root (R2), solid line, arecompletely dissipated and only the (R3) root, dotted line,survives.

In Fig. 10a, we illustrate the behavior of γ max for the(R2) root, as a function of varying U O+6 for fixed U α = 0.2.In Fig. 10b we do the same as in Fig. 10a, but for U α = 0.3.From Fig. 10 one can clearly see the stabilization of the

(R3) root of the system for U O+6 ≥ U α.Finally, in Fig. 11, we illustrate the effect of increasing

U α while keeping U O+6 = 0.2 fixed. Thus, in Figs 11a-d wehave taken U α = 0.1, 0.15, 0.18, 0.19 respectively. The un-stable roots correspond to (R3), dotted line, in Fig. 11a and11b, and to (R3) and (R2), solid line, in Figs. 11c and 11d.In Fig. 11c the unstable root corresponding to (R2) does notappear, because γ < 10−5. As it follows from Fig. 11a, theeffect of increasing the drift velocity of the alpha particlesis to stabilize the high frequency branch of the spectrum.However, if U α is further increased to U α = 0.15, the highfrequency branch reappears as shown in Fig. 11b. In Fig.11c we have raised U α = 0.18. As a consequence, there isa large increase in the maximum growth rate of the high

frequency branch, and also another root becomes unstable(R2). In Fig. 11d, U α has been raised a little further toU α = 0.19, and a new instability branch appears in betweenthe two larger peaks of the previous figure. This new insta-bility also corresponds to the root denoted by (R2).

5. Summary

We have continued the study of the combined effectof heavy ion thermal anisotropy and drifting velocity onthe ion-cyclotron instability. This study was initiated byGomberoff and Valdivia [2002], in an attempt to explain theproton heating observed at the base of solar coronal holes,

by making use of the large oxygen ion thermal anisotropyobserved there [Kohl et al ., 1998, 1999a, 1999b; Cranmer ,2000].

To this end, we have first considered the effect of alphaparticles thermal anisotropy and drift velocity on the ion-cyclotron instability. It is shown that as compared to oxygenions, a much smaller thermal anisotropy is required to trig-ger a much larger proton-cyclotron instability growth rate.

This effect is illustrated in Fig. 1b and 1c. A three di-mensional study of the maximum growth rate, including thesimultaneous effects of drift velocity U α, and β α, is shownin Fig. 2. The unstable branch of the dispersion relationcorresponds to the upper branch of Fig. 1a, denoted by R2.

We have also shown that the lower branch of the dis-persion relation, R1, can become unstable for frequenciesbeyond the proton gyrofrequency. This occurs for large val-ues of the alpha particle thermal anisotropy, and very lowvalues of β α. These results are illustrated in Fig. 4.

Finally, we have studied the combined effect of oxygenions and alpha particles. Since there are several parametersinvolved, a systematic analysis is difficult. In an attempt todo this, we have done the following. We have started with

a situation corresponding to the three species at rest rela-tive to each other. In this case, as it is well known, thereare two unstable branches below the corresponding gyro-frequencies. The corresponding marginal modes are givenby xm,j = (Ωj/Ωp)(Aj/(Aj + 1), with j = O+5, α [see, e.g., Gomberoff and Valdivia , 2002]. This is shown in Fig. 7b.We have then increased the U O+6 drift velocity. As it isexpected, a new branch appears close to the proton gyrofre-quency. Then, in Fig. 8 we have considered several valuesof U O+6 while keeping U α = 0.2 fixed. The system shows acomplicated interplay among the branches of the dispersionrelation. Both the (R2) and (R3) roots are now unstable.The effect of increasing the O+6 drift velocity beyond theα-particle drift velocity is to stabilize the waves belonging to

the (R2) root. Finally, in Fig. 11 we illustrate the behaviorof the growth rate for varying U α while keeping U O+6 = 0.2fixed. In this case the unstable root is (R3) in Figs. 11a and11b, and (R3) and (R2) in Figs. 11c and 11d.

As pointed out above, a systematic study that would al-low for an exact prediction of the effect of changing thevarious parameters involved, is difficult. However, one canconclude that, in general, heavy ion thermal anisotropyand drift velocity (relative to the protons) can trigger ion-cyclotron waves above their corresponding gyrofrequencyleading, thereby, to resonant effects with ion species hav-ing larger charge to mass ratios. Also, the combined effectdue to several heavy ion components can lead to a large en-hancement of the ion-cyclotron instability, both for low andhigh frequencies. Due to ion-cyclotron interaction heavyions become anisotropic.

Since high frequency ion-cyclotron waves are not observedat the base of the corona nor in the fast solar wind, theresults of this paper naturally suggest a possible cascademechanism that utilizes low frequency ion-cyclotron wavesin the coronal holes to heat anisotropically larger and largerq/m ions. First, low frequency ion-cyclotron waves can heatanisotropically ion components with small q/m ratio. Thisanisotropy, complemented with the ion drift velocity, cangenerate instabilities for larger frequencies, which in turncan heat anisotropically ions of larger q/m ratio. Oncethese last ions are anisotropically heated, they can gener-ate instabilities of even larger frequencies, in a process that




may repeat until it reaches the protons. This is the phe-nomenon that we denominate a ”cascade mechanism”, andof course we do not pretend that this is the unique phe-nomenon responsible for the observations, but it may pro-vide a possible contribution. However, at larger heliocentricdistances where β p becomes much larger than 10−4, fromFig. 2b it follows that the proton-cyclotron instability iscompletely stabilized (for Aα = 6) and, therefore, this is an

unlikely mechanism for proton-cyclotron wave generation atsuch distances.

Acknowledgments. This work has been partially supportedby FONDECYT grant N o 1020152 and FONDECYT grant N o

1000808.

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Lee, L. C., and B. H. Wu, Heating and acceleration of protonsand minor ions by fast shocks in the solar corona, Astrophys.J.,

537, 535, 2000.Li, X., and S, Habbal., Damping of fast and ion-cyclotron obliquewaves in the multi-ion fast solar wind, J. Geophys. Res., 106 ,10.669, 2001

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Markovskii, S.A., Generation of ion cyclotron waves in coronaholes by a global resonant magnetohydrodynamic mode, As-trophys.J., 557,, 337, 2001.

Marsch, E., Close of multi-fluid and kinetic equations forcyclotron-resonant interactions of solar wind ions with Alfvenwaves, Nonlinear Proc. Geophys., 5 , 11, 1998.

Ofman, L., A. Vinas, S. P. Gary, Constraints on the O+5

anisotropy in the solar corona, Astrophys. J., 547 , L175, 2001.Tu, C. Y., and E. Marsch, MHD structures, waves, and turbu-lence in the solar wind, Space Sci. Rev., 73 , 1, 1995.

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L. Gomberoff, Depto. de Fısica, Facultad de Ciencias, Univer-sidad de Chile, Casilla 653, Santiago, Chile. [email protected]

J. A. Valdivia, Depto. de Fısica, Facultad de Cien-cias, Universidad de Chile, Casilla 653, Santiago, [email protected]

(Received .)




0 .25 0 .5 0. 75 1 1.2 5 1.5 1 .75 2

x

2

4

6

8

10

12

14

R2

R1

a

0.2 0.4 0.6 0.8 1

x

105

104

103

102

101

1

Γ

b

0.2 0.4 0.6 0.8 1

x

105

104

103

102

101

1

Γ

c

Figure 1. (a) Dispersion relation Eq. (6), y = kV A/Ωp vs.

x = ω/Ωp, for ηα = 0.05, U α = 0.2. The two relevant rootsare marked as (R1) and (R2) respectively. (b) Correspond-ing growth rate for (R2), Eq. (7), for β α = 0.004, Aα = 6,

and β p = 10−6. (c) The same as (b), but for β p = 10−4.




a

0

0.2

0.4

0.6

0.8

1

UΑ

104

103

102

101

1

ΒΑ

105

104

103

102

101

1

max

0

0.2

0.4

0.6

0.8UΑ

0

0.1

0.2

0.3

0.4

0.5

UΑ

106

105

104

103

102

ΒP

105

104

103

102

101

1

Γmax

0

0.1

0.2

0.3

0.4UΑ

Figure 2. (a) Three dimensional plot of γ max vs. U α andβ α for the (R2) root, using β p = 10−6. (b) Three dimen-sional plot of γ max vs. U α and β p for the (R2) root, usingβ α = 0.004. We take Aα = 6.






0.5 1 1.5 2 2.5 3

x

0.5

1

1.5

2

.

1 0 3

Γ

UΑ0.6

0.5 1 1.5 2 2.5 3

x

0.5

1

1.5

2

.

1 0 3

Γ

UΑ0.45

0.5 1 1.5 2 2.5 3

x

0.5

1

1.5

2

.

1 0 3

Γ

UΑ0.24

Figure 4. (a) Growth rate γ vs. normalized frequency, x,

for Aα = 500, β α = β p = 10−6

, and U α = 0.6. (b) Same as(a), but for U α = 0.45. (c) Same as (a), but for U α = 0.24.




a

0

100

200

300

400

500

AΑ

106

105

104

103

102

101

1

ΒΑ

105

104

103

102

101

1

Γmax

0

100

200

300

400AΑ

0

100

200

300

400

500

AΑ

0

0.2

0.4

0.6

0.8

1

uΑ

105

104

103

102

101

1

Γmax

0

100

200

300

400AΑ

Figure 5. (a) Three dimensional plot of γ max vs. Aα andβ α using β p = 10−6 and U α = 0.5. (b) Three dimensional

plot of γ max vs. Aα and U α using β p = β α = 10−6.




100 200 300 400 500

AΑ

105

104

103

102

101

Γ m a x

UΑ 0.2

100 200 300 400 500

AΑ

105

104

103

102

101

Γ m a x

UΑ 0.3

100 200 300 400 500

AΑ

105

104

103

102

101

Γ m a x

UΑ 0.4

100 200 300 400 500

AΑ

105

104

103

102

101

Γ m a x

UΑ 0.6

100 200 300 400 500

AΑ

105

104

103

102

101

1

Γ m a x

UΑ 0.8

Figure 6. Cross sections of Fig. 5b for several values of U α = 0.2, 0.3, 0.4, 0.6, 0.8.

0.2 0.4 0.6 0.8 1

x

1

2

3

4

R1

R3

R2

0.2 0.4 0.6 0.8 1

x

105

104

103

102

101

Γ

UO 0

Figure 7. (a) Dispersion relation, Eq. (8), y vs. x forηO+6 = 0.0002, ηα = 0.05 for U O+6 = U α = 0. The rootsare denoted by (R1), (R2), (R3) and (R4), but only the firstthree belong to the region of interest. (b) Correspondinggrowth rates, Eq. (9), vs. x for AO+6 = 100, Aα = 6,β O+6 = 0.0004 β α = 0.005, and β p = 10−6. The unstableroots are (R2), solid line, and (R3), dotted line.




0.2 0.4 0.6 0.8 1

x

105

104

103

102

101

1

Γ

UO 0

0.2 0.4 0.6 0.8 1

x

105

104

103

102

101

1

Γ

UO0.1

0.2 0.4 0.6 0.8 1

x

105

104

103

102

101

Γ

UO0.15

0.2 0.4 0.6 0.8 1

x

1

2

3

4

5

6

7

R1

R3 R2

Figure 8. Growth rate vs. x for the case of Fig. 7, but forU α = 0.2 and (a) U O+6 = 0.0, (b) U O+6 = 0.1,(c) U O+6 =0.15. The unstable roots are (R2), solid line, and (R3),dotted line. (d)The dispersion relation for the case U O+6 =0.1.

0.2 0.4 0.6 0.8 1

x

105

104

103

102

101

1

Γ

UO0.2

0.2 0.4 0.6 0.8 1

x

105

104

103

102

101

1

Γ

UO0.25

Figure 9. (a) Same as Fig. 8, but for U O+6 = 0.2. (b)Same as (a) but for U O+6 = 0.25. The unstable roots are(R2), solid line, and (R3), dotted line.




0.1 0.2 0.3 0.4 0.5

UO

105

104

103

102

101

Γ m a x

UΑ0.2

0.1 0.2 0.3 0.4 0.5

UO

105

104

103

102

101

Γ m a x

UΑ0.3

Figure 10. (a) The maximum growth rate γ max as a func-tion of U O+6 for the same parameters as in Fig. 9, but forU α = 0.2. The same as in (a) but for U α = 0.3.

0.2 0.4 0.6 0.8 1

x

105

104

103

102

101

Γ

UΑ0.1

0.2 0.4 0.6 0.8 1

x

105

104

103

102

101

Γ

UΑ0.15

0.2 0.4 0.6 0.8 1

x

105

104

103

102

101

Γ

UΑ0.18

0.2 0.4 0.6 0.8 1

x

105

104

103

102

101

Γ

UΑ0.19

Figure 11. Growth rate vs. x for the case of Fig. 7, but forU O+6 = 0.2 and (a) U α = 0.1, (b) U α = 0.15, (c) U α = 0.18,(d) U α = 0.19. The unstable roots are (R2), solid line, and(R3), dotted line.

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