Self-induced Scattering of Strahl Electrons in the Solar Wind

Daniel Verscharen1,2 , Benjamin D. G. Chandran2,3 , Seong-Yeop Jeong1 , Chadi S. Salem4, Marc P. Pulupa4 , andStuart D. Bale4,5,6

1 Mullard Space Science Laboratory, University College London, Dorking, RH56NT, UK; d.verscharen@ucl.ac.uk, s.jeong.17@ucl.ac.uk2 Space Science Center, University of New Hampshire, Durham, NH 03824, USA; benjamin.chandran@unh.edu

3 Department of Physics and Astronomy, University of New Hampshire, Durham, NH 03824, USA4 Space Sciences Laboratory, University of California, Berkeley, CA 94720, USA; salem@ssl.berkeley.edu, pulupa@ssl.berkeley.edu, bale@berkeley.edu

5 Physics Department, University of California, Berkeley, CA 94720, USA6 The Blackett Laboratory, Imperial College London, SW72AZ, UK

Received 2019 June 6; revised 2019 October 5; accepted 2019 October 7; published 2019 November 29

Abstract

We investigate the scattering of strahl electrons by microinstabilities as a mechanism for creating the electron haloin the solar wind. We develop a mathematical framework for the description of electron-driven microinstabilitiesand discuss the associated physical mechanisms. We find that an instability of the oblique fast-magnetosonic/whistler (FM/W) mode is the best candidate for a microinstability that scatters strahl electrons into the halo. Wederive approximate analytic expressions for the FM/W instability threshold in two different βc regimes, where βcis the ratio of the core electrons’ thermal pressure to the magnetic pressure, and confirm the accuracy of thesethresholds through comparison with numerical solutions to the hot-plasma dispersion relation. We find that thestrahl-driven oblique FM/W instability creates copious FM/W waves under low-βc conditions when U w30s c,where U0s is the strahl speed and wc is the thermal speed of the core electrons. These waves have a frequency ofabout half the local electron gyrofrequency. We also derive an analytic expression for the oblique FM/Winstability for βc∼1. The comparison of our theoretical results with data from the Wind spacecraft confirms therelevance of the oblique FM/W instability for the solar wind. The whistler heat-flux, ion-acoustic heat-flux,kinetic-Alfvén-wave heat-flux, and electrostatic electron-beam instabilities cannot fulfill the requirements for self-induced scattering of strahl electrons into the halo. We make predictions for the electron strahl close to the Sun,which will be tested by measurements from Parker Solar Probe and Solar Orbiter.

Key words: instabilities – plasmas – solar wind – Sun: corona – turbulence – waves

1. Introduction

The solar wind is a plasma consisting of electrons, protons,and other ion species. Since the mass of an electron me is byabout a factor of 1836 smaller than the mass of a proton mp, theelectron contribution to the momentum flux in the solar wind isnegligible. However, electrons are important to ensure quasi-neutrality, and their pressure gradient generates a substantialelectrostatic field. Furthermore, the skewness provided bysuperthermal features in their distribution function supplies thesolar wind with a significant heat flux (Gary et al. 1999; Scimeet al. 1999; Pagel et al. 2005; Marsch 2006; Verscharen et al.2019).

Observations show that typical solar-wind electron distributionfunctions consist of three main populations: a thermal core, asuperthermal halo, and a field-aligned beam, which is usuallycalled the strahl (German for beam; Feldman et al. 1975;Rosenbauer et al. 1977; Pilipp et al. 1987a, 1987b; Hammondet al. 1996; Fitzenreiter et al. 1998; Lin 1998; Maksimovic et al.2000; Gosling et al. 2001; Salem et al. 2003a; Wilson et al.2018). The thermal core typically exhibits temperatures compar-able to the proton temperature and contains about 95% of theelectrons. The halo is a tail in the distribution function extendingto large velocities, which can be well modeled by a κ-distribution(Maksimovic et al. 1997, 2005). The halo is present in alldirections with respect to the field; however, relative drifts andtemperature anisotropies of both the halo and the core have beenobserved (Štverák et al. 2008; Bale et al. 2013). The electronstrahl forms a “shoulder” in the distribution function. Its bulkvelocity is shifted with respect to the electron core either parallel

or antiparallel to the magnetic field, and its radial velocitycomponent is almost always greater than the core’s radial velocitycomponent. The nonthermal features of the distribution functionare more distinctive in the fast solar wind. This observation isattributed to the typically weaker collisionality of the fast solarwind (Scudder & Olbert 1979a, 1979b; Phillips & Gosling 1990;Lie-Svendsen et al. 1997; Landi & Pantellini 2003; Salem et al.2003b; Gurgiolo & Goldstein 2017).When binary collisions among electrons are sufficiently

rare, instabilities can reduce the skewness of the distributionfunction and thereby limit the heat flux (Hollweg 1974;Gary et al. 1975a, 1975b; Feldman et al. 1976; Lakhina 1977;Ramani & Laval 1978; Lazar et al. 2011). Candidates for suchinstabilities include the electromagnetic whistler heat-fluxinstability (Gary & Feldman 1977; Gary et al. 1994; Gary &Li 2000; Lazar et al. 2013), fan instabilities of the lower-hybridmode (Lakhina 1979; Omelchenko et al. 1994; Krafft et al.2005; Krafft & Volokitin 2006; Shevchenko & Galinsky2010), the kinetic-Alfvén-wave (KAW) heat-flux instability(Gary et al. 1975a), and a family of electrostatic instabilities(Gary 1978; Gary & Saito 2007; Pavan et al. 2013) includingthe ion-acoustic heat-flux instability (Gary 1979; Detering et al.2005).In strahl-driven instabilities, the instability thresholds for

bulk speed of the strahl component, U0s, in the reference framein which the protons are at rest scale approximately as theelectron Alfvén speed pºv B n m4Ae 0 0e e , where n0e is theequilibrium electron number density and B0 is the backgroundmagnetic field. Since vAe decreases with heliocentric distancein the inner heliosphere, the thresholds also decrease. This

The Astrophysical Journal, 886:136 (11pp), 2019 December 1 https://doi.org/10.3847/1538-4357/ab4c30© 2019. The American Astronomical Society. All rights reserved.

1

https://orcid.org/0000-0002-0497-1096https://orcid.org/0000-0002-0497-1096https://orcid.org/0000-0002-0497-1096https://orcid.org/0000-0003-4177-3328https://orcid.org/0000-0003-4177-3328https://orcid.org/0000-0003-4177-3328https://orcid.org/0000-0001-8529-3217https://orcid.org/0000-0001-8529-3217https://orcid.org/0000-0001-8529-3217https://orcid.org/0000-0002-1573-7457https://orcid.org/0000-0002-1573-7457https://orcid.org/0000-0002-1573-7457https://orcid.org/0000-0002-1989-3596https://orcid.org/0000-0002-1989-3596https://orcid.org/0000-0002-1989-3596mailto:d.verscharen@ucl.ac.ukmailto:s.jeong.17@ucl.ac.ukmailto:benjamin.chandran@unh.edumailto:salem@ssl.berkeley.edumailto:pulupa@ssl.berkeley.edumailto:bale@berkeley.eduhttps://doi.org/10.3847/1538-4357/ab4c30https://crossmark.crossref.org/dialog/?doi=10.3847/1538-4357/ab4c30&domain=pdf&date_stamp=2019-11-29https://crossmark.crossref.org/dialog/?doi=10.3847/1538-4357/ab4c30&domain=pdf&date_stamp=2019-11-29

situation, therefore, leads to a quasi-continuous excitation ofunstable wave modes during the solar wind’s passage throughthe inner heliosphere beyond the point at which U0s reaches theinstability threshold for the first time. The exact location of thispoint depends on the properties of the innermost heliosphereand is thus still unknown. The unstable wave modes causepitch-angle scattering of electrons, which reduces the strahlheat flux while transferring strahl electrons into the halo.

All electron-driven instabilities generate waves with wave-lengths between the electron and the ion inertial length scalesand inject these unstable waves into the fluctuation spectrumbetween these scales. Therefore, the understanding of electronmicroinstabilities is important for the understanding of the natureof fluctuations in the ion-to-electron spectral range, which is stillunder debate (Alexandrova et al. 2009, 2012; Sahraoui et al.2010, 2012; Chen et al. 2012; He et al. 2012; Salem et al. 2012).

In Section 2, we present the scenario we have in mind forhow the electron distribution function evolves as the electronsstream away from the Sun. Section 3 introduces themathematical framework of the quasilinear theory of wave–particle interactions and a conceptual discussion of whyoblique fast-magnetosonic/whistler (FM/W) waves are apromising candidate for scattering strahl electrons into thehalo. In Section 4, we present analytic expressions for theinstability threshold of oblique FM/W waves in two differentβc regimes. Section 5 compares our theoretical results within situ electron observations in the solar wind. In Section 6, wediscuss other electron-heat-flux instabilities and relate them toour predictions. Section 7 summarizes and concludes ourtreatment.

2. Radial Evolution of the Strahl and Halo

Measurements of the core, halo, and strahl densities (nc, nh, andns, respectively) at different heliocentric distances show that ns/nedecreases with distance, while nh/ne increases, where ne=nc+nh+ns (Maksimovic et al. 2005; Graham et al. 2017). Thequantity +n n nh s e( ) , however, remains almost constant between0.3au and 2au (Štverák et al. 2009). This observation is strikingevidence for the notion that the halo and strahl populations areclosely related to each other, and that the electron halo is the resultof a scattering of strahl electrons into the halo. These observationshave given rise to the following paradigm for the radial evolutionof superthermal electrons in the solar wind:

1. electrons are heated to superthermal energies in (or veryclose to) the solar corona (Vocks & Mann 2003; Owenset al. 2008);

2. magnetic-moment conservation in the expanding magn-etic field focuses the superthermal particles into the strahl;

3. microinstabilities, which constrain the strahl to stableregions in parameter space, regulate the strahl and theassociated heat flux and generate waves that scatter thestrahl into the halo during the passage of the plasmathrough the heliosphere, restricting the strahl to stableregions of parameter space.

In this article, we address point 3 of this scenario.

3. Resonant Wave–Particle Interactions

We consider a plasma consisting of protons (index p),core electrons (index c), and strahl electrons (index s) witha background magnetic field of the form =B B0, 0,0 0( ). We

neglect the halo for the sake of simplicity. We perform allcalculations in the reference frame that moves with the protonbulk velocity. The plasma fulfills the quasi-neutrality condition,

= +n n n , 10p 0c 0s ( )

and carries no field-parallel currents on average:

+ =n U n U 0, 20c 0c 0s 0s ( )

where n0j and U0j are the equilibrium number density and theequilibrium bulk velocity of species j, respectively.

3.1. Quasilinear Theory of Resonant Wave–ParticleInteractions

Quasilinear theory describes the evolution of a plasma underthe effects of resonant wave–particle interactions. Prerequisitesfor the application of this description are small amplitudes andsmall growth or damping rates (i.e., g wk kr∣ ∣ ∣ ∣ ) of theresonant waves, where ωkr is the real part of the frequency ωkat wavevector k, and γk is the imaginary part. Theseassumptions imply that the background distribution functionand the wave amplitudes change on a timescale that is muchlonger than the wave periods. We use a cylindrical coordinatesystem for the velocity with the components v⊥ and vPperpendicular and parallel to B0. In the same coordinatesystem, the wavevector components are given by k⊥ and kP.We denote the azimuthal angle of the wavevector as f.Resonant particles of species j diffuse in velocity space

according to the equation (Stix 1992)

òå pd w y

¶

¶=

´ - - W

¥ =-¥

+¥

^^

f

t

q

m VvGv

k v n Gf d k

lim8

1

, 3

j

V n

j

j

k j kj n

j

2

2 2

r, 2 3

ˆ

( )∣ ∣ ˆ ( )

where fj is the distribution function of species j,

w wº -

¶¶

+¶¶^

^G

k v

v

k v

v1 , 4

k kr r

ˆ ( )⎛⎝⎜

⎞⎠⎟

and

y º +

+

f f+

--

^

E e J x E e J x

v

vE J x

1

2

. 5

kj n

ki

n j ki

n j

kz n j

,,r 1 ,l 1[ ( ) ( )]

( ) ( )

The argument of the νth-order Bessel function Jν(xj) is given byº W^ ^x k vj j, and Ωj≡qjB0/mjc is the (signed) cyclotron

frequency of species j, where qj and mj are the particle chargeand mass. The quantities

òº -E k E xt t e d x, , 6k xk Vi 3( ) ( ) ( )·

and

òº -B k B xt t e d x, , 7k xk Vi 3( ) ( ) ( )·

are the Fourier transforms of the electric and magnetic fieldsE x t,( ) and B x t,( ), after these fields have been multiplied by a

2

The Astrophysical Journal, 886:136 (11pp), 2019 December 1 Verscharen et al.

window function of volume V.7 The left and right circularlypolarized components of the electric field are given by

º +E E iE 2k kx ky,l ( ) and º -E E iE 2k kx ky,r ( ) .When quasilinear diffusion occurs according to Equation (3),

particles diffuse in velocity space along curves of constant energyin the wave frame (i.e., the reference frame that moves with thespeed w kkr along B0). This diffusive flux of particles is locallytangent to semicircles centered on the parallel phase velocity

wºv kkph r (see also Figure 1), which satisfy the equation

- + =^v v v constant. 8ph 22( ) ( )

At the same time, Equation (3) allows for diffusion only fromhigher phase-space densities to lower phase-space densities.Only waves and particles fulfilling the resonance condition

w = + Wk v n 9k jr ( )

participate in the resonant wave–particle interaction due to theδ-function in Equation (3).

Complementary to Equation (3), Kennel & Wong (1967)calculated the growth/damping rate of waves with g wk kr∣ ∣ ∣ ∣in quasilinear theory and found that the contribution of species

j to γk is given by

åg g==-¥

+¥

, 10kj

nkj n, ( )

where

ò

ò

gw

p w ww

dw y

=

´ -- W

¥

^ ^

+¥

-¥

n kdv v

dv vn

k

Gf

W

8

,

11

kj n

k j

k j

k

k j kj n

j

k

,

r 0

r p

r

2

0

2

r, 2

0^

∣ ∣

∣ ∣( )

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

p wweº +

¶¶ w w=

B B E EW1

16, 12k k k k kh

kr

* *· · ( ) ( )⎡⎣⎢

⎤⎦⎥

eh denotes the Hermitian part of the dielectric tensor, and f0j isthe background distribution function of species j.8 The plasma

frequency of species j is defined by w pº n q m4 j j jpj 02 . We

assume that the background distribution functions f0j arerepresented by drifting Maxwellians,

p= -

+ -^fn

w

v v U

wexp , 13j

j

j

j

j0

0

3 2 3

20

2

2

( )( )

⎛⎝⎜⎜

⎞⎠⎟⎟

where ºw k T m2j j jB 0 is the thermal speed of species j, kB isthe Boltzmann constant, and T0j is the equilibrium temperatureof species j. The set of Equations (10) and (3) couple theevolution of the waves and particles in the presence of resonantwave–particle interactions and fulfill energy conservation(Kennel & Wong 1967; Chandran et al. 2010).

3.2. Conceptual Predictions of Quasilinear Diffusion

Figure 1 shows quasilinear diffusion paths for resonant strahlparticles in velocity space. The relative alignment of thegradients of f0j in velocity space and the semicircles given byEquation (8) determines if the resonant particles lose or gainkinetic energy during the quasilinear diffusion process. In case(a), 0

is a necessary condition in order for strahl electrons to losekinetic energy through a wave–particle resonance. At thispoint, we limit ourselves to ωkr>0 and kP>0 without loss ofgenerality.9

The frequency ωkr is associated with the wavevector through thelinear dispersion relation. Figure 2 shows two representative plotsfor the FM/W-wave dispersion relation (for details, see Section 4)with ωkr>0 and kP>0. In addition, we plot the resonancecondition in Equation (9) for n=−1 and n=+1. Resonantinteractions only occur for waves and particles for which the linecorresponding to the resonance condition intersects the plot of thedispersion relation. Figure 2 illustrates that strahl electrons with

>v 0 only fulfill the resonance condition with FM/W waveswith ωkr>0 and kP>0 through resonance with n 0. Theycannot fulfill the resonance condition with these wavesthrough resonances with -n 1 since w < Wkr e∣ ∣ for allk and Ωe 0 at the resonance speed),which is not observed (see also Section 6.4). Moreover,instabilities driven via the n=0 strahl resonance are unable toaccount for the halo formation since they cause particles to diffuseonly in vP and not in v⊥.

Although we will allow for oblique modes, we still limitourselves to rk̂ 1e to avoid cyclotron damping by the core,where r º Wwe e e∣ ∣, a point that we discuss further inSection 4.1 below.

When rk̂ 1e and n>0, the only non-negligible term inEquation (5) is Ek,l e

−if Jn−1(xj) which, moreover, is nonzeroonly when n=+1, because nJ x 0j( ) for x 0j for alln ¹ 0. Therefore, in order for electrons to undergo an n>0resonance with FM/W waves with rk̂ 1e , the FM/Wwaves must have a left-circularly polarized component (i.e.,

¹E 0k,l ). This requirement rules out the parallel-propagating

FM/W wave, which is purely right-circularly polarized (i.e.,Ek,l= 0), forcing us to consider obliquely propagating FM/Wwaves.10

Figure 1(b) shows a case that satisfies all of theserequirements for the instability of the oblique FM/W wave.The diffusing particles increase in v⊥ and (slightly) decrease invP. These particles are the seed for the halo population.However, scattering by the initially excited FM/W waves doesnot fully describe the formation of the observed halo since it isrestricted to particles in a certain range in vP that fulfill theresonance condition. The scattered seed population, however,represents a strong deformation of the electron distributionfunction, which may eventually relax through secondaryinstabilities into a more symmetric halo about the electroncore. A detailed study of the nonlinear evolution of the electronsystem is, however, beyond the scope of this work.

4. Instability of the Oblique Fast-magnetosonic/Whistler Mode

In this section, we derive approximate analytical expressionsfor the instability thresholds of the oblique FM/W mode in aplasma containing an electron strahl. Gary et al. (1975a, 1975b)refer to this instability as the “magnetosonic instability.” Aninstability of the oblique FM/W mode has recently beendiscussed in the context of the solar wind (Horaites et al. 2018;Vasko et al. 2019). This instability is also a candidate toexplain heat-flux regulation in other astrophysical plasmas suchas the intracluster medium (Roberg-Clark et al. 2016, 2018)and in solar flares (Roberg-Clark et al. 2019).

4.1. Instability Mechanism and Dispersion Relations

We assume that the instability drive by strahl electrons ismost efficient at the center of the strahl distribution function,i.e., by particles with =v U0s and =v̂ 0. Through then=+1 resonance, the oblique FM/W instability only occursat frequencies of about

w = - Wk U 15r 0s e∣ ∣ ( )

according to Equation (9).We discuss the properties of the FM/W mode at different

angles of propagation in the wr-kP plane in Figure 3, where weshow four solutions from the full dispersion relation of a hotelectron–proton plasma. We use the linear Vlasov–Maxwellsolver NHDS (for details on the numerics, see Verscharen et al.2013b; Verscharen & Chandran 2018).In addition, we show Equation (15). The intersection between

this line and a plot of the dispersion relation indicates awavenumber and frequency at which the resonance conditionbetween the wave and an electron with =v U0s is fulfilled. Sincehighly oblique modes cease to exist in plasmas with Maxwellianbackground distributions at large b pº n T B8p 0p 0p 0

2 due toincreasing Landau damping, we apply lower βp for the highlyoblique cases in Figure 3.11 We note that k⊥ in the case withq = 89 is by a factor q ~tan 57 greater than kP. The FM/W

Figure 2. Dispersion relation of the FM/W wave for θ=0° (solid black) andθ=60° (dashed black) from Equation (16), as well as the strahl resonanceconditions according to Equation (9) with =v v1.5 Ae for n=+1 (red) andn=−1 (blue). Strahl electrons with >v 0 can only resonate with waves withvph>0 through resonances with n0.

9 Our arguments for instability also apply to a configuration in which U0s <vph

mode exists in two regimes in the wavenumber range in whichEquation (15) can be fulfilled—i.e., where the line of the strahlresonance intersects with the corresponding plot of thedispersion relation. Regime 1 is the whistler regime in whichthe angle of propagation fulfills q 0 70 , and

w »W +

kk v

k d1, 16kr

Ae2

e2

e2∣ ∣( )

( )

where º Wd ve Ae e∣ ∣ is the electron inertial length.Equation (16) follows from the cold-plasma dispersion relationin a plasma with a single electron species under the assumptionthat kd 1p , where dp is the proton inertial length, and isapproximately valid for small βc and βp. The dispersionrelation asymptotes toward q~ W cose∣ ∣ for large kP providedthat q m mcos2 e p. In the highly oblique limit (regime 2; i.e.,

q m mcos2 e p), the wave propagates in the lower-hybridregime. Its frequency asymptotes toward a frequency of orderthe lower-hybrid frequency,

ww

º+

w

W1

, 17LHpp

pe2

e2

( )

as long as thermal corrections are small (Verdon et al. 2009).

4.2. Analytical Instability Thresholds

To determine whether the oblique FM/W wave is unstable,we must consider not only the instability drive provided by thestrahl, but also the possibility of damping by the core electrons.According to Equations (9) and (13), cyclotron damping by thecore with n=−1 occurs at wavenumbers and frequencies thatfulfill

w- + W + W k w k w . 18c e r c e∣ ∣ ∣ ∣ ( ) Landau damping by the core with n=0 occurs at wavenum-bers and frequencies that fulfill

w- k w k w . 19c r c ( )

Strahl driving with n=+1 occurs at wavenumbers andfrequencies that fulfill Equation (15).Figure 4 shows the plot of the dispersion relation from

Equation (16), the strahl-resonance line from Equation (15),and the parameter space in which core Landau damping andcore cyclotron damping act from Equations (18) and (19).For this plot, we have assumed that w vc Ae . To a firstapproximation, the FM/W wave is unstable if there is awavenumber range in which Equation (15) is fulfilled andEquations (18) and (19) are not fulfilled. The resonance line inFigure 4 represents the minimum U0s for which strahl drivingcan occur in a wavenumber range in which neither core cyclotrondamping nor core Landau damping acts. In order for the dispersionrelation to intersect this resonance line within the white triangle,two conditions must be met: (1) βe must be small (otherwisethe dispersion relation will lie within the Landau-damped region—see Figure 4), and (2) θ must be 60 . When these conditions aresatisfied, the frequency of the resonant waves is given by

w » W1

2. 20kr e∣ ∣ ( )

Furthermore, it follows from Equations (18) and (19) that thewavenumber of the resonant waves satisfies

= ºW

k kw

1

2, 21crit

e

c

∣ ∣ ( )

where kcrit is the minimum wavenumber at which the green andblue regions in Figure 4 overlap; i.e., the minimumwavenumber at which both core Landau damping and corecyclotron damping can occur. Combining Equations (15), (20),and (21) leads to the instability criterion

U w3 . 220s c ( )For βc∼1, the dispersion relation lies below the Landau-

damping threshold from =k 0 to =k kcrit . This situation isillustrated in Figure 5. We now determine the instabilitythreshold in this regime by balancing the destabilizing effectsof the strahl against the stabilizing effects of core Landaudamping. Under these assumptions, the FM/W mode is

Figure 3. Hot-plasma dispersion relations for the FM/W mode at differentangles of propagation in an electron–proton plasma with T0p=T0e. We neglectthe influence of the strahl on the dispersion relation. For the parallel-propagating mode and for the mode with θ=60°, we use βp=1. For the othertwo modes, βp=0.001 in order to avoid strong linear damping. The strahl-resonance line shows Equation (15) for U0s=3vAe.

Figure 4. Dispersion relation and resonance conditions for the FM/W modewith θ=60° in the low-βc case. The black line shows Equation (16). The blueand green areas show Equations (18) and (19), respectively, and the red lineshows Equation (15) with U0s=3wc. We use wc=0. 2vAe. This situationrepresents a marginally stable state for the oblique FM/W instability.

5

The Astrophysical Journal, 886:136 (11pp), 2019 December 1 Verscharen et al.

unstable if there is a wavenumber range in which

g g+ >=+ = 0. 23kn

kns, 1 c, 0 ( )

Using Equation (11), we derive the instability criterion in theAppendix. We find that the FM/W mode is unstable if

qq q

+-

U nn

T

Tv w2

1 cos

1 cos cos240s

0c

0s

0s

0cAe2

c2

1 4( )( )

( )⎡⎣⎢

⎤⎦⎥

for the βc∼1 case. The minimum of the right-hand side ofEquation (24) suggests that the lowest threshold occurs for

q = - +cos 1 2 , i.e., for θ≈65°.6. The transition betweenthe low-βc case and the βc∼1 case occurs when wc is largeenough that the inequality in Equation (19) encompasses theentire plot of the dispersion relation. By combiningEquation (16) for θ=60° and Equation (19), we find thatthis transition occurs when

w v2

. 25cAe ( )

We compare our analytical thresholds from Equations (22)and (24) with numerical solutions of the hot-plasma dispersionrelation from our NHDS code in Figure 6.

Our analytical instability thresholds agree well with ournumerical solutions. The transition between the low-βc caseand the βc∼1 case occurs at wc≈0.2vAe, which is slightlybelow our analytical finding in Equation (25). We attribute thisdifference to inaccuracies based on our assumption of a discreteonset of Landau damping as soon as ωkrkPwc.

5. Comparison with Observations

We use data from the 3DP instrument on board the Windspacecraft (Lin et al. 1995), obtained between 1995 and 1998.With an automated routine (Pulupa et al. 2014), we fit thedistribution function as a combination of a bi-Maxwelliandistribution (core) and a bi-κ-distribution (halo):

= +f f f 260e 0c 0h ( )

with

p= -

--

-

^

^ ^

^

fn

w w

v U

w

v U

wexp

27

0c0c

3 2c

2c

0 c2

c2

0 c2

c2

( ) ( )

( )

⎛⎝⎜⎜

⎞⎠⎟⎟

and

p kk

k

k

=-

G +G -

´ +-

-+

-k

^

^ ^

^

- +

fn

w w

v U

w

v U

w

2

2 3

1

1 2

12

2 3,

28

0h0h

h2

h

3 2

0 h2

h2

0 h2

h2

1

( )( )

( )

( ) ( )

( )

( )⎪

⎪

⎪

⎪

⎡⎣⎢

⎤⎦⎥

⎧⎨⎩

⎡⎣⎢⎢

⎤⎦⎥⎥

⎫⎬⎭

where Γ(x) is the Γ-function, and the fit parameters are n0c, ŵ c,w c , ^U0 c,U0 c , n0h, ŵ h, w h , ^U0 h,U0 h , and κ. We determinethe strahl bulk parameters as the result from subtracting theobserved total electron distribution from the fit result andcalculating the numerical moments of the remaining strahldistribution. In this way, we obtain the densities, relative driftspeeds, temperatures, and temperature anisotropies of allelectron species as well as the κ-index of the halo distribution.We bin the data distribution in the n0s/n0c versus U0s/vAe planeand count the number of data points in each bin. We show theresult in Figure 7. We limit ourselves to cases in which

n n 0.0050s 0c since our automated method for the determi-nation of the observed strahl bulk parameters suffers a loss ofaccuracy for smaller relative strahl densities.We overplot numerical results for the instability threshold of

the oblique FM/W instability for a maximum growth rate ofg = W-10m

3e∣ ∣ from NHDS. We evaluate the threshold at the

angle of propagation that leads to the maximum growth rate γm.This angle varies between 51° and 67° in the shown range. Weuse the following free parameters: βp=1, T0c=T0p,T0s=T0p, and vAp/c=10

−4. All species are isotropic. These

Figure 5. Dispersion relation and resonance conditions for the FM/W modewith θ=60° in the βc∼1 case. The black line shows Equation (16). The blueand green areas show Equations (18) and (19), respectively, and the red lineshows Equation (15) with U0s=4wc. We use wc=vAe.

Figure 6. Comparison of Equations (22) and (24) with numerical solutions ofthe hot-plasma dispersion relation from our NHDS code. The blue and orangelines show Equations (22) and (24), except that the signs have been replacedwith equal signs. We use =w w2s c and T0p=T0c. For the numerical solutions,we show isocontours of constant maximum growth. The analytical solutionsuse θ=65°. 5, while the numerical solutions are evaluated at the angle forwhich the lowest U0s leads to a maximum growth rate of g = W-10m

3e∣ ∣.

6

The Astrophysical Journal, 886:136 (11pp), 2019 December 1 Verscharen et al.

choices represent typical values for these parameters at 1 auconsistent with our data set. In addition, we overplotEquation (24) for θ=65°.5, and wc=vAe=ws. We considerthe parameter space to the lower left of the plotted instabilitythreshold as the stable parameter space, while we consider theparameter space to the upper right of this curve as the unstableparameter space. The instability threshold restricts the data inthis parameter space to stable values, while only an insignif-icant number of data points populate the unstable parameterspace. This finding is broadly consistent with our argument thatthe oblique FM/W instability sets the upper limit to U0s in thesolar wind. We also note that the numerical solution and ouranalytical solution agree reasonably well, especially at smalln0s/n0c, the regime in which the strahl effect on the dispersionrelation is negligible as assumed in our derivation. We note thatsome of the parameter combinations shown in Figure 7 exhibita bump-on-tail configuration when the model Maxwellian coreand strahl distributions are summed, which can be unstable toother instabilities (see Section 6.4). However, we neglectbump-on-tail instabilities in this paper, because, in a morerealistic model, the total electron distribution function would bea monotonically decreasing function of v∣ ∣ . Under realisticsolar-wind conditions, the number of halo electrons that are inresonance with the unstable FM/W waves is small. Therefore,we conjecture that our simple core/strahl model of the electrondistribution provides a reasonable approximation of the obliqueFM/W instability threshold. However, future investigationsbased on more realistic core/halo/strahl electron distributionswill be needed in order to test this conjecture.

6. Relation to Other Electron-driven Instabilities

In this section, we discuss the relevance of other electron-driven instabilities to the evolution of the electron strahl in thesolar wind. We specifically address the consistency of thesealternative instabilities with the strahl-scattering scenariodescribed in Section 2. Our reasoning relies on the

considerations presented in Section 3 of the diffusion pathswith respect to the phase speed as well as a careful analysiswith NHDS.

6.1. Whistler Heat-flux Instability

The parallel-propagating FM/Wwave is purely right-handedin polarization (i.e., Ek,l= Ekz= 0). Equation (5) for =k̂ 0,therefore, requires that the only contributing resonant interac-tion is the cyclotron resonance with n=−1. The resonantparticles driving this instability must move in the oppositedirection along the magnetic field as the wave (i.e., vP< 0 inour convention) in order to fulfill the resonance condition,Equation (9). This property characterizes the halo rather thanthe strahl. If unstable, this mode corresponds to the whistlerheat-flux instability (Gary & Feldman 1977; Gary et al. 1999;Wilson et al. 2009, 2013; Shaaban et al. 2018a). The diffusionpaths for this instability are shown in Figure 8 (see alsoShaaban et al. 2019).The resonant particles diffuse toward smaller values of v⊥

and form a tail-like structure in the distribution. This behaviordoes not agree with the scenario that strahl scattering forms thehalo distribution function as discussed in Section 2. For thesereasons, we exclude the parallel whistler heat-flux instability asa candidate for a plasma instability that scatters the strahl intothe halo. This instability can, however, be relevant for theregulation of the halo heat flux in the solar wind. Its thresholdthen also depends on the halo anisotropy (Shaaban et al.2018b). We note that a reduction of U0c can lead to an indirectreduction of U0s by the fulfillment of quasi-neutrality accordingto Equation (2). This indirect effect is also relevant for the ion-acoustic heat-flux instability and for the KAW heat-fluxinstability. Evidence for the whistler heat-flux instability wasfound in measurements of solar-wind core and halo electrons(Tong et al. 2019).

6.2. Lower-hybrid Fan Instability

In the limit q < m mcos2 e p, the FM/W-mode branchcorresponds to the lower-hybrid mode as shown in Section 4.1.The fan instability of the lower-hybrid mode is driven by then=+1 resonance of strahl electrons like the oblique FM/Winstability (Omelchenko et al. 1994; Shevchenko & Galinsky2010). It scatters particles about the parallel phase speed ofthe lower-hybrid wave, w=v kph LH , and is, thus, capable of

Figure 7. Data distribution of the analyzed solar-wind interval in the n0s/n0cversus U0s/vAe plane. The color-coding shows the probability density (i.e., thenumber of counts per bin, normalized by the corresponding bin area to accountfor the logarithmic scaling of both plot axes) in arbitrary units. We determinethe observed values for n0s and U0s by taking moments of the strahldistribution. The black line shows the isocontour of maximum growth rateg = W-10m

3e∣ ∣ for the oblique FM/W instability from our NHDS solutions.

The red dashed line shows Equation (24) for θ=65°. 5, and wc=vAe=ws.

Figure 8. Diffusion paths for the parallel whistler heat-flux instability. Theelectron beam must fulfill U0s > vph and be hot enough to have a sufficientnumber of electrons at vP

scattering strahl electrons into the halo. The highly oblique lower-hybrid mode has a strong electrostatic component and relativelylow frequencies compared to the moderate-θ FM/W mode.Therefore, it is prone to strong core Landau damping if wc is largeenough that the core provides a significant number of electrons at

»v vph . In this case, the resonance speed for Landau-resonantcore electrons lies deep within the core distribution function. Withincreasing wc, the phase speed of the lower-hybrid wave increasesslightly due to thermal corrections to its dispersion relation. Thiseffect can overcompensate the increasing number of Landau-resonant core electrons at the strahl resonance speed. However, thegrowth rate of the highly oblique lower-hybrid mode is still lessthan the growth rate of the oblique FM/W instability by about twoorders of magnitude under typical solar-wind conditions. As U0sincreases, U0c∣ ∣ must increase so that the parallel current vanishesas per Equation (2). In cases with very low βc, the lower-hybrid faninstability is a good candidate for self-induced strahl scattering.This situation is illustrated in Figure 9.

The instability is related to the oblique FM/W instabilitysince it is driven by the same resonance effect and transitionsinto the oblique FM/W mode for smaller θ. We note that someauthors use a broader definition of the term fan instabilities forall instabilities driven by an n=+1 resonance (e.g., Krafft &Volokitin 2003).

6.3. Ion-acoustic Heat-flux Instability and Kinetic-Alfvén-waveHeat-flux Instability

The ion-acoustic heat-flux instability (Gary 1978) has acomparable instability threshold to the threshold of the obliqueFM/W instability under certain conditions. However, thisinstability acts on particles with »v vph . The parallel phasespeed of the ion-acoustic mode is much less than both wc andthe typical strahl speed in the solar wind. Therefore, it is morelikely that this instability is driven by the core drift, which isdirected in the opposite direction of the strahl speed accordingto Equation (2). For this case, we illustrate the quasilineardiffusion of particles under the action of the ion-acousticinstability in Figure 10. The ion-acoustic heat-flux instability isdriven by the Landau resonance of the core electrons, whichleads to a diffusion of core electrons at vP=vph toward smallerv∣ ∣ . This instability is not a candidate to explain strahlscattering into the halo since it does not increase v⊥ of strahlelectrons. We also note that large T T0c 0p are required inorder to increase vph of the ion-acoustic mode and to avoid

proton Landau damping, which would otherwise efficientlysuppress this instability. The maximum growth rate of the ion-acoustic heat-flux instability occurs in parallel propagation.The same instability mechanism is responsible for the

kinetic-Alfvén-wave (KAW) heat-flux instability (Gary et al.1975a). In the parameter range explored by Gary et al. (1975a),the KAW heat-flux instability requires low b - 10c 3 to havethe lowest threshold of all electron-drift driven instabilities. Itpropagates at very large angles θ with respect to B0. Thenonzero Ekz of the KAW allows for the electron core to drivethe mode unstable through the Landau resonance as in the caseof the ion-acoustic heat-flux instability. Likewise, thisinstability does not scatter strahl particles into the halo.

6.4. Electrostatic Electron-beam Instability

The electrostatic electron-beam instability (Gary 1978) is alow-βc instability that propagates into the direction of the strahland has maximum growth parallel to B0. Its typical phase speedis vphU0s. For βc0.1, it has lower thresholds than theother instabilities, as long as the strahl forms a bump-on-tailconfiguration rather than a shoulder of the distribution while allother parameters are kept at representative solar-wind values. Itis driven by the Landau resonance of the strahl electrons. Weillustrate this instability in Figure 11.The Landau-resonant diffusion of strahl electrons leads to an

excitation of this instability only if the diffusion paths down thegradients of the strahl distribution function are directed to

Figure 9. Diffusion paths in a core–strahl electron distribution function for thelower-hybrid fan instability. The core distribution is shown as blue dashedsemicircles and the strahl distribution as a red shaded semicircle. The particlediffusion path (red arrow) is locally circular (indicated by black semicircles)about the parallel phase speed vph=ωkr/kP. The green circle segment indicatesconstant kinetic energy. βc is small so that the distribution functions of core andstrahl do not overlap sufficiently for strong core Landau damping to suppressthe instability.

Figure 10. Diffusion paths in a core–strahl electron distribution function for theion-acoustic heat-flux instability and for the KAW heat-flux instability. Thecore distribution is shown as blue dashed semicircles and the strahl distributionas a red shaded semicircle. The particle diffusion path (blue arrow) is purelyparallel to the magnetic field and occurs at vP=vph. In the case of the ion-acoustic heat-flux instability, T0c?T0p so that the phase speed is large enoughto overcome proton Landau damping. For the KAW heat-flux instability, thecore temperature is typically small βc 10−3.

Figure 11. Diffusion paths in a core–strahl electron distribution function for theelectrostatic electron-beam instability. The core distribution is shown as bluedashed semicircles and the strahl distribution as a red shaded semicircle. Theparticle diffusion path (red arrow) is purely parallel to the magnetic field andoccurs at vP=vph. The green circle segment indicates constant kinetic energy.βc is small so that core-Landau damping is inefficient in suppressing theinstability. The strahl speed has to be slightly greater than vph in order to yieldan kinetic-energy loss of strahl electrons during the diffusion.

8

The Astrophysical Journal, 886:136 (11pp), 2019 December 1 Verscharen et al.

lower kinetic energies. For this reason, U0s has to be slightlygreater than vph as illustrated in Figure 11. This instability onlyreduces vP and does not increase v⊥ of the resonant strahlelectrons and is thus not a candidate mechanism to scatter strahlelectrons into the halo. We also note that, if this instability wereto operate in the solar wind, it would lead to a quasilinearflattening of the electron distribution function in at least somenarrow range of parallel velocities near the strahl velocity.However, the measured electron distribution functions arestrongly decreasing functions of vP near the strahl velocity(Pilipp et al. 1987a; Marsch 2006).

7. Conclusions

Electron distribution functions in the solar wind consist ofthree components: a core, halo, and strahl. The relative driftsamong these populations carry a significant heat flux from thesolar corona into the heliosphere. It is, therefore, of greatimportance for global solar-wind models to understand theregulation of the relative drifts between these electroncomponents. Observations indicate that strahl electrons arecontinuously transferred into the halo. We propose a mech-anism that explains this scattering as the consequence of a self-induced excitation of the oblique FM/W instability. In thisscenario, we assume that electrons are accelerated to highenergies in the solar corona. We furthermore assume that theconservation of the magnetic moment in the wideningmagnetic-field structure of a coronal hole focuses theseenergetic electrons into the antisunward direction, formingthe electron strahl. Based on these assumptions, we find that thestrahl itself then quasi-continuously excites an instability of theoblique FM/W wave that scatters strahl electrons into the haloand generates plasma waves with wavelengths between the ionand electron kinetic scales. This instability reduces the strahldensity, increases the halo density, and limits the strahlheat flux.

In Section 4, we derive analytical expressions for thethresholds of the oblique FM/W instability in both the low-βcregime and the βc∼1 regime. In the low-βc regime, the strahlexcites FM/W waves when U0s is large enough that the strahlcan resonate with the waves at wavenumbers and frequencies atwhich core Landau damping and core cyclotron damping arenegligible. At βc∼1, on the other hand, core Landau dampingof FM/W waves cannot be avoided, and cyclotron driving bythe strahl must overcome this core Landau damping in order tomake the FM/W waves unstable.

In Section 5, we compare the instability thresholds of theoblique FM/W instability with direct in situ measurementsfrom the Wind spacecraft. We find that the instability limits thedata distribution to the stable regime. This finding corroboratesour hypothesis that the oblique FM/W instability indeed limitsthe strahl speed and heat flux in the solar wind. In the future,we will study the instability of observed electron distributionswith our ALPS code (Verscharen et al. 2018) without relyingon the assumption of a Maxwellian shape of the electroncomponents’ distribution in Equation (13), which is also madein our NHDS solutions.

Other electron-driven instabilities (whistler heat-flux, lower-hybrid fan, ion-acoustic heat-flux, kinetic-Alfvén-wave heat-flux, and electrostatic electron-beam instabilities) are either notcapable of scattering strahl electrons into the halo or (in thecase of the lower-hybrid fan instability) have growth ratesmuch smaller than the oblique FM/W instability.

Fully kinetic simulations, such as Vlasov or particle-in-cellsimulations, and simulations of the quasilinear diffusionequations (e.g., using the methods presented by Pongkitiwa-nichakul & Chandran 2014) will allow us to model thenonlinear evolution of the strahl–halo system in a futureanalysis. Parker Solar Probe (PSP) measures electrondistribution functions and waves in the close vicinity of theSun. We predict that PSP will encounter the low-bc regime ofthe oblique FM/W instability in which its threshold is given byEquation (22). In this case, we predict the presence of copiousFM/W waves with θ≈60° and w » W0.5r e∣ ∣. PSP may alsoencounter the point at which the strahl speed crosses thethreshold of the FM/W instability for the first time. Thisdiscovery will help us understand the range of distances wherethe oblique FM/W instability is relevant. The Solar Orbiterspacecraft will link the observed high-cadence and high-resolution in situ electron properties with the associated sourceregions in the corona in order to improve our understanding ofthe global evolution of electrons in the solar wind.

We appreciate helpful discussions with Sofiane Bourouaine,Georgie Graham, Allan Macneil, Rob Wicks, and Chris Owen.This work was discussed at the 2019 ESAC Solar WindElectron Workshop, which was supported by the Faculty of theEuropean Space Astronomy Centre (ESAC). D.V.is supportedby the STFC Ernest Rutherford Fellowship ST/P003826/1 andSTFC Consolidated Grant ST/S000240/1. This work issupported in part by NASA contract NNN06AA01C, NASAgrant NNX16AG81G, NASA grant NNX17AI18G, NASAgrant 80NSSC19K0829, and NSF SHINE grant 1460190.Work at UC Berkeley is supported in part by NASA grantNNX14AC09G, NASA grant NNX16AI59G, and NSF SHINEgrant 1622498. S.D.B.acknowledges support from the Lever-hulme Trust Visiting Professorship program.

AppendixAnalytical Instability Criterion for βc∼1

Our calculation of the instability threshold is based onEquation (10) for the case in which strahl driving balances withthe stabilizing effects of core Landau damping. We firstcalculate expressions for y = Wk

nk

c, 0 2∣ ∣ and y =+ Wk n ks, 1 2∣ ∣ . Wethen use these expressions to derive γk

c,n=0 and γks,n=+1. In order

to simplify Equation (10), we approximate ωkr using the cold-plasma dispersion relation for an electron–proton plasma(Stix 1992), recognizing that this will introduce some errorinto our results.

A.1. Polarization of Oblique FM/W Waves

In the cold-plasma dispersion relation (Stix 1992),

=-iE

E

n S

D29kx

ky

2( )

and

q qq q

=- + + -iE

E

n n S S D

Dn

cos 1 cos

cos sin, 30kz

ky

4 2 2 2 2 2

2

( ) ( )

9

The Astrophysical Journal, 886:136 (11pp), 2019 December 1 Verscharen et al.

where wºn kc k is the refractive index and, in the whistler-wave regime,

w

wW -S 31

k

pe2

e2 2

( )

and

w-

WD S . 32

k

e ( )

Using Equation (16), Equations (29) and (30) simplify to

qq

+iEE

k d1 sin

cos33kx

ky

2e2 2

( )

and

qiE

Ek d sin . 34kz

ky

2e2 ( )

We choose the coordinate system in which f=0 so that=k̂ kx and ky=0. With the use of Equations (5), (16), (33),

and (34), we find

yq +

=

^W

v

vk d J x J x

E

Wsin . 35k

n

k

ky

k

c, 0 22

e2

0 c 1 c

2 2∣ ∣( ) ( )

∣ ∣( )

⎡⎣⎢

⎤⎦⎥

Since iEkx/Eky ? iEkz/Eky and rk̂ 1e , we retain only theterm proportional to J0 in ψk

s,n=+1. With the use ofEquations (5), (16), (33), and (34), we then obtain

y q qq

- +=+

W

k dJ x

E

W

1

4

1 cos sin

cos.

36

kn

k

ky

k

s, 1 2 2e2 2 2

2 02

s

2∣ ∣ ( ) ( )∣ ∣

( )

A.2. Dispersion Relation and Resonance Condition

By combining Equations (15) and (16), we find for theresonance condition

qq

+- + =

k d

k dkd

U

v

cos

1cos 1 0. 37

2e2

2e2 e

0s

Ae( )

We make the simplifying approximation that

º vU

1. 38Ae

0s( )

Solving Equation (37) using the method of dominant balance(Bender & Orszag 1999), we obtain

q» + +k d

v

U

v

U1

1

cos... . 39e

Ae

0s

Ae2

0s2

( )⎛⎝⎜

⎞⎠⎟

Upon substituting Equation (39) into Equation (16), we findthat the parallel phase velocity of the resonant wave is

wq q

q» + - +k

v

U

v

Ucos1

coscos 1 ... 40kr Ae

2

0s

Ae2

0s2 2

( ) ( )⎡⎣⎢

⎤⎦⎥

at the resonant wavenumber and frequency.

A.3. Evaluation of Growth and Damping Rates

We simplify the integrals in Equation (11) by exploiting theδ-function and the Bessel-function identities12

ò lW =nl

n¥

^ ^^ ^ - -^dv v J

k ve

we I

2, 41

j

v w jj

0

22

j j2 2 ( ) ( )

⎛⎝⎜

⎞⎠⎟

òl l l l

W=

´ - - ¢

nl

n n n

¥

^ ^^ ^ - -^dv v J

k ve

we

I I I

2

, 42

j

v w j

j j j j

0

3 24

j j2 2

{ ( ) [ ( ) ( )]} ( )

⎛⎝⎜

⎞⎠⎟

and

òl l

W W=

W

´ -

l

¥

^ ^^ ^ ^ ^ - ^ -^dv v J

k vJ

k ve

k we

I I

4

, 43

j j

v w j

j

j j

0

20 1

4

1 0

j j2 2

[ ( ) ( )] ( )

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

where Iν is the modified Bessel function of order ν andl º Wk̂ w 2j j j

2 2 2. Equations (41)–(43) follow from the identity(Watson 1922)

ò = -+

n n n¥

-dt tJ at J bt ep

a b

pI

ab

p

1

2exp

4 2

44

p t

0 2

2 2

2 2

2 2( ) ( )

( )

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

in the way presented by Stix (1992), which is valid forRe(ν)>−1 and p0, >k 0 , we find that to leading orderin ò,

gw

w ww p

l-=

k w

E W

845k

n

k

k

k

ky kc, 0

r

r

c

pc

r

2 2

c∣ ∣

( )⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

and

gw

w ww p

wqq

´W -

=+

k w

E W

n

n

T

T

8

4

1 cos

cos. 46

kn

k

k

k

ky k

k

s, 1

r

r

c

pc

r

2 2

0s

0c

0c

0s

e

r

2

2

∣ ∣

∣ ∣ ( ) ( )

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

Using these closed expressions for γkc,n=0 and γk

s,n=+1, theinstability condition in Equation (23) then translates to

qq

q- n

n

T

T

U

v

w

U4

1 cos

cos 2tan , 470s

0c

0c

0s

0s2

Ae2

2c2

0s2

2( ) ( )

which furthermore simplifies to

qq q

+-

Uv

n

n

T

T

w

v2

1 cos

1 cos cos. 480s

4

Ae4

0c

0s

0s

0c

c2

Ae2

( )( )

( )

This criterion can be rewritten as Equation (24).

12 Our result in Equations (45) and (46) can also be obtained by firstapproximating the Jν(x) term using their small-x expansions, which avoids theuse of the Bessel-function identities in Equations (41) through (43).

10

The Astrophysical Journal, 886:136 (11pp), 2019 December 1 Verscharen et al.

ORCID iDs

Daniel Verscharen https://orcid.org/0000-0002-0497-1096Benjamin D. G. Chandran https://orcid.org/0000-0003-4177-3328Seong-Yeop Jeong https://orcid.org/0000-0001-8529-3217Marc P. Pulupa https://orcid.org/0000-0002-1573-7457Stuart D. Bale https://orcid.org/0000-0002-1989-3596

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stract

1. Introduction2. Radial Evolution of the Strahl and Halo3. Resonant Wave–Particle Interactions3.1. Quasilinear Theory of Resonant Wave–Particle Interactions3.2. Conceptual Predictions of Quasilinear Diffusion

4. Instability of the Oblique Fast-magnetosonic/Whistler Mode4.1. Instability Mechanism and Dispersion Relations4.2. Analytical Instability Thresholds

5. Comparison with Observations6. Relation to Other Electron-driven Instabilities6.1. Whistler Heat-flux Instability6.2. Lower-hybrid Fan Instability6.3. Ion-acoustic Heat-flux Instability and Kinetic-Alfvén-wave Heat-flux Instability6.4. Electrostatic Electron-beam Instability

7. ConclusionsAppendixAnalytical Instability Criterion for βc ∼ 1A.1. Polarization of Oblique FM/W WavesA.2. Dispersion Relation and Resonance ConditionA.3. Evaluation of Growth and Damping Rates

References