Inertial Alfven wave acceleration of solar flare electrons

K. G. McClements

k.g.mcclements@ukaea.org.uk

UKAEA Culham Division, Culham Science Centre, Abingdon, Oxfordshire, OX14 3DB, UK

and

L. Fletcher

lyndsay@astro.gla.ac.uk

University of Glasgow, Department of Physics and Astronomy, Glasgow, G12 8QQ, UK

ABSTRACT

The possibility that electrons could be accelerated by inertial Alfven waves

to hard X-ray-emitting energies in the low solar corona during flares is investi-

gated theoretically. This investigation is prompted in part by recent microwave

observations indicating that the coronal magnetic field is strong enough that the

Alfven velocity cA above active regions could be of the order of a tenth of the

speed of light or more; electrons can be accelerated to velocities in excess of cA

on collisionless timescales via reflection by a single inertial Alfven wave pulse. It

is shown that the fraction of particles accelerated is a sensitive function of the

initial electron temperature and the transverse length scale δx of the shear Alfven

wave pulse; under typical pre-flare coronal conditions, a significant fraction of the

electron population can be accelerated if δx is of the order of a few metres or

less.

Subject headings: acceleration of particles - MHD - plasmas - Sun: flares - Sun:

corona - waves

1. Introduction

It has been proposed recently that the energy released in solar flares is transported in the

first instance via Alfven waves rather than fast particles (Fletcher & Hudson 2008). Due to

two-fluid and/or kinetic effects it is possible for the electric field associated with such waves

– 2 –

to have a component Ez that is aligned with the ambient magnetic field; particle acceleration

can then occur. Alfven waves with finite Ez are described as inertial (rather than kinetic)

when the plasma beta β is less than the ratio of electron to ion mass; Fletcher & Hudson

(2008) have argued that β values as low as this are likely to occur in the low solar corona

in the core of active regions, and particularly above sunspots. There is strong evidence that

inertial Alfven waves accelerate electrons in polar regions of the Earth’s magnetophere and

ionosphere (Thompson & Lysak 1996; Stasiewicz et al. 2000; Chaston et al. 2002; Chaston

et al. 2008); waves of this type have also been generated and detected in the laboratory,

specifically in the LArge Plasma Device (LAPD) at UCLA (Gekelman et al. 1994).

In the inertial regime Ez is smaller than the perpendicular electric field Ex of the wave

by a factor of the order of the square of the electron collisionless skin depth δe divided by the

product of the parallel and perpendicular wavelengths (Chaston et al. 1996). Given that δe

is less than a metre in the solar corona whereas magnetohydrodynamic (MHD) length scales,

including the wavelengths of MHD modes, are typically measured in Mm (1 Mm= 106m),

it is tempting to assume that the parallel electric fields of Alfven waves are too small to be

of any importance in this environment. However, as noted by Fletcher & Hudson (2008),

the perpendicular electric field of an Alfven wave in the solar corona can be very large (up

to around 106Vm−1 for coronal magnetic fields of the order of 0.05T and electron number

densities of 1015m−3), and the macroscopic length scales in flares are such that parallel

electric fields of less than 1 Vm−1 can be sufficient to accelerate electrons to relativistic

energies. Moreover, it is likely that waves are present in the solar atmosphere whose parallel

and perpendicular wavelengths are too small to be resolved using existing detectors, which

are currently restricted to spatial resolutions of around 150 km in the lower atmosphere,

achievable with the Hinode Solar Optical Telescope. Therefore Ez/Ex could be significantly

larger than estimates of this ratio based on typical MHD length scales. Indeed Chaston et

al. (2008) have recently presented evidence from data obtained using the FAST spacecraft

that terrestrial aurora are powered by a turbulent cascade of Alfven wave energy from MHD

scales down to perpendicular wavelengths of the order of δe. In these circumstances it is

appropriate to examine quantitatively the issue of whether inertial Alfven waves could play

a significant role in the acceleration of electrons in solar flares; this is the purpose of the

present paper. In previous work on this topic, Kletzing (1994) demonstrated the Fermi-like

acceleration in inertial Alfven waves of a small fraction of electrons, up to speeds of twice the

Alfven speed. He argued that this is a plausible scenario for auroral electron acceleration.

In the solar context, Stasiewicz & Ekeberg (2008) investigated solitary wave solutions to the

one- and two-fluid MHD equations, finding that electrostatic potentials of up to hundreds

of kilovolts could be generated under coronal conditions, but did not explicitly address the

conditions for electron acceleration to occur.

– 3 –

In Sect. 2 we discuss a pulse-like solution of the equations describing shear Alfven waves,

and establish the conditions in which this solution is modified significantly by the Hall term

in the generalized Ohm’s law. This analysis demonstrates that the Hall term can be neglected

provided that the parallel scale length of the pulse is greater than the ion skin depth δi, and

the perpendicular scale length is less than the product of the relative perturbation amplitude

with δi. In Sect. 3 we use our model inertial Alfven wave profile to derive a simple condition

for electrons to be reflected and hence accelerated by the pulse. Expressions are obtained

for the fraction of electrons accelerated and the electron energy spectrum as functions of the

model parameters. We conclude in Sect. 4 that inertial Alfven waves could indeed play a

significant role in flare electron acceleration, and thus warrant further investigation in the

solar context. We also comment briefly on other possible astrophysical applications of this

mechanism.

2. Inertial Alfven wave fields

We consider a uniform equilibrium plasma with magnetic field B = B0z, z being the

unit vector in the z-direction. The equilibrium electrostatic potential and magnetic vector

potential can be taken to be, respectively, φ = 0 and A = xB0y. The plasma beta is assumed

to be smaller than the electron to ion (proton) mass ratio me/mp, which means that the

typical ion Larmor radius is smaller than the electron collisionless skin depth δe = c/ωpe, c

being the speed of light and ωpe the electron plasma frequency. As discussed by Stasiewicz et

al. (2000), Alfven wave perturbations to φ and A under these conditions satisfy the equations

(1 − δ2

e∇2⊥) ∂Az

∂t= −∂φ

∂z, (1)

and

∂Az

∂z= − 1

c2A

∂φ

∂t, (2)

where

∇2⊥ =

∂2

∂x2+

∂2

∂y2, (3)

where cA is the Alfven speed, generalized to include the effect of a finite displacement current

term in Ampere’s law:

cA =c√

1 + µ0ρc2/B20

. (4)

– 4 –

Here, µ0 is the permeability of free space and ρ is the equilibrium plasma density. Although

recent microwave observations suggest that cA in the low corona could be as high as a few

tenths of c (Fletcher & Hudson 2008), the MHD approximation cA � B0/(µ0ρ)1/2 remains

fairly accurate even in such cases.

Fourier analysing in the x-direction and assuming invariance in the y-direction, i.e.

considering perturbations of the form exp(ik⊥x), we find that equation (1) becomes

(1 + k2

⊥δ2e

) ∂Az

∂t= −∂φ

∂z. (5)

Equations (2) and (5) yield the wave equation

c2A

∂2Az

∂z2= (1 + k2

⊥δ2e)

∂2Az

∂t2, (6)

which has solutions of the form

Az = g (z − cAt) sin(k⊥x), (7)

where

cA =cA√

1 + k2⊥δ2

e

, (8)

and the function g can be chosen arbitrarily. We may, for example, choose a Gaussian profile

of the form

g = −B1

k⊥exp

[−

(z − cAt

δz

)2]

, (9)

where B1 is the maximum absolute value of the magnetic field perturbation and δz is a

constant that characterizes the parallel length scale of the perturbation. With this choice of

wave profile it is straightforward to show from equations (2) or (5) that

φ = −B1cA(1 + k2⊥δ2

e)

k⊥sin(k⊥x) exp

[−

(z − cAt

δz

)2]

. (10)

Using the standard relations between potentials and fields

B = ∇× A, (11)

and

E = −∇φ − ∂A

∂t, (12)

– 5 –

we deduce finally that

Bx = 0, (13)

By = B1 cos(k⊥x) exp

[−

(z − cAt

δz

)2]

, (14)

Bz = B0, (15)

Ex = B1cA(1 + k2⊥δ2

e) cos(k⊥x) exp

[−

(z − cAt

δz

)2]

, (16)

Ey = 0, (17)

Ez =2B1cAk⊥δ2

e sin(k⊥x)

δz2(z − cAt) exp

[−

(z − cAt

δz

)2]

. (18)

It is straightforward to verify that in the MHD limit (k⊥ → 0) these expressions describe a

propagating shear Alfven wave pulse. The crucial departure from this limit is indicated by

the presence of a longitudinal electric field component Ez. This is typically several orders

of magnitude smaller than the transverse component Ex, but the latter merely produces an

ambipolar E× B drift and is thus irrelevant for particle acceleration.

The wave fields described above depend on the electron skin depth δe but not on the ion

skin depth δi, despite the latter being larger than the former by a factor of (mp/me)1/2 � 43.

Essentially this means that we are neglecting the Hall term in the generalized Ohm’s law

while retaining the electron inertia term. Before applying the above analysis to the problem

of electron acceleration in flares, it is useful to determine the exact parameter regime in which

it is permissible to neglect the Hall term. We first note that the current associated with the

magnetic field perturbation produces a Lorentz force which, in the linear approximation that

we are using, is given by

j× B =Bz

µ0

∂By

∂zy. (19)

Substituting this expression in the generalized Ohm’s law, neglecting resistive and pressure

gradient terms,

E + v × B =1

enj ×B +

me

ne2

∂j

∂t, (20)

where e is proton charge, v is single-fluid velocity and n is electron number density, we

deduce the existence of a hitherto-neglected electric field component

Ey =Bz

enµ0

∂By

∂zy, (21)

– 6 –

Noting that ∂By/∂z ∼ By/δz and using equation (16), we find that

Ey

Ex∼ δi

δz. (22)

Similarly, by considering the contribution of ∇× (j × B) to the induction equation, it can

be shown that

Bx

By

∼ δi

δz. (23)

Thus, it is permissible to neglect the Hall term when evaluating the transverse components of

the wave fields provided that the ion skin depth is small compared to the parallel scale length

of the wave pulse, irrespective of the perpendicular scale length δx ≡ π/(2k⊥). Clearly the

Hall term cannot affect Ez, but we need to take account of the fact that ∇× (j× B) has a

longitudinal component that modifies Bz. Denoting this modification by Bz, we find that

Bz

B0

∼ B1

B0

δi

δx. (24)

Thus, for perturbation amplitudes B1/B0 of, say 10%, the approximation Bz � B0 is only

valid if δx is a significant fraction of δi (although it is important to note that it does not

have to exceed δi).

3. Electron acceleration

The behavior of electrons moving in the fields described above can be deduced as follows.

The longitudinal component of the electric field is an odd function of z − cAt; any electron

that is not reflected by the wave pulse thus undergoes both acceleration and deceleration, and

does not gain significant energy. The phase space trajectories of such particles behind the

pulse will not be exact mirror images of the trajectories ahead of the pulse, due to particle

drifts, but the grad-B and curvature drift velocities are too small to have any significant

effect (due to the low electron mass) and the E×B drift lies almost entirely in the invariant

y-direction and can thus also be ignored.

If an electron has insufficient kinetic energy to surmount the electrostatic potential

barrier in the pulse front, it will be reflected and accelerated via a single-interaction Fermi-

type process. To determine the criterion for this to occur, we first recall that since (φ,A) is

a 4-vector the potential in the pulse frame is given by

φ′ = γ(φ − cAAz), (25)

– 7 –

where γ = (1− c2A/c2)−1/2. Taking the limit γ → 1 and substituting for φ, Az using equations

(7), (9) and (10) we obtain

φ′ = −B1k⊥δ2e cA sin(k⊥x) exp

[−

(z − cAt

δz

)2]

, (26)

where z′ = z − cAt. The total height of the potential barrier for a particle approaching from

z′ � δz is evidently

∆φ′ = B1k⊥δ2e cA sin(k⊥x). (27)

For a particle at rest in the unprimed frame, v′z = −cA and hence reflection will occur if

e∆φ′ ≥ mec2A/2. Taking sin(k⊥x) � 1, and assuming that k2

⊥δ2e � 1, we find that this

inequality can be expressed in the form

δx ≤ πB1

B0

δi. (28)

Putting B1/B0 = 0.1 and n = 1015m−3 we infer from equation (28) that reflection (and

hence acceleration) will occur when δx ≤ 4.5m. Since the fields are time-independent in the

pulse frame, particle energy is conserved in the absence of collisions in this frame. Hence,

for an electron that begins and ends its trajectory far from the pulse, i.e. in regions where

φ′ � 0, the final kinetic energy in the pulse frame is mec2A/2. Assuming that almost all of

the electron’s final momentum lies in the positive z-direction, i.e. v′ = cA, we deduce that

the final energy in the unprimed frame is 2mec2A.

An important point to note here is that both the final particle energy and the criterion

for the particle to be accelerated are independent of the parallel length scale of the pulse,

δz. Thus, the value of this parameter is irrelevant provided that it is larger than the ion skin

depth [cf. equations (22) and (23)] and sufficiently small that the acceleration time τacc ∼2δz/cA is less than the thermal electron collision time τc, so that energy is approximately

conserved in the pulse frame. Given that δi � 7 m whereas τccA/2 � 200−2000 km for typical

coronal densities of n = 1015m−3, B0 = 0.1 T and temperatures in the range (1− 4)× 106K,

this condition can be easily satisfied. Indeed, it can also hold for conditions corresponding

to the upper chromosphere, i.e. T ∼ 105 K and n a few times 1016m−3, so that acceleration

in this denser part of the atmosphere, with its larger electron reservoir, is also possible.

Equation (24), when combined with equation (28), provides a somewhat tighter con-

straint on the model parameters. In fact these two equations indicate that there is at best

only a narrow regime in which δx is both sufficiently small for the bulk of the electron

population to be accelerated and sufficiently large that the Hall perturbation to Bz can be

– 8 –

neglected. The analysis would not be strictly valid in this scenario anyway, since the wave

fields were calculated on the premise that the electrons could be treated as a cold fluid

carrying zero net current ahead of the pulse.

It is more self-consistent to consider a scenario in which only a fraction of the electrons

are reflected by the pulse. If the initial velocity distribution is f , the number of electrons

reflected is given by

n1 = 2π

∫∫R

fv⊥dv⊥dv‖, (29)

where the domain of integration R in (v‖, v⊥) space is the quarter-circle centred on v‖ = cA,

v⊥ = 0 with radius

v1 =

(2eB1k⊥δ2

e cA sin(k⊥x)

me

)1/2

, (30)

and v‖ < cA (electrons with initial v‖ ≥ cA never encounter the pulse and hence cannot be

reflected by it). Electrons lying inside this quarter-circle have insufficient kinetic energy in

the pulse frame

E ′ =1

2me

[(v‖ − cA)2 + v2

⊥], (31)

to penetrate a potential barrier with height given by equation (27). The obvious choice for

f is an isotropic Maxwellian

f =n

(2π)3/2v3e

exp

[−v2

‖ + v2⊥

2v2e

], (32)

where ve = (kBT/me)1/2, T being the initial electron temperature and kB Boltzmann’s

constant. In this case equation (29) yields

n1

n=

1

2

[erf

(cA√2ve

)− erf

(cA − v1√

2ve

)]− ve√

2πcA

e−(c2A+v21)/2v2

e

[1 − e−v1cA/v2

e

], (33)

where

erf(x) =2√π

∫ x

0

e−x2

dx, (34)

is the error function. Of the free parameters in equation (33) the least well-determined

observationally is the perpendicular scale length δx. We have established that this quantity

must be of the order of the ion skin depth or less (i.e. much smaller than the spatial resolution

– 9 –

of any solar coronal diagnostic) in order for significant particle acceleration to occur. Figure

1 shows n1/n as a function of δx for four different values of T ; in each case x was set equal

to −δx. As before, the density was taken to be 1015 m−3, while B0, B1 were set equal to

0.1 T and B1 = 0.01 T respectively. It is apparent that the accelerated fraction is extremely

sensitive to the perpendicular scale length of the inertial Alfven wave pulse in a narrow range

of values of this parameter, particularly at low temperature. In fact as T → 0 we find that

n1/n becomes a step function of δx, with the step occuring at δx = πB1/B0 [cf. equation

(28)]. As discussed above, our analysis is not strictly valid when n1/n is predicted to be of

order unity, but it is reasonable to assume that a large fraction of the electron population

would be found to be accelerated in this regime if a more exact (nonlinear) analysis were to

be carried out.

We can compute the velocity distribution of accelerated electrons as follows. For a

particle beginning and ending its trajectory in regions unperturbed by the wave, conservation

of magnetic moment requires that v⊥ be unchanged, and therefore we only need to consider

the change in v‖. Denoting the accelerated electron distribution by f and the final parallel

velocity of a given electron by v‖, we note that, as a consequence of Liouville’s theorem,

f(v‖)dv‖ = f(v‖)dv‖. (35)

Following reflection by an inertial or kinetic Alfven wave pulse, an electron with initial

parallel velocity v‖ in the unprimed frame (v‖ − cA in the pulse frame) has final parallel

velocity (Kletzing 1994)

v‖ = (cA − v‖) + cA = 2cA − v‖. (36)

Using this result to express v‖ as a function of v‖, it follows from equation (35) that for the

case of an initially Maxwellian distribution, dropping the tildes on v‖,

f =n

(2π)3/2v3e

exp

[− v2

⊥2v2

e

− (2cA − v‖)2

2v2e

]H(v‖ − v1 − cA), (37)

where H is the unit step function and v⊥ has maximum value

v⊥,max =[v2

1 − (v‖ − cA)2]1/2

, (38)

The total number of acclerated electrons per unit volume per unit v‖ is then given by

F (v‖) = 2π

∫ v⊥.max

0

v⊥fdv⊥, (39)

=n

(2π)1/2ve

{1 − exp

[−v2

1 − (v‖ − cA)2

2v2e

]}e−(2cA−v‖)2/2v2

e H(v‖ − v1 − cA). (40)

– 10 –

In the parameter regime that we are considering, the final particle energy E is essentially

equal to mev2‖/2 since all of the acceleration occurs in the parallel direction, and so v⊥ is

of the order of ve, which is typically much smaller than the values of v‖ for which F is not

negligibly small. Hence equation (40) gives essentially the electron energy spectrum, with

v‖ � (2E/me)1/2. Spectra are plotted in Figure 2 for δx = 3 m, B0 = 0.1 T, B1 = 0.01 T,

n = 1015m−3, x = −δx and T = (1−4)×106K. The units of F are arbitrary, but the spectra

are correctly normalized with respect to each other. The width of the spectrum increases

rapidly with initial electron temperature and, in each case, F peaks at an energy lying

slightly below the upper cutoff at me(cA + v1)2/2, which is about 47 keV for the parameters

used here. This peaking arises because electrons reflected by the pulse with the lowest initial

energies undergo the greatest acceleration. We note that qualitatively similar results were

obtained by Kletzing (1994) in a study of electron acceleration by kinetic Alfven waves.

The spectrum of accelerated electrons is fairly sensitive to the model parameters. This

is illustrated by Figure 3, which shows F when both the equilibrium magnetic field and the

perturbation to this field are increased by 50%. The transverse scale length is 2.5 m, and

the other parameters are identical to those used to generate Figure 2. The width of the

spectrum is again sensitive to T , while the cutoff energy has increased from 47 keV to 115

keV. The spectrum can be modified in a similar fashion by changing the values of k⊥ and n

[cf. equation (30)].

Solar flare hard X-ray spectra (measured in photons m−2 s−1 keV−1) can be generally

represented as power law functions of photon energy, with spectral indices γ typically ly-

ing in the range 3-5 (Dennis 1985). The thick target model of hard X-ray production in

flares (Brown 1971) then implies that the injected electron flux spectrum (electrons m−2 s−1

keV−1), which is proportional to the quantity F defined above, is a power law function of

electron energy, with index α = γ + 1. Clearly the spectra in Figures 2 and 3 cannot be

characterized as power laws, but it should be noted that we have considered only one per-

pendicular Fourier component of the inertial Alfven wave field. A spectrum of such Fourier

components, for example of the type observed using the FAST spacecraft in the Earth’s

auroral zones (Chaston et al. 2008), would be expected to generate fast electrons with a

more extended range of energies than those of the distributions plotted in Figures 2 and 3.

The electron spectrum would also be broadened by spatial variations in B0, B1, n and T .

4. Conclusions and discussion

We have investigated theoretically the possibility that electrons could be accelerated

during solar flares to hard X-ray-emitting energies via their interaction with inertial Alfven

– 11 –

waves. This possibility arises from the fact that evidence has emerged of magnetic fields

in the low corona in excess of 0.1 T; for typical coronal densities, the corresponding Alfven

speed cA is of the order of c/10. The significance of this for particle acceleration in flares lies

in the fact that electrons reflected from inertial Alfven wave pulses are accelerated to field-

aligned velocities in excess of cA, and hence energies in the tens of keV range if cA ∼ c/10.

We have demonstrated that the fraction of particles accelerated depends critically on the

initial electron temperature and the transverse length scale δx of the shear Alfven wave pulse;

under conditions typical of the corona and upper chromosphere, assuming wave perturbation

amplitudes of 10%, we have shown that a significant fraction of the electron population is

energized if δx is of the order of a few metres or less. Such length scales lie far below the

spatial resolution of existing observational data, but there are no theoretical reasons for

supposing that they cannot occur in the pre-flare corona. Indeed, in situ measurements of

Alfven waves in the Earth’s auroral zones indicate that δx can be of the order of the electron

skin depth δe (Stasiewicz et al. 2000; Chaston et al. 2008), which is well below a metre for

typical coronal densities. If shear Alfven waves with perpendicular wavelengths of the order

of δe were to be present in the solar corona, substantial electron acceleration could occur for

wave amplitudes much lower than the 10% assumed previously [cf. equation (30)], although

the maximum electron energy would be somewhat reduced due to the k⊥ dependence of

cA. We have shown that the distribution of electrons accelerated by a single perpendicular

Fourier component of an inertial Alfven wave pulse is strongly-peaked at an energy that

depends on B0, B1, n and T as well as δx; a broader distribution, more closely resembling

the power law-like spectra inferred from hard X-ray observations, would be expected from

a source region with spatially varying values of these parameters, and inertial Alfven waves

with a spectrum of perpendicular wavelengths.

It is well-known that the inefficiency of hard X-ray production in the thick-target model

requires a substantial fraction of the total number of electrons in the corona to be acceler-

ated in order to account for the X-ray fluxes produced in flares; any candidate acceleration

mechanism proposed in the framework of this model should therefore include nonlinear ef-

fects. The linear analysis in the present paper constitutes a proof-of-principle that inertial

Alfven waves could play an important role in the acceleration of electrons in flares; we hope

that we have provided sufficient evidence of the efficacy of this wave mode as a particle

accelerator in the flare context to stimulate further research. A relatively straightforward

extension of the work described above would be to compute the trajectories of test electrons

in inertial Alfven wave fields with a spectrum of k⊥ values, for example using the CUEBIT

full orbit code (Hamilton et al. 2003), and determine the conditions, if any, in which such

spectra produce power law fast electron distributions. However, given the limitations of this

approach in the solar flare context, noted above, it would be more satisfactory to perform a

– 12 –

fully self-consistent, nonlinear simulation of the electron-wave interaction, using for example

the drift-kinetic formalism (Watt et al. 2004; Watt & Rankin 2008).

We note finally that for arbitrary values of γ = (1 − c2A/c2)−1/2 the final kinetic energy

of an initially stationary electron reflected by an inertial Alfven wave pulse can be easily

shown to be 2γ2mec2A. The factor of γ2 arises from two successive Lorentz transformations,

to the pulse frame and back again. In this regime inertial Alfven wave acceleration is rather

analogous to inverse Compton scattering (see e.g. Rybicki & Lightman 1979), with the

roles of the electron and the wave quantum reversed. It is possible that electrons could

be accelerated to highly relativistic energies via this mechanism in astrophysical plasma

environments with B20 � µ0ρc2, such as pulsar magnetospheres (Michel 1991) and, depending

on the exact values of the plasma parameters, supernova remnants (Biermann & Cassinelli

1993). Further investigation of inertial Alfven wave acceleration of electrons to relativistic

energies in the wider astrophysical context is, we believe, strongly merited.

This work was funded by the United Kingdom Science and Technology Facilities Council

and the United Kingdom Engineering and Physical Sciences Research Council. LF is pleased

to acknowledge support from the European Commission through the SOLAIRE Network

(MTRN-CT-2006-035484).

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This preprint was prepared with the AAS LATEX macros v5.2.

– 14 –

acce

lera

ted

elec

tron

frac

tion

δx (m)

Fig. 1.— Fraction of electrons accelerated by inertial Alfven wave pulse, plotted versus δx

for initial electron temperatures T = 1 MK (solid curve), 2 MK (short dashed curve), 3 MK

(dotted curve) and 4 MK (long dashed curve). In every case the electrons initially lie at

x = −δx and the parameter values are n = 1015 m−3, B0 = 0.1 T and B1 = 0.01 T.

– 15 –

F(E

)(a

rbit

rary

unit

s)

E (keV)

Fig. 2.— Energy spectra of electrons accelerated by inertial Alfven wave pulse for initial

electron temperatures T = 1 MK (solid curve), 2 MK (short dashed curve), 3 MK (dotted

curve) and 4 MK (long dashed curve). In every case the electrons initially lie at x = −δx

and the parameter values are n = 1015 m−3, B0 = 0.1 T, B1 = 0.01 T and δx = 3 m.

– 16 –

F(E

)(a

rbit

rary

unit

s)

E (keV)

Fig. 3.— As Figure 2 except that B0 = 0.15 T, B1 = 0.015 T and δx = 2.5 m.