Interaction of plasma waves and particles in the non-uniform magnetosphere and acceleration of auroral particles

Interaction of plasma waves and particles in the non-uniform magnetosphere and acceleration of auroral particles

Planet. Space Sci. 1968. Vol. 16, pp. 1297to 1303. PergamottPress. Printed in Northern Ireland INTERACTION OF PLASMA WAVES AND PARTICLES IN THE NON-U...

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Planet. Space Sci. 1968. Vol. 16, pp. 1297to 1303. PergamottPress. Printed in Northern Ireland


of Geophysics,

University of British Columbia, Vancouver,


(Received in final form 27 May 1968) Abstract-A plasma wave propagating across a field-aligned sheet with a steep density gradient is considered. The wave may be modified inside the sheet in such way that a strong electric field appears parallel to the unperturbed magnetic field. This field component may deliver energy to ‘hot’ particles at a rate comparable to or even much higher than the maximum possible rate of energy exchange between waves and particles in a uniform plasma due to the cyclotron instability mechanism. It is suggested that this energization of particles in the regions of abrupt change of the plasma density (such as the plasmapause or the ducts of whistler mode waves) might be responsible for discrete spatially limited events of aurora1 precipitation. 1. INTRODUCTION

The interaction of high-energy particles with plasma waves has been investigated many times during the past decade. Various forms of the interaction have been proven to be useful for explaining several phenomena such as VLF emissions and hydromagnetic whistlers. Many aspects of the interaction are comprehensively discussed in the recent compilation by Kimura. (l) The wave-particle interaction has been investigated under the assumption that the plasma density is uniform. This is not always fulfilled. Whistler studies (cf. e.g. Carpenter, c2)Angerami and Carpenter, t3)Smith and Angerami(*)) show that rather sharp variations of cold plasma density exist in the magnetosphere and that the density changes considerably when going across the field-lines over a distance L, which may be comparable with the wave length of the whistler mode wave. For example if the thickness of the plasmapause is about 0.1 RE (Earth radii) or smaller and if the ratio of the density outside the plasmapause to the density on the inner boundary of the plasmapause is of the order of 10e2, then the density inside the plasmapause changes by a factor of the order of unity on a length-scale of a few kilometers or smaller. We may ask how plasma disturbances are modified by the sharp density gradients and whether this modification may be a cause of an interaction between the disturbance and ‘hot’ (high energy) particles which is quantitatively and/or qualitatively different from the interaction in a uniform plasma. It cannot be expected that this problem can be solved rigorously. The analytical solution of the equations describing disturbances in an inhomogeneous plasma is usually mathematically untenable and even the simplest (linearized) description of the interaction complicates the solution still further. Nevertheless, we shall be able to make several order of magnitude estimates of the e&.ziency of the interaction inside sheets of steep density gradient. It will be shown that the parallel electric field associated with a disturbance propagating across the field-aligned density inhomogeneity can contribute considerably to the energization of ‘hot’ particles, giving rise to high-energy particles in spatially restricted * Permanent address:




vakia. 1297

Academy of Sciences, Prague, Czechoslo-



regions. The investigated mechanism may be responsible for some types of aurora1 precipitation of particles. Basic equations describing the plasma disturbances are summarized in Section 2. The modification of a wave in an ambient (cold) plasma by density gradients is demonstrated in Section 3 in a very simple, though not typical, example of magnetodynamic waves. The parallel electric field associated with a plasma wave of higher frequency is estimated in Section 4. The estimates of the energy gain of ‘hot’ particles in the regions of sharp density gradients are made in Section 5, and finally, the last section is devoted to a short discussion of the acceleration of aurora1 particles. 2. BASIC EQUATIONS

Let the difference between the velocity of ions and electrons be denoted by v, the plasma frequency of electrons by p and the cyclotron frequency of ions and electrons by wi and CO,respectively. The equations of motion of ions and electrons in a zero temperature collisionless plasma yield the relations 477iwen




47ricoen c2

v,, = -

P2 ; E,,. 0

Here 1 = B,,IB,j, B, is the unperturbed curl-free field, the symbols of parallelism and perpendicularity refer to the direction of 1, cc)= i a/&, and finally, 2 cc) U= (we2





Co2)(Coi2 - w2) ’

(co,2 -

d)(Wi2 -

(COG has been neglected with respect to w, in the last two expressions.) curl curl E = 4ricoenc-2v + (co/c)~E.

0.P) *

Further, (3)

The Equations (I), (2) and (3) re p resent the general set of equations describing waves in a cold plasma. Two more special forms of this set will be needed in the following sections. Let the frequency of the investigated disturbance be much smaller than the ion cyclotron frequency and the plasma frequency, let 1 be directed along the x1 -axis of the curvilinear, rectangular, right-handed coordinate system (x,, x2, x3) with scale factors h,, h,, /z3and let the geometry of the field B, and all quantities be independent of the coordinate x,, say. We get from the set 1, 2 and 3 the relations

-’ a (32 (h&J h, as1h, asI --’


a (” a W.J) asI h, as,




+ ;

(6s, = h,dx,


k = 1,2,3).




A more general form of the plasma wave equations is necessary for higher frequencies; however, in the magnetosphere, the wave-length of these waves is much smaller than the length scale of the variation of the magnetic field B, and that of the density as measured along 1. Thus, the zero order eikonal approximation is adequate for the description of these waves. Putting ---~(a/&,) = k = constant and (&,/a~~) = constant (i,i = 1, 2, 3) we obtain

(9) Generally, all three components of E are coupled. In a uniform plasma, the directions of propagation of a wave, for which IEl1 > 1E, I, may be found for a chosen frequency. In this paper we shall search for such regions in a non-uniform plasma where El increases considerably for waves over an extended range of frequencies, and where this component exceeds the value of E, corresponding to a wave incident on the region. 3. MAGNETODYNAMIC

The Equations


(4) and (5) yield cl

If the length scales of variation L1, and L,, respectively, then





p2 Ge

of all quantities El I

and the second term in the brackets if



both parallel





on the left-hand-side L,/LI]





and perpendicular

to 1 are

(11) of Equation

(5) may be neglected WI

Relation 12 is a condition for the validity of the classical concept of toroidal magnetodynamic waves and, because of the smallness of the ratio IX~/W,W,, it is fulfilled in the majority of situations we meet in geophysical applications. The guided mode equation (13) is valid under the condition 12 and describes correctly the magnetodynamic toroidal mode if the density and magnetic field B, vary smoothly with position. Radoski and McClay(5) analyzed one model of the field and density distribution, imposing the boundary conditions of perfect reflection on the solution of the guided mode equation. They found the expected result, namely that E, varies with x2 as the Dirac function 6(x, - x.~‘) where x2’ is the value of x2 corresponding to a resonant frequency at a chosen field-line. Further, they found that this form of E, implies the necessary existence of non-zero El and concluded that the magnetodynamic concept oftoroidal waves is physically not tractable. Clearly, the condition



12 has been violated in their treatment by imposing on the solution of the equation of type 13 the boundary conditions implying the &form of E,. It will be shown elsewhere(6) that, in the cases of geophysical interest, the boundary conditions of perfect reflection, though they are apparently reasonable in some cases and provide sufficiently reliable information on the resonance frequencies of magnetodynamic oscillations, represent certain limiting procedures which should be avoided in a more careful analysis. Here rather, we are interested in the situations in which the parallel electric field is actually not negligible. Consider a wave with angular frequency w N 10-l and length scale L,, N 1Og,5cm propagating near the plasmapause characterized by the length scale L, - 1W5 cm. We find that at the plasmapause position near the equatorial plane O/(OiO,)l” N lo4 and thus LJ LII- 0/(wico3 u2 * The parallel electric field must be taken into account. It will be shown in the following section that IE,,/E, 1may even be much higher than unity for higher values of o. 4. THE RATIO


In this paper, we are considering propagating waves only i.e. the waves whose wave vector fulfils the inequality k2 > 0 in the hologeneous portion of the plasma. Further, we restrict our discussion to the analysis of waves with frequencies smaller than p and not close to Oi, o, and (w~cJ.#/~in any point of space, we also assume thatp2u > 1. To estimate the ratio of parallel to perpendicular electric field in the region of sharp density gradient, let us consider a plasma of constant density for x2 > X2 = L, and for x2 < 0 and nonuniform in the interval (0, X2). Let the angle between the wave vector and the direction 1 be small for a wave propagating in the region x2 > X2. The value of k, which is approximately equal to the wave vector corresponding to the propagation along B,, is a constant, as the properties of the medium do not vary with x1; k2 = (WP/C)2(Wi& u))-~(o, F co)-l, where P is the value ofp in the region x2 > X2. We may use the order of magnitude estimate k2 - (P/c)~(w/o~) - (p/~)~(o/cuJ for oi G o < co,. l&l < lE21, lEaI for propagating waves in the region x2 > X2 and the perpendicular components of the electric field are, under our assumptions, of a comparable order of magnitude with the exception of the low-frequency limit u < wi, discussed in the previous section (cf. Stix(‘) for details). The value of El may be expressed as a function of E3 using the Equations (7), (8) and (9) ,+?a

k;+ej+_ ~2 as2 [(


a2 - b2



1 a a2E, k b i%,


l,, k;+&-~k

a2 - b2 (




and a -=kb

1 W2- u)iO, 1 k coca, -ii’

(q -6 w < 0,).

The components El and E3 are continuous at the interfaces x2 = 0, X2 and change considerably inside the interval (0, X2). This is because of the sudden drop of the typical length scale from a large value, (corresponding to a small perpendicular component of the wave vector in the region x2 > X,), to the value of the order of magnitude of L,. If R denotes the order of magnitude of the ratio IE,,/E, I, then inside the sheet 0 < x2 < X2 the relation (15)




is valid for sufficiently small L,, (L, G c/p - c/P G k-l), and for o’s which are not too close to frequencies CC)~, w, and (e~~c(),)~/~ and are also much smaller than p. (If X2 + 0 then, of course, Ei(xI, X2 + E, t) + Ej(x,, -E, t) for E -+ 0, but &(x1, 0, t) may be considerably different from E,(xl, A’, + E, t); herej = 1 or 3.) Clearly, the value of R may be comparable with or higher than unity and the disturbance propagating across a sheet of a sharp density gradient may be associated with a relatively strong parallel electric field. This may not be expected in the case of waves propagating at a small angle to B, in a homogeneous plasma. The first component of the electric field is the dominant component for sufficiently small L, in a wide range of frequencies. 5. PARTICLE ACCELERATION

The effect of a disturbance, propagating in the ambient plasma, on ‘hot’ particles may be characterized by the average value of the time-rate of change of energy, q, per ‘hot’ particle. The exact calculation of 7 is impossible due to the complicated electromagnetic fields which are expected in sheets of sharp density gradient, but the rate of change of energy of particles, Q, due to the parallel electric field, and the rate of change of energy of particles, ‘~;l~,due to a circularly polarized wave propagating along B, in a homogeneous plasma may be found relatively easily. Consider particles with a velocity distribution function F =f(vl)S(v,2), where 6 is the Dirac function. u1 fulfils, in the linear approximation, the relation


= & i Er(x,) exp (i(kx, - wt)), dt where x1 = (v&t + (x&, (unperturbed value of q,, is given by (Stix”)), V’ = -

where f,, =

J F dv I’



are denoted

by subscript 0).



[ au, uc=O/b’

If the distribution of parallel velocities is Maxwellian, then rY2e2 k -q3e-qaE12. “’ = m [kl co



Kis the Boltzmann constant and T,, the parallel temperature. In agreement with the second law of thermodynamics, the ‘organized’ form of energy of the plasma waves is transformed into energy of random motions of the particles. To estimate the efficiency of acceleration of the particles by a parallel electric field we compare q,, with the rate of energy exchange qL between particles and waves propagating along B, in a homogeneous ambient plasma. We find (HruSka(8))

where the angular brackets denote the average value, upper and lower signs refer to left and right-polarized waves respectively and w, is the cyclotron frequency of the investigated



‘hot’ particles with mass m. The value of 1~~ 1reaches a maximum 1qI IM which is independent of the anisotropy of the velocity distribution function and we get for a Maxwellian distribution of parallel velocities the expression (17) combining

(16), (17) and (15), we find


= t


[email protected],


For q of the order of unity, the value of E may be considerably larger than unity and, in this sense, we may say that the acceleration of particles inside sheets of steep density gradient is effective. Of course, it should be stressed that for a given disturbance propagating across the sheet, the value of E is not the ratio of the energy exchange inside the sheet to the energy exchange outside the sheet, but rather the ratio of these energy exchanges if the value of E3 were the same in both regions. Nevertheless, it is clear from Equation (14) that IEJ changes much more rapidly than IEJ inside the sheet. Both quantities El and E, are continuous at the point X2, but El at the point x2 = X2 - E (E + 0) may reach a relatively high value in spite of the fact that El M 0 at x2 = X2. Inspection of Equation (16) shows that electrons can be more energized than protons, that the parallel component of particle velocity is affected, and that the acceleration is energy-selective, i.e. the particles with random velocities of the order of magnitude of the phase velocity are accelerated most efficiently. Numerical estimates show that the most influenced electrons in the plasmapause are those electrons with an energy of several tens of kev. 6. AURORAL


The importance of steady potential electric fields, associated with the plasma motion in the magnetosphere and in the magnetotail, for acceleration of aurora1 particles has been emphasized several times from various points of view (cf. e.g. Axford and Hines,tg) Taylor and Hones,(l’J) Brice,ol) Speiser (12m3)). The aim of these theories is the explanation of systematic large-scale features of particle precipitation, rather than the explanation of discrete spatially and/or temporary limited events, cf. e.g. 0’Brien,04) Mozer and Bruston,(15)(16) Mozer.07) As Taylor and Hones noted, some sort of turbulence or instability must be invoked in order to explain the small scale structure of many aurora1 precipitations and emissions. The characteristic feature of the mechanism investigated here is the spatial variability of the rate of particle energization; this variability depends on the small scale structure of ambient plasma in the magnetosphere. The field-aligned plasmapause certainly represents the geometrical locus of points where the investigated mechanism works. If the magnetosphere is a medium with whistler mode ducts of enhanced density, superimposed on a smooth density distribution then particles are (Carpenter, (i2) Smith and Angeramit4)) also preferably accelerated inside these spatially limited tubes during the geomagnetic disturbances. This gives rise to jets of high-energy particles. Their scale L, is as small as a few kilometers down to hundreds or tens of meters at ionospheric levels (if the thickness of







the ducts is of the order of magnitude of the wave length of whistler mode). This is regardless of whether the particles have been already trapped or are just coming into the magnetospere during its ‘opening’ (Dungey(ls)). Note also that according to the whistler studies,(2)(3)(4)the magnetospheric density is most irregular in the region around 5 f IRE in the late evening hours of local time. This might correspond to the enhanced discrete electron precipitation at the same time. One could also speculate whether the rapid temporal changes of the precipitation might be connected with a large scale motion of the fieldaligned irregularities of density, or with some other mechanism. At present, we are not prepared to discuss this question. REFERENCES 1. 2. 3. 4. 5.

I. KIMURA,Planet. Space Sci. 15, 1427 (1967). D. L. CARPENTER, J. geophys. Res. 71,693 (1966). J. J. ANGERAMIand D. L. CARPENTER, J. geophys. Res. 71,711 (1966). R. L. SMITHand J. J. ANGERAMI,J. geophys. Res. 73, 1 (1968). H. R. RADOSKIand J. F. MCCLAY, J. geophys. Res. 72,4899 (1967).

6. A. HRU~KA,Planet. Space Sci. 16, 1305 (1968). 7. T. H. Snx, The Theory ofPlasmu Waves. McGraw-Hill, New York (1962). 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18.

A. HRU;KA, J. geophys. Res. 71, 1377 (1966). W. I. AXFORDand C. 0. HINES, Can. J. Phys. 39, 1433 (1961). H. E. TAYLORand E. W. HONES,J. geophys. Res. 70, 3605 (1965). N. M. BRICE,J.geophys. Res. 72, 5193 (1967). T. W. SPEISER, J. geophys. Res. 70,4219 (1965), T. W. SPEISER, J. geophys. Res. 72, 3919 (1967). B. J. O’BRIEN,J.geophys. Res. 69, 13 (1964). F. S. MOZER and P. BRUSTON,J.geophys. Res. 71,220l (1966). F. S. MOZER and P. BRUSTON,J.geophys. Res. 71,445l (1966). F. S. MOZFR, J. geophys. Res. 73, 999 (1968). J. W. DUNGEY,Phys. Rev. Lett. 6, 47 (1961).