Nonlocal kinetic theory of Alfve´n waves on dipolar …ham.space.umn.edu/~bob/kinetic-dipole-paper.pdfNonlocal kinetic theory of Alfve´n waves on dipolar field lines Robert L. Lysak

Nonlocal kinetic theory of Alfven waves on dipolar field lines

Robert L. Lysak and Yan SongSchool of Physics and Astronomy, University of Minnesota, Minneapolis, Minnesota, USA

Received 24 January 2003; revised 21 May 2003; accepted 3 June 2003; published 26 August 2003.

[1] Recent observations have indicated that in addition to the quasi-static acceleration ofelectrons in inverted V structures, auroral electrons frequently have a spectrum that isbroad in energy and confined to parallel pitch angles, indicative of acceleration in low-frequency waves. Test particle models have indicated that these electrons may beaccelerated by the parallel electric fields in kinetic Alfven waves. However, such modelsare not self-consistent, in that the wave structure is not influenced by the acceleratedparticles. A nonlocal kinetic theory of electrons along auroral field lines is necessary toprovide this self-consistency. Results from such a model based on electron motions ondipole field lines are presented. For a typical Alfven speed profile, kinetic effects lead tosignificant energy dissipation when the electron temperature exceeds �100 eV. Thedissipation generally occurs near the peak of the Alfven speed profile. This dissipationgenerally increases with increasing temperature and decreasing perpendicular wavelengthup to �1 keV and 10 km, respectively. At larger temperatures and smaller perpendicularwavelengths the dissipation begins to decrease and the ionospheric Joule dissipationgoes to zero, indicating that the wave is reflected from the dissipation region. Dissipationin the 0.1–1.0 Hz band is structured by the modes of the ionospheric Alfvenresonator. INDEX TERMS: 2704 Magnetospheric Physics: Auroral phenomena (2407); 2736

Magnetospheric Physics: Magnetosphere/ionosphere interactions; 2772 Magnetospheric Physics: Plasma

waves and instabilities; 7827 Space Plasma Physics: Kinetic and MHD theory; 7867 Space Plasma Physics:

Wave/particle interactions; KEYWORDS: auroral particle acceleration, Alfven waves, magnetosphere-

ionosphere coupling, nonlocal kinetic theory, wave-particle interactions

Citation: Lysak, R. L., and Y. Song, Nonlocal kinetic theory of Alfven waves on dipolar field lines, J. Geophys. Res., 108(A8), 1327,

doi:10.1029/2003JA009859, 2003.

1. Introduction

[2] Observations from the FAST satellite [Chaston et al.,1999, 2000, 2002a, 2002b] have indicated that precipitatingauroral electrons often are field aligned and have a broaddistribution in energy, in contrast to the typical auroral‘‘inverted V’’ type precipitation, which have a characteristicenergy but are broad in pitch angle. Regions of this type offield-aligned acceleration have been observed throughoutthe auroral zone but are often seen at the polar cap boundaryof the auroral zone. Similar observations of field-alignedelectrons have been seen over many years, particularly fromsounding rocket missions [Johnstone and Winningham,1982; Arnoldy et al., 1985; Robinson et al., 1989;McFaddenet al., 1986, 1987, 1990, 1998; Clemmons et al., 1994; Lynchet al., 1994, 1999; Knudsen et al., 1998]. These field-aligneddistributions are not consistent with the usual picture of aplasma sheet distribution that has been accelerated in aquasi-static potential drop. Rather, it has been suggestedthat low-frequency waves are responsible for this accelera-tion, whether in electromagnetic ion cyclotron waves[Temerin et al., 1986] or Alfven waves [Kletzing, 1994;

Thompson and Lysak, 1996; Chaston et al., 1999, 2000,2002a, 2002b; Kletzing and Hu, 2001].[3] These studies have used test particle models to

describe the interaction of Alfven waves with the electronpopulation. In most of these cases the electron inertial effectwas used to determine the parallel electric field that accel-erates these electrons. Thus the electron acceleration processdid not modify the wave fields self-consistently. Suchmodifications require the use of a kinetic theory to describethe electron acceleration. Since the electrons can travel largedistances during a wave period, such a kinetic theory mustbe nonlocal, i.e., the parallel electric field at one locationalong the field line can influence the current at otherlocations along the field line. A first attempt at such anonlocal theory in the context of field line resonances hasbeen developed by Rankin et al. [1999] and Tikhonchuk andRankin [2000, 2002]. These authors determined that thekinetic effect of bouncing electrons could enhance theparallel electric field to values much greater than that givenby electron inertia.[4] A model that extended this type of calculation to the

higher frequencies of waves trapped in the ionosphericAlfven resonator has recently been presented by Lysakand Song [2003]. This work considered a simplified modelconsisting of a straight magnetic field line with an idealizedAlfven speed profile typical of the ionospheric Alfven

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 108, NO. A8, 1327, doi:10.1029/2003JA009859, 2003

Copyright 2003 by the American Geophysical Union.0148-0227/03/2003JA009859$09.00

SMP 9 - 1

resonator that becomes constant at high altitudes. In addi-tion to the recovery of the uniform plasma Landau dampingat high altitudes, this model showed that an enhancement ofthe dissipation due to the wave-particle resonance occurredjust below the point at which the Alfven speed began todecrease.[5] Although these papers indicate the importance of the

nonuniformity of the field line in wave-particle interactions,none of them included the self-consistent modification of thewave fields due to the kinetic effects. This approximation isjustified when the kinetic effects are weak, which occurs forlarger values of the perpendicular wavelength. However, thewaves that may be responsible for auroral arcs are expectedto have perpendicular wavelengths comparable to the elec-tron skin depth or smaller. At such wavelengths the wavemode structure itself will be modified by the kinetic effects,and so a self-consistent model should be developed.[6] The purpose of this paper is to present such a model.

We generalize the model of Lysak and Song [2003] to use amore realistic profile that includes the dipole geometry of anauroral flux tube. We will still concentrate on the higherfrequency waves associated with the ionospheric Alfvenresonator and so will model only the lower regions of thefield line where the Alfven speed gradients are strongest andthe local kinetic theory breaks down. The model results forthe relative amounts of dissipation due to wave-particleinteractions will be presented for a variety of plasmaparameters.

2. Theoretical Development

[7] The development of the model follows the resultspresented by Rankin et al. [1999], Tikhonchuk and Rankin[2000, 2002], and Lysak and Song [2003]. The details of themodel can be found in those works; here we will brieflysketch the theory behind these calculations.[8] The basic concept of this model is to consider the

wave equations for Alfven waves along auroral field lines,including a fully kinetic treatment for the electron motionalong the field line. The wave equations for the kineticAlfven wave can be expressed by assuming a wave ofconstant frequency and perpendicular wave number (whichis assumed to be in the x direction). In this case, Faraday’slaw and the perpendicular and parallel components ofAmpere’s law can be written as

@Ex

@z¼ iwBy þ ik?Ez ð1Þ

@By

@z¼ iw

c21þ c2

V 2A

1� �0

mi

� �Ex ð2Þ

�iwe0Ez þZ

dz0s z; z0ð ÞEz z0ð Þ ¼ ik?

m0

B0

B0I

By: ð3Þ

It should be noted that in these equations, Ex, By and k? aremapped to their ionospheric values using an isotropic dipolarmapping. Thus the only mapping factor that appearsexplicitly is the ratio of the background magnetic fieldstrength to that in the ionosphere, B0/B0I, that appears in

equation (3). The ion gyroradius correction is included inequation (2) through the Bessel function �0 which is afunction of the parameter mi = k?

2ri2. It may be noted

that this term is sometimes approximated by the relation(1 � �0(mi))/mi 1/(1 + mi), as in the work of Johnsonand Cheng [1997]; however, in this work we retain thefull Bessel function expression.[9] The electron kinetic effects enter into the wave equa-

tions above through the nonlocal conductivity relation in theparallel Ampere’s law, equation (3). This relation can bedetermined by solving the Vlasov equation for electrons in adipolar magnetic field subject to the propagation of a kineticAlfven wave. Since these waves are at very low frequencies,the magnetic moment of the electrons will be conserved. Inaddition, the total energy of an electron, given by

W ¼ 1

2mv2z þ mB zð Þ þ q� zð Þ; ð4Þ

is conserved in the absence of the wave; however, theparallel electric field of the wave will modify this totalenergy. While equation (4) includes the possibility of abackground parallel potential drop, this term will not beincluded in the results presented in this paper. Thus thelinearized Vlasov equation will take the form

@

@tþ vz

@

@z

� �df z; m;W ; tð Þ þ qdEzvz

@f0 m;Wð Þ@W

¼ 0; ð5Þ

where f0± give the unperturbed distribution functions andthe plus or minus refers to the direction of the parallelvelocity (where we take the positive z direction to beupward). This equation can be solved in the usual way byintegrating along the unperturbed orbits defined by constantmagnetic moment and energy

df z; tð Þ ¼ df z0; t0ð Þ � q@f0@W

Z t

t0

dt0dEz z0; t0ð Þvz t0ð Þ: ð6Þ

Note that here the dependence of the distribution functionson m and W is implicit. The first term in this equation givesthe initial value of the particle that leaves a boundaryposition z0 at a time t0. Note that since dz0 = vzdt

0, theintegral can be written in terms of the z coordinate, giving

df z; tð Þ ¼ df z0; t � t z0; zð Þð Þ � q@f0@W

Zz

z0

dz0dEz z0; t � t z0; zð Þð Þ;

ð7Þwhere the function t refers to the travel time between twopositions in z

t z1; z2ð Þ ¼Zz2z1

dz0

vz z0ð Þ ¼ Zz2z1

dz0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2=mð Þ W � mB z0ð Þ � q� z0ð Þð Þ

p :

ð8Þ

Note that the plus or minus sign here again refers to theparticle’s direction of motion: for z1 < z2, the plus sign is

SMP 9 - 2 LYSAK AND SONG: NONLOCAL ALFVEN WAVE KINETIC THEORY

taken, with the minus sign taken otherwise, so that t isdefined to be positive. Again note that this travel time is animplicit function of m and W. Assuming that the wave fieldoscillates at a frequency w, we can write

df zð Þ ¼ df z0ð Þeiwt z0;zð Þ � q@f0@W

Zz

z0

dz0dEz z0ð Þeiw z0;zð Þ; ð9Þ

where a factor of e�iwt has been cancelled from each term.[10] Now we must consider the types of particle trajec-

tories that are present. In general, there are a number ofparticle populations on an auroral field line, due to theinterplay between the magnetic mirror force and the parallelelectric field [e.g., Whipple, 1977; Chiu and Schulz, 1978].However, for the initial results presented here we willassume that there is no parallel electric field, i.e., � = 0in equations (4) and (8). In this case, there are only threetypes of electron trajectory, each with their own backgrounddistribution: (1) ionospheric electrons moving upward;(2) magnetospheric electrons that precipitate into the iono-sphere; and (3) magnetospheric electrons that mirror atsome altitude and return to the magnetosphere. The iono-spheric electrons begin at the ionospheric boundary (z0 = 0)and are described by f+. The equilibrium value of thisdistribution f0+ is given by the upgoing half of a Maxwell-ian, and it is assumed that the perturbation in this distribu-tion function at the ionosphere is zero, i.e., df+(0) = 0. Theprecipitating and mirroring electrons begin at an assumedupper boundary (z0 = L) and are described by f�. Theequilibrium value of this distribution f0� is described by thedowngoing half of a Maxwellian, and the density andtemperature of this distribution can be different from theionospheric distribution. The boundary term df�(L) is set tothe value given by the local kinetic distribution function atthis location, as in Lysak and Song [2003]. The mirroringelectrons also begin at L; however, at some altitude zm(which is a function of m and W) their parallel velocitybecomes zero and they begin moving upward, contributingto the upgoing f+ distribution. Since the density of theseparticles is conserved, at the mirror point we have df+(zm) =df�(zm).[11] The perturbation in the field-aligned current by the

wave is required to solve the wave equations. This currentcan be defined in the usual way by integrating over theperturbed distribution function

djz zð Þ ¼ q

Z10

dvv dfþ z; vð Þ � df� z;�vð Þ½ : ð10Þ

Since the perturbation in the distribution function can bewritten in terms of an integral over the parallel electric field,equation (10) can be written as a nonlocal conductivityrelation

djz zð Þ ¼ZL0

dz0s z; z0ð ÞdEz z0ð Þ: ð11Þ

The detailed calculation of the conductivity kernel is givenin Appendix A.

[12] The wave mode structure of the Alfven wave withthe electron kinetic effects included can be found by solvingequations (1)–(3) with the conductivity kernel found bysolving the Vlasov equation, as in equations (9)–(11).These equations are solved as follows. First, the equilibriumdistributions are determined, which give the Alfven speedprofile and the nonlocal conductivity kernel. The explicitform of this kernel is described in Appendix A and is givenby equation (A10). Then, equations (1) and (2) are solvedby integrating upward from the ionosphere, where the twofields are related by the ionospheric boundary condition,By + m0�PEx = 0. For this first approximation the parallelelectric field is determined from the two-fluid model. Thevalue of By obtained in this calculation is then used to solvethe integral equation (3) using the Nystrom method withGaussian integration and zero-order regularization as de-scribed by Delves and Mohamed [1985]. The new value ofEz obtained from this solution is then used to integrateequations (1) and (2) again. This procedure is repeated untilconvergence is reached. The solution is checked by moni-toring the conservation of energy in the system: The inputPoynting flux must equal the sum of the Poynting fluxreflected out the upper boundary, the ionospheric Jouledissipation, and the parallel dissipation due to the wave-particle interaction, or symbolically

Sinc ¼ Sref þ1

2

XpExIj j2þ 1

2

Zdz Re jzEz*ð Þ: ð12Þ

[13] It should be noted that there are a number of differ-ences between the approach presented here and those givenby Rankin et al. [1999] and Tikhonchuk and Rankin [2000,2002]. The most important of these is that these authors haveconcentrated on wave-particle interactions for long-periodfield line resonances (periods of 10 min or more), while ouremphasis is on the higher frequency waves in the iono-spheric Alfven resonator. Thus their model considers anentire closed field line, while our model only treats the lowerportion of the field line. Our approach can also considerprocesses that occur on open or very extended field linessuch as those in the plasma sheet boundary layer, where thesolution can be matched to the local dispersion relation asdiscussed in Lysak and Song [2003]. Their field line reso-nance model considers electrons that bounce between theirmirror points at a rate much faster than the period of the fieldline resonances. In contrast, our model does not considersuch bouncing since we emphasize shorter-period waves thatare not strongly affected by bouncing electrons, and, indeed,on open field lines such as those in the plasma sheetboundary layer these bouncing electrons would not bepresent. A second distinction between these models is thatthe field line resonance models consider an envelope ap-proximation, i.e., the kinetic effects are not considered tochange the wave profile along the field line, whereas ourmodel includes the self-consistent modification of the waveprofile by the kinetic particles. Finally, their model includesthe effect of a background equilibrium parallel electric field,while our model does not. While such an equilibrium field iscertainly present on auroral field lines once a discrete auroralarc has been established, our interest here is in the earlystages of arc formation before such a parallel potential dropis present. Future work will include such a potential drop,

LYSAK AND SONG: NONLOCAL ALFVEN WAVE KINETIC THEORY SMP 9 - 3

which will allow the incorporation of additional particlepopulations, e.g., particles that leave the ionosphere, reflectfrom the parallel electric field, and return to the ionosphere.

3. Results

[14] We present results from a variety of cases in thissection. Since there are a number of parameters that can bevaried we will start from a benchmark case and thendetermine the variation of other parameters taken one at atime. Our benchmark will consist of a cold plasma popula-tion in the topside ionosphere that has an ionosphericdensity of n0 = 105 cm�3 and a scale height of 600 km thatis assumed to be composed of oxygen ions. The coldelectrons will not be treated with the full kinetic formalismbut rather with the cold plasma conductivity, which is givenby

scold zð Þ ¼incold zð Þe2

mew: ð13Þ

It is this conductivity that corresponds to the electroninertial effect. The ‘‘hot’’ plasma, by which in this contextwe mean the part of the distribution that will be treatedkinetically, is assumed to have a density of 0.5 cm�3 and anisotropic electron temperature of 500 eV, although thistemperature will be varied in some of the cases shownbelow. The perpendicular ion temperature is assumed to be3 keV for calculating the ion gyroradius. The ion populationthat neutralizes these electrons is assumed to be composedof protons. These parameters are typical of plasma sheetparameters, although, of course, they may vary in differentcircumstances. Figure 1 plots the density distribution andAlfven speed profile for these parameters. Note that thisprofile gives a rather large density in the auroral accelera-tion region, e.g., �5 cm�3 at 2 RE. This is consistent withmeasurements of background density along auroral fieldlines outside of the auroral density cavity region [e.g.,

Figure 1. (a) Plot of the density profile used in the runs.(b) Plot of the Alfven speed profile used in the runs. Thedotted curve gives the correction due to ion gyroradiuseffects, and the dashed curves give the velocity of electronsat 10 keV, 1 keV, 100 eV, 10 eV, and 1 eV, from top tobottom, for comparison.

Figure 2. Plots of the relative magnitudes of the terms inthe energy conservation equation (12) as a function offrequency for (a) �P = 1.9 mho and (b) �P = 10 mho. Thesolid curve gives the reflected power (Ref ), the dotted curvethe ionospheric Joule dissipation (Ionos), and the dashedcurve the dissipation in the wave-particle interaction (WPI).A perpendicular wavelength of 10 km is assumed. Theplasma sheet density is 0.5 cm�3, the electron temperature is500 eV, and the ion temperature is 3 keV.


Mozer et al., 1979; Kletzing et al., 1998; Johnson et al.,2001]. Since our model does not include a backgroundparallel electric field, it is reasonable to use such a densityprofile. The horizontal dashed curves in Figure 1b indicatethe velocity of an electron with a parallel energy of 10 keV,1 keV, 100 eV, 10 eV, and 1 eV, running from top to bottom.The curves in Figure 1b serve as an indicator of wherewave-particle resonances would occur in a uniform plasma.However, as we shall see below, the locations of maximumdissipation do not strictly follow these curves due to thenonlocal nature of the wave-particle interaction. The dottedcurve in Figure 1b that closely follows the Alfven speedprofile gives the correction to the wave phase speed due tothe ion gyroradius, as given in equation (2), assuming a3 keV ion population and a perpendicular wavelength of10 km, mapped to the ionosphere. It can be seen that thiscorrection is minimal.[15] The other important parameters that will be varied are

the perpendicular wavelength and frequency of the wave,and the ionospheric Pedersen conductivity. For the bench-mark run we take the perpendicular wavelength mapped tothe ionosphere to be 10 km. We consider two values for the

Pedersen conductivity: 1.9 mho and 10 mho. The value of1.9 mho is chosen since it matches the Alfven admittance atthe ionospheric end of the field line, while the other caseillustrates the effects of higher conductivity. It should also benoted that the top end of the computational box is placed at4 RE geocentric altitude. This location is somewhat arbitrary;however, we shall see that most of the dissipation occursbelow this altitude for the runs considered.[16] Figure 2 shows the magnitude of the terms on the

right-hand side of the energy balance equation (12) normal-ized to the input Poynting flux for a series of runs with theparameters as given above as a function of frequency. InFigure 2 the solid curve gives the ratio of reflected toincident Poynting flux (labeled ‘‘Ref’’), the dotted curvegives the ratio of ionospheric dissipation to input Poyntingflux (labeled ‘‘Ionos’’), and the dashed curve gives thenormalized dissipation due to the wave-particle interaction(labeled ‘‘WPI’’). These results show that a broad maximumexists in the wave-particle dissipation around 0.2 Hz, inwhich almost all of the incident wave power is dissipated.Below �0.1 Hz the ionospheric dissipation goes to zeroindependent of the value of the Pedersen conductivity,

Figure 3. Results from the mode structure of a wave with a frequency of 0.22 Hz and a perpendicularwavelength of 10 km. A plasma density of 0.5 cm�3, electron temperature of 500 eV, ion temperature of3 keV, and ionospheric conductivity of 1.9 mho are assumed. (a) Magnitude of the Ex/By ratio, withdotted curve giving the local Alfven speed. (b) Parallel electric field, showing real (solid curves) andimaginary (dotted curves) parts. Dashed and dash-dotted curves give the results from cold plasma theory.(c) Phase shift between Ex and By fields (in degrees). (d) Dissipation due to the wave-particle interaction.


indicating that the wave reflects from the parallel electricfield region rather than from the ionosphere. The modestructure of the ionospheric Alfven resonator is clearly seen,with the resonator modes corresponding to peaks in theionospheric Joule heating at frequencies of �0.16, 0.36,0.56, 0.76, and 0.97 Hz. It should be noted that these peaksoccur at the same frequencies as they do in the cold plasmaapproximation. In the high-conductivity case where theionospheric dissipation is lower, the peaks are sharper. Notethat in this range there is a point plotted with at least 0.01 Hzresolution (0.005 Hz in the neighborhood of the peaks) sothat the structure in these ratios are well resolved.

3.1. Wave Mode Structure

[17] Next let us consider the wave mode structure alongthe magnetic field line. In the remainder of this paper we willconcentrate on two frequencies, 0.22 Hz and 0.35 Hz, whichcorrespond to a local maximum and minimum, respectively,in the Landau dissipation, as can be seen from Figure 2. Wewill also assume a Pedersen conductivity, �P = 1.9 mho,perpendicular wavelength l? = 10 km, and the same plasmaparameters as above. The curves are normalized so that the

field-aligned current at the ionosphere is 10 mA m�2, whichis of course arbitrary since the calculation is linear.[18] Figures 3 and 4 show the wave mode structure for

the 0.22 Hz and 0.35 Hz cases. Figures 3a and 4a show theratio of the perpendicular electric and magnetic field varia-tions. The dotted curves in Figures 3a and 4a are the localvalue of the Alfven speed. Note that the electric-to-magneticfield ratio is lower than the local Alfven speed in the regionof the Alfven speed peak, while it becomes larger than thelocal Alfven speed at higher altitudes. This results from thefull wave solution of the wave in the ionospheric resonator,as has been seen in earlier studies [Lysak, 1998]. Figures 3cand 4c show the phase shift between Ex and By. This isgenerally 180, indicative of the downward Poynting flux;however, there are fluctuations in this phase shift due to thepartial reflections of the wave in the Alfven resonator, withthe phase shift becoming close to 90 at 2.5 RE in the0.22 Hz case, indicating a standing wave.[19] Figures 3b and 4b show the parallel electric fields for

these two cases. In Figures 3b and 4b the solid and dottedcurves give the real and imaginary parts of the parallelelectric field, respectively, while the dotted and dot-dash

Figure 4. Same as Figure 3, except that a wave frequency of 0.35 Hz is assumed. Note that these plotsare plotted to the same scale as the corresponding panels of Figure 3. (a) Magnitude of the Ex/By ratio,with dotted curve giving the local Alfven speed. (b) Parallel electric field, showing real (solid curves) andimaginary (dotted curves) parts. Dashed and dash-dotted curves give the results from cold plasma theory.(c) Phase shift between Ex and By fields (in degrees). (d) Dissipation due to the wave-particle interaction.


curves give the real and imaginary parts calculated using coldplasma theory for comparison. At low altitudes, where theplasma is dense, the high cold plasma conductivity essentiallyshorts out the parallel electric field, and it stays nearly zero.The parallel electric field becomes significant between 2 and3 RE where the cold plasma disappears. In this region theelectron distribution must be significantly modified so thatthe warm plasma can carry the necessary current required bythe curl of the magnetic field. Figures 3d and 4d show thedissipation, given by Re( jzE*z) (see equation (12)). Figures 3dand 4d show a region of negative dissipation; thus theparticles are locally giving energy to the wave at that point.However, the total dissipation integrated over the wholeregion is still positive, indicating damping of the wave.

3.2. Temperature Dependence

[20] Figure 5 shows the temperature dependence of theterms in the energy conservation equation for the 0.22 Hzand 0.35 Hz cases, for a Pedersen conductivity of 1.9 mho,and for a perpendicular wavelength of 10 km. As inFigure 2, the solid curve gives the fraction of reflectedenergy, the dotted curve is the Joule dissipation in theionosphere and the dashed curve is the dissipation due tothe wave-particle interaction. It can be seen in both casesthat there is very little wave-particle dissipation at lowtemperatures, consistent with cold plasma theory. At0.22 Hz this dissipation increases with temperature until�1 keV then decreases. The peak occurs at a temperature of�2.3 keV for the 0.35 Hz case. This suggests that the peaktemperature corresponds to an electron transit time effect,since the thermal velocity of the electrons at the peaktemperature roughly scales with the wave frequency, witha/w � 2 RE in both cases, where a is the electron thermalspeed. At higher temperatures the ionospheric Joule heatingalso decreases, indicating that the wave is mostly reflectedbefore reaching the ionosphere. The amount of reflection inboth the cold and hot ranges is more for the 0.22 Hz casethan for the 0.35 Hz case, since this latter case correspondsto a mode of the ionospheric Alfven resonator, as defined bythe enhanced Joule dissipation [Lysak, 1991].[21] Figure 6 shows the profiles of the parallel electric

fields for temperatures of 1, 10, 100, and 1000 eV. As inFigures 3b and 4b, the solid and dotted curves in Figure 6are the real and imaginary part of the parallel electric field,respectively, while the dashed and dash-dotted curves arethe real and imaginary part of the parallel electric fieldcalculated from the cold plasma model. It can be seen thatthe 1 eV case closely follows the cold plasma results, aswould be expected. As the temperature increases, theparallel electric field in the 2–3 RE region increases andbecomes dominant at large temperatures.

3.3. Conductivity Dependence

[22] Figure 7 shows the dependence of the energy balanceon the Pedersen conductivity. In these plots, there is a cleardistinction between the frequency away from the Alfvenresonator mode (0.22 Hz) and the frequency at the resonatormode (0.35 Hz). In both cases the ionospheric dissipationpeaks at an intermediate conductivity; however, the peak isat larger conductivity (�10 mho) in the 0.35 Hz case than inthe 0.22 Hz case. At higher conductivities the 0.22 Hz caseabsorbs most of the energy in the wave-particle resonance

while in the 0.35 Hz case the wave is mostly reflected athigh frequencies. The distinction between the two casesdecreases at the lowest conductivities, where the wave ismostly absorbed in both cases. These differences arise fromthe sharpening of the resonance structure at higher conduc-tivities, as can be seen in Figure 2.

3.4. Perpendicular Wavelength Dependence

[23] Finally, Figure 8 shows the dependence of the energybalance on perpendicular wavelength. Both frequencies givesimilar behavior, with a maximum absorption by the wave-particle resonance at wavelengths of <10 km. At largerwavelengths the wave-particle resonance becomes lessimportant and the Joule dissipation in the ionospherebecomes larger. In contrast, at short wavelengths, theabsorption of the wave begins to decrease and the reflectionincreases. In this regime, the wave is mostly reflected from

Figure 5. Plots of the relative magnitudes of the terms inthe energy conservation equation (12) as a function oftemperature for a frequency of (a) 0.22 Hz and (b) 0.35 Hz.The solid curve gives the reflected power (Ref ), the dottedcurve the ionospheric Joule dissipation (Ionos), and thedashed curve the dissipation in the wave-particle interaction(WPI). An ionospheric conductivity of 1.9 mho and aperpendicular wavelength of 10 km are assumed. A hotplasma density of 0.5 cm�3 and an ion-to-electrontemperature ratio of 6 are used.


the wave-particle resonance region rather than the iono-sphere, as is evidenced by the lack of ionospheric Jouledissipation. This is consistent with the general behavior ofthe Alfven waves in a dissipative medium, in which thewave becomes reflected from a resistive layer at shortwavelengths [Lysak and Dum, 1983]. An analysis of thereflection of Alfven waves from an auroral accelerationregion gives similar results [Vogt and Haerendel, 1998].These results all point to the difficulty of transmittingAlfven wave energy through the auroral acceleration regionat short perpendicular wavelengths.[24] It may be noted that the results of Figure 8 are

shown only down to 1 km wavelength. At shorter wave-lengths, the iterative procedure used to solve the system ofequations fails to converge. This is due to the increasedimportance of the parallel electric field term in Faraday’slaw, equation (1), and the increase of the magnetic fielddriver term in the parallel Ampere’s law, equation (3).Both of these terms become larger as k? is increased,leading to the divergence of the method. However, thetrends as the wavelength becomes shorter are clear, with

the wave reflection increasing and the ionospheric dissi-pationgoing to zero.

4. Discussion

[25] The results shown above represent a first calculationof self-consistent kinetic effects on Alfven waves on dipolarauroral field lines. The model is based on a scenario inwhich an Alfven wave of fixed frequency and perpendicularwavelength is launched from the outer magnetospheretoward the auroral zone. Recent observations from Polarshow that significant Poynting flux is present, particularly atthe plasma sheet boundary layer, to provide such a sourcefor these Alfven waves [e.g., Wygant et al., 2000]. Thesecalculations give the fate of the wave energy as it passesinto the auroral zone. The fraction of the energy reflected,dissipated by Joule heating in the ionosphere, and absorbedby the wave-particle interaction are shown to be a functionof the frequency and perpendicular wavelength of the wave,the ionospheric Pedersen conductivity, and the temperatureof the plasma sheet electrons.

Figure 6. The parallel electric field for four cases with plasma sheet temperatures of (a) 1 eV, (b) 10 eV,(c) 100 eV, and (d) 1 keV. The solid curve gives the real part and the dotted curve the imaginary part ofEk. The dashed and dash-dotted curves give the real and imaginary parts of Ek, respectively, from coldplasma theory. The wave frequency is 0.35 Hz, the perpendicular wavelength is 10 km, and theionospheric conductivity is 1.9 mho for all cases.


[26] One consistent result of these parameter studies isthat there is often a transition from a state in which kineticeffects are not important, through a region of maximumabsorption, and finally to a state in which the wave isreflected from the interaction region before reaching theionosphere. This transition takes place as the temperature isincreased, as the perpendicular wavelength is decreased,and as the wave frequency is decreased. This behavior isconsistent with the simplified model of Alfven wave inter-actions with a resistive layer presented by Lysak and Dum[1983]. Figure 2 shows that Alfven waves are mostlytransmitted through the resistive layer at long perpendicularwavelength, go through a transition region where theabsorption maximizes, and then are mostly reflected at shortwavelengths. Vogt and Haerendel [1998] showed a similarbehavior based on a model in which the auroral accelerationregion was modeled as a Knight [1973] current-voltagerelationship. Pilipenko et al. [2002] have recently empha-

sized the role of this reflection in trapping waves in theresonator region. Although they distinguish between wavesreflected by the Alfven speed gradient and those reflectedby the parallel electric field, these two phenomena cannotreally be separated and must be considered together. In anycase, from a number of different theoretical approaches,there is a consistent theme that Alfven waves with shortperpendicular wavelengths are reflected by the auroralacceleration region.[27] This result has important consequences for the

evolution of auroral arcs. Auroral arcs can have thicknessesof <1 km [e.g., Maggs and Davis, 1968; Borovsky andSuszcynsky, 1993; Trondsen and Cogger, 2001; Knudsen etal., 2001]. These results would suggest that such narrow-scale auroras, if they are related to field-aligned currentsand Alfven waves on the same scale, must be producedwithin or below the auroral acceleration region and not

Figure 7. Plots of the relative magnitudes of the terms inthe energy conservation equation (12) as a function ofionospheric conductivity for a frequency of (a) 0.22 Hz and(b) 0.35 Hz. The solid curve gives the reflected power (Ref ),the dotted curve the ionospheric Joule dissipation (Ionos),and the dashed curve the dissipation in the wave-particleinteraction (WPI). A perpendicular wavelength of 10 km isassumed. Other plasma parameters are as in Figure 2.

Figure 8. Plots of the relative magnitudes of the terms inthe energy conservation equation (12) as a function ofperpendicular wavelength for a frequency of (a) 0.22 Hzand (b) 0.35 Hz. The solid curve gives the reflected power(Ref ), the dotted curve the ionospheric Joule dissipation(Ionos), and the dashed curve the dissipation in the wave-particle interaction (WPI). An ionospheric conductivity of1.9 mho is assumed. Other plasma parameters are as inFigure 2.


imposed from a small-scale generator in the outer magne-tosphere. It should be noted that Knudsen et al. [2001]showed a bimodal distribution of auroral arc widths, with apeak in the subkilometer range as well as a peak near10 km. It is tempting to speculate that the larger-widthpeak is associated with the direct absorption of Alfvenwaves from the outer magnetosphere while the narrowerpeak is associated with other processes at low altitude.Processes such as the ionospheric feedback instability [e.g.,Atkinson, 1970; Sato, 1978; Lysak, 1991; Lysak and Song,2002; Pokhotelov et al., 2002] could produce such small-scale structures, as could nonlinear interactions betweenAlfven waves trapped in the ionospheric resonator [Songand Lysak, 2001].[28] Although results have been shown for a wide

variety of plasma parameters, there are many other casesthat could be considered. For example, the ionosphericscale height for the density plays an important role indetermining the frequencies of the ionospheric Alfvenresonator. Runs (not shown) that have a scale height of300 km rather than 600 km show similar features asidefrom the fact that the resonant frequencies are higher(scaling as VAI/2h, as noted by Lysak [1991]). Runs atvarious frequencies as in Figure 2 but with a 300 km scaleheight show a maximum in the wave-particle dissipation at0.30 Hz, with a peak absorption of 0.87. A major differ-ence with smaller scale height is that the parallel electricfield is at lower altitudes and is stronger in magnitude. InFigure 3b the maximum magnitude of the parallel electricfield is 0.46 mV m�1 at a radial distance of 2.65 RE, whilefor the 0.30 Hz run with 300 km scale height the maximumparallel electric field is 1.3 mV m�1 at 1.87 RE. Neverthe-less, the total dissipation in wave-particle interactions issimilar in these two runs.[29] Another parameter is the ion-to-electron temperature

ratio, which is taken to be 6 in the runs above. Smaller ratiosof this parameter affect results primarily at small scales,where the ion gyroradius effect is most important. As notedin Figure 1, an ion temperature of 3 keV has only a minimaleffect on the Alfven speed at a perpendicular wavelength of10 km, but at smaller wavelengths it can have a greatereffect. As noted by Lysak and Lotko [1996], a higher Ti/Teratio leads to less damping in a uniform plasma. In thisnonuniform situation, runs like those in Figure 8 but withTi = Te show more reflection and less dissipation at wave-lengths <7 km than the runs shown with Ti = 6Te. This isconsistent with the uniform plasma results since the strongerdissipation regions generally lead to more reflection as wasfound.[30] The calculations presented here describe the linear

damping of Alfven waves in the auroral region; however,they do not explicitly address the fate of the dissipatedenergy. The energy lost in the wave-particle interaction willgo into the heating of electrons in the auroral zone, whichmay either precipitate into the ionosphere or go up the fieldline into the magnetosphere. Preliminary calculations solv-ing the Vlasov equation for these electrons indicate thatonly �10% of the energy goes into precipitating electrons.However, these calculation only consider the hot plasmasheet electrons and, as test particle studies have shown, coldionospheric electrons can be readily accelerated along thefield lines by Alfven waves to produce aurora. Thus the

present work is only the first step in producing a fully self-consistent theory of the interactions of electrons and Alfvenwaves in the auroral region.

Appendix A: Details of Conductivity Kernel

[31] As we noted previously, the perturbation in thedistribution function can be written as

df zð Þ ¼ df z0ð Þeiwt z0 ;zð Þ � q@f0@W

Zz

z0

dz0dEz z0ð Þeiwt z0 ;zð Þ: ðA1Þ

We must multiply this distribution by the parallel velocityand integrate over velocity space in order to get theconductivity. We first note that the unperturbed distributionshould be written as a function of the constants of motion mand W. Assuming a bi-Maxwellian distribution with thepossibility of parallel and perpendicular temperatures, sucha distribution is written as

f0 m;Wð Þ ¼ nm

2p

� �3=2 1

T?T1=2k

exp �W

Tk� mBT?

1� T?

Tk

� � :

ðA2Þ

In the usual cylindrical coordinates we have d3v =2pv?dv?dv0. To transfer to the constants of motion, wemust calculate the Jacobian

J ¼ det

@W=@vk @W=@v?

@m=@vk @m=@v?

24

35¼ det

mvk mv?

0 mv?=B

24

35¼ m2vkv?

B:

ðA3ÞThen we have

d3v ¼ 2pv?J

dWdm ¼ 2pBm2vk

dWdm: ðA4Þ

Thus the current density is given by

jz ¼2pqBm2

ZdWdm dfþ vð Þ � df� vð Þ½ ; ðA5Þ

where v ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2=mð Þ W � mBð Þ

p. Now let us consider the three

particle populations and for the moment neglect the boundaryvalue term (i.e., the first term in (A1)). The ionosphericparticles give a contribution to df+, given by

dfIþ ¼ �q@f0I@W

Zz

0

dz0Ez z0ð Þeiwt z0;zð Þ: ðA6Þ

It should be noted that ionospheric particles all have W >mBI, where BI is the magnetic field strength at the ionosphere.[32] Next we consider downgoing magnetospheric elec-

trons. These particles reach an altitude where the magneticfield strength is B only if W > mB. Their contribution,assuming that z = L is the top boundary, is given by

dfM� ¼ �q@f0M@W

Zz

L

dz0Ez z0ð Þeiwt z;z0ð Þ ¼ q

@f0M@W

ZLz

dz0Ez z0ð Þeiwt z;z0ð Þ:

ðA7Þ


Note the change in sign due to the switching of the limits ofintegration.[33] The third population is upgoing particles that have

mirrored below the location z. For these particles, theintegration must go from z = L down to the mirror pointzm, which is the point zm where the magnetic field is givenby Bm =W/m, and then back up to the location z. Then again,switching the limits of integration for the downgoing part ofthe integration, we can write

dfMþ ¼q@f0M@W

ZLzm

dz0Ez z0ð Þeiw t zm;zð Þþt z0;zð Þð Þ �

Zz

zm

dz0Ez z0ð Þeiwt z0;zð Þ

24

35:

ðA8Þ

There is no contribution to the integral from these particlesunless the field point z0 is also above the mirror point. Thusif we denote B0 = B(z0) and B = B(z), Bm must be larger thanboth B and B0 for a contribution, or W > mBmax, where Bmax

is the larger of B and B0. The second of these integrals isonly defined when z0 < z or B0 > B, so Bmax = B for thisintegral. In addition, these particles must not be in the losscone; therefore we must have W < mBI.[34] We can recover the form of the conductivity integral

by taking these distributions and inserting them in equation(A5), noting the respective accessibility conditions. Thisprocedure leads to

jz zð Þ ¼ � 2pq2Bm2

ZL0

dz0Ez z0ð ÞZ10

dW

(� z� z0ð Þ

ZW=BI

0

� dm @f0I@W

eiwt z0; zð Þ þ� z0 � zð ÞZW=B

0

dm@f0M@W

eiwt z;z0ð Þ

�ZW=Bmax

W=BI

dm@f0M@W

eiw t zm;zð Þþt zm;z0ð Þð Þ þ� z� z0ð Þ

�ZW=B0

W=BI

dm@f0M@W

eiwt z0;zð Þ

); ðA9Þ

where �(x) is the step function that is 1 for positiveargument and 0 for negative argument. This function takesaccount of the limits of integration in equations (A6)–(A8).Comparing equation (A9) with the general form of thenonlocal conductivity relation (11), we can write

s z; z0ð Þ ¼ � 2pq2Bm2

Z10

dW � z� z0ð ÞZW=BI

0

8<: dm

@f0I@W

eiwt z0;zð Þ

þ� z0 � zð ÞZW=B

0

dm@f0M@W

eiwt z;z0ð Þ

�ZW=Bmax

W=BI

dm@f0M@W

eiw t zm;zð Þþ zm;z0ð Þð Þ

þ� z� z0ð ÞZW=B0

W=BI

dm@f

@Weiwt z0;zð Þ

9>=>;: ðA10Þ

[35] We must next take into account the boundary termin equation (A1). Since we assume no perturbation in theionospheric upgoing distribution, we need only to considerthe downgoing magnetospheric particles. This can bewritten in a manner analogous to equation (A9) withz0 = L. Note that the first and fourth terms in equation(A9) refer to the effect of fields below the observationpoint z. Thus only the second and third terms willcontribute. Writing the current due to the boundary termin a manner similar to equation (A5), we have

jL zð Þ ¼ 2pqBm2

Z10

dW

ZW=B

W=BI

dmdf� Lð Þeiw t zm;Lð Þþt zm;zð Þð Þ

264

�ZW=B

0

dmdf� Lð Þeiwt z;Lð Þ

375: ðA11Þ

These two terms correspond to the third and second terms inequation (A9), respectively. As in Lysak and Song [2003],we determine the distribution function at L from the localdispersion relation, including the finite gyroradius correc-tion (which is small for electrons)

df� Lð Þ ¼ � iq

wJ 20

k?v?0

�0

� �vk0VM

vk0 � VM

@f0M@W

Ez Lð Þ; ðA12Þ

where �0 = qB0/m is the gyrofrequency, v?0 = (2mB0/m)1/2

is the perpendicular velocity, vk0 = [(2/m)(W � mB0)]1/2 is

the parallel velocity, B0 is the magnetic field at L, and VM isthe parallel phase velocity found by solving the localdispersion relation. It can be seen that from equations (A11)and (A12) we can write the contribution of the boundaryterm to the current as

jL zð Þ ¼ sL zð ÞEz Lð Þ: ðA13Þ

[36] As a final point, it should be noted that the traveltime integral can be written in the form

t z1; z2ð Þ ¼ffiffiffiffiffiffiffim

2W

r Zz2z1

dz0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� mB z0ð Þ=W

p : ðA14Þ

When m = 0, this integral is simple; when it is not 0, theintegral can be converted to an integral over magnetic fieldsince B(z) = BE/(1 + z/RE)

3. Then (A14) can be written


2W

rREB

1=3E

3

ZB1

B1

dB0

B4=3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� mB0=W

p : ðA15Þ

If we now make a change in variable to x = mB0/W, thisintegral can be written


2W

rRE

3

mBE

W

� �1=3 Zx1x2

dx

x 4=3ffiffiffiffiffiffiffiffiffiffiffi1� x

p : ðA16Þ


The dimensionless integral in (A16) can be calculatednumerically and stored for efficient computation of thetravel times.

[37] Acknowledgments. This work has benefited from discussionswith R. Rankin, V. Tikhonchuk, R. Marchand, W. Lotko, and A. Streltsov.The work is supported in part by NASA grant NAG5-9254 and NSF grantATM-0201703.[38] Arthur Richmond thanks Craig A. Kletzing and Robert Rankin for

their assistance in evaluating this manuscript.

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��R. L. Lysak and Y. Song, School of Physics and Astronomy, University

of Minnesota, Minneapolis, MN 55455, USA. ([email protected]; [email protected])


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Nonlocal kinetic theory of Alfve´n waves on dipolar …ham.space.umn.edu/~bob/kinetic-dipole-paper.pdfNonlocal kinetic theory of Alfve´n waves on dipolar field lines Robert L. Lysak