Nonlocal kinetic theory of Alfvén waves on dipolar field lines

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 18, NO. A8, 137, doi:1.19/3ja9859, 3 Nonlocal kinetic theory of Alfvén waves on dipolar field lines Robert L. Lysak and Yan Song School of Physics and Astronomy, University of Minnesota, Minneapolis, Minnesota, USA Received 4 January 3; revised 1 May 3; accepted 3 June 3; published 6 August 3. [1] Recent observations have indicated that in addition to the quasi-static acceleration of electrons in inverted V structures, auroral electrons frequently have a spectrum that is broad in energy and confined to parallel pitch angles, indicative of acceleration in lowfrequency waves. Test particle models have indicated that these electrons may be accelerated by the parallel electric fields in kinetic Alfvén waves. However, such models are not self-consistent, in that the wave structure is not influenced by the accelerated particles. A nonlocal kinetic theory of electrons along auroral field lines is necessary to provide this self-consistency. Results from such a model based on electron motions on dipole field lines are presented. For a typical Alfvén speed profile, kinetic effects lead to significant energy dissipation when the electron temperature exceeds 1 ev. The dissipation generally occurs near the peak of the Alfvén speed profile. This dissipation generally increases with increasing temperature and decreasing perpendicular wavelength up to 1 kev and 1 km, respectively. At larger temperatures and smaller perpendicular wavelengths the dissipation begins to decrease and the ionospheric Joule dissipation goes to zero, indicating that the wave is reflected from the dissipation region. Dissipation in the.1 1. Hz band is structured by the modes of the ionospheric Alfvén resonator. INDEX TERMS: 74 Magnetospheric Physics: Auroral phenomena (47); 736 Magnetospheric Physics: Magnetosphere/ionosphere interactions; 77 Magnetospheric Physics: Plasma waves and instabilities; 787 Space Plasma Physics: Kinetic and MHD theory; 7867 Space Plasma Physics: Wave/particle interactions; KEYWORDS: auroral particle acceleration, Alfvén waves, magnetosphereionosphere coupling, nonlocal kinetic theory, wave-particle interactions Citation: Lysak, R. L., and Y. Song, Nonlocal kinetic theory of Alfvén waves on dipolar field lines, J. Geophys. Res., 18(A8), 137, doi:1.19/3ja9859, Introduction [] Observations from the FAST satellite [Chaston et al., 1999,, a, b] have indicated that precipitating auroral electrons often are field aligned and have a broad distribution in energy, in contrast to the typical auroral inverted V type precipitation, which have a characteristic energy but are broad in pitch angle. Regions of this type of field-aligned acceleration have been observed throughout the auroral zone but are often seen at the polar cap boundary of the auroral zone. Similar observations of field-aligned electrons have been seen over many years, particularly from sounding rocket missions [Johnstone and Winningham, 198; Arnoldy et al., 1985; Robinson et al., 1989; McFadden et al., 1986, 1987, 199, 1998; Clemmons et al., 1994; Lynch et al., 1994, 1999; Knudsen et al., 1998]. These field-aligned distributions are not consistent with the usual picture of a plasma sheet distribution that has been accelerated in a quasi-static potential drop. Rather, it has been suggested that low-frequency waves are responsible for this acceleration, whether in electromagnetic ion cyclotron waves [Temerin et al., 1986] or Alfvén waves [Kletzing, 1994; Copyright 3 by the American Geophysical Union /3/3JA9859 Thompson and Lysak, 1996; Chaston et al., 1999,, a, b; Kletzing and Hu, 1]. [3] These studies have used test particle models to describe the interaction of Alfvén waves with the electron population. In most of these cases the electron inertial effect was used to determine the parallel electric field that accelerates these electrons. Thus the electron acceleration process did not modify the wave fields self-consistently. Such modifications require the use of a kinetic theory to describe the electron acceleration. Since the electrons can travel large distances during a wave period, such a kinetic theory must be nonlocal, i.e., the parallel electric field at one location along the field line can influence the current at other locations along the field line. A first attempt at such a nonlocal theory in the context of field line resonances has been developed by Rankin et al. [1999] and Tikhonchuk and Rankin [, ]. These authors determined that the kinetic effect of bouncing electrons could enhance the parallel electric field to values much greater than that given by electron inertia. [4] A model that extended this type of calculation to the higher frequencies of waves trapped in the ionospheric Alfvén resonator has recently been presented by Lysak and Song [3]. This work considered a simplified model consisting of a straight magnetic field line with an idealized Alfvén speed profile typical of the ionospheric Alfvén SMP 9-1

2 SMP 9 - LYSAK AND SONG: NONLOCAL ALFVÉN WAVE KINETIC THEORY resonator that becomes constant at high altitudes. In addition to the recovery of the uniform plasma Landau damping at high altitudes, this model showed that an enhancement of the dissipation due to the wave-particle resonance occurred just below the point at which the Alfvén speed began to decrease. [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 the wave fields due to the kinetic effects. This approximation is justified when the kinetic effects are weak, which occurs for larger values of the perpendicular wavelength. However, the waves that may be responsible for auroral arcs are expected to have perpendicular wavelengths comparable to the electron skin depth or smaller. At such wavelengths the wave mode 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 [3] to use a more realistic profile that includes the dipole geometry of an auroral flux tube. We will still concentrate on the higher frequency waves associated with the ionospheric Alfvén resonator and so will model only the lower regions of the field line where the Alfvén speed gradients are strongest and the local kinetic theory breaks down. The model results for the relative amounts of dissipation due to wave-particle interactions will be presented for a variety of plasma parameters.. Theoretical Development [7] The development of the model follows the results presented by Rankin et al. [1999], Tikhonchuk and Rankin [, ], and Lysak and Song [3]. The details of the model can be found in those works; here we will briefly sketch the theory behind these calculations. [8] The basic concept of this model is to consider the wave equations for Alfvén waves along auroral field lines, including a fully kinetic treatment for the electron motion along the field line. The wave equations for the kinetic Alfvén wave can be expressed by assuming a wave of constant frequency and perpendicular wave number (which is assumed to be in the x direction). In this case, Faraday s law and the perpendicular and parallel components of Ampere s law can be written ¼ iw c Z iwe E z ¼ iwb y þ ik? E z 1 þ c V A 1 m i E x dz sðz; z ÞE z ðz Þ ¼ ik? B B y : m B I It should be noted that in these equations, E x, B y and k? are mapped to their ionospheric values using an isotropic dipolar mapping. Thus the only mapping factor that appears explicitly is the ratio of the background magnetic field strength to that in the ionosphere, B /B I, that appears in ð1þ ðþ ð3þ equation (3). The ion gyroradius correction is included in equation () through the Bessel function which is a function of the parameter m i = k? r i. It may be noted that this term is sometimes approximated by the relation (1 (m i ))/m i 1/(1 + m i ), as in the work of Johnson and Cheng [1997]; however, in this work we retain the full Bessel function expression. [9] The electron kinetic effects enter into the wave equations above through the nonlocal conductivity relation in the parallel Ampere s law, equation (3). This relation can be determined by solving the Vlasov equation for electrons in a dipolar magnetic field subject to the propagation of a kinetic Alfvén wave. Since these waves are at very low frequencies, the magnetic moment of the electrons will be conserved. In addition, the total energy of an electron, given by W ¼ 1 mv z þ mbðþþq z ðþ; z is conserved in the absence of the wave; however, the parallel electric field of the wave will modify this total energy. While equation (4) includes the possibility of a background parallel potential drop, this term will not be included in the results presented in this paper. Thus the linearized Vlasov equation will take ðm; WÞ z df ðz; m; W ; tþþqde z v z ¼ where f ± give the unperturbed distribution functions and the plus or minus refers to the direction of the parallel velocity (where we take the positive z direction to be upward). This equation can be solved in the usual way by integrating along the unperturbed orbits defined by constant magnetic moment and energy df ðz; tþ ¼ df ðz ; t Þ dt de z ðz ; t Þv z ðt Þ: t Note that here the dependence of the distribution functions on m and W is implicit. The first term in this equation gives the initial value of the particle that leaves a boundary position z at a time t. Note that since dz = v z dt,the integral can be written in terms of the z coordinate, giving df ðz; t Z t Þ ¼ df ðz ; t tðz ; zþþ Z z z ð4þ ð5þ ð6þ dz de z ðz ; t tðz ; zþþ; ð7þ where the function t refers to the travel time between two positions in z Z z dz Z z tðz 1 ; z Þ ¼ v z ðz Þ ¼ dz p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ð=mþðw mbðz Þ qðz ÞÞ z 1 z 1 ð8þ Note that the plus or minus sign here again refers to the particle s direction of motion: for z 1 < z, the plus sign is

3 LYSAK AND SONG: NONLOCAL ALFVÉN WAVE KINETIC THEORY SMP 9-3 taken, with the minus sign taken otherwise, so that t is defined to be positive. Again note that this travel time is an implicit function of m and W. Assuming that the wave field oscillates at a frequency w, we can write df ðþ¼df z ðz Þe iwt ð z;z Z z Þ dz de z z z ð Þe iw ð z ;z Þ ; ð9þ where a factor of e iwt has been cancelled from each term. [1] Now we must consider the types of particle trajectories that are present. In general, there are a number of particle populations on an auroral field line, due to the interplay between the magnetic mirror force and the parallel electric field [e.g., Whipple, 1977; Chiu and Schulz, 1978]. However, for the initial results presented here we will assume that there is no parallel electric field, i.e., = in equations (4) and (8). In this case, there are only three types of electron trajectory, each with their own background distribution: (1) ionospheric electrons moving upward; () magnetospheric electrons that precipitate into the ionosphere; and (3) magnetospheric electrons that mirror at some altitude and return to the magnetosphere. The ionospheric electrons begin at the ionospheric boundary (z =) and are described by f +. The equilibrium value of this distribution f + is given by the upgoing half of a Maxwellian, and it is assumed that the perturbation in this distribution function at the ionosphere is zero, i.e., df + () =. The precipitating and mirroring electrons begin at an assumed upper boundary (z = L) and are described by f. The equilibrium value of this distribution f is described by the downgoing half of a Maxwellian, and the density and temperature of this distribution can be different from the ionospheric distribution. The boundary term df (L) issetto the value given by the local kinetic distribution function at this location, as in Lysak and Song [3]. The mirroring electrons also begin at L; however, at some altitude z m (which is a function of m and W) their parallel velocity becomes zero and they begin moving upward, contributing to the upgoing f + distribution. Since the density of these particles is conserved, at the mirror point we have df + (z m )= df (z m ). [11] The perturbation in the field-aligned current by the wave is required to solve the wave equations. This current can be defined in the usual way by integrating over the perturbed distribution function dj z ðþ¼q z Z 1 dvv½df þ ðz; vþ df ðz; vþš: ð1þ Since the perturbation in the distribution function can be written in terms of an integral over the parallel electric field, equation (1) can be written as a nonlocal conductivity relation dj z ðþ¼ z Z L dz sðz; z ÞdE z ðz Þ: ð11þ The detailed calculation of the conductivity kernel is given in Appendix A. [1] The wave mode structure of the Alfvén wave with the electron kinetic effects included can be found by solving equations (1) (3) with the conductivity kernel found by solving the Vlasov equation, as in equations (9) (11). These equations are solved as follows. First, the equilibrium distributions are determined, which give the Alfvén speed profile and the nonlocal conductivity kernel. The explicit form of this kernel is described in Appendix A and is given by equation (A1). Then, equations (1) and () are solved by integrating upward from the ionosphere, where the two fields are related by the ionospheric boundary condition, B y + m P E x =. For this first approximation the parallel electric field is determined from the two-fluid model. The value of B y obtained in this calculation is then used to solve the integral equation (3) using the Nystrom method with Gaussian integration and zero-order regularization as described by Delves and Mohamed [1985]. The new value of E z obtained from this solution is then used to integrate equations (1) and () again. This procedure is repeated until convergence is reached. The solution is checked by monitoring the conservation of energy in the system: The input Poynting flux must equal the sum of the Poynting flux reflected out the upper boundary, the ionospheric Joule dissipation, and the parallel dissipation due to the waveparticle interaction, or symbolically S inc ¼ S ref þ 1 X j E p xij þ 1 Z dz Reðj z E z * Þ: ð1þ [13] It should be noted that there are a number of differences between the approach presented here and those given by Rankin et al. [1999] and Tikhonchuk and Rankin [, ]. The most important of these is that these authors have concentrated on wave-particle interactions for long-period field line resonances (periods of 1 min or more), while our emphasis is on the higher frequency waves in the ionospheric Alfvén resonator. Thus their model considers an entire closed field line, while our model only treats the lower portion of the field line. Our approach can also consider processes that occur on open or very extended field lines such as those in the plasma sheet boundary layer, where the solution can be matched to the local dispersion relation as discussed in Lysak and Song [3]. Their field line resonance model considers electrons that bounce between their mirror points at a rate much faster than the period of the field line resonances. In contrast, our model does not consider such bouncing since we emphasize shorter-period waves that are not strongly affected by bouncing electrons, and, indeed, on open field lines such as those in the plasma sheet boundary layer these bouncing electrons would not be present. A second distinction between these models is that the field line resonance models consider an envelope approximation, i.e., the kinetic effects are not considered to change the wave profile along the field line, whereas our model includes the self-consistent modification of the wave profile by the kinetic particles. Finally, their model includes the effect of a background equilibrium parallel electric field, while our model does not. While such an equilibrium field is certainly present on auroral field lines once a discrete auroral arc has been established, our interest here is in the early stages of arc formation before such a parallel potential drop is present. Future work will include such a potential drop,

4 SMP 9-4 LYSAK AND SONG: NONLOCAL ALFVÉN WAVE KINETIC THEORY which will allow the incorporation of additional particle populations, e.g., particles that leave the ionosphere, reflect from the parallel electric field, and return to the ionosphere. 3. Results [14] We present results from a variety of cases in this section. Since there are a number of parameters that can be varied we will start from a benchmark case and then determine the variation of other parameters taken one at a time. Our benchmark will consist of a cold plasma population in the topside ionosphere that has an ionospheric density of n =1 5 cm 3 and a scale height of 6 km that is assumed to be composed of oxygen ions. The cold electrons will not be treated with the full kinetic formalism but rather with the cold plasma conductivity, which is given by in coldðþe z s cold ðþ¼ z m e w : ð13þ Figure. Plots of the relative magnitudes of the terms in the energy conservation equation (1) as a function of frequency for (a) P = 1.9 mho and (b) P = 1 mho. 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). A perpendicular wavelength of 1 km is assumed. The plasma sheet density is.5 cm 3, the electron temperature is 5 ev, and the ion temperature is 3 kev. Figure 1. (a) Plot of the density profile used in the runs. (b) Plot of the Alfvén speed profile used in the runs. The dotted curve gives the correction due to ion gyroradius effects, and the dashed curves give the velocity of electrons at 1 kev, 1 kev, 1 ev, 1 ev, and 1 ev, from top to bottom, for comparison. It is this conductivity that corresponds to the electron inertial effect. The hot plasma, by which in this context we mean the part of the distribution that will be treated kinetically, is assumed to have a density of.5 cm 3 and an isotropic electron temperature of 5 ev, although this temperature will be varied in some of the cases shown below. The perpendicular ion temperature is assumed to be 3 kev for calculating the ion gyroradius. The ion population that neutralizes these electrons is assumed to be composed of protons. These parameters are typical of plasma sheet parameters, although, of course, they may vary in different circumstances. Figure 1 plots the density distribution and Alfvén speed profile for these parameters. Note that this profile gives a rather large density in the auroral acceleration region, e.g., 5 cm 3 at R E. This is consistent with measurements of background density along auroral field lines outside of the auroral density cavity region [e.g.,

5 LYSAK AND SONG: NONLOCAL ALFVÉN WAVE KINETIC THEORY SMP 9-5 Figure 3. Results from the mode structure of a wave with a frequency of. Hz and a perpendicular wavelength of 1 km. A plasma density of.5 cm 3, electron temperature of 5 ev, ion temperature of 3 kev, and ionospheric conductivity of 1.9 mho are assumed. (a) Magnitude of the E x /B y ratio, with dotted curve giving the local Alfvén speed. (b) Parallel electric field, showing real (solid curves) and imaginary (dotted curves) parts. Dashed and dash-dotted curves give the results from cold plasma theory. (c) Phase shift between E x and B y fields (in degrees). (d) Dissipation due to the wave-particle interaction. Mozer et al., 1979; Kletzing et al., 1998; Johnson et al., 1]. Since our model does not include a background parallel electric field, it is reasonable to use such a density profile. The horizontal dashed curves in Figure 1b indicate the velocity of an electron with a parallel energy of 1 kev, 1 kev, 1 ev, 1 ev, and 1 ev, running from top to bottom. The curves in Figure 1b serve as an indicator of where wave-particle resonances would occur in a uniform plasma. However, as we shall see below, the locations of maximum dissipation do not strictly follow these curves due to the nonlocal nature of the wave-particle interaction. The dotted curve in Figure 1b that closely follows the Alfvén speed profile gives the correction to the wave phase speed due to the ion gyroradius, as given in equation (), assuming a 3 kev ion population and a perpendicular wavelength of 1 km, mapped to the ionosphere. It can be seen that this correction 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 benchmark run we take the perpendicular wavelength mapped to the ionosphere to be 1 km. We consider two values for the Pedersen conductivity: 1.9 mho and 1 mho. The value of 1.9 mho is chosen since it matches the Alfvén admittance at the ionospheric end of the field line, while the other case illustrates the effects of higher conductivity. It should also be noted that the top end of the computational box is placed at 4 R E geocentric altitude. This location is somewhat arbitrary; however, we shall see that most of the dissipation occurs below this altitude for the runs considered. [16] Figure shows the magnitude of the terms on the right-hand side of the energy balance equation (1) normalized to the input Poynting flux for a series of runs with the parameters as given above as a function of frequency. In Figure the solid curve gives the ratio of reflected to incident Poynting flux (labeled Ref ), the dotted curve gives the ratio of ionospheric dissipation to input Poynting flux (labeled Ionos ), and the dashed curve gives the normalized dissipation due to the wave-particle interaction (labeled WPI ). These results show that a broad maximum exists in the wave-particle dissipation around. Hz, in which almost all of the incident wave power is dissipated. Below.1 Hz the ionospheric dissipation goes to zero independent of the value of the Pedersen conductivity,

6 SMP 9-6 LYSAK AND SONG: NONLOCAL ALFVÉN WAVE KINETIC THEORY Figure 4. Same as Figure 3, except that a wave frequency of.35 Hz is assumed. Note that these plots are plotted to the same scale as the corresponding panels of Figure 3. (a) Magnitude of the E x /B y ratio, with dotted curve giving the local Alfvén speed. (b) Parallel electric field, showing real (solid curves) and imaginary (dotted curves) parts. Dashed and dash-dotted curves give the results from cold plasma theory. (c) Phase shift between E x and B y fields (in degrees). (d) Dissipation due to the wave-particle interaction. indicating that the wave reflects from the parallel electric field region rather than from the ionosphere. The mode structure of the ionospheric Alfvén resonator is clearly seen, with the resonator modes corresponding to peaks in the ionospheric Joule heating at frequencies of.16,.36,.56,.76, and.97 Hz. It should be noted that these peaks occur at the same frequencies as they do in the cold plasma approximation. In the high-conductivity case where the ionospheric dissipation is lower, the peaks are sharper. Note that in this range there is a point plotted with at least.1 Hz resolution (.5 Hz in the neighborhood of the peaks) so that the structure in these ratios are well resolved Wave Mode Structure [17] Next let us consider the wave mode structure along the magnetic field line. In the remainder of this paper we will concentrate on two frequencies,. Hz and.35 Hz, which correspond to a local maximum and minimum, respectively, in the Landau dissipation, as can be seen from Figure. We will also assume a Pedersen conductivity, P = 1.9 mho, perpendicular wavelength l? = 1 km, and the same plasma parameters as above. The curves are normalized so that the field-aligned current at the ionosphere is 1 ma m, which is of course arbitrary since the calculation is linear. [18] Figures 3 and 4 show the wave mode structure for the. Hz and.35 Hz cases. Figures 3a and 4a show the ratio of the perpendicular electric and magnetic field variations. The dotted curves in Figures 3a and 4a are the local value of the Alfvén speed. Note that the electric-to-magnetic field ratio is lower than the local Alfvén speed in the region of the Alfvén speed peak, while it becomes larger than the local Alfvén speed at higher altitudes. This results from the full wave solution of the wave in the ionospheric resonator, as has been seen in earlier studies [Lysak, 1998]. Figures 3c and 4c show the phase shift between E x and B y. This is generally 18, indicative of the downward Poynting flux; however, there are fluctuations in this phase shift due to the partial reflections of the wave in the Alfvén resonator, with the phase shift becoming close to 9 at.5 R E in the. 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 dotted curves give the real and imaginary parts of the parallel electric field, respectively, while the dotted and dot-dash

7 LYSAK AND SONG: NONLOCAL ALFVÉN WAVE KINETIC THEORY SMP 9-7 curves give the real and imaginary parts calculated using cold plasma theory for comparison. At low altitudes, where the plasma is dense, the high cold plasma conductivity essentially shorts out the parallel electric field, and it stays nearly zero. The parallel electric field becomes significant between and 3 R E where the cold plasma disappears. In this region the electron distribution must be significantly modified so that the warm plasma can carry the necessary current required by the curl of the magnetic field. Figures 3d and 4d show the dissipation, given by Re( j z E* z ) (see equation (1)). Figures 3d and 4d show a region of negative dissipation; thus the particles are locally giving energy to the wave at that point. However, the total dissipation integrated over the whole region is still positive, indicating damping of the wave. 3.. Temperature Dependence [] Figure 5 shows the temperature dependence of the terms in the energy conservation equation for the. Hz and.35 Hz cases, for a Pedersen conductivity of 1.9 mho, and for a perpendicular wavelength of 1 km. As in Figure, the solid curve gives the fraction of reflected energy, the dotted curve is the Joule dissipation in the ionosphere and the dashed curve is the dissipation due to the wave-particle interaction. It can be seen in both cases that there is very little wave-particle dissipation at low temperatures, consistent with cold plasma theory. At. Hz this dissipation increases with temperature until 1 kev then decreases. The peak occurs at a temperature of.3 kev for the.35 Hz case. This suggests that the peak temperature corresponds to an electron transit time effect, since the thermal velocity of the electrons at the peak temperature roughly scales with the wave frequency, with a/w R E in both cases, where a is the electron thermal speed. At higher temperatures the ionospheric Joule heating also decreases, indicating that the wave is mostly reflected before reaching the ionosphere. The amount of reflection in both the cold and hot ranges is more for the. Hz case than for the.35 Hz case, since this latter case corresponds to a mode of the ionospheric Alfvén resonator, as defined by the enhanced Joule dissipation [Lysak, 1991]. [1] Figure 6 shows the profiles of the parallel electric fields for temperatures of 1, 1, 1, and 1 ev. As in Figures 3b and 4b, the solid and dotted curves in Figure 6 are the real and imaginary part of the parallel electric field, respectively, while the dashed and dash-dotted curves are the real and imaginary part of the parallel electric field calculated from the cold plasma model. It can be seen that the 1 ev case closely follows the cold plasma results, as would be expected. As the temperature increases, the parallel electric field in the 3 R E region increases and becomes dominant at large temperatures Conductivity Dependence [] Figure 7 shows the dependence of the energy balance on the Pedersen conductivity. In these plots, there is a clear distinction between the frequency away from the Alfvén resonator mode (. Hz) and the frequency at the resonator mode (.35 Hz). In both cases the ionospheric dissipation peaks at an intermediate conductivity; however, the peak is at larger conductivity (1 mho) in the.35 Hz case than in the. Hz case. At higher conductivities the. Hz case absorbs most of the energy in the wave-particle resonance Figure 5. Plots of the relative magnitudes of the terms in the energy conservation equation (1) as a function of temperature for a frequency of (a). Hz and (b).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 of 1.9 mho and a perpendicular wavelength of 1 km are assumed. A hot plasma density of.5 cm 3 and an ion-to-electron temperature ratio of 6 are used. while in the.35 Hz case the wave is mostly reflected at high frequencies. The distinction between the two cases decreases at the lowest conductivities, where the wave is mostly absorbed in both cases. These differences arise from the sharpening of the resonance structure at higher conductivities, as can be seen in Figure Perpendicular Wavelength Dependence [3] Finally, Figure 8 shows the dependence of the energy balance on perpendicular wavelength. Both frequencies give similar behavior, with a maximum absorption by the waveparticle resonance at wavelengths of <1 km. At larger wavelengths the wave-particle resonance becomes less important and the Joule dissipation in the ionosphere becomes larger. In contrast, at short wavelengths, the absorption of the wave begins to decrease and the reflection increases. In this regime, the wave is mostly reflected from

8 SMP 9-8 LYSAK AND SONG: NONLOCAL ALFVÉN WAVE KINETIC THEORY Figure 6. The parallel electric field for four cases with plasma sheet temperatures of (a) 1 ev, (b) 1 ev, (c) 1 ev, and (d) 1 kev. The solid curve gives the real part and the dotted curve the imaginary part of E k. The dashed and dash-dotted curves give the real and imaginary parts of E k, respectively, from cold plasma theory. The wave frequency is.35 Hz, the perpendicular wavelength is 1 km, and the ionospheric conductivity is 1.9 mho for all cases. the wave-particle resonance region rather than the ionosphere, as is evidenced by the lack of ionospheric Joule dissipation. This is consistent with the general behavior of the Alfvén waves in a dissipative medium, in which the wave becomes reflected from a resistive layer at short wavelengths [Lysak and Dum, 1983]. An analysis of the reflection of Alfvén waves from an auroral acceleration region gives similar results [Vogt and Haerendel, 1998]. These results all point to the difficulty of transmitting Alfvén wave energy through the auroral acceleration region at short perpendicular wavelengths. [4] It may be noted that the results of Figure 8 are shown only down to 1 km wavelength. At shorter wavelengths, the iterative procedure used to solve the system of equations fails to converge. This is due to the increased importance of the parallel electric field term in Faraday s law, equation (1), and the increase of the magnetic field driver 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, the trends as the wavelength becomes shorter are clear, with the wave reflection increasing and the ionospheric dissipationgoing to zero. 4. Discussion [5] The results shown above represent a first calculation of self-consistent kinetic effects on Alfvén waves on dipolar auroral field lines. The model is based on a scenario in which an Alfvén wave of fixed frequency and perpendicular wavelength is launched from the outer magnetosphere toward the auroral zone. Recent observations from Polar show that significant Poynting flux is present, particularly at the plasma sheet boundary layer, to provide such a source for these Alfvén waves [e.g., Wygant et al., ]. These calculations give the fate of the wave energy as it passes into the auroral zone. The fraction of the energy reflected, dissipated by Joule heating in the ionosphere, and absorbed by the wave-particle interaction are shown to be a function of the frequency and perpendicular wavelength of the wave, the ionospheric Pedersen conductivity, and the temperature of the plasma sheet electrons.

9 LYSAK AND SONG: NONLOCAL ALFVÉN WAVE KINETIC THEORY SMP 9-9 [6] One consistent result of these parameter studies is that there is often a transition from a state in which kinetic effects are not important, through a region of maximum absorption, and finally to a state in which the wave is reflected from the interaction region before reaching the ionosphere. This transition takes place as the temperature is increased, as the perpendicular wavelength is decreased, and as the wave frequency is decreased. This behavior is consistent with the simplified model of Alfvén wave interactions with a resistive layer presented by Lysak and Dum [1983]. Figure shows that Alfvén waves are mostly transmitted through the resistive layer at long perpendicular wavelength, go through a transition region where the absorption maximizes, and then are mostly reflected at short wavelengths. Vogt and Haerendel [1998] showed a similar behavior based on a model in which the auroral acceleration region was modeled as a Knight [1973] current-voltage relationship. Pilipenko et al. [] have recently emphasized the role of this reflection in trapping waves in the resonator region. Although they distinguish between waves reflected by the Alfvén speed gradient and those reflected by the parallel electric field, these two phenomena cannot really be separated and must be considered together. In any case, from a number of different theoretical approaches, there is a consistent theme that Alfvén waves with short perpendicular wavelengths are reflected by the auroral acceleration region. [7] This result has important consequences for the evolution of auroral arcs. Auroral arcs can have thicknesses of <1 km [e.g., Maggs and Davis, 1968; Borovsky and Suszcynsky, 1993; Trondsen and Cogger, 1; Knudsen et al., 1]. These results would suggest that such narrowscale auroras, if they are related to field-aligned currents and Alfvén waves on the same scale, must be produced within or below the auroral acceleration region and not Figure 7. Plots of the relative magnitudes of the terms in the energy conservation equation (1) as a function of ionospheric conductivity for a frequency of (a). Hz and (b).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). A perpendicular wavelength of 1 km is assumed. Other plasma parameters are as in Figure. Figure 8. Plots of the relative magnitudes of the terms in the energy conservation equation (1) as a function of perpendicular wavelength for a frequency of (a). Hz and (b).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 waveparticle interaction (WPI). An ionospheric conductivity of 1.9 mho is assumed. Other plasma parameters are as in Figure.

10 SMP 9-1 LYSAK AND SONG: NONLOCAL ALFVÉN WAVE KINETIC THEORY imposed from a small-scale generator in the outer magnetosphere. It should be noted that Knudsen et al. [1] showed a bimodal distribution of auroral arc widths, with a peak in the subkilometer range as well as a peak near 1 km. It is tempting to speculate that the larger-width peak is associated with the direct absorption of Alfvén waves from the outer magnetosphere while the narrower peak is associated with other processes at low altitude. Processes such as the ionospheric feedback instability [e.g., Atkinson, 197; Sato, 1978; Lysak, 1991; Lysak and Song, ; Pokhotelov et al., ] could produce such smallscale structures, as could nonlinear interactions between Alfvén waves trapped in the ionospheric resonator [Song and Lysak, 1]. [8] Although results have been shown for a wide variety of plasma parameters, there are many other cases that could be considered. For example, the ionospheric scale height for the density plays an important role in determining the frequencies of the ionospheric Alfvén resonator. Runs (not shown) that have a scale height of 3 km rather than 6 km show similar features aside from the fact that the resonant frequencies are higher (scaling as V AI /h, as noted by Lysak [1991]). Runs at various frequencies as in Figure but with a 3 km scale height show a maximum in the wave-particle dissipation at.3 Hz, with a peak absorption of.87. A major difference with smaller scale height is that the parallel electric field is at lower altitudes and is stronger in magnitude. In Figure 3b the maximum magnitude of the parallel electric field is.46 mv m 1 at a radial distance of.65 R E, while for the.3 Hz run with 3 km scale height the maximum parallel electric field is 1.3 mv m 1 at 1.87 R E. Nevertheless, the total dissipation in wave-particle interactions is similar in these two runs. [9] Another parameter is the ion-to-electron temperature ratio, which is taken to be 6 in the runs above. Smaller ratios of this parameter affect results primarily at small scales, where the ion gyroradius effect is most important. As noted in Figure 1, an ion temperature of 3 kev has only a minimal effect on the Alfvén speed at a perpendicular wavelength of 1 km, but at smaller wavelengths it can have a greater effect. As noted by Lysak and Lotko [1996], a higher T i /T e ratio leads to less damping in a uniform plasma. In this nonuniform situation, runs like those in Figure 8 but with T i = T e show more reflection and less dissipation at wavelengths <7 km than the runs shown with T i =6T e. This is consistent with the uniform plasma results since the stronger dissipation regions generally lead to more reflection as was found. [3] The calculations presented here describe the linear damping of Alfvén waves in the auroral region; however, they do not explicitly address the fate of the dissipated energy. The energy lost in the wave-particle interaction will go into the heating of electrons in the auroral zone, which may either precipitate into the ionosphere or go up the field line into the magnetosphere. Preliminary calculations solving the Vlasov equation for these electrons indicate that only 1% of the energy goes into precipitating electrons. However, these calculation only consider the hot plasma sheet electrons and, as test particle studies have shown, cold ionospheric electrons can be readily accelerated along the field lines by Alfvén waves to produce aurora. Thus the present work is only the first step in producing a fully selfconsistent theory of the interactions of electrons and Alfvén waves in the auroral region. Appendix A: Details of Conductivity Kernel [31] As we noted previously, the perturbation in the distribution function can be written as df ðþ¼ z df ðz Þe iwt ð z;z Z z Þ dz de z z z ð Þe iwt ð z ;z Þ : ða1þ We must multiply this distribution by the parallel velocity and integrate over velocity space in order to get the conductivity. We first note that the unperturbed distribution should be written as a function of the constants of motion m and W. Assuming a bi-maxwellian distribution with the possibility of parallel and perpendicular temperatures, such a distribution is written as f ðm; W Þ ¼ n m 3= 1 exp W mb 1 T? p T? T 1= T k T? T k k : ðaþ In the usual cylindrical coordinates we have d 3 v = pv? dv? dv. To transfer to the constants of motion, we must calculate the Jacobian 3 3 =@v k =@v? mv k mv? J ¼ det4 5¼ det4 5¼ m v k v? B? mv? =B ða3þ Then we have d 3 v ¼ pv? dwdm ¼ pb J m dwdm: v k ða4þ Thus the current density is given by j z ¼ pqb Z dwdm½df þ ðþ df v ðþ v Š; ða5þ m p where v ¼ ð ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi =mþðw mbþ. Now let us consider the three particle populations and for the moment neglect the boundary value term (i.e., the first term in (A1)). The ionospheric particles give a contribution to df +, given by df Iþ ¼ Z z I dz E z ðz Þe iwt ð z ;z Þ : ða6þ It should be noted that ionospheric particles all have W > mb I, where B I is the magnetic field strength at the ionosphere. [3] Next we consider downgoing magnetospheric electrons. These particles reach an altitude where the magnetic field strength is B only if W > mb. Their contribution, assuming that z = L is the top boundary, is given by df M ¼ M Z z L dz E z ðz Þe iwtðz;z Þ ¼ M Z L z dz E z ðz iwtðz;z Þe Þ : ða7þ

11 LYSAK AND SONG: NONLOCAL ALFVÉN WAVE KINETIC THEORY SMP 9-11 Note the change in sign due to the switching of the limits of integration. [33] The third population is upgoing particles that have mirrored below the location z. For these particles, the integration must go from z = L down to the mirror point z m, which is the point z m where the magnetic field is given by B m = W/m, and then back up to the location z. Then again, switching the limits of integration for the downgoing part of the integration, we can write 3 df Mþ M Z L 4 dz E z ðz Þe iwtzm;z ð z m Z z ð Þþt ð z ;zþ Þ z m dz E z z ð Þe iwt ð z ;z Þ ða8þ There is no contribution to the integral from these particles unless the field point z is also above the mirror point. Thus if we denote B = B(z ) and B = B(z), B m must be larger than both B and B for a contribution, or W > mb max, where B max is the larger of B and B. The second of these integrals is only defined when z < z or B > B, sob max = B for this integral. In addition, these particles must not be in the loss cone; therefore we must have W < mb I. [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. This procedure leads to j z ðþ¼ z pq B m Z L dz E z ðz Þ Z 1 I eiwt ð z ; z Þ þ ðz zþ W Z=B max W=B I ZW=B W=B M M dm ( dw ðz z Þ Z W=B tzm;z eiw ð Þþtðzm;zÞ eiwt ð z ;zþ M ZW=B I ðz;zþ eiwt ð Þ þ ðz z Þ ) ; ða9þ where (x) is the step function that is 1 for positive argument and for negative argument. This function takes account of the limits of integration in equations (A6) (A8). Comparing equation (A9) with the general form of the nonlocal conductivity relation (11), we can write 8 Z sðz; z Þ ¼ pq B 1 < ZW=B I m dw ðz z Þ I : eiwt ð z ;zþ þ ðz zþ W=B Z max W=B I þ ðz z Þ Z M dm ZW=B W=B I M ðz;zþ eiwt ðtzm;z eiw ð dm eiwt ð z ;zþ 9 >= >; : ða1þ 5: [35] We must next take into account the boundary term in equation (A1). Since we assume no perturbation in the ionospheric upgoing distribution, we need only to consider the downgoing magnetospheric particles. This can be written in a manner analogous to equation (A9) with z = L. Note that the first and fourth terms in equation (A9) refer to the effect of fields below the observation point z. Thus only the second and third terms will contribute. Writing the current due to the boundary term in a manner similar to equation (A5), we have j L ðþ¼ z pqb m Z W=B Z 1 6 dw 4 Z W=B W=B I iwt z;l dmdf ðlþe dmdf ðlþe iwtzm;l ð ð 3 ð Þ 7 Þþt ð zm;z ÞÞ 5: ða11þ These two terms correspond to the third and second terms in equation (A9), respectively. As in Lysak and Song [3], we determine the distribution function at L from the local dispersion relation, including the finite gyroradius correction (which is small for electrons) df ðlþ ¼ iq w J k? v? vk V M v k V M E zðlþ; ða1þ where = qb /m is the gyrofrequency, v? =(mb /m) 1/ is the perpendicular velocity, v k = [(/m)(w mb )] 1/ is the parallel velocity, B is the magnetic field at L, and V M is the parallel phase velocity found by solving the local dispersion relation. It can be seen that from equations (A11) and (A1) we can write the contribution of the boundary term to the current as j L ðþ¼ z s L ðþe z z ðlþ: ða13þ [36] As a final point, it should be noted that the travel time integral can be written in the form tðz 1 ; z Þ ¼ rffiffiffiffiffiffiffi m W Z z z 1 dz p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ða14þ 1 mbðz Þ=W When m =, this integral is simple; when it is not, the integral can be converted to an integral over magnetic field since B(z) =B E /(1 + z/r E ) 3. Then (A14) can be written tðz 1 ; z Þ ¼ rffiffiffiffiffiffiffi m RE B 1=3 W E 3 Z B1 B 1 B 4=3 db p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ða15þ 1 mb =W If we now make a change in variable to x = mb /W, this integral can be written tðz 1 ; z Þ ¼ rffiffiffiffiffiffiffi m W RE 3 mb E W 1=3 Z x1 x x 4=3 dx p ffiffiffiffiffiffiffiffiffiffi : ða16þ 1 x

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(bob@aurora.space.umn. edu; yan@aurora.space.umn.edu)