Whistler interactions with density gradients in the magnetosphere

JOURNAL OF GEOPHYSICAL RESEARCH: SPACE PHYSICS, VOL. 118, 167 17, doi:1.19/1ja1838, 13 Whistler interactions with density gradients in the magnetosphere J. R. Woodroffe 1 and A. V. Streltsov 1 Received 18 September 1; revised 18 October 1; accepted 1 November 1; published 1 January 13. [1] Very-low frequency whistler-mode waves are frequently observed in the equatorial magnetosphere. Large amplitude whistlers, capable of rapidly accelerating electrons to large energies, are typically observed to propagate at highly oblique angles relative to the bacground magnetic field. In this paper, we discuss a potential mechanism for the conversion of moderately oblique whistlers to highly oblique whistlers on a transverse gradient of the bacground plasma. We offer a theoretical explanation for this process and illustrate its operation using numerical simulations. This effect has been previously observed in laboratory experiments and it also provides a straightforward explanation for observations of highly oblique whistlers in magnetospheric plasmas. Citation: Woodroffe, J. R., and A. V. Streltsov (13), Whistler interactions with density gradients in the magnetosphere, J. Geophys. Res. Space Physics, 118, 167 17, doi:1.19/1ja1838. 1. Introduction [] Naturally occurring density variations of a few to tens of percent from the bacground are a ubiquitous feature of magnetospheric plasmas. Observations indicate the presence of such variations across a variety of spatial scales, from a few meters to hundreds of ilometers [Sonwalar, 6]. VLF whistler mode waves (whistlers), which have wavelengths in the middle of this range (l < 1 m at r = 4.9R E for f = 1 Hz and n = 1 cm 3 ), are particularly affected by the presence of these density variations. At subwavelength scales, these variations are responsible for localized enhancements of wave magnetic and/or electric fields [Reiniusson et al., 6] and mode conversion to electrostatic lower hybrid waves [Bamber et al., 1995]. At scales much larger than the wavelength, these variations can be responsible for guiding or confining whistlers [Helliwell, 1965; Inan and Bell, 1977; Streltsov et al., 7]. At scales comparable to the wavelength, however, it is not immediately clear what behavior should be expected. The physical optics approximation is not valid and plane wave-lie behavior is not guaranteed. Consequently, we must use numerical methods to study whistler dynamics in these intermediate-scale gradients. [3] Recently, large-amplitude whistlers have been observed in the Earth s radiation belts [Cattell et al., 8; Breneman et al., 11]. Unlie the more frequently observed whistlermode chorus, which propagate primarily parallel to the bacground magnetic field (although there are exceptions, see, e.g., Goldstein and Tsurutani [1974]; Li et al. [11]), these waves were found to propagate at highly oblique angles. This observation led Cattell et al. [8] to postulate that 1 Department of Physical Sciences, Embry-Riddle Aeronautical University, Daytona Beach, Florida, USA. Corresponding author: J. R. Woodroffe, Department of Physical Sciences, Embry-Riddle Aeronautical University, Daytona Beach, FL 318, USA. (woodrofj@erau.edu) 1. American Geophysical Union. All Rights Reserved. 169-938/13/1JA1838 some sort of propagation effect associated with inhomogeneous plasmas might be occurring. [4] Density inhomogeneities are also present in laboratory plasmas. Stenzel [1976] observed that a whistler that enters a density gradient while traveling nearly parallel to the bacground magnetic field would end up highly oblique after passing through an intermediate transition region where interesting mode coupling effects are observed.specifically, two different waves in the transition region were observed, with one having θ 15 (where θ is the angle between the wave vector and the bacground magnetic field) and the other having θ 6 the nearly-parallel incident wave and a highly oblique outgoing wave. It was further noted that Wave reflection on density gradients may play a role in the transition region, but cannot explain the stable far-zone pattern which appears to be an eigenmode of the nonuniform plasma. [5] The purpose of this paper is to provide an explanation of this inhomogeneity-associated wave effect using the theoretical framewor of electron magnetohydrodynamics (EMHD), which we will discuss in the next section.. Theoretical Bacground [6] When the wave angular frequency, o, satisfies the condition o LH < o < min(ω ce,o e ) (where o LH is the lower hybrid frequency, o is the wave frequency, Ω ce is the electron gyrofrequency, and o e is the electron plasma frequency), whistlers can be described with the quasi-longitudinal approximation of EMHD. The equations of EMHD include Faraday s Law, the equation of motion for a cold electron fluid, and a generalized Ohm s law (derived from Ampere s Law with displacement current omitted) including the effects of electron inertia (see, e.g., Streltsov et al. [1] for further details). @B ¼ re; (1) @t @ @t þ u r u ¼ e ðe þ u BÞ; () m e l e rreþe ¼ ub m e u ru (3) e 167

n/n 1.8.6.4. Figure 1. (Adapted from Streltsov et al. [6]). Electron density as a function of whistler perpendicular wave number for o =.13Ω ce. The horizontal dashed line indicates n/n 1 =.78, which corresponds to θ =6 for the 1 root (θ = tan 1 ( / ) defines the wave normal angle); the corresponding wave numbers for this mode, 1 and, are indicated on the x-axis. The perpendicular wave number at which 1 = is labeled as, * and the maximum perpendicular wave number is labeled R. where u is the electron fluid velocity, e is the fundamental charge, m e is the electron mass, and l e = c/o e is the electron inertial length. [7] If these equations are linearized and solved in a uniform plasma with a bacground magnetic field, the EMHD whistler dispersion relation is obtained, Ω ce 1 ¼ ; (4) q where ¼ l e ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ o is the magnitude of the wave vector, is the component of the wave vector perpendicular to the bacground magnetic field, and is the component of the wave vector parallel to the bacground magnetic field. [8] It is traditional to classify whistlers by frequency, with whistlers having o < Ω ce / being labeled low-frequency (LF) while whistlers having o > Ω ce / are labeled highfrequency (HF). This classification has physical importance, because whistler phenomenology is different in the LF and HF cases. [9] The dispersion relation, equation (4), can be solved for the perpendicular wave number,, giving us s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ω rffiffiffiffiffiffiffiffiffiffiffiffiffi ce ¼ 1 1 n 1; (5) o n 1 where n 1 ¼ m e = ðm e ÞðΩ ce =oþ is a cutoff density (if n > n 1, becomes complex). [1] Streltsov et al. [6] introduced a density-based formalism for understanding the trapping of whistlers by ducts. Requiring that solutions of equation (5) be real reveals a pair of critical densities: n ¼ m e Ω ce m e o 1 n 1 ¼ m e Ω ce (6) m e o [11] The existence of different solutions to equation (5) depends on the magnitude of the density relative to these two critical values. Interestingly, it can also be shown that perpendicular whistler mode group velocity is exactly zero when n = n or n = n 1. (The latter case n = n 1 corresponds to the so-called Gendrin mode [Gendrin, 1961], where energy is propagated exactly parallel to the bacground magnetic field despite having a nonzero perpendicular wave number.) [1] Equation (5) actually defines a pair of perpendicular wave numbers, 1 and, where 1. For HF whistlers, 1 is imaginary and is real only if n < n (since n < n 1, the cutoff at n = n 1 is not important in this case). For LF whistlers, 1 is real if n < n < n 1 and is real so long as n < n 1. Thus, there is at most one nonevanescent whistler ( = ) in the HF case, but there may be two nonevanescent whistlers ( = 1 and/or = )in the LF case. [13] Solving the dispersion relation, equation (4), for the density (contained in the inertial length) and plotting the result for fixed o and allows for a simple graphical interpretation of the existence and multiplicity of whistler wave modes, an example of which can be seen in Figure 1. [14] Regions of either enhanced or depleted density, which can confine and guide whistlers, are nown as ducts. Density enhancements that trap whistlers are referred to as high-density ducts (HDDs) and density depletions that trap whistlers are referred to as low-density ducts (LDDs). Both HDDs and LDDs are able to trap LF whistlers, with the 1 mode best confined by HDDs and the mode best confined by LDDs. On the other hand, only LDDs are able to confine HF whistlers because the 1 whistler is evanescent in this frequency range. In general, HDDs are able to trap LF whistlers if the densities inside and outside the duct, n in and n out, satisfy the inequality n out < n < n in < n 1 ; similarly, LDDs are able to trap LF whistlers if n in < n 1 < n out and HF whistlers if n in < n < n out. [15] The effectiveness of HDDs for the 1 and LDDs for the whistlers is due not only to the satisfiability of the trapping criteria, but also to a difference in the propagation behavior. The 1 mode propagates preferentially toward regions of higher density while the mode propagates toward regions of lower density. This difference has important consequences for the trapping and confinement of whistler-mode waves. 3. Simulation [16] We model a small region of the equatorial magnetosphere, 1 m across the magnetic field and 4 m along it. The curvature of the magnetic field can be neglected at this scale, allowing us to wor in the simpler Cartesian (x,z) geometry. The simulation region is discretized using 1 points across the field (^x -direction) and 8 along the field ( ^z -direction), giving us spatial resolutions of Δ =.1 m and Δ =.5 m. The bacground magnetic field and electron density values are chosen to be appropriate for the equatorial plasmapause region near L = 4.9, with the magnetic field being set to 7 nt and the electron density set to 15 cm 3. 168

Electric Field (mv/m) 1 8 E X - -.5..5 e b 6 a 4 c Distance Across B (m) 1 8 6 4 1 E Y d f 8 E Z 6 4 5 1 15 5 3 35 4 Distance Along B (m) 5 1 15 n (cm -3 ) Figure. Electric fields for a whistler with θ =6 and o =.13Ω ce in an asymmetrical duct with wall thicness Δ = 1 m. The top, middle, and bottom panels show the electric field components and the cross-sectional density profile. Dar horizontal lines in each panel indicate the expected boundaries of the density duct. In the top panel, white lines correspond to the various spectra shown in Figure 3, with horizontal lines corresponding to parallel spectra and vertical lines corresponding to perpendicular spectra. In the bottom panel, the directions of the wave vector and the associated wave angles are indicated at locations before the gradient interaction (left) and afterward (right). [17] The uniform bacground density is modified using an asymmetrical duct profile on one side of the domain, the density decreases to 5 cm 3 ; on the opposite side, the density increases to 5 cm 3. The transition between density regions is modeled using hyperbolic tangent functions, so the thicness of these duct walls, Δ, is the scale length of the hyperbolic tangent. The width of the uniform duct region is set to 6 m and we initially tae the wall thicness to be Δ = 1 m. A 1 Hz whistler is introduced to the system using an antenna, which localizes a uniform plasma solution to a portion of the domain boundary (details of this solution are given in Appendix A of Streltsov et al. [6]). This antenna produces a pair of waves with identical but opposite. This driver, combined with the asymmetrical duct profile, allows us to simultaneously probe whistler behavior at both positive (LDD-lie) and negative (HDD-lie) density gradients. [18] The spatial derivatives in the EMHD equations (1) (3) are approximated using sixth-order finite differences, and the evolution equations (1) () are advanced in time using a fourth-order predictor-corrector method [Burden and Faires, 1993]. The electric field is obtained from the Ohm s law (3) at each step using the method of successive overrelaxation. 4. Results and Discussion [19] The conditions for the confinement of whistlers to ducts have been well-established by previous studies [Streltsov et al., 6, 7]. However, these studies focused on identifying the loss-free eigenmodes of the duct. One particular condition for matching the theoretical formalism to simulations was the use of very sharply defined duct boundaries (Δ l ). This is not the case for the ducts being considered in this paper, as we are considering ducts whose widths are greater than l with wavelengthscale gradients (Δ l ). [] In Figure, we show the three components of the electric field, both before and after interacting with the gradient. From these components it can be seen that there is a clear difference between how the wave interacts with the negative density gradient (top edge of duct) and the positive density gradient (bottom edge of duct). Whereas the wave incident on the upper boundary is reflected (i.e., changes 169

Spectral Power Spectral Power.5.5.5.5.5.5 1 direction but not magnitude), the wave incident on the lower boundary undergoes some sort of transformation at the boundary, with increasing and the wave becoming increasingly oblique. The wave vector directions and wave angles for these two states are indicated in Figure (bottom). A change in is accompanied by a change in the magnitude of the electric field components, with the wave becoming increasingly electrostatic for large ; the relative amplitudes of the field components in Figure are consistent with the whistler amplitude ratios given in Appendix A of Streltsov et al. [6] for the two different values of. [1] Spectral analysis of the wave fields in the duct reveals that both o (not shown) and (Figures 3a 3c) are the same for both the primary wave and secondary waves, with f = 1 Hz and l = 9.14 m throughout. However, as shown in Figures 3d 3f, the perpendicular wave numbers Perpendicular Wavenumber (m -1 ) Parallel Wavenumber (m -1 ) Figure 3. Perpendicular power spectrum of E x for θ =6. Vertical dashed lines indicate values expected from the dispersion relation. (a) Parallel power spectrum for x = 5 m, z m; (b) parallel power spectrum for x = 7 m, z 4 m; (c) parallel power spectrum for x = 3 m, z 4 m; (d) perpendicular power spectrum for z = 75 m, m x 1 m; (e) perpendicular power spectrum for z = 5 m,5 m x 1 m; and (f) perpendicular power spectrum for z = 5 m, m x 5 m. The constancy of is demonstrated by Figures 3a 3c and the switching between modes is clearly demonstrated by Figures 3d 3f. Where the peas of the power spectra occur is entirely consistent with values predicted by the dispersion relation. a b c d e f for these waves are considerably different but still consistent with the whistler mode dispersion relation, equation (4). This result indicates that a process associated with the positive density gradient is causing waves to transition from a moderately oblique mode to a highly oblique mode. Since mode coupling is generally used to describe the transition between two entirely different wave modes, we refer to this whistler-whistler transition as mode switching. [] Additional simulations, the perpendicular spectra of which are shown in Figures 4, 5, reveal that mode switching is a robust occurrence for LF whistlers in the presence of positive density gradients at wavelength scales. The particular cause of the switching is not obvious for the results shown in Figure, but in a duct with thicer walls it becomes clear that it is due to phase velocity gradient in the finite-thicness duct wall. Specifically, the gradient traps a portion of incident wave and this trapped wave propagates inside the gradient, with the component on the inner edge of the gradient moving faster than the component on the outer edge due to the density dependence of the phase velocity (v / l e / n 1/ ). The wave becomes increasingly inclined toward the density gradient, thereby increasing while maintaining constant due to the uniformity of the density in the parallel direction. Although the wave might initially be aligned relatively parallel to the bacground magnetic field, the gradient-trapped phase fronts are sheared until the perpendicular wave number becomes sufficiently large that it matches the highly oblique whistler solution in the uniform duct region and the wave leas bac out of the density gradient. It should be noted that the thicness of the wall (inverse steepness of the gradient) does not affect the resulting mode, only the speed at which the switching occurs namely, decreasing the wall thicness increases the rate of switching because of its larger phase velocity gradient. [3] This mode switching process provides a direct explanation for the previously discussed observations of Stenzel [1976]. The two waves observed in the transition region, with θ 15 and θ 6, are consistent with the 1 and in equation (5) for a whistler with o =.85Ω ce (the parameters from their study). [4] Additionally, this switching mechanism may have implications for the magnetospheric distribution of whistler-mode chorus. Haque et al. [11] noted that lower-band chorus is typically observed to be highly field-aligned in the equatorial source region, but the distribution of source chorus reported by Santolí et al. [9] is concentrated near the resonance cone ( ¼ R in Figure 1). A possible explanation for this observation is that the whistlers were nearly field-aligned, as expected, but propagated through a positive density gradient that transitioned the whistler from the fieldaligned 1 to the oblique mode. This mechanism could also be responsible for the obliqueness of the large-amplitude, o =.Ω ce whistlers observed by Cattell et al. [8]; if these whistlers were primarily field-aligned when generated, mode switching would result in a concentration of power near the resonance cone. [5] The rate of switching is strongly dependent upon the rate of energy transport from the duct into the gradient. Consequently, the effects of switching should be minimal when the speed of perpendicular energy transport (i.e., the 17

1 1 1 Spectral Power.75.5.5.5.5.75 Perpendicular Wavenumber (m -1 ) Figure 4. Spectral power (normalized to pea) as a function of inverse perpendicular wavelength for three different wave normal angles. The spectral peas are paired, corresponding to the solutions from equation (5) for an o =.13Ω ce whistler in a 15 cm 3 plasma. The values expected from the dispersion relation are indicated by vertical lines. Note that unlie the spectra shown in Figure 3, these correspond to a cross-section across the entire simulation domain. Switching occurs most slowly for the θ =7 case (note the presence of power at spatial frequencies between the two peas that is absent at smaller θ) because of its relatively low perpendicular group velocity. perpendicular group velocity) is close to zero. As discussed previously, the perpendicular group velocity is a minimum when n n or n n 1, equivalent to θ or θ θ G, where θ G =cos 1 (o/ω ce ) is the wave angle for the parallelpropagating Gendrin mode. Consequently, the wave fields associated with nonducted, guided whistlers are least affected by this mode switching. Additional simulations (not shown) indicate these waves may propagate a considerable distance without undergoing a considerable degree of distortion from the whistler-gradient interaction. This is demonstrated in Figure 6, which shows the E x wave fields of a highly parallel whistler. The parameters of this run differ slightly from those of the previous run, as we have increased the width of the duct from 6 m to 8 m and have used a symmetrical LDD rather than an asymmetrical duct; the purpose of maing these changes was to increase the effective perpendicular wavelength while emphasizing the gradient coupling effects. [6] Modes with small v g, are not the only case where mode switching is minimally effective, however. It has also been convincingly demonstrated that LDDs can support loss-free eigenmodes Streltsov et al. [6], which show no signs of mode switching. However, the eigenmodes of a given duct are very sensitively dependent on for a given value of o and are not generally representative of ducted whistler propagation. [7] It should be noted that HF whistlers would not (and, according to other simulation results not shown, do not) exhibit this behavior. The reason for this is the difference in their dispersion properties, since the 1 mode is imaginary and HF whistler wave normals divert toward regions 1 Figure 5. Spectral power of a θ =6 whistler with o =.13Ω ce for ducts with different wall thicnesses. Each pea corresponds to an individual spectrum taen across one half of the domain at z = 4 m, and each of these spectra has been normalized to its pea. Values expected from the dispersion relation are indicated by vertical dotted lines. From these results, we can infer that duct wall thicness does not affect which modes are being switched between. 171

1 WOODROFFE AND STRELTSOV: WHISTLER-GRADIENT INTERACTIONS Distance Along B (m) 8 6 4 Distance Across B (m) Figure 6. The electric field of a highly parallel propagating whistler in an LDD. The top panel shows the E x electric field and the density cross-section; the inner and outer edges of the density gradients are indicated by solid horizontal lines. The bottom panel shows the axial electric field profile taen along the dashed white line in the top panel. Only minor distortion of the wave fronts are observed, and these are only at the extreme edges near the positive gradient. of lower density. Thus, although high-frequency whistlers are very effectively confined by an LDD, they do not undergo any type of switching effect at negative density gradients. 5. Conclusions [8] We present the following conditions for the occurrence of whistler mode switching: LF, nearly-parallel to moderately oblique (θ < θ G ) waves, propagating in a plasma featuring a positive density gradient (r n > ), which either satisfies the LDD trapping criteria or is broad enough to allow for sufficient phase front shearing; HF and highly oblique (θ > θ G ) whistlers do not have the correct dispersion properties to permit trapping of wave power in the gradient necessary to produce mode switching. [9] The identification of the whistler mode switching mechanism provides a new tool for the interpretation of whistler phenomena in both laboratory and space plasmas. We have argued that this effect explains previous laboratory observations and may also explain magnetospheric observations of highly oblique whistlers. Future research will consider how this mechanism contributes to the energization of electrons by large-amplitude whistlers and the self-consistent generation of additional whistler phenomena including banded chorus and hiss. [3] Acnowledgments. The research was supported by NSF award AGS-1141676 and DARPA contract HR11-9-C-99. References Bamber, J. F., Maggs, J. E., and W. Geelman (1995), Whistler wave interaction with a density striation: A laboratory investigation of an auroral process, J. Geophys. Res., 1(A1), 3795 381. Burden, R. L., and J. D. Faires (1993), Numerical Analysis, Fifth Edition, Boston: PWS-Kent Publishing Company. Breneman, A., C. Cattell, J. Wygant, K. Kersten, L. B. Wilson III, S. Schreiner, P. J. Kellogg, and K. Goetz (11), Large-amplitude transmitter-associated and lightning-associated whistler waves in the Earth s inner plasmasphere at L <, J. Geophys. Res., 116, A631, doi:1.19/ 1JA1688. Cattell, C., et al. (8), Discovery of very large amplitude whistlermode waves in Earth s radiation belts, Geophys. Res. Lett., 35, L115, doi:1.19/7gl39. Gendrin, R. (1961), Le guidage des whistlers par le champ magnetique, Planet. Space Sci., 5(4), 74. Goldstein, B. E., and B. T. Tsurutani (1974), Wave normal directions of chorus near the equatorial source region, J. Geophys. Res., 89(A5), doi:1.19/ja89ia5p789. Haque, N., U. S. Inan, T. F. Bell, J. S. Picett, J. G. Trotignon, and G. Facsó (11), Cluster observations of whistler mode ducts and banded chorus, Geophys. Res. Lett., 38, L1817, doi:1.19/11gl4911. Helliwell, R. A. (1965), Whistlers and Related Ionospheric Phenomena, Stanford: Stanford University Press. Inan, U. S., and T. F. Bell (1977), The Plasmapause as a VLF Wave Guide, J. Geophys. Res., 8(19), 819 87. Li, W., R. M. Thorne, J. Bortni, Y. Y. Shrpits, Y. Nishimura, V. Angelopoulos, C. Chaston, O. Le Contel, and J. W. Bonnell (11), Typical properties of rising and falling tone chorus waves, Geophys. Res. Lett., 38, L1413, doi:1.19/11gl4795. Reiniusson, A., G. Stenberg, P. Norqvist, A. I. Erisson, and K. Rönnmar (6), Enhancement of electric and magnetic wave fields at density gradients, Ann. Geophys., 4, 367 379. Santolí, O., D. A. Gurnett, J. S. Picett, J. Chum, and N. Cornilleau- Wehrlin (9), Oblique propagation of whistler mode waves in the chorus source region, J. Geophys. Res., 114, AF3, doi:1.19/ 9JA14586. Sonwalar, V. S. (6), The Influence of Plasma Density Irregularities on Whistler-Mode Wave Propagation, Lect. Notes Phys. 687, 141 191. Stenzel, R. L. (1976), Whistler Propagation in a Large Magnetoplasma, Phys. Fluids, 19, 857 864. Streltsov, A. V., M. Lampe, W. Manheimer, G. Ganguli and G. Joyce (6), Whistler Propagation in an Inhomogeneous Plasma, J. Geophys. Res., 111, A316, doi:1.19/5ja11357. Streltsov, A. V., M. Lampe, and G. Ganguli (7), Whistler propagation in nonsymmetrical density channels, J. Geophys. Res., 11, A66, doi:1.19/7ja193. Streltsov, A. V., J. R. Woodroffe, W. Geelman, and P. Pribyl (1), Modeling the propagation of whistler-mode waves in the presence of field-aligned density irregularities, Phys. Plasmas., 19, 514, doi:1.163/1.471971. 17