Modeling the wave normal distribution of chorus waves

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH: SPACE PHYSICS, VOL. 8, 74 88, doi:.9/ja8343, 3 Modeling the wave normal distribution of chorus waves Lunjin Chen, Richard M. Thorne, Wen Li, and Jacob Bortnik Received 8 September ; revised 3 November ; accepted December ; published 8 March 3. [] The propagation and attenuation characteristics of lower band and upper band chorus waves are investigated by ray tracing, and the evaluation of Landau damping based on an empirical suprathermal electron model derived from Time History of Events and Macroscale Interactions during Substorms (THEMIS) data. The rate of Landau damping is found to increase at larger L-shell, for more oblique wave normal angles, and for higher geomagnetic activity. Damping is also larger on the nightside than on the dayside and is more pronounced in the upper band than in the lower band. These features can account for the statistical pattern of chorus waves observed away from the equatorial source region. A physical model of the wave normal angle distribution along a field line is presented, which provides insight on how the wave normal angle distribution varies as chorus waves propagate away from the equatorial source region. Our modeling shows that wave emissions at low latitudes (l 3 )come predominately from the equatorial source at the same L-shell and that their wave normal angles increase with increasing latitude due to wave refraction caused by magnetic gradients and curvature. However, at high latitudes (l 3 ), the wave normal angle distribution along a particular field line is affected by chorus waves that arrive from an equatorial source at lower L because of significant cross-l propagation. As a consequence, the lower band wave normal angle tends to decrease with increasing latitudes, while the upper band wave normal angle can either increase or decrease depending on the equatorial source at lower L. The effect of cross-l propagation might also explain why observed wave normal angle distribution tends to become more field-aligned at high latitudes. Interestingly, the upper band chorus at such high latitudes originates from lower band waves originating near the equator at lower L. A global model of wave normal variation along a field line constructed in this study is not currently available from observations but is nonetheless critically important for evaluating bounce-averaged diffusion coefficients for future radiation belt modeling. Citation: Chen L., R. M. Thorne, W. Li, and J. Bortnik (3), Modeling the wave normal distribution of chorus waves, J. Geophys. Res. Space Physics, 8, 74 88, doi:.9/ja Introduction [] Chorus waves are intense, naturally occurring electromagnetic whistler mode emissions in the magnetosphere [e.g., Burtis and Helliwell, 969; Hayakawa et al., 99]. They are typically observed over the frequency range..8 f ce [e.g., Tsurutani and Smith, 977], where f ce is the equatorial electron gyrofrequency, and often exhibit a pronounced power minimum near f ce / [Tsurutani and Smith, 974], separating two distinct frequency bands known as lower band and upper band chorus. Chorus waves are generated near the geomagnetic equator outside the plasmasphere [e.g., LeDocq et al., 998; Lauben et al., ; Santolík et al., 5] due to cyclotron resonant interactions [e.g., Omura et al., 8; Li et al., 9b] with energetic (greater kiloelectron volt) electrons injected into All Supporting Information may be found in the online version of this article. Department of Atmospheric and Oceanic Sciences, University of California, Los Angeles, California, USA. Corresponding author: Lunjin Chen, Department of Atmospheric Sciences, University of California, Los Angeles, CA 94, USA. (clj@atmos.ucla.edu). American Geophysical Union. All Rights Reserved /3/JA8343 the inner magnetosphere during storms and substorms. Pitch angle and energy diffusion induced by resonant interactions with chorus waves are considered important for loss and acceleration of radiation belt electrons [e.g., review by Thorne, ] and also for the formation of the diffuse aurora [Thorne, ] and the pulsating aurora [Nishimura et al.,, ). Chorus waves are also the dominant embryonic source of the plasmaspheric hiss emission [Bortnik et al., 8], which has gained supporting evidence both from satellite observations [e.g., Bortnik et al., 9; Wang et al., ] and from detailed numerical modeling [Bortnik et al., a, b], which is able to reproduce many observed features of hiss [Chen et al., a, b, c, d]. Hiss, in turn, also controls the quiet time two-zone structure of the radiation belts [e.g., Lyons and Thorne, 973; Meredith et al., 7] and quiet time decay of radiation belt electrons [e.g., Lyons et al., 97; Meredith et al., 6; Summers et al., 7]. [3] Accurate evaluation of resonant diffusion coefficients requires a global model of the wave normal angle distribution. A widely adopted wave normal angle distribution is a Gaussian distribution of X = tanc centered along the fieldaligned direction [e.g., Lyons et al., 97; Glauert and Horne, 5; Albert, 7], where c is the wave normal 74

2 angle with respect to the ambient magnetic field. Although the wave normal is often quasi-field-aligned near the equatorial source region [e.g., Li et al., ], this is not generally true in the region away from the equator. The observed wave normal angle of chorus waves can attain values that extend up to the resonance cone [Hayakawa et al., 984; Santolík et al., 9], which has also been confirmed by a recent statistical study based on the Time History of Events and Macroscale Interactions during Substorms (THEMIS) wave observation [Li et al., ]. Statistics describing the wave normal angle distribution of VLF whistler emissions were also obtained based on a 9 year of Cluster measurement [Agapitov et al., ], showing approximately field-aligned wave normals in the vicinity of the geomagnetic equator and more oblique wave normals with increasing latitudes, which could be due to wave refraction during propagation [e.g., Thorne and Kennel, 967; Breuillard et al., ]. However, high-latitude wave measurements by the Polar spacecraft indicate that both the lower and upper band wave normal angles tend to decrease with increasing latitudes at latitudes greater than ~5 [Haque et al.,, Figures 6 and ]. A few attempts have been made to investigate the dependence of diffusion coefficients with respect to wave normal angle [Shprits and Ni, 9; Ni et al., ], and most recently, Artemyev et al. [b] showed that inclusion of oblique wave normals, based on an empirical model derived from a Cluster observation [Agapitov et al., ], produces a significant increase in the electron pitch angle diffusion rate, especially for electrons with a small pitch angle, and therefore could lead to about an order of magnitude decrease of the electron lifetime. [4] In an attempt to understand various characteristics of the wave normal observations and to better prescribe the wave normal distribution for modeling the dynamics of energetic electrons, we present a physical model of wave normal angle distribution, based on ray tracing, which considers wave propagation in a heterogeneous magnetosphere and Landau damping due to suprathermal electrons. For the first time, variations of the wave normal angle distribution along a particular field line are modeled. This model will be invaluable for bounce-averaged diffusion coefficient calculations, which generally require knowledge of the wave spectrum and wave normal distribution along the field line. In section, chorus wave normal observations from the THEMIS spacecraft near the equatorial region are presented, which are used to characterize the wave normal distribution of the equatorial source emission. In section 3, an empirical model of suprathermal electron velocity distribution is introduced, based on THEMIS statistical observations, as a function of L-shell, magnetic local time (MLT), and the geomagnetic activity represented by the Kp index. This model is subsequently used for Landau damping rate calculations. Propagation and Landau damping characteristics of both lower and upper band chorus waves will be examined in section 4, followed by a description of the wave normal angle model, whose results are presented in section 5. Finally, we summarize our main conclusions followed by discussion in section 6.. Chorus Wave Normal Angle Observation [5] The THEMIS spacecraft, consisting of five probes in near-equatorial orbits with perigees below R E and apogees above R E [Angelopoulos, 8], are well situated to measure the near-equatorial chorus emissions. The search-coil magnetometer (SCM) [Le Contel et al., 8; Roux et al., 8] measures magnetic field fluctuations in three orthogonal directions over a frequency range from. to 4 khz, with a sampling frequency up to ~8 khz. The high-resolution waveform data are recorded several times (about 43 s) over a 4 h - orbit, where each waveform data set normally lasts 6 8s,and are distributed widely between the perigee and the apogee [Angelopoulos, 8]. The magnetic field waveforms are used to obtain the polarization properties of the chorus waves, including calculation of wave normal angles [Li et al., ]. All available waveform data collected at L from 5 to within < of the equator, following Li et al. [] but over the extended 4 years from June 8 to May, are used to investigate the statistics of chorus wave normal angles. The wave normal angle results are sorted into wave normal angle bins from to 9 with a bin width of 5, for two local time ranges, nightside ( 6 MLT) and dayside (6 5 MLT), and for the lower and upper bands separately. Figure shows the occurrence rate, defined as the ratio between the number of chorus wave events in each wave normal angle bin and the total number of chorus wave events in each of the four categories, dayside/nightside lower band and dayside/nightside upper band. For the lower band on both the dayside and nightside (Figure a), most of chorus waves are observed within c <, with a secondary peak, which is more pronounced on the dayside than nightside, at c ~7 near the resonance cone (e.g., the resonance cone angle c res 76 for a wave frequency of.5f ce ). The upper band waves (Figure b) occurance % occurance % 5 5 dayside lower band nightside lower band dayside upper band nightside upper band Figure. Occurrence rates of wave normal angles for (a) lower band and (b) upper band chorus obtained from 4 years of waveform measurements made by the THEMIS spacecraft within < of the equator, on the nightside ( 6 MLT) and the dayside (6 5 LT). 75

3 also have the highest occurrence rate at low c (< ) but appear to be more spread to the oblique angles than the lower band, especially on the dayside where occurrence rates remain high up to ~5. The question of why the observed wave normal angle appears this way, which is beyond our current study, may involve the physics of the wave generation mechanism and also partially the wave propagation due to uncertainty in the determination of the geomagnetic equator. The occurrence peak at lower c is consistent with the cyclotron instability [Kennel and Thorne, 967], which is favorable for fieldaligned waves. The secondary peak for the lower band indicates propagation toward the resonance cone, or more likely, an instability favoring excitation near the resonance cone. It should also be noted that the apparent minimum of the occurrence rate near might be associated with the nature of 3-D wave normal directions [e.g., Breuillard et al., ] or with the propagation effect inside the source region that has a finite latitudinal width of <. In this study, however, we will simply use the wave normal angle occurrence rates to characterize the wave normal angle distributions of chorus emission in the equatorial source region as described in section 4 and then investigate how the wave normal angle distribution evolves away from the equatorial source. 3. Suprathermal Electron Model [6] To model the evolution of chorus waves away from the equatorial source, it is necessary to take into account the Landau damping of waves, which is most severe, due to electrons typically in the suprathermal energy range ( ev kev). We derive an empirical model of the suprathermal electron velocity distribution, based on measurements made over a 4 year period (from June 8 to May ) by the ESA [McFadden et al., 8] on board the THEMIS spacecraft A, D, and E, which constantly monitor the electron distribution in the energy range from a few electron volts to ~3 kev throughout the magnetosphere. Electron data are collected over six energy channels, 9, 35, 975, 98, 8789, and 6,389 ev, which are typically representative of electron Landau resonance energy with whistler mode chorus in the magnetosphere. The data are then binned in L from.5 to with a spacing of ΔL =.5 and all MLT with a spacing of h and further sorted into locations inside/outside the plasmapause, based on the method used in Li et al. [], and three levels of geomagnetic activity, Kp < (quiet), Kp = (moderate), and Kp 3 (active). Finally, an isotropic power law velocity distribution Av n (A and n are parameters to be fitted, and v is electron velocity) is used to fit the averaged phase space density data at each L and MLT bin for all the categories, similar to the method used by Bortnik et al. [7], to obtain the global empirical model of statistical suprathermal electron distributions with Kp dependence. Thesuprathermalelectronvelocitydistributionisthenusedto evaluate the wave attenuation during propagation using the temporal damping rate formula ([Chen et al.,, equation ()], with m set to, representing the Landau resonance. Suprathermal electron energy fluxes outside the plasmapause, which generally tend to increase at larger L, are higher on the nightside than on the dayside and are larger during geomagnetically active times than during quiet times, although activity dependence is more pronounced on the nightside than on the dayside (see details in Li et al. []). The flux inside the plasmasphere is generally less than that outside [e.g., Chen et al., 9, a]. Figure shows the dependence of the damping rate on Kp, L, MLT, wave frequency f, and ambient plasma density N e. The calculated Landau damping rate outside the plasmapause follows the same trend of the electron flux, being higher at higher L, and higher Kp, and on the nightside than on the dayside. In addition, the Landau damping rate is also larger for chorus waves with higher frequency, with more oblique wave normal angles, and for lower ambient plasma density. These Landau damping characteristics are naturally imbedded in the modeling results in section Wave Normal Angle Model [7] The propagation characteristics of whistler mode chorus waves in the heterogeneous magnetospheric plasma are investigated using HOTRAY [Horne, 989], a ray tracing code which is capable of tracing any type of wave mode in a cold magnetized plasma or a hot magnetized plasma with weak growth or damping. Cold ray tracing in the meridian plane is performed since we are not concerned with the wave generation in the current study; instead, we use observed characteristics to model the chorus waves at the equator. The background magnetospheric plasma medium on the dayside (MLT = ) used in the ray tracing follows Bortnik et al. [a], where a dipole magnetic field is assumed and the cold plasma density distribution is based on a modified diffusive equilibrium model with a plasmapause structure. For the nightside (MLT = ), the values w =., R c = 4 km, and R u = 4 km are used, and other parameters are the same (see details of parameters in Bortnik et al. [a]) to better fit observation-based equatorial density profile at this local time [Carpenter and Anderson, 99]. The inner edge of the plasmapause L pp is set to be as low as 3.5 R E in order to exclude the effects of the density gradient Nominal case Kp 3 L=5 MLT= f= Hz N e = cm -3 Nominal case MLT= N e =5 cm -3 L=6 f= Hz Wave normal angle, deg Figure. Dependence of temporal damping rate at the equator on Kp, L, MLT, f, and N e. Thick black line shows the calculation for the nominal case, while other lines for the cases with different value for one of the five parameters. 76

4 associated with plasmapause, whose effect will be discussed in section 6. After a chorus ray is launched at a specified L and MLT with a specified wave frequency f and wave normal angle c, ray tracing equations [Horne, 989, equations () and ()] are solved to obtain the ray path and variation of wave normal angle along the ray path. Note that the wave frequency f remains unchanged along the ray path. In addition, we also evaluate the path-integrated damping due to Landau resonance with the suprathermal electrons as described in section 3. Rays are terminated when path-integrated damping reaches 3 db, which means that wave power is reduced by three orders of magnitude than the equatorial source region. [8] Figure 3 shows examples of ray tracing of chorus waves launched at the equator at L = 6 and MLT = for Kp 3 over a range of initial wave normal angles (c ) within the resonance cone and for f = and.4 khz, corresponding to lower (f =.5f ce ) and upper (f =.6f ce ) bands at the launch location (f ce = 4 khz), respectively. It should be emphasized that the lower band and upper band are defined by comparing the wave frequency at any point along the field line to one half of equatorial electron gyrofrequency. In our study, the wave normal angle is assigned a positive or negative sign, where the positive sign denotes the wave normal direction pointing outward (toward increasing L), while the negative sign denotes the wave normal angle pointing inward (toward decreasing L). Since we are only following (a). khz (b).4 khz R XY, R E Figure 3. Ray paths of chorus waves launched at L = 6 with (a) f =.5f ce and (b) f =.6f ce over a range of initial wave normal angles c (color-coded). Each ray is terminated when its power is Landau-damped by 3 db. Black dashed lines denote dipolar field lines with L from to 8 with spacing of and black dotted lines represents latitudes with spacing of. The shaded areas represent the high density plasmasphere. Suprathermal electron model for Kp 3andMLT = is used for the Landau damping calculation. the wave normal away from the equator in the Northern Hemisphere, the wave normal angles are launched in the following range throughout the manuscript: 9 < c < 9. For the lower band waves, the spread of ray paths over L is small within of the equator but increases with latitude l and exceeds ΔL ~atl =3, indicating that chorus emissions observed near l ~3 can originate from a significant range of L-shells (ΔL ~.5) from the observation point. At higher l (~4 ), the spread of ray paths over L, occurring on both sides of the rays original field line (L ), is even greater (~), indicating that the field line mapping method to find the equatorial source emission might produce an error of R E. A simple explanation of the increasing spread of ray paths over L with l is that the separation of two field lines becomes smaller at higher latitudes and that wave normal can (but does not necessarily have to) become more oblique as l increases. Compared to the lower band, the spread over L for the upper band is smaller, and the waves are confined within ~5 of the equator because of the stronger damping at higher wave frequency. [9] The initial wave normal angles c clearly affect both the ray paths and how far the waves can propagate away from the equator by controlling the path-integrated damping. Lower band rays with intermediate negative c ( 5 c 3 ) can propagate to high latitude (>3 ), i.e., they are subject to weaker damping than other values of c.rayswithc 48 can propagate into the plasmasphere and contribute to the formation of another emission known as the plasmaspheric hiss [e.g., Bortnik et al., a; Chen et al., c]. Upper band rays with 34 c 5 can propagate up to relatively high latitude (> ) than other values of c.figure4 shows refractive index surfaces (n = kc/o), which describe the relationship between n and n for a fixed wave frequency. The direction of the ray group velocity v g is geometrically represented by the normal to the surface [e.g., Stix, 99, pp. 79 8]. For lower band waves, there are two characteristic wave normal angles: the Gendrin angle c G = cos (f/f ce ) (= 6 for f/f ce =.5), above which v g changes sign from having the same sign as k to having the opposite sign to k, and the resonance cone angle c res = cos (f/f ce ), above which the waves become evanescent in a cold plasma. For upper fce 5 G resonance cone fce.5 fce Figure 4. Refractive index surfaces for whistler mode waves of frequency f =.5,.5, and.6 f ce. Group velocity v g vectors, normal to each surface, are denoted by arrows. Characteristic wave normal angles, such as the Gendrin angle c G and resonance cone angle c res, are also illustrated for reference. Angle values are c G =6 for f =.5f ce,andc res =76,6, and 53 for.5,.5, and.6 f ce respectively. 77

5 band waves, v g and k always have opposite signs. Figure 3b illustrates that upper band waves with positive c propagate toward smaller L, while those with negative c propagate toward larger L, as suggested by the topology of n surface. In contrast, since the lower band waves can propagate much longer distances to higher latitudes, the refraction due to the inhomogeneity of the plasma medium, especially the magnetic field gradient and curvature which refract the wave normal direction outward (increasing c), needs to be considered to understand the propagation characteristics of lower band waves. It should be emphasized that increasing c due to wave refraction does not necessarily mean that waves become more oblique during propagation. Rays launched with c < actually become more field-aligned during the propagation until c = is reached, and then c becomes positive and increases with l. Therefore, rays launched with intermediate negative c are subject to less overall path-integrated damping and therefore propagate toward higher latitudes than those with c > (Figure 3). This is true for both lower and upper band waves. Figures 5a 5g show seven examples of ray paths that correspond to those in Figure 3a. Rays with c < c G (Figure 5a) and c > c G (Figure 5g) propagate outward and inward, respectively (Figure 4), and damp out quickly, within a few degrees of the equator, because of strong damping prevalent at such oblique angles. Rays with c = c G (Figure 5b) and c =+c G (Figure 5f) more or less follow the same field line although the former ray experiences weaker damping and propagates toward slightly higher latitudes. Rays with c G < c < (Figure 5c) will be refracted continuously toward increasing c. As a consequence, the ray will pass through c = at l ~, during which the ray path reaches the lowest L and the ray propagation changes from inward to outward, then crosses L at l ~35,andfinally damps out at higher latitude. Comparison between Figure 5c and Figures 5d and 5e clearly shows that increasing c due to wave refraction produces weaker path-integrated damping for intermediate negative c than for positive c. Although rays with c = (Figure 5d) and c = (Figure 5e) have a lower damping rate than the ray with c = 44 (Figure 5c) at the equator, the latter ray can propagate toward higher latitudes with less overall damping. The ray with c = (Figure 5d) does not always follow the field line during propagation, and instead deviates from L toward higher L. Three examples of upper band waves are shown in Figures 5h 5j, which also demonstrate the same wave = R XY, R E R XY, R E Figure 5. Ray tracing details of the ray examples shown in Figure 3. (a g) LB and (h j) UB denote the lower band and the upper band wave frequency,.5f ce ( khz) and.6f ce (.4 khz), respectively. Red lines represent ray paths, along which the directions of the wave normals are indicated by short black segments. The varying width of shaded gray area represents the variation of wave amplitude during propagation, which is progressively reduced due to Landau damping. More examples are shown in the movie files (rtkamp_lb.mov for lower band and rtkamp_ub.mov for upper band) in the Supporting Information. 78

6 refraction effect that rotates the wave normal progressively outward (increasing c) and that results in less path-integrated damping for intermediate negative c (comparing Figure 5h with Figures 5i and 5j). Interestingly, as c increases, upper band waves tend to propagate toward slightly lower L (e.g., Figure 5i) because of the opposite sign between v g and k (Figure 4). Because upper band waves only propagate up to l ~,the wave refraction effect on the upper band is less pronounced than that in the lower band. More examples of ray tracings can be found in the Supporting Information. [] Figure 5 demonstrates that wave refraction due to the inhomogeneity of the plasma medium results in a change of both wave normal angles and L-shell as the waves propagate away from the equator. The basic capability of the ray tracing technique to keep track of the wave normal angle variation is fundamental for modeling the wave normal angles off the equator. However, the cross-l propagation increases the difficulty of modeling the wave normal angle variation along a given field line, which requires ray tracing of rays launched over a range of L-shells. The wave normal angle distribution along the field line is directly relevant to understand the wave-induced resonant diffusion of energetic ring current and radiation belt particles bouncing back and forth along field lines. Specifically, test particle simulations in a dipole field [e.g., Tao et al., ] and bounce-averaged diffusion coefficient of radiation belt electrons [e.g., Glauert and Horne, 5; Summers et al., 7; Shprits et al., 9] require information on the wave normal angle distribution along field lines. To overcome the difficulty introduced by the cross-l propagation, we adopt a backward ray tracing method here, i.e., we solve the ray tracing equation as a function of time in the reversed direction. Rays are launched with a specific frequency f and specific wave normal angle c at a given latitude l off the equator along a specific field line L,and then the equatorial crossing values L eq, c eq,and(f/f ce ) eq are identified. Based on the specified wave power distribution at the equator and the path-integrated damping between the launching point and the equatorial crossing point, the power of the ray being launched is obtained. The detailed steps for creating our wave normal angle model along a field line are described as follows: []. Specify the Kp and MLT to determine the suprathermal electron velocity distribution (with L-dependence) for calculating the Landau damping rate g. The need to include the L-dependence comes about due to the cross-l propagation of the rays. []. Specify the field line of interest L, along which the wave normal angle distribution will be modeled. [3] 3. Rays are launched at a given latitude l along the field line (L ), with a given frequency f and a range of initial wave normal angles, c res < c < c res. equatorial crossing c eq (c ), L eq (c ), and (f/f ce ) eq (c ) are recorded, and path-integrated damping between the launching point and the equatorial crossing [Chen et al.,, equation (6)]: PID, in db = log (exp( R gdt)) are recorded as a function of initial wave normal angle c too. [5] 5. Specify the wave power at the equatorial crossing according to a prescribed equatorial wave power distribution P eq (L, c, f/f ce ), which contains dependences on L, c, andf. [6] 6. Reconstruct the wave normal angle distribution at l as follows: P c ; l ; f ; L ; MLT; Kp ¼ P eq L eq ðc Þ; c eq ðc Þ; ðf =f ce Þ eq ðc Þ PID ð c Þ= : [7] 7. Repeat steps 3 6 for different l to obtain the wave normal angle distribution along the given field line L. [8] 8. Repeat steps 7 for other choices of Kp, MLT, and L. [9] It is worthwhile to note that the c range, 9 < c <9, in step 3 covers the full range of wave normal directions pointing away from the equator. Therefore, no wave reflection that results in waves propagating toward the equator is considered. Note that few rays can be reflected toward the equator without being subject to significant damping unless those rays access into the high-density plasmasphere where the suprathermal electron fluxes diminish substantially [e.g., Chen et al., a, b]. [] The above steps can be applied to arbitrary forms of the equatorial wave power distribution P eq (L,c,f). Once the mapping between (L, l, c, f)and(l eq, c eq,(f/f ce ) eq,pid) is found, the resulting wave normal angle distribution can be rescaled by choice of P eq (L,c,f) according to equation () without re-running the ray tracing simulation. In the current study, the equatorial wave power distribution P eq is modeled as follows: () P eq ¼ f L ðlþf f ðf =f ce ÞW c ðcþ; () where f L (L), f f (f/f ce )andw c (c)arel-shell dependence, wave frequency dependence, and wave normal angle dependence respectively. The spatial dependence f L is assumed to have a Gaussian shape: f L ¼ exp ðl L m Þ = ðdlþ for L m dl L L m þ dl; (3) representing a chorus wave power source that is spatially centered at L m with spreading width dl. The frequency dependence f f is modeled as follows: 8 >< exp ðf =f ce :5Þ =:5 f f ¼ :exp ðf =f ce :6Þ =: >: for lower band ð:75 f =f ce :5Þ for upper bandð:5 < f =f ce :8Þ otherwise (4) [4] 4. Backward ray tracing of these rays is performed until the rays reach the equator or the rays are damped by 3 db, whichever occurs first. If an equatorial crossing is found, representing a typical chorus two-band frequency distribution, lower band and upper band, the latter of which has a weaker power at least by one order of magnitude [Li et al., 79

7 ; Meredith et al., ], with the power minimum near f ce / [Tsurutani and Smith, 974]. The wave normal angle dependence of the equatorial chorus power is constrained according to the observed wave normal angle occurrence rate (Figure ). The W c (c) takes one of the four forms, which represents the four combinations of lower/upper band and nightside/dayside, respectively (shown in Figure ). For example, if f/f ce <.5 and 6 < MLT < 5, then W c will be scaled according to the dayside lower band wave normal occurrence rate. For all the four different forms, W c is assumed to be symmetric at about c =, i.e., W c (c)=w c ( c), and is normalized such that Z þp= p= dcw c ¼ : (5) [] Figure 6 shows examples of the backward ray tracing of rays launched at four different latitudes along L =6 at MLT = when Kp 3. The frequency chosen ( khz) corresponds to.5 f ce in the lower band at the equator at L =6. The backward ray tracing is performed over all possible c from 9 to +9 (i.e., excluding the angles greater than the resonance cone angle) with a spacing of. Note that the number of rays increases at higher launching latitudes because of the increasing c res due to the decreasing f/f ce. The ray paths are color-coded according to the path-integrated damping from the launching point and the equatorial crossing point so that the less-damped ray paths can be easily identified by their red color. These red ray paths indicate the equatorial source L that contributes most to the emission at the launching latitude. Waves at low latitude l < ~ (Figures 6a and 6b) come nearly from the same field line L ~ 6. The dominant equatorial source contributing to wave power at higher latitudes shifts toward lower L (Figures 6c and 6d; where red ray paths link to lower L at the equator as l increases). The primary reasons why less path-integrated damping is experienced by the equatorial chorus at lower L are that damping rate decreases due to the decreasing suprathermal electron flux compared to the cold plasma density and that the length of ray path between the launching point and the equatorial crossing is shorter for the equatorial crossing at lower L. Polar plots of path-integrated damping in decibel versus the launching c are superimposed as subplots, showing that the wave normal angle of minimum damping tends to shift toward larger positive angles as l increases, as expected due to the wave refraction effect. It is worth noting that path-integrated damping at l =4 (Figure 6d) can be less than that experienced by rays at lower latitudes (Figure 6b) because the equatorial source shifts toward lower L, where the suprathermal flux responsible for damping is lower. [] Following the same procedure shown in Figure 6, backward ray tracing is performed at finer latitude grids from to 6 with a latitude spacing of. The resulting path-integrated damping is color-mapped against the initial c and l in Figures 7a 7c, all of which show identical color maps. Contours of equatorial crossing L eq,(f/f ce ) eq, and c eq are over-plotted as black solid lines in Figures 7a 7c, respectively. Significant damping is indicated by the bluecolored regions of l and c during propagation from the corresponding equatorial source, whose information is indicated by the black contour lines. In contrast, the orangered color regions correspond to the ray paths with small path-integrated damping; those ray paths provide an effective means of transferring the equatorial wave power to locations off the equator. Wave emissions within l 3 are dominated damping from the equator, db o -3 o 3 o -6 o 6 o <- - [db] (a) -6 o -3 o o 3 o 6 o (b) o o 3 o (c) 3-6 o 6 o o -3 o o 3 o 6 o (d) R XY, R E R XY, R E Figure 6. Backward ray tracing of chorus waves of fixed frequency khz (~.5 f ce at the equator at L =6) at latitudes of (a),(b),(c)3, and (d) 4. For each panel, ~8 rays with varying initial wave normal angles ( 9 < c < 9 ) are launched. Ray paths are color-coded by the amount of damping experienced when rays reach the equator. A small semicircular panel shows such damping in decibels versus c in polar plots. The suprathermal electron distribution for Kp 3 and MLT = is used. 8

8 λ, deg λ, deg (a) 6 (d) 5.5 L eq contours CHEN ET AL.: CHORUS WAVE NORMAL ANGLE (b) (e) f/f ce,eq contours ψ, deg ψ, deg ψ, deg (f) -6 ψ eq contours -4 - Figure 7. Path-integrated damping in decibel (color-coded) of chorus waves [(a c) khz, (d f).4 khz] from the initial latitude of backward ray tracing to the equatorial crossing point, when Kp 3, MLT =, L = 6. Contour lines of L eq,(f/f ce ) eq, and c eq of the equatorial crossing are over-plotted on top of pathintegrated damping color map, in left, middle, and right column, respectively. The color maps in (Figures a c) are identical, and so are in (Figures d f). -6 (c) db by equatorial chorus waves between L =6.5 (Figure 7a). On the other hand, the emission at higher latitudes (> 3 ) largely originate from the equatorial emission at lower L ~ 5 (Figure 7a) with lower f/f ce <. (Figure 7b) and intermediate negative wave normal angle channel ~ 4 < c eq < ~ (Figure 7c). Figures 7d 7f show similar plots for a higher frequency, f =.4 khz, corresponding to upper band waves (.6 f ce )attheequatorofl = 6. Interestingly, there are two separated regions where less-damped off-equatorial emissions are possible. The first region is within ~ of the equator, whose equatorial source is centered at L ~ 6 in the upper band with wave normal angle just greater than 4 ; the second region is above l =4, contributed by an equatorial source at the lower L (< 5.5) in the lower band (f/f ce <.4) with wave normal angles 4 < c eq <. Path-integrated damping calculations show that the damping at higher l for both lower and upper bands becomes smaller when the dominant equatorial source shifts toward L-shell lower than the field line of interest. Ray paths corresponding to lower path-integrated damping at high latitude are those having intermediate negative c eq so that wave normal direction is kept almost aligned with the magnetic field over a longer distance (e.g., Figure 5c). The path-integrated damping calculation provides useful information on the propagation distance and the variation of wave power, but modeling of wave power distribution off the equator still requires the power distribution of equatorial source emissions, which can be treated as a weighting function according to equation (). Modeling results that take into account the equatorial source chorus power distribution are presented in the section Results [3] In section 4, we presented our modeling of the wave normal angle distribution along a given field line, based on backward ray tracing and an empirical model of equatorial chorus wave power distribution (equation ()) where L m and dl contained in the spatial source distribution f L (equation (3)) was yet to be specified. Therefore, the input for modeling the wave normal angle distribution along a field line includes the effects of MLT, Kp, thefield line of interest L,the frequency of interest f, and the spatial distribution of equatorial chorus source L m and dl.wesetl m = L and f to be either.5 f ce or.6 f ce to represent the lower or upper band, respectively, where f ce is the electron gyrofrequency at the equator of the field line L so that the wave power at the dominant frequency is traced along the field line. Figure 8 shows the wave normal angle distribution of the lower band (Figures 8d 8f) and upper band (Figures 8g 8i) chorus along L =6forKp 3and MLT =, with three different values of dl, (Figures 8a, 8d, and 8g),.5 (Figures 8b, 8e, and 8h), and. (Figures 8c, 8f, and 8i). The wave normal angle distribution for the lower 8

9 (a) (b) (c) f L L L L Pw 5 (d) (e) (f) λ along L=6, deg λ along L=6, deg (g) 4 3 (h) (i) ψ, deg ψ, deg ψ, deg Figure 8. The effect of L-width of the equatorial source chorus on the wave normal angle distribution along a field line for Kp 3, MLT =, L =6.Thef L for three values of dl = (a), (b).5, (c).. The wave normal angle distribution along the field line for a typical (d f) lower (f =khz) and(g i) upper band frequency (f =.4 khz). band and the upper band within l < ~3 depends little on the choice of dl because the emissions at low latitudes predominately originate from the equator over a narrow range of L near L (also seen in Figures 7a and 7d). The high-latitude emission (l > 3 ), however, is greatly affected by cross-l propagation from the equatorial region at lower L to the locations where the emission is seen. The difference between Figure 8d and Figure 8f at l > 3 stems from the equatorial chorus source located at L < 5.5 (f/f ce <.) (Figure 7a), which, although having weaker power than L = 6 because of the L and f/f ce dependence, is subject to less path-integrated damping. This effect of cross-l propagation is clearer for the upper band in Figure 8g, where the secondary peak at high latitude is due to the equatorial emissions of the lower band (f/f ce <.4) at lower L < ~5.5 with weak path-integrated damping (Figures 7d and 7e). Note that these equatorial sources in the lower band at lower L are not necessarily weaker in power than the sources in the upper band at L =6 for the same frequency f because of the f/f ce dependence (equation (6)). [4] For the remainder of this section, we assume that dl =.5, bearing in mind that the increase or decrease of dl only enhances or weakens the high-latitude emissions, with little effect at low latitude, and that the high-latitude emission predominately originates from the equatorial emission at lower L. Now we examine the characteristics of the wave normal angle distribution for the case dl =.5 (Figures 8b, 8e, and 8h). The wave normal angle of peak power of the lower band wave (Figure 8e), c peak,startsat~ near the equator and evolves toward ~+5 at l =3 with the increase rate of c peak given by Δc peak / Δl. Oblique waves ( c > 6 ) are strongly confined within a few degrees of the equator because of the larger damping rate. Although waves at such oblique angles can also be produced by propagation toward l 3, those waves can only last for a few degrees too before being damped out. Therefore, the majority of emissions observed off the equator (at l < 3 )areconfined within c < 5, with stronger power at positive c as l increases. The wave normal distribution at high latitude l > 3 shows decreasing c peak as l increases, Δc peak / Δl. For the upper band waves (Figure 8h), emissions are confined closer to the equator (l ) than those of the lower band which can extend up to l 5 with appreciable power. Upper band wave power with c > 3 only lasts over ~ in latitude just off the equator due to the large damping rate. The c peak also increases as a function of l with Δc peak / Δl 3. [5] Next, we investigate the dependence of the wave normal angle distribution on L, MLT, and Kp. Figure 9 shows the results of the wave normal angle distribution of lower band (Figures 9a 9c) and upper band (Figures 9d 9f) along L =6 8

10 5 (a) Kp, MLT=, L=6 (b) Kp =, MLT=, L=6 (c) Kp 3, MLT=, L=6 Pw 4 - λ, deg λ, deg (d) (e) (f) ψ, deg ψ, deg ψ, deg Figure 9. Kp dependence of the wave normal angle distribution along the field line for (a c) lower band (f = khz) and (d f) upper band (f =.4 khz), with MLT = and L =6. on the dayside (MLT = ) with three levels of geomagnetic activity represented by Kp, where Kp (Figures 9a and 9d), Kp = (Figures 9b and 9e), and Kp 3 (Figures 9c and 9f). As Kp increases, the suprathermal electron flux and therefore the Landau damping rate increase. The increasing pathintegrated damping is more pronounced at higher latitudes because of the longer ray path exposed to the Landau damping electrons. As a consequence, the wave power at high latitudes decreases, and waves become more confined to the equator. Similarly, upper band waves are confined closer to the equator from ~5 for Kp to~ for Kp 3. The increase in Landau damping due to the increase in Kp does not, however, change the general characteristics of wave normal angle distribution along the field line. [6] The results for nightside (MLT = ) chorus along L = 6 with varying Kp are shown in Figure. Clearly, the nightside wave normal angle distribution is more sensitive to the Kp index than the dayside. The Landau damping electron flux increases greatly from Kp tokp 3, causing the lower band emission to be confined within ~ of the equator for Kp 3 while the lower band emission can extend to l > 4 for Kp. Upper band emissions are confined within ~5 for Kp to within ~5 for Kp 3. Furthermore, the width of the wave normal distribution of both the lower and upper bands off the equator becomes narrower at higher Kp also due to stronger damping. [7] Wave normal angle distributions along the field line at lower L = 5 on the dayside (MLT = ) and nightside (MLT = ) are shown in Figures and, respectively. Comparing results of the two L values on the dayside (Figures 9 and ) yields a similar tendency. As Kp increases, Landau damping increases slightly for both the lower and upper bands, resulting in the confinement of wave power closer to the equator and the narrowing of the wave normal distribution. There are two major differences between the results at L = 5 and 6. First, the wave power at lower L can extend to higher latitudes for all Kp conditions because the Landau damping flux decreases as L decreases, and the ray path between L = 5 and the equatorial source is generally shorter than L =6, both of which lead to smaller path-integrated damping. Second, there is a more pronounced secondary peak of upper band waves at high latitudes near l 4 at L =5 due to more significant cross-l propagation. There are also two reasons behind the enhanced cross-l propagation: () for the same dl (=.5), the separation of field lines at high latitudes for lower L is smaller, and () the path-integrated damping is reduced at lower L. The results on the nightside (L = 5 in Figure and L = 6 in Figure ) also share similar trends to the dayside. For example, the increase in the Kp index leads to larger damping, and therefore enhanced equatorial confinement, and narrowing of the wave normal angle distribution. A notable difference between Figures and, besides the weaker damping at L = 5, is the appearance of a pronounced secondary peak of upper band waves at high latitudes l > 4 (Figures d f), suggesting the importance of cross-l propagation for waves observed at lower L. [8] Interestingly, although Landau damping flux varies with Kp, L, and MLT, the modeled wave normal angle distributions show fairly robust signatures. For the lower band, the wave normal angle increases from roughly c < near the equator toward c 6 at intermediate latitudes l 3,as expected from wave refraction which leads to increasing c. 83

11 CHEN ET AL.: CHORUS WAVE NORMAL ANGLE Kp, MLT=, L=6 (a) Kp 3, MLT=, L=6 Kp =, MLT=, L=6 (b) Pw (c) 5 - λ, deg (d) (e) (f) 5 λ, deg ψ, deg ψ, deg ψ, deg Figure. Same as Figure 9 except for MLT = and L = 6. Kp, MLT=, L=5 (a) Kp 3, MLT=, L=5 Kp =, MLT=, L=5 (b) Pw (c) 5 - λ, deg (d) (e) (f) 5 λ, deg ψ, deg ψ, deg Figure. Same as Figure 9 except for MLT = and L = ψ, deg

12 5 (a) Kp, MLT=, L=5 (b) Kp =, MLT=, L=5 (c) Kp 3, MLT=, L=5 Pw 4 - λ, deg λ, deg (d) (e) (f) ψ, deg ψ, deg ψ, deg Figure. Same as Figure 9 except for MLT = and L =5. Beyond l 3, c decreases with l and can be nearly fieldaligned at higher latitudes, which is attributed to rays that have intermediate negative c eq and then reach high latitudes with nearly field-aligned wave normal (see Figures 5c and 7c). Upper band emissions start with a broader wave normal angle distribution at the equator and then quickly become nearly field-aligned ( c < 4 ) at just a few degrees off the equator because of significant damping at more oblique angles. The wave normal becomes more oblique as l increases, and the waves are damped out by l. At a higher latitude above 3, the upper band emission appears again due to the cross-l propagation of lower band waves at lower L-shell (Figures 7d and 7e), which also have intermediate negative c eq (Figure 7f). This secondary peak at high latitude has different characteristics from those near the equator in that the wave normal angle can increase or decrease, depending on the equatorial source distribution. These wave normal features of the lower and upper bands are consistent with those of the Polar observations [Haque et al.,, Figures 6 and ], showing that upper and lower bands tend to be less oblique at l =5 5 than at l = 5. [9] To track the latitudinal variation of wave power, P i is defined by integrating P with respect to c, which is presented as follows: P i P i MLT=,L=6 MLT=,L=5 MLT=, L=6 MLT=, L=5 a) lower band b) upper band P i ¼ Z þp= p= dcp: (6) [3] Figure 3 shows the calculated P i (l) of lower band waves (Figure 3a) and upper band waves (Figure 3b) for the four cases when MLT = or and L = 6 or 5 during λ, deg Figure 3. Latitudinal variation of wave power, P i versus l, for (a) lower band and (b) upper band chorus waves when Kp 3, MLT = or, L = 5 or 6. 85

13 Kp 3 (Figures 9c, c, c, and c and Figures 9f, f, f, and f). The damping rate of lower band emissions P i (l) for the nightside and for larger L (solid blue line of Figure 3a), ΔP i / Δl.7 db/deg of l is much higher than that for the dayside, and lower L with ΔP i / Δl.3 db/deg of l (dashed red line). The latitudinal variation of the upper band P i (Figure 3b) shows a similar trend to that of the lower band except that the upper band damping rate over latitude is stronger than the lower band within l, where ΔP i / Δl 5 db/deg for MLT = and L =6, and ~.3 db/deg for MLT = and L = 5. As discussed above, the secondary peak power at high latitude l > ~3, especially for the cases L = 5 on both the dayside and nightside (dashed lines in Figure 3b), is due to cross-l propagation from the lower band emission originating at a lower L. Recently, Meredith et al. [] utilized wave observations from multiple spacecraft, DE, CRRES, Cluster, TC and THEMIS, to obtain a global distribution of chorus wave intensity for both lower and upper bands for various geomagnetic conditions represented by the AE index. They found that during active conditions, () the lower band on the dayside can extend to higher latitudes than that on the nightside, which can be explained by the lower damping flux on the dayside (Figure 3a), and () the upper band is more confined to the equator than the lower band for both dayside and nightside, which can be explained by the higher damping rate in the upper band (Figures 3a and 3b). In the work of Meredith et al. [, Figures 6 and 7], the images also indicate () that intense wave emissions can extend toward higher latitude at lower L compared to higher L for both lower and upper band waves for both dayside and nightside, which is consistent with the larger damping ΔP i /Δl at larger L (Figures 3a and 3b), and () that lower band waves can extend toward high latitudes with less damping during quiet conditions than during active conditions, which is also in agreement with our Kp-dependent Landau damping fluxes (Figures 9 ), although the equatorial emission is generally one order of magnitude weaker than active conditions in power. 6. Conclusions and Discussion [3] The propagation and Landau damping characteristics of chorus waves have been investigated using ray tracing simulations and a suprathermal electron model based on THEMIS measurements. A new model of chorus wave normal angle distribution along a field line is presented and is used to investigate the characteristics of lower and upper band wave normal angle distribution along field lines at different L, on the dayside and nightside, for different levels of the Kp index. The principal conclusions in this study can be summarized as follows: [3]. The rate of Landau damping, based on an empirical suprathermal electron model, increases for more oblique wave normal angles, is larger in the upper band than in the lower band, increases at larger L, is larger on nightside than on dayside, and increases during more active geomagnetic conditions. These Landau damping characteristics, which are consistent with those of Bortnik et al. [7], can account for most features of chorus wave observations from multiple spacecraft on the dayside or nightside at different geomagnetic activities [Meredith et al.,, Figures 6 and 7]. [33]. The wave normal angle distribution of equatorial lower band waves peaks at c <, with a secondary peak near the resonance cone, particularly for the dayside. Equatorial upper band waves show a broader wave normal angle distribution than the lower band chorus, especially on the dayside. [34] 3. Chorus wave normals tend to rotate outward, i.e., clockwise in our plots (but not necessarily become more oblique) due to wave refraction caused by the magnetic gradient and curvature. Those waves with intermediate wave normal angles pointing toward the Earth are generally able to propagate to higher latitude than other directions because of the lower path-integrated damping. [35] 4. For both lower and upper band chorus, emissions below l < predominately originate from an equatorial chorus source along a nearby field line. As l increases, the lower band chorus emissions originate from an equatorial source at lower L-shells with a source wave normal angle lying in the range of 4 < c <, while high-latitude upper band chorus emission originates from an equatorial source in the lower band at lower L-shells with source c in the range of 4 < c <. [36] 5. At low latitudes (l < ~3 ), the wave normal directions of both lower band and upper band tend to refract outward because of the magnetic gradient and curvature, resulting in increasing wave normal angles. At higher latitudes, the wave normal angle of the most intense lower band chorus wave tends to decrease with l and becomes more field-aligned, while the wave normal angle of high-latitude upper band chorus can increase toward + 9 or decrease toward being field-aligned with l, depending on the equatorial source distribution. [37] Chorus waves observed at high latitudes (l > ~3 )are subject to cross-l propagation from an equatorial source at lower L. This is particularly true for high-latitude upper band waves, which originate from lower band waves near the equator at lower L. There can therefore be some ambiguity in the classification into upper or lower band emission for high latitudes. The effect of cross-l propagation depends on the spatial location and width of the chorus source. If two bands of chorus are generated over a broad L-shell range, the observed high-latitude chorus emission will be enhanced by the cross-l propagation. If the equatorial source moves earthward (or anti-earthward), the high-latitude chorus emission becomes stronger (or weaker). When cross-l propagation is significant, the bounce-averaged diffusion coefficient calculation might need equatorial wave information at other field lines for resonance occurring at high latitudes. [38] No generation mechanism is included in our chorus wave normal angle modeling. Instead, we assume a typical two-band chorus wave with a normally distributed spread over L-shell and assume a source wave normal angle distribution based on wave measurement from THEMIS near the equator. The generation mechanism is considered to take place within 5 of the equator [Lauben et al., ], which also justifies our assumption of the geomagnetic equator being the source region. The latitudinal extent of equatorial source could introduce an offset of ~5 in our modeling results, which is especially true for oblique chorus waves. To be observed, wave growth of those oblique chorus waves should overcome Landau damping in the equatorial source region rather than having waves damped out rapidly over the region less than a 86