The importance of amplitude modulation in nonlinear interactions between electrons and large amplitude whistler waves

Journal of Atmospheric and Solar-Terrestrial Physics 99 (2013) 67–72

Contents lists available at SciVerse ScienceDirect

Journal of Atmospheric and Solar-Terrestrial Physics journal homepage: www.elsevier.com/locate/jastp

The importance of amplitude modulation in nonlinear interactions between electrons and large amplitude whistler waves X. Tao a,n, J. Bortnik a, J.M. Albert b, R.M. Thorne a, W. Li a a b

Department of Atmospheric and Oceanic Sciences, UCLA, Los Angeles, CA, USA Air Force Research Laboratory, Space Vehicles Directorate, Kirtland Air Force Base, NM, USA

a r t i c l e i n f o

abstract

Article history: Received 17 February 2012 Received in revised form 25 April 2012 Accepted 23 May 2012 Available online 2 June 2012

The effects of amplitude modulation on nonlinear interactions between a parallel propagating whistler wave and electrons in a dipole field are investigated in this work using a test particle code. Here we first use the test particle simulation to validate a previous single-wave nonlinear theory. Then we adopt a simple two-wave model to represent the recently observed amplitude modulation of a whistler wave field. By varying the frequency spacing between the two waves, we investigate the effects of different modulation frequencies on the nonlinear interactions. We demonstrate that when the resonance overlap condition is satisfied, the resulting change in the electron pitch angle and energy could be very different from what has been predicted by ideal single-wave nonlinear theories. Using a previously observed probability distribution of the subpacket modulation frequency of a chorus event, we obtain the probability distribution of different types of electron response. Our results indicate that the observed subpacket distribution produces particle responses in both non-overlapping and overlapping regimes. Our results suggest that the observed amplitude modulation should be considered when quantitatively treating interactions between electrons and recently observed large amplitude whistler waves or chorus waves. & 2012 Elsevier Ltd. All rights reserved.

Keywords: Nonlinear interaction Phase trapping Amplitude modulation Resonance overlap

1. Introduction Research on nonlinear interactions between electrons and a large amplitude whistler wave in an inhomogeneous magnetic field has a long history, partly due to their important roles in the excitation of discrete Very Low Frequency (VLF) emissions (e.g., Dysthe, 1971; Helliwell, 1967). Various test particle theories of the nonlinear interaction have been developed (Albert, 1993; Bell, 1984; Dysthe, 1971; Inan et al., 1978; Omura et al., 2007; Tao and Bortnik, 2010). These nonlinear theories predict that when the wave amplitude is large enough, trajectories of resonant charged particles will be significantly different from predictions of linear theory. Charged particles are then phase bunched by the wave field and show advective changes of their energy and pitch angle. Some particles could also be phase trapped and stay resonant with the wave for an extended period of time (e.g., Albert, 2000). The nonlinear processes have recently gained strong interest in the radiation belt community because of its possibly important role in the dynamics of energetic particles. For example, through the calculation of nonlinear transport coefficients, Albert (2002) n

Corresponding author. Tel.: þ1 310 825 1659. E-mail addresses: [email protected] (X. Tao), [email protected] (J. Bortnik), [email protected] (J.M. Albert), [email protected] (R.M. Thorne), [email protected] (W. Li). 1364-6826/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jastp.2012.05.012

demonstrated that coherent VLF waves could accelerate a 100 keV electrons to MeV energies in about 1 min. Assuming a single wave with a constant amplitude, Omura et al. (2007) showed that a coherent whistler wave packet that lasts sufficiently long ð  1 sÞ could accelerate a few hundred keV electrons to MeV range through a single relativistic turning acceleration, which is a special form of phase trapping. These results have also been obtained by other analyses (e.g., Bortnik et al., 2008a; Yoon, 2011), after the recent discovery of large amplitude whistler waves in the radiation belt region (Cattell et al., 2008; Cully et al., 2008). Another important reason for the interest in nonlinear interactions is because of whistler mode chorus waves. Chorus waves are electromagnetic emissions with discrete rising or falling tones, and they have been shown to be very important in controlling the electron dynamics in the inner magnetosphere (Bortnik et al., 2008b; Horne et al., 2005; Lakhina et al., 2010; Lorentzen et al., 2001; Nishimura et al., 2010; O’Brien et al., 2004; Summers et al., 1998; Thorne et al., 2005, 2007). Quasilinear theory, which assumes that the waves are broadband and incoherent, was generally used to model the effects of various plasma waves on energetic electrons (Albert et al., 2009; Tao et al., 2011a; Xiao et al., 2009, 2010, 2012). However, the discrete and coherent nature of chorus waves has led to questions regarding the applicability of the quasilinear theory to the modeling of interactions between energetic electrons and chorus waves. Thus various

68

X. Tao et al. / Journal of Atmospheric and Solar-Terrestrial Physics 99 (2013) 67–72

authors have attempted to use the nonlinear theories to study the effects of chorus waves on electrons (Bortnik et al., 2008a; Furuya et al., 2008; Omura and Summers, 2006). It should be noted, however, that the nonlinear theories cited above generally assume a single wave with a constant amplitude. While the observed large amplitude whistler waves or chorus waves have been reported to be coherent or quasi-coherent, they nevertheless show amplitude modulation. For example, highresolution observations of chorus waves (Santolı´k et al., 2004; Tsurutani et al., 2009) have shown the modulation of chorus amplitude, which leads to structures called chorus subpackets (Santolı´k et al., 2004). Observations of Cattell et al. (2008) and Cully et al. (2008) also show similar modulation of the wave amplitude of large amplitude whistler waves (see Fig. 1 of Cattell et al. (2008) and Fig. 2 of Cully et al. (2008)). The amplitude modulation inevitably introduces additional frequency components to the wave–particle interactions in the above single-wave models, which could affect the nonlinear interaction process and the efficiency of phase trapping as an acceleration mechanism. For example, Dowden (1982) and Matsumoto and Omura (1981) have shown the de-trapping of phase trapped electrons by an additional wave in the interaction process. Thus the effects of amplitude modulation of the observed waves on the nonlinear phase bunching and trapping derived from the ideal single-wave theories need to be carefully investigated. Here we use a test particle simulation to explore the effects of including amplitude modulation in nonlinear wave particle interactions. We first describe our simulation model in Section 2, and we validate the single-wave nonlinear theory of Albert (1993) in Section 3. In Section 4, we use a simple two-wave model to represent the amplitude modulation of observed whistler waves, and focus on the effects of amplitude modulation on nonlinear interactions between electrons and large amplitude whistler waves in Section 5. We demonstrate that, depending on the modulation frequency, interactions between electrons and a whistler wave field can be very different from or very similar to those predicted by previous single-wave nonlinear theories. Our work is then summarized in Section 6.

Rz where wave phase Fj  0 kj ðz0 Þ dz0 oj t þ Fj0 . Here kj is the wave number from the cold plasma wave theory (Stix, 1992) and Oe0 ¼ qB0 ðl ¼ 0Þ=m is the signed non-relativistic equatorial electron cyclotron frequency, with q being the charge and m being the mass. In simulations below, we keep Byj constant between l ¼ 01 and 301 and we set Byj ¼ 0 outside this region. Other field w w components (Ew xj ,Eyj and Bxj ) are calculated using the cold plasma dispersion relation at each location along the field line and the given Byj (Tao and Bortnik, 2010). We assume a latitude dependent cold electron density given by ne ¼ ne0 cos4 l, with ne0 ¼ 10 cm3 , following Denton et al. (2002). Initial wave phases are chosen to be random between ½0; 2p. These parameters are chosen following Bortnik et al. (2008a), but note that we use only parallel waves in this paper, in order to avoid the additional complexities introduced by nonzero wave normal angles (Bell, 1984; Tao and Bortnik, 2010). In simulations shown below, we always choose the frequency of the first wave o1 in Eqs. (1) and (2) to be 0:39Oe0 9 to represent whistler waves in the magnetosphere, and this is the approximate frequency where lower-band whistler-mode chorus waves tend to maximize in power (Burtis and Helliwell, 1976). We then calculate trajectories of electrons with the same initial equatorial pitch angle a0 ¼ 301 but having initial gyrophases uniformly distributed between 0 and 2p. The energy E of each electron is E¼236.4 keV, so that the cyclotron resonance condition okvJ ¼ Oe =g with o ¼ o1 is satisfied at latitude lres  231, which was chosen following Bortnik et al. (2008a). Here Oe ¼ qB0 ðlÞ=m is the signed non-relativistic local electron cyclotron frequency and g is the relativistic factor. Each electron is launched from a random latitude between lmin and lmax , moving toward the equatorial plane. Here lmin and lmax are randomly chosen to be lmin ¼ 301 and lmax ¼ 321. The simulation time, which is one unperturbed bounce period, is chosen to be long enough so that the phase trapped electrons without relativistic turning acceleration (Omura et al., 2007) can reach the equatorial plane. Some electrons might encounter a second resonance with the wave field in the simulation, but only the first resonance between an electron and the wave field at lres is presented and analyzed below.

2. The simulation model

3. Comparison with a single-wave nonlinear theory

The Lorentz equation for each test particle is solved to simulate the interaction with a whistler wave field (Tao et al., 2011b). We use a simplified dipole field without field line curvature as the background field B0 (Bell, 1984), whose z-component B0z is a function of z only, with z being the distance along the field line from the equatorial plane. Latitude ðlÞ is also used below to facilitate the interpretation of our simulation results, with dz ¼ LRE ð1 þ 3 sin2 lÞ1=2 cos l dl and B0z ðlÞ ¼ B0z ðl ¼ 0Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ3 sin2 l=cos6 l. Here L is the L-shell value and RE is the Earth radius. We choose L¼5 in this work and correspondingly B0z ðl ¼ 0Þ ¼ 2:50  107 T to represent the Earth’s dipole field in the outer radiation belt. The x and y components of B0 are chosen so that r  B0 ¼ 0 is satisfied. In this work, we simply use B0x ¼ xðdB0z =dzÞ=2 and B0y ¼ yðdB0z =dzÞ=2. The whistler wave field, assumed to be parallel propagating from l ¼ 0 1 to 30 1, is modeled by

We begin by using one wave (Nw ¼ 1 in Eqs. (1) and (2)) in the numerical model to compare with the single-wave nonlinear theory developed by Albert (1993). The wave amplitude Bw y1 ¼ 1 nT following Bortnik et al. (2008a). A nonlinear parameter r has been derived in the previous work (e.g., Albert, 1993; Bell, 1984; Omura et al., 2008; Tao and Bortnik, 2010) to indicate whether the interaction is nonlinear. For a parallel propagating wave, we use r  1=9S9, where S is defined by Eq. (28) of Omura et al. (2008) as ! # (" ) 1 kgv2? d2 Oe þ go @Oe þ 1þ S¼ v , ð3Þ res 2Oe 2 Oe þ o @z o2t d2

Bw ¼

Nw X

w Bw xj cos Fj ex Byj sin Fj ey ,

ð1Þ

w Ew xj sin Fj ex Eyj cos Fj ey ,

ð2Þ

j¼1

Ew ¼

Nw X j¼1

2

where d  11=n2 with n being the refractive index, vres being the resonance velocity, and o2t  kv? Ow with Ow  9qBw =m9. Here we have used the fact that we are considering waves of constant frequency to set do=dt ¼ 0, and that Oe is the signed cyclotron frequency in this work. The parameter r represents the relative importance of the wave-induced motion and the adiabatic motion due to background field. A value of r larger than one means that the wave amplitude is large enough to cause nonlinear interactions of electrons at resonance. For our chosen parameters, r is about 3.5 at lres , indicating that the interaction is nonlinear. There are generally two classes of nonlinear behavior when the wave amplitude is large enough (e.g., Albert, 1993).

X. Tao et al. / Journal of Atmospheric and Solar-Terrestrial Physics 99 (2013) 67–72

where E ¼ 9q9By1 =kmc with c being the speed of light in vacuum, the refractive index n ¼ kc=o, p is particle’s momentum and pJ its component parallel to the background magnetic field. Here a is the local pitch angle and is evaluated at the resonant latitude in Eqs. (4) and (5). For phase trapped particles, changes of pitch angle and energy can be evaluated by enforcing the resonant condition and the constant of motion of the interaction defined by Eqs. (12) and (13) in Albert (1993). For a parallel propagating wave, the two conditions can be written as

okJ pJ =gm ¼ 9Oe 9=g, rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  p 2 o  p? 2 29Oe 9 mc



1þ

mc

ð6Þ ¼ const,

ð7Þ

where p? is the momentum of the particle perpendicular to the background magnetic field, and the two equations can be reduced to a quadratic equation for p2? . Changes of pitch angle and energy can then be solved from pJ and p? . The comparison between the theory and the test particle simulation is shown in Fig. 1. We use 24 electrons in the run. Changes of pitch angle and energy from the simulation are shown in black. We also calculate the theoretical changes of pitch angle and energy using Eqs. (4) and (5) for the phase bunched electrons and Eqs. (6) and (7) for the phase trapped electrons. The resulting pitch angle and energy are plotted in green for the phase bunched particles and in red for the phase trapped particles. Fig. 1 demonstrates that the theoretical estimate of Da0 and DE for phase bunched particles given by Albert (1993) (or Eqs. (4) and (5) above) gives approximately the maximum changes of pitch

90 350

70 simulation

50 40

theory: trapping theory: bunching

300

250

30 20 30 20 10 0 -10 -20 Latitude (degree)

1400 0.5 fce 1200 0.5 fce 1000 800 600 0.4 0.2 0.0 -0.2 -0.4 LAT 2.5 2.5 L 6.7 6.7 MLT 6.4 6.4 Seconds .400 .600 2008 Oct 23 0650: 51

10 10 10 10 10 10 10 Bx By Bz

0.15 0.10 0.05 0.00 -0.05 -0.10 -0.15 .100 Seconds 2008 Oct 23 0650:52

2.5 6.7 6.4 .800

2.5 6.7 6.4 .000 52

2.5 6.7 6.4 .200

2.5 6.7 6.4 .400

2.5 6.7 6.4 .600

2.5 6.7 6.4 .800

30 20 10 0 -10 -20 Latitude (degree)

Fig. 1. Comparison between test particle simulation results (black) and the theoretical estimates of the pitch angle and energy of electrons after interacting with a large amplitude whistler wave. The theoretical estimates of pitch angle and energy after the resonant interaction at the chosen resonant latitude lres for phase bunched electrons are shown in green and the phase trapped electrons in red. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this article.)

2.5 6.7 6.4 .000 53

Bx By Bz

Bw (nT)

60

E (keV)

α0 (degree)

80

Having established the relationship between the analytical estimates of single-wave large amplitude scattering and our test-particle results, we now examine the effects of amplitude modulation upon the particle scattering. The amplitude modulation of the recently reported large amplitude whistler waves can be seen from Fig. 1 of Cattell et al. (2008) and Fig. 2 of Cully et al. (2008). We show in this work a high-resolution waveform observation of chorus wave packets from the Search Coil Magnetometer (Roux et al., 2008) on board THEMIS (Angelopoulos, 2008) in Fig. 2, where the modulation of the wave amplitude, or the subpacket structure (Santolı´k et al., 2003), is evident. This modulation clearly indicates the presence of additional frequency components with generally different amplitudes, which are not included in the previous single-wave numerical and theoretical analysis. The simplest way to represent the amplitude modulation is to use two waves with equal amplitudes. The use of a simple twowave model also permits a detailed analytical analysis. Here we use the two-wave model to investigate the effects of varying the amplitude modulation frequency on interactions between electrons and large amplitude whistler waves. The amplitude modulation frequency in the two-wave model is simply the difference between the two wave frequencies. This can be seen by considering two cosine waves, cosðo1 t þ j01 Þ and cosðo2 t þ j02 Þ. The superposition of these two waves gives 2 cosðdot=2 þ dj0 =2Þ cosð/oSt þ /j0 SÞ, where do ¼ o1 o2 and /oS ¼ ðo1 þ o2 Þ=2, similarly for dj0 and /j0 S. If do 5 /oS, this could be regarded as a wave with frequency /oS and its amplitude modulated at frequency do.

nT /Hz

cos a ½ðnp=mccos aÞ=ðp=mcÞ2  DE , sin a cos a0 mc2 ð5Þ

4. A two-wave model

Frequency (Hz)

Da0 ¼ ð1 þ3 sin2 lÞ1=4 cos3 l

angle and energy due to phase bunching. The evolution of the pitch-angle and energy of phase trapped particles after the chosen resonant latitude lres is well described by Eqs. (6) and (7).

Bw (nT)

Most particles are phase bunched and show advective changes of pitch angle and energy. A small number of particles might also become phase trapped by the wave field and stay resonant with the wave field subsequently for an extended period of time. Changes in the equatorial pitch angle ðDa0 Þ and energy ðDEÞ for an electron after the phase bunching process can be estimated by Eq. (40) of Albert (1993), which are rewritten here for a parallel propagating wave using our notations as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi DE 8 E tan2 a pJ ¼ , ð4Þ p mc2 n2 1 mc

69

.120

.140

.160

.180

.200

Fig. 2. Observations from THEMIS D showing (a) frequency–time spectrogram of wave magnetic field spectral density, with the white line representing half of the equatorial electron cyclotron frequency, (b) three components of wave magnetic field (bandpass filtered over the frequency range of 500–1500 Hz), where z is parallel to the background magnetic field. (c) The same as panel (b), but during a shorter time interval (0.1 s), which is indicated by two vertical red dashed lines in panels (a) and (b). (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this article.)

X. Tao et al. / Journal of Atmospheric and Solar-Terrestrial Physics 99 (2013) 67–72

A criterion for determining whether df  do=2p is large enough for the effect of the second wave to be negligible on the resonant interaction between an electron and the first wave can be obtained from the resonance overlap condition (Lichtenberg and Lieberman, 1983). Here we derive the resonance overlap condition for non-relativistic particles with a uniform background field for simplicity. From Karimabadi et al. (1990) and expressed in terms of parallel momentum, the half width of the resonant island near resonance with a parallel wave at frequency o is, assuming the refractive index n b1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi w DpJ p? 9Oe 9 By 1 ¼2 : ð8Þ mc mc o B0 n From the non-relativistic resonance condition with a parallel propagating wave, the distance between resonances with o1 and o2 is   dpJ 1 o1 þ Oe o2 þ Oe  ¼   : ð9Þ c mc k1 k2  The resonance overlap parameter K is defined as K ¼ ðDpJ1 þ DpJ2 Þ=dpJ , where DpJj is the half resonant island width from the jth wave, calculated using Eq. (8). When do is small, dpJ =mc  ðdk=kcÞ9vg vres 9, where vg is the group velocity, vres is the resonance velocity, and dk ¼ k2 k1 . The condition for resonance overlap is then K Z 1 (Lichtenberg and Lieberman, 1983). On the other hand, when K 5 1, the two resonances are far apart from each other indicating the second wave would not be important to the resonant interaction between an electron and the first wave. Fig. 3(a) demonstrates the case of K 5 1 (the ‘‘non-overlap’’ region), where we set df ¼ 3000 Hz, corresponding to a modulation period of about 0.3 ms and K  0:23. Nonlinear resonant scattering occurs at two distinct latitudes, one for each wave, because of relatively large df . After each resonance, electrons behave quantitatively as predicted by the previous one-wave nonlinear theories (e.g., Albert, 2000) exhibiting phase trapping and bunching. Electrons that are phase trapped by the second wave around l ¼ 291 also show the relativistic turning acceleration (RTA) phenomenon (Omura et al., 2007). At each resonance latitude, the effect of the non-resonant wave is negligible with respect to the resonant nonlinear interaction. Fig. 3(b) shows the case with df ¼ 350 Hz, corresponding to K  1:3 (the ‘‘overlap’’ region), indicating that the second wave should strongly affect the interactions between electrons and the first wave. Changes in both a0 and E are more symmetric with respect to their initial values, compared with predictions from previous nonlinear theories using an ideal single wave shown in Fig. 1. Electrons with positive changes of a0 and E show signs of phase trapping but they quickly get de-trapped at different times,

E (keV)

60 40

RTA

1500 1000 500

20

0

55

270 260 E (keV)

45 35 25

250 240 230

15

220 300

10

60

8

50

6 40 4 30

2

20 30

20

10

0

Latitude (degree)

0 -10

280

theory: trapping theory: bunching

10 8 6

260

ρ

5.1. Three types of interactions

2000

80

E (keV)

5. Effects of amplitude modulation on nonlinear interactions

α0 (degree)

sponding to By1 ¼ 0:71 nT is about 2.5, which is strong enough to cause nonlinear interactions according to the previous singlewave nonlinear theories at lres for the given pitch angle and energy. Even though, as we demonstrate below, the resulting behavior of electrons in the two-wave model strongly depends on the frequency spacing between the two waves.

2500

100

α0 (degree)

In the runs below with the two-wave model, we add a second wave with frequency o2 ¼ o1 þ do in Eqs. (1) and (2). We have w chosen Bw y1 ¼ By2  0:71 nT, being constant along the z-direction, to simulate a whistler wave packet with averaged magnitude qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi w 2 2 /Bw S ¼ ðBw y1 Þ þ ðBy2 Þ  1 nT. The nonlinear parameter r corre-

α0 (degree)

70

4 240 220

2 0 30 20 10 0 -10 Latitude (degree)

Fig. 3. Effects of frequency spacing on changes of a0 and E due to nonlinear interaction between 24 electrons and chorus subpackets. (Top) The ‘‘non-overlap’’ region with df ¼ 3000 Hz. The arrow in the right figure identifies electrons with relativistic turning acceleration (RTA). (Middle) The ‘‘overlap’’ region with df ¼ 350 Hz. (Bottom) The ‘‘degeneracy’’ region with df ¼ 1 Hz. The right (blue) axes show the nonlinear parameter r assuming electrons to be always phase trapped. Also, theoretical changes of pitch angle and energy using the modulated wave amplitudes are presented in green for the phase bunched electrons and in red for phase trapped electrons. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this article.)

leading to different values of positive Da0 and DE. Hence for df ¼ 350 Hz, the resulting overall effect on the particle distribution bears a qualitative resemblance to diffusion as a result of resonance overlap (Lichtenberg and Lieberman, 1983). If two waves are very close in frequency so that df is close to 0, however, one would expect the two waves to behave like a single wave with a slowly varying amplitude. Thus generally only the pffiffiffiffiffiffiffi case of 1 r K r N w is considered to be the ‘‘overlap’’ region, where the resulting behavior of particles are stochastic. If pffiffiffiffiffiffiffi K 4 N w , resonances start to degenerate and interaction becomes gradually like that with a single wave as K increases (or df decreases) (Zaslavsky, 1985, p. 133), even though the transition from the overlap region to the degenerate region might not be as sharp as the transition from the non-overlap region to the overlap region. In Fig. 3(c), we show an extreme case of degeneracy, where df ¼ 1 Hz, corresponding to K  417. Indeed, the motion of electrons looks like that from a single wave as shown in Fig. 1, but with one significant difference: the originally trapped electrons get de-trapped around the same latitude. This can be understood by considering the two waves as a single wave with a slowly modulated amplitude; i.e., we rewrite Eq. (1) as Bw ¼ 2Bw x1 cos ðdF=2Þ cos/FSex 2Bw y1 cosðdF=2Þ sin /FSey , where dF  F1 F2 and /FS  ðF1 þ F2 Þ=2. Eqs. (4)–(7) are then applied to estimate the theoretical changes of pitch angle and energy due to nonlinear interactions with wave amplitude equal to 2Bw y1 cosðdF=2Þ, which slowly varies with both time and latitude. Good agreement

X. Tao et al. / Journal of Atmospheric and Solar-Terrestrial Physics 99 (2013) 67–72

is found between the simulation results and the analytical estimates. The nonlinear parameter r (shown in Fig. 3(c) by the blue axes) is also estimated for the phase trapped electrons assuming that electrons would stay phase trapped until the equatorial plane. Albert (2000) shows that electrons stay trapped because of the growth of the resonant island. In the two-wave model, however, because the amplitude of the wave is modulated, so is the resonant island width and the nonlinear parameter r. The shrinking of the resonant islands thus causes de-trapping of electrons in the two-wave model. Fig. 3(c) shows that the detrapping occurs around the local maximum of r; detailed conditions for trapping and de-trapping will be analyzed in future work. 5.2. Variation of the electron behavior with the change of df To show a systematic variation of the electron behavior when

df changes, we perform 100 runs with each run using a df

10−1 10−3 10−5 10−7 50 40 30 20 10 0 150

50

Number of electrons

100

K=1

250

K = 1.4

Δ E (keV)

Δα0 (°)

PD

logarithmically distributed between 1 Hz and 3000 Hz. We now increase the number of electrons to 400, while keeping the other settings of the simulation the same as those used in the above two-wave simulations. In the bottom two panels of Fig. 4, we plot the histogram of Da0 and DE as a function of df from 100 runs. The range of the vertical axis of the histogram of DE is limited to be less than 150 keV to show the details of the main region of scattering. The distributions of Da and DE are similar to each other, and we will only use Da0 to simplify the description below. We plot two vertical dashed lines indicating the locations of K  1 ðdf  pffiffiffiffiffiffiffi 450 HzÞ and K  N w with N w ¼ 2 ðdf  320 HzÞ. We see that when K o 1 ðdf 4 450 HzÞ, corresponding to the non-overlap region, changes of Da0 are grouped into three categories, corresponding to two groups with positive Da0 due to phase trapping by the two waves at different latitudes and one group with mainly negative Da0 due to phase bunching, consistent with what has been shown in Fig. 3(a). The transition from the non-overlap region to the overlap region occurs at K  1. Now the distribution of Da0 extends almost continuously from about 101 to 201, consistent with Fig. 3(b). Changes of Da0 in the overlap region are more stochastic, and more distinct from the single-wave results shown in Fig. 1. The transition from overlap region to the degenerate region, which is not very sharp, is marked by the dashed line indicating K  1:4. When df o 10 Hz pffiffiffiffiffiffiffi (K b N w with Nw ¼ 2), the two waves behave like a single wave

200 150 100 50

0 1

10

100

1000

δ f (Hz) Fig. 4. (Top) Probability density (PD) distribution of subpackets with different frequency separations (df ), obtained using data from Fig. 5 of Santolı´k et al. (2004). Histogram of Da0 (middle) and DE (bottom) due to interactions with chorus subpackets as a function of df . The total number of electrons used in each run is 400. The dashed lines show df with K  1 (right) and K  1:4 (left).

71

similar to what was shown in Figs. 3(c) and 1. The large positive changes of Da0 are due to phase trapping, while the small negative changes are due to phase bunching. The results obtained above could be used to infer the importance of amplitude modulation, given the distribution of amplitude modulation rate of observed waves. While this distribution for large amplitude whistler waves reported by Cully et al. (2008) and Cattell et al. (2008) is unknown, the study of Santolı´k et al. (2004) shows the distribution of the time delay between two neighboring maxima for a chorus event observed on Cluster 3. The time delay ranges from less than 1 ms to more than 1000 ms, with the highest probability between about 1 ms and 10 ms. If this time delay is interpreted as the amplitude modulation period (1=df ) in the above two-wave model, it corresponds to a range of df from about less than 1 Hz to more than 1 kHz. We reproduce the distribution of df here in the top panel of Fig. 4 using data from Fig. 5 of Santolı´k et al. (2004) for a reference. Fig. 4 demonstrates that a significant fraction of chorus subpackets in the event considered by Santolı´k et al. (2004) results in electron responses in the ‘‘overlap’’ or slightly ‘‘non-overlap’’ regime, and the resulting changes of pitch angle and energy are more chaotic than what have been predicted by single-wave nonlinear theories, given the simulation parameters we used. Since the above physical processes causing the change in the behavior of electrons are very general, it indicates that the interaction between electrons and large amplitude whistler waves or chorus waves might not be correctly described by single-wave nonlinear theories because of the amplitude modulation. Thus when evaluating the effects of nonlinear interactions on the whole population of energetic electrons, the realistic amplitude modulation should be considered.

6. Summary In this work, a full test-particle simulation was used to investigate the interaction between electrons and large amplitude whistler waves. We presented the comparison between changes of pitch angle and energy from the theory of Albert (1993) and the test particle simulation and demonstrated that there is good agreement between the analytical estimates and the test particle theory, lending greater validity to the single-wave theory. However, the ideal single-wave nonlinear theory might not be directly applicable to the interaction between electrons and large amplitude whistler waves or chorus waves because of the amplitude modulation of these waves shown by the high-resolution waveform observations. A two-wave model, which is simple to analyze and contains the essential physics of the interaction, was used to represent the amplitude modulation. The effects of amplitude modulation frequency on nonlinear interactions between electrons and large amplitude whistler waves were investigated. We demonstrated that the nonlinear interactions between electrons and large amplitude whistler waves could be divided into three types according to the frequency spacing between the two waves. We categorized interactions using the resonance overlap parameter K. When K o 1, the interaction is in the non-overlap region, where the second wave only weakly affects the interaction between pffiffiffiffiffiffiffi electrons and the first wave. While for 1 r K r Nw , we have the overlap region, where resonances between two waves are strongly coupled, leading to more stochastic behavior of electrons than that from the interaction with a single wave. Finally, when pffiffiffiffiffiffiffi K 4 N w , the two waves start to behave like a single wave with changing amplitude and this is in the ‘‘degeneracy’’ region. We roughly estimated the importance of amplitude modulation on radiation belt electron dynamics using a distribution

72

X. Tao et al. / Journal of Atmospheric and Solar-Terrestrial Physics 99 (2013) 67–72

of chorus subpacket modulation frequency df from the observation of a chorus event by Santolı´k et al. (2004). Our results indicate that the majority of the chorus subpackets observed by Santolı´k et al. (2004) result in electron responses in the ‘‘overlap’’ or slightly ‘‘non-overlap’’ regime, given the simulation parameters we used, and the amplitude modulation should be taken into consideration to correctly describe interactions between chorus waves and electrons. On the other hand, observations indicate that a chorus wave packet is generally more complicated than what is represented by our two wave model; e.g., the wave frequency inside a chorus wave packet should also vary with time. These features should be also included in a realistic quantitative calculation of transport coefficients of interactions between electrons and chorus waves that could be used in the modeling of radiation belts. References Albert, J.M., 1993. Cyclotron resonance in an inhomogeneous magnetic field. Physics of Fluids B 5, 2744–2750, http://dx.doi.org/10.1063/1.860715. Albert, J.M., 2000. Gyroresonant interactions of radiation belt particles with a monochromatic electromagnetic wave. Journal of Geophysical Research 105 (A9), 21191–21209. Albert, J.M., 2002. Nonlinear interaction of outer zone electrons with VLF waves. Geophysical Research Letters 29 (8), 1275, http://dx.doi.org/10.1029/ 2001GL013941. Albert, J.M., Meredith, N.P., Horne, R.B., 2009. Three-dimensional diffusion simulation of outer radiation belt electrons during the 9 October 1990 magnetic storm. Journal of Geophysical Research 114, A09214, http://dx.doi.org/ 10.1029/2009JA014336. Angelopoulos, V., 2008. The THEMIS mission. Space Science Reviews 141, 5–34 http://dx.doi.org/10.1007/s11214-008-9336-1. Bell, T.F., 1984. The nonlinear gyroresonance interaction between energetic electrons and coherent VLF waves propagating at an arbitrary angle with respect to the Earth’s magnetic field. Journal of Geophysical Research 89 (A2), 905–918. Bortnik, J., Thorne, R.M., Inan, U.S., 2008a. Nonlinear interaction of energetic electrons with large amplitude chorus. Geophysical Research Letters 35, L21102, http://dx.doi.org/10.1029/2008GL035500. Bortnik, J., Thorne, R.M., Meredith, N.P., 2008b. The unexpected origin of plasmaspheric hiss from discrete chorus emissions. Nature 452, http://dx.doi.org/ 10.1038/nature06741. Burtis, W.J., Helliwell, R.A., 1976. Magnetospheric chorus: occurrence patterns and normalized frequency. Planetary and Space Science 24 (11), 1007–1010, http://dx.doi.org/10.1016/0032-0633(76)90119-7. Cattell, C., et al., 2008. Discovery of very large amplitude whistler-mode waves in Earth’s radiation belts. Geophysical Research Letters 35, L01105, http://dx.doi. org/10.1029/2007GL032009. Cully, C.M., Bonnell, J.W., Ergun, R.E., 2008. THEMIS observations of long-lived regions of large-amplitude whistler waves in the inner magnetosphere. Geophysical Research Letters 35, L17S16, http://dx.doi.org/10.1029/2008GL033643. Denton, R.E., Goldstein, J., Menietti, J.D., 2002. Field line dependence of magnetospheric electron density. Geophysical Research Letters 29 (24), 2205, http://dx. doi.org/10.1029/2002GL015963. Dowden, R.L., 1982. Detrapping by an additional wave of wave-trapped electrons. Journal of Geophysical Research 87 (A8), 6237–6242, http://dx.doi.org/ 10.1029/JA087iA08p06237. Dysthe, K.B., 1971. Some studies of triggered whistler emissions. Journal of Geophysical Research 76 (28), 6915–6931. Furuya, N., Omura, Y., Summers, D., 2008. Relativistic turning acceleration of radiation belt electrons by whistler mode chorus. Journal of Geophysical Research 113, A04224, http://dx.doi.org/10.1029/2007JA012478. Helliwell, R.A., 1967. A theory of discrete VLF emissions from the magnetosphere. Journal of Geophysical Research 72 (19), 4773–4790. Horne, R.B., et al., 2005. Wave acceleration of electrons in the Van Allen radiation belts. Nature 437, http://dxdoi.org/10.1038/nature03939. Inan, U.S., Bell, T.F., Helliwell, R.A., 1978. Nonlinear pitch angle scattering of energetic electrons by coherent VLF waves in the magnetosphere. Journal of Geophysical Research 83 (A7), 3235–3253. Karimabadi, H., Akimoto, K., Omidi, N., Menyuk, C.R., 1990. Particle acceleration by a wave in a strong magnetic field: regular and stochastic motion. Physics of Fluids B 2 (3), 606–628, http://dx.doi.org/10.1063/1.859296.

Lakhina, G.S., Tsurutani, B.T., Verkhoglyadova, O.P., Pickett, J.S., 2010. Pitch angle transport of electrons due to cyclotron interactions with the coherent chorus subelements. Journal of Geophysical Research 115, A00F15, http://dx.doi.org/ 10.1029/2009JA014885. Lichtenberg, A., Lieberman, M., 1983. Regular and Chaotic Dynamics, second ed. Springer Verlag, New York. Lorentzen, K.R., Blake, J.B., Inan, U.S., Bortnik, J., 2001. Observations of relativistic electron microbursts in association with VLF chorus. Journal of Geophysical Research 106 (A4), 6017–6027. Matsumoto, H., Omura, Y., 1981. Cluster and channel effect phase bunchings by whistler waves in the nonuniform geomagnetic field. Journal of Geophysical Research 86 (A2), 779–791. Nishimura, Y., et al., 2010. Identifying the driver of pulsating aurora. Science 330 (6000), 81–84, http://dx.doi.org/10.1126/science.1193130. O’Brien, T.P., Looper, M.D., Blake, J.B., 2004. Quantification of relativistic electron microburst losses during the GEM storms. Geophysical Research Letters 31, L04802, http://dx.doi.org/10.1029/2003GL018621. Omura, Y., Summers, D., 2006. Dynamics of high-energy electrons interacting with whistler mode chorus emissions in the magnetosphere. Journal of Geophysical Research 111, A09222, http://dx.doi.org/10.1029/2006JA011600. Omura, Y., Furuya, N., Summers, D., 2007. Relativistic turning acceleration of resonant electrons by coherent whistler mode waves in a dipole magnetic field. Journal of Geophysical Research 112, A06236, http://dx.doi.org/10.1029/ 2006JA012243. Omura, Y., Katoh, Y., Summers, D., 2008. Theory and simulation of the generation of whistler-mode chorus. Journal of Geophysical Research 113, A04223, http:// dx.doi.org/10.1029/2007JA012622. Roux, A., Contel, O.L., Coillot, C., Bouabdellah, A., de la Porte, B., Alison, D., Ruocco, S., Vassal, M.C., 2008. The search coil magnetometer for THEMIS. Space Science Reviews 141, 265–275, http://dx.doi.org/10.1007/s11214-008-9455-8. Santolı´k, O., Gurnett, D.A., Pickett, J.S., Parrot, M., Cornilleau-Wehrlin, N., 2003. Spatio-temporal structure of storm-time chorus. Journal of Geophysical Research 108 (A7), 1278, http://dx.doi.org/10.1029/2002JA009791. Santolı´k, O., Gurnett, D.A., Pickett, J.S., Parrot, M., Cornilleau-Wehrlin, N., 2004. A microscopic and nanoscopic view of storm-time chorus on 31 March 2001. Geophysical Research Letters 31, L02801, http://dx.doi.org/10.1029/2003GL018757. Stix, T.H., 1992. Waves in Plasmas. American Institute of Physics. Summers, D., Thorne, R.M., Xiao, F., 1998. Relativistic theory of wave particle resonant diffusion with application to electron acceleration in the magnetosphere. Journal of Geophysical Research 103, 20,487. Tao, X., Bortnik, J., 2010. Nonlinear interactions between relativistic radiation belt electrons and oblique whistler mode waves. Nonlinear Processes in Geophysics 17, 599–604, http://dx.doi.org/10.5194/npg-17-599-2010. Tao, X., Thorne, R.M., Li, W., Ni, B., Meredith, N.P., Horne, R.B., 2011a. Evolution of electron pitch-angle distributions following injection from the plasma sheet. Journal of Geophysical Research 116, A04229, http://dx.doi.org/10.1029/ 2010JA016245. Tao, X., Bortnik, J., Albert, J.M., Liu, K., Thorne, R.M., 2011b. Comparison of quasilinear diffusion coefficients for parallel propagating whistler mode waves with test particle simulations. Geophysical Research Letters 38, L06105, http:// dx.doi.org/10.1029/2011GL046787. Thorne, R.M., O’Brien, T.P., Shprits, Y.Y., Summers, D., Horne, R.B., 2005. Timescale for MeV electron microburst loss during geomagnetic storms. Journal of Geophysical Research 110, A09202, http://dx.doi.org/10.1029/2004JA010882. Thorne, R.M., Shprits, Y.Y., Meredith, N.P., Horne, R.B., Li, W., Lyons, L.R., 2007. Refilling of the slot region between the inner and outer electron radiation belts during geomagnetic storms. Journal of Geophysical Research 112, A06203 http://dx.doi.org/10.1029/2006JA012176. Tsurutani, B.T., Verkhoglyadova, O.P., Lakhina, G.S., Yagitani, S., 2009. Properties of dayside outer zone chorus during HILDCAA events: loss of energetic electrons. Journal of Geophysical Research 114, A03207, http://dx.doi.org/10.1029/ 2008JA013353. Xiao, F., Su, Z., Zheng, H., Wang, S., 2009. Modeling of outer radiation belt electrons by multidimensional diffusion process. Journal of Geophysical Research 114, A03201, http://dx.doi.org/10.1029/2008JA013580. Xiao, F., Su, Z., Zheng, H., Wang, S., 2010. Three-dimensional simulations of outer radiation belt electron dynamics including cross-diffusion terms. Journal of Geophysical Research 115, A05216, http://dx.doi.org/10.1029/2009JA014541. Xiao, F., Zhang, S., Su, Z., He, Z., Tang, L., 2012. Rapid acceleration of radiation belt energetic electrons by Z-mode waves. Geophysical Research Letters 39, L03103, http://dx.doi.org/10.1029/2011GL050625. Yoon, P.H., 2011. Large-amplitude whistler waves and electron acceleration. Geophysical Research Letters 38, L12105, http://dx.doi.org/10.1029/2011GL047893. Zaslavsky, G.M., 1985. Chaos in Dynamic Systems. Harwood Academic Publishers, New York.