Theoretical Model of the Nonlinear Resonant Interaction of Whistler-Mode Waves and Field-Aligned Electrons

Theoretical model of the nonlinear resonant interaction of whistler-mode waves and field-aligned electrons. A. V. Artemyev,1, a) A. I. Neishtadt,2, b) J. M. Albert,3 L. Gan,4 W. Li,4 and Q. Ma4 1)Institute of Geophysics and Planetary Physics, University of California, Los Angeles, CA 90024, USA. 2)Department of Mathematical Sciences, Loughborough University, Loughborough LE11 3TU, United Kingdom. 3)Air Force Research Laboratory, Kirtland Air Force Base, Albuquerque, NM 87123, USA. 4)Center for Space Physics, Boston University, Boston, MA 02215, USA. (Dated: 9 April 2021) The nonlinear resonant interaction of intense whistler-mode waves and energetic electrons in the Earth’s radiation belts is traditionally described by theoretical models based on the consideration of slow-fast resonant systems. Such models reduce the electron dynamics around the resonance to the single pendulum equation, that provides solutions for the electron nonlinear scattering (phase bunching) and phase trapping. Applicability of this approach is limited to not-too-small electron pitch-angles (i.e., sufficiently large electron magnetic moments), whereas model predictions contradict to the test particle results for small pitch-angle electrons. This study is focused on such field-aligned (small pitch-angle) electron resonances. We show that the nonlinear resonant interaction can be described by the slow-fast Hamiltonian system with the separatrix crossing. For the first cyclotron resonance, this interaction results in the electron pitch-angle increase for all resonant electrons, contrast to the pitch-angle decrease predicted by the pendulum equation for scattered electrons. We derive the threshold value of the magnetic moment of the transition to a new regime of the nonlinear resonant interaction. For field-aligned electrons the proposed model provides the magnitude of magnetic moment changes in the nonlinear resonance. This model supplements existing models for not-too- small pitch-angles and contributes to the theory of the nonlinear resonant electron interaction with intense whistler-mode waves.

I. INTRODUCTION Although oblique whistler-mode waves represent a significant fraction of whistlers in the Earth’s radiation 25 26–28 The wave-particle resonant interaction is the key pro- belts , the most intense are field-aligned whistlers cess for energy exchange between different particle pop- (e.g., lower band chorus waves with the wave frequency ulations in collisionless plasma1. Particle scattering by ω below a half of the electron gyrofrequency, Ωce), which 29 waves is responsible for losses in magnetic traps2, e.g. often resonate with electrons nonlinearly . There is in Earth’s radiation belts3,4 where whistler-mode cho- only the first cyclotron resonance available for field- rus and hiss waves together with electromagnetic ion cy- aligned whistler-mode waves interacting with electrons: p 2 clotron waves control electron precipitation into Earth’s γω −kc γ − 1 cos α = Ωce (k is the wavevector, α is an atmosphere5–7. The basic concept describing such scat- electron pitch-angle, and γ is an electron Lorentz factor). tering is the quasi-linear theory8,9 that assumes electron The phase trapping for this resonance results in elec- resonant interaction with a broad-band spectrum of low tron acceleration with the electron pitch-angle increase coherence, low amplitude waves10. In an inhomogeneous for energetic particles (with the γ < Ωce/ω) and with the ambient magnetic field the requirement for a low coher- electron pitch-angle decrease for ultra-relativistic parti- ence is significantly relaxed11,12, and electron scattering cles (with γ > Ωce/ω), see Refs. 30–32. The nonlinear by the monochromatic low amplitude waves can be de- scattering (phase bunching) results in decrease of elec- 33,34 scribed by the quasi-linear diffusion13,14. However, ef- tron energy and pitch-angle . Although effects of 35–39 fects of electron resonances with intense waves, e.g. phase realistic wave frequency drift and wave amplitude 40–43 trapping and nonlinear scattering (phase bunching)15–20, modulation alter the electron nonlinear resonant in- are well beyond the quasi-linear theory and require a sep- teraction, the basic concept remains the same: trapping arate consideration21–24. results in electron transport away from the loss-cone and nonlinear scattering results in electron transport toward the loss-cone. A competition of these two nonlinear processes determines electron acceleration and losses. a) Also atSpace Research Institute of the Russian Academy of Sci- The theory of nonlinear electron resonances with ences (IKI), 84/32 Profsoyuznaya Str, Moscow, 117997, Russia; Author to whom correspondence should be addressed; Electronic whistler-mode waves is based on individual orbit anal- mail: [email protected] ysis, that reduces the electron motion equation to the b)Also atSpace Research Institute of the Russian Academy of Sci- pendulum equation with torque15,44–47. Such analysis ences (IKI), 84/32 Profsoyuznaya Str, Moscow, 117997, Russia describes well both phase trapping and nonlinear scat- 2 tering effects and provides typical amplitudes of energy wave frequency ω (i.e., ∂φ/∂s = k, ∂φ/∂t = −ω). Hamil- and pitch-angle changes, ∆γ and ∆α. The basic idea tonian equations for (1) are behind this analysis is the separation of time-scales of p ∂U fast variations of resonant phase (the inverse time scale s˙ = k + w sin (φ + ψ) is ∼ φ˙ ∼ ω) and slow variations of the ambient mag- meγ ∂pk netic fields along electron trajectories (the inverse time- I ∂Ω ∂U p˙ = − x ce − kU cos (φ + ψ) − w sin (φ + ψ) scale is ∼ cp1 − γ−2/R ω, R is a typical inhomo- k γ ∂s w ∂s geneity scale). This separation provides a single small ˙ Ωce ∂Uw parameter c/Rω ∼ 1/kR 1. For the nonlinear wave- ψ = + sin (φ + ψ) (2) γ ∂Ix particle interaction this parameter is about the ratio of I˙x = −Uw cos (φ + ψ) a wave amplitude Bw and ambient magnetic field magnitude B0, i.e. a wave force ∼ kBw can compete with a where φ˙ = ks˙ − ω. Equations (2) show that in absence mirror force ∼ B0/R and temporally trap electrons into of wave (U = 0) and for I of the order of p2/m Ω the resonance17,47,48. However, this theoretical concept is w x k e ce (for not too small pitch-angles), phases φ and ψ change invalid for systems with the second small parameter, e.g. with the rate ∼ Ω (we consider whistler-mode waves for very small pitch-angle (almost field-aligned) electrons. ce with ω of the order of Ω ), whereas (s, p ) change with This effect has been found in Ref. 49: the resonant inter- ce k the rate (p /γ)∂Ω /∂s ∼ c/R where R is a spatial scale action cannot result in decrease of electron pitch-angles k ce of B gradient (for the Earth radiation belts R ≈ R L, below zero, and for sufficiently small pitch-angles such 0 E R ≈ 6380 km and L is the distance from the Earth in interaction would increase pitch-angles. Therefore, the E R ). Comparing these rates, we obtain c/RΩ 1, i.e., nonlinear scattering model, predicting ∆α < 0, meets E ce φ and ψ change much faster than (s, p ) do. For intense difficulties in describing small pitch-angle electron reso- k whistler waves in the radiation belts B /B ≥ c/RΩ nances (see discussion in Refs. 42 and 50). Test parti- w 0 ce despite that B /B 1 (see Ref. 27). Therefore, cle simulations show that ∆α due to nonlinear resonant w 0 ∼ U term in Eqs. (2) does not modify the rates of interaction becomes positive for all electrons with suffi- w φ, ψ and (s, p ) change: phases are fast variables and ciently small initial α (such electron repulsion from the k field-aligned coordinate, momentum are slow variables. loss-cone is the so-called anomalous electron trapping50). Hamiltonian (1) with such time-scale separation has been The similar effect of the absence of pitch-angle scatter- studied both numerically and analytically34,41,45,47. Fig- ing with α decrease around the loss-cone is observed for ure 1(a) shows several fragments of electron trajectories the electron resonant interaction with electromagnetic ˙ ˙ ion cyclotron waves (see Refs. 51–53). So, the actual around the resonance φ + ψ = 0 for this Hamiltonian. question is: can the theoretical model of the nonlinear Resonant electrons can be either trapped (and acceler- wave-particle interaction be modified to account for such ated) by the wave or scattered (with energy decrease). electron repulsion from the loss-cone? We address this Trapping increases Ix (and increases electron equatorial 2 2 2 question below. pitch-angle αeq; Ix = mec γ − 1 sin αeq/2Ωce(0)), whereas nonlinear scattering decreases Ix and αeq. Am- plitudes of energy and Ix (or pitch-angle) changes for II. BASIC EQUATIONS such trapping and scattering are well described by ana- lytical equations33,54,56. For electron nonlinear scattering, the amplitude of I We start with the Hamiltonian of a relativistic electron x changes is about ∼ pB /B (see details in, e.g., Refs. (m is the rest mass, −e is the charge, energy is compa- w 0 e 23, 24, and 57 and references therein). However, if I is rable to m c2 where c is the speed of light) describing x e sufficiently small (i.e., for field-aligned electrons), these two pairs of conjugate variables: the field-aligned coordi- changes can be larger than the initial I . This would nate and momentum (s, p ), gyrophase ψ and momentum x k break the theory, because I is a positively defined vari- I = cµ/e where µ is the classical magnetic moment. x x able that cannot become negative. Figure 1(b) shows In presence of a field-aligned whistler-mode wave, this several fragments of field-aligned electron trajectories: Hamiltonian can be written as (see, e.g., Refs. 23 and there are only positive I changes due to trapping and 54): x absence of scattering (compare with Fig. 1(a)). Thus, a H = m c2γ + U (s, I ) sin (φ + ψ) new model should be developed to describe this regime e w x p s of 100% trapping for small I ≤ B /B . p2 x w 0 k 2IxΩce Let us introduce new phase ϕ and conjugate mo- γ = 1 + 2 2 + 2 (1) mec mec mentum I through the generating function W = R k(˜s)ds˜ − ωt + ψ I + sP : where Ωce = eB0/mec is the electron gyrofrequency 2 (B0(s) is the background magnetic√ field given by, e.g., HI = −ωI + mec γ + Uw sin ϕ reduced dipole model44), U = 2I Ω m eB /γm ck w x ce e w e s 2 with Bw the wave amplitude. The wave number k(ω, s) (P + kI) 2IΩce 55 γ = 1 + 2 2 + 2 (3) is given by the cold plasma dispersion for a constant mec mec 3

80 130 (a) 75 (b) trapped V V e e 70

k 120 k

, , y y 65 g g r r

e 110 e 60 n n e e 100 scattered 55 50 0.5 1 1.5 2 2.5 1.5 2 2.5 3 3.5 4 80 80 70 trapped 70 60 o o

50 , , q q e e 60 α α 40 30 50 scattered 20 40 10 0.5 1 1.5 2 2.5 1.5 2 2.5 3 3.5 4 0.3 0.15 0.125 0.25 trapped 2 2 c c e

e 0.1 m

0.2 m / / q

q 0.075 e e Ω Ω x 0.15 x 0.05 I I scattered 0.025 0.1 0 0.5 1 1.5 2 2.5 1.5 2 2.5 3 3.5 4 tc/R tc/R

FIG. 1: Changes of electron energy, pitch-angle, and Ix (Ωeq = Ωce(0) is the electron equatorial gyrofrequency) due to nonlinear scattering (black; blue shows averaged energy of scattered particles) and trapping (red). Left and right columns show results for 45◦ and 15◦ initial equatorial pitch-angles. The time interval of one resonant interaction is shown. For these trajectories we consider the curvature-free dipole magnetic field44 with the radial distance from the Earth R = 6RE (i.e., L = 6). The wave frequency is 0.35Ωeq, and plasma frequency equals to 6Ωeq. To evaluate the wave number k we use the cold plasma dispersion of whistler-mode waves55. Wave amplitude is 500 pT, i.e. this is 26,28,29,58 an intense wave . The distribution of the wave amplitude along magnetic field lines, Bw(s), is modeled by 2 2 p 2 function tanh((λ/δλ1) ) exp(−(λ/δλ2) ) with λ being the magnetic latitude (ds = Rdλ 1 + sin λ cos λ) and ◦ ◦ 59 δλ1 = 2 , δλ2 = 20 . This function fits the observed whistler-mode wave intensity distribution . To simplify the simulations, we consider waves only in one hemisphere, Bw = 0 for s < 0, and thus there is only one resonance within one bounce period. Waves are moving away from the equatorial plane, s = 0, to large s, i.e. only k > 0 are included.

Hamiltonian (3) does not depend on time, i.e. HI = III. EXPANSION AROUND SMALL I 2 const and mec γ − ωI is the integral of motion. Far from the resonanceϕ ˙ = ∂HI /∂I = 0, particles move in 2D Let us consider Hamiltonian (3) for small kI values: surface formed by intersection of HI (s, P, I) = const and 2 2 I = const. The standard procedure suggests expansion Ωce + kP/me K I 2 HI ≈ − ω I + + mec γ0 of Hamiltonian (3) around the resonance I = Ires deﬁned γ0 2me by ∂H /∂I = 0. However, for small momentum I such I r2IΩ eB an approach is not applicable. To follow an alternative + ce w sin ϕ (4) approach, we start with the expansion for small I and me ckγ0 then consider the resonance. where 2 2 2 2 2 ∂ γ 1 2 kP Ωce Ωce K = mec 2 = 3 k − 2 2 − 2 (5) ∂I I=0 γ0 mec c

q 2 and γ0 = 1 + (P/mec) . Note Eq. (4) shows that 2 HI ≈ mec γ0 +O(I) is the integral of motion. We rewrite 4

Hamiltonian (4) as approach is applicable for description of such system24,57. We are interested in system with sufficiently small I, r 1−3β/2 1 2 2 2IΩce eBw such that the perturbation magnitude ε ∼ 1 (i.e. HI = Λ + K (I − IR) + sin ϕ 2me me ckγ0 β ∼ 2/3) and Hamiltonian is 2 2 me Ωce + kP/me √ Λ = m c γ − ω − 1 2 e 0 2 H = (Y − Y ) + 2Y u sin ϕ (10) 2K γ0 Y 2 R me Ωce + kP/me IR = 2 ω − (6) Hamiltonian (10) does not contain small parameters, but K γ0 coefficients u, YR depend on slow variables. To describe where IR = Ires is the resonant momentum, because dynamics of Hamiltonian√ system√ (10), we introduce new ∂HI /∂I = 0 has a solution I = IR. The Hamiltonian variables p = 2Y cos ϕ, q = 2Y sin ϕ with equations for I and ϕ are ∂q ∂p ∂q ∂p p BwΩce − = 1 I˙ = − 2IΩceme cos ϕ (7) ∂ϕ ∂Y ∂Y ∂ϕ B0kγ0 r Thus, (q, p) are new canonical coordinate and momen- 1 2 Ωceme BwΩce ϕ˙ = K (I − IR) + sin ϕ tum, and new Hamiltonian takes the form me 2I B0kγ0 2 where coefficients depend on slowly changing (s, P ). In 1 1 2 1 2 √ F = p + q − YR + uq (11) the second equation of Eq. (7) the term ∼ (Bw/ I) sin ϕ 2 2 2 is the Lorentz force of the wave field (i.e. a bunching force). For the standard consideration of the wave- System with Hamiltinian (11) is called the second fun- particle resonant interaction21–24 this force is weaker damental model for resonance, and the principal dynam- 2 ics of this system has been described in Ref. 60 (see also than the generalized inertial force ∼ K (I −IR), whereas √ Ref. 61). In application to the wave-particle resonant in- we focus on systems with so small I, that ∼ Bw/ I be- 2 teraction this system has been considered in, e.g., Refs. comes comparable to ∼ K (I − IR). To compare these two terms we introduce a small parameter 19 and 20. Let us consider a profile of Hamiltonian (11) on the 2 2 eB axis p = 0: U = Fp=0 = (1/2)(q /2 − YR) + uq. Equa- ε = w 2 tion determining extrema of U(q) function is dU/dq = mec keq 3 (1/2)q − YRq + u = 0, that can be rewritten as 3 ∗ 1/3 and functions (1/2)˜q − (2/3)˜q(YR/YR) + 1 = 0 withq ˜ = q/u and ∗ 2/3 ∗ s YR = (3/2)u . Figure 2(a) shows that for YR < YR k Ω 1 K2c2 ∗ eq ce there is only one extremum and for YR > YR there are u = , w = 2 k Ωeq wγ0 Ωeq three extrema of U(q). Therefore, the phase portrait of Hamiltonian (11) have two types shown in Fig 2(b): for ∗ with Ωeq = Ωce(0), keq = k(0). Then we introduce new YR < YR there is only one O-point in the phase plane 2 β 2 β variable Y = IΩeq/mec ε , YR = IRΩeq/mec ε , and phase trajectories rotate around this point, whereas ∗ for YR > YR there are two O-points and one X-point Z t β 0 0 (saddle point), and two separatrices `1,2 demarcate the τ = Ωeqε w(t )dt , phase portrait onto three domains Gouter, Ginner, Ginter (see also Ref. 60 and 61 and examples in Refs. 19 and β (i.e., (w(t)Ωeqε )d/dτ = d/dt and integration is along 20). electron trajectories) and rewrite Eqs. (7) as For constant u, YR system (11) is integrable one, whereas for slowly changing u, YR we can introduce dY √ −1 H = −ε1−3β/2 2Y u cos ϕ (8) an adiabatic invariant Ip = (2π) pdq (because all dτ phase trajectories in the portrait shown in Fig. 2(b) are 1−3β/2 dϕ ε closed; see Ref. 62). In absence of separatrix (for YR < = (Y − YR) + √ u sin ϕ ∗ dτ 2Y YR), Ip would conserve with the exponential accuracy ∼ exp −ε−1/3 where ε1/3 1 separates time-scales of Equations (8) are Hamiltonian equations for 62,63 u, YR change (s, P change) and p, q change . For √ conserved Ip the system becomes integrable and Y well 1 2 1−3β/2 HY = (Y − YR) + ε 2Y u sin ϕ (9) before the resonance equals to Y well after the resonance. 2 −1 H −1 H 2 Note Ip = (2π) pdq = 2Y (2π) cos ϕdϕ = Y far The β parameter is controlled by smallness of I (and from `1,2 or in absence of `1,2. 1−3β/2 IR). If ε 1 (i.e., β < 2/3), then I is sufficiently For phase trajectories crossing the separatrices in the large, the Hamiltonian system resembles the general sys- phase portrait from Fig. 2(b), there is a change of Ip tem with small perturbations ∼ sin ϕ and the standard (see Refs. 64–66). This change can be separated into 5 so-called dynamical jump ∼ ε1/3 ln ε1/3 and geometrical where jump ∼ O(1) (see details in reviews 67 and 68). The π π geometrical jump is much larger than dynamical one (and Z Z 2 2 2 1 2 effects of such jumps for application to the wave-particle ρ±dη˜ = 4qc cos ηdη˜ − 4 qc − YR (π − η±) resonant interaction in system (11) have been considered 2 η± η± in, e.g., Refs. 19 and 20). π The separatrix crossing results in particle transition r Z 1 2 2 ∓ 8qc YR − q cosηd ˜ η˜ = 4YR (π − η±) − q sin 2η± from the region with area 2πIp,init (Ip,init is the ini- 2 c c tial value of Ip) to the region with another area S. η± At the moment of the separatrix crossing the invari- r 1 ant I becomes equal to S/2π. Thus, there is a jump ± 8q Y − q2 sin η = 4Y (π − η ) + 3q2 sin 2η p c R 2 c ± R ± c ± ∆Ip = ∆S/2π = S/2π − Ip,init. This jump directly re- 4Y (π − η ) + 3q2 sin 2η η = η lates to the jump of Ix of the initial system (1) R c c c + c = 2 4YRηc − 3qc sin 2ηc η− = π − ηc m c2∆Y m c2∆S eB 2/3 ∆I = ∆I = e = e w x −2/3 2 Ωeqε 2πΩeq mec keq and q cos η = pY − q2/2 (solution of ρ = 0, i.e. we (12) c c R c + take ηc < π/2, see Fig. 3(a)). Therefore, to describe electron resonances in system (1) 1/3 ∗ with small I (small I), we should describe I change due Introducingq ˜c = qc/u and yR = YR/YR (where x p ∗ 2/3 to separatrix crossing on the phase portrait shown in Fig. YR = (3/2)u ), we rewrite Eqs. (16) as 2(b). ˜ ∗ ˜ 2 Sinner = Sinner4YR, Sinner = yRηc − q˜c sin ηc cos ηc ˜ ∗ ˜ 2 IV. DYNAMICS OF SYSTEM (11) Souter = Souter4YR, Souter = yR (π − ηc) +q ˜c sin ηc cos ηc ˜ ∗ ˜ 2 Sinter = Sinter4YR, Sinter = yR (π − 2ηc) +q ˜c sin 2ηc Let us study properties of Hamiltonian (11) that are s 3 yR 1 −3/2 important for evaluation of the ∆Ip jump. First, we ηc = arcos 2 − = arcos q˜c (17) 2 q˜c 2 are interested in areas of regions Ginner, Ginter, Gouter. Coordinates of X-point C in the (q, p) plane are (qc, 0) 3 where qc is the maximum root of equation dU/dq = andq ˜c − 3˜qcyR + 2 = 0. Figures 3(b,c) showq ˜c, cos(ηc), (1/2)q3 − Y q + u = 0. We introduce polar coordinates R and areas S˜inner, Sõuter, S˜inter as function of yR param- (ρ, η) as q = qc + ρ cos η, p = ρ sin η and rewrite Hamil- eter. tonian (11) as 1 1 1 2 1 F = ρ2 + q2 − Y + q2ρ2 cos2 η 2 2 2 c R 2 c V. ELECTRON DYNAMICS 1 + ρ3q cos η + uq (13) 2 c c Figure 3(c) shows that areas Sinner, Souter, Sinter Then, separatrices `1,2 are defined by equation F = Fρ=0: ∗ evolve with yR = YR/YR, but yR is the function of slow q2 time (slow variables) and evolves along the electron tra- ρ2 + 4ρq cos η + 4q2 cos2 η + 4 c − Y = 0 (14) c c 2 R jectory: with the solution r YR 2 IRΩeq 1 yR = = (18) 2 ∗ 2 2/3 ρ = −2q cos η ± 2 Y − q (15) YR 3 m c (uε) ± c R 2 c e 2/3 2 2 2 κ k c IRΩce Areas of Ginner, Ginter, and Gouter are given by equa- = 3 4/3 2 2 2/3 tions: γ0 Ωce mec (Bw/B) π 1 I Z S = ρ2 dη = ρ2 dη = 4Y η − 3q2 sin 2η where inner 2 − − R c c c η− K2c2 κ k2c2 Ω2 Ω P π = ; κ = 1 − ce − 2 ce 1 I Z 2 3 2 2 2 S = ρ2 dη = ρ2 dη = 4Y (π − η ) + 3q2 sin 2η Ωce γ0 Ωce k c kc mec outer 2 + + R c c c Ω q 2 η+ = γ2 − ce + γ2 − 1 (19) I 0 kc 0 1 2 2 Sinter = ρ+ − ρ− dη = Souter − Sinner 2 γ4/3 Ω2 eB 2/3 Ω (uε)2/3 = 0 ce w eq 2 2/3 2 2 2 = 4YR (π − 2ηc) + 6qc sin 2ηc (16) κ k c mec k Ωce 6

(a) (b) 4 4 4 ℓ 2 1 2 2 q d / 0 p 0 p 0 U d * C −2 YRY* −4 R R −4 −4 −3 −2 −1 0 1 2 3 −4 −3 −2 −1 0 1 2 3 −4 −3 −2 −1 0 1 2 3 q/u1/3 q q

3 ∗ ∗ 2/3 1/3 FIG. 2: (a) Proﬁles of dU/dq =q ˜ /2 − (3/2)˜q(YR/YR) + 1 for diﬀerent YR; YR = (3/2)u andq ˜ = q/u . (b) ∗ ∗ Phase portraits of system (11) with u = 2 for YR > YR (left) and YR < YR (right). Bold red curve shows the separatrices `1,2.

2 6 0.6 5 c (a) 3

/ (b) Gouter 0.5 o 1

4 s

u 0.4 η /

c 0.3 G 3 c inter q 0.2 1 2 0.1 1 0 S

* R inner 10 (c) Souter Y

Ginner ηc 4 π-η / c 5 S

p 0 Sinter 0 C 0 2 4 6 8 10 yR −1 5 5 3 / 1

u 0 0 0 / p −5 −5 −2 −1 −0.5 0 0.5 1 1.5 2 3 −8 −4 0 4 −8 −4 0 4 q q/u1/3 q/u1/3

FIG. 3: (a) Schematic of Ginner, Ginter, and Gouter regions; angle ηc is shown. (b) and (c) Proﬁles ofq ˜c, cos ηc, ˜ ∗ ˜ ∗ ˜ ∗ Sinner = Sinner/4YR, Souter = Souter/4YR, and Sinter = Sinter/4YR; bottom panels show phase portraits for two ∗ yR = YR/YR values.

The resonant interaction starts with IR = Ix (initial I is described by Eq. (18) with IR given by Eq. (6). value is Ix), and for this moment we can write Figure 4(a) shows yR variation with the magnetic lati- 2/3 2 2 2 κ k c IxΩce tude for particles moving from high latitudes toward the yR = (20) 3 4/3 2 2 2/3 equatorial plane. Along the resonant trajectories yR in- γ0 Ωce mec (Bw/B) creases, i.e. normalized areas S˜inner, Sõuter, and S˜inter 2 2 κ = γ0 1 − (ω/kc) should grow (see Fig. 3(c)). However, expression of ∗ ˜ ∗ 2/3 (21) S = 4YRS includes the wave amplitude u (YR ∼ u ) that generally drops to zero at the equatorial plane. 2 2/3 The factor ∼ 2IxΩce/mec (Bw/B) is of the order of Thus, S dynamics along the resonant trajectories is de- ˜ ∗ one due to smallness of Ix. In the resonance yR variation termined by the competition of S increase and YR de- 7 crease. Figure 4(b) shows that Sinter decreases, whereas Ix change) would mean Ix decrease, i.e. this is classical 22–24 Sinner increases. Taking into account that yR grows nonlinear electron scattering on the resonance . from values smaller than one (i.e. from values for which Sinner = 0 and there is only Souter; see Fig. 3(c)), we describe resonant particle dynamics. C. Electron drift in pitch-angle space 2 2/3 Far from the resonance, Y equals to ΩeqIx/mec ε −1 H and equals to Ip = (2π) pdq for Hamiltonian system Let us summarize results of Ip (Y ) changes. The sep- (11). Thus, for given Ix, in the (p, q) plane√ electrons are aratrix crossing by trajectory in the (q, p) plane occurs distributed along the circle with the radius 2Y ; the en- when Sinter = 2πIp, and thus the moment of crossing tire circle is inside the region Gouter, because Ginner and depends on Ip (i.e. on the initial Y or, equivalently, on Ginter do not exist (see schematic in Fig. 4(c), moment initial Ix). There are two possible situations shown in #1). With time (or with yR increase) regions Ginner, Fig. 5. Ginter form (when yR exceeds one), whereas trajectories If the initial Ix (initial Ip = Sinter/2π) is sufficiently on the (q, p) plane deform, but save their areas. Then small, then the separatrix crossing occurs when elec- two scenarios are possible. tron orbits are in Ginter and Sinter < Sinner (this moment is quite far from the moment of Ginter forming, i.e., particles spend a significant time ∼ O(ε1/3) in the A. Scenario for small Y resonance before the separatrix crossing). Thus, jump ∆Ip = (Sinner − Sinter)/2π > 0 and Ix increases due If initial Y is sufficiently small, at the moment of to the electron resonant interaction with the waves (see Fig. 5(a) and black trajectories in Fig. 1, right panels). Ginner,inter formation trajectories appear inside Ginter (Fig. 4(c), moment #2 shows such red trajectory within Magnitude of Ix change is defined by Eq. (22). If the initial Ix (initial Ip) is sufficiently large, then Ginte). From this moment and up to the sepatrix crossing the particle is in the resonance with the wave. When the separatrix crossing occurs when electron orbits are decreasing S /2π reaches I , electrons cross the sep- in Gouter. This happens right around the resonance, inter p i.e. particles do not spend a long time being trapped in aratrix `2 and appear in Ginner region (Souter increases Gouter. Thus, jump ∆Ip = (Sinner − Souter)/2π < 0 (be- with Sinter, and electrons cannot come to Gouter) with cause Souter > Sinner) and Ix decreases due to the elec- growing Sinner (see schematic in Fig. 4(c), moment #3). This separatrix crossing results in a jump of the adia- tron resonant interaction with the waves (see Fig. 5(b) and black trajectories in Fig. 1, left panels). batic invariant ∆Ip = (Sinner − Sinter)/2π where Sinner These two situations explain classical scattering with is evaluated at the moment when Sinter = 2πIp (escape Ix decrease for electrons having not-too-small Ix (see from the resonance). A new Ip would be conserved, and far from the separatrix crossing (far from the resonance) Refs. 22 and 23) and anomalous trapping with Ix increase for all electrons with initial very small Ix (see Refs. new Y is equal to new Ip. Thus, the jump of Y equals ∗ ˜ ˜ 42, 49, and 50). to ∆Y = (Sinner − Sinter)/2π = 4YR(Sinner − Sinter)/2π ∗ where YR is evaluated at the moment of the separatrix crossing (that can be quite far from the resonance when VI. DISCUSSION Ginter forms). Jump ∆Y can be rewritten as a jump of Ix, see Eq. (13): Whistler-mode waves nonlinearly resonate with field- ( ∗ 2 2/3 ) ( 2 ) aligned (small-Ix) electrons and transport them away S˜ 4Y mec ε 3∆S˜ mec ∆I = ∆ R = ∆ (uε)2/3 from the loss-cone49, and this effect is called anomalous x 2π Ω π Ω eq eq electron trapping50, when 100% of resonant electrons are ( ˜ 2 4/3 2 ) phase trapped. Such trapping of field-aligned electrons 3S mec γ0 Ωce 2/3 = ∆ (Bw/B) (22) differs significantly from whistler-mode wave interaction π Ω κ2/3k2c2 ce with moderate pitch-angle electrons, that results in scattering toward the loss-cone of the majority of resonant where all variables changing along magnetic field line electrons. should be evaluated at the separatrix crossing moment. Similar effect of the change of trapping/scattering relation has been found for electromagnetic ion cyclotron waves. These waves nonlinearly scatter electrons away B. Scenario for large Y from the loss-cone, whereas electron phase trapping results in pitch-angle decrease.52,53,69. For field-aligned If initial Y is sufficiently large, then at the moment (small-Ix) electrons the trapping disappears and there of Ginner,inter formation the electron trajectories in the is only nonlinear scattering with the pitch-angle increase (q, p) plane appear in Gouter. For such trajectories the (see detailed analysis in Ref. 70, and references therein). separatrix crossing would results in ∆Y = (Sinner − As we show in this study, such modification of the nonlin- Souter)/2π. Because Souter > Sinner, this Y change (and ear resonant interaction for small Ix can be described an- 8

100 100 3 / (b) 2 (a) 2 - 80 ) Souter q

10 e ○ 60 R B

R 10 1 / y y

w S 15○ 40 inner 1 B ○ ( 5 0 ⋅ 20 S 21 20.5 20 S inter 0.1 0 20 15 10 5 0 20 15 10 5 0 λ, o λ, o 6 6 6 (c1) (c2) Souter (c3) 4 G 4 2πIp 4 2πI outer 2 p 2 Sinter 2 3 3 / / 1 1 u u 0 0 0 new 2πIp / / 1/2 Sinner p p (2Y) −2 −2 −2 −4 S −4 −4 outer Sinter Sinner −6 −6 −6 −6 −4 −2 0 2 4 6 −6 −4 −2 0 2 4 6 −6 −4 −2 0 2 4 6 q/u1/3 q/u1/3 q/u1/3

FIG. 4: (a) yR as a function of magnetic latitude for 50 keV energy and three pitch-angles. (b) Areas ◦ Sinner,inter,outer as functions of magnetic latitude for 50 keV, 10 pitch-angle. Details of wave model are the same as in the caption of Fig.1. (c) three schematic views of particle trajectories in the (q, p) plane. Phase portraits change from c1 to c3 along particle trajectory.

8 30 (a) (b) small to keep derived equations valid. Change of Ix 25 2 2 S Souter can be written as ∆Ix = ∆αeq sin αeq cos αeqmec (γ − I6p outer 20 2 Sinner 1)/(Ωeq − sin αeqωγ) where αeq is the electron equato- π π Sinner 2 2 I ∗ / / 4 15 p rial pitch-angle. We denote α value of pitch-angle for S

S eq ∗ 10 which I = I (for given energy γ), and write ∆α = ΔIp>0 ΔIp<0 x x eq 2 2∆I Ω /m c2 sin 2α ≈ (α − α∗ )ν0 where ν0 = Ip 5 Sinter x eq e eq eq eq Sinter 2 ∂ν/∂α | ∗ and ν = 2∆I (α )Ω /m c sin 2α 0 0 eq αeq =αeq x eq eq e eq 10 1 0.1 30 29.5 29 28.5 28 ∗ λ,○ λ,○ with ν(αeq) = 0. Figure 6 shows two possible variants of field-aligned FIG. 5: Schematic of Ip jumps due to separatrix (small Ix) electron resonant interaction. Independently crossing. (a) System with positive ∆Ip; (b) System with 0 ∗ of ν , electrons with αeq < α experience the pitch-angle negative ∆Ip. For simplicity we use IR ≈ Ix here. eq increase, i.e. transported away from the loss-cone αLC . Therefore, the transport into loss-cone (electron precipi- ∗ tations) would require that electrons with αeq > αeq are alytically, because even for small Ix there is a separation scattered with sufficiently large ∆αeq: αeq+∆αeq < αLC . of time-scales of (ϕ, I) variations and (s, pk) variations. 0 If ν < −1, αeq + ∆αeq < αLC can be rewritten as Let us discuss and summarize obtained results. 0 ∗ 0 αeq ≥ (|ν |αeq − αLC )/(|ν | − 1), and this inequality ∗ is always satisfied for αeq > αeq and sufficiently small 0 loss-cone αLC < 1. Thus, for ν < −1 there are elec- A. Electron dynamics around loss-cone tron nonlinear scattering into loss-cone and precipita- 0 tions. If ν > −1, αeq + ∆αeq < αLC can be rewritten as ∗ 0 ∗ 0 Figure 5 shows that for some initial Ix = Ix, the sepa- αeq ≤ (αLC −|ν |αeq)/(1−|ν |), and this inequality is sat- ∗ ∗ ratrix crossing occurs with ∆Ix = 0, i.e. Sinter ≈ Sinner. isfied for αeq > αeq only for the exotic case αLC > αeq. ∗ 0 Thus, electrons with this Ix would not experience pitch- Thus, if ν > −1 there are no electron scattering into angle change (in the leading approximation). Let us loss-cone and no precipitations (this is anomalous elec- ∗ 2/3 consider a realistic system with Ix ∼ ε sufficiently tron trapping, see Ref. 50). These estimates demon- 9

n‘<-1 n‘>-1 B. Two types of phase portraits

We consider nonlinear resonant interaction of waves loss-cone loss-cone and electrons with small Ix, for which the resonant Hamiltonian takes the form given by Eq. (10) or Eq. 0 aLC a* 0 aLC a* (11). Let us compare this Hamiltonian with one describ- FIG. 6: Schematic view of pitch-angle jumps for system ing the classical problem of charged particle scattering for 0 with electron losses (left panel, ν < −1) and system moderate Ix values (see Refs. 22 and 23). Instead of ex- without electron losses (right panel, ν0 > −1). pansion of Hamiltonian (3) around I = 0, we can expand it around the resonant value I = Ires(s, P ) determined from ∂H/∂I = 0 equation:

1 2 2 H = −ωIres + γres + Kres (I − Ires) 80 80 2m (a) 60 (b)60 e 40 40 r 30 30 2IresΩce eBw 20 20 + sin ϕ (23) o o m ckγ

e res , , q q 10 10 e e α

α 2 2 5 5 where Kres = ∂ H/∂I |I=Ires and γres = γ|I=Ires . The main diﬀerence of this Hamiltonian and one from Eq. (4) loss-cone loss-cone is that the eﬀective wave amplitude does not depend on 0 1 2 3 4 0 1 2 3 4 I in Eq. (23). The phase portrait of Hamiltonian (23) tc/R tc/R with frozen slow variables is shown in Fig. 8(a). 80 80 (c) 60 (d)60 Instead of introducing (q, p) coordinates it is more con- 40 40 30 30 venient to introduce Pφ = I −Ires through the generation ∗ 20 20 function W = (I − Ires)ϕ + P s with new slow variables o o

, , ∗ ∗ q 10 q 10 s = s + (∂I /∂P )ϕ, P = P − (∂I /∂P )ϕ. Expand- e e res res α 5 α 5 ing Ires(s, P ) and γres(s, P ) over (∂Ires/∂P ), (∂Ires/∂s), we rewrite Hamiltonian (23) as loss-cone loss-cone H ≈ −ωIres + γres (24) 0 1 2 3 4 0 1 2 3 4 2 2 r tc/R tc/R KresPϕ 2IresΩce eBw + − {γres,Ires} ϕ + sin ϕ 2me me ckγres FIG. 7: Pitch-angle change along electron trajectories for L = 6 (see details of the wave model in the caption where {... } are Poisson brackets, and all function in Eq. ∗ ∗ of Fig. 1). Panels (a) and (b) show trajectories for (24) depend on (s ,P ). The phase portrait of Hamilto- ∗ ∗ Bw = 500 pT; panels (c) and (d) shows trajectories for nian (24) with the frozen slow variables (s ,P ) is shown Bw = 3 nT. Left and right panels show results for initial in Fig. 8(b). This is the classical portrait of the pen- 67 α = 15◦ and α = 20◦. The loss-cone is shown for dulum with torque with three main phase space re- ◦ illustration, αLC ≈ 5 . gions: before resonance Pϕ = 0 crossing particles are in Gouter, and resonance crossing can result in trapping (particles appear in Ginter) or scattering (particles appear in Ginner). Therefore, there is direct relation between three regions Gouter,inner,inter of Hamiltonian of (11) and Hamiltonian (24). strate that the nonlinear scattering of small pitch-angle For the initial system given by Eq. (3) nonlinear scat- electrons with ∆Ix > 0 does not necessary stop electron tering (transition from Gouter to Ginner) always appears precipitations. with Ix decrease, and this effect is well seen in the phase portrait (c) of Fig. 8: the area Souter is always larger The pitch-angle jump ∆αeq is proportional to ∆Ix than the area Sinner. But when the initial invariant Ix −1 H given by Eq. (22), and thus there is a scaling ∆αeq ∼ (Ip or Iϕ = (2π) Pϕdϕ) is sufficiently small, parti- 2/3 ∗ Bw . If αeq depends on Bw sufficiently weakly, then cles become trapped within region Ginter as soon as this the wave amplitude increase would result in |ν0| increase. region appears during particle motion along their tra- Therefore, for sufficiently high Bw we should expect the jectories. In phase portrait (c) of Fig. 8 this trapping regime with electron nonlinear scattering into loss-cone means that the area surrounded by the particle trajec- (see Fig. 6, left panels). To check this scenario, we in- tory 2πIp ≈ 2πY is smaller than Sinter at the moment tegrate numerically set of electron trajectories given by when Ginter appears (when yR becomes larger than one, Hamiltonian equations (2) for the same initial γ and two see Fig. 3(c)). For yR = 1 we get ηc = 0,q ˜c = 1, ∗ wave amplitudes. Figure 7 shows that for large Bw elec- and Sinter = 4πYR. Thus, the threshold Ix value is 2 2 2/3 trons can be scattered into loss-cone and precipitate. 2IxΩce/mec = 3(Ωce/kc) (Bw/κB) . 10

C. Small Ix for electrons resonating with electrostatic waves (a) 4 G oGutoeurter 2 For resonances in inhomogeneous plasma, there is

s the direct analogy between electron interaction with e

r 15,74–77

I 0 whistler-mode waves and with electrostatic waves . - Ginter I Therefore, we can expect a change of the regime of −2 electron nonlinear resonant interaction with electrostatic −4 G waves for field-aligned electrons. The main difference inner between the cyclotron resonance with electromagnetic 0.2 0.4 0.6 0.8 1 1.2 waves (e.g., whistler-mode waves) and the Landau resonance with electrostatic waves is that for electrostatic φ/2π waves the wave amplitude Uw = eΦw is determined by 4 the electric field potential Φw and does not depend on (b) I . Thus, Eqs. (7) would take the form: Gouter x 2 ˙ 1 2 φ 0 I = −eΦw cos ϕ, ϕ˙ = K (I − IR) (25) P Ginter me

−2 2 Deﬁning a small parameter ε = eΦw/mec (the analog of 2 −4 Ginner eBw/mec keq used through the paper), we can estimate the threshold value of Ix for which there is no time sep- 0 0.2 0.4 0.6 0.8 1 aration of I and ϕ variations in Eqs. (25). For τ ∼ tεβ and Y ∼ Iε−β, Eqs. (25) give β = 1/2. Thus, instead of φ/2π 2/3 the scaling Ix ∼ (eBw/k) for the cyclotron resonance, 1/2 (c) 4 we obtain the scaling Ix ∼ (eΦw) for the Landau res- Gouter onance. 2 Electrons can resonate with whistler mode waves through the Landau resonance, if waves are obliquely

p 0 propagating, i.e. if there is a ﬁnite angle θ between a Ginter Ginner wavevector and a background magnetic ﬁeld. For a gen- −2 eral case of θ 6= 0, the whistler-mode wave amplitude in Hamiltonian (3) takes the form47,56,78,79: −4 s  r 2 2IxΩce eBw X cos θ ± C1 2Ixk −4 −2 0 2 4 U = J sin θ w m c2 k 2γ n±1  m Ω  q e ± e ce (26) FIG. 8: Phase portraits of Hamiltonian (23), panel (a), s  2 of Hamiltonian (24), panel (b), and Hamiltonian (11), eBw pk 2Ixk + + C2 Jn  sin θ sin θ panel (c). k γmec meΩce

where n is the resonance number, C1,2 are functions of wave dispersion and θ, and Jn are Bessel functions. Equation (26) shows that for the cyclotron√ resonance n = −1 and small Ix we have Uw ∼ Ix, and thus 2/3 This is so-called autoresonance phenomena of the 100% Ix ∼ (eBw/k) scaling works. For the Landau reso- trapping in the resonant systems (see, e.g., Refs. 71–73 nance n = 0 and small Ix we have Uw ∼ sin θ · (const + I ) ∼ O(I ), and thus I ∼ (eB /k)1/2 scaling works. and references therein). Being trapped into Ginter, parti- x x x w cles can both increase or decrease their Ip (and Ix) during the transition from Ginter to Ginner: the Ix change de- D. Model applicability and verification pends on the ratio of Sinter/Sinner at the moment when Sinter becomes equal to 2πIp. For sufficiently small Ip (small Ix), the ratio Sinter/Sinner = 2πIp/Sinter will be The proposed theoretical model predicts the para- below one, and particles will increase their Ip (and Ix) metrical boundary in the energy/pitch-angle space 2 2/3 due to the resonant interaction. Formally, this interac- (2IxΩce/mec ∼ ε or, more precise, yR = 1 in Eq. tion cannot be called scattering, because particles are (20)) where the nonlinear scattering (phase bunching) trapped into Ginter from the beginning. disappears and the nonlinear wave-particle interaction is 11 totally dominated by the phase trapping that transports ACKNOWLEDGMENTS resonant electrons away from the loss-cone. This para- metrical boundary depends on wave and background filed A.V.A. are thankful to M. Kitahara and Y. Katoh for characteristics, and can be evaluated for each specific pointing on the problem of the nonlinear resonant in- plasma system. Thus, this boundary can be implemented teraction of small pitch-angle electrons50, to V. Grach into numerical models utilizing the generalized Fokker- and A. Demekhov for very useful discussion of their 39,56,80 81 Plank equations and mapping techniques . Such results on EMIC waves scattering52,53,70, to D. Shkl- models require boundary conditions for the nonlinear yar for discussion of initial research of this problem49, wave-particle interaction domain (in the energy/pitch- and to D. Mourenas for useful suggestions. Work of angle space), and our model shows that the low pitch- A.V.A. and A.I.N. was supported by the Russian Sci- angle boundary of this domain is the boundary of 100% entific Foundation, project 19-12-00313. A.I.N. also of trapping. was supported by the Leverhulme Trust, project RPG- The most straightforward verification of the derived 2018-143. W.L. and Q.M. would like to acknowledge theoretical mode consists in comparison of theoretical the NASA grants 80NSSC19K0845, 80NSSC20K0698, predictions with measurements of low-altitude spacecraft 80NSSC20K0196, and NSF grants AGS-1723588 and probing energetic electron precipitations from the Earth’s AGS-1847818. L.G. would like to thank the NASA FI- radiation belts82–85. Such measurements, being com- NESST grant 80NSSC20K1506. bined with the equatorial measurements of wave characteristics, can provide the energy range of electrons trapped around loss-cone and not precipitating due to DATA AVAILABILITY the effect 100% phase trapping. This is theoretical study, and all figures are plotted using numerical solutions of equations provided with the VII. CONCLUSIONS paper. The data used for figures and findings in this study are available from the corresponding author upon reasonable request. 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