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 res- onant 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 sig- nificant 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 ! 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 γ! −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=!) 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=!), 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 pro- cesses 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 ! (i.e., @φ/∂s = k, @φ/∂t = −!). 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 + ) is ∼ φ_ ∼ !) 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 + ) − w sin (φ + ) scale is ∼ cp1 − γ−2=R !, R is a typical inhomo- k γ @s w @s geneity scale). This separation provides a single small _ Ωce @Uw parameter c=R! ∼ 1=kR 1. For the nonlinear wave- = + sin (φ + ) (2) γ @Ix particle interaction this parameter is about the ratio of I_x = −Uw cos (φ + ) a wave amplitude Bw and ambient magnetic field mag- nitude B0, i.e. a wave force ∼ kBw can compete with a where φ_ = ks_ − !. 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 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 ! 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 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, 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 + = 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 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.