Velocity-space Signatures of Resonant Energy Transfer between Whistler Waves and Electrons in the Earth’s Magnetosheath

Wave–particle interactions play a crucial role in transferring energy between electromagnetic fields and charged particles in space and astrophysical plasmas. Despite the prevalence of different electromagnetic waves in space, there is still a lack of understanding of fundamental aspects of wave–particle interactions, particularly in terms of energy flow and velocity-space characteristics. In this study, we combine a novel quasilinear model with observations from the Magnetospheric Multiscale mission to reveal the signatures of resonant interactions between electrons and whistler waves in magnetic holes, which are coherent structures often found in the Earth’s magnetosheath. We investigate the energy transfer rates and velocity-space characteristics associated with Landau and cyclotron resonances between electrons and slightly oblique propagating whistler waves. In the case of our observed magnetic hole, the loss of electron kinetic energy primarily contributes to the growth of whistler waves through the n = −1 cyclotron resonance, where n is the order of the resonance expansion in linear Vlasov–Maxwell theory. The excitation of whistler waves leads to a reduction of the temperature anisotropy and parallel heating of the electrons. Our study offers a new and self-consistent understanding of resonant energy transfer in turbulent plasmas.


INTRODUCTION
Electromagnetic fluctuations in space and astrophysical plasmas expand across an extensive range of spatial and temporal scales (Tu & Marsch 1995;Bruno & Carbone 2013;Alexandrova et al. 2013;Verscharen et al. 2019b;Sahraoui et al. 2020).The interactions between charged particles and electromagnetic fluctuations play crucial roles for the energy conversion and dissipation in astrophysical plasma environments such as the solar wind, planetary magnetospheres, and the interstellar medium (Marsch 2006;Schekochihin et al. 2009).Resonant wave-particle interactions include Landau-resonant and cyclotron-resonant processes.
In the Earth's magnetosheath, enhanced electromagnetic fluctuations at kinetic scales such as whistler waves are sometimes localized near coherent structures like current sheets, magnetic islands, and magnetic holes (Zhang et al. 1998;Tsurutani et al. 2011;Karimabadi et al. 2014;Ahmadi et al. 2018;Breuillard et al. 2018;Kitamura et al. 2020;Behar et al. 2020;Li et al. 2021).These whistler waves are frequently observed as right-hand polarized electromagnetic waves with a small propagation angle with respect to the background magnetic field.Micro-instabilities driven by unstable butterfly or beam-like VDFs are key candidates to explain the occurrence of these waves (Zhima et al. 2015;Ahmadi et al. 2018;Ren et al. 2019;Huang et al. 2020;Zhang et al. 2021).Although the direct observation of the energy transfer via cyclotron resonance is sometimes possible through data from the Magnetospheric Multiscale (MMS) mission (Kitamura et al. 2022), the understanding of the associated velocity-space signatures and the timedependent properties of the resonant energy transfer between electrons and whistler waves is still lacking.
In this letter, we focus on an interval previously studied by Jiang et al. (2022) and use a novel quasilinear model to numerically solve the quasilinear impact of wave-particle interactions on the temporal and energy evolution of the plasma.We discuss the quantified signatures of energy transfer between whistler waves and electrons under the action of three different wave-particle resonance mechanisms.Finally, we provide suggestions for direct in-situ observations of such resonant wave-particle interactions.

QUASILINEAR EVOLUTION OF THE DISTRIBUTION FUNCTION
Quasilinear theory describes the collective and slow (compared to the wave frequency) response of the VDF to fluctuating electromagnetic fields in resonant wave-particle interactions (Shapiro & Shevchenko 1962;Kennel & Engelmann 1966;Rowlands et al. 1966).In quasilinear theory, the time evolution of electron VDF via resonant interactions between electromagnetic fields and electrons (denoted by subscript e) follows a diffusion in velocity space: where f e is the electron VDF,

Ĝ[k
q e is the charge of an electron, m e is the mass of an electron, k ∥ is the wavevector component parallel to the background magnetic field ⃗ B 0 so that the full wavevector is decomposed as √ 2 (see also Verscharen et al. 2019a).

Numerical Model for the Quasilinear Evolution
To investigate the time-dependent nature of quasilinear diffusion, we use a numerical model to solve the time evolution of electron VDFs according to Eq. (1) (Jeong et al. 2020).Using a Crank-Nicolson scheme, the Jeong et al. (2020) model is a novel and generalized method to solve the time evolution of 2-dimensional VDFs under the action of a dominant resonant wave-particle resonance.
In this model, we define approximate window functions to reflect the distribution of wave energy over k ∥ (see Jeong et al. 2020) as where is the nth order resonance velocity, v g0 is the group velocity, k ∥0 is the central parallel wave number, ω k0 is the central frequency, σ ∥0 is the half width of the window function, v Ae = B 0 / √ 4πn e m e is the electron Alfvén speed, and n e is the electron number density.Eq. 4 determines the region in v ∥ space in which the quasilinear diffusion through the nth order resonance with unstable whistler waves is effective.
The width of the window function is determined by the unstable whistler-wave spectrum calculated with the Arbitrary Linear Plasma Solver (ALPS; Verscharen et al. 2018;Klein et al. 2023a).To determine the window functions, we implement a non-Maxwellian VDF model into ALPS to evaluate the stability of whistler waves.The VDF model is based on realistic electron VDF data from MMS1 on 2017 January 25 from 00:25:44.38 to 00:26:44.80UT (more details see Jiang et al. 2022).Figure 1b shows isocontours of the initial electron VDF.ALPS predicts an unstable spectrum of whistler waves, i.e., γ k > 0 in the range 0.5 < k ∥ d e < 0.8, where d e denotes the electron inertial length.Therefore, we set σ ∥0 = 0.08d e , k ∥0 d e = 0.64 and v g0 = 0.07v Ae in our numerical model.The unstable whistler waves have a propagation angle slightly oblique to the background magnetic field (∼ 10 • ) and the real part of the frequency at maximum growth is ω k0 = 0.26Ω e , which is in agreement with the observed wave properties (Jiang et al. 2022).We determine the magnetic-field amplitude of whistler waves as 0.0018 nT from direct MMS observations (Torbert et al. 2016).In our model, the electron VDF is discretized into a 120 × 240 grid on the (v ∥ , v ⊥ ) plane, where

Trajectories of Quasilinear Diffusion
In Figure 1a, we show the relevant W n , which are defined to have their maximum at the resonance velocities according to Eq. ( 5).We determine the directions of the diffusive flux of particles according to Eq. ( 1) based on the local gradients of the velocity distribution function.The diffusive flux of particles in velocity space is locally tangent to concentric elliptical/hyperbolic curves around the point given by Jeong et al. (2020): Figure 1b shows the trajectories according to Eq. ( 6  Particles at different pitch angles and different energy can experience opposite effects given a specific fine structure of the VDF in velocity space.For example, around v ⊥ ≈ 0.7v Ae along the n = 0 resonance, f e has a negative gradient in the v ∥ direction, suggesting that the direction of the diffusive particle flux points into the negative v ∥ direction.The corresponding electrons lose kinetic energy, which thus is transferred into the wave energy.However, electrons at v ⊥ ≈ 1.5v Ae diffuse in the opposite direction, thus absorbing wave energy.For n = 1, the diffusive particle flux is clockwise at v ∥ ≈ v +1 ∥res , leading to an increase in the kinetic energy of these electrons.This diffusion contributes to the damping of the whistler waves.For n = −1, the diffusive particle flux is counter-clockwise at v ∥ ≲ v −1 ∥res , leading to a decrease in the kinetic energy of these electrons.This diffusion contributes to the growth of the whistler waves.Electrons at v ∥ ≳ v −1 ∥res diffuse towards larger kinetic energy and thus contribute to wave damping.The net gain or loss of energy of all electrons defines whether the corresponding resonant mode undergoes growth or damping.

Resonant Energy Transfer in Velocity Space
Upon obtaining VDFs at different quasilinear evolution times, we evaluate the relative contributions of the three resonances to the energy conversion using the method presented by Howes et al. (2017).We first calculate the velocity-space energy density and the two-dimensional energy transfer rate By performing partial integration, we obtain the one-dimensional energy transfer rates both in the parallel direction and in the perpendicular direction Moreover, we obtain the net energy transfer rate by a second integration over the remaining direction: To compare contributions from different resonances, we define the energy transfer rates for different orders n as where w n is the half-width of the effective v ∥ range for the resonant interaction of nth order.Both one-dimensional energy conversion rates show a strong dependence on v ∥ and v ⊥ .According to Figure 2a, C(v ⊥ , v ∥ ) shows an alternating pattern in velocity space.For n = 0, C(v ⊥ , v ∥ ) shows the expected bipolar double-band signature along the v ∥ direction, which when integrated contributes to the damping of the whistler waves.However, this bipolar signature reverses at v ⊥ ≈ v Ae due to reversed velocity gradients of the electron VDFs along v ∥ (see also Figure 2c).The reversed bipolar part of contributes to the growth of the whistler waves.As shown by the blue circles in Figure 2d, the n = 0 resonance overall makes a damping contribution to the whistler-wave evolution.
shows a velocity-space pattern consistent with the diffusive paths shown in Figure 1b.At v ∥ ≈ v +1 ∥res , the diffusive flux is directed towards larger v 2 , suggesting an increase in kinetic energy of the resonant electrons.
While |C +1 | ≪ |C 0 |, the n = 1 resonance also contributes to the damping of the whistler instability.
The n = −1 resonance produces a triple-band signature in C(v ⊥ , v ∥ ).This signature indicates that the electrons with v ∥ ≈ v −1 ∥res diffuse along both possible directions shown in Figure 1b and Figure 2a.Inspecting C ∥ (v ∥ ) in Figure 2c suggests that the electrons with v ∥ ≈ v −1 ∥res overall lose kinetic energy, contributing to the growth of whistler waves.The loss of kinetic energy through the n = −1 resonance is greater than the combined gain of kinetic energy through the n = 0 and n = 1 resonances.According to Figure 2d, the contribution of the n = −1 resonance is the dominant source for the growth of whistler instability.

Quasilinear Saturation and Stabilization
During the time evolution according to Eq. ( 1), the magnitudes of C ⊥ (v ⊥ ) and C ∥ (v ∥ ) decrease.This result suggests that the diffusion caused by whistler waves slows down, which is a result of the decrease in the local velocity gradients of the VDFs.This secular effect is the quasilinear saturation mechanism of the whistler-wave instability under consideration.By integrating over all velocities, we obtain the net energy transfer rates at t|Ω e | = 2 as C −1 ≈ −1.87 × 10 −12 kg m −1 s −3 , C 0 ≈ 1.65 × 10 −13 kg m −1 s −3 , and C +1 ≈ 2.45 × 10 −14 kg m −1 s −3 .We obtain a net energy transfer rate of C net ≈ 1.68 × 10 −12 kg m −1 s −3 .
As shown in Figure 2d, the net energy transfer rates gradually decrease with time, indicating quasilinear saturation of the system.Therefore, we conclude that the n = −1 cyclotron resonance drives the growth of the whistler waves, while the n = 0 Landau resonance and the n = 1 cyclotron resonance lower their growth rate.as the overall velocity gradients in the resonance regions decrease.The increase in β ∥e with time is directly related to the increase in T ∥e , which is the result of the quasilinear diffusion in velocity space.The velocityspace morphology of the electron beam configuration at energies between 200 and 300 eV is noticeably weakened and evolves toward a more isotropic distribution (see Figure 3b).The major contribution to the energy transfer is from regions with high phase space densities.While the contribution of the n = 1 resonance to the total energy transfer is small, its role in reshaping the distribution in regions of velocity space with small phase space densities is still important.

Virtual Observations of Velocity-space Signatures
To guide future in-situ observations of similar signatures, we interpolate our result from Figure 2a onto energy-angle grids with a logarithmic energy table of a virtual spacecraft.As an example, we demonstrate our results for two sets of energy-angle resolutions (∆θ = 11.25 • , ∆E/E = 13%; and ∆θ = 4 • , ∆E/E = 14%), which are shown in Figure 4.For the virtual instrument, we define the pitch-angle as θ and the energy as E. We assume that the background magnetic field direction is aligned with the axis when best pitch-angle resolution is achieved.The chosen energy-angle resolutions are selected according to typical values for the electrostatic analyzer onboard Solar Orbiter Electron Analyzer System (Figure 4a) (Owen et al. 2020) and MMS Fast Plasma Investigation (Figure 4b) (Pollock et al. 2016).
As ∆E/E and ∆θ decrease, the signatures of the three resonances in velocity space become more pronounced and thus easier to identify.With decreasing pitch-angle resolution, the morphological features of the two-dimensional energy transfer rate become increasingly difficult to distinguish.Nevertheless, the range of wave-particle resonances are variable depending on the wave parameters.Furthermore, our results reveal significant dependencies of quasilinear diffusion on the local velocity-space structures of the VDFs, which gives rise to complicated patterns of energy transfer in velocity space.
Our results present complex velocity-space signatures of resonant energy transfer between unstable whistler waves and electrons.We interpolate these signatures into finite phase-space bins similar to the operating principle of recent spacecraft instrumentation such as those onboard MMS (Pollock et al. 2016) and Solar Orbiter (Owen et al. 2020).The angular and energy resolution of the two instruments is sufficient for direct measurement of resonant energy transfer in velocity space.However, a significant under-sampling issue would affect the measurement if the time scale of resonant diffusion is much smaller than the time resolution of the instrument (Wilson et al. 2022;Verniero et al. 2021;Horvath et al. 2022).We also acknowledge that the used angle/energy resolutions in Section 4.3 are simplified to some extent.The resolution of pitch angles depends on the orientation of the magnetic field direction with respect to the the instrument frame due to the conversion of instrument-frame angles (azimuth/elevation) into pitch angles.In the case of MMS, for example, the pitch-angle resolution varies between 4 • and 11.25 • depending on the orienta-tion of the magnetic field.The direct measurement of fast diffusion created by high-frequency whistler waves is still a major challenge that can potentially benefit from novel instrument concepts that focus on electron-scale wave-particle interactions.Our method, applied to high-resolution particle data from present and future missions such as Solar Orbiter, MMS, and HelioSwarm (Klein et al. 2023b), is a helpful tool to investigate wave-particle interactions and their impact on VDFs both for electrons and ions. The the real part of the wave frequency, n is an integer (n = 0 represents the Landau resonance and n ̸ = 0 represents cyclotron resonances), J n is the Bessel function, ρ e = k ⊥ v ⊥ /Ω e , Ω e = q e B 0 /(m e c) is the electron cyclotron frequency, and c is the speed of light.In quasilinear theory, it is assumed that |γ k | ≪ |ω k |, where γ k is the imaginary part of the wave frequency.We define the Fourier transformation of the electric field as Figure 1b shows the trajectories according to Eq. (6) using colored curves with arrows for n = 0 (blue), n = 1 (green), and n = −1 (red).The arrows represent the directions of diffusive fluxes, which are always directed from larger values of f e towards smaller values of f e .Variations in the local gradients cause the direction of the diffusive flux for n = 0 to change at different v ⊥ as shown by the alternating blue arrows.

Figure 1 .
Figure 1.Quasilinear diffusion in velocity space and window functions for the unstable wave energy spectrum.(a) Window functions W n as a function of v ∥ for n = 0 (blue), n = 1 (green), and n = −1 (red) according to Eq. 4. (b) Isocontours of our electron VDF fit in velocity space.The black (blue) vertical dashed line represents the group (phase) velocity of the whistler waves.The green and red vertical dashed lines represent v +1 ∥res and v −1 ∥res .The colored arrows show the local direction of the flux of diffusing particles (blue: n = 0, green: n = 1, and red: n = −1) according to Eq. 6 and the local gradient of the VDF.
Velocity-space Signature of Energy Transfer

Figure 2 .
Figure 2. Energy transfer rates of resonant electron-whistler interactions predicted by our numerical model.(a) C(v ∥ , v ⊥ ) at t|Ω e | = 2 as a function of v ∥ and v ⊥ .The background isocontours show the initial electron VDF.(b) C ⊥ (v ⊥ ) as a function of v ⊥ .(c) C ∥ (v ∥ ) as a function of v ∥ .(d) Time plots of the total energy transfer rate (black crosses) and the contributions from n = −1 (red circles), n = 1 (green circles), and n = 0 (blue circles).Blue curves in (b) and (c) represent the energy transfer rates at t|Ω e | = 2, and orange curves at t|Ω e | = 8.

Figure
Figure 2a shows C(v ⊥ , v ∥ ) in the (v ⊥ , v ∥ )-plane.The isocontours represent the initial VDF, while the color represents the magnitude of C(v ⊥ , v ∥ ) at time t|Ω e | = 2.We show C ⊥ (v ⊥ ) in Figure 2b and C ∥ (v ∥ ) in

Figure 3 .
Figure 3. Joint time evolution of ∆A e and ∆β ∥e .(a) Evolution of ∆A e and ∆β ∥e at different times.(b) Pitch-angle distributions for 200 eV (red) and 300 eV (blue) electrons.Solid and dotted curves represent distributions at t|Ω e | = 0 and t|Ω e | = 30.

Figure
Figure 3a presents the time evolution of ∆A e and ∆β ∥e , which are the differences between A e and β ∥e and their initial values.A e = T ⊥e /T ∥e and β ∥e are calculated as the moments of the VDFs in our quasilinear model.From t|Ω e | = 0 to t|Ω e | = 30, A e decreases and β e∥ increases as a result of the action of the discussed resonant wave-particle interactions.The parametric variation of A e and β ∥e gradually decreases
5. DISCUSSION AND CONCLUSIONSBased on our MMS observations of the electron VDF in a magnetic hole, we combine linear Vlasov-Maxwell theory and a quasilinear numerical model to investigate the properties of wave-particle interactions between electrons and whistler waves.Our method reveals the velocity-space signatures of energy transfer in these resonant interactions.We quantify the relative contributions of different resonances (n = 0, n = −1, and n = 1) to the energy transfer rate in velocity space.The energy conversion rate for the n = −1 cyclotron resonance is the dominant contribution to the whistler wave instability, followed by the damping contributions from the n = 0 Landau resonance and the n = +1 cyclotron resonance.The net energy transfer from the three resonances is C net ≈ 1.30 × 10 −12 kg m −1 s −3 .The net energy transfer from electron kinetic energy to whistler-wave energy leads to the growth of the observed electron-whistler wave instability.
Arbitrary Linear Plasma Solver (ALPS) code is publicly available at https://github.com/danielver02/ALPS, and the ALPS documentation can be found at http://alps.space.The ALPS project received support from UCL's Advanced Research Computing Centre through the Open Source Software Sustainability Funding scheme.This research was supported by the International Space Science Institute (ISSI) in Bern, through ISSI International Team project #563 (Ion Kinetic Instabilities in the Solar Wind in Light of Parker Solar Probe and Solar Orbiter Observations) led by L. Ofman and L. Jian.This work is supported by NNSFC grants (42022032, 42374198, 41874203, 42188101), project of Civil Aerospace "13th Five Year Plan" Preliminary Research in Space Science (D020301).D.V. is supported by the STFC Ernest Rutherford Fellowship ST/P003826/1 and STFC Consolidated Grants ST/S000240/1 and ST/W001004/1.H.L. is supported by the International Partnership Program of CAS (Grant No. 183311KYSB20200017) and in part by the Specialized Research Fund for State Key Laboratories of China.K.G.K. is supported by grant DE-SC0020132 and NASA ECIP 80NSSC19K0912.