The nonlinear evolution of whistler-mode chorus:modulation instability as the source of tones

We review the modulation stability of parallel-propagating/field-aligned whistler-mode chorus (WMC) waves propagating in a warm plasma from a formal perspective with a focus on wave–particle interactions via ponderomotive forces. The modulation instability criteria are characterised by the group velocity dispersion, dcg/dk, for whistler-mode waves and a condition on the ratio between the group velocity cg and the electron sound speed cs,e. We also demonstrate that in order to investigate the spatiotemporal evolution of the envelope and the formation of packets (according to this mechanism), one necessarily needs to account for the motion of ions within the system, leading to an ionic influence on the modulation instability threshold determined by the ion fraction of the plasma. Finally, we demonstrate that chirping may be captured when higher-order effects are included within the spatiotemporal evolution of the amplitude. This yields not only an explicit expression for the sweep rate but also identifies a possible origin for the power band gap that occurs at half the electron gyrofrequency. Numerical validation demonstrates that the interaction between wave packets is a source for the emergence of tones observed within mission data, and such interactions may be a major source of the electron energisation which WMC are responsible for

Theory of one-dimensional Vlasov-Maxwell equilibria: with applications to collisionless current sheets and flux tubes

Vlasov-Maxwell equilibria are characterised by the self-consistent descriptions of the steady-states of collisionless plasmas in particle phase-space, and balanced macroscopic forces. We study the theory of Vlasov-Maxwell equilibria in one spatial dimension, as well as its application to current sheet and flux tube models. The ‘inverse problem’ is that of determining a Vlasov-Maxwell equilibrium distribution function self-consistent with a given magnetic field. We develop the theory of inversion using expansions in Hermite polynomial functions of the canonical momenta. Sufficient conditions for the convergence of a Hermite expansion are found, given a pressure tensor. For large classes of DFs, we prove that non-negativity of the distribution function is contingent on the magnetisation of the plasma, and make conjectures for all classes. The inverse problem is considered for nonlinear ‘force-free Harris sheets’. By applying the Hermite method, we construct new models that can describe sub-unity values of the plasma beta (βpl) for the first time. Whilst analytical convergence is proven for all βpl, numerical convergence is attained for βpl = 0.85, and then βpl = 0.05 after a ‘re-gauging’ process. We consider the properties that a pressure tensor must satisfy to be consistent with ‘asymmetric Harris sheets’, and construct new examples. It is possible to analytically solve the inverse problem in some cases, but others must be tackled numerically. We present new exact Vlasov-Maxwell equilibria for asymmetric current sheets, which can be written as a sum of shifted Maxwellian distributions. This is ideal for implementations in particle-in-cell simulations. We study the correspondence between the microscopic and macroscopic descriptions of equilibrium in cylindrical geometry, and then attempt to find Vlasov-Maxwell equilibria for the nonlinear force-free ‘Gold-Hoyle’ model. However, it is necessary to include a background field, which can be arbitrarily weak if desired. The equilibrium can be electrically non-neutral, depending on the bulk flows

Weak turbulence and quasilinear diffusion for relativistic wave-particle interactions via a Markov approach

Funding: OA acknowledges financial support from the University of Exeter and from the United Kingdom Natural Environment Research Council (NERC) Independent Research Fellowship NE/V013963/1. OA and CW acknowledge financial support from the NERC Highlight Topic Grant NE/P017274/1 (Rad-Sat), and from United Kingdom Science and Technology Facilities Council (STFC) via Consolidated Grant ST/W000369/1. TE acknowledges financial support from an Early Career Fellowship, split jointly by the Leverhulme Trust (ECF2019-155) and the University of Leicester in the first instance (2019-21), but presently the University of Glasgow (2021-). TN acknowledges financial support from the STFC via Consolidated Grant ST/S000402/1. The University of Exeter cover the Open Access Publication Fee via a UKRI block grant.We derive weak turbulence and quasilinear models for relativistic charged particle dynamics in pitch-angle and energy space, due to interactions with electromagnetic waves propagating (anti-)parallel to a uniform background magnetic field. We use a Markovian approach that starts from the consideration of single particle motion in a prescribed electromagnetic field. This Markovian approach has a number of benefits, including: (i) the evident self-consistent relationship between a more general weak turbulence theory and the standard resonant diffusion quasilinear theory (as is commonly used in e.g. radiation belt and solar wind modelling); (ii) the general nature of the Fokker-Planck equation that can be derived without any prior assumptions regarding its form; (iii) the clear dependence of the form of the Fokker-Planck equation and the transport coefficients on given specific timescales. The quasilinear diffusion coefficients that we derive are not new in and of themselves, but this concise derivation and discussion of the weak turbulence and quasilinear theories using the Markovian framework is physically very instructive. The results presented herein form fundamental groundwork for future studies that consider phenomena for which some of the assumptions made in this manuscript may be relaxed.Publisher PDFPeer reviewe

Radial Transport in the Earth’s Radiation Belts: Linear, Quasi-linear, and Higher-order Processes

Observational studies of the Earth’s radiation belts indicate that Alfvénic fluctuations in the frequency range of 2–25 mHz accelerate electrons to relativistic energies. For decades, statistical models of radiation belts have quantified the impact of Alfvénic waves in terms of quasi-linear diffusion. However, quasi-linear models are inadequate to quantify Alfvénic radial transport occurring on timescales comparable to the azimuthal drift period of 0.1–10 MeV electrons. With recent advances in observational methodologies offering coverage of the Earth’s radiation belts on fast timescales, a theoretical framework that distinguishes between fast and diffusive radial transport can be tested for the first time in situ. In this report, we present a drift-kinetic description of radial transport for planetary radiation belts. We characterize fast linear processes and determine the conditions under which higher-order effects become dynamically significant. In the linear regime, wave–particle interactions are categorized in terms of resonant and nonresonant responses. We demonstrate that the phenomenon of zebra stripes is nonresonant and can originate from injection events in the inner radiation belts. We derive a radial diffusion coefficient for a field model that satisfies Faraday’s law and that contains two terms: one scaling as L ^10 independent of the azimuthal number m , and a second scaling as m ^2 L ^6 . In the higher-order regime, azimuthally symmetric waves with properties consistent with in situ measurements can energize 10–100 keV electrons in less than a drift period. This process provides new evidence that acceleration by Alfvénic waves in radiation belts cannot be fully contained within diffusive models

Zooming through the MIST

Mathew Owens, Oliver Allanson and Megan Maunder report on the 51st annual Magnetosphere, Ionosphere, and Solar–Terrestrial (MIST) meetin

Force-free collisionless current sheets : a systematic method for adding asymmetries

Funding: LN acknowledges financial support by the University of St Andrews, and TN acknowledges support by the United Kingdom’s Science and Research Council (STFC) via Consolidated Grant ST/W001195/1.Recent observations have shown that current sheets in the solar wind can have systematic spatial asymmetries in their particle density and temperature while the pressure remains constant. For one-dimensional current sheets the magnetic field has to be force-free, but known self-consistent equilibrium particle distribution functions for force-free current sheets usually lead to spatial density and temperature structures that are either constant or vary symmetrically in space. Using a specific ad hoc example, Neukirch et al. (2020) showed that it is possible to introduce spatial asymmetries into the density and temperature profiles without changing the magnetic field structure. In this contribution, a systematic method will be presented that can in principle be used to construct particle distribution functions leading to density and temperature asymmetries of the form given in Neukirch et al. (2020). We will show how it explains why the known examples work and present some results of our attempts to find new examples.Publisher PDFPublisher PDFNon peer reviewe

Particle-in-cell experiments examine electron diffusion by whistler-mode waves: 1. Benchmarking with a cold plasma

Using a particle-in-cell code, we study the diffusive response of electrons due to wave particle interactions with whistler-mode waves. The relatively simple configuration of field-aligned waves in a cold plasma is used in order to benchmark our novel method, and to compare with previous works that used a different modelling technique. In this boundary-value problem, incoherent whistler-mode waves are excited at the domain boundary, and then propagate through the ambient plasma. Electron diffusion characteristics are directly extracted from particle data across all available energy and pitch-angle space. The ‘nature’ of the diffusive response is itself a function of energy and pitch-angle, such that the rate of diffusion is not always constant in time. However, after an initial transient phase, the rate of diffusion tends to a constant, in a manner that is consistent with the assumptions of quasilinear diffusion theory. This work establishes a framework for future investigations on the nature of diffusion due to whistler-mode wave-particle interactions, using particle-in-cell numerical codes with driven waves as boundary value problems