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Ultra-Low Frequency (ULF) waves are critical drivers of particle acceleration and loss in the Earth's magnetosphere. While statistical models of ULF-induced radial transport have traditionally assumed that the waves are uniformly distributed across magnetic local time (MLT), decades of observational evidence show significant MLT localization of ULF waves in the Earth's magnetosphere. This study presents, for the first time, a quasi-linear radial diffusion coefficient accounting for localized ULF waves. We demonstrate that even though quasi-linear radial diffusion is averaged over drift orbits, MLT localization significantly alters the efficiency of particle transport. Our results reveal that when ULF waves cover more than 30\% of the MLT, the radial diffusion efficiency is comparable to that of uniform wave distributions. However, when ULF waves are confined within 10\% of the drift orbit, the diffusion coefficient is enhanced by 10 to 25\%, indicating that narrowly localized ULF waves are efficient drivers of radial transport., Comment: 16 pages, 4 figures. Currently under review. Feedback and suggestions are welcome

We review the modulation stability of parallel propagating/field aligned Whistler Mode Chorus waves propagating in a warm plasma from a formal perspective with a focus on wave-particle interactions. The modulation instability criteria is characterised by a curvature of the dispersion relation for Whistler mode waves and a condition on the ratio between the group velocity $c_g$ and the electron sound speed $c_{s,e}$. We also demonstrate the in order to investigate the spatiotemporal evolution of the envelope and the formation of packets, 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 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 Whistler-Mode chorus are responsible for.

Observational studies of the Earth's radiation belts indicate that Alfv\'enic fluctuations in the frequency range of 2-25 mHz accelerate magnetically trapped electrons to relativistic energies. For decades, statistical models of the Earth's radiation belts have quantified the impact of Alfv\'enic waves in terms of quasi-linear diffusive models. However, quasi-linear models are inadequate to quantify Alfv\'enic radial transport occurring on timescales comparable to the azimuthal drift period of $0.1- 10$ MeV electrons. With recent advances in observational methodologies offering spatial and temporal coverage of the Earth's radiation belts on fast timescales, a theoretical framework that distinguishes between fast and diffusive radial transport can also be tested for the first time with in situ measurements. In this report, we present a drift kinetic description of radial transport for planetary radiation belts. We characterize linear processes that are too fast to be modelled by quasi-linear models and determine the conditions under which nonlinearities become dynamically significant. In the linear regime, wave-particle interactions are categorized in terms of resonant and non-resonant responses. We demonstrate that the phenomenon of zebra stripes is non-resonant and can originate from the injection of particles 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 one scaling as $m^2 L^6$. In the nonlinear regime, we show that azimuthally symmetric waves with properties consistent with in situ measurements can energize 10-100 keV electrons in less than a drift period. This coherent process provides new evidence that acceleration by Alfv\'enic waves in radiation belts cannot be fully contained within diffusive models., Comment: 54 pages and 14 figures

Jupiter has the most energetic and complex radiation belts in our solar system. Their hazardous environment is the reason why so many spacecraft avoid rather than investigate them, and explains how they have kept many of their secrets so well hidden, despite having been studied for decades. In this White Paper we argue why these secrets are worth unveiling. Jupiter's radiation belts and the vast magnetosphere that encloses them constitute an unprecedented physical laboratory, suitable for both interdisciplinary and novel scientific investigations: from studying fundamental high energy plasma physics processes which operate throughout the universe, such as adiabatic charged particle acceleration and nonlinear wave-particle interactions; to exploiting the astrobiological consequences of energetic particle radiation. The in-situ exploration of the uninviting environment of Jupiter's radiation belts present us with many challenges in mission design, science planning, instrumentation and technology development. We address these challenges by reviewing the different options that exist for direct and indirect observation of this unique system. We stress the need for new instruments, the value of synergistic Earth and Jupiter-based remote sensing and in-situ investigations, and the vital importance of multi-spacecraft, in-situ measurements. While simultaneous, multi-point in-situ observations have long become the standard for exploring electromagnetic interactions in the inner solar system, they have never taken place at Jupiter or any strongly magnetized planet besides Earth. We conclude that a dedicated multi-spacecraft mission to Jupiter's radiation belts is an essential and obvious way forward and deserves to be given a high priority in ESA's Voyage 2050 programme., Comment: 28 pages, 3 Tables, 11 Figures

So far, only one distribution function giving rise to a collisionless nonlinear force-free current sheet equilibrium allowing for a plasma beta less than one is known (Allanson et al., 2015; 2016). This distribution function can only be expressed as an infinite series of Hermite functions with very slow convergence and this makes its practical use cumbersome. It is the purpose of this paper to present a general method that allows us to find distribution functions consisting of a finite number of terms (therefore easier to use in practice), but which still allow for current sheet equilibria that can, in principle, have an arbitrarily low plasma beta. The method involves using known solutions and transforming them into new solutions using transformations based on taking integer powers ($N$) of one component of the pressure tensor. The plasma beta of the current sheet corresponding to the transformed distribution functions can then, in principle, have values as low as $1/N$. We present the general form of the distribution functions for arbitrary $N$ and then, as a specific example, discuss the case for $N=2$ in detail.

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 polynomials 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 $(\beta_{pl})$ for the first time. Whilst analytical convergence is proven for all $\beta_{pl}$, numerical convergence is attained for $\beta_{pl}= 0.85$, and then $\beta_{pl} = 0.05$ 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, 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, 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., Comment: PhD thesis (264 pages). PhD research was conducted in the Solar & Magnetospheric Theory Group, in the School of Mathematics & Statistics, at The University of St Andrews. The research project was supervised by Professor Thomas Neukirch, and successfully defended in August 2017 on examination by Dr Andrew Wright (internal, St Andrews) and Professor Alexander Schekochihin (external, Oxford)