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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

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.

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

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)

The NASA Magnetospheric Multiscale mission has made in situ diffusion region and kinetic-scale resolution measurements of asymmetric magnetic reconnection for the first time, in the Earth's magnetopause. The principal theoretical tool currently used to model collisionless asymmetric reconnection is particle-in-cell simulations. Many particle-in-cell simulations of asymmetric collisionless reconnection start from an asymmetric Harris-type magnetic field but with distribution functions that are not exact equilibrium solutions of the Vlasov equation. We present new and exact equilibrium solutions of the Vlasov-Maxwell system that are self-consistent with one-dimensional asymmetric current sheets, with an asymmetric Harris-type magnetic field profile, plus a constant nonzero guide field. The distribution functions can be represented as a combination of four shifted Maxwellian distribution functions. This equilibrium describes a magnetic field configuration with more freedom than the previously known exact solution and has different bulk flow properties., Final Author version. The Open Access Geophysical Review Letters article is at http://onlinelibrary.wiley.com/doi/10.1002/2017GL074168/full

We consider the theory and application of a solution method for the inverse problem in collisionless equilibria, namely that of calculating a Vlasov-Maxwell equilibrium for a given macroscopic (fluid) equilibrium. Using Jeans' Theorem, the equilibrium distribution functions are expressed as functions of the constants of motion, in the form of a Maxwellian multiplied by an unknown function of the canonical momenta. In this case it is possible to reduce the inverse problem to inverting Weierstrass transforms, which we achieve by using expansions over Hermite polynomials. A sufficient condition on the pressure tensor is found which guarantees the convergence and the boundedness of the candidate solution, when satisfied. This condition is obtained by elementary means, and it is clear how to put it into practice. We also argue that for a given pressure tensor for which our method applies, there always exists a positive distribution function solution for a sufficiently magnetised plasma. Illustrative examples of the use of this method with both force-free and non-force-free macroscopic equilibria are presented, including the full verification of a recently derived distribution function for the force-free Harris Sheet (Allanson $\textit{et al., Phys. Plasmas}$, vol. 22 (10), 2015, 102116). In the effort to model equilibria with lower values of the plasma beta, solutions for the same macroscopic equilibrium in a new gauge are calculated, with numerical results presented for $\beta_{pl}=0.05$., Comment: Published version is Open Access in Journal of Plasma Physics, 82, 03 (2016). arXiv admin note: substantial text overlap with arXiv:1508.07925

We present a first discussion and analysis of the physical properties of a new exact collisionless equilibrium for a one-dimensional nonlinear force-free magnetic field, namely the Force-Free Harris Sheet. The solution allows any value of the plasma beta, and crucially below unity, which previous nonlinear force-free collisionless equilibria could not. The distribution function involves infinite series of Hermite Polynomials in the canonical momenta, of which the important mathematical properties of convergence and non-negativity have recently been proven. Plots of the distribution function are presented for the plasma beta modestly below unity, and we compare the shape of the distribution function in two of the velocity directions to a Maxwellian distribution., Comment: 18 pages, 7 figures (2 colour)