Theory of one-dimensional Vlasov-Maxwell equilibria: with applications to collisionless current sheets and flux tubes
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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.
Thesis, PhD Doctor of Philosophy
Attribution-NonCommercial-NoDerivatives 4.0 Internationalhttp://creativecommons.org/licenses/by-nc-nd/4.0/
Description of related resourcesAn exact collisionless equilibrium for the Force-Free Harris Sheet with low plasma beta: O. Allanson, T. Neukirch, F. Wilson and S. Troscheit, Physics of Plasmas, 22, 102116 (2015)
From one-dimensional fields to Vlasov equilibria: theory and application of Hermite Polynomials: O. Allanson, T. Neukirch, S. Troscheit and F. Wilson, Journal of Plasma Physics, 82, 905820306 (2016)
Neutral and non-neutral collisionless plasma equilibria for twisted flux tubes: The Gold-Hoyle model in a background field: O. Allanson, F. Wilson and T. Neukirch, Physics of Plasmas, 23, 092106 (2016)
Exact Vlasov-Maxwell equilibria for asymmetric current sheets: O. Allanson, F. Wilson, T. Neukirch, Y.-H. Liu and J.D.B. Hodgson, Geophysical Research Letters, 44, 17, 8685-8695 (2017)
The inverse problem for collisionless plasma equilibria: O. Allanson, S. Troscheit and T. Neukirch, Invited paper for The IMA Journal of Applied Mathematics (submitted)
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