Show simple item record

Files in this item


Item metadata

dc.contributor.authorAllanson, Oliver Douglas
dc.contributor.authorTroscheit, Sascha
dc.contributor.authorNeukirch, Thomas
dc.identifier.citationAllanson , O D , Troscheit , S & Neukirch , T 2018 , ' On the inverse problem for Channell collisionless plasma equilibria ' , IMA Journal of Applied Mathematics , vol. 83 , no. 5 , pp. 849-873 .
dc.identifier.otherPURE: 253098531
dc.identifier.otherPURE UUID: fa563130-284d-48c0-8849-02ecb55eb43e
dc.identifier.otherORCID: /0000-0002-7597-4980/work/45744971
dc.identifier.otherScopus: 85061976476
dc.identifier.otherWOS: 000453665500004
dc.descriptionFunding: Science and Technology Facilities Council Consolidated Grant Nos. ST/K000950/1 and ST/N000609/1, the Science and Technology Facilities Council Doctoral Training Grant No. ST/K502327/1, and the Natural Environment Research Council Grant No. NE/P017274/1 (Rad-Sat) (OA). Engineering and Physical Sciences Research Council Doctoral Training Grant No. EP/K503162/1 (ST).en
dc.description.abstractVlasov–Maxwell equilibria are described by the self-consistent solutions of the time-independent Maxwell equations for the real-space dynamics of electromagnetic fields and the Vlasov equation for the phase-space dynamics of particle distribution functions (DFs) in a collisionless plasma. These two systems (macroscopic and microscopic) are coupled via the source terms in Maxwell’s equations, which are sums of velocity-space ‘moment’ integrals of the particle DF. This paper considers a particular subset of solutions of the broad plasma physics problem: ‘the inverse problem for collisionless equilibria’ (IPCE), viz. ‘given information regarding the macroscopic configuration of a collisionless plasma equilibrium, what self-consistent equilibrium DFs exist?’ We introduce the constants of motion approach to IPCE using the assumptions of a ‘modified Maxwellian’ DF, and a strictly neutral and spatially one-dimensional plasma, and this is consistent with ‘Channell’s method’ (Channell, 1976, Exact Vlasov-Maxwell equilibria with sheared magnetic fields. Phys. Fluids, 19 , 1541–1545). In such circumstances, IPCE formally reduces to the inversion of Weierstrass transformations (Bilodeau, 1962, The Weierstrass transform and Hermite polynomials. Duke Math. J., 29 , 293–308). These are the same transformations that feature in the initial value problem for the heat/diffusion equation. We discuss the various mathematical conditions that a candidate solution of IPCE must satisfy. One method that can be used to invert the Weierstrass transform is expansions in Hermite polynomials. Building on the results of Allanson et al. (2016, From one-dimensional fields to Vlasov equilibria: Theory and application of Hermite polynomials. Journal of Plasma Physics, 82 , 905820306), we establish under what circumstances a solution obtained by these means converges and allows velocity moments of all orders. Ever since the seminal work by Bernstein et al. (1957, Exact nonlinear plasma oscillations. Phys. Rev., 108 , 546–550) on ‘stationary’ electrostatic plasma waves, the necessary quality of non-negativity has been noted as a feature that any candidate solution of IPCE will not a priori satisfy. We discuss this problem in the context of Channell equilibria, for magnetized plasmas.
dc.relation.ispartofIMA Journal of Applied Mathematicsen
dc.rights© 2018, the Author(s). This work has been made available online in accordance with the publisher’s policies. This is the author created, accepted version manuscript following peer review and may differ slightly from the final published version. The final published version of this work is available at
dc.subjectKinetic physicsen
dc.subjectCollisionless plasmaen
dc.subjectHermite polynomialen
dc.subjectInverse problemen
dc.subjectMaxwell's equationsen
dc.subjectVlasov equationen
dc.subjectQA Mathematicsen
dc.subjectQC Physicsen
dc.titleOn the inverse problem for Channell collisionless plasma equilibriaen
dc.typeJournal articleen
dc.contributor.sponsorScience & Technology Facilities Councilen
dc.contributor.sponsorScience & Technology Facilities Councilen
dc.contributor.institutionUniversity of St Andrews. Applied Mathematicsen
dc.contributor.institutionUniversity of St Andrews. School of Mathematics and Statisticsen
dc.contributor.institutionUniversity of St Andrews. Pure Mathematicsen
dc.description.statusPeer revieweden

This item appears in the following Collection(s)

Show simple item record