Communications in Mathematical Sciences

Volume 10 (2012)

Number 1

Special Issue on the Occasion of C. David Levermore’s Sixtieth Birthday

Euler-Poincaré formulation of hybrid plasma models

Pages: 191 – 222



Darryl D. Holm (Department of Mathematics, Imperial College, London, United Kingdom)

Cesare Tronci (Section de Mathématiques, École Polytechnique Fédérale de Lausanne, Switzerland)


Three different hybrid Vlasov-fluid systems are derived by applying reduction by symmetry to Hamilton’s variational principle. In particular, the discussion focuses on the Euler- Poincaré formulation of three major hybridMHD models, which are compared in the same framework. These are the current-coupling scheme and two different variants of the pressure-coupling scheme. The Kelvin-Noether theorem is presented explicitly for each scheme, together with the Poincaré invariants for its hot particle trajectories. Extensions of Ertel’s relation for the potential vorticity and for its gradient are also found in each case, as well as new expressions of cross helicity invariants.


variational principles, Euler-Poincaré reduction, magnetohydrodynamics, MHD, Vlasov kinetic equations

2010 Mathematics Subject Classification

35Q83, 76M30, 76M60, 76W05, 82D10

Published 14 October 2011