A geometric theory of selective decay with applications in MHD

Abstract : Modifications of the equations of ideal fluid dynamics with advected quantities are introduced that allow selective decay of either the energy h or the Casimir quantities C in the Lie-Poisson (LP) formulation. The dissipated quantity (energy or Casimir, respectively) is shown to decrease in time until the modified system reaches an equilibrium state consistent with ideal energy-Casimir equilibria, namely d(h + C) = 0. The result holds for LP equations in general, independently of the Lie algebra and the choice of Casimir. This selective decay process is illustrated with a number of examples in 2D and 3D magnetohydrodynamics. © 2014 IOP Publishing Ltd & London Mathematical Society.