A geometric theory of selective decay with applications in MHD (Q2877441)

Modifications of the fluid dynamical equations are introduced in order to allow for selective decay of either the energy \(h\) or the Casimir quantity \(C\) in its Lie-Poisson formulation. (The description of \(C-s\) is briefly summarized.) The quantity selected to be described in dissipation (the energy or a concrete \(C\)) is shown to decrease in time until the modified system reaches a set of equilibrium states, satisfying \(\delta(h+ C)= 0\). The results hold for the Lie-Poisson equations in general, independently of the Lie algebra and the choice of \(C\). The equilibrium conditions are derived and it is observed that they describe the equilibrium states of the selected decay process, and some important restrictive remarks are made.NEWLINENEWLINE The physical background is considered in the form that the selected decay modifications of the equations correspond to dynamical nonlinear parametrizations of the interactions among different scales which is the outcome of the new nonlinear pathways to dissipation. As immediate applications, the process is illustrated on 2D and 3D MHD, and some selective decay forms are shown (cross helicity, magnetic helicity, energy-dissipative case (magnetic Lamb surfaces), potential magnetic intensity, etc.).