Concomitant Hamiltonian and topological structures of extended magnetohydrodynamics

•Common Hamiltonian structure of the extended MHD models presented.•The generalized helicities of extended MHD shown to be topological invariants analogous to fluid/magnetic helicity.•Generalized helicities can be studied through powerful topological and knot-theoretic methods such as the Jones polynomial.•Each extended MHD model shown to possess two Lie-dragged 2-forms, which are interpreted as the generalized vorticity fluxes. The paper describes the unique geometric properties of ideal magnetohydrodynamics (MHD), and demonstrates how such features are inherited by extended MHD, viz. models that incorporate two-fluid effects (the Hall term and electron inertia). The generalized helicities, and other geometric expressions for these models are presented in a topological context, emphasizing their universal facets. Some of the results presented include: the generalized Kelvin circulation theorems; the existence of two Lie-dragged 2-forms; and two concomitant helicities that can be studied via the Jones polynomial, which is widely utilized in Chern–Simons theory. The ensuing commonality is traced to the existence of an underlying Hamiltonian structure for all the extended MHD models, exemplified by the presence of a unique noncanonical Poisson bracket, and its associated energy.