Hamiltonian description of Hall and sub-electron scales in collisionless plasmas with reduced fluid models

In MHD magnetic helicity has been shown to represent Gauss linking numbers of magnetic field lines by Moffatt and others; thus it is endowed with topological meaning. The noncanonical Hamiltonian formulation of extended MHD models (that take two-fluid effects into account) has been used to arrive at their common mathematical structure, which manifests itself via the existence of two generalized helicities and two Lie-dragged 2-forms. The helicity invariants play an important role in the second part of thesis dedicated to understanding the directionality of turbulent cascades.

Generally speaking, invariants (such as energy) can flow in two directions in a turbulent cascade: forward (towards small scales, leading to dissipation) and inverse (towards large scales), leading to the formation of a condensate. This directionality in extended MHD models is estimated using analytical considerations as well as tests involving 2D numerical simulations. The cascade reversal (transition) of the square magnetic vector potential is found, viz. when the forcing wavenumber exceeds the inverse electron skin depth the square magnetic vector potential starts to flow towards large wavenumbers, as opposed to the typical MHD behavior. In addition, the numerics suggest a simultaneous transition to the inverse cascade of energy in this inertial MHD regime. This is accompanied by the appearance of large scale structures in the velocity field, as opposed to the magnetic field as in the MHD case.

Final chapters of the thesis are devoted to devising the action principle for the relativistic extended MHD. First the special relativistic version is discussed, where the covariant noncanonical Poisson bracket is found. This is followed by a short recourse towards describing relativistic collisionless reconnection mediated by the electron thermal inertia (purely relativistic effect). Next, 3+1 splitting inside the Poisson bracket is performed, while only non-relativistic terms are retained. Thus one arrives at nonrelativistic extended MHD bracket with arbitrary ion to electron mass ratio. In conclusion, it is outlined how the Hamiltonian 3+1 formalism can be developed for general relativistic Hall MHD using canonical Clebsch parametrization and some comments are added on possible issues regarding the quasi-neutrality assumption in the model that is used throughout the chapter.