Hamiltonian and Action Principle formulations of plasma fluid models

Lingam, Manasvi
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The Hamiltonian and Action Principle (HAP) formulations of plasmas and fluids are explored in a wide variety of contexts. The principles involved in the construction of Action Principles are presented, and the reduction procedure to obtain the associated noncanonical Hamiltonian formulation is delineated. The HAP formulation is first applied to a 2D magnetohydrodynamics (MHD) model, and it is shown that one can include Finite Larmor Radius effects in a transparent manner. A simplified 2D limit of the famous Branginskii gyroviscous tensor is obtained, and the origins of a powerful tool - the gyromap - are traced to the presence of a gyroviscous term in the action. The noncanonical Hamiltonian formulation is used to extract the Casimirs of the model, and an Energy-Casimir method is used to derive the equilibria and stability; the former are shown to be generalizations of the Grad-Shafranov equation, and possess both flow and gyroviscous effects. The action principle of 2D MHD is generalized to encompass a wider class of gyroviscous fluids, and a suitable gyroviscous theory for liquid crystals is constructed. The next part of the thesis is devoted to examining several aspects of extended MHD models. It is shown that one can recover many such models from a parent action, viz. the two-fluid model. By performing systematic orderings in the action, extended MHD, Hall MHD and electron MHD are recovered. In order to obtain these models, novel techniques, such as non-local Lagrange-Euler maps which enable a transition between the two fluid frameworks, are introduced. A variant of extended MHD, dubbed inertial MHD, is studied via the HAP approach in the 2D limit. The model is endowed with the effects of electron inertia, but is shown to possess a remarkably high degree of similarity with (inertialess) ideal MHD. A reduced version of inertial MHD is shown to yield the famous Ottaviani-Porcelli model of reconnection. Similarities in the mathematical structure of several extended MHD models are explored in the Hamiltonian framework, and it is hypothesized that these features emerge via a unifying action principle. Prospects for future work, reliant on the HAP formulation, are also presented.