Applications of Hamiltonian theory to plasma models
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Three applications of Hamiltonian Methods in Plasma Physics are presented. The first application is the development of a new, five-field, Hamiltonian gyrofluid model. It is comprised by evolution equations for the ion density, pressure and parallel temperature and electron density and pressure. It contains curvature and compressibility effects. The model is shown to satisfy a conserved energy and a Lie-Poisson bracket for it is given. Casimir invariants are calculated and through them, the normal fields of the system are recovered. Later, the model is linearized and shown to possess modes that are identified with the slab ITG, toroidal ITG and KBM modes. Both an electrostatic and an electromagnetic study are performed. Growth rates and critical parameters for instability are computed and compared to their fluid and kinetic counterparts. The accuracy of the model is shown to be between the fluid and the kinetic results, as was expected. Dissipation is added to the ideal system via the use of non-local terms that mimic Landau damping. The modes of the system are shown to undergo Krein bifurcations and their behavior once dissipation is turned on, strongly suggests that they are negative energy modes. A connection between the marginal stability condition of the ITG mode at high k┴ and the (missing) equation of perpendicular pressure is conjectured opening an interesting possibility for future research. The second application is a method for the derivation of reduced fluid models through the use of an action principle. The importance of the method lies in the fact that since all approximations are made directly at the level of the action, the models that result from the action minimization are guaranteed to retain the Hamiltonian character of their parent-model. The two-fluid action is given in Lagrangian variables and the two-fluid equations of motion are recovered by it's minimization. The Eulerian (field) equations of motion are retrieved through the Lagrange-to-Euler (L-E) map. New, single-fluid variables are defined but instead of being implemented at the level of the equations of motion, they are implemented directly in the action. The action is subjected to approximations. Different approximations lead to different models with the models of Lust, Extended MHD, Hall MHD and electron MHD being retrieved. The passing from Lagrangian to Eulerian variables in the single-fluid description requires a non-trivial modification of the E-L map. A note about the importance of quasineutrality in single-fluid models and its ramifications in the Lagrangian framework is given. Several invariants of the models are calculated via Noethers' Theorem. The third application concerns the imposition of constraints in Hamiltonian systems. Two worked examples of the method of Dirac are presented. The first one is on an electrostatic model which has the Hasegawa-Mima and RMHD as distinct limits. The constraint that leads to the Hasegawa-Mima is investigated. The calculations are demonstrated in detail and the reduced system is produced. A brief discussion of the dispersion relation of the reduced system concludes the first example. The second example is the imposition of quasineutrality and divergence-free current on the bracket of the two-fluid model. The various steps of the method are displayed and the example is completed with the verification that the new bracket satisfies the constraints. The possibility of performing the same calculation with single-fluid variables remains open for future research.