Abstract
We consider a simple electromagnetic gyrokinetic model for collisionless plasmas and show that it possesses a Hamiltonian structure. Subsequently, from this model we derive a two-moment gyrofluid model by means of a procedure which guarantees that the resulting gyrofluid model is also Hamiltonian. The first step in the derivation consists of imposing a generic fluid closure in the Poisson bracket of the gyrokinetic model, after expressing such bracket in terms of the gyrofluid moments. The constraint of the Jacobi identity, which every Poisson bracket has to satisfy, selects then what closures can lead to a Hamiltonian gyrofluid system. For the case at hand, it turns out that the only closures (not involving integro/differential operators or an explicit dependence on the spatial coordinates) that lead to a valid Poisson bracket are those for which the second order parallel moment, independently for each species, is proportional to the zero order moment. In particular, if one chooses an isothermal closure based on the equilibrium temperatures and derives accordingly the Hamiltonian of the system from the Hamiltonian of the parent gyrokinetic model, one recovers a known Hamiltonian gyrofluid model for collisionless reconnection. The proposed procedure, in addition to yield a gyrofluid model which automatically conserves the total energy, provides also, through the resulting Poisson bracket, a way to derive further conservation laws of the gyrofluid model, associated with the so called Casimir invariants. We show that a relation exists between Casimir invariants of the gyrofluid model and those of the gyrokinetic parent model. The application of such Hamiltonian derivation procedure to this two-moment gyrofluid model is a first step toward its application to more realistic, higher-order fluid or gyrofluid models for tokamaks. It also extends to the electromagnetic gyrokinetic case, recent applications of the same procedure to Vlasov and drift- kinetic systems.
Highlights
Fluid models represent a widespread and effective tool for investigating important phenomena in fusion plasmas such as instabilities, turbulence and reconnection events
In this paper we have shown that the electromagnetic gyrokinetic model (1)-(4) possesses a noncanonical Hamiltonian structure
The derivation, is new and shows, by taking advantage of the results of Ref. [11], that the constraint of the Jacobi identity selects this model as a Hamiltonian model derived consistently from the Poisson bracket and the Hamiltonian of the parent gyrokinetic model
Summary
Fluid models represent a widespread and effective tool for investigating important phenomena in fusion plasmas such as instabilities, turbulence and reconnection events. [9]), which implies the existence of additional conserved quantities, denoted as Casimirs Their knowledge can provide useful information on the nonlinear dynamics of the fluid model. In the presence of dissipation, the decay rate and the cascade of such invariants can provide a further way to characterize plasma turbulence They can be used as additional invariants to test the conservation properties of numerical codes. Because with Hamiltonian plasma models one has to deal with noncanonical Poisson brackets, a delicate point in preserving the Hamiltonian structure throughout the derivation, is that of not violating the Jacobi identity. The latter is one of the properties defining a Poisson bracket. It is always satisfied for canonical Poisson brackets, showing its validity for noncanonical Poisson brackets is often far from obvious
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