Abstract
Effects of collisions on conservation laws for toroidal plasmas are investigated based on the gyrokinetic field theory. Associating the collisional system with a corresponding collisionless system at a given time such that the two systems have the same distribution functions and electromagnetic fields instantaneously, it is shown how the collisionless conservation laws derived from Noether's theorem are modified by the collision term. Effects of the external source term added into the gyrokinetic equation can be formulated similarly with the collisional effects. Particle, energy, and toroidal momentum balance equations including collisional and turbulent transport fluxes are systematically derived using a novel gyrokinetic collision operator, by which the collisional change rates of energy and canonical toroidal angular momentum per unit volume in the gyrocenter space can be given in the conservative forms. The ensemble-averaged transport equations of particles, energy, and toroidal momentum given in the present work are shown to include classical, neoclassical, and turbulent transport fluxes which agree with those derived from conventional recursive formulations.
Highlights
Conservation of the total energy and momentum is naturally obtained in the gyrokinetic field theory, where all governing equations for the distribution functions and the electromagnetic fields are derived from the Lagrangian which describes the whole system consisting of particles and fields.[13,14,15,16,17]
We examine how the collision and external source terms added into the gyrokinetic equations influence the conservation laws derived from Noether’s theorem in the gyrokinetic field theory for collisionless systems
∇s · ÊL ÊL + BB + (∇ × B)Â · eζ dU dμ dξ Da Sa ma U bζ + O(δ 3 ), The expressions for the toroidal momentum fluxes shown in Eqs. (104)–(107) agree with those given by conventional recursive formulations in Refs. 33–35. [Since the so-called high-flow ordering is used in Refs. 33 and 34, the expressions for the toroidal momentum fluxes in it reduce to those in the present work in the low-flow-speed limit.] As argued in Refs. 18 and 35, when there exists the up-down symmetry of the background magnetic field, all toroidal momentum fluxes vanish to O(δ 2 ) and the nontrivial toroidal momentum balance equation is of O(δ 3 )
Summary
Gyrokinetic theories and simulations are powerful means to investigate microinstabilities and turbulent transport processes in magnetically confined plasmas.[1,2,3,4]. In our previous work,[18] using the gyrokinetic Vlasov-Poisson-Ampère system of equations, conservation laws of particles, energy, and toroidal angular momentum are obtained for collisionless toroidal plasmas, in which the slow temporal variation of the background magnetic field is taken into account in order to enable self-consistent treatment of physical processes on transport time scales. Based on these results, the particle, energy, and toroidal angular momentum balance equations for the collisional plasma are derived from the gyrokinetic Boltzmann-Poisson-Ampère system of equations in Secs. Principle using the action I defined by Eq (1) in Ref. 29 where its variation δI associated with infinitesimal transformations of independent and dependent variables [see
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