Dielectric response and thermal fluctuations in gyrokinetic plasma

A gyrokinetic fluctuation–dissipation theorem is deduced and used to predict unusual thermal-equilibrium fluctuation spectra for both electrostatic and weakly electromagnetic gyrokinetic plasmas in the limit of weak coupling. The results are in agreement with an application of the test-particle superposition principle. They are interpreted physically in terms of the concept of the ‘‘gyrokinetic vacuum,’’ whose large (tensor) dielectric constant embodies the shielding effects of the polarization drift and the inductive electric field. The calculations are performed entirely with gyrokinetic response functions, although the relations to the conventional Vlasov dielectric are also described. The previous heuristic results of Krommes et al. [Phys. Fluids 29, 2421 (1986)] are recovered systematically. A nonrelativistic covariant formalism is used to derive the finite-β results (where β is the plasma pressure). The wave-number power spectra of the charge density, electrostatic potential, and perpendicular electric field are independent of β, while those of the parallel current, parallel vector potential, and perpendicular magnetic field are increasing functions of β. These results are derived both by exact integrations of the frequency spectra and, in the appropriate limit, by resonance integrations over weakly damped normal modes. To interpret the latter results, a new form for the unusual energy conservation law of gyrokinetic plasma is exhibited and a new formula for the gyrokinetic plasmon action 𝒩(k) valid at finite β is derived. The fluctuation spectra for the normal modes can then be derived by setting the plasmon energy ωk𝒩(k) to T/2. The power spectrum of the inductive component of the parallel electric field is dominated by high-frequency, non-normal-mode noise.