Abstract
Starting with a conventional bump on tail problem, which is equivalent to finding a solution of the Vlasov/Fokker-Planck equation in the presence of the phase space island, we obtain a primary equilibrium state. The stability of this state is investigated as a function of the effective velocity-space drag and diffusion, as well as the width of these phase space islands. The secondary instabilities have been found in a certain range of plasma parameters and wave numbers. Solving the full Vlasov/Fokker-Planck – Poisson system, we obtain the dispersion function, which provides information about the secondary mode onset and allows an estimate of the secondary mode growth rate for different input plasma parameters.
Highlights
Energetic particles (EPs), generated by Resonance Heating or Neutral Beam Injection, as well as fusion alpha particles can excite Alfven modes, resonating with plasma waves in a tokamak [1]
Once this new equilibrium state is obtained, we investigate its stability by solving the Vlasov/FokkerPlanck – Poisson system
We have found secondary modes, which correspond to γ > 0 for a certain range of plasma parameters. γ has been calculated as a function of the diffusion and the slowing down rates, as well as l = k/k0, based on a full dispersion function, Eqs. (15,16,17)
Summary
Energetic particles (EPs), generated by Resonance Heating or Neutral Beam Injection, as well as fusion alpha particles can excite Alfven modes, resonating with plasma waves in a tokamak [1]. Working in the wave reference frame, we first seek the time independent solution of the initial Fokker-Planck equation, localised to the island vicinity Once this distribution function is found, we address the Vlasov/Fokker-Planck – Poisson system to explore the stability of this new obtained equilibrium. Other types of the functional behaviour close to the island edge are allowed and, in general, depend on a contribution of barely passing particles, i.e. particles in the vicinity of the separatrix from the outer island region Another solution is the Zakharov and Karpman solution [10], found in the presence of strong velocity diffusion in steady state, and exploited by Berk et al in [11]. From the physical point of view, we keep only one primary mode, as for instance only one toroidal Alfven mode would appear first in the context of EP driven modes
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