We derive the bounce-averaged quasilinear operator in axisymmetric toroidal geometry using the standard quasilinear formalism. When specialized to resonant ion cyclotron harmonic interactions, this operator predicts the same radio-frequency-induced radial diffusion as the orbit-averaged operator (L.-G. Eriksson and P. Helander, Phys. Plasmas 1 (1994) 308) obtained using action-angle variables. By assuming the wave field known as a superposition of toroidal and poloidal Fourier modes, the quasilinear diffusion coefficients are written in a form which can be directly evaluated using the output of a spectral full-wave solver of Maxwell equations in toroidal plasmas.
-Introduction.The fact that in toroidal geometry resonant cyclotron interactions inevitably cause radial diffusion of the heated ions is known since a long time [1]. As pointed out by Ryopulos et al. [2] and Hellsten [3], this is a consequence of the fact that the variation of the magnetic moment µ suffered by an ion transiting through a cyclotron resonance also causes a variation of its toroidal momentum P ϕ . This, in turn, causes the mean radius of the drift orbit to shift radially outwards or inwards, depending on the sign of the toroidal wavenumber of the hf wave. The effect has been demonstrated experimentally in the JET tokamak by launching a toroidally asymmetric spectrum of fast waves [4].An elegant form of the quasilinear (QL) kinetic equation including a term describing rf-induced radial diffusion has been derived by Eriksson and Helander [5]. A Monte Carlo method for its solution has been developed [5][6][7], and has since been successfully used for the interpretation of high-power ion cyclotron (IC) heating experiments in JET [8][9]. A review of recent work along these lines can be found in [10].The derivations of [1] and [5] are based on the action-angle variables formalism of Kaufman [11]. This approach has been called 'orbit-averaging', to distinguish it from conventional 'bounce-averaging ' [12], which is thought to be valid only in the limit of thin banana orbits. In this note we show, however, that the standard bounce-averaging procedure predicts just the same radial diffusion as orbit-averaging, unless it is explicitly or tacitly assumed that all charged particles are tied to magnetic field lines. We argue, therefore, that the real difference between orbit and bounce averaging lies in the treatment of collisions, which in Montecarlo codes, but not in bounce-averaged Fokker-Planck solvers, can be evaluated at the true position of the ions, independently from the shape and thickness of their orbit.Our derivation differs from previous ones also in an other minor technical aspect, which is, however, of practical relevance. Most derivations of the QL operator assume that the wave field in the vicinity of IC resonances can be represented in Eikonal form. Here the wave electric field is supposed instead to be given as a superposition of toroidal and poloidal Fourier modes. This is the representation used to solve Maxwell equations ...