The analytical expression for the frequency of the unperturbed radial electrostatic lower-hybrid (LH) ray oscillations is derived, which constitutes the first analytically calculable result in nonlinear LH ray dynamics. For LH current drive (CD) conditions, and after comparing the frequency of the unperturbed LH ray motion with the frequency of the ripple perturbation, it is concluded that the electrostatic LH ray dynamics is generally regular in cylindrical plasmas with magnetic ripple. It is also found analytically that including the magnetic ripple in ray-tracing calculations does not lead to a final closure of the spectral-gap problem for LHCD in circular-poloidal-cross-section tokamaks having sufficiently low electron density, in addition to a high enough aspect ratio and safety factor, in which case electrostatic LH wave propagation turns out to be independent of the latter, and the unperturbed LH ray oscillations become basically linear. This is an important null result that helps to show that the conventional ray-tracing picture widely used in LHCD modeling (that is to say, a standardly coupled LH spectrum propagating according to geometrical optics in an established tokamak equilibrium) is not to be taken as final. The analysis presented, carried out within an explicitly Hamiltonian formalism and addressing the role of resonances between the frequency of the unperturbed LH ray motion and the frequencies of the perturbations due to magnetic ripple and toroidicity, is detailed and careful, with analytical results and conclusions being supported by numerical calculations.

Effects of the shear of the magnetic field on the stochastic acceleration of ions due to an electrostatic wave with a frequency in the lower-hybrid range are considered. An appropriate Hamiltonian formalism is used to analyze the equations of motion numerically and theoretically. The surface of section method is used to visualize the solutions and to compare these with the theoretical predictions. From this analysis it appears that there exists an upper adiabatic barrier for the stochastic region which seems to be responsible for the formation of a hot tail in the ion velocity distribution. In addition to lowering the threshold for the onset of stochasticity, the effect of shear is to shift the tail structure to lower values of the velocities. Consequently, these results might help to improve the efficiency of heating by external radiation in the lower-hybrid frequency range.

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