Ion acceleration by beating electrostatic waves: Domain of allowed acceleration

The conditions under which a magnetized ion can be accelerated through a nonlinear interaction with a pair of beating electrostatic waves are explored. It has been shown [Benisti et al., Phys. Plasma 5, 3224 (1998)] that the electric field of the beating waves can, under some conditions, accelerate ions from arbitrarily low initial velocity in stark contrast with the well-known nonlinear threshold criteria for ion acceleration by a single wave. It is shown here that the previously found condition is necessary but not sufficient for acceleration to occur. The sufficient and necessary conditions are identified in terms of the location of the critical points of the motion on the Poincaré section. A second-order perturbation analysis was carried out to approximate the location of these critical points and define the domains of allowed and forbidden acceleration. It is shown that for an ion to be significantly energized, the Hamiltonian must be outside the energy barrier defined by the location of the elliptic and hyperbolic critical points. Despite the restriction on the Hamiltonian, an ion with arbitrarily low initial velocity may benefit from this acceleration mechanism.