Ion Stochastic Heating by Low-frequency Alfv\'en Wave Spectrum

Finite-amplitude low-frequency Alfv\'en waves are commonly found in plasma environments, such as space plasmas, and play a crucial role in ion heating. The nonlinear interaction between oblique Alfv\'en wave spectra and ions has been studied. As the number of wave modes increases, ions are more likely to exhibit chaotic motion and experience stochastic heating. The stochastic heating threshold in the parameter space can be characterized by a single parameter, the effective relative curvature radius $P_{{eff.}}$. The results show excellent agreement with the chaotic regions identified through test particle simulations. The anisotropic characteristics of stochastic heating are explained using a uniform solid angle distribution model. The stochastic heating rate $Q=\dot{T}$ is calculated, and its relationship with wave conditions is expressed as $Q/(\Omega_i m_i v_A^2) = H(\alpha) \tilde{v}^3 \tilde{B}_w^2 \tilde{\omega}_1$, where $\alpha$ is propagating angle, $\Omega_i$ is the gyrofrequency, $m_i$ is the ion mass, $v_A$ is the Alfv\'en speed, $\tilde{v}$ is the dimensionless speed, $\tilde{B}_w$ is the dimensionless wave amplitude, and $\tilde{\omega}_1$ is the lowest dimensionless wave frequency.