  We consider ion heating by turbulent Alfven waves (AWs) and kinetic Alfven
waves (KAWs) with perpendicular wavelengths comparable to the ion gyroradius
and frequencies smaller than the ion cyclotron frequency. When the turbulence
amplitude exceeds a certain threshold, an ion's orbit becomes chaotic. The ion
then interacts stochastically with the time-varying electrostatic potential,
and the ion's energy undergoes a random walk. Using phenomenological arguments,
we derive an analytic expression for the rates at which different ion species
are heated, which we test by simulating test particles interacting with a
spectrum of randomly phased AWs and KAWs. We find that the stochastic heating
rate depends sensitively on the quantity epsilon = dv/vperp, where vperp is the
component of the ion velocity perpendicular to the background magnetic field
B0, and dv (dB) is the rms amplitude of the velocity (magnetic-field)
fluctuations at the gyroradius scale. In the case of thermal protons, when
epsilon << eps1, where eps1 is a constant, a proton's magnetic moment is nearly
conserved and stochastic heating is extremely weak. However, when epsilon >
eps1, the proton heating rate exceeds the cascade power that would be present
in strong balanced KAW turbulence with the same value of dv, and
magnetic-moment conservation is violated. For the random-phase waves in our
test-particle simulations, eps1 is approximately 0.2. For protons in low-beta
plasmas, epsilon is approximately dB/B0 divided by the square root of beta, and
epsilon can exceed eps1 even when dB/B0 << eps1. At comparable temperatures,
alpha particles and minor ions have larger values of epsilon than protons and
are heated more efficiently as a result. We discuss the implications of our
results for ion heating in coronal holes and the solar wind.
