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Stochastic heating refers to an increase in the average magnetic moment ofcharged particles interacting with electromagnetic fluctuations whosefrequencies are smaller than the particles' cyclotron frequencies. This type ofheating arises when the amplitude of the gyroscale fluctuations exceeds acertain threshold, causing particle orbits in the plane perpendicular to themagnetic field to become stochastic rather than nearly periodic. We considerthe stochastic heating of protons by Alfv\'en-wave (AW) andkinetic-Alfv\'en-wave (KAW) turbulence, which may make an importantcontribution to the heating of the solar wind. Using phenomenologicalarguments, we derive the stochastic-proton-heating rate in plasmas in which$\beta_{\rm p} \sim 1-30$, where $\beta_{\rm p}$ is the ratio of the protonpressure to the magnetic pressure. (We do not consider the $\beta_{\rm p}\gtrsim 30$ regime, in which KAWs at the proton gyroscale becomenon-propagating.) We test our formula for the stochastic-heating rate bynumerically tracking test-particle protons interacting with a spectrum ofrandomly phased AWs and KAWs. Previous studies have demonstrated that at$\beta_{\rm p} \lesssim 1$, particles are energized primarily by timevariations in the electrostatic potential and thermal-proton gyro-orbits arestochasticized primarily by gyroscale fluctuations in the electrostaticpotential. In contrast, at $\beta_{\rm p} \gtrsim 1$, particles are energizedprimarily by the solenoidal component of the electric field and thermal-protongyro-orbits are stochasticized primarily by gyroscale fluctuations in themagnetic field.

Stochastic proton heating by kinetic-Alfv\'en-wave turbulence in moderately high-$\beta$ plasmas