Evan L. Yerger, Benjamin D. G. Chandran, Vincent David, Trevor A. Bowen, Stuart D. Bale

The $\textit{Parker Solar Probe}$ ($\textit{PSP}$) mission has observed near-continuous power in parallel ion cyclotron waves (PICWs) in the young, fast solar wind. These waves are unlikely to be directly produced by the turbulent cascade and are likely born of a local instability; yet, they are observed to both cool -- and heat -- the plasma. We propose that these observations can be self-consistently explained as the natural consequence of PICWs propagating in the inhomogeneous solar wind after they have been driven unstable. In this work, we argue that strong proton heating by a turbulent cascade of oblique ICWs will result in PICWs being driven unstable in a process known as quasi-linear focusing. Because the power in the turbulent cascade is concentrated at scales above the turbulent transition region, PICWs will be driven unstable within a range of wave numbers parallel to the background magnetic field, $k_\parallel$, that is bounded from above by $k_{\parallel\rm P}^*$, corresponding to the start of the transition region. As unstable PICWs propagate away from the sun to regions of lower proton density, their $k_\parallel$, multiplied by the proton inertial length $d_{\rm p}$, increases. Eventually, the $k_\parallel d_{\rm p}$ of the PICWs becomes larger than $k_{\parallel\rm P}^*d_{\rm p}$ and the waves damp, heating the solar wind. We call this effect `cyclotron breaking', in analogy with ocean waves breaking on the shore. We then discuss the testable predictions of the theory, including a distinct heating signature in which PICWs cool fast protons and heat slow protons at any given heliocentric distance $r$. Finally, we conjecture that cyclotron breaking can lead to net heating by PICWs if the power emitted as PICWs decreases sufficiently rapidly with $r$ that local emission of PICWs is overwhelmed by the local damping of PICWs generated closer to the sun.