Authors: Evan L. Yerger (Space Science Center, University of New Hampshire, Durham, NH 03824, USA), Michael F. Zhang (Physics Department, University of Otago, 730 Cumberland St, Dunedin 9016, New Zealand), Benjamin D. G. Chandran (Department of Physics, University of New Hampshire, Durham, NH 03824, USA), Matthew W. Kunz (Department of Astrophysical Sciences, Princeton University, Peyton Hall, Princeton, NJ 08544, USA), Vincent David (Department of Physics and Astronomy, Dartmouth College, Hanover, NH 03755), Kristopher G. Klein (Lunar and Planetary Laboratory, University of Arizona, Tucson, AZ 85721, USA), Jonathan Squire (Physics Department, University of Otago, 730 Cumberland St, Dunedin 9016, New Zealand)

The exact mechanism behind proton heating in imbalanced turbulence in the fast solar wind is still not fully understood. One clue towards the identity of this mechanism is the near-continuous observation of parallel ion cyclotron waves (PICWs) by Parker Solar Probe. It has been proposed that the observed PICWs can be explained by a process called quasi-linear focusing, in which the emission of PICWs is the result of strong cyclotron heating by oblique ion cyclotron waves (ICWs) from the wave-vector-anisotropic turbulent cascade. Quasi-linear heating by oblique turbulence pushes the proton velocity distribution function (VDF) to be constant along “quasi-linear contours”, or contours of constant energy in the frame of the oblique waves. PICWs have relatively slower phase speeds — and shallower quasi-linear contours — than oblique ICWs. As such, a proton VDF that is constant along oblique ICW contours is unstable to PICWs because the emission of PICWs allow protons to diffuse along their shallower contours toward lower energy. Once the unstable PICWs reach finite amplitude, they will start to feed back on the proton VDF. However, the exact effect of PICW feedback on the proton VDF is difficult to determine because it is complicated by a number of factors, including continuous heating by the oblique cascade. To provide insight into PICW feedback on proton heating by imbalanced turbulence, we present two hybrid-particle-in-cell (hybrid-PIC) simulations of forced, imbalanced, Alfvénic turbulence at plasma beta 1/16 and 3/10. We show that both simulations reach a quasi-steady state when the turbulent fluctuations reach sufficiently high frequency that they heat thermal protons by cyclotron resonance. We then compute full linear dispersion relations from the simulation VDFs using the Arbitrary Linear Plasma Solver (ALPS). These dispersion relations show show that cyclotron heating of the turbulent cascade drives PICWs unstable in a way that is consistent with quasi-linear focusing. To conclude, we present a number of diagnostics that quantify the extent to which PICWs feed back on the quasi-linear heating in our simulations and discuss the implications of our findings to observations of PICWs in the solar wind.