Mind the Gap: Nonlocal Cascades and Preferential Heating in High-$\beta$ Alfv\'enic Turbulence

Characterizing the thermodynamics of turbulent plasmas is key to decoding observable signatures from astrophysical systems. In magnetohydrodynamic (MHD) turbulence, nonlinear interactions between counter-propagating Alfv\'en waves cascade energy to smaller spatial scales where dissipation heats the protons and electrons. When the thermal pressure far exceeds the magnetic pressure, linear theory predicts a spectral gap at perpendicular scales near the proton gyroradius where Alfv\'en waves become non-propagating. For simple models of an MHD turbulent cascade that assume only local nonlinear interactions, the cascade halts at this gap, preventing energy from reaching smaller scales where electron dissipation dominates, leading to an overestimate of the proton heating rate. In this work, we demonstrate that nonlocal contributions to the cascade, specifically large scale shearing and small scale diffusion, can bridge the non-propagating gap, allowing the cascade to continue to smaller scales. We provide an updated functional form for the proton-to-electron heating ratio accounting for this nonlocal energy transfer by evaluating a nonlocal weakened cascade model over a range of temperature and pressure ratios. In plasmas where the thermal pressure dominates the magnetic pressure, we observe that the proton heating is moderated compared to the significant enhancement predicted by local models.