Proton-acoustic wave effects on the relaxation of proton transverse heating in magnetized plasmas

Transverse electromagnetic and electrostatic plasma wave modes propagating along a background magnetic field B⃗₀ are independent according to linear kinetic theory. However, resonant interactions and energy exchange between waves and particles break this linear decoupling. This work tracks the coupled evolution of Alfvén-cyclotron (ACWs) and Ion-acoustic waves (IAWs) by solving moment-based quasilinear equations for a collisionless plasma of bi-Maxwellian protons and Maxwellian electrons. Unlike earlier quasilinear studies that adopt the cold-electron limit, our formulation retains the full kinetic response of both species, treating the electrons as a thermal reservoir to isolate proton heating. A parameter survey over 0.01≤β∥ p≤10 and 1≤ Tₑ/Tₚ≤10 shows that IAWs can raise the proton temperature anisotropy to T⊥ p/T∥ p from isotropic conditions at low β∥ p0.1, thereby triggering an ACW instability. The quasilinear evolution self-regulates the ACW, driving the system toward a quasi-stationary state with γ/Ωₚ<10⁻¹ and reduced anisotropy. For β∥ p0.3 or Tₑ/Tₚ5, IAWs lose the ability to generate appreciable transverse heating. The results explain how modest electrostatic activity in low-β environments such as the inner heliosphere and planetary magnetosheaths, can seed, but not indefinitely sustain, cyclotron instabilities. While the bi-Maxwellian closure neglects phase-space diffusion and BGK trapping, the framework cleanly separates the roles of IAWs and ACWs and can be extended to kinetic electron sub-populations or other instabilities.