Evolution of an Alfvén Wave-Driven Proton Beam in the Expanding Solar Wind

TL;DR

Problem: how proton beams originate and evolve in the expanding solar wind while interacting with Alfvénic fluctuations. Approach: 1D hybrid expanding-box simulations implemented in CAMELIA, across a range of $\beta_{\parallel}$ and $T_\perp/T_\parallel$, driven by a large-amplitude Alfvén wave; kinetic-instability diagnostics are provided by NHDS and ALPS and comparisons are made to Helios and Ulysses data. Findings: beams form at the steepened edge of the Alfvén wave with $v_d \approx v_a$, drift grows sub-linearly with radius due to instability-mediated braking, and firehose-driven heating reshapes the VDF; perpendicular heating is underrepresented due to 1D geometry. Significance: the results support a kinetic-instability–regulated mechanism for proton-beam evolution in the solar wind and inform interpretations of solar wind heating estimates, while underscoring the need for 3D simulations to capture perpendicular cascades.

Abstract

We investigate the self-consistent formation and long-term evolution of proton beams in the expanding solar wind using an ensemble of one-dimensional hybrid expanding box simulations. Initial conditions are chosen to represent a range of plasma states observed by the Helios spacecraft at 0.3 AU, including an amplitude-modulated Alfvén wave that nonlinearly drives a proton beam aligned with the magnetic field. We compare simulation results with solar wind data out to 1.5 AU and show that our model reproduces key observed features of proton beams on average, such as the radial evolution of the drift and the relative core-to-beam density ratio. These findings support the theory that the observed evolution of the proton beam drift in the solar wind is determined by kinetic instabilities. More broadly, our results indicate that the interplay between nonlinear Alfvén wave dynamics, expansion effects and kinetic instabilities plays a fundamental role in solar wind dynamics, with implications for interpreting solar wind heating rate estimates.

Figures (10)

• Figure 1: Snapshots taken from simulation RB displaying the Alfvén wave collapse and proton beam formation. Top row: magnetic field components $B_y(x)$ and $B_z(x)$; middle row: fluctuations of the magnetic field magnitude $B^2(x)$, field-aligned electric field $E_x(x)$, and density $\rho(x)$; bottom row: contour plot of the VDF in the $(v_\perp,v_\parallel)$ space.
• Figure 2: Normalized cross-helicity $\sigma$ as a function of time and radial distance for all simulations. The left panel shows a close-up view of $\sigma$ at early times. The blue profile in the right panel shows $\sigma$ averaged over all runs. The vertical blue dashed line indicates time $t=1000\;\Omega_{ci}^{-1}$ as a reference marker for the left panel. The x's represent the times just before each run became firehose unstable. The right panel shows $v_a$, B, and $\rho$ averaged simulation profiles as a function of time and radial distance.
• Figure 3: The time evolution of the distribution function $F(v_x)$ from $t=0$ to $t=1500$ for R3, showing the formation of a trapped population and a beam that slows down to the local Alfvén speed.
• Figure 4: The normalized drift speed $v_d/v_a$ for each run as a function of time and radial distance. The vertical blue line in the right panel indicates $t$ = 1000 $\Omega_{ci}^{-1}$ which marks the end of the initial transient stage. The black line shows the best fit to the whole ensemble of data points and the dark blue line represents the CGL profile. On the left is a zoom into early times.
• Figure 5: The two adiabatic invariants (Eqn. \ref{['eqn:cglpars']}) calculated for each run.