Coupled hydromagnetic wave excitation and ion acceleration at interplanetary traveling shocks

Abstract

A self-consistent theory is presented for the excitation of hydromagnetic waves and the acceleration of ions upstream of interplanetary traveling shocks. The waves are excited by the ions by virtue of ion streaming relative to the solar wind; the ions are accelerated by being coupled to the compression of the shock via pitch angle scattering on the upstream waves and the downstream turbulence. Diffusion equations describing the ion transport and wave kinetic equations describing the hydromagnetic wave transport are solved self-consistently to yield analytical expressions for the differential wave intensity spectrum as a function of wave number k and distance z upstream of the shock and for the ion omnidirectional distribution functions and anisotropies as functions of energy E and z. In quantitative agreement with observations the theory predicts, for example, (1) power law ion spectra at the shock •E -r with 2 •< F •< 3, (2) a decrease in intensity and hardening of the ion spectra with increasing z, (3) upstream ion anisotropies (,-•0.3 fo 30-keV protons) away from the shock front in the frame of the solar wind, (4) an unpolarized enhanced wave intensity spectrum in the wave number range corresponding to 0.4-3 x 10-2 Hz in the spacecraft frame, and (5) a decrease in the wave intensity spectrum with increasing z.

Proton transport through self-generated waves in impulsive flares

“…The full quasi-linear resonance condition would involve a spectrum of waves even in case of mono-energetic particle distribution. The artificially sharpened resonance condition α 0 kp = mΩ p , where α 0 is a numerical constant of the order unity, is, however, a commonly used approximation in diffusive particle transport and acceleration theories involving self-generated waves (e.g., Skilling 1975;Bell 1978;Lee 1983). It introduces an inaccuracy to the scattering rates, but for a power-law spectrum of particles it can be tuned by choosing α 0 properly to yield correct growth rates for waves (Skilling 1975).…”

On the “injection problem” at the solar wind termination shock

“…The shock parameter γ sh in Bamert et al (2004) is γ sh ≈ 4.9, close to what one may expect for the termination shock, γ sh ≈ 4.5. The amplification G A (k) of Alfvén waves was found as G A (k) = 15 (k/k 1 MeV ) γ sh −13/3 and is thought to be caused by an anisotropic flux of energetic particles (Lee 1983). Anisotropies are in fact observed at 85 AU (Krimigis et al 2003), however the phase space density of energetic ions is a factor of 3000 lower than F p during the Bastille Day event (Fig.…”

“…As Γ p scales with ρ 3 and P (k, ∞) does not vary with ρ, one may expect a wave amplification near the termination shock of perhaps 15 × 85 3 × 10 −2 ≈ 10 5 for a quasi-parallel shock. The Alfvén wave generation scales as cos Ψ (Lee 1983), where Ψ is the angle of the ambient magnetic field to the shock normal. This yields typically G A (k) ≈ 10 5 cos Ψ (k/k 1 MeV ) γ sh −13/3 at the termination shock.…”

The Acceleration of High-energy Protons at Coronal Shocks: The Effect of Large-scale Streamer-like Magnetic Field Structures

“…Particle acceleration at a nearly perpendicular shock is much faster than that at a parallel shock (Jokipii 1987). Note, however, that self-excited waves produced by streaming energetic particles may further enhance particle acceleration at parallel shocks (e.g., Lee 1983Lee , 2005Li et al 2003;Rice et al 2003). When a CME-driven shock develops in the corona, the nonplanar shock front sweeps through the coronal magnetic field with a range of different shock angles, which has been proposed to have significant effects on particle acceleration (e.g., Giacalone 2005aGiacalone , 2005bTylka et al 2005).…”