Parametric Instability, Inverse Cascade, and the $1/f$ Range of Solar-Wind Turbulence

TL;DR

This paper uses weak turbulence theory to show that parametric decay in solar wind plasmas naturally produces the observed $f^{-1}$ turbulence spectrum, linking nonlinear wave interactions to in situ measurements.

Contribution

It analytically demonstrates the inverse cascade of Alfvén wave quanta due to parametric instability and connects this process to observed solar wind spectra at 0.3 AU.

Findings

Parametric decay leads to an $f^{-1}$ spectrum in Alfvén waves.

The $f^{-1}$ spectrum matches Helios observations at 0.3 AU.

Predicted narrowing of the $f^{-1}$ range closer to the Sun.

Abstract

In this paper, weak turbulence theory is used to investigate the nonlinear evolution of the parametric instability in 3D low-$\beta$ plasmas at wavelengths much greater than the ion inertial length under the assumption that slow magnetosonic waves are strongly damped. It is shown analytically that the parametric instability leads to an inverse cascade of Alfv\'en wave quanta, and several exact solutions to the wave kinetic equations are presented. The main results of the paper concern the parametric decay of Alfv\'en waves that initially satisfy $e^+ \gg e^-$, where $e^+$ and $e^-$ are the frequency ($\,f$) spectra of Alfv\'en waves propagating in opposite directions along the magnetic field lines. If $e^+$ initially has a peak frequency $f_0$ (at which $f e^+$ is maximized) and an "infrared" scaling $f^p$ at smaller $f$ with $-1 < p < 1$, then $e^+$ acquires an $f^{-1}$ scaling throughout a range of frequencies that spreads out in both directions from $f_0$. At the same time, $e^-$ acquires an $f^{-2}$ scaling within this same frequency range. If the plasma parameters and infrared $e^+$ spectrum are chosen to match conditions in the fast solar wind at a heliocentric distance of 0.3 astronomical units (AU), then the nonlinear evolution of the parametric instability leads to an $e^+$ spectrum that matches fast-wind measurements from the Helios spacecraft at 0.3 AU, including the observed $f^{-1}$ scaling at $f \gtrsim 3 \times 10^{-4} \mbox{ Hz}$. The results of this paper suggest that the $f^{-1}$ spectrum seen by Helios in the fast solar wind at $f \gtrsim 3\times 10^{-4} \mbox{ Hz}$ is produced in situ by parametric decay and that the $f^{-1}$ range of $e^+$ extends over an increasingly narrow range of frequencies as $r$ decreases below 0.3 AU. This prediction will be tested by measurements from the Parker Solar Probe.