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

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 Alfvén wave quanta, and several exact solutions to the wave kinetic equations are presented. The main results of the paper concern the parametric decay of Alfvén waves that initially satisfy $e^{+}\gg e^{-}$, where $e^{+}$ and $e^{-}$ are the frequency ($f$) spectra of Alfvén 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.

25 pages, 4 figures, accepted for publication in the Journal of Plasma Physics

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• Frequency-dependent Alfvén-wave propagation in the solar wind: Onset and suppression of parametric decay instability
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• The Near-Sun Streamer Belt Solar Wind: Turbulence and Solar Wind Acceleration
• Multiscale Solar Wind Turbulence Properties inside and near Switchbacks measured by Parker Solar Probe
• New Observations of Solar Wind 1/f Turbulence Spectrum from Parker Solar Probe
• The Evolution of the 1/f Range Within a Single Fast-Solar-Wind Stream Between 17.4 and 45.7 Solar Radii
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• Anisotropic magnetohydrodynamic turbulence driven by parametric decay instability: the onset of phase mixing and Alfvén wave turbulence
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• Particle-in-cell simulations of Alfvén wave parametric decay in a low-beta plasma
• Phase mixing and the 1/f spectrum in the solar wind
• Three-dimensional simulations of the inhomogeneous Low Solar Wind
• Evidence for Parameteric Decay Instability in the Lower Solar Atmosphere
• How Alfvén waves energize the solar wind: heat vs work
• Near subsonic solar wind outflow from an active region
• Kappa-Maxwellian electrons and Bi-Maxwellian protons in a two-fluid model for fast solar wind
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• Interplanetary magnetic correlation and low-frequency spectrum over many solar rotations
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