ABSTRACT We investigate how the proton distribution function evolves when the protons undergo stochastic heating by strong, low-frequency, Alfvén-wave turbulence under the assumption that β is small. ...We apply our analysis to protons undergoing stochastic heating in the supersonic fast solar wind and obtain proton distributions at heliocentric distances ranging from 4 to 30 solar radii. We find that the proton distribution develops non-Gaussian structure with a flat core and steep tail. For , the proton distribution is well approximated by a modified Moyal distribution. Comparisons with future measurements from Solar Probe Plus could be used to test whether stochastic heating is occurring in the solar-wind acceleration region.
Abstract
Motivated by recent Parker Solar Probe (PSP) observations of “switchbacks” (abrupt, large-amplitude reversals in the radial magnetic field, which exhibit Alfvénic correlations), we examine ...the dynamics of large-amplitude Alfvén waves in the expanding solar wind. We develop an analytic model that makes several predictions: switchbacks should preferentially occur in regions where the solar wind plasma has undergone a greater expansion, the switchback fraction at radii comparable to PSP should be an increasing function of radius, and switchbacks should have their gradients preferentially perpendicular to the mean magnetic field direction. The expansion of the plasma generates small compressive components as part of the wave’s nonlinear evolution: these are maximized when the normalized fluctuation amplitude is comparable to
sin
θ
, where
θ
is the angle between the propagation direction and the mean magnetic field. These compressive components steepen the primary Alfvénic waveform, keeping the solution in a state of nearly constant magnetic field strength as its normalized amplitude
δB/B
grows due to expansion. The small fluctuations in the magnetic field strength are minimized at a particular
θ
-dependent value of
β
, usually of order unity, and the density and magnetic-field-strength fluctuations can be correlated or anticorrelated depending on
β
and
θ
. Example solutions of our dynamical equation are presented; some do indeed form magnetic-field reversals. Our predictions appear to match some previously unexplained phenomena in observations and numerical simulations, providing evidence that the observed switchbacks result from the nonlinear evolution of the initially small-amplitude Alfvén waves already known to be present at the coronal base.
We perform a statistical study of the turbulent power spectrum at inertial and kinetic scales observed during the first perihelion encounter of the Parker Solar Probe. We find that often there is an ...extremely steep scaling range of the power spectrum just above the ion-kinetic scales, similar to prior observations at 1 A.U., with a power-law index of around − 4 . Based on our measurements, we demonstrate that either a significant ( > 50 % ) fraction of the total turbulent energy flux is dissipated in this range of scales, or the characteristic nonlinear interaction time of the turbulence decreases dramatically from the expectation based solely on the dispersive nature of nonlinearly interacting kinetic Alfvén waves.
We investigate the conditions under which parallel-propagating Alfven/ion-cyclotron (A/IC) waves and fast-magnetosonic/whistler (FM/W) waves are driven unstable by the differential flow and ...temperature anisotropy of alpha particles in the solar wind. We focus on the limit in which w sub(|| alpha ) > ~ 0.25v sub(A), where w sub(|| alpha ) is the parallel alpha-particle thermal speed and v sub(A) is the Alfven speed. We derive analytic expressions for the instability thresholds of these waves, which show, e.g., how the minimum unstable alpha-particle beam speed depends upon w sub(|| alpha )/v sub(A), the degree of alpha-particle temperature anisotropy, and the alpha-to-proton temperature ratio. We validate our analytical results using numerical solutions to the full hot-plasma dispersion relation. Consistent with previous work, we find that temperature anisotropy allows A/IC waves and FM/W waves to become unstable at significantly lower values of the alpha-particle beam speed U sub( alpha ) than in the isotropic-temperature case. Likewise, differential flow lowers the minimum temperature anisotropy needed to excite A/IC or FM/W waves relative to the case in which U sub( alpha ) = 0. We discuss the relevance of our results to alpha particles in the solar wind near 1 AU.
We develop a one-dimensional solar-wind model that includes separate energy equations for the electrons and protons, proton temperature anisotropy, collisional and collisionless heat flux, and an ...analytical treatment of low-frequency, reflection-driven, Alfven-wave (AW) turbulence. To partition the turbulent heating between electron heating, parallel proton heating, and perpendicular proton heating, we employ results from the theories of linear wave damping and nonlinear stochastic heating. We account for mirror and oblique firehose instabilities by increasing the proton pitch-angle scattering rate when the proton temperature anisotropy exceeds the threshold for either instability. We numerically integrate the equations of the model forward in time until a steady state is reached, focusing on two fast-solar-wind-like solutions. These solutions are consistent with a number of observations, supporting the idea that AW turbulence plays an important role in the origin of the solar wind.
An ion beam can destabilize Alfven/ion-cyclotron waves and magnetosonic/whistler waves if the beam speed is sufficiently large. Numerical solutions of the hot-plasma dispersion relation have ...previously shown that theminimum beam speed required to excite such instabilities is significantly smaller for oblique modes with k x B sub(0) not = 0 than for parallel-propagating modes with k x B sub(0) = 0, where k is the wavevector and B sub(0) is the background magnetic field. In this paper, we explain this difference within the framework of quasilinear theory, focusing on low- beta plasmas. We begin by deriving, in the cold-plasma approximation, the dispersion relation and polarization properties of both oblique and parallel-propagating waves in the presence of an ion beam. We then show how the instability thresholds of the different wave branches can be deduced from the wave-particle resonance condition, the conservation of particle energy in the wave frame, the sign (positive or negative) of the wave energy, and the wave polarization. We also provide a graphical description of the different conditions under which Landau resonance and cyclotron resonance destabilize Alfven/ion-cyclotron waves in the presence of an ion beam. We draw upon our results to discuss the types of instabilities that may limit the differential flow of alpha particles in the solar wind.
The Solar Wind Electrons Alphas and Protons (SWEAP) Investigation on Solar Probe Plus is a four sensor instrument suite that provides complete measurements of the electrons and ionized helium and ...hydrogen that constitute the bulk of solar wind and coronal plasma. SWEAP consists of the Solar Probe Cup (SPC) and the Solar Probe Analyzers (SPAN). SPC is a Faraday Cup that looks directly at the Sun and measures ion and electron fluxes and flow angles as a function of energy. SPAN consists of an ion and electron electrostatic analyzer (ESA) on the ram side of SPP (SPAN-A) and an electron ESA on the anti-ram side (SPAN-B). The SPAN-A ion ESA has a time of flight section that enables it to sort particles by their mass/charge ratio, permitting differentiation of ion species. SPAN-A and -B are rotated relative to one another so their broad fields of view combine like the seams on a baseball to view the entire sky except for the region obscured by the heat shield and covered by SPC. Observations by SPC and SPAN produce the combined field of view and measurement capabilities required to fulfill the science objectives of SWEAP and Solar Probe Plus. SWEAP measurements, in concert with magnetic and electric fields, energetic particles, and white light contextual imaging will enable discovery and understanding of solar wind acceleration and formation, coronal and solar wind heating, and particle acceleration in the inner heliosphere of the solar system. SPC and SPAN are managed by the SWEAP Electronics Module (SWEM), which distributes power, formats onboard data products, and serves as a single electrical interface to the spacecraft. SWEAP data products include ion and electron velocity distribution functions with high energy and angular resolution. Full resolution data are stored within the SWEM, enabling high resolution observations of structures such as shocks, reconnection events, and other transient structures to be selected for download after the fact. This paper describes the implementation of the SWEAP Investigation, the driving requirements for the suite, expected performance of the instruments, and planned data products, as of mission preliminary design review.
In this paper, weak-turbulence theory is used to investigate the nonlinear evolution of the parametric instability in three-dimensional low-
$\unicodeSTIX{x1D6FD}$
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
$fe^{+}$
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
$f\gtrsim 3\times 10^{-4}~\text{Hz}$
. The results of this paper suggest that the
$f^{-1}$
spectrum seen by Helios in the fast solar wind at
$f\gtrsim 3\times 10^{-4}~\text{Hz}$
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.
Spacecraft measurements show that protons undergo substantial perpendicular heating during their transit from the Sun to the outer heliosphere. In this paper, we use Helios 2 measurements to ...investigate whether stochastic heating by low-frequency turbulence is capable of explaining this perpendicular heating. We analyze Helios 2 magnetic field measurements in low- beta fast-solar-wind streams between heliocentric distances r = 0.29 AU and r = 0.64 AU to determine the rms amplitude of the fluctuating magnetic field, delta B sub(p), near the proton gyroradius scale rho sub(p). We then evaluate the stochastic heating rate Q sub(perpendicularstoch) using the measured value of delta B sub(p) and a previously published analytical formula for Q sub(perpendicularstoch). Using Helios measurements we estimate the "empirical" perpendicular heating rate (ProQuest: Formulae and/or non-USASCII text omitted) that is needed to explain the T sub(perpendicularp) profile. We find that Q sub(perpendicularstoch) ~ Q sub(perpendicularemp), but only if a key dimensionless constant appearing in the formula for Q sub(perpendicularstoch) lies within a certain range of values. This range is approximately the same throughout the radial interval that we analyze and is consistent with the results of numerical simulations of the stochastic heating of test particles in reduced magnetohydrodynamic turbulence. These results support the hypothesis that stochastic heating accounts for much of the perpendicular proton heating occurring in low- beta fast-wind streams.
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