arXiv subjects

Benjamin D. G. Chandran

Publications and source records attributed to Benjamin D. G. Chandran.

At least 55 records · Page 3Linked to original sources

arXiv2016

Evolution of The Proton Velocity Distribution due to Stochastic Heating in the Near-Sun Solar Wind

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 $r >5 \ R_{\rm S}$, the proton distribution is well approximated by a modified Moyal distribution. Comparisons with future measurements from \emph{Solar Probe Plus} could be used to test whether stochastic heating is occurring in the solar-wind acceleration region.

astro-ph.SR↗

arXiv2015

On the Conservation of Cross Helicity and Wave Action in Solar-Wind Models with Non-WKB Alfven Wave Reflection

The interaction between Alfven-wave turbulence and the background solar wind affects the cross helicity in two ways. Non-WKB reflection converts outward-propagating Alfven waves into inward-propagating Alfven waves and vice versa, and the turbulence transfers momentum to the background flow. When both effects are accounted for, the total cross helicity is conserved. In the special case that the background density and flow speed are independent of time, the equations of cross-helicity conservation and total-energy conservation can be combined to recover a well-known equation derived by Heinemann and Olbert that has been interpreted as a non-WKB generalization of wave-action conservation. This latter equation (in contrast to cross-helicity and energy conservation) does not hold when the background varies in time.

astro-ph.SR↗

arXiv2015

Deceleration of Alpha Particles in the Solar Wind by Instabilities and the Rotational Force: Implications for Heating, Azimuthal Flow, and the Parker Spiral Magnetic Field

Protons and alpha particles in the fast solar wind are only weakly collisional and exhibit a number of non-equilibrium features, including relative drifts between particle species. Two non-collisional mechanisms have been proposed for limiting differential flow between alpha particles and protons: plasma instabilities and the rotational force. Both mechanisms decelerate the alpha particles. In this paper, we derive an analytic expression for the rate $Q_{\mathrm{flow}}$ at which energy is released by alpha-particle deceleration, accounting for azimuthal flow and conservation of total momentum. We show that instabilities control the deceleration of alpha particles at $r< r_{\mathrm{crit}}$, and the rotational force controls the deceleration of alpha particles at $r> r_{\mathrm{crit}}$, where $r_{\mathrm{crit}} \simeq 2.5 \,\mathrm{AU}$ in the fast solar wind in the ecliptic plane. We find that $Q_{\mathrm{flow}}$ is positive at $r<r_{\mathrm{crit}}$ and $Q_{\mathrm{flow}} = 0$ at $r\geq r_{\mathrm{crit}}$, consistent with the previous finding that the rotational force does not lead to a release of energy. We compare the value of~$Q_{\mathrm{flow}}$ at $r< r_{\mathrm{crit}}$ with empirical heating rates for protons and alpha particles, denoted $Q_{\mathrm{p}}$ and $Q_α$, deduced from in-situ measurements of fast-wind streams from the \emph{Helios} and \emph{Ulysses} spacecraft. We find that $Q_{\mathrm{flow}}$ exceeds $Q_α$ at $r < 1\,\mathrm{AU}$, and that $Q_{\mathrm{flow}}/Q_{\rm p}$ decreases with increasing distance from the Sun from a value of about one at $r=0.29 - 0.42\,\mathrm{AU}$ to about 1/4 at 1 AU. We conclude that the continuous energy input from alpha-particle deceleration at $r< r_{\mathrm{crit}}$ makes an important contribution to the heating of the fast solar wind.

physics.space-ph↗

arXiv2015

A Modified Version of Taylor's Hypothesis for Solar Probe Plus Observations

The Solar Probe Plus (SPP) spacecraft will explore the near-Sun environment, reaching heliocentric distances less than $10 R_{\odot}$. Near Earth, spacecraft measurements of fluctuating velocities and magnetic fields taken in the time domain are translated into information about the spatial structure of the solar wind via Taylor's "frozen turbulence" hypothesis. Near the perihelion of SPP, however, the solar-wind speed is comparable to the Alfvén speed, and Taylor's hypothesis in its usual form does not apply. In this paper, we show that, under certain assumptions, a modified version of Taylor's hypothesis can be recovered in the near-Sun region. We consider only the transverse, non-compressive component of the fluctuations at length scales exceeding the proton gyroradius, and we describe these fluctuations using an approximate theoretical framework developed by Heinemann and Olbert. We show that fluctuations propagating away from the Sun in the plasma frame obey a relation analogous to Taylor's hypothesis when $V_{\rm sc,\perp} \gg z^-$ and $z^+ \gg z^-$, where $V_{\rm sc,\perp}$ is the component of the spacecraft velocity perpendicular to the mean magnetic field and $\bm{z}^+$ ($\bm{z}^-$) is the Elsasser variable corresponding to transverse, non-compressive fluctuations propagating away from (towards) the Sun in the plasma frame. Observations and simulations suggest that, in the near-Sun solar wind, the above inequalities are satisfied and $\bm{z}^+$ fluctuations account for most of the fluctuation energy. The modified form of Taylor's hypothesis that we derive may thus make it possible to characterize the spatial structure of the energetically dominant component of the turbulence encountered by SPP.

physics.space-ph↗

arXiv2014

Acceleration of Relativistic Electrons by MHD Turbulence: Implications for Non-thermal Emission from Black Hole Accretion Disks

We use analytic estimates and numerical simulations of test particles interacting with magnetohydrodynamic (MHD) turbulence to show that subsonic MHD turbulence produces efficient second-order Fermi acceleration of relativistic particles. This acceleration is not well-described by standard quasi-linear theory but is a consequence of resonance broadening of wave-particle interactions in MHD turbulence. We provide momentum diffusion coefficients that can be used for astrophysical and heliospheric applications and discuss the implications of our results for accretion flows onto black holes. In particular, we show that particle acceleration by subsonic turbulence in radiatively inefficient accretion flows can produce a non-thermal tail in the electron distribution function that is likely important for modeling and interpreting the emission from low luminosity systems such as Sgr A* and M87.

astro-ph.HE↗

arXiv2014

Stochastic Acceleration of Electrons by Fast Magnetosonic Waves in Solar Flares: the Effects of Anisotropy in Velocity andWavenumber Space

We develop a model for stochastic acceleration of electrons in solar flares. As in several previous models, the electrons are accelerated by turbulent fast magnetosonic waves ("fast waves") via transit-time-damping (TTD) interactions. (In TTD interactions, fast waves act like moving magnetic mirrors that push the electrons parallel or anti-parallel to the magnetic field). We also include the effects of Coulomb collisions and the waves' parallel electric fields. Unlike previous models, our model is two-dimensional in both momentum space and wavenumber space and takes into account the anisotropy of the wave power spectrum $F_k$ and electron distribution function $f_{\rm e}$. We use weak turbulence theory and quasilinear theory to obtain a set of equations that describes the coupled evolution of $F_k$ and $f_{\rm e}$. We solve these equations numerically and find that the electron distribution function develops a power-law-like non-thermal tail within a restricted range of energies $E\in (E_{\rm nt}, E_{\rm max})$. We obtain approximate analytic expressions for $E_{\rm nt}$ and $E_{\rm max}$, which describe how these minimum and maximum energies depend upon parameters such as the electron number density and the rate at which fast-wave energy is injected into the acceleration region at large scales. We contrast our results with previous studies that assume that $F_k$ and $f_{\rm e}$ are isotropic, and we compare one of our numerical calculations with the time-dependent hard-x-ray spectrum observed during the June 27, 1980 flare. In our numerical calculations, the electron energy spectra are softer (steeper) than in models with isotropic $F_k$ and $f_{\rm e}$ and closer to the values inferred from observations of solar flares.

astro-ph.SR↗

arXiv2014

Magnetohydrodynamic Slow Mode with Drifting He$^{++}$: Implications for Coronal Seismology and the Solar Wind

The MHD slow mode wave has application to coronal seismology, MHD turbulence, and the solar wind where it can be produced by parametric instabilities. We consider analytically how a drifting ion species (e.g. He$^{++}$) affects the linear slow mode wave in a mainly electron-proton plasma, with potential consequences for the aforementioned applications. Our main conclusions are: 1. For wavevectors highly oblique to the magnetic field, we find solutions that are characterized by very small perturbations of total pressure. Thus, our results may help to distinguish the MHD slow mode from kinetic Alfvén waves and non-propagating pressure-balanced structures, which can also have very small total pressure perturbations. 2. For small ion concentrations, there are solutions that are similar to the usual slow mode in an electron-proton plasma, and solutions that are dominated by the drifting ions, but for small drifts the wave modes cannot be simply characterized. 3. Even with zero ion drift, the standard dispersion relation for the highly oblique slow mode cannot be used with the Alfvén speed computed using the summed proton and ion densities, and with the sound speed computed from the summed pressures and densities of all species. 4. The ions can drive a non-resonant instability under certain circumstances. For low plasma beta, the threshold drift can be less than that required to destabilize electromagnetic modes, but damping from the Landau resonance can eliminate this instability altogether, unless $T_{\mathrm e}/T_{\mathrm p}\gg1$.

physics.space-ph↗

arXiv2014

Intermittency and Alignment in Strong RMHD Turbulence

We develop an analytic model of intermittent, three-dimensional, strong, reduced magnetohydrodynamic turbulence with zero cross helicity. We take the fluctuation amplitudes to have a log-Poisson distribution and incorporate into the model a new phenomenology of scale-dependent dynamic alignment. The log-Poisson distribution in our model is characterized by two parameters. To calculate these parameters, we make use of three assumptions: that the energy cascade rate is independent of scale within the inertial range, that the most intense coherent structures at scale $λ$ are sheet-like with a volume filling factor proportional to $λ$, and that most of the cascade power arises from interactions between exceptionally intense fluctuations and much weaker fluctuations. We then compute the scalings of the power spectrum, the kurtosis, higher-order structure functions, and three different average alignment angles. These scalings appear to be consistent with numerous results from direct numerical simulations.

astro-ph.SR↗

arXiv2013

Limits on Alpha Particle Temperature Anisotropy and Differential Flow from Kinetic Instabilities: Solar Wind Observations

Previous studies have shown that the observed temperature anisotropies of protons and alpha particles in the solar wind are constrained by theoretical thresholds for pressure-anisotropy-driven instabilities such as the Alfvén/ion-cyclotron (A/IC) and fast-magnetosonic/whistler (FM/W) instabilities. In this letter, we use a long period of in-situ measurements provided by the {\em Wind} spacecraft's Faraday cups to investigate the combined constraint on the alpha-proton differential flow velocity and the alpha-particle temperature anisotropy due to A/IC and FM/W instabilities. We show that the majority of the data are constrained to lie within the region of parameter space in which A/IC and FM/W waves are either stable or have extremely low growth rates. In the minority of observed cases in which the growth rate of the A/IC (FM/W) instability is comparatively large, we find relatively higher values of $T_{\perpα}/T_{\perp p}$ ($T_{\parallelα}/T_{\parallel p}$) when alpha-proton differential flow velocity is small, where $T_{\perpα}$ and $T_{\perp p}$ ($T_{\parallelα}$ and $T_{\parallel p}$) are the perpendicular (parallel) temperatures of alpha particles and protons. We conjecture that this observed feature might arise from preferential alpha-particle heating which can drive the alpha particles beyond the instability thresholds.

astro-ph.SR↗

arXiv2013

Direct Numerical Simulations of Reflection-Driven, Reduced MHD Turbulence from the Sun to the Alfven Critical Point

We present direct numerical simulations of inhomogeneous reduced magnetohydrodynamic (RMHD) turbulence between the Sun and the Alfvén critical point. These are the first such simulations that take into account the solar-wind outflow velocity and the radial inhomogeneity of the background solar wind without approximating the nonlinear terms in the governing equations. RMHD turbulence is driven by outward-propagating Alfvén waves ($z^+$ fluctuations) launched from the Sun, which undergo partial non-WKB reflection to produce sunward-propagating Alfvén waves ($z^-$ fluctuations). We present ten simulations with different values of the correlation time $τ_{\rm c\,\sun}^+$ and perpendicular correlation length $L_{\perp \sun}$ of outward-propagating Alfvén waves (AWs) at the coronal base. We find that between 15% and 33% of the $z^+$ energy launched into the corona dissipates between the coronal base and Alfvén critical point. Between 33% and 40% of this input energy goes into work on the solar-wind outflow, and between 22% and 36% escapes as $z^+$ fluctuations through the simulation boundary at $r=r_{\rm A}$. The $z^\pm$ power spectra scale like $k_\perp^{-α^\pm}$, where $k_\perp$ is the wavenumber in the plane perpendicular to $\vec{B}_0$. In our simulation with the smallest value of $τ_{\rm c\,\sun}^+$ ($\sim 2 \mbox{min}$) and largest value of $L_{\perp \sun}$ ($2\times 10^4 \mbox{km}$), we find that $α^+$ decreases approximately linearly with increasing $\ln(r)$, reaching a value of 1.3 at $r=11.1 R_{\sun}$. Our simulations with larger values of $τ_{\rm c\,\sun}^+$ exhibit alignment between the contours of constant $ϕ^+$, $ϕ^-$, $Ω_0^+$, and $Ω_0^-$, where $ϕ^\pm$ are the Elsässer potentials and $Ω_0^\pm$ are the outer-scale parallel Elsässer vorticities.

astro-ph.SR↗

arXiv2013

Perpendicular Ion Heating by Reduced Magnetohydrodynamic Turbulence

Recent theoretical studies argue that the rate of stochastic ion heating in low-frequency Alfvén-wave turbulence is given by $Q_\perp = c_1 [(δu)^3 /ρ] \exp(-c_2/ε)$, where $δu$ is the rms turbulent velocity at the scale of the ion gyroradius $ρ$, $ε= δu/v_{\perp \rm i}$, $v_{\perp \rm i}$ is the perpendicular ion thermal speed, and $c_1$ and $c_2$ are dimensionless constants. We test this theoretical result by numerically simulating test particles interacting with strong reduced magnetohydrodynamic (RMHD) turbulence. The heating rates in our simulations are well fit by this formula. The best-fit values of $c_1$ are $\sim 1$. The best-fit values of $c_2$ decrease (i.e., stochastic heating becomes more effective) as the grid size and Reynolds number of the RMHD simulations increase. As an example, in a $1024^2 \times 256$ RMHD simulation with a dissipation wavenumber of order the inverse ion gyroradius, we find $c_2 = 0.21$. We show that stochastic heating is significantly stronger in strong RMHD turbulence than in a field of randomly phased Alfvén waves with the same power spectrum, because coherent structures in strong RMHD turbulence increase orbit stochasticity in the regions where ions are heated most strongly. We find that $c_1$ increases by a factor of $\sim 3$ while $c_2$ changes very little as the ion thermal speed increases from values $\ll v_{\rm A}$ to values $\sim v_{\rm A}$, where $v_{\rm A}$ is the Alfvén speed. We discuss the importance of these results for perpendicular ion heating in the solar wind.

astro-ph.SR↗

arXiv2013

The Efficiency of Second-Order Fermi Acceleration by Weakly Compressible MHD Turbulence

We investigate the effects of pitch-angle scattering on the efficiency of particle heating and acceleration by MHD turbulence using phenomenological estimates and simulations of non-relativistic test particles interacting with strong, subsonic MHD turbulence. We include an imposed pitch-angle scattering rate, which is meant to approximate the effects of high frequency plasma waves and/or velocity space instabilities. We focus on plasma parameters similar to those found in the near-Earth solar wind, though most of our results are more broadly applicable. An important control parameter is the size of the particle mean free path lambda_{mfp} relative to the scale of the turbulent fluctuations L. For small scattering rates, particles interact quasi-resonantly with turbulent fluctuations in magnetic field strength. Scattering increases the long-term efficiency of this resonant heating by factors of a few-10, but the distribution function does not develop a significant non-thermal power-law tail. For higher scattering rates, the interaction between particles and turbulent fluctuations becomes non-resonant, governed by particles heating and cooling adiabatically as they encounter turbulent density fluctuations. Rapid pitch-angle scattering can produce a power-law tail in the proton distribution function but this requires fine-tuning of parameters. Moreover, in the near-Earth solar wind, a significant power-law tail cannot develop by this mechanism because the particle acceleration timescales are longer than the adiabatic cooling timescale set by the expansion of the solar wind. Our results thus imply that MHD-scale turbulent fluctuations are unlikely to be the origin of the v^{-5} tail in the proton distribution function observed in the solar wind.

astro-ph.SR↗

arXiv2013

Instabilities Driven by the Drift and Temperature Anisotropy of Alpha Particles in the Solar Wind

We investigate the conditions under which parallel-propagating Alfvén/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 $w_{\parallel α} \gtrsim 0.25 v_{\mathrm A}$, where $w_{\parallel α} $ is the parallel alpha-particle thermal speed and $v_{\mathrm A}$ is the Alfvén 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 $w_{\parallel α}/v_{\mathrm A}$, 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 $U_α$ 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 $U_α=0$. We discuss the relevance of our results to alpha particles in the solar wind near 1 AU.

physics.space-ph↗

arXiv2013

A Parallel-Propagating Alfvénic Ion-Beam Instability in the High-Beta Solar Wind

We investigate the conditions under which parallel-propagating Alfvén/ion-cyclotron waves are driven unstable by an isotropic ($T_{\perp α} = T_{\parallelα}$) population of alpha particles drifting parallel to the magnetic field at an average speed $U_α$ with respect to the protons. We derive an approximate analytic condition for the minimum value of $U_α$ needed to excite this instability and refine this result using numerical solutions to the hot-plasma dispersion relation. When the alpha-particle number density is $\simeq 5%$ of the proton number density and the two species have similar thermal speeds, the instability requires that $β_{\rm p} \gtrsim 1$, where $β_{\rm p}$ is the ratio of the proton pressure to the magnetic pressure. For $1\lesssim β_{\mathrm p}\lesssim 12$, the minimum $U_α$ needed to excite this instability ranges from $0.7v_{\mathrm A}$ to $0.9v_{\mathrm A}$, where $v_{\mathrm A}$ is the Alfvén speed. This threshold is smaller than the threshold of $\simeq 1.2v_{\mathrm A}$ for the parallel magnetosonic instability, which was previously thought to have the lowest threshold of the alpha-particle beam instabilities at $β_{\mathrm p}\gtrsim 0.5$. We discuss the role of the parallel Alfvénic drift instability for the evolution of the alpha-particle drift speed in the solar wind. We also analyze measurements from the \emph{Wind} spacecraft's Faraday cups and show that the $U_α$ values measured in solar-wind streams with $T_{\perp α}\approx T_{\parallelα}$ are approximately bounded from above by the threshold of the parallel Alfvénic instability.

physics.space-ph↗

arXiv2013

Observational Test of Stochastic Heating in Low-$β$ Fast Solar Wind Streams

Spacecraft measurements show that protons undergo substantial perpendicular heating during their transit from the Sun to the outer heliosphere. In this paper, we use {\em Helios~2} measurements to investigate whether stochastic heating by low-frequency turbulence is capable of explaining this perpendicular heating. We analyze {\em Helios~2} magnetic-field measurements in low-$β$ 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, $δB_{\rm p}$, near the proton gyroradius scale $ρ_{\rm p}$. We then evaluate the stochastic heating rate $Q_{\perp \rm stoch}$ using the measured value of $δB_{\rm p}$ and a previously published analytical formula for $Q_{\perp \rm stoch}$. Using {\em Helios} measurements we estimate the `empirical' perpendicular heating rate $Q_{\perp \rm emp} = (k_{\rm B}/m_{\rm p}) B V (d/dr) (T_{\perp \rm p}/B)$ that is needed to explain the $T_{\perp \rm p}$ profile. We find that $Q_{\perp \rm stoch} \sim Q_{\perp \rm emp}$, but only if a key dimensionless constant appearing in the formula for $Q_{\perp \rm stoch}$ 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-$β$ fast-wind streams.

astro-ph.SR↗

arXiv2013

The Dispersion Relations and Instability Thresholds of Oblique Plasma Modes in the Presence of an Ion Beam

An ion beam can destabilize Alfvén/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 the minimum beam speed required to excite such instabilities is significantly smaller for oblique modes with $\vec k \times \vec B_0\neq 0$ than for parallel-propagating modes with $\vec k \times \vec B_0 = 0$, where $\vec k$ is the wavevector and $\vec B_0$ is the background magnetic field. In this paper, we explain this difference within the framework of quasilinear theory, focusing on low-$β$ 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 Alfvén/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.

physics.space-ph↗

arXiv2012

Resonance Broadening and Heating of Charged Particles in Magnetohydrodynamic Turbulence

The heating, acceleration, and pitch-angle scattering of charged particles by MHD turbulence are important in a wide range of astrophysical environments, including the solar wind, accreting black holes, and galaxy clusters. We simulate the interaction of high-gyrofrequency test particles with fully dynamical simulations of subsonic MHD turbulence, focusing on the parameter regime with beta ~ 1, where beta is the ratio of gas to magnetic pressure. We use the simulation results to calibrate analytical expressions for test particle velocity-space diffusion coefficients and provide simple fits that can be used in other work. The test particle velocity diffusion in our simulations is due to a combination of two processes: interactions between particles and magnetic compressions in the turbulence (as in linear transit-time damping; TTD) and what we refer to as Fermi Type-B (FTB) interactions, in which charged particles moving on field lines may be thought of as beads spiralling around moving wires. We show that test particle heating rates are consistent with a TTD resonance which is broadened according to a decorrelation prescription that is Gaussian in time. TTD dominates the heating for v_s >> v_A (e.g. electrons), where v_s is the thermal speed of species s and v_A is the Alfven speed, while FTB dominates for v_s << v_A (e.g. minor ions). Proton heating rates for beta ~ 1 are comparable to the turbulent cascade rate. Finally, we show that velocity diffusion of collisionless, large gyrofrequency particles due to large-scale MHD turbulence does not produce a power-law distribution function.

astro-ph.SR↗

arXiv2011

Incorporating Kinetic Physics into a Two-Fluid Solar-Wind Model with Temperature Anisotropy and Low-Frequency Alfven-Wave Turbulence

We develop a 1D 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 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 Alfven-wave turbulence plays an important role in the origin of the solar wind.

astro-ph.SR↗