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Abstract Analytical solutions for 2D and slab turbulence energies in the solar corona are presented, including a derivation of the corresponding correlation lengths, with implications for the proton and electron temperatures in the solar corona. These solutions are derived by solving the transport equations for 2D and slab turbulence energies and their correlation lengths, as well as proton and electron pressures. The solutions assume background profiles for the solar wind speed, solar wind mass density, and Alfvén velocity. Our analytical solutions can be related to those obtained from joint Parker Solar Probe and Solar Orbiter Metis coronagraph observations, as reported in Telloni et al. We find that the solution for 2D turbulence energy in the absence of nonlinear dissipation decreases more slowly compared to the dissipative solution. The solution for slab turbulence energy with no dissipation exhibits a more rapid increase compared to the dissipative solution. The proton heating rate is found to be about 82% of the total plasma heating rate at 6.3R_⊙, which gradually decreases with increasing distance, eventually becoming ∼80% of the total plasma heating rate at ∼13R_⊙, consistent with that found by Bandyopadhyay et al. (2023). These analytical solutions provide valuable insight for our understanding of turbulence, and its effect on proton and electron heating rates, in the solar corona. We compare the numerically solved turbulent transport equations for the 2D and slab turbulence energies, correlation lengths, and proton and electron pressures with the analytical solutions, finding good agreement between them. 

Abstract We study solar wind turbulence anisotropy in the inertial and energy-containing ranges in the inbound and outbound directions during encounters 1–9 by the Parker Solar Probe (PSP) for distances between ∼21 and 65R_⊙. Using the Adhikari et al. approach, we derive theoretical equations to calculate the ratio between the 2D and slab fluctuating magnetic energy, fluctuating kinetic energy, and the outward/inward Elsässer energy in the inertial range. For this, in the energy-containing range, we assume a wavenumberk⁻¹power law. In the inertial range, for the magnetic field fluctuations and the outward/inward Elsässer energy, we consider that (i) both 2D and slab fluctuations follow a power law ofk^−5/3, and (ii) the 2D and slab fluctuations follow the power laws withk^−5/3andk^−3/2, respectively. For the velocity fluctuations, we assume that both the 2D and slab components follow ak^−3/2power law. We compare the theoretical results of the variance anisotropy in the inertial range with the derived observational values measured by PSP, and find that the energy density of 2D fluctuations is larger than that of the slab fluctuations. The theoretical variance anisotropy in the inertial range relating to thek^−5/3andk^−3/2power laws between 2D and slab turbulence exhibits a smaller value in comparison to assuming the same power lawk^−5/3between 2D and slab turbulence. Finally, the observed turbulence energy measured by PSP in the energy-containing range is found to be similar to the theoretical result of a nearly incompressible/slab turbulence description. 

Abstract The nature and radial evolution of solar wind electrons in the suprathermal energy range are studied. A wave–particle interaction tensor and a Fokker–Planck Coulomb collision operator are introduced into the kinetic transport equation describing electron collisions and resonant interactions with whistler waves. The diffusion tensor includes diagonal and off-diagonal terms, and the Coulomb collision operator applies to arbitrary electron velocities describing collisions with both background protons and electrons. The background proton and electron densities and temperatures are based on previous turbulence models that mediate the supersonic solar wind. The electron velocity distribution functions and electron heat flux are calculated. Comparison and analysis of the numerical results with analytical solutions and observations in the near-Sun region are made. The numerical results reproduce well the creation of the sunward electron deficit observed in the near-Sun region. The deficit of the electron velocity distribution function below the core Maxwellian fit at low velocities results from Coulomb collisions, and the excess part above the core Maxwellian fit at high velocities is determined by strong wave–particle interactions. 

Abstract Electron beams that are commonly observed in the corona were discovered to be associated with solar flares. These “coronal” electron beams are found ≥300 Mm above the acceleration region and have velocities ranging from 0.1 c up to 0.6 c . However, the mechanism for producing these beams remains unclear. In this paper, we use kinetic transport theory to investigate how isotropic suprathermal energetic electrons escaping from the acceleration region of flares are transported upwardly along the magnetic field lines of flares to develop coronal electron beams. We find that magnetic focusing can suppress the diffusion of Coulomb collisions and background turbulence and sharply collimate the suprathermal electron distribution into beams with the observed velocity within the observed distance. A higher bulk velocity is produced if energetic electrons have harder energy spectra or travel along a more rapidly expanding coronal magnetic field. By modeling the observed velocity and location distributions of coronal electron beams, we predict that the temperature of acceleration regions ranges from 5 × 10 6 to 2 × 10 7 K. Our model also indicates that the acceleration region may have a boundary where the temperature abruptly decreases so that the electron beam velocity can become more than triple (even up to 10 times) the background thermal velocity and produce the coronal type III radio bursts. 

Abstract Observations of Type III radio bursts discovered that electron beams with power-law energy spectra are commonly produced during solar flares. The locations of these electron beams are ~ 300 Mm above the particle acceleration region of the photosphere, and the velocities range from 3 to 10 times the local background electron thermal velocity. However, the mechanism that can commonly produce electron beams during the propagation of energetic electrons with power-law energy spectra in the corona remains unclear. In this paper, using kinetic transport theory, we find for the first time that the magnetic focusing effect governs the formation of electron beams over the observational desired distance in the corona. The magnetic focusing effect can sharply increase the bulk velocity of energetic electrons to the observed electron beam velocity within 0.4 solar radii (300 Mm) as they escape from the acceleration region and propagate upward along magnetic field lines. In more rapidly decreasing magnetic fields, energetic electrons with a harder power-law energy spectrum can generate a higher bulk velocity, producing type III radio bursts at a location much closer to the acceleration region. During propagation, the spectral index of the energetic electrons is unchanged. 

Abstract The electron VDF in the solar wind consists of a Maxwellian core, a suprathermal halo, a field-aligned component strahl, and an energetic superhalo that deviates from the equilibrium. Whistler wave turbulence is thought to resonantly scatter the observed electron velocity distribution. Wave–particle interactions that contribute to Whistler wave turbulence are introduced into a Fokker–Planck kinetic transport equation that describes the interaction between the suprathermal electrons and the Whistler waves. A recent numerical approach for solving the Fokker–Planck kinetic transport equation has been extended to include a full diffusion tensor. Application of the extended numerical approach to the transport of solar wind suprathermal electrons influenced by Whistler wave turbulence is presented. Comparison and analysis of the numerical results with observations and diagonal-only model results are made. The off-diagonal terms in the diffusion tensor act to depress effects caused by the diagonal terms. The role of the diffusion coefficient on the electron heat flux is discussed.