We present a stochastic field line mapping model where the interplanetary magnetic field lines are described by a density distribution function satisfying a Fokker–Planck equation that is solved numerically. Due to the spiral geometry of the nominal Parker field and to the evolving nature of solar wind turbulence, the heliospheric diffusion of the magnetic field lines is both heterogeneous and anisotropic, including a radial component. The longitudinal distributions of the magnetic field lines are shown to be close to circular Gaussian distributions, although they develop a noticeable skewness. The magnetic field lines emanating from the Sun are found to differ, on average, from the spirals predicted by Parker. Although the spirals remain close to Archimedean, they are here underwound, on average. Our model predicts a spiral angle that is smaller by ∼5° than the Parker spiral angle at Earth’s orbit for the same solar wind speed of V _sw = 400 km s ^−1 . It also predicts an angular position on the solar disk of the best magnetically connected footpoint to an observer at 1 au that is shifted westward by ∼10° with respect to the Parker’s field model. This significantly changes the angle of the most probable magnetic connection between possible sources on the Sun and observers in the inner heliosphere. The results have direct implications for the heliospheric transport of “scatter-free” electrons accelerated in the aftermath of solar eruptions.

Turbulent rotations of the magnetic field vector are observed in the Alfvénic streams of the solar wind where the magnetic field strength remains close to a constant. They can lead to reversals of the radial magnetic field component or switchbacks. It is not ruled out from the data that the rotations are divisible into the sum of small random angular deflections. In this work, we develop tools aimed at the analysis of the one-point statistical properties of the directional fluctuations of the magnetic field vector in the solar wind. The angular fluctuations are modeled by a drift-diffusion process which admits the exponential distribution as steady-state solution. Realizations of the stochastic process are obtained by solving the corresponding Langevin equation. It is shown that the cumulative effects of consecutive small-angle deflections can yield frequent reversals of the magnetic field vector even when the concentration parameter of the directional data is large. The majority of the rotations are associated with nearly transverse magnetic field fluctuations in this case.

Lagrangian perspectives on the small-scale structure of anisotropic Alfvénic turbulence are adopted. We are interested in relating the statistical properties of the Eulerian field increments evaluated along the fluid particle trajectories, in the direction perpendicular to the guiding magnetic field and along the magnetic field lines. We establish the basis for a unified multifractal phenomenology of Eulerian and Lagrangian Alfvénic turbulence. The critical balance condition is generalized to structure functions of an order different than 2. A Lagrangian perspective is not only useful for investigating the small-scale structure of Alfvénic turbulence, it is also tailored to the modeling of large-scale turbulent transport. Therefore, we develop Lagrangian stochastic models for the dispersion of fluid particles and magnetic field lines in the solar wind. The transport models are based on the integrated Ornstein–Uhlenbeck process that is not Markov, yielding smooth stochastic fluid particle trajectories and magnetic field lines. Brownian diffusion is recovered by tending the integral scale parameter to zero while keeping the diffusivity finite.

In this work, we extend Leighton’s diffusion model describing the turbulent mixing of magnetic footpoints on the solar wind source surface. The present Lagrangian stochastic model is based on the spherical Ornstein–Uhlenbeck process with drift that is controlled by the rotation frequency Ω of the Sun, the Lagrangian integral timescale τ L , and the root-mean-square footpoint velocity V rms . The Lagrangian velocity and the positions of magnetic footpoints on the solar wind source surface are obtained from the solutions of a set of stochastic differential equations, which are solved numerically. The spherical diffusion model of Leighton is recovered in the singular Markov limit when the Lagrangian integral timescale tends to zero while keeping the footpoint diffusivity finite. In contrast to the magnetic field lines driven by standard Brownian processes on the solar wind source surface, the interplanetary magnetic field lines are smooth differentiable functions with finite path lengths in our model. The path lengths of the boundary-driven interplanetary magnetic field lines and their probability distributions at 1 au are computed numerically, and their dependency with respect to the controlling parameters is investigated. The path-length distributions are shown to develop a significant skewness as the width of the distributions increases.

It was recently shown that, owing to the turbulent nature of the solar wind, the interplanetary magnetic field lines can be well described by stochastic Parker spirals. These are realizations of Brownian diffusion on a sphere of increasing radius, superimposed on the angular drift due to the solar rotation. In this work, we present a model for the transport of solar energetic particles along stochastic Parker spirals in the inner heliosphere. The transport model is governed by a set of four stochastic differential equations for the heliographic position ( r , α = cos θ , ϕ ) of the guiding centers and the cosine of the pitch angle between the velocity vector and the Parker field. The model accounts for the role played by the combination of pitch angle scattering and magnetic focusing in the interplanetary medium. The effects of the dynamical evolution of the turbulence are included in the model by taking the field line angular diffusivity to be a function of the radial distance from the Sun. The heliolongitudinal distribution of particles propagating along stochastic Parker spirals is given by the wrapped Gaussian distribution. This angular distribution can also well be represented by the von Mises distribution that interpolates between the Gaussian distribution at small angular spread and the uniform distribution at large distances from the acceleration region of energetic particles in the aftermath of a solar eruption.

We present a Langevin model describing the local structure of the interplanetary magnetic field lines. It is established on the basis of the analysis of the Lagrangian properties of strong Alfvénic turbulence, which provides a new perspective on the critical balance condition. The model is consistent with the k∥−2 spectrum of magnetic fluctuations derived from in situ measurements. We show that the magnetic field line diffusivity at the spacecraft position can be inferred from the wavelet analysis of one-point measurements of the fluctuating magnetic fields in the solar wind independently of the three-dimensional nature of the anisotropy.

Energetic electrons in impulsive events can serve as an ideal probe of solar wind magnetic field. Using a recently developed Fractonal Velocity Dispersion Analysis (FVDA), the release time at the Sun and the path length of interplanetary magnetic field can be obtained with very small uncertainties in many impulsive events. Further knowing the source location, one can examine how much do the field lines deviate from the Parker spiral. In this work, we present an analytic model for the angular dispersion of magnetic field lines that results from the turbulence in the solar wind and at the solar source surface. The heliospheric magnetic field lines in this model is derived from a Hamiltonian $H_{\rm m}(\mu, \phi, r)$ in which the pair of canonically conjugated variables the cosine of the heliographic colatitude $\mu$ and the longitude $\phi$. This model naturally incorporates the effect of a random footpoint motion on the source surface since such a motion is due to the zero-frequency component of the solar wind turbulence. Assuming the footpoint motion is also diffusive, it is shown that the angular diffusivity of the stochastic Parker spirals is given by the angular diffusivity of the footpoints divided by the solar wind speed and is controlled by a unique parameter which is the Kubo number. We also present some model calculations of meandering field lines resulting from stochastic footpoint motion and statistical results of the field line path length from observations. Our model and statistical results can shed lights on observations made by Parker Solar Probe and Solar Orbiter.

An analytic model for the angular dispersion of magnetic field lines resulting from the turbulence in the solar wind and at the solar source surface is presented. The heliospheric magnetic field lines in our model are derived from a Hamiltonian with the pair of canonically conjugated variables the cosine of the heliographic colatitude μ and the longitude ϕ. In the diffusion approximation, the Parker spirals are modeled by a set of stochastic differential equations for θ and ϕ as functions of r. These stochastic Parker spirals are realizations of a standard random walk on a sphere of increasing radius, superimposed on an angular drift due to solar rotation. The Green function solution of the Fokker–Planck equation describing the angular diffusion of the field line density is obtained in terms of spherical harmonics. Magnetic field lines traced from an observer back to the Sun are realizations of a Brownian bridge. Our model incorporates the effect of the random footpoint motions at the source surface, which is associated with the zero-frequency component of the solar wind turbulence. Assuming that the footpoint motion is diffusive, its contribution to the angular diffusivity of the stochastic Parker spirals is then given by the angular diffusivity of the footpoints divided by the solar wind speed and is controlled by a unique parameter, which is the Kubo number.

Combining in situ measurements of energetic electrons and remote sensing observations of hard X‐rays and type III radio bursts, we examine the release times of energetic electrons in the July 23, 2016 event. We find that the releases of in situ energetic electrons from the Sun are delayed from those electrons that are responsible for the hard X‐rays. We further find that the release time of in situ electrons is a function of electron energy. Under the assumption that the acceleration mechanism for the upward propagating electrons is of Fermi‐type and is controlled by an energy‐dependent diffusion coefficient, we fit these release times by a simple functional form, related to the turbulence spectral index. Implications of our study on the underlying electron acceleration mechanisms and the magnetic reconnection process in solar flares are discussed. Our results demonstrate the power of the recently developed fractional velocity dispersion analysis (FVDA) method in solar flare studies.

The solar atmosphere is dominated by loops of magnetic fluxes that connect the multi-million degree corona to the much cooler chromosphere. The temperature and density structure of quasi-static loops are determined by the continuous flow of energy from the hot corona to the lower solar atmosphere. Loop scaling laws provide relationships between global properties of the loop (such as the peak temperature, pressure, and length); they follow from the physical variable dependencies of various terms in the energy equation, and, hence, the form of the loop scaling law provides insight into the key physics that control the loop structure. Traditionally, scaling laws have been derived under the assumption of collision-dominated thermal conduction. Here, we examine the impact of different regimes of thermal conduction-collision-dominated, turbulence-dominated, and free-streaming-on the form of the scaling laws relating the loop temperature and heating rate to its pressure and half-length. We show that the scaling laws for turbulence-dominated conduction are fundamentally different than those for collision-dominated and free-streaming conduction, inasmuch as the form of the scaling laws now depend primarily on conditions at the low-temperature, rather than high-temperature, part of the loop. We also establish regimes in the temperature and density space in which each of the applicable scaling laws prevail.

We present an analytic model for the intensity-time profile of solar energetic particle events, based on considerations related to the delay-time distribution that results from interplanetary scattering of a focused distribution of energetic particles. For instantaneous injection, the intensity-time profile is shown to be well described by a Lévy distribution at times up to and just past the time of peak intensity. This allows a fitting procedure that uses velocity dispersion analysis of both the onset time and the peak time to estimate the time of particle release, the magnetic connection length, and the scattering length (and its dependence on velocity). Furthermore, the delay-time distribution can be used as a Green function response when taking into account particle injections of finite duration, in order to infer the release time profile in extended events.

Electromagnetic wave scattering off density inhomogeneities in the solar corona is an important process that determines both the apparent source size and the time profile of radio bursts observed at 1 au. Here we model the scattering process using a Fokker–Planck equation and apply this formalism to several regimes of interest. In the first regime the density fluctuations are considered quasi-static and diffusion in wavevector space is dominated by angular diffusion on the surface of a constant energy sphere. In the small-angle (“pencil-beam”) approximation, this diffusion further occurs over a small solid angle in wavevector space. The second regime corresponds to a much later time, by which scattering has rendered the photon distribution near-isotropic, resulting in a spatial diffusion of the radiation. The third regime involves time-dependent fluctuations and, therefore, Fermi acceleration of photons. Combined, these results provide a comprehensive theoretical framework within which to understand several important features of propagation of radio burst waves in the solar corona: emitted photons are accelerated in a relatively small inner region and then diffuse outward to larger distances. En route, angular diffusion results both in source sizes that are substantially larger than the intrinsic source and in observed intensity-versus-time profiles that are asymmetric, with a sharp rise and an exponential decay. Both of these features are consistent with observations of solar radio bursts.

The heat flux in a plasma is determined by the degree of anisotropy in the particle distribution function, which is in turn driven by gradients in the ambient density and temperature. When the mean free path at the thermal speed is substantially smaller than the scale length associated with the temperature variation, the heat flux simply depends on the local value of the temperature gradient. However, when the temperature scale length and mean free path are comparable, heat conduction becomes substantially non-local in character: the magnitude of the heat flux now depends on the overall temperature profile and is generally smaller than the locally determined value. In the presence of angular scattering associated with turbulence, the mean free path (and its velocity dependence) can be significantly smaller than its collisional value; this makes the expression for the heat flux more local in character, but also results in a heat flux that is lower than that obtained through a purely collisional analysis. Therefore, whether or not turbulence is present, the heat flux is generally smaller than the value obtained from a local collisional analysis. We here present an analytic expression for the conductive heat flux in terms of a convolution of the local heat flux with a non-local kernel function that incorporates both Coulomb collisions and turbulent scattering. We comment on the need to include both non-local and turbulent scattering effects in the modeling of quasi-static active region loops and in the conductive cooling of post-flare loops.

A considerable fraction of the energy in a solar flare is released as suprathermal electrons; such electrons play a major role in energy deposition in the ambient atmosphere, and hence the atmospheric response to flare heating. Historically, the transport of these particles has been approximated through a deterministic approach in which first-order secular energy loss to electrons in the ambient target is treated as the dominant effect, with second-order diffusive terms (in both energy and angle) being generally either treated as a small correction or neglected. However, it has recently been pointed out that while neglect of diffusion in energy may indeed be negligible, diffusion in angle is of the same order as deterministic scattering and hence must be included. Here we therefore investigate the effect of angular scattering on the energy deposition profile in the flaring atmosphere. A relatively simple compact expression for the spatial distribution of energy deposition into the ambient plasma is presented and compared with the corresponding deterministic result. For unidirectional injection there is a significant shift in heating from the lower corona to the upper corona; this shift is much smaller for isotropic injection. We also compare the heating profiles due to return current ohmic heating in the diffusional and deterministic models.

Using the "enthalpy-based thermal evolution of loops" (EBTEL) model, we investigate the hydrodynamics of the plasma in a flaring coronal loop in which heat conduction is limited by turbulent scattering of the electrons that transport the thermal heat flux. The EBTEL equations are solved analytically in each of the two (conduction-dominated and radiation-dominated) cooling phases. Comparison of the results with typical observed cooling times in solar flares shows that the turbulent mean free path λT lies in a range corresponding to a regime in which classical (collision-dominated) conduction plays at most a limited role. We also consider the magnitude and duration of the heat input that is necessary to account for the enhanced values of temperature and density at the beginning of the cooling phase and for the observed cooling times. We find through numerical modeling that in order to produce a peak temperature ≃1.5 × 107 K and a 200 s cooling time consistent with observations, the flare-heating profile must extend over a significant period of time; in particular, its lingering role must be taken into consideration in any description of the cooling phase. Comparison with observationally inferred values of post-flare loop temperatures, densities, and cooling times thus leads to useful constraints on both the magnitude and duration of the magnetic energy release in the loop, as well as on the value of the turbulent mean free path λT .

The original version of this article contained errors in Refs 15, 27, 32, 33 and 43, which were incorrectly given with the wrong journal name “ Solid Phys .” rather than the correct “ Sol. Phys .”. This has now been corrected in the PDF and HTML versions of the article.

The transport of the energy contained in suprathermal electrons in solar flares plays a key role in our understanding of many aspects of flare physics, from the spatial distributions of hard X-ray emission and energy deposition in the ambient atmosphere to global energetics. Historically the transport of these particles has been largely treated through a deterministic approach, in which first-order secular energy loss to electrons in the ambient target is treated as the dominant effect, with second-order diffusive terms ( in both energy and angle) generally being either treated as a small correction or even neglected. Here, we critically analyze this approach, and we show that spatial diffusion through pitch-angle scattering necessarily plays a very significant role in the transport of electrons. We further show that a satisfactory treatment of the diffusion process requires consideration of non-local effects, so that the electron flux depends not just on the local gradient of the electron distribution function but on the value of this gradient within an extended region encompassing a significant fraction of a mean free path. Our analysis applies generally to pitch-angle scattering by a variety of mechanisms, from Coulomb collisions to turbulent scattering. We further show that the spatial transport of electrons along the magnetic field of a flaring loop can be modeled rather effectively as a Continuous Time Random Walk with velocity-dependent probability distribution functions of jump sizes and occurrences, both of which can be expressed in terms of the scattering mean free path.

We study the influence of kinetic-scale Alfvénic turbulence on the generation of plasma radio emission in the solar coronal regions where the ratio \(\upbeta\) of plasma to magnetic pressure is lower than the electron-to-ion mass ratio \(m_{\mathrm{e}}/m_{\mathrm{i}}\). The present study is motivated by the phenomenon of solar type I radio storms that are associated with the strong magnetic field of active regions. The measured brightness temperature of the type I storms can be up to \(10^{10}~\mbox{K}\) for continuum emission, and can exceed \(10^{11}~\mbox{K}\) for type I bursts. At present, there is no generally accepted theory explaining such high brightness temperatures and some other properties of the type I storms. We propose a model with an imbalanced turbulence of kinetic-scale Alfvén waves that produce an asymmetric quasi-linear plateau on the upper half of the electron velocity distribution. The Landau damping of resonant Langmuir waves is suppressed and their amplitudes grow spontaneously above the thermal level. The estimated saturation level of Langmuir waves is high enough to generate observed type I radio emission at the fundamental plasma frequency. Harmonic emission does not appear in our model because the backward-propagating Langmuir waves undergo strong Landau damping. Our model predicts \(100\%\) polarization in the sense of the ordinary (o-) mode of type I emission.

The energy released in solar flares derives from a reconfiguration of magnetic fields to a lower energy state, and is manifested in several forms, including bulk kinetic energy of the coronal mass ejection, acceleration of electrons and ions, and enhanced thermal energy that is ultimately radiated away across the electromagnetic spectrum from optical to x rays. Using an unprecedented set of coordinated observations, from a suite of instruments, we here report on a hitherto largely overlooked energy component-the kinetic energy associated with small-scale turbulent mass motions. We show that the spatial location of, and timing of the peak in, turbulent kinetic energy together provide persuasive evidence that turbulent energy may play a key role in the transfer of energy in solar flares. Although the kinetic energy of turbulent motions accounts, at any given time, for only ∼(0.5-1)% of the energy released, its relatively rapid (∼1-10  s) energization and dissipation causes the associated throughput of energy (i.e., power) to rival that of major components of the released energy in solar flares, and thus presumably in other astrophysical acceleration sites.

We investigate the impact of turbulent suppression of parallel heat conduction on the cooling of post-flare coronal loops. Depending on the value of the mean free path lambda(T) associated with the turbulent scattering process, we identify four main cooling scenarios. The overall temperature evolution, from an initial temperature in excess of 10(7) K, is modeled in each case, highlighting the evolution of the dominant cooling mechanism throughout the cooling process. Comparison with observed cooling times allows the value of lT to be constrained, and interestingly this range corresponds to situations where collision-dominated conduction plays a very limited role, or even no role at all, in the cooling of post-flare coronal loops.