In this review we will focus on a topic of fundamental importance for both plasma physics and astrophysics, namely the occurrence of large-amplitude low-frequency fluctuations of the fields that describe the plasma state. This subject will be treated within the context of the expanding solar wind and the most meaningful advances in this research field will be reported emphasizing the results obtained in the past decade or so. As a matter of fact, Ulysses’ high latitude observations and new numerical approaches to the problem, based on the dynamics of complex systems, brought new important insights which helped to better understand how turbulent fluctuations behave in the solar wind. In particular, numerical simulations within the realm of magnetohydrodynamic (MHD) turbulence theory unraveled what kind of physical mechanisms are at the basis of turbulence generation and energy transfer across the spectral domain of the fluctuations. In other words, the advances reached in these past years in the investigation of solar wind turbulence now offer a rather complete picture of the phenomenological aspect of the problem to be tentatively presented in a rather organic way.

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The Solar Wind as a Turbulence Laboratory

This paper presents a theoretical framework for understanding plasma turbulence in astrophysical plasmas. It is motivated by observations of electromagnetic and density fluctuations in the solar wind, interstellar medium and galaxy clusters, as well as by models of particle heating in accretion disks. All of these plasmas and many others have turbulent motions at weakly collisional and collisionless scales. The paper focuses on turbulence in a strong mean magnetic field. The key assumptions are that the turbulent fluctuations are small compared to the mean field, spatially anisotropic with respect to it and that their frequency is low compared to the ion cyclotron frequency. The turbulence is assumed to be forced at some system-specific outer scale. The energy injected at this scale has to be dissipated into heat, which ultimately cannot be accomplished without collisions. A kinetic cascade develops that brings the energy to collisional scales both in space and velocity. The nature of the kinetic cascade in various scale ranges depends on the physics of plasma fluctuations that exist there. There are four special scales that separate physically distinct regimes: the electron and ion gyroscales, the mean free path and the electron diffusion scale. In each of the scale ranges separated by these scales, the fully kinetic problem is systematically reduced to a more physically transparent and computationally tractable system of equations, which are derived in a rigorous way. In the inertial range above the ion gyroscale, the kinetic cascade separates into two parts: a cascade of Alfvenic fluctuations and a passive cascade of density and magnetic-field-strength fluctuations. The former are governed by the reduced magnetohydrodynamic (RMHD) equations at both the collisional and collisionless scales; the latter obey a linear kinetic equation along the (moving) field lines associated with the Alfvenic component (in the collisional limit, these compressive fluctuations become the slow and entropy modes of the conventional MHD). In the dissipation range below ion gyroscale, there are again two cascades: the kinetic-Alfven-wave (KAW) cascade governed by two fluid-like electron reduced magnetohydrodynamic (ERMHD) equations and a passive cascade of ion entropy fluctuations both in space and velocity. The latter cascade brings the energy of the inertial-range fluctuations that was Landau-damped at the ion gyroscale to collisional scales in the phase space and leads to ion heating. The KAW energy is similarly damped at the electron gyroscale and converted into electron heat. Kolmogorov-style scaling relations are derived for all of these cascades. The relationship between the theoretical models proposed in this paper and astrophysical applications and observations is discussed in detail.

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Astrophysical Gyrokinetics: Kinetic and Fluid Turbulent Cascades in Magnetized Weakly Collisional Plasmas

We present a series of models for the plasma properties along open magnetic flux tubes rooted in solar coronal holes, streamers, and active regions. These models represent the first self-consistent solutions that combine (1) chromospheric heating driven by an empirically guided acoustic wave spectrum; (2) coronal heating from Alfven waves that have been partially reflected, then damped by anisotropic turbulent cascade; and (3) solar wind acceleration from gradients of gas pressure, acoustic wave pressure, and Alfven wave pressure. The only input parameters are the photospheric lower boundary conditions for the waves and the radial dependence of the background magnetic field along the flux tube. We have not included multifluid or collisionless effects (e.g., preferential ion heating), which are not yet fully understood. For a single choice for the photospheric wave properties, our models produce a realistic range of slow and fast solar wind conditions by varying only the coronal magnetic field. Specifically, a two-dimensional model of coronal holes and streamers at solar minimum reproduces the latitudinal bifurcation of slow and fast streams seen by Ulysses. The radial gradient of the Alfven speed affects where the waves are reflected and damped, and thus whether energy is deposited below or above the Parker critical point. As predicted by earlier studies, a larger coronal "expansion factor" gives rise to a slower and denser wind, higher temperature at the coronal base, less intense Alfven waves at 1 AU, and correlative trends for commonly measured ratios of ion charge states and FIP-sensitive abundances that are in general agreement with observations. These models offer supporting evidence for the idea that coronal heating and solar wind acceleration (in open magnetic flux tubes) can occur as a result of wave dissipation and turbulent cascade.

http://iopscience.iop.org/article/10.1086/518001/pdf

Self-consistent Coronal Heating and Solar Wind Acceleration from Anisotropic Magnetohydrodynamic Turbulence

We present a comprehensive model of the global properties of Alfven waves in the solar atmosphere and the fast solar wind. Linear non-WKB wave transport equations are solved from the photosphere to a distance past the orbit of the Earth, and for wave periods ranging from 3 s to 3 days. We derive a radially varying power spectrum of kinetic and magnetic energy fluctuations for waves propagating in both directions along a superradially expanding magnetic flux tube. This work differs from previous models in three major ways. (1) In the chromosphere and low corona, the successive merging of flux tubes on granular and supergranular scales is described using a two-dimensional magnetostatic model of a network element. Below a critical flux-tube merging height the waves are modeled as thin-tube kink modes, and we assume that all of the kink-mode wave energy is transformed into volume-filling Alfven waves above the merging height. (2) The frequency power spectrum of horizontal motions is specified only at the photosphere, based on prior analyses of G-band bright point kinematics. Everywhere else in the model the amplitudes of outward and inward propagating waves are computed with no free parameters. We find that the wave amplitudes in the corona agree well with off-limb nonthermal line-width constraints. (3) Nonlinear turbulent damping is applied to the results of the linear model using a phenomenological energy loss term. A single choice for the normalization of the turbulent outer-scale length produces both the right amount of damping at large distances (to agree with in situ measurements) and the right amount of heating in the extended corona (to agree with empirically constrained solar wind acceleration models). In the corona, the modeled heating rate differs by more than an order of magnitude from a rate based on isotropic Kolmogorov turbulence.

https://iopscience.iop.org/article/10.1086/426507/pdf

On the generation, propagation, and reflection of alfven waves from the solar photosphere to the distant heliosphere

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.

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Solar Wind Electrons Alphas and Protons (SWEAP) Investigation: Design of the Solar Wind and Coronal Plasma Instrument Suite for Solar Probe Plus

Identifying intermittency events in the solar wind

Direct evidence for the presence of an inertial energy cascade, the most characteristic signature of hydromagnetic turbulence (MHD), is observed in the solar wind by the Ulysses spacecraft. After a brief rederivation of the equivalent of Yaglom's law for MHD turbulence, a linear relation is indeed observed for the scaling of mixed third-order structure functions involving Elsasser variables. This experimental result firmly establishes the turbulent character of low-frequency velocity and magnetic field fluctuations in the solar wind plasma.

Observation of inertial energy cascade in interplanetary space plasma.

Plasma and magnetic field measurements by Ulysses are used to investigate the radial evolution of hourly-scale Alfvenic fluctuations in the polar wind. The data span from 1.4 to 4.3 AU in heliocentric distance. Different radial regimes at different distances emerge. Inside about 2.5 AU the large outward traveling fluctuations decrease faster, in terms of energy per unit mass, than the small inward ones. This is in agreement with previous low-latitude observations inside 1 AU within the trailing edge of fast streams. As a result of this different gradient the ratio of inward to outward fluctuation energy rises to about 0.5 near 2.5 AU. Beyond this distance the radial gradient of the inward fluctuations becomes increasingly steeper, while that of the outward ones does not vary appreciably. A state is quickly reached where both populations decline at almost the same rate. These results on the behavior of outward and inward Alfvenic fluctuations are new and represent a constraint for models of turbulence evolution in steadily expanding flows like the polar wind. Finally, an extrapolation to regions near the Sun would suggest that Alfvenic fluctuations at hourly scale should not play a relevant role in solar wind heating and acceleration. Obviously, this last conclusion may be invalidated by non-WKB effects and by compressive and dissipative processess close to the Sun.

https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/1999JA000276

On the evolution of outward and inward Alfvénic fluctuations in the polar wind

Solar wind plasma and magnetic field measurements by Ulysses have been used to study magnetohydrodynamic turbulence in different heliospheric regions. Four intervals of six solar rotations have been analyzed. Two of them are on the ecliptic around 2 and 5 AU, respectively, one is at midlatitude near 5 AU, and the last one is at high latitude around 3 AU. Conditions on the ecliptic are those typical of high solar activity periods. The midlatitude interval is characterized by very strong gradients in the wind speed, due to an intermittent appearance of the wind coming from the polar coronal hole. In the high-latitude interval, fully inside the polar wind, the speed is steadily high. We investigated at three different scales (1, 4, and 12 hours) the level of correlation between velocity and magnetic field fluctuations, as given by the normalized cross-helicity, and the sharing of the fluctuation energy between its kinetic and magnetic component, as measured by the normalized residual energy. The observations on the ecliptic, while confirming previous findings based on Voyagers data, clearly indicate that the normalized cross-helicity is well different from zero also at distances as large as 5 AU. The midlatitude turbulence, when compared to that at low and high heliographic latitudes, appears much more evolved, with a remarkably lower normalized cross-helicity (in absolute value). This unambiguously highlights that processes at velocity gradients are an important factor in the turbulence evolution. For all the analyzed intervals the residual energy values indicate an imbalance in favor of magnetic fluctuations, in agreement with previous results. The strongest imbalance is observed for the high-latitude sample, where the turbulence is comparatively the least evolved. This is a quite unexpected result, probably related to the presence of interstellar pickup ion populations. In conclusion, our analysis indicates that (1) velocity gradients play a dominant role in driving the turbulence evolution in the solar wind and (2) pickup ion effects might be significant.

https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/97JA03029

Cross‐helicity and residual energy in solar wind turbulence: Radial evolution and latitudinal dependence in the region from 1 to 5 AU

Intermittency is a well-established feature of interplanetary MHD fluctuations and, we show the effect of intermittency on the radial evolution of solar wind velocity and magnetic field fluctuations anisotropy. On one hand we confirm results obtained by previous investigations which showed that magnetic fluctuation anisotropy increases with distance and, on the other hand, we prove that much of this trend is due to intermittency. Once intermittency has been reduced, thanks to a technique based on wavelet transform for the identification of the intermittent events, the radial trend vanishes. Similar analysis performed on velocity fluctuations, showed that intermittency although altering the anisotropy, does not markedly change its radial trend.

https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/1999GL010668

Ermanno Pietropaolo

Papers

Identifying intermittency events in the solar wind

Observation of inertial energy cascade in interplanetary space plasma.

On the evolution of outward and inward Alfvénic fluctuations in the polar wind

Cross‐helicity and residual energy in solar wind turbulence: Radial evolution and latitudinal dependence in the region from 1 to 5 AU

Effects of intermittency on interplanetary velocity and magnetic field fluctuations anisotropy