The anisotropy of magnetophydrodynamic turbulence is investigated by using solar wind data from the Helios 2 spacecraft. We investigate the behaviour of the complete high-order moment tensors of magnetic field increments and we compare the usual longitudinal structure functions which have both isotropic and anisotropic contributions, to the fully anisotropic contribution. Scaling exponents have been extracted by an interpolation scaling function. Unlike the usual turbulence in fluid flows, small-scale magnetic fluctuations remain anisotropic. We discuss the radial dependence of both anisotropy and intermittency and their relationship.
A relationship is derived between power anisotropy and wavevector anisotropy in turbulent fluctuations. This can be used to interpret plasma turbulence measurements, for example in the solar wind. If fluctuations are anisotropic in shape then the ion gyroscale break point in spectra in the directions parallel and perpendicular to the magnetic field would not occur at the same frequency, and similarly for the electron gyroscale break point. This is an important consideration when interpreting solar wind observations in terms of anisotropic turbulence theories. Model magnetic field power spectra…
The field-aligned anisotropy of the solar wind turbulence, which is quantified by the ratio of the parallel to the perpendicular correlation (and Taylor) length scales, is determined by simultaneous two-point correlation measurements during the time period 2001-2017. Our results show that the correlation scale along the magnetic field is the largest, and the correlation scale in the field-perpendicular directions is the smallest, at both solar maximum and solar minimum. However, the Taylor scale reveals inconsistent results for different stages of the solar cycles. During the years 2001-2004, …
To investigate the power and spectral index anisotropy in the inertial range of solar wind turbulence, we use 70 intervals of electric field data accumulated by Cluster spacecraft in the free solar wind. We compute the electric field fluctuation power spectra using wavelet analysis technique and study its spectral index variation with the change in angle between the heliocentric radial direction and the local mean magnetic field. We find clear power and spectral index anisotropy in the frequency ranging from 0.01 Hz to 0.1 Hz, with more power in parallel fluctuations than perpendicular. We als…
We present the first measurement of the scale-dependent power anisotropy of Elsasser variables in imbalanced fast solar wind turbulence. The dominant Elsasser mode is isotropic at lower spacecraft frequencies but becomes increasingly anisotropic at higher frequencies. The sub-dominant mode is anisotropic throughout, but in a scale-independent way (at higher frequencies). There are two distinct subranges exhibiting different scalings within what is normally considered the inertial range. The low Alfven ratio and shallow scaling of the sub-dominant Elsasser mode suggest an interpretation of the …
We observe the anisotropy of the power spectral tensor of magnetic field fluctuations in the fast solar wind for the first time. In heliocentric RTN coordinates the power in each element of the tensor has a unique dependence on the angle between the magnetic field and velocity of the solar wind (θ) and the angle of the vector in the plane perpendicular to the velocity (ϕ). We derive the geometrical effect of the high speed flow of the solar wind past the spacecraft on the power spectrum in the frame of the plasma P(k) to arrive at the observed power spectrum P(f,θ,ϕ) based on a scalar field de…
Using the Parker Solar Probe data taken in the inner heliosphere, we investigate the power and spatial anisotropy of magnetic-field spectra at kinetic scales (i.e., around sub-ion scales) in solar wind turbulence in the inner heliosphere. We find that strong anisotropy of magnetic spectra occurs at kinetic scales with the strongest power in the perpendicular direction with respect to the local magnetic field (forming an angle theta_B with the mean flow velocity). The spectral index of magnetic spectra varies from -3.2 to -5.8 when the angle theta_B changes from 90 to 180 (or 0) deg, indicating…
In this paper, we study the development of anisotropy in strong MHD turbulence in the presence of a large scale magnetic field B 0 by analyzing the results of direct numerical simulations. Our results show that the developed anisotropy among the different components of the velocity and magnetic field is a direct outcome of the inverse cascade of energy of the perpendicular velocity components u? and a forward cascade of the energy of the parallel component u k . The inverse cascade develops for a strong B0, where the flow exhibits a strong vortical structure by the suppression of fluctuations …
Particle velocity distributions measured in the weakly collisional solar wind are frequently found to be non-Maxwellian, but how these non-Maxwellian distributions impact the physics of plasma turbulence in the solar wind remains unanswered. Using numerical solutions of the linear dispersion relation for a collisionless plasma with a bi-Maxwellian proton velocity distribution, we present a unified framework for the four proton temperature anisotropy instabilities, identifying the associated stable eigenmodes, highlighting the unstable region of wavevector space, and presenting the properties o…
Correlation anisotropy emerges dynamically in magnetohydrodynamics (MHD), producing stronger gradients across the large-scale mean magnetic field than along it. This occurs both globally and locally, and has significant implications in space and astrophysical plasmas, including particle scattering and transport, and theories of turbulence. Properties of local correlation anisotropy are further documented here by showing through numerical experiments that the effect is intensified in more localized estimates of the mean field. The mathematical formulation of this property shows that local aniso…
Recent modelling of solar energetic particles (SEPs) propagation through the heliospheric turbulence, also discussed in this workshop, has investigated the role of the pitch-angle scattering and the perpendicular transport in spreading particles in heliolongitude, as shown by multi-spacecraft measurements (STEREO A/B, ACE, SOHO, etc.) at 1 AU in various energy ranges. In some events the first-order pitch-angle anisotropy of the particles distribution is not-negligible. We calculate the average perpendicular displacement due to the gradient/curvature drift in an inhomogeneous turbulence account…
We investigate the anisotropy of Alfvénic turbulence in the inertial range of slow solar wind and in both driven and decaying reduced magnetohydrodynamic simulations. A direct comparison is made by measuring the anisotropic second-order structure functions in both data sets. In the solar wind, the perpendicular spectral index of the magnetic field is close to -5/3. In the forced simulation, it is close to -5/3 for the velocity and -3/2 for the magnetic field. In the decaying simulation, it is -5/3 for both fields. The spectral index becomes steeper at small angles to the local magnetic field d…
We measure the power and spectral index anisotropy of high speed solar wind turbulence from scales larger than the outer scale down to the ion gyroscale, thus covering the entire inertial range. We show that the power and spectral indices at the outer scale of turbulence are approximately isotropic. The turbulent cascade causes the power anisotropy at smaller scales manifested by anisotropic scalings of the spectrum: close to k^{-5/3} across and k^{-2} along the local magnetic field, consistent with a critically balanced Alfvenic turbulence. By using data at different radial distances from the…
The measure of the third-order structure function, Y, is employed in the solar wind to compute the cascade rate of turbulence. In the absence of a mean field B0=0, Y is expected to be isotropic (radial) and independent of the direction of increments, so its measure yields directly the cascade rate. For turbulence with mean field, as in the solar wind, Y is expected to become more two dimensional (2D), that is, to have larger perpendicular components, loosing the above simple symmetry. To get the cascade rate one should compute the flux of Y, which is not feasible with single-spacecraft data, t…
When magnetohydrodynamic turbulence evolves in the presence of a large-scale mean magnetic field, an anisotropy develops relative to that preferred direction. The well-known tendency is to develop stronger gradients perpendicular to the magnetic field, relative to the direction along the field. This anisotropy of the spectrum is deeply connected with anisotropy of estimated timescales for dynamical processes, and requires reconsideration of basic issues such as scale locality and spectral transfer. Here analysis of high-resolution three-dimensional simulations of unforced magnetohydrodynamic t…
The anisotropy of turbulence in the fast solar wind, between the ion and electron gyroscales, is directly observed using a multispacecraft analysis technique. Second order structure functions are calculated at different angles to the local magnetic field, for magnetic fluctuations both perpendicular and parallel to the mean field. In both components, the structure function value at large angles to the field S_perp is greater than at small angles S_par: in the perpendicular component S_perp/S_par = 5 +- 1 and in the parallel component S_perp/S_par > 3, implying spatially anisotropic fluctuation…
Turbulent behavior at sub-proton scales in magnetized plasmas is important for a full understanding of the energetics of astrophysical flows such as the solar wind. We study the formation of electron temperature anisotropy due to reconnection in the turbulent decay of sub-proton scale fluctuations using two dimensional, particle-in-cell (PIC) plasma simulations with realistic electron-proton mass ratio and a guide field out of the simulation plane. A fluctuation power spectrum with approximately power law form is created down to scales of order the electron gyroradius. In the dynamic magnetic …
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 pro…
Hybrid Vlasov-Maxwell simulations are employed to investigate the role of kinetic effects in a two-dimensional turbulent multi-ion plasma, composed of protons, alpha particles and fluid electrons. In the typical conditions of the solar-wind environment, and in situations of decaying turbulence, the numerical results show that the velocity distribution functions of both ion species depart from the typical configuration of thermal equilibrium. These non-Maxwellian features are quantified through the statistical analysis of the temperature anisotropy, for both protons and alpha particles, in the …
The anisotropy of solar wind turbulence is a critical issue in understanding the physics of energy transfer between scales and energy conversion between fields and particles in the heliosphere. Using the measurement of \emph{Parker Solar Probe} (\emph{PSP}), we present an observation of the anisotropy at kinetic scales in the slow, Alfvénic, solar wind in the inner heliosphere. \textbf{The magnetic compressibility behaves as expected for kinetic Alfvénic turbulence below the ion scale.} A steepened transition range is found between the inertial and kinetic ranges in all directions with respect…
The nature of MHD turbulence is analyzed through both temporal and spatial magnetic fluctuation spectra. A magnetically turbulent plasma is produced in the MHD wind-tunnel configuration of the Swarthmore Spheromak Experiment (SSX). The power of magnetic fluctuations is projected into directions perpendicular and parallel to a local mean field; the ratio of these quantities shows the presence of variance anisotropy which varies as a function of frequency. Comparison amongst magnetic, velocity, and density spectra are also made, demonstrating that the energy of the turbulence observed is primari…
We propose a simple statistical model of three-dimensionally anisotropic, intermittent, strong Alfvénic turbulence, incorporating both critical balance and dynamic alignment. Our model is based on log-Poisson statistics for Elsasser-field increments along the magnetic field. We predict the scalings of Elsasser-field conditional two-point structure functions with point separations in all three directions in a coordinate system locally aligned with the direction of the magnetic field and of the fluctuating fields and obtain good agreement with numerical simulations. We also derive a scaling of t…
We study the variation of average powers and spectral indices of electric field fluctuations with respect to the angle between average flow direction and the mean magnetic field in solar wind turbulence. Cluster spacecraft data from the years 2002 and 2007 are used for the present analysis. We perform a scale dependent study with respect to the local mean magnetic field using wavelet analysis technique. Prominent anisotropies are found for both the spectral index and power levels of the electric power spectra. Similar to the magnetic field fluctuations, the parallel (or antiparallel) electric …
``Exact'' laws for evaluating cascade rates, tracing back to the Kolmogorov ``4/5'' law, have been extended to many systems of interest including magnetohydrodynamics (MHD), and compressible flows of the magnetofluid and ordinary fluid types. It is understood that implementations may be limited by the quantity of available data and by the lack of turbulence symmetry. Assessment of the accuracy and feasibility of such ``third-order'' (or Yaglom) relations is most effectively accomplished by examining the von Karman-Howarth equation in increment form, a framework from which the third-order laws …
Observations and modelling suggest that the fluctuations in magnetised plasmas exhibit scale-dependent anisotropy, with more energy in the fluctuations perpendicular to the mean magnetic field than in the parallel fluctuations and the anisotropy increasing at smaller scales. The scale-dependence of the anisotropy has not been studied in full-orbit simulations of particle transport in turbulent plasmas so far. In this paper, we construct a model of critically balanced turbulence, as suggested by \cite{GoSr1995}, and calculate energetic particle spatial diffusion coefficients using full-orbit si…
Turbulence in space plasmas usually exhibits two regimes separated by a spectral break that divides the so called inertial and kinetic ranges. Large scale magnetic fluctuations are dominated by non-linear MHD wave-wave interactions following a -5/3 or -2 slope power-law spectrum. After the break, at scales in which kinetic effects take place, the magnetic spectrum follows a steeper power-law $k^{-α}$ shape given by a spectral index $α> 5/3$. Despite its ubiquitousness, the possible effects of a turbulent background spectrum in the quasilinear relaxation of solar wind temperatures are usually n…
We propose that pressure anisotropy causes weakly collisional turbulent plasmas to self-organize so as to resist changes in magnetic-field strength. We term this effect "magneto-immutability" by analogy with incompressibility (resistance to changes in pressure). The effect is important when the pressure anisotropy becomes comparable to the magnetic pressure, suggesting that in collisionless, weakly magnetized (high-$β$) plasmas its dynamical relevance is similar to that of incompressibility. Simulations of magnetized turbulence using the weakly collisional Braginskii model show that magneto-im…
Solar wind turbulence is anisotropic with respect to the mean magnetic field. Anisotropy leads to ambiguity when interpreting in-situ turbulence observations in the solar wind because an apparent change in the measurements could be due either to the change of intrinsic turbulence properties or to a simple change of the spacecraft sampling direction. We demonstrate the ambiguity using the spectral index and magnetic compressibility in the inertial range observed by the Parker Solar Probe during its first seven orbits ranging from 0.1 to 0.6 AU. To unravel the effects of the sampling direction, …
Magnetic reconnection is a process that changes magnetic field topology in highly conducting fluids. Within the standard Sweet-Parker model, this process would be too slow to explain observations (e.g. solar flares). In reality, the process must be ubiquitous as astrophysical fluids are magnetized and motions of fluid elements necessarily entail crossing of magnetic frozen-in field lines and magnetic reconnection. In the presence of turbulence, the reconnection is independent of microscopic plasma properties, and may be much faster than previously thought, as proposed in Lazarian & Vishniac (1…
In the lower solar coronal regions where the magnetic field is dominant, the Alfven speed is much higher than the wind speed. In contrast, the near-Earth solar wind is strongly super-Alfvenic, i.e., the wind speed greatly exceeds the Alfven speed. The transition between these regimes is classically described as the "Alfven point" but may in fact occur in a distributed Alfven critical region. NASA's Parker Solar Probe (PSP) mission has entered this region, as it follows a series of orbits that gradually approach more closely to the sun. During its 8th and 9th solar encounters, at a distance of …
Small-scale magnetic turbulence observed by the Cluster spacecraft in the plasma sheet is investigated by means of a wavelet estimator suitable for detecting distinct scaling characteristics even in noisy measurements. The spectral estimators used for this purpose are affected by a frequency dependent bias. The variances of the wavelet coefficients, however, match the power-law shaped spectra, which makes the wavelet estimator essentially unbiased. These scaling characteristics of the magnetic field data appear to be essentially non-steady and intermittent. The scaling properties of bursty bul…
We present observations of the power spectral anisotropy in wave-vector space of solar wind turbulence, and study how it evolves in interplanetary space with increasing heliocentric distance. For this purpose we use magnetic field measurements made by the Helios-2 spacecraft at three positions between 0.29 and 0.9 AU. To derive the power spectral density (PSD) in $(k_\parallel, k_\bot)$-space based on single-satellite measurements is a challenging task not yet accomplished previously. Here we derive the spectrum $\rm{PSD}_{\rm{2D}}$($\rm{k}_\parallel$, $\rm{k}_\bot$) from the spatial correlati…
Anisotropy of MHD turbulence has been studied extensively for many years, most prominently by measurements in the solar wind and high resolution simulations. The spectrum parallel to the local magnetic field was observed to be steeper than perpendicular spectrum, typically $k^{-2}$, consistent with the widely accepted Goldreich-Sridhar (1995) model. In this Letter I looked deeper into the nature of the relation between parallel and perpendicular spectra and argue that this $k^{-2}$ scaling has the same origin as the $ω^{-2}$ scaling of Lagrangian frequency spectrum in strong hydrodynamic turbu…
This paper summarises some of the recent progress that has been made in understanding astrophysical plasma turbulence in the solar wind, from in situ spacecraft observations. At large scales, where the turbulence is predominantly Alfvenic, measurements of critical balance, residual energy, and 3D structure are discussed, along with comparison to recent models of strong Alfvenic turbulence. At these scales, a few percent of the energy is also in compressive fluctuations, and their nature, anisotropy, and relation to the Alfvenic component is described. In the small scale kinetic range, below th…
We present results from a three-dimensional particle-in-cell simulation of plasma turbulence, resembling the plasma conditions found at kinetic scales of the solar wind. The spectral properties of the turbulence in the subion range are consistent with theoretical expectations for kinetic Alfv\' en waves. Furthermore, we calculate the local anisotropy, defined by the relation $k_{\parallel}(k_{\perp})$, where $k_{\parallel}$ is a characteristic wave number along the local mean magnetic field at perpendicular scale $l_{\perp}\sim 1/k_{\perp}$. The subion range anisotropy is scale dependent with …
A statistical survey of spectral anisotropy of space plasma turbulence is performed using five years measurements from MMS in the magnetosheath. By measuring the five-point second-order structure functions of the magnetic field, we have for the first time quantified the three-dimensional anisotropies and scalings at sub-ion-scales ($<$ 100 km). In the local reference frame $(\hat L_{\perp}, \hat l_{\perp}, \hat l_{\parallel})$ defined with respect to local mean magnetic field $\boldsymbol {B}_0$ (Chen et al. 2012), the "statistical eddies" are found to be mostly elongated along $\boldsymbol {B…
We present a measurement of the scale-dependent, three-dimensional structure of the magnetic field fluctuations in inertial range solar wind turbulence with respect to a local, physically motivated coordinate system. The Alfvenic fluctuations are three-dimensionally anisotropic, with the sense of this anisotropy varying from large to small scales. At the outer scale, the magnetic field correlations are longest in the local fluctuation direction, consistent with Alfven waves. At the proton gyroscale, they are longest along the local mean field direction and shortest in the direction perpendicul…
In order to resolve and characterize anisotropy in turbulent plasma flows a proper estimation of the background magnetic field is crucially important. Various approaches to calculate the background magnetic fields, ranging from local fields to globally averaged fields, are commonly used in the analysis of turbulent data. Here we investigate how the uncertainty in the orientation of a scale dependent background magnetic field influences the ability to resolve anisotropy. Therefore we introduce a quantitative measure, the angle uncertainty, which characterizes the uncertainty of the orientation …
Kinetic plasma processes have been investigated in the framework of solar wind turbulence, employing Hybrid Vlasov-Maxwell (HVM) simulations. The dependency of proton temperature anisotropy T_{\perp}/T_{\parallel} on the parallel plasma beta β_{\parallel}, commonly observed in spacecraft data, has been recovered using an ensemble of HVM simulations. By varying plasma parameters, such as plasma beta and fluctuation level, the simulations explore distinct regions of the parameter space given by T_{\perp}/T_{\parallel} and β_{\parallel}, similar to solar wind sub-datasets. Moreover, both simulati…
Intensive studies have been conducted to understand the anisotropy of solar wind turbulence. However, the anisotropy of Elsässer variables ($\textbf{Z}^\pm$) in 2D wave-vector space has yet to be investigated. Here we first verify the transformation based on the projection-slice theorem between the power spectral density PSD$_{2D}(k_\parallel,k_\perp )$ and the spatial correlation function CF$_{2D} (r_\parallel,r_\perp )$. Based on the application of the transformation to the magnetic field and the particle measurements from the WIND spacecraft, we investigate the spectral anisotropy of Elsäss…
