Wave Vector Anisotropy Research Articles

Non-universality of the Turbulent Spectra at Sub-ion Scales in the Solar Wind: Dispersive Effects versus the Doppler Shift

Large surveys of the power spectral density of the magnetic fluctuations in the solar wind have reported different slope distributions at MHD, sub-ion and sub-electron scales: the smaller the scale, the broader the distribution. Here, we review briefly some of the most relevant explanations of the broadening of the slopes at sub-ion scales. Then, we present a new one that has been overlooked in the literature, which is based on the relative importance of the dispersive effects with respect to the Doppler shift due to the mean flow speed. We build a toy model based on a dispersion relation of a linear mode that matches at high frequency (ω ≳ ω ci) the Alfvén (respectively whistler) mode at high oblique (respectively quasi-parallel) propagation angles θ kB . Starting with a double power-law spectrum of turbulence in the inertial range and at the sub-ion scales, the transformed spectrum (in frequency f) as it would be measured in the spacecraft reference frame shows a broad range of slopes at the sub-ion scales that depend both on the angle θ kB and the flow speed V. Varying θ kB in the range 4°–106° and V in the range 400−800 km s−1 the resulting distribution of slopes at the sub-ion scales reproduces quite well the observed one in the solar wind. Fluctuations in the solar wind speed and the wavevector anisotropy of the turbulence may explain (or at least contribute to) the variability of the spectral slopes reported from spacecraft observations in the solar wind.

Whistler turbulence vs. whistler anisotropy instability: Particle-in-cell simulation and statistical analysis

Particle-in-Cell simulations and statistical analysis are carried out to study the dynamic evolution of a collisionless, magnetized plasma with co-existing whistler turbulence and electron temperature anisotropy as the initial condition, and the competing consequences of whistler turbulence cascade and whistler anisotropy instability growth. The results show that the operation of the whistler instability within whistler turbulence has almost no effects on the fluctuating magnetic field energy and intermittency generated by turbulence. However, it leads to a small reduction of the magnetic field wavevector anisotropy and a major reduction of the intermittency of electron temperature anisotropy. Hence, while the overall effect from whistler instability is minor as compared to that of whistler turbulence due to its much smaller field energy, the whistler instability may act as a regulation mechanism for kinetic-range turbulence through wave-particle interactions.

The Gary Picture of Short-Wavelength Plasma Turbulence—The Legacy of Peter Gary

Collisionless plasmas in space often evolve into turbulence by exciting an ensemble of broadband electromagnetic and plasma fluctuations. Such dynamics are observed to operate in various space plasmas such as in the solar corona, the solar wind, as well as in the Earth and planetary magnetospheres. Though nonlinear in nature, turbulent fluctuations in the kinetic range (small wavelengths of the order of the ion inertial length or smaller) are believed to retain some properties reminiscent of linear-mode waves. In this paper we discuss what we understand, to the best of our ability, was Peter Gary’s view of kinetic-range turbulence. We call it the Gary picture for brevity. The Gary picture postulates that kinetic-range turbulence exhibits two different channels of energy cascade: one developing from Alfvén waves at longer wavelengths into kinetic Alfvén turbulence at shorter wavelengths, and the other developing from magnetosonic waves into whistler turbulence. Particle-in-cell simulations confirm that the Gary picture is a useful guide to reveal various properties of kinetic-range turbulence such as the wavevector anisotropy, various heating mechanisms, and control parameters that influence the evolution of turbulence in the kinetic range.

High-latitude Observations of Inertial-range Turbulence by the Ulysses Spacecraft During the Solar Minimum of 1993–96

Abstract We have examined Ulysses magnetic field measurements for the years 1993 through 1996 as the spacecraft moved sunward from 5 au at high southern latitudes, passing through perihelion during the first fast-latitude scan to achieve high northern latitudes, and finally returning to 5 au. These years represent near-solar-minimum activity, providing a clear measure of high-latitude solar-wind turbulence. We apply a series of tests to the data, examining both the magnetic variance anisotropy and the underlying wavevector anisotropy, finding them to be consistent with past 1 au observations. The variance anisotropy depends upon both the thermal proton temperature parameter and the amplitude of the magnetic power spectrum, while the underlying wavevector anisotropy is dominated by the component perpendicular to the mean magnetic field. We also examine the amplitude of the magnetic power spectrum as well as the associated turbulent transport of energy to small scales that results in the heating of the thermal plasma. The measured turbulence is found to be stronger than that seen at low latitudes by the Voyager spacecraft as it traverses the distance from 1 to 5 au during the years approaching solar maximum. If the high- and low-latitude sources are comparable, this would indicate that while the heating processes are active in both regions, the turbulence has had less decay time in the transport of energy to small scales. Alternatively, it may also be that the high-latitude source is stronger.

Inertial-range Magnetic-fluctuation Anisotropy Observed from Parker Solar Probe’s First Seven Orbits

Abstract 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 to either 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, we assess whether the wave-vector anisotropy is consistent with a two-dimensional (2D) plus slab turbulence transport model and determine the fraction of power in the 2D versus slab component. Our results confirm that the 2D plus slab model is consistent with the data and the power ratio between 2D and slab components depends on radial distance, with the relative power in 2D fluctuations becoming smaller closer to the Sun.

Three-Dimensional Anisotropy and Scaling Properties of Solar Wind Turbulence at Kinetic Scales in the Inner Heliosphere: Parker Solar Probe Observations

Abstract We utilize the data from the Parker Solar Probe mission at its first perihelion to investigate the three-dimensional (3D) anisotropies and scalings of solar wind turbulence for the total, perpendicular, and parallel magnetic-field fluctuations at kinetic scales in the inner heliosphere. By calculating the five-point second-order structure functions, we find that the three characteristic lengths of turbulence eddies for the total and the perpendicular magnetic-field fluctuations in the local reference frame ( L ˆ ⊥ , l ˆ ⊥ , l ˆ ∣ ∣ ) defined with respect to the local mean magnetic field B local feature as l ∣∣ > L ⊥ > l ⊥ in both the transition range and the ion-to-electron scales, but l ∣∣ > L ⊥ ≈ l ⊥ for the parallel magnetic-field fluctuations. For the total magnetic-field fluctuations, the wave-vector anisotropy scalings are characterized by l ∣ ∣ ∝ l ⊥ 0.78 and L ⊥ ∝ l ⊥ 1.02 in the transition range, and they feature as l ∣ ∣ ∝ l ⊥ 0.44 and L ⊥ ∝ l ⊥ 0.73 in the ion-to-electron scales. Still, we need more complete kinetic-scale turbulence models to explain all these observational results.

Anisotropy of Solar Wind Turbulence in the Inner Heliosphere at Kinetic Scales: PSP Observations

Abstract 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 Parker Solar Probe (PSP), we present an observation of the anisotropy at kinetic scales in the slow, Alfvénic, solar wind in the inner heliosphere. 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 to the local background magnetic field direction. The anisotropy of k ⊥ ≫ k ∥ is found evident in both transition and kinetic ranges, with the power anisotropy P ⊥/P ∥ > 10 in the kinetic range leading over that in the transition range and being stronger than that at 1 au. The spectral index varies from α t∥ = −5.7 ± 1.0 to α t⊥ = −3.7 ± 0.3 in the transition range and α k∥ = −3.12 ± 0.22 to α k⊥ = −2.57 ± 0.09 in the kinetic range. The corresponding wavevector anisotropy has the scaling of k ∥ ∼ k ⊥ 0.71 ± 0.17 in the transition range, and changes to k ∥ ∼ k ⊥ 0.38 ± 0.09 in the kinetic range, consistent with the kinetic Alfvénic turbulence at sub-ion scales.

Solar Wind Turbulence from 1 to 45 au. III. Anisotropy of Magnetic Fluctuations in the Inertial Range Using Voyager and ACE Observations

Abstract We examine both Voyager and Advanced Composition Explorer magnetic field measurements at frequencies that characterize the inertial range and evaluate the anisotropy of the fluctuations as they relate to both the compressive component and underlying wavevector anisotropy of the turbulence. The magnetic fluctuation anisotropy as it relates to the compressive component is directly dependent upon both the plasma beta of the thermal proton component and the ratio of magnetic fluctuation magnitude to the strength of the mean magnetic field. This has been seen before at 1 au. The magnetic fluctuation anisotropy in the plane perpendicular to the mean magnetic field, which is a measure of the anisotropy of the underlying wavevector distribution, should depend on the angle between the mean magnetic field and the radial direction and should be confined to values between one and the index of the power spectrum, which is typically 5/3. Our results show that the average of this anisotropy exceeds the value of the spectral index and is out of bounds with the theory. Although the results are suggestive of past analyses, we find that spherical expansion of the turbulence may offer at least a partial explanation of the apparent amplification of this measured anisotropy.

Space\u2013time structure and wavevector anisotropy in space plasma turbulence

Space and astrophysical plasmas often develop into a turbulent state and exhibit nearly random and stochastic motions. While earlier studies emphasize more on understanding the energy spectrum of turbulence in the one-dimensional context (either in the frequency or the wavenumber domain), recent achievements in plasma turbulence studies provide an increasing amount of evidence that plasma turbulence is essentially a spatially and temporally evolving phenomenon. This review presents various models for the space–time structure and anisotropy of the turbulent fields in space plasmas, or equivalently the energy spectra in the wavenumber–frequency domain for the space–time structures and that in the wavevector domain for the anisotropies. The turbulence energy spectra are evaluated in different one-dimensional spectral domains; one speaks of the frequency spectra in the spacecraft observations and the wavenumber spectra in the numerical simulation studies. The notion of the wavenumber–frequency spectrum offers a more comprehensive picture of the turbulent fields, and good models can explain the one-dimensional spectra in the both domains at the same time. To achieve this goal, the Doppler shift, the Doppler broadening, linear-mode dispersion relations, and sideband waves are reviewed. The energy spectra are then extended to the wavevector domain spanning the directions parallel and perpendicular to the large-scale magnetic field. By doing so, the change in the spectral index at different projections onto the one-dimensional spectral domain can be explained in a simpler way.

Transference of Fermi Contour Anisotropy to Composite Fermions.

There has been a surge of recent interest in the role of anisotropy in interaction-induced phenomena in two-dimensional (2D) charged carrier systems. A fundamental question is how an anisotropy in the energy-band structure of the carriers at zero magnetic field affects the properties of the interacting particles at high fields, in particular of the composite fermions (CFs) and the fractional quantum Hall states (FQHSs). We demonstrate here tunable anisotropy for holes and hole-flux CFs confined to GaAs quantum wells, via applying insitu in-plane strain and measuring their Fermi wave vector anisotropy through commensurability oscillations. For strains on the order of 10^{-4} we observe significant deformations of the shapes of the Fermi contours for both holes and CFs. The measured Fermi contour anisotropy for CFs at high magnetic field (α_{CF}) is less than the anisotropy of their low-field hole (fermion) counterparts (α_{F}), and closely follows the relation α_{CF}=sqrt[α_{F}]. The energy gap measured for the ν=2/3 FQHS, on the other hand, is nearly unaffected by the Fermi contour anisotropy up to α_{F}∼3.3, the highest anisotropy achieved in our experiments.

Tuning of Fermi contour anisotropy in GaAs (001) 2D holes via strain

We demonstrate tuning of the Fermi contour anisotropy of two-dimensional (2D) holes in a symmetric GaAs (001) quantum well via the application of in-plane strain. The ballistic transport of high-mobility hole carriers allows us to measure the Fermi wavevector of 2D holes via commensurability oscillations as a function of strain. Our results show that a small amount of in-plane strain, on the order of 10−4, can induce significant Fermi wavevector anisotropy as large as 3.3, equivalent to a mass anisotropy of 11 in a parabolic band. Our method to tune the anisotropy in situ provides a platform to study the role of anisotropy in phenomena such as the fractional quantum Hall effect and composite fermions in interacting 2D systems.

Ion-Scale Sideband Waves and Filament Formation: Alfvénic Impact on Heliospheric Plasma Turbulence

A mini-review is given about two recent discoveries in the solar wind turbulence research on ion-kinetic scales: the existence of sideband waves (or breakdown of the linear mode wave picture) and the wavevector anisotropy leading to a persistent filament formation in a wide range of plasma beta.

KINETIC EXTENSION OF CRITICAL BALANCE TO WHISTLER TURBULENCE

ABSTRACT Kinetic extension of the critical balance for plasma turbulence is presented on the spatial scales around the ion inertial length by considering electron fluid turbulence in the directions around the mean magnetic field and obliquely propagating whistler waves. The wavevector anisotropy scaling and the energy spectra for the flow velocity, the magnetic field, and the electric field are discussed.

Non-elliptic wavevector anisotropy for magnetohydrodynamic turbulence

Abstract. A model of non-elliptic wavevector anisotropy is developed for the inertial-range spectrum of magnetohydrodynamic turbulence and is presented in the two-dimensional wavevector domain spanning the directions parallel and perpendicular to the mean magnetic field. The non-elliptic model is a variation of the elliptic model with different scalings along the parallel and the perpendicular components of the wavevectors to the mean magnetic field. The non-elliptic anisotropy model reproduces the smooth transition of the power-law spectra from an index of −2 in the parallel projection with respect to the mean magnetic field to an index of −5/3 in the perpendicular projection observed in solar wind turbulence, and is as competitive as the critical balance model to explain the measured frequency spectra in the solar wind. The parameters in the non-elliptic spectrum model are compared with the solar wind observations.

ION-SCALE TURBULENCE IN THE INNER HELIOSPHERE: RADIAL DEPENDENCE

The evolution of the ion-scale plasma turbulence in the inner heliosphere is studied by associating the plasma parameters for hybrid-code turbulence simulations to the radial distance from the Sun via a Solar wind model based mapping procedure. Using a mapping based on a one-dimensional solar wind expansion model, the resulting ion-kinetic scale turbulence is related to the solar wind distance from the Sun. For this purpose the mapping is carried out for various values of ion beta that correspond to the heliocentric distance. It is shown that the relevant normal modes such as ion cyclotron and ion Bernstein modes will occur first at radial distances of about 0.2–0.3 AU, i.e., near the Mercury orbit. This finding can be used as a reference, a prediction to guide the in situ measurements to be performed by the upcoming Solar Orbiter and Solar Probe Plus missions. Furthermore, a radial dependence of the wave-vector anisotropy was obtained. For astrophysical objects this means that the spatial scales of filamentary structures in interstellar media or astrophysical jets can be predicted for photometric observations.

WHISTLER TURBULENCE FORWARD CASCADE VERSUS INVERSE CASCADE: THREE-DIMENSIONAL PARTICLE-IN-CELL SIMULATIONS

We present the results of the first fully three-dimensional particle-in-cell simulations of decaying whistler turbulence in a magnetized, homogeneous, collisionless plasma in which both forward cascades to shorter wavelengths, and inverse cascades to longer wavelengths are allowed to proceed. For the electron beta β e = 0.10 initial value considered here, the early-time rate of inverse cascade is very much smaller than the rate of forward cascade, so that at late times the fluctuation energy in the regime of the inverse cascade is much weaker than that in the forward cascade regime. Similarly, the wavevector anisotropy in the inverse cascade regime is much weaker than that in the forward cascade regime.

Wavevector anisotropy of plasma turbulence at ion kinetic scales: solar wind observations and hybrid simulations

Abstract. Wavevector anisotropy of ion-scale plasma turbulence is studied at various values of ion beta. Two complementary methods are used. One is multi-point measurements of magnetic field in the near-Earth solar wind as provided by the Cluster spacecraft mission, and the other is hybrid numerical simulation of two-dimensional plasma turbulence. Both methods demonstrate that the wavevector anisotropy is reduced with increasing values of ion beta. Furthermore, the numerical simulation study shows the existence of a scaling law between ion beta and the wavevector anisotropy of the fluctuating magnetic field that is controlled by the thermal or hybrid particle-in-cell simulation noise. Likewise, there is weak evidence that the power-law scaling can be extended to the turbulent fluctuating cascade. This fact can be used to construct a diagnostic tool to determine or to constrain ion beta using multi-point magnetic field measurements in space.

THREE-DIMENSIONAL HYBRID SIMULATION STUDY OF ANISOTROPIC TURBULENCE IN THE PROTON KINETIC REGIME

Three-dimensional numerical hybrid simulations with particle protons and quasi-neutralizing fluid electrons are conducted for a freely decaying turbulence that is anisotropic with respect to the background magnetic field. The turbulence evolution is determined by both the combined root-mean-square (rms) amplitude for fluctuating proton bulk velocity and magnetic field and by the ratio of perpendicular to parallel wavenumbers. This kind of relationship had been considered in the past with regard to interplanetary turbulence. The fluctuations nonlinearly evolve to a turbulent phase whose net wave vector anisotropy is usually more perpendicular than the initial one, irrespective of the initial ratio of perpendicular to parallel wavenumbers. Self-similar anisotropy evolution is found as a function of the rms amplitude and parallel wavenumber. Proton heating rates in the turbulent phase vary strongly with the rms amplitude but only weakly with the initial wave vector anisotropy. Even in the limit where wave vectors are confined to the plane perpendicular to the background magnetic field, the heating rate remains close to the corresponding case with finite parallel wave vector components. Simulation results obtained as a function of proton plasma to background magnetic pressure ratio β p in the range 0.1-0.5 show that the wave vector anisotropy also weakly depends on β p .

Spatial structure of ion-scale plasma turbulence

Spatial structure of small-scale plasma turbulence is studied under different conditions of plasma parameter beta directly in the three-dimensional wave vector domain. Two independent approaches are taken: observations of turbulent magnetic field fluctuations in the solar wind measured by four Cluster spacecraft, and direct numerical simulations of plasma turbulence using the hybrid code AIKEF, both resolving turbulence on the ion kinetic scales. The two methods provide independently evidence of wave vector anisotropy as a function of beta. Wave vector anisotropy is characterized primarily by an extension of the energy spectrum in the direction perpendicular to the large-scale magnetic field. The spectrum is strongly anisotropic at lower values of beta, and is more isotropic at higher values of beta. Cluster magnetic field data analysis also provides evidence of axial asymmetry of the spectrum in the directions around the large-scale field. Anisotropy is interpreted as filament formation as plasma evolves into turbulence. Axial asymmetry is interpreted as the effect of radial expansion of the solar wind from the corona.

Whistler anisotropy instability: Spectral transfer in a three‐dimensional particle‐in‐cell simulation

A three‐dimensional (3‐D) particle‐in‐cell (PIC) simulation of the whistler anisotropy instability is carried out for a collisionless, homogeneous, magnetized plasma with βe=0.10. This is the first 3‐D PIC simulation of the evolution of enchanced fluctuations from this growing mode driven by an anisotropic electron velocity distribution with T⟂e/T∥e>1 where ⟂ and ∥ represent directions perpendicular and parallel to the background magnetic field Bo, respectively. The early‐time magnetic fluctuation spectrum grows with properties reflecting the predictions of linear theory with narrowband maxima at kc/ωe≃ 1 and k×Bo=0, and a wave vector anisotropy in the sense of k⟂<<k∥. Here ωe represents the electron plasma frequency. At later times the fluctuations undergo both a forward transfer to shorter wavelengths, also with k⟂<<k∥, and an inverse transfer to longer wavelengths with wave vector anisotropy k⟂>>k∥. The inverse transfer is consistent with a prediction of nonlinear three‐wave coupling theory.