Isotropy restoration toward high-beta space plasmas

Isotropy restoration toward high-beta space plasmas H. Comişel, Y. Narita, and U. Motschmann Institut für Theoretische Physik, Technische Universität Braunschweig, Mendelssohnstr. 3, 38106, Germany Institute for Space Sciences, Atomiştilor 409, P.O. Box MG-23, Bucharest-Măgurele, 77125, Romania Space Research Institute, Austrian Academy of Sciences, Schmiedlstr. 6, 8042 Graz, Austria Institut für Geophysik und extraterrestrische Physik, Technische Universität Braunschweig, Mendelssohnstr. 3, 38106, Germany Deutsches Zentrum für Luftund Raumfahrt, Institut für Planetenforschung, Rutherfordstr. 2, 12489 Berlin, Germany


Introduction
Wavevector anisotropy appears in various plasma systems whenever a large-scale magnetic field is imposed.Anisotropy is characterized by extension or elongation of the energy spectrum in the direction parallel or perpendicular to the large-scale field.Examples of wavevector anisotropy can be found in near-Earth solar wind (e.g.Matthaeus et al., 1990;Chen et al., 2012), astrophysical systems such as diffusion of galactic cosmic ray (Bieber et al., 1994(Bieber et al., , 1996;;Ahlers, 2014) and magnetic field decay process in the neutron star crust (Cumming et al., 2004), as well as in laboratory plasmas (Howes et al., 2012;Drake et al., 2013).Development in wavevector anisotropy leads to a structure formation while the plasma evolves into a turbulent state, which is markedly different from fluid turbulence.Evidence of anisotropy in plasma turbulence has also been presented in numerical simulations using different schemes for plasma dynamics such as magnetohydrodynamic (MHD) treatment (Matthaeus et al., 1996;Matthaeus and Gosh, 1999) , 2011), and full-particle treatment (Chang et al., 2013;Gary et al., 2010;Saito et al., 2010).
Here we propose a scenario that plasma beta is the primary control parameter such that wavevector anisotropy is diminished toward higher values of beta, namely, isotropy restoration.By extending the method and the result obtained in Narita et al. (2014) (hereafter NCM14), we find a transition in the wavevector anisotropy as a function of beta.While the qualitative picture of beta dependence is demonstrated in NCM14, this manuscript presents a more systematic survey of beta dependence both in spacecraft measurements and direct numerical simulations.Our goal is to search for a scaling law of the wavevector anisotropy as a function of beta.Since waves and instabilities in plasmas are known to be dependent on the value of beta, it is natural to assume the existence of mapping or relation between beta and anisotropy.Interestingly, such an idea will lead us to determine or constrain plasma beta using magnetic field measurements only.
The wavevector anisotropy can be quantitatively measured by employing the anisotropy index that compares two covariance quantities, one between the wavevector spectrum and the parallel components of the wavevector and the other between the spectrum and the perpendicular components of the wavevector.The anisotropy index A is defined after Shebalin et al. (1983) as where k and k ⊥ denote the wavevector components parallel and perpendicular to the large-scale magnetic field, respectively, and E (k ⊥ , k ) the wavevector spectrum.tool in this work, and we take two independent and complementary approaches to determine the wavevector anisotropy.One is solar wind observations using four-point magnetic field and the other is numerical simulations.

Multi-spacecraft measurements
Four-point magnetic field data sampled by Cluster fluxgate magnetometer in the near-Earth solar wind (Balogh et al., 2001) are used to determine the wavevector spectra.Three time intervals are added to that used in the previous analysis in NCM14: 9 February 2002, 02:10-02:40 UT;12 February 2002, 14:15-14:45 UT;18 March 2002, 21:30-22:00 UT.The intervals are taken from the solar wind intervals of Cluster measurements under the conditions of nearly regular tetrahedral formation with the minimum inter-spacecraft distance.The magnetic field magnitude, the ion density, the ion bulk speed, and the plasma parameter beta of the three time intervals are listed in Table 1.Ion data are obtained from the ion spectrometry on board Cluster (Rème et al., 2001).With the three added intervals and that used in NCM14, the wavevector spectra are determined at seven different values of beta in the range 0.58-3.66 in the solar wind.
The spectral estimator MSR (Multi-point Signal Resonator) (Narita et al., 2011) is used extensively in the data analysis.The MSR technique is a high-resolution spectral estimator in the wavevector domain using four-point magnetic field data.The method assumes only a set of plane waves, and no additional assumption is needed such as wave modes or Taylor's frozen-in flow hypothesis.Time series of four-point magnetic field data are first transformed into frequencies using the Fast Fourier Transform algorithm.Cross spectral density matrix is then constructed, which is a 12-by-12 matrix, consisting of three components of magnetic field and four spatial points, in the frequency domain.This matrix is reduced into a 3-by-3 matrix by projecting onto the Figures wavevector domain.The matrix projection is a combination of the minimum variance projection with the MUSIC (Multiple Signal Classification) algorithm (Schmidt, 1986).
To complete the projection method, the eigenvalue decomposition of the cross spectral density matrix (Choi et al., 1993) is incorporated.
The wavevector spectra are determined in the following steps.First, the energy spectra for the magnetic field fluctuations are evaluated in the four-dimensional Fourier domain spanned by the frequencies and the wavevectors.Second, the spectra are integrated over the frequencies extended to negative values of frequencies to guarantee the frame-invariance of the wavevector dependence after the frequency integration.Third, the spectra are further reduced into the two-dimensional wavevector domain spanned by the parallel components of wavevector and the perpendicular one by integrating over the directions around the large-scale magnetic field and then summing over the positive and negative components of the parallel wavevector.The two-dimensional wavevector spectra for the three time intervals are displayed in Fig. 1 using the normalization to the proton inertial length by multiplying the Alfvén speed and Ω p the proton cyclotron frequency).Except for the case at beta 3.66, the measured wavevector spectra are determined at the wavevector components up to kV A Ω p = 2.5.At beta 3.66, the spectrum was determined up to The wavevector anisotropy is then evaluated in the reduced spectra using the method of anisotropy index (Eq.1).The lower limit and the higher limit of the wavenumbers are set to respectively, in the computation of anisotropy index.
The lower limit is determined such that the pump waves set in the simulation work (see below) are not counted as anisotropy (the pump waves have the wavenumbers up to kV A Ω p = 0.2).The upper limit is determined such that all intervals (except for beta 3.66) have the same range in the wavenumbers.At beta 3.66, we use the upper limit

Direct numerical simulation
Direct numerical simulation serves as an independent and complementary approach of the anisotropy study.The wavevector anisotropy is studied under six different conditions of beta (0.05, 0.1, 0.2, 0.5, 1.0, and 2.0) covering nearly two orders of magnitude.Following the procedure in the observational approach, the wavevector spectra are determined from the numerical simulation of plasma turbulence, and the anisotropy is evaluated using the spectra.However, due to the computational load, we have to limit ourselves to the two-dimensional space spanned by the parallel and the perpendicular directions to the large-scale magnetic field.In this setup, eddies which are intrinsic to fluid-mechanical or gas-dynamic nature of plasmas are suppressed due the lack of the degree of freedom around the large-scale magnetic field.In other words, though the spatial dimensions are two, this numerical setup is advantageous in studying the wave dynamics of plasmas in detail without being influenced by eddies.
Turbulence is produced in the simulation box by solving dynamics of plasma and fields in a step-by-step fashion.We use the hybrid plasma code AIKEF (Adaptive Ion Kinetic Electron Fluid).This code solves a set of Newtonian equation of motion for ions as super-particles under the Coulomb and Lorenz forces together with the Maxwell equations.Electrons are treated as a massless charge-neutralizing fluid (see details on the code in Müller et al. (2011).In contrast to the analytic approach of solving plasma dynamics, the simulation approach does not require any a-priori knowledge on the statistical property such as a Gaussian distribution of the fluctuating fields nor assumption of the wave modes.The hybrid plasma treatment is suitable particularly for resolving ion kinetic effects as far as electron gyromotion can be neglected, i.e., on the spatial scales between the electron and ion gyroradii (cf.10-100 km in the solar wind).A disadvantage is that the strong electrostatic effects due to charge localization cannot develop due to the massless electron fluid, as the electrostatic field is immediately canceled out by electrons.However, the charge localization can safely be neglected in our purpose of anisotropy study here.Implementation of the AIKEF code to our simulation study follows that presented by Verscharen et al. (2012);Comişel et al. (2013) and NCM14.The simulation box has the size 250 × 250 proton inertial lengths, and is spanned by the parallel to the large-scale magnetic field B 0 (the z direction) and the perpendicular direction (the y direction).The large-scale field is constant in space and time.The spatial configuration is two-dimensional, but the vectorial quantities are treated as three-dimensional, e.g., fluctuating magnetic field, electric field, particle velocities.The x direction is pointing out of the simulation plane.Plasma is modeled as the electron-proton plasma.
As the initial condition, a superposition of one-thousand Alfvén waves with random initial phases are set to the system.These pump waves are limited to the wavevectors in the range The simulation box is divided into 1024 × 1024 computational cells.The number of super-particles N per cell are spanning in the range from N = 400 to N = 1600 per cell while the beta values are in the range between 0.05 and 2.0.We note that the higher value of beta one uses in the simulation, the more particles need to be put into the simulation box.Otherwise many particles may leave the computation cell due to their thermal mobility in the high-beta plasma.This process will eventually lead to a failure in solving the equations of the electromagnetic field when the so-called vacuum cell is Introduction

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Full formed in which no ion is stored instantaneously.To carry out a successful run at the value of beta 2.0, a large number of super-particles is set, N = 1600, which is a highlydemanding computation.The wavevector spectra are determined at 2000 ion gyroperiods (tΩ p = 2000) by extending the iterations in the simulation run (cf.anisotropy study in NCM14 uses the evolution time at tΩ p = 1000).This represents the longest possible simulation run for plasma turbulence using the AIKEF code, and the fluctuations reach a quasi-stationary stage at which the fluctuation amplitude, the spectrum, and the anisotropy do not evolve substantially any more.The spectra are obtained by applying the Fast Fourier Transform algorithm to the spatial distribution of magnetic field fluctuations (Fig. 2).
Again, the anisotropy index is computed at each value of beta using Eq. ( 1).The anisotropy-beta relation is displayed in Fig. 3.

Results and discussion
The added solar wind intervals of the Cluster measurements show a diversity (including both similarities and differences) in the contour shapes of the wavevector spectra.First of all, all the three cases exhibit the anisotropy that the spectrum is extended primarily in the direction perpendicular to the large-scale magnetic field.That is, the spectral decay in the wavevector domain is flatter along the perpendicular wavevector axis, while it is steeper along the parallel wavevector axis.Closer look at the spectra yields the following detailed structures. in the case of beta 1.05 (left panel in Fig. 1), the contours of the spectrum are nearly diagonal, connecting a larger value of the parallel component of wavevector to a lager value of the perpendicular one.In the case of beta 1.18, the spectral extension appears not only in the perpendicular direction but also in oblique direction (about 60 • from the direction of the large-scale magnetic field).While these two cases show the monotonous spectral decay toward larger values of the wavevector components, the spectrum in the case of beta 3.66 exhibits a formation of the secondary peak along the perpendicular wavevector axis at about and the third peak in the oblique direction at about ( 2).The spectral decay in the wavevector domain is rather flat in the third case.The evaluated anisotropy index is about 1.8 (at beta 1.05), 2.3 (at beta 1.18), and 2.1 (at beta 3.66).These values are plotted in Fig. 3 together with that already obtained in NCM14 (5.7 at beta 0.58, 3.3 at beta 0.76, 2.3 at beta 1.66, and 2.3 at beta 2.53).
The wavevector spectra obtained from the numerical simulations provide another kind of diversity in the contour shapes.Again, the primary extension of the spectrum appears along the perpendicular wavevector direction (the spectral decay is flatter in the perpendicular direction, and steeper in the parallel direction).Still, the secondary peak is formed along the parallel wavevector axis (e.g., 1 < 3.5 at beta 1.0).It is also interesting to note that the contour shape at smaller values of the perpendicular wavevector component exhibits a similarity in that there is a hump or a spectral extension, e.g., the transitions from ( ) = (0.5, 4.0) at beta 0.2, and further to ( ) = (0.5, 5.0) at beta 0.5.In contrast, at larger values of the perpendicular wavevector component, the spectral contours do not vary across the parallel wavevector component.The anisotropy index is obtained for the wavevector spectra from the numerical simulations as 4.3 (at beta 0.05), 4.0 (at beta 0.1), 3.4 (at beta 0.2), 2.5 (at beta 0.5), 2.1 (at beta 1.0), and 1.7 (at beta 2.0) (also displayed in Fig. 3).
The anisotropy index plotted as a function of beta on the logarithmic scale (Fig. 3) is suggestive of isotropy restoration at higher values of beta.Both the spacecraft measurements and the numerical simulations show that the anisotropy is stronger (i.e., the spectral decay is steeper along the parallel wavevector axis) when beta is below unity, and is moderate and asymptotes slowly to isotropy (A → 1) when beta is above unity.However, the quantitative picture of the beta-dependence is different between the observations and the simulations.In the spacecraft measurements in the solar wind, the anisotropy index falls down rather abruptly in the range of beta from 0.5 1321 Figures

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Full to 1, and then forms a plateau with only moderate decrease in the anisotropy index.Furthermore, the anisotropy index measured in the solar wind tends to higher than that from the simulations.The wavevector anisotropy from the simulations shows a smooth transition from the low-beta to the high-beta regime.The anisotropy index from the simulations exhibits a nearly power-law scaling in the form A ∝ β −α .The slope in the scaling can be determined by the fitting procedure, and we obtain the empirical scaling law of anisotropy as Namely, the slope value is close to −0.3.The scaling law β −0.3 is superposed to the anisotropy index from the simulations in Fig. 3 for comparison.Except for the smallest value of beta (0.05), the scaling law explains the simulation results with only small deviation.
What is the reason of the difference in the anisotropy index profile (or slope) between the observations and the simulations?We interpret that this difference comes from the spatial dimensions in plasma dynamics.The simulations are limited to twodimensional spatial setup, and the motion around the large-scale magnetic field is forbidden.Eddies are suppressed in the simulation in the both two planes, the plane perpendicular to the large-scale field (or the plane spanned by the x and y directions) and that spanned by the parallel and perpendicular directions (the y and z directions).
Therefore, fluctuations in the simulations represent either electromagnetic waves that propagate or non-eddy spatial structures that do not propagate.In such an ideal setup, two-dimensional plasma exhibits a self-organization or systematic transition of the wavevector anisotropy into isotropy toward higher values of beta.Introduction

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Conclusions
Our work provides evidence that the wavevector anisotropy is stronger in low-beta plasmas, and weaker in high-beta plasmas, which justifies the earlier study in NCM14.Furthermore, in the case of two-dimensional plasmas, there exists a scaling law between the anisotropy index and the plasma parameter beta.Magnetized plasma exhibits a tendency to restore isotropy in the wavevector domain gradually toward higher values of beta.This fact, however, should not be taken as surprising when considering that the role of magnetic field is weaker in high-beta plasmas.We observe the primary extension of the wavevector spectrum in the perpendicular direction to the large-scale magnetic field, which supports the notion of filamentation process in structure formation.What determines the scaling slope α 0.3 will be a subject of theoretical studies.It may be worth while to compare with the other observational fact that the variance anisotropy (which is a measure of compressibility in magnetic field fluctuations) also exhibits a power-law scaling to beta with the slope −0.56 (Smith et al., 2006).
We point out that the logics in our work can be reversed, that is, the scaling law can be applied to measurements of magnetic field as a diagnostic tool of beta.At the current stage, such a method is applicable in two-dimensional plasmas.Still, the method can be used to constrain the value of beta in three-dimensional plasmas as well.An interesting extension would be an application to astrophysical systems.If such a scaling exists in the magnetohydrodynamic picture of plasmas, one may obtain or constrain the value of beta by analyzing photometry data of filament structures in interstellar media or astrophysical jets.Introduction

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Full , ion kinetic or hybrid treatment (Comişel et al., Discussion Paper | Discussion Paper | Discussion Paper | et al.
According to this relation, a spectrum is regarded isotropic if the index is unity.Larger values of the index implies that the fluctuation energy is extended to the axis of perpendicular wavevector, while smaller values of the index implies the spectral extension along the axis of parallel wavevector.This index is used as the analysis Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | kV A Ω p = 1.7.For error estimate of the anisotropy index, these wavenumbers are varied by 10 % and this effect is transmitted to computation of the index.The anisotropy profile as a function of beta is displayed in Fig. 3 in the following section.Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | which the fluid picture of plasma is valid.Wavevectors are randomly and isotropically chosen.The amplitude of the pump waves follows Kolmogorov's inertial-range scaling for fluid turbulence, that is, the spectral energy density is set to proportional to |k| −5/3 .In addition, the total fluctuation amplitude is set to 1 % of the large-scale magnetic field, δB B 0 = 0.01.To generate the magnetohydrodynamic Alfvén waves, the plasma velocity is set to correlated to the pump wave magnetic field at the initial time.The periodic boundaries are set to the simulation.No fluctuation energy is given externally into the system, neither the fluid-scale range ( kV A Ω p < 1) nor the kinetic range ( kV A Ω p > 1) during the simulation.Thus, fluctuations (or waves) in the ion-kinetic domain evolve as a consequence of largescale Alfvén waves.

Figure 1 .Figure 2 .Figure 3 .
Figure 1.Magnetic energy spectra in the plane spanned by the perpendicular and parallel components of the wavevectors with respect to the large-scale magnetic field measured by Cluster spacecraft in the solar wind.

Table 1 .
Narita et al. (2014)tude B 0 , ion number density n i , flow speed V , and plasma beta for ions of the solar wind data added to that inNarita et al. (2014).Items are sorted in ascending order of beta.