Magnetic Field Correlation Length Research Articles

Solar Wind Magnetic Field Correlation Length: Correlation Functions versus Cross-field Displacement Diffusivity Test

Abstract The estimate of the solar wind magnetic fields’ parallel correlation length, λ, be it from the measured fields’ correlation functions or their spectral power at “zero” frequency, have long pointed toward short values on the order of 0.01 au. Evaluation of the mean cross-field displacements (CFDs), however, fails to show the decorrelation and resulting diffusion at the expected scales, pointing instead toward λ values on the order of 0.1 au or more. In an effort to understand this “order-of-magnitude” discrepancy and reconcile the approaches using correlation functions and the CFD diffusivity test, both approaches are applied here, with renewed attention to the “details” as well as the broader sense of the calculations, to a large, 20 yr long set of magnetic field and flow data from the ACE spacecraft. It is found that solar wind intervals too short relative to λ are a likely reason for some underestimate through the correlation-function approach, causing a premature drop of the correlation functions. Once converged to their long-time limit, however, the correlation functions produce magnetic field correlation lengths very much consistent with the magnetic-field-line (MFL) correlation lengths of the diffusivity test, with nearly matching distributions of the correlation lengths corrected by the proper ratio of their theoretical estimates. The fields’ correlation lengths mostly range from 0.03 to 0.08 au, and the MFL correlation lengths from 0.04 to 0.3 au, with peaks at 0.075 and 0.15 au, likely due to nonlinear and quasilinear regimes of MFL wandering. As for the power-at-zero-frequency approach, it is doomed by the solar rotation.

Nonlinear Turbulent Dynamo during Gravitational Collapse

Abstract Via amplification by turbulent dynamo, magnetic fields can be potentially important for the formation of the first stars. To examine the dynamo behavior during the gravitational collapse of primordial gas, we extend the theory of the nonlinear turbulent dynamo to include the effect of gravitational compression. The relative importance between dynamo and compression varies during contraction, with the transition from dynamo- to compression-dominated amplification of magnetic fields with the increase of density. In the nonlinear stage of magnetic field amplification with the scale-by-scale energy equipartition between turbulence and magnetic fields, reconnection diffusion of magnetic fields in ideal magnetohydrodynamic turbulence becomes important. It causes the violation of the flux-freezing condition and accounts for (a) the small growth rate of the nonlinear dynamo, (b) the weak dependence of magnetic energy on density during contraction, (c) the saturated magnetic energy, and (d) the large correlation length of magnetic fields. The resulting magnetic field structure and the scaling of magnetic field strength with density are radically different from the expectations of flux freezing.

Viable inflationary magnetogenesis with helical coupling

We consider helical coupling to electromagnetism and present a simple scenario of evolution of the coupling function leading to a viable inflationary magnetogenesis without the problem of back-reaction. In this scenario, helical magnetic fields of strength of order up to 10− 7 G, when extrapolated to the current epoch, can be generated in a narrow spectral band centered at any reasonable wavenumber by adjusting the model parameters. We discuss implications of this model for baryogenesis, which impose additional constraints on the strength and correlation length of magnetic field.

Diffusion of charged particles in a stochastic force-free magnetic field

We study diffusion of charged particles in stationary stochastic magnetic field ${\bf B}$ with zero mean, $\langle {\bf B} \rangle = 0 $. In the case when electric current is carried by electrons, the field is force-free, $\mathrm{curl} \,{\bf B} = \alpha{\bf B} $, where $\alpha({\bf r})$ is an arbitrary scalar function. In a small region where the function $\alpha $ and the field magnitude $|{\bf B}|$ are approximately constant, the equations of motion of charged particles are integrated and reduced to the equation of mathematical pendulum. The transition from trapped to untrapped particles is continuously traced. Averaging over the magnetic field spectrum gives the spatial diffusion coefficient $D$ of particles as a function of the Larmor radius $r_L$ in the large-scale magnetic fields ($B_{LS}$) and magnetic field correlation length $L_0$. The diffusion coefficient turns out to be proportional to the Larmor radius, $D\propto r_L $, for $r_L <L_0 / 2\pi $, and to the Larmor radius squared, $ D \propto r_L^2 $, for $ r_L> L_0 /2\pi $. We apply obtained results to the diffusion of cosmic rays in the Galaxy, which contains a large number of independent regions with parameters $L_0$ and $B_{LS}$ varying in wide range. We average over $B_{LS}$ with the Kolmogorov spectrum and over $L_0$ with the distribution function $f(L_0)\propto L_0^{- 1+ \sigma}$. For the practically flat spectrum $\sigma = 1/15$, we have $ D\propto r_m^{0.7}$, which is consistent with observations.

Statistical properties of scale-invariant helical magnetic fields and applications to cosmology

We investigate the statistical properties of isotropic, stochastic, Gaussian distributed, helical magnetic fields characterized by different shapes of the energy spectra at large length scales and study the associated realizability condition. We discuss smoothed magnetic fields that are commonly used when the primordial magnetic field is constrained by observational data. We are particularly interested in scale-invariant magnetic fields that can be generated during the inflationary stage by quantum fluctuations. We determine the correlation length of such magnetic fields and relate it to the infrared cutoff of perturbations produced during inflation. We show that this scale determines the observational signatures of the inflationary magnetic fields on the cosmic microwave background. At smaller scales, the scale-invariant spectrum changes with time. It becomes a steeper weak-turbulence spectrum at progressively larger scales. We show numerically that the critical length scale where this happens is the turbulent-diffusive scale, which increases with the square root of time.

Influence of the Solar Cycle on Turbulence Properties and Cosmic-Ray Diffusion

Abstract The solar cycle dependence of various turbulence quantities and cosmic-ray (CR) diffusion coefficients is investigated by using OMNI 1 minute resolution data over 22 years. We employ Elsässer variables z ± to calculate the magnetic field turbulence energy and correlation lengths for both the inwardly and outwardly directed interplanetary magnetic field (IMF). We present the temporal evolution of both large-scale solar wind (SW) plasma variables and small-scale magnetic fluctuations. Based on these observed quantities, we study the influence of solar activity on CR parallel and perpendicular diffusion using quasi-linear theory and nonlinear guiding center theory, respectively. We also evaluate the radial evolution of the CR diffusion coefficients by using the boundary conditions for different solar activity levels. We find that in the ecliptic plane at 1 au (1), the large-scale SW temperature T, velocity V sw, Alfvén speed V A , and IMF magnitude B 0 are positively related to solar activity; (2) the fluctuating magnetic energy density , residual energy E D , and corresponding correlation functions all have an obvious solar cycle dependence. The residual energy E D is always negative, which indicates that the energy in magnetic fluctuations is larger than the energy in kinetic fluctuations, especially at solar maximum; (3) the correlation length λ for magnetic fluctuations does not show significant solar cycle variation; (4) the temporally varying shear source of turbulence, which is most important in the inner heliosphere, depends on the solar cycle; (5) small-scale fluctuations may not depend on the direction of the background magnetic field; and (6) high levels of SW fluctuations will increase CR perpendicular diffusion and decrease CR parallel diffusion, but this trend can be masked if the background IMF changes in concert with turbulence in response to solar activity. These results provide quantitative inputs for both turbulence transport models and CR diffusion models, and also provide valuable insight into the long-term modulation of CRs in the heliosphere.

Resolution Dependence of Magnetorotational Turbulence in the Isothermal Stratified Shearing Box

Abstract Magnetohydrodynamic turbulence driven by the magnetorotational instability can provide diffusive transport of angular momentum in astrophysical disks, and a widely studied computational model for this process is the ideal, stratified, isothermal shearing box. Here we report results of a convergence study of such boxes up to a resolution of N = 256 zones per scale height, performed on blue waters at NCSA with ramses-gpu. We find that the time and vertically integrated dimensionless shear stress , i.e., the shear stress is resolution dependent. We also find that the magnetic field correlation length decreases with resolution, . This variation is strongest at the disk midplane. We show that our measurements of are consistent with earlier studies, and we discuss possible reasons for the lack of convergence.

Magnetogenesis from axion inflation

In this work we compute the production of magnetic fields in models of axion inflation coupled to the hypercharge sector of the Standard Model through a Chern-Simons interaction term. We make the simplest choice of a quadratic inflationary potential and use lattice simulations to calculate the magnetic field strength, helicity and correlation length at the end of inflation. For small values of the axion-gauge field coupling strength the results agree with no-backreaction calculations and estimates found in the literature. For larger couplings the helicity of the magnetic field differs from the no-backreaction estimate and depends strongly on the comoving wavenumber. We estimate the post-inflationary evolution of the magnetic field based on known results for the evolution of helical and non-helical magnetic fields. The magnetic fields produced by axion inflation with large couplings to U(1)Y can reach Beff ≳ 10−16 G, exhibiting a field strength Bphys ≈ 10−13 G and a correlation length λphys ≈10 pc. This result is insensitive to the exact value of the coupling, as long as the coupling is large enough to allow for instantaneous preheating. Depending on the assumptions for the physical processes that determine blazar properties, these fields can be found consistent with blazar observations based on the value of Beff. Finally, the intensity of the magnetic field for large coupling can be enough to satisfy the requirements for a recently proposed baryogenesis mechanism, which utilizes the chiral anomaly of the Standard Model.

Cosmological magnetic fields: their generation, evolution and observation

We review the possible mechanisms for the generation of cosmological magnetic fields, discuss their evolution in an expanding Universe filled with the cosmic plasma and provide a critical review of the literature on the subject. We put special emphasis on the prospects for observational tests of the proposed cosmological magnetogenesis scenarios using radio and gamma-ray astronomy and ultra high energy cosmic rays. We argue that primordial magnetic fields are observationally testable. They lead to magnetic fields in the intergalactic medium with magnetic field strength and correlation length in a well defined range. We also state the unsolved questions in this fascinating open problem of cosmology and propose future observations to address them.

Evolution of helical cosmic magnetic fields as predicted by magnetohydrodynamic closure theory

We extend our recent derivation of the time evolution equations for the energy content of magnetic fields and turbulent motions for incompressible, homogeneous, and isotropic turbulence to include the case of nonvanishing helicity. These equations are subsequently numerically integrated in order to predict the present day primordial magnetic field strength and correlation length, depending on its initial helicity and magnetic energy density. We find that all prior analytic predictions for helical magnetic fields, such as the epoch when they become maximally helical and their subsequent growth of correlation length L ~ a^{1/3} and decrease of magnetic field strength B ~ a^{-1/3} with scale factor a, are well confirmed by the simulations. An initially fully helical primordial magnetic field is a factor 4 10^4 stronger at the present epoch then its nonhelical counterpart when generated during the electroweak epoch.

Measuring the correlation length of intergalactic magnetic fields from observations of gamma-ray induced cascades

Context. The imaging and timing properties of {\gamma}-ray emission from electromagnetic cascades initiated by very-high-energy (VHE) {\gamma}-rays in the intergalactic medium depend on the strength B and correlation length {\lambda}B of intergalactic magnetic fields (IGMF). Aims. We study the possibility of measuring both B and {\lambda}B via observations of the cascade emission with {\gamma}-ray telescopes. Methods. For each measurement method, we find two characteristics of the cascade signal, which are sensitive to the IGMF B and {\lambda}B values in different combinations. For the case of IGMF measurement using the observation of extended emission around extragalactic VHE {\gamma}-ray sources, the two characteristics are the slope of the surface brightness profile and the overall size of the cascade source. For the case of IGMF measurement from the time delayed emission, these two characteristics are the initial slope of the cascade emission light curve and the overall duration of the cascade signal. Results. We show that measurement of the slope of the cascade induced extended emission and/or light curve can both potentially provide measure of the IGMF correlation length, provided it lies within the range 10 kpc< {\lambda}B <1 Mpc. For correlation lengths outside this range, gamma-ray observations can provide upper or lower bound on {\lambda}B. The latter of the two methods holds great promise in the near future for providing a measurement/constraint using measurements from present/next-generation {\gamma}-ray-telescopes. Conclusions. Measurement of the IGMF correlation length will provide an important constraint on its origin. In particular, it will enable to distinguish between an IGMF of galactic wind origin from an IGMF of cosmological origin.

GLOBAL GENERAL RELATIVISTIC MAGNETOHYDRODYNAMIC SIMULATIONS OF BLACK HOLE ACCRETION FLOWS: A CONVERGENCE STUDY

Global, general relativistic magnetohydrodynamic (GRMHD) simulations of nonradiative, magnetized disks are widely used to model accreting black holes. We have performed a convergence study of GRMHD models computed with HARM3D. The models span a factor of 4 in linear resolution, from 96x96x64 to 384x384x256. We consider three diagnostics of convergence: (1) dimensionless shell-averaged quantities such as plasma \beta; (2) the azimuthal correlation length of fluid variables; and (3) synthetic spectra of the source including synchrotron emission, absorption, and Compton scattering. Shell-averaged temperature is, except for the lowest resolution run, nearly independent of resolution; shell-averaged plasma \beta\ decreases steadily with resolution but shows signs of convergence. The azimuthal correlation lengths of density, internal energy, and temperature decrease steadily with resolution but show signs of convergence. In contrast, the azimuthal correlation length of magnetic field decreases nearly linearly with grid size. We argue by analogy with local models, however, that convergence should be achieved with another factor of 2 in resolution. Synthetic spectra are, except for the lowest resolution run, nearly independent of resolution. The convergence behavior is consistent with that of higher physical resolution local model (shearing box) calculations and with the recent nonrelativistic global convergence studies of Hawley et al. (2011).

ACCELERATION, MAGNETIC FLUCTUATIONS, AND CROSS-FIELD TRANSPORT OF ENERGETIC ELECTRONS IN A SOLAR FLARE LOOP

Plasma turbulence is thought to be associated with various physical processes involved in solar flares, including magnetic reconnection, particle acceleration and transport. Using Ramaty High Energy Solar Spectroscopic Imager ({\it RHESSI}) observations and the X-ray visibility analysis, we determine the spatial and spectral distributions of energetic electrons for a flare (GOES M3.7 class, April 14, 2002 23$:$55 UT), which was previously found to be consistent with a reconnection scenario. It is demonstrated that because of the high density plasma in the loop, electrons have to be continuously accelerated about the loop apex of length $\sim 2\times 10^9$cm and width $\sim 7\times 10^8$cm. Energy dependent transport of tens of keV electrons is observed to occur both along and across the guiding magnetic field of the loop. We show that the cross-field transport is consistent with the presence of magnetic turbulence in the loop, where electrons are accelerated, and estimate the magnitude of the field line diffusion coefficient for different phases of the flare. The energy density of magnetic fluctuations is calculated for given magnetic field correlation lengths and is larger than the energy density of the non-thermal electrons. The level of magnetic fluctuations peaks when the largest number of electrons is accelerated and is below detectability or absent at the decay phase. These hard X-ray observations provide the first observational evidence that magnetic turbulence governs the evolution of energetic electrons in a dense flaring loop and is suggestive of their turbulent acceleration.

The Variation of Solar Wind Correlation Lengths Over Three Solar Cycles

We present the results of a study of solar wind velocity and magnetic field correlation lengths over the last 35 years. The correlation length of the magnetic field magnitude lambda(|B|) increases on average by a factor of two at solar maxima compared to solar minima. The correlation lengths of the components of the magnetic field lambda(Bxyz) and of the velocity lambda(Vyz) do not show this change and have similar values, indicating a continual turbulent correlation length of around 1.4 x 10^6 km. We conclude that a linear relation between lambda(|B|), VB^2 and Kp suggests that the former is related to the total magnetic energy in the solar wind and an estimate of the average size of geo-effective structures which is in turn proportional to VB^2. By looking at the distribution of daily correlation lengths we show that the solar minimum values of lambda(|B|) correspond to the turbulent outer scale. A tail of larger lambda(|B|) values is present at solar maximum, causing the increase in mean value.

The Hanle effect in a random magnetic field

Context. The Hanle effect is used to determine weak turbulent magnetic fields in the solar atmosphere, usually assuming that the angular distribution is isotropic, the magnetic field strength constant, and that micro-turbulence holds, i.e. that the magnetic field correlation length is much less than a photon mean free path. Aims. To examine the sensitivity of turbulent magnetic field measurements to these assumptions, we study the dependence of Hanle effect on the magnetic field correlation length, its angular, and strength distributions. Methods. We introduce a fairly general random magnetic field model characterized by a correlation length and a magnetic field vector distribution. Micro-turbulence is recovered when the correlation length goes to zero and macro-turbulence when it goes to infinity. Radiative transfer equations are established for the calculation of the mean Stokes parameters and they are solved numerically by a polarized approximate lambda iteration method. Results. We show that optically thin spectral lines and optically very thick ones are insensitive to the correlation length of the magnetic field, while spectral lines with intermediate optical depths (around 10–100) show some sensitivity to this parameter. The result is interpreted in terms of the mean number of scattering events needed to create the surface polarization. It is shown that the single-scattering approximation holds good for thin and thick lines but may fail for lines with intermediate thickness. The dependence of the polarization on the magnetic field vector probability density function (PDF) is examined in the micro-turbulent limit. A few PDFs with different angular and strength distributions, but equal mean value of the magnetic field, are considered. It is found that the polarization is in general quite sensitive to the shape of the magnetic field strength PDF and somewhat to the angular distribution. Conclusions. The mean field derived from Hanle effect analysis of polarimetric data strongly depends on the choice of the field strength distribution used in the analysis. It is shown that micro-turbulence is in general a safe approximation.

Magnetic Field Production during Preheating at the Electroweak Scale

We study the generation of magnetic fields during preheating within a scenario of hybrid inflation at the electroweak scale. We find that the nonperturbative and strongly out-of-equilibrium process of generation of magnetic fields with a nontrivial helicity occurs along the lines predicted by Vachaspati many years ago. The magnitude (rho_{B}/rho_{EW} approximately 10{-2}) and correlation length of these helical magnetic fields grow linearly with time during preheating and are consistent with the possibility that these seeds gave rise to the microgauss fields observed today in galaxies and clusters of galaxies.

The Hanle effect in a random medium

This paper considers the Hanle effect produced by a turbulent magnetic field. To overcome the simplified microturbulent treatment whereby the Hanle phase matrix is locally averaged over some magnetic field distribution, we consider a turbulent magnetic field with a finite correlation length. We assume that the magnetic field along each individual photon path can be represented by a Kubo-Anderson process (KAP) and study the stationary solution as time goes to infinity. A KAP is a discontinuous Markov process. The random magnetic field is characterized by a correlation length and a distribution function of the magnetic field vector; both can be chosen arbitrarily. The microturbulent limit is recovered when the correlation length goes to zero. A non-stochastic integral equation of the Wiener-Hopf type is obtained for a mean conditional source vector. This integral equation yields explicit expressions for the mean Stokes parameters, provided one makes physically realistic approximations, namely neglect the effect of the magnetic field on Stokes I , keep only the contributions from I and Q in the source terms for Stokes Q and Stokes U and solve the integral equation for Q with a two-scattering approximation. The final expressions involve mean values and correlation functions of some of the elements of the Hanle phase matrix and show the dependence on the correlation length of the random magnetic field. The combined effects of a turbulent velocity field and a turbulent magnetic field with finite correlation lengths is also studied. The velocity field is represented by a KAP with the same correlation length as the magnetic field. Some of the velocity field effects are treated with an effective medium approximation as in Frisch & Frisch (1976, MNRAS, 175, 157). Explicit expressions are obtained for the mean Stokes parameters. They can account for correlations between velocity field and magnetic field fluctuations.

Anomalous and classical diffusion of cosmic rays due to nonlinear two‐dimensional structures and random magnetic fields

Magnetic field turbulence in the heliosphere is modeled by two‐dimensional coherent vortex structures and random‐phase linear wave fields added to a background magnetic field. Energetic particle trapping and acceleration by interaction with the turbulent fields are studied numerically. The corresponding radial and field‐aligned diffusion coefficients are estimated. Different diffusion regimes (subdiffusion, superdiffusion, classical diffusion) are controlled by the following parameters: the ratio between amplitudes of the random component and the regular vortex field, ε1, and the ratio between the average particle gyroradius and the magnetic field correlation length, ε2. The anomalous diffusion regime occurs when the correlation length of magnetic field disturbances is about the same or larger than the particle gyroradius (ε2 ≤ 1). Applications of our results to cosmic ray transport are discussed.

Evolution of cosmic magnetic fields: From the very early Universe, to recombination, to the present

A detailed numerical and analytical examination of the evolution of stochastic magnetic fields between a putative magnetogenesis era at high cosmic temperatures $T\ensuremath{\sim}100\text{ }\mathrm{M}\mathrm{e}\mathrm{V}--100\text{ }\mathrm{G}\mathrm{e}\mathrm{V}$ and the present epoch is presented. The analysis includes all relevant dissipation processes, such as neutrino- and photon-induced fluid viscosities as well as ambipolar and hydrogen diffusion. A simple and intuitive analytical model matching the results of the three-dimensional MHD simulations allows for the prediction of prerecombination and present day magnetic field correlation lengths and energy densities as a function of initial magnetic field energy density, helicity, and spectral index. Our conclusions are multifold. (a) Initial primordial fields with only a small amount of helicity are evolving into maximally helical fields at the present. Furthermore, the simulations show a self-similarity in the evolution of maximally helical fields implying a seemingly acausual amplification of magnetic fields on large scales is observed. (b) There exists a correlation between the strength of the magnetic field $B$ at the peak of its spectrum and the location of the peak, given at the present epoch by $B\ensuremath{\approx}5\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}12}\text{ }\mathrm{G}(L/\mathrm{k}\mathrm{p}\mathrm{c})$, where $L$ is the magnetic field correlation length determined by the initial properties of the magnetic field. (c) Concerning studies of the generation of cosmic microwave background (CMBR) anisotropies due to primordial magnetic fields of $B\ensuremath{\sim}{10}^{\ensuremath{-}9}\text{ }\mathrm{G}$ on $\ensuremath{\gtrsim}10\text{ }\mathrm{M}\mathrm{p}\mathrm{c}$ scales, such fields are not only impossible to generate in early causal magnetogenesis scenarios but also seemingly ruled out by distortions of the CMBR spectrum due to magnetic field dissipation on smaller scales and the overproduction of cluster-magnetic fields. (d) The most promising detection possibility of CMBR distortions due to primordial magnetic fields may be on much smaller scales at higher multipoles $l\ensuremath{\sim}{10}^{6}$ where the signal is predicted to be the strongest. (e) It seems possible that magnetic fields in clusters of galaxies are entirely of primordial origin, without invoking dynamo amplification. Such fields would be of (precollapse) strength ${10}^{\ensuremath{-}12}--{10}^{\ensuremath{-}11}\text{ }\mathrm{G}$ with correlation lengths in the kpc range and would also exist in voids of galaxies.

Faraday Rotation of the Cosmic Microwave Background Polarization and Primordial Magnetic Field Properties

Measurements of the Faraday rotation of the cosmic microwave background radiation (CMBR) polarization could provide evidence for the existence of primordial magnetic fields. The Faraday rotation could also allow the study of some properties of these fields. In this paper, we calculate the angular dependence of the Faraday rotation correlator for different assumptions about the spectral index and correlation length of the magnetic field. We show that the helical part of the magnetic field does not make any contribution to the correlator. We stress the importance of the angular resolution of the detector in the Faraday rotation measure, showing that it could severely reduce the effect, even for a relatively large magnetic field correlation length.