Supersonic Turbulence Research Articles

Evidence for large scale compressible turbulence in the ism of CSWA13, a star-Forming Lensed Galaxy at z = 1.87 with outflowing wind

Recently, Keerthi Vasan G. et al. (2024) presented spatially resolved observations of a wind outflow in CSWA13, a gravitationally lensed Star-Forming galaxy at z = 1.87. The gravitational lensing allowed for a substantially improved spatial and kinematic resolution of the wind and of the nebular gas. In this paper we take advantage of the resolved data to test for the existence of turbulence and to study its nature. We obtained the autocorrelation functions of the two velocity fields, and derived the spatial structure functions of the residual nebular and wind velocities along the major axis of the galaxy. The structure functions, of both velocity fields, reveal the existence of an underlying k −2 power spectrum scaling. This scaling suggests the existence of supersonic compressible turbulence. The autocorre- lation functions exhibit correlations over scales comparable to the total extent of the velocity fields. Thus, the turbulence is a large scale one. The turbulent timescale corresponding to the largest scale is about 200 Myr, an order of magnitude larger than the estimated age of the wind and of the young stars. This implies that the turbulence in the ISM formed well before the wind and the young stars. Given the large spatial scale of the turbulence, it is plausible that the source of the turbulence is a large scale one e.g. a merger or tidal event that triggered the formation of molecular clouds, in the cores of which, the young stars formed. A steepening of the structure functions on the smaller scales provides an estimate for the effective depth along the line of sight of the turbulent layer. The latter turns out to be ∼ 2kpc

Detached-Eddy Simulation of Supersonic Turbulent Flow over a Cylinder/Skewed Flare Configuration

The computational campaign reported in this article focuses on a series of experiments at Mach 2.85 carried out in the 1980s at NASA Ames Research Center on a set of cylinder/skewed flare configurations designed to produce highly three-dimensional shock-wave/boundary-layer interactions in the absence of end-wall effects. Computations carried out in that era were unable to match the experimental results using the numerical techniques, turbulence models, and grid resolution available at the time. In the present work, newer Reynolds-averaged Navier–Stokes and detached-eddy simulation methods have been applied to these flows, and relatively good agreement has been obtained with the experimental data. Difficulty in capturing the correct separation-bubble size was encountered with initial detached- eddy simulations, but the introduction of resolved turbulence via a boundary-layer trip produced much better results. The present paper reports on results obtained for four inclination angles (0, 5, 10, and 23 deg) of the skewed 30-deg flare. Detached-eddy simulation is seen to be an economical alternative to large-eddy simulation for capturing many features of large-scale separation unsteadiness over long time intervals at true Reynolds number.

Understanding density fluctuations in supersonic, isothermal turbulence.

Supersonic turbulence occurs in many environments, particularly in astrophysics. In the crucial case of isothermal turbulence, the probability density function (PDF) of the logarithmic density, s, is well measured, but a theoretical understanding of the processes leading to this distribution remains elusive. We investigate these processes using Lagrangian tracer particles to track s and [Formula: see text] in direct numerical simulations, and we show that their evolution can be modeled as a stochastic differential process with time-correlated noise. The temporal correlation functions of s and [Formula: see text] decay exponentially, as predicted by the model, and the decay timescale is ≈1/6 the eddy turnover time. The behaviors of the conditional averages of [Formula: see text] and [Formula: see text] are also well explained by the model, which shows that the density PDF arises from a balance between stochastic compressions/expansions, which tend to broaden the PDF, and the acceleration/deceleration of shocks by density gradients, which tends to narrow it.

Exploring the intermittency of magnetohydrodynamic turbulence by synchrotron polarization radiation

Context. Magnetohydrodynamic (MHD) turbulence plays a critical role in many key astrophysical processes, such as star formation, acceleration of cosmic rays, and heat conduction. However, its properties are still poorly understood. Aims. In this work, we explore how to extract the intermittency of compressible MHD turbulence from synthetic and real observations. Methods. We used three statistical methods, namely the probability distribution function, kurtosis, and scaling exponent of the multi-order structure function, to reveal the intermittency of MHD turbulence. Results. Our numerical results demonstrate that: (1) the synchrotron polarization intensity statistics can be used to probe the intermittency of magnetic turbulence, by which we can distinguish different turbulence regimes; (2) the intermittency of MHD turbulence is dominated by the slow mode in the sub-Alfvénic turbulence regime; and (3) the Galactic interstellar medium (ISM) in the low latitude region corresponds to the sub-Alfvénic and supersonic turbulence regime. Conclusions.We have successfully measured the intermittency of the Galactic ISM from synthetic and realistic observations.

Filamentary Molecular Cloud Formation via Collision-induced Magnetic Reconnection in a Cold Neutral Medium

We have investigated the possibility of molecular cloud formation via the collision-induced magnetic reconnection (CMR) mechanism of the cold neutral medium (CNM). Two atomic gas clouds with conditions typical of the CNM were set to collide at the interface of reverse magnetic fields. The cloud–cloud collision triggered magnetic reconnection and produced a giant 20 pc filamentary structure that was not seen in the control models without CMR. The cloud, with rich fiber-like substructures, developed a fully molecular spine at 5 Myr. Radiative transfer modeling of dust emission at far-infrared wavelengths showed that the middle part of the filament contained dense cores over a span of 5 pc. Some of the cores were actively forming stars and typically exhibited both connecting fibers in dust emission and high-velocity gas in CO line emission, indicative of active accretion through streamers. Supersonic turbulence was present in and around the CMR filament due to inflowing gas moving at supersonic velocities in the collision midplane. The shocked gas was condensed and transported to the main filament piece by piece by reconnected fields, making the filament and star formation a bottom-up process. Instead of forming a gravitationally bounded cloud that then fragments hierarchically (top-down) and forms stars, the CMR process creates dense gas pieces and magnetically transports them to the central axis to constitute the filament. Since no turbulence is manually driven, our results suggest that CMR is capable of self-generating turbulence. Finally, the resulting helical field should show field reversal on both sides of the filament from most viewing angles.

Probing the physics of star formation (ProPStar)

Context. Turbulence is a key component of molecular cloud structure. It is usually described by a cascade of energy down to the dissipation scale. The power spectrum for subsonic incompressible turbulence is ∝k−5/3, while for supersonic turbulence it is ∝k−2. Aims. We determine the power spectrum in an actively star-forming molecular cloud, from parsec scales down to the expected magnetohydrodynamic (MHD) wave cutoff (dissipation scale). Methods. We analyzed observations of the nearby NGC 1333 star-forming region in three different tracers to cover the different scales from ∼10 pc down to 20 mpc. The largest scales are covered with the low-density gas tracer 13CO (1–0) obtained with a single dish, the intermediate scales are covered with single-dish observations of the C18O (3–2) line, while the smallest scales are covered in H13CO+ (1–0) and HNC (1–0) with a combination of NOEMA interferometer and IRAM 30m single-dish observations. The complementarity of these observations enables us to generate a combined power spectrum covering more than two orders of magnitude in spatial scale. Results. We derive the power spectrum in an active star-forming region spanning more than 2 decades of spatial scales. The power spectrum of the intensity maps shows a single power-law behavior, with an exponent of 2.9 ± 0.1 and no evidence of dissipation. Moreover, there is evidence that the power spectrum of the ions to have more power at smaller scales than the neutrals, which is opposite to the theoretical expectations. Conclusions. We show new possibilities for studying the dissipation of energy at small scales in star-forming regions provided by interferometric observations.

Simulated analogues II: a new methodology for non-parametric matching of models to observations

ABSTRACT Star formation is a multiscale problem, and only global simulations that account for the connection from the molecular cloud-scale gas flow to the accreting protostar can reflect the observed complexity of protostellar systems. Star-forming regions are characterized by supersonic turbulence, and as a result, it is not possible to simultaneously design models that account for the larger environment and in detail reproduce observed stellar systems. Instead, the stellar inventories can be matched statistically, and the best matches found that approximate specific observations. Observationally, a combination of single-dish telescopes and interferometers are now able to resolve the nearest protostellar objects on all scales from the protostellar core to the inner $10\, \mathrm{au}$. We present a new non-parametric methodology which uses high-resolution simulations and post-processing methods to match simulations and observations using deep learning. Our goal is to perform a down-selection from large data sets of synthetic images to a ranked list of best-matching candidates with respect to the observation. This is particularly useful for binary and multiple stellar systems that form in turbulent environments. The objective is to accelerate the rate at which we can do such comparisons, remove biases from hand-picking matches, and contribute to identifying the underlying physical processes that drive the creation and evolution of observed protostellar systems.

Supersonic turbulence simulations with GPU-based high-order Discontinuous Galerkin hydrodynamics

ABSTRACT We investigate the numerical performance of a Discontinuous Galerkin (DG) hydrodynamics implementation when applied to the problem of driven, isothermal supersonic turbulence. While the high-order element-based spectral approach of DG is known to efficiently produce accurate results for smooth problems (exponential convergence with expansion order), physical discontinuities in solutions, like shocks, prove challenging and may significantly diminish DG’s applicability to practical astrophysical applications. We consider whether DG is able to retain its accuracy and stability for highly supersonic turbulence, characterized by a network of shocks. We find that our new implementation, which regularizes shocks at subcell resolution with artificial viscosity, still performs well compared to standard second-order schemes for moderately high-Mach number turbulence, provided we also employ an additional projection of the primitive variables on to the polynomial basis to regularize the extrapolated values at cell interfaces. However, the accuracy advantage of DG diminishes significantly in the highly supersonic regime. Nevertheless, in turbulence simulations with a wide dynamic range that start with supersonic Mach numbers and can resolve the sonic point, the low-numerical dissipation of DG schemes still proves advantageous in the subsonic regime. Our results thus support the practical applicability of DG schemes for demanding astrophysical problems that involve strong shocks and turbulence, such as star formation in the interstellar medium. We also discuss the substantial computational cost of DG when going to high order, which needs to be weighted against the resulting accuracy gain. For problems containing shocks, this favours the use of comparatively low DG order.

Planck Dust Polarization Power Spectra Are Consistent with Strongly Supersonic Turbulence

The polarization of the cosmic microwave background is rich in information but obscured by foreground emission from the Milky Way’s interstellar medium (ISM). To uncover relationships between the underlying turbulent ISM and the foreground power spectra, we simulated a suite of driven, magnetized, turbulent models of the ISM, varying the fluid properties via the sonic Mach number, MS , and magnetic (Alfvén) Mach number, MA . We measure the power spectra of density (ρ), velocity (v), magnetic field (H), total projected intensity (T), parity-even polarization (E), and parity-odd polarization (B). We find that the slopes of all six quantities increase with MS . Most increase with MA , while the magnetic field spectrum steepens with MA . By comparing spectral slopes of E and B to those measured by Planck, we infer typical values of MS and MA for the ISM. As the fluid velocity increases, MS>4 , the ratio of BB power to EE power increases to approach a constant value near the Planck-observed value of ∼0.5, regardless of the magnetic field strength. We also examine correlation coefficients between projected quantities, and find that r TE ≈ 0.3, in agreement with Planck, for appropriate combinations of MS and MA . Finally, we consider parity-violating correlations r TB and r EB.

Statistics of Energy in Isothermal Supersonic Turbulence

Supersonic isothermal turbulence is a common process in astrophysical systems. In this work, we explore the energy in such systems. We show that the conserved energy is the sum of the kinetic energy (K) and Helmholtz free energy (F). We develop analytic predictions for the probability distributions, P(F) and P(K), as well as their nontrivial joint distribution, P(F, K). We verify these predictions with a suite of driven turbulence simulations, finding excellent agreement. The turbulence simulations were performed at Mach numbers ranging from 1 to 8, and three modes of driving: purely solenoidal, purely compressive, and mixed. We find that P(F) is discontinuous at F = 0, with the discontinuity increasing with Mach number and compressive driving. P(K) resembles a lognormal with a negative skew. The joint distribution, P(F, K), shows a bimodal distribution, with gas either existing at high F and high K or at low F and low K.

CO in the Draco nebula: the atomic–molecular transition

ABSTRACT This paper presents maps of the J=2-1 transition of CO towards the Draco nebula intermediate-velocity cloud (IVC). The maps cover 8500 square arcmin with a velocity resolution of 0.33 km s$^{-1}$ and angular resolution of 38 ${\rm arcsec}$, or 0.11 pc at the cloud distance of 600 pc. The mapped area includes all the emission detected by the Herschel satellite with 250 $\mu$m intensity >5 MJy sr−1. Previously published observations of the far-infrared (far-IR) emission and the 21 cm line of HI are used to derive the column density distribution of H$_2$ and the abundance ratio CO/H$_2$, as well as the distribution of the molecular fraction of hydrogen, which approaches 90 per cent over much of the brighter parts of the nebula. The CO emission is highly clumpy and closely resembles the structures seen in far-IR images. The kinematics of the CO show supersonic motions between clumps but near-thermal to trans-sonic motions within clumps, consistent with model predictions that the scale length for dissipation of supersonic turbulence should be $\sim 0.1$ pc, mediated by kinematic viscosity and/or ambipolar diffusion. Different parts of the nebula show evidence for a spread of molecular formation time-scales of a few 10$^5$ yr, comparable with the dynamical time-scale of the infalling gas. The IVC will likely merge with the Galactic interstellar medium in $\sim 10^7$ yr, and the densest clumps may form an unbound cluster of low-mass stars.

The ALMA-QUARKS Survey. II. The ACA 1.3 mm Continuum Source Catalog and the Assembly of Dense Gas in Massive Star-Forming Clumps

Leveraging the high resolution, sensitivity, and wide frequency coverage of the Atacama Large Millimeter/submillimeter Array (ALMA), the QUARKS survey, standing for “Querying Underlying mechanisms of massive star formation with ALMA-Resolved gas Kinematics and Structures”, is observing 139 massive star-forming clumps at ALMA Band 6 (λ ∼ 1.3 mm). This paper introduces the Atacama Compact Array (ACA) 7 m data of the QUARKS survey, describing the ACA observations and data reduction. Combining multi-wavelength data, we provide the first edition of QUARKS atlas, offering insights into the multiscale and multiphase interstellar medium in high-mass star formation. The ACA 1.3 mm catalog includes 207 continuum sources that are called ACA sources. Their gas kinetic temperatures are estimated using three formaldehyde transitions with a non-LTE radiation transfer model, and the mass and density are derived from a dust emission model. The ACA sources are massive (16–84 percentile values of 6–160 M ⊙), gravity-dominated (M ∝ R 1.1) fragments within massive clumps, with supersonic turbulence ( M>1 ) and embedded star-forming protoclusters. We find a linear correlation between the masses of the fragments and the massive clumps, with a ratio of 6% between the two. When considering fragments as representative of dense gas, the ratio indicates a dense gas fraction (DGF) of 6%, although with a wide scatter ranging from 1% to 10%. If we consider the QUARKS massive clumps to be what is observed at various scales, then the size-independent DGF indicates a self-similar fragmentation or collapsing mode in protocluster formation. With the ACA data over four orders of magnitude of luminosity-to-mass ratio (L/M), we find that the DGF increases significantly with L/M, which indicates clump evolutionary stage. We observed a limited fragmentation at the subclump scale, which can be explained by a dynamic global collapse process.

Development and Validation of a Compressible Reacting Gas-Dynamic Flow Solver for Supersonic Combustion

An approach based on the OpenFOAM library has been developed to solve a high-speed, multicomponent mixture of a reacting, compressible flow. This work presents comprehensive validation of the newly developed solver, called compressibleCentralReactingFoam, with different supersonic flows, including shocks, expansion waves, and turbulence–combustion interaction. The comparisons of the simulation results with experimental and computational data confirm the fidelity of this solver for problems involving multicomponent high-speed reactive flows. The gas dynamics of turbulence–chemistry interaction are modeled using a partially stirred reactor formulation and provide promising results to better understand the complex physics involved in supersonic combustors. A time-scale analysis based on local Damköhler numbers reveals different regimes of turbulent combustion. In the core of the jet flow, the Damköhler number is relatively high, indicating that the reaction time scale is smaller than the turbulent mixing time scale. This means that the combustion is controlled by turbulent mixing. In the shear layer, where the heat release rate and the scalar dissipation rate have the highest value, the flame is stabilized due to finite rate chemistry with small Damköhler numbers and a limited fraction of fine structure. This solver allows three-dimensional gas dynamic simulation of high-speed multicomponent reactive flows relevant to practical combustion applications.

Amplification and Saturation of Turbulent Magnetic Fields in Collapsing Primordial Gas Clouds

Recent numerical studies suggest that magnetic fields play an important role in primordial star formation in the early Universe. However, the detailed evolution of the magnetic field in the collapse phase still has uncertainties because of the complicated physics associated with turbulence in a collapsing magnetized system. Here, we perform a suite of numerical MHD simulations that follow the collapse of magnetized, turbulent primordial gas clouds to investigate the evolution of the magnetic field associated with the turbulence, assuming a polytropic equation of state with exponent γ eff and with various numerical resolutions. In addition, we generalize the analytic theory of magnetic field growth/saturation so that it can deal with various exponents γ eff and turbulence energy spectra. We find that the numerical results are well reproduced by the theory for various γ eff through the collapse phase during the formation of the first stars. The magnetic field is eventually amplified by a factor of 1012–1015 due to kinematic and nonlinear turbulent dynamo effects and reaches 3%–100% of the equipartition level, depending on γ eff. We also find that the transition between the kinematic and nonlinear stages can be analytically estimated. These results indicate that the strong magnetic field accompanied by supersonic turbulence is a general property and suggest that it can play a crucial role in the formation of the first stars.

Formaldehyde observations of the Perseus Molecular Cloud

ABSTRACT Large-scale observations of the Perseus Molecular Cloud (MC) with Nanshan 26-m telescope are presented using the 6 cm ortho-H2CO (110–111) transition. As a probe of dense gas at low temperatures, the H2CO absorption extends over the main parts of the Perseus MC. A comparison of the H2CO absorption line with the 12CO and 13CO (J = 1–0) emissions shows that the H2CO and CO are similarly distributed over the Perseus but that H2CO correlates better with 13CO. Comparison with the Herschel-derived H2 column density shows that both the 13CO and H2CO column densities vary linearly with the H2 column density. The main parameters of H2CO absorption line data show a log-normal distribution, which suggests that the strong non-thermal line-broadening results from large-scale supersonic turbulence related to the star formation. Formaldehyde absorption serves well as a tracer of star formation activity and also the H2CO-to-13CO and H2CO-to-H2 abundances systematically trace the star formation activity in the six subregions of Perseus MC. The H2CO abundances anticorrelate with the number of prestellar and protostellar cores and the IRIS 12 μm flux in the six subregions and reveal the star formation history in the Perseus MC.

Necessary conditions for the formation of filaments and star clusters in the cold neutral medium

ABSTRACT Star formation takes place in filamentary molecular clouds which arise by physical processes that take place in the cold neutral medium (CNM). We address the necessary conditions for this diffuse (n ≈ 30 cm−3), cold (T ≈ 60 K), magnetized gas undergoing shock waves, and supersonic turbulence, to produce filamentary structures capable of fragmenting into cluster forming regions. Using ramses and a magnetized CNM environment as our initial conditions, we simulate a 0.5 kpc turbulent box to model a uniform gas with magnetic field strength of 7 μG, varying the 3D velocity dispersion via decaying turbulence. We use a surface density of 320 M⊙ pc−2, representative of the inner 4.0 kpc central molecular zone of the Milky Way and typical luminous galaxies. Filamentary molecular clouds are formed dynamically via shocks within a narrow range of velocity dispersions in the CNM of 5–10 km s−1 with a preferred value at 8 km s−1. Cluster sink particles appear in filaments which exceed their critical line mass, occurring optimally for velocity dispersions of 8 km s−1. Tracking the evolution of magnetic fields, we find that they lead to double the dense star-forming gas than in purely hydro runs. Perpendicular orientations between magnetic field and filaments can increase the accretion rates onto filaments and hence their line masses. Because magnetic fields help support gas, magnetohydrodynamic runs result in average temperatures an order of magnitude higher than unmagnetized counterparts. Finally, we find magnetic fields delay the onset of cluster formation by ∝ 0.4 Myr.

The role of turbulence in high-mass star formation: Subsonic and transonic turbulence are ubiquitously found at early stages

Context. Traditionally, supersonic turbulence is considered to be one of the most likely mechanisms slowing the gravitational collapse in dense clumps, thereby enabling the formation of massive stars. However, several recent studies have raised differing points of view based on observations carried out with sufficiently high spatial and spectral resolution. These studies call for a re-evaluation of the role turbulence plays in massive star-forming regions. Aims. Our aim is to study the gas properties, especially the turbulence, in a sample of massive star-forming regions with sufficient spatial and spectral resolution, which can both resolve the core fragmentation and the thermal line width. Methods. We observed NH3 metastable lines with the Very Large Array (VLA) to assess the intrinsic turbulence. Results. Analysis of the turbulence distribution histogram for 32 identified NH3 cores reveals the presence of three distinct components. Furthermore, our results suggest that (1) sub- and transonic turbulence is a prevalent (21 of 32) feature of massive star-forming regions and those cold regions are at early evolutionary stage. This investigation indicates that turbulence alone is insufficient to provide the necessary internal pressure required for massive star formation, necessitating further exploration of alternative candidates; and (2) studies of seven multi-core systems indicate that the cores within each system mainly share similar gas properties and masses. However, two of the systems are characterized by the presence of exceptionally cold and dense cores that are situated at the spatial center of each system. Our findings support the hub-filament model as an explanation for this observed distribution.

A Two-Component Probability Distribution Function Describes the Mid-IR Emission from the Disks of Star-forming Galaxies

High-resolution JWST-MIRI images of nearby spiral galaxies reveal emission with complex substructures that trace dust heated both by massive young stars and the diffuse interstellar radiation field. We present high angular (0.″85) and physical resolution (20–80 pc) measurements of the probability distribution function (PDF) of mid-infrared (mid-IR) emission (7.7–21 μm) from 19 nearby star-forming galaxies from the PHANGS-JWST Cycle 1 Treasury. The PDFs of mid-IR emission from the disks of all 19 galaxies consistently show two distinct components: an approximately lognormal distribution at lower intensities and a high-intensity power law component. These two components only emerge once individual star-forming regions are resolved. Comparing with locations of H ii regions identified from Very Large Telescope/MUSE Hα mapping, we infer that the power-law component arises from star-forming regions and thus primarily traces dust heated by young stars. In the continuum-dominated 21 μm band, the power law is more prominent and contains roughly half of the total flux. At 7.7–11.3 μm, the power law is suppressed by the destruction of small grains (including PAHs) close to H ii regions, while the lognormal component tracing the dust column in diffuse regions appears more prominent. The width and shape of the lognormal diffuse emission PDFs in galactic disks remain consistent across our sample, implying a lognormal gas column density N(H) ≈ 1021 cm−2 shaped by supersonic turbulence with typical (isothermal) turbulent Mach numbers ≈5−15. Finally, we describe how the PDFs of galactic disks are assembled from dusty H ii regions and diffuse gas and discuss how the measured PDF parameters correlate with global properties such as star formation rate and gas surface density.

Numerical Simulation of Supersonic Turbulent Separated Flows Based on k–ω Turbulence Models with Different Compressibility Corrections

The accurate prediction of supersonic turbulent separated flows involved in aerospace vehicles is a great challenge for current numerical simulations. Based on the k–ω equations, several different compressibility corrections are incorporated in turbulence models to improve their prediction capabilities. Two benchmark test cases, namely the ramped cavity and the compression corner, are adopted for the numerical validation. Detailed comparisons between simulations and experiments are conducted to evaluate the effect of compressibility corrections on turbulence models. The computed results indicate that compressibility corrections have a significant impact on turbulence model performance. The compressibility correction, considering the effects of both dilatation dissipation and pressure dilatation, is suitable for the prediction of compressible free shear layers, but it may have a negative impact on the prediction of low-speed flows in the near-wall region due to the severe underprediction of the wall skin friction coefficient. In comparison, the compressibility correction only considering the effects of dilatation dissipation is conservative, with decreased predictability of free shear layers in supersonic flows, although it improves the predictions of the original models without corrections.

Numerical Testing of Mirror Diffusion of Cosmic Rays

The tension between recent observations and theories on cosmic-ray (CR) diffusion necessitates exploration of new CR diffusion mechanisms. We perform the first numerical study on the mirror diffusion of CRs that is recently proposed by Lazarian & Xu. We demonstrate that the perpendicular superdiffusion of turbulent magnetic fields and magnetic mirroring that naturally arise in magnetohydrodynamic (MHD) turbulence are the two essential physical ingredients for the mirror diffusion to happen. In supersonic, subsonic, and incompressible MHD turbulence, with the pitch angles of CRs repeatedly crossing 90° due to the mirror reflection, we find that the mirror diffusion strongly enhances the confinement of CRs, and their pitch-angle-dependent parallel mean free path can be much smaller than the injection scale of turbulence. With the stochastic change of pitch angles due to gyroresonant scattering, CRs stochastically undergo slow mirror diffusion at relatively large pitch angles and fast scattering diffusion at smaller pitch angles, resulting in a Lévy-flight-like propagation.