Power-law Energy Distribution Research Articles

Constraining the average magnetic field in galaxy clusters with current and upcoming CMB surveys

Galaxy clusters that host radio halos indicate the presence of population(s) of non-thermal electrons. These electrons can scatter low-energy photons of the Cosmic Microwave Background, resulting in the non-thermal Sunyaev-Zeldovich (ntSZ) effect. We measure the average ntSZ signal from 62 radio-halo hosting clusters using the Planck multi-frequency all-sky maps. We find no direct evidence of the ntSZ signal in the Planck data. Combining the upper limits on the non-thermal electron density with the average measured synchrotron power collected from the literature, we place lower limits on the average magnetic field strength in our sample. The lower limit on the volume-averaged magnetic field is 0.01–0.24 μG, depending on the assumed power-law distribution of electron energies. We further explore the potential improvement of these constraints from the upcoming Simons Observatory and Fred Young Submillimeter Telescope (FYST) of the CCAT-prime collaboration. We find that combining these two experiments, the constraints will improve by a factor oftwo, which can be sufficient to rule out some power-law models.

Survey of non-thermal electrons around supermassive black holes through polarization flips

Abstract Optically thick non-thermal synchrotron sources notably produce linear polarization vectors that are parallel to projected magnetic field lines on the observer’s screen, although they are perpendicular in well-known optically thin cases. To elucidate the complex relationship between the vectors and fields, and to investigate the energy and spatial distribution of non-thermal electrons through the images, we perform polarization radiative transfer calculations at submillimeter wavelengths. Here the calculations are based on semi-analytic force-free jet models with non-thermal electrons with a power-law energy distribution. In the calculated images, we find a $90{^\circ }$-flip of linear polarization vectors at the base of counter-side (receding) jet near a black hole, which occurs because of large optical depths for the synchrotron self-absorption effect. The $90{^\circ }$-flip of LP vectors is also seen on the photon ring at a high frequency, since the optical depth along the rays is large there due to the light bending effect. In addition, we see the flip of the sign of circular polarization components on the counter-jet and photon ring. Furthermore, we show that these polarization flips are synthesized with large values in the spectral index map, and also give rise to outstanding features in the Faraday Rotation Measure map. Since the conditions of flipping depend on the magnetic field strength and configuration and the energy distribution of electrons, we can expect that the polarization flips will provide us with observational evidence for the presence of non-thermal electrons around the black hole, and a clue to the magnetic driving mechanism of plasma jets.

Gravitational Self-lensing of Fast Radio Bursts in Neutron Star Magnetospheres. I. The Model

Fast radio bursts (FRBs) are cosmological subsecond bursts of coherent radio emission, whose source is still unknown. To date, the Galactic magnetar SGR 1935 + 2154 is the only astrophysical object known to emit radio bursts akin to FRBs, albeit less powerful, supporting suggestions that FRBs originate from magnetars. Many remarkable properties of FRBs—e.g., the dichotomy between repeaters and one-off sources, and their power-law energy distributions (with typical index ∼2–3)—are not well understood yet. Moreover, the huge radio power released by the most active repeaters is challenging even for the magnetic energy reservoir of magnetars. Here, we assume that FRBs originate from corotating hotspots anchored in neutron star (NS) magnetospheres and occasionally get amplified by large factors via gravitational self-lensing in the strong NS field. We evaluate the probability of amplification and show that: (i) a power-law energy distribution of events ∝E −(2−3) is generally expected; (ii) all FRB sources may be regarded as repeating, their appearance as one-off sources or repeaters being determined by the critical dependence of the amplification probability on the emission geometry and source orientation relative to Earth; and (iii) the most active repeaters, in particular, correspond to extremely rare and finely tuned orientations (∼1 in 106), leading to large probabilities of amplification that make their bursts frequently detectable. At the same time, their power release appears enhanced, typically by factors ≳10, easing their energy budget problem.

Revisiting γ-Ray Emission of the Supernova Remnant RCW 103

We analyze more than 15 yr of γ-ray data obtained with the Large Area Telescope (LAT) on board the Fermi Gamma-ray Space Telescope (Fermi) for the region of the young supernova remnant (SNR) RCW 103, since the nearby source 4FGL J1616.2−5054e, counterpart to HESS J1616−518 and ≃13′ away from the SNR, is determined to be extended in the more recent Fermi-LAT source catalog. Different templates for 4FGL J1616.2−5054e and RCW 103 are tested, and we find that a point source (PS) with a power-law (PL) spectrum at the southern limb of the SNR best describes the detected γ-ray emission. The photon index of the PL emission is Γ ≃ 2.31 (or α ≃ 2.4 in a log-parabola model), softer than the previously reported Γ ≃ 2.0 when the counterpart to HESS J1616−518 was considered to be a PS (which likely caused misidentification of extended emission at RCW 103). In order to produce the γ-ray emission in a hadronic scenario, we estimate that protons with an index ∼2.4 PL energy distribution are needed. These results fit with those from multiwavelength observations that have indicated the remnant at the southern limb is interacting with a molecular cloud.

Insights into the physics of gamma-ray bursts from the high-energy extension of their prompt emission spectra

The study of the high-energy part (MeV-GeV) of the spectrum of gamma-ray bursts (GRBs) can play a crucial role in investigating the physics of prompt emission, but it is often hampered by low statistics and the paucity of GeV observations. In this work, we analyze the prompt emission spectra of the 22 brightest GRBs which have been simultaneously observed by Fermi/GBM and Fermi/LAT, spanning six orders of magnitude in energy. The high-energy photon spectra can be modeled with a power-law N(E)∝E−β possibly featuring an exponential cutoff. We find that, with the inclusion of the LAT data, the spectral index β is softer than what is typically inferred from the analysis of Fermi/GBM data alone. Under the assumption that the emission is synchrotron, we derived a median value of the index p ∼ 2.79 of the power-law energy distribution of accelerated particles (N(γ)∝γ−p). In nine out of 22 GRB spectra, we find a significant presence of an exponential cutoff at high energy, ranging between 14 and 298 MeV. By interpreting the observed cutoff as a sign of pair-production opacity, we estimate the jet bulk Lorentz factor Γ, finding values in the range 130–330. These values are consistent with those inferred from the afterglow light curve onset time. Finally, by combining the information from the high-energy prompt emission spectrum with the afterglow light curve, we exploited a promising method to derive the distance R from the central engine where the prompt emission occurs. The distances (R > 1013 − 15 cm) inferred for the only two GRBs in our sample that are suitable for the application of this method, which have only lower limits on their cutoff energies, suggest large emitting regions, although they are still compatible with the standard model. Larger samples of GRBs with measured cutoff energies and afterglow deceleration time will allow for more informative values to be derived. These results highlight the importance of including high-energy data, when available, in the study of prompt spectra and their role in addressing the current challenges of the GRB standard model.

High-energy Radiation and Ion Acceleration in Three-dimensional Relativistic Magnetic Reconnection with Strong Synchrotron Cooling

We present the results of 3D particle-in-cell simulations that explore relativistic magnetic reconnection in pair plasma with strong synchrotron cooling and a small mass fraction of nonradiating ions. Our results demonstrate that the structure of the current sheet is highly sensitive to the dynamic efficiency of radiative cooling. Specifically, stronger cooling leads to more significant compression of the plasma and magnetic field within the plasmoids. We demonstrate that ions can be efficiently accelerated to energies exceeding the plasma magnetization parameter, ≫σ, and form a hard power-law energy distribution, f i ∝ γ −1. This conclusion implies a highly efficient proton acceleration in the magnetospheres of young pulsars. Conversely, the energies of pairs are limited to either σ in the strong cooling regime or the radiation burnoff limit, γ syn, when cooling is weak. We find that the high-energy radiation from pairs above the synchrotron burnoff limit, ε c ≈ 16 MeV, is only efficiently produced in the strong cooling regime, γ syn < σ. In this regime, we find that the spectral cutoff scales as ε cut ≈ ε c (σ/γ syn) and the highest energy photons are beamed along the direction of the upstream magnetic field, consistent with the phenomenological models of gamma-ray emission from young pulsars. Furthermore, our results place constraints on the reconnection-driven models of gamma-ray flares in the Crab Nebula.

A Model for Nonthermal Particle Acceleration in Relativistic Magnetic Reconnection

The past decade has seen an outstanding development of nonthermal particle acceleration in magnetic reconnection in magnetically dominated systems, with clear signatures of power-law energy distributions as a common outcome of first-principles kinetic simulations. Here we propose a semianalytical model for systematically investigating nonthermal particle acceleration in reconnection. We show particle energy distributions are well determined by particle injection, acceleration, and escape processes. Using a series of kinetic simulations, we accurately evaluate the energy- and time-dependent model coefficients. The resulting spectral characteristics, including the spectral index and lower and upper bounds of the power-law distribution, agree well with the simulation results. Finally, we apply the model to predict the power-law indices and break energies in astrophysical reconnection systems.

Large-scale Evolution of Seconds-long Relativistic Jets from Black Hole–Neutron Star Mergers

We present the first numerical simulations that track the evolution of a black hole–neutron star (BH–NS) merger from premerger to r ≳ 1011 cm. The disk that forms after a merger of mass ratio q = 2 ejects massive disk winds (3–5 × 10−2 M ⊙). We introduce various postmerger magnetic configurations and find that initial poloidal fields lead to jet launching shortly after the merger. The jet maintains a constant power due to the constancy of the large-scale BH magnetic flux until the disk becomes magnetically arrested (MAD), where the jet power falls off as L j ∼ t −2. All jets inevitably exhibit either excessive luminosity due to rapid MAD activation when the accretion rate is high or excessive duration due to delayed MAD activation compared to typical short gamma-ray bursts (sGRBs). This provides a natural explanation for long sGRBs such as GRB 211211A but also raises a fundamental challenge to our understanding of jet formation in binary mergers. One possible implication is the necessity of higher binary mass ratios or moderate BH spins to launch typical sGRB jets. For postmerger disks with a toroidal magnetic field, dynamo processes delay jet launching such that the jets break out of the disk winds after several seconds. We show for the first time that sGRB jets with initial magnetization σ 0 > 100 retain significant magnetization (σ ≫ 1) at r > 1010 cm, emphasizing the importance of magnetic processes in the prompt emission. The jet–wind interaction leads to a power-law angular energy distribution by inflating an energetic cocoon whose emission is studied in a companion paper.

Avalanches and the Distribution of Reconnection Events in Magnetized Circumstellar Disks

Cosmic rays produced by young stellar objects can potentially alter the ionization structure, heating budget, chemical composition, and accretion activity in circumstellar disks. The inner edges of these disks are truncated by strong magnetic fields, which can reconnect and produce flaring activity that accelerates cosmic radiation. The resulting cosmic rays can provide a source of ionization and produce spallation reactions that alter the composition of planetesimals. These reconnection and particle acceleration processes are analogous to the physical processes that produce flaring in and the heating of stellar coronae. Flaring events on the surface of the Sun exhibit a power-law distribution of energy, reminiscent of those measured for earthquakes and avalanches. Numerical lattice reconnection models are capable of reproducing the observed power-law behavior of solar flares under the paradigm of self-organized criticality. One interpretation of these experiments is that the solar corona maintains a nonlinear attractor—or “critical”—state by balancing energy input via braided magnetic fields and output via reconnection events. Motivated by these results, we generalize the lattice reconnection formalism for applications in the truncation region of magnetized disks. Our numerical experiments demonstrate that these nonlinear dynamical systems are capable of both attaining and maintaining criticality in the presence of Keplerian shear and other complications. The resulting power-law spectrum of flare energies in the equilibrium attractor state is found to be nearly universal in magnetized disks. This finding indicates that magnetic reconnection and flaring in the inner regions of circumstellar disks occur in a manner similar to the activity on stellar surfaces. These results, in turn, have ramifications for the spallation-driven injection of radionuclides in planetesimals, disk ionization, and the subsequent planetary formation process.

BEYONDPLANCK

We constrained the level of polarized anomalous microwave emission (AME) on large angular scales usingPlanckLow-Frequency Instrument (LFI) and WMAP polarization data within a Bayesian cosmic microwave background (CMB) analysis framework. We modeled synchrotron emission with a power-law spectral energy distribution, as well as the sum of AME and thermal dust emission through linear regression with thePlanckHigh-Frequency Instrument (HFI) 353 GHz data. This template-based dust emission model allowed us to constrain the level of polarized AME while making minimal assumptions on its frequency dependence. We neglected CMB fluctuations, but show through simulations that these fluctuations have a minor impact on the results. We find that the resulting AME polarization fraction confidence limit is sensitive to the polarized synchrotron spectral index prior. In addition, for prior meansβs &lt; −3.1 we find an upper limit ofpAMEmax ≲ 0.6% (95% confidence). In contrast, for meansβs = −3.0, we find a nominal detection ofpAME = 2.5 ± 1.0% (95% confidence). These data are thus not strong enough to simultaneously and robustly constrain both polarized synchrotron emission and AME, and our main result is therefore a constraint on the AME polarization fraction explicitly as a function ofβs. Combining the currentPlanckand WMAP observations with measurements from high-sensitivity low-frequency experiments such as C-BASS and QUIJOTE will be critical to improve these limits further.

GRB 221009A afterglow from a shallow angular structured jet

ABSTRACT Exceptionally bright gamma-ray burst (GRB) afterglows can reveal the angular structure of their jets. GRB jets appear to have a narrow core (of half-opening angle θc), beyond which their kinetic energy drops as a power-law with angle θ from the jet’s symmetry axis, $E_{\mathrm{ k},\rm iso}(\theta)\propto [1+(\theta /\theta _\mathrm{ c})^2]^{-a/2}$. The power-law index a reflects the amount of mixing between the shocked jet and confining medium, which depends on the jet’s initial magnetization. Weakly magnetized jets undergo significant mixing, leading to shallow (a ≲ 2) angular profiles. We use the exquisite multiwaveband afterglow observations of GRB 221009A to constrain the jet angular structure using a dynamical model that accounts for both the forward and reverse shocks, for a power-law external density profile, next ∝ R−k. Both the forward shock emission, that dominates the optical and X-ray flux, and the reverse shock emission, that produces the radio afterglow, require a jet with a narrow core (θc ≈ 0.021) and a shallow angular structure (a ≈ 0.8) expanding into a stellar wind (k ≈ 2). Moreover, these data appear to favour a small fraction (ξe ≈ 10−2) of shock heated electrons forming a power-law energy distribution in both shocks.

Ultra-High-Energy Cosmic Rays from a Population of Non-identical Sources

Astrophysical candidates for the sources of ultra-high-energy cosmic rays (UHECRs) exhibit a large diversity in terms of their properties relevant for the acceleration of charged particles, such as luminosity, Lorentz factor, size and magnetic field. Yet, fits of the observed UHECR spectrum and composition often assume identical sources. Here we investigate a population of sources with a power-law distribution of maximum energies. We show that the allowed source-to-source variance of the maximum energy must be small to describe the UHECR data. Even in the most extreme scenario, with a very sharp cutoff of individual source spectra and negative redshift evolution of the accelerators, the maximum energies of 90% of sources must be identical within a factor of three – in contrast to the variance expected for astrophysical sources. However, the overall population variance can be large when maximum rigidities are distributed as a broken power law, with a steep decline above the break and with hard source spectra. In this scenario, most of the observed UHECR flux is produced by sources near the break.

Constraints on electron acceleration in gamma-ray bursts afterglows from radio peaks

ABSTRACT Studies of gamma-ray bursts (GRBs) and their multiwavelength afterglows have led to insights in electron acceleration and emission properties from relativistic, high-energy astrophysical sources. Broad-band modelling across the electromagnetic spectrum has been the primary means of investigating the physics behind these sources, although independent diagnostic tools have been developed to inform and corroborate assumptions made in particle acceleration simulations and broad-band studies. We present a methodology to constrain three physical parameters related to electron acceleration in GRB blast waves: the fraction of shock energy in electrons, ϵe; the fraction of electrons that gets accelerated into a power-law distribution of energies, ξe; and the minimum Lorentz factor of the accelerated electrons, γm. These parameters are constrained by observations of the peaks in radio afterglow light curves and spectral energy distributions. From a sample of 49 radio afterglows, we are able to find narrow distributions for these parameters, hinting at possible universality of the blast wave microphysics, although observational bias could play a role in this. Using radio peaks and considerations related to the prompt gamma-ray emission efficiency, we constrain the allowed parameter ranges for both ϵe and ξe to within about one order of magnitude, 0.01 ≲ ϵe ≲ 0.2 and 0.1 ≲ ξe ≲ 1. Such stringent constraints are inaccessible for ξe from broad-band studies due to model degeneracies.

Monte Carlo simulations of nuclear de-excitation gamma-ray line spectra from solar flares

Recently, we have demonstrated that the Monte Carlo package FLUKA can be used as an effective tool for simulating nuclear processes which occur in solar flares and that it is capable to provide a self-consistent treatment of all typical components of the γ-ray spectra observed in those events. In this work, we have employed a new simulation strategy that allows to improve statistics and resolution in energy of the generated γ-ray spectra. Using this new strategy, we have calculated spectra of γ-ray nuclear de-excitation lines produced by solar flare primary accelerated ions with typical power-law energy distributions.

FAST helps push the study of fast radio bursts into statistical regimes

<p indent="0mm">The 2007 discovery of fast radio bursts (FRBs), bright and millisecond-duration astronomical radio pulses, presented to the astronomical community a tantalizing opportunity into the transient Universe. Understanding the origins, mechanisms and applications of FRBs is currently one of the most rapidly evolving sub-fields of astrophysics. Most FRBs originate from far outside the Milky Way and are many orders of magnitude more luminous than other known bursting radio sources, particularly pulsars (neutron stars). The total FRB population now stands at approximately 600 published sources, 24 repeaters, and 19 with localized host galaxies. Most FRBs appear to be one-off, only about four percent of the FRBs are now known to repeat. While the volume of observations grows at a breakneck pace, the overarching question of the field remains: “Where do FRBs come from?” Interestingly, FRBs can be hosted in a variety of galaxy types and environments. FRB 20121102 was successfully located in a low-mass and low-metallicity dwarf galaxy, coincident with a persistent radio source. VLBI observations have demonstrated that this source is compact. With the Five-hundred-meter Aperture Spherical radio Telescope (FAST), FRB 20121102 has been extensively observed, resulting in the discovery of 1652 independent bursts with a peak burst rate of 122 per hour and over three orders of magnitude variation in energy. For the first time, the burst energy distribution can be fully recovered and is found to be bimodal, suggesting that there may be more than one emission mechanisms or emission sites or beam shapes. In contrast to the single power-law energy distribution commonly found for FRBs, a characteristic peak in the burst rate energy distribution is found at <sc>4.8×10³⁷ erg.</sc> The estimated total isotropic energy, <sc>6.4×10⁴⁶ erg,</sc> released during the 47-day observational campaign, corresponds to approximately 38% of the available magnetar spin energy. Despite the high burst rate, no periodicity has been found. Higher time-resolution data, sustained monitoring, and better localization from the last <sc>2–3 years</sc> have provided a wealth of information and significant discoveries regarding the origin of FRBs. A supremely bright FRB-like burst (FRB 20200428) has been detected from a Galactic magnetar SGR 1935+2154 by STARE2 and CHIME/FRB. It provides strong evidence linking magnetars to sources of FRBs. The explosively growing number of FRBs and studies of individual FRBs start to reveal the source and radiation mechanisms of FRBs. Significant breakthroughs rely on the emergence of new observational facilities as well as a growing breadth of expertise joining the field. With ever-growing capabilities of observing facilities and processing power, the FRB sample in the next <sc>5 years</sc> will supersede that of radio pulsars from the past half century. We are witnessing the unveiling of a dynamic Universe.

Particle Acceleration in Magnetic Reconnection with Ad Hoc Pitch-angle Scattering

Particle acceleration during magnetic reconnection is a long-standing topic in space, solar, and astrophysical plasmas. Recent 3D particle-in-cell simulations of magnetic reconnection show that particles can leave flux ropes due to 3D field-line chaos, allowing particles to access additional acceleration sites, gain more energy through Fermi acceleration, and develop a power-law energy distribution. This 3D effect does not exist in traditional 2D simulations, where particles are artificially confined to magnetic islands due to their restricted motions across field lines. Full 3D simulations, however, are prohibitively expensive for most studies. Here, we attempt to reproduce 3D results in 2D simulations by introducing ad hoc pitch-angle scattering to a small fraction of the particles. We show that scattered particles are able to transport out of 2D islands and achieve more efficient Fermi acceleration, leading to a significant increase of energetic particle flux. We also study how the scattering frequency influences the nonthermal particle spectra. This study helps achieve a complete picture of particle acceleration in magnetic reconnection.

Non-thermal particle acceleration from maximum entropy in collisionless plasmas

Dissipative processes cause collisionless plasmas in many systems to develop non-thermal particle distributions with broad power-law tails. The prevalence of power-law energy distributions in space/astrophysical observations and kinetic simulations of systems with a variety of acceleration and trapping (or escape) mechanisms poses a deep mystery. We consider the possibility that such distributions can be modelled from maximum-entropy principles, when accounting for generalizations beyond the Boltzmann–Gibbs entropy. Using a dimensional representation of entropy (related to the Renyi and Tsallis entropies), we derive generalized maximum-entropy distributions with a power-law tail determined by the characteristic energy scale at which irreversible dissipation occurs. By assuming that particles are typically energized by an amount comparable to the free energy (per particle) before equilibrating, we derive a formula for the power-law index as a function of plasma parameters for magnetic dissipation in systems with sufficiently complex topologies. The model reproduces several results from kinetic simulations of relativistic turbulence and magnetic reconnection.

Variation of the micromechanics of impact fracture of MgAl-=SUB=-2-=/SUB=-O-=SUB=-4-=/SUB=- in response to the applied hot isostatic pressing

The MgAl2O4 ceramics produced by the preliminary clinkering of synthesized powder was exposed to the hot isostatic pressing (HIP). Samples were subjected to a pointed impact loading, which excited the acoustical and electromagnetic emissions (AE and EME, respectively). In non-HIP-treated samples, the energy distribution in time series of AE and EME pulses followed an exponential law, while after an application of the hot isostatic pressing procedure, the emissions of both types exhibited the power law energy distributions in time sweeps. A variation of the specificity of the impact-induced AE/EME energy release was explained by a transition from the nucleation/decay of microcracks in non-HIP-treated ceramics to the self-organized dislocation folding directed to grain boundaries in the material characterized by the density that was close to that of a MgAl2O4 single crystal. Keywords: MgAl2O4 ceramics, hot isostatic pressing, impact loading, acoustic emission, electromagnetic emission.

Particle Energization in Relativistic Plasma Turbulence: Solenoidal versus Compressive Driving

Abstract Many high-energy astrophysical systems contain magnetized collisionless plasmas with relativistic particles, in which turbulence can be driven by an arbitrary mixture of solenoidal and compressive motions. For example, turbulence in hot accretion flows may be driven solenoidally by the magnetorotational instability or compressively by spiral shock waves. It is important to understand the role of the driving mechanism on kinetic turbulence and the associated particle energization. In this work, we compare particle-in-cell simulations of solenoidally driven turbulence with similar simulations of compressively driven turbulence. We focus on plasma that has an initial beta of unity, relativistically hot electrons, and varying ion temperature. Apart from strong large-scale density fluctuations in the compressive case, the turbulence statistics are similar for both drives, and the bulk plasma is described reasonably well by an isothermal equation of state. We find that nonthermal particle acceleration is more efficient when turbulence is driven compressively. In the case of relativistically hot ions, both driving mechanisms ultimately lead to similar power-law particle energy distributions, but over a different duration. In the case of nonrelativistic ions, there is significant nonthermal particle acceleration only for compressive driving. Additionally, we find that the electron-to-ion heating ratio is less than unity for both drives, but takes a smaller value for compressive driving. We demonstrate that this additional ion energization is associated with the collisionless damping of large-scale compressive modes via perpendicular electric fields.

Kinetic simulations of imbalanced turbulence in a relativistic plasma: Net flow and particle acceleration

ABSTRACT Turbulent high-energy astrophysical systems often feature asymmetric energy injection: for instance, Alfvén waves propagating from an accretion disc into its corona. Such systems are ‘imbalanced’: the energy fluxes parallel and antiparallel to the large-scale magnetic field are unequal. In the past, numerical studies of imbalanced turbulence have focused on the magnetohydrodynamic regime. In this study, we investigate externally driven imbalanced turbulence in a collision-less, ultrarelativistically hot, magnetized pair plasma using 3D particle-in-cell (PIC) simulations. We find that the injected electromagnetic momentum efficiently converts into plasma momentum, resulting in net motion along the background magnetic field with speeds up to a significant fraction of lightspeed. This discovery has important implications for the launching of accretion disc winds. We also find that although particle acceleration in imbalanced turbulence operates on a slower time-scale than in balanced turbulence, it ultimately produces a power-law energy distribution similar to balanced turbulence. Our results have ramifications for black hole accretion disc coronae, winds, and jets.