Disk Corona Research Articles

Correlation between relativistic reflection fraction and photon index in NuSTAR sample of Seyfert 1 AGN

ABSTRACT The primary X-ray emission from active galactic nuclei (AGNs), described by a power-law, irradiates the accretion disc producing reflection features in the spectrum. The reflection features arising from the inner regions of the disc can be significantly modified by the relativistic effects near the black hole. We investigate the relationship between the relativistic reflection fraction Rf, defined as the ratio of the coronal intensity that illuminates the accretion disc to the coronal intensity observed directly, and the hard X-ray photon index Γ of a Nuclear Spectroscopic Telescope Array (NuSTAR) sample of Seyfert 1 galaxies. The X-ray spectra are modelled using relxill code that helps to directly obtain the reflection fraction of a relativistically smeared reflection component. The parameter Rf depends on the amount of Comptonized X-ray emission intercepted by the inner accretion disc. We found a positive correlation between Γ and Rf in our sample. Seed photons from a larger area of an accretion disc entering the corona will result in increased cooling of the coronal plasma, giving rise to steeper X-ray spectrum. The corona irradiating the larger area of the disc will result in higher reflection fraction. Thus, the observed Rf –Γ relation is most likely related to the variations in the disc–corona geometry of AGNs.

Magnetic-reconnection-heated corona in active galactic nuclei: refined disc–corona model and application to broad-band radiation

ABSTRACT A long-standing question in active galactic nucleus (AGN) research is how the corona is heated up to produce X-ray radiation much stronger than that arising from the viscous heating within the corona. In this paper, we carry out detailed investigations of magnetic-reconnection heating to the corona, specifically, studying how the disc and corona are self-consistently coupled with the magnetic field, and how the emergent spectra depend on the fundamental parameters of AGN. It is shown that diverse spectral shapes and luminosities over a broad bandpass from optical to X-ray can be produced from the coupled disc and corona within a limited range of the black hole mass, accretion rate, and magnetic field strength. The relative strength of X-ray emission with respect to optical/ultraviolet (UV) depends on the strength of the magnetic field in the disc, which, together with accretion rate, determines the fraction of accretion energy transported and released in the corona. This refined disc–corona model is then applied to reproduce the broad-band spectral energy distributions (SEDs) of a sample of 20 bright local AGNs observed simultaneously in X-ray and optical/UV. We find that, in general, the overall observed broad-band SEDs can be reasonably reproduced, except for rather hard X-ray spectral shapes in some objects. The radiation pressure-dominant region, as previously predicted for the standard accretion disc in AGN, disappears for strong X-ray sources, revealing that AGN accretion discs are indeed commonly stable as observed. Our study suggests the disc–corona coupling model involving magnetic fields to be a promising approach for understanding the broad-band spectra of bright AGNs.

The Persistent Radio Jet Coupled to Hard X-Rays in the Soft State of Cyg X-1

Abstract We study long-term radio/X-ray correlations in Cyg X-1. We find the persistent existence of a compact radio jet in its soft state. This represents a new phenomenon in black hole binaries, in addition to compact jets in the hard state and episodic ejections of ballistic blobs in the intermediate state. While the radio emission in the hard state is strongly correlated with both the soft and hard X-rays, the radio flux in the soft state is not directly correlated with the flux of the dominant disk blackbody in soft X-rays, but instead it is lagged by about 100 days. We interpret the lag as occurring in the process of advection of the magnetic flux from the donor through the accretion disk. On the other hand, the soft-state radio flux is very tightly correlated with the hard X-ray, 15–50 keV, flux without a measurable lag and at the same rms. This implies that the X-ray-emitting disk corona and the soft-state jet are powered by the same process, probably magnetically.

On the Jet Production Efficiency in a Sample of the Youngest Radio Galaxies

Abstract Here we discuss the jet production efficiency in a sample of 17 young radio galaxies with measured redshifts, kinematic ages, and nuclear X-ray fluxes, for which the observed luminosities of compact jets/lobes and accretion disks correspond to the same episode of AGN activity. For the targets, we analyze the available optical data, estimating the bolometric luminosities of the accretion disks L bol, and the black hole masses; we also derive the minimum jet kinetic luminosities, P j. With this information we investigate the distribution of our sample in the three-dimensional space of the accretion rate λ Edd ≡ L bol/L Edd, the nuclear X-ray luminosity L X considered here as a limit for the emission of the disk coronae, and P j, expressing the latter two parameters either in the Eddington units, or in the units of the disk luminosity. We find that (i) the accretion rate λ Edd in our sample is distributed within a narrow range λ Edd ∼ 0.01–0.2; (ii) the normalized jet power P j/L Edd formally correlates with the accretion rate λ Edd, with some saturation at the largest values λ Edd > 0.05; (iii) the jet production efficiency spans a range from η jet ≲ 10−3 up to ∼0.2 at maximum, which is below the level expected for magnetically arrested disks around maximally spinning black holes; and (iv) there is a diversification in η jet on the hardness–intensity diagram L X/L bol − λ Edd, with the jets being produced most efficiently during the high/hard states, and suppressed during the soft states.

Properties of Trans-fast Magnetosonic Jets in Black Hole Magnetospheres

Abstract Traveling across several orders of magnitude in distance, relativistic jets from strong gravity regions to asymptotic flat spacetime regions are believed to consist of several general relativistic magnetohydrodynamic (GRMHD) processes. We present a semianalytical approach for modeling the global structures of a trans-fast magnetosonic relativistic jet, which should be ejected from a plasma source near a black hole in a funnel region enclosed by dense accreting flow and a disk corona around the black hole. Our model consistently includes the inflow and outflow part of the GRMHD solution along the magnetic field lines penetrating the black hole horizon. After the rotational energy of the black hole is extracted electromagnetically by the negative energy GRMHD inflow, the huge electromagnetic energy flux propagates from the inflow to the outflow region across the plasma source, and in the outflow region, the electromagnetic energy converts to the fluid kinetic energy. Eventually, the accelerated outflow must exceed the fast magnetosonic wave speed. We apply the semianalytical trans-fast magnetosonic flow model to the black hole magnetosphere for both parabolic and split-monopole magnetic field configurations and discuss the general flow properties, that is, jet acceleration, jet magnetization, and the locations of some characteristic surfaces of the black hole magnetosphere. We have confirmed that, at large distances, the GRMHD jet solutions are in good agreement with the previously known trans-fast special relativistic magnetohydrodynamic jet properties, as expected. The flexibility of the model provides a prompt and heuristic way to approximate the global GRMHD trans-fast magnetosonic jet properties.

Evolution of the Accretion Disk–Corona during the Bright Hard-to-soft State Transition: A Reflection Spectroscopic Study with GX 339–4

Abstract We present the analysis of several observations of the black hole binary GX 339–4 during its bright intermediate states from two different outbursts (2002 and 2004), as observed by the RXTE/Proportional Counter Array. We perform a consistent study of its reflection spectrum by employing the relxill family of relativistic reflection models to probe the evolutionary properties of the accretion disk including the inner disk radius (R in), ionization parameter (ξ), temperatures of the inner disk (T in), corona (kT e), and its optical depth (τ). Our analysis indicates that the disk inner edge approaches the innermost stable circular orbit during the early onset of the bright hard state, and that the truncation radius of the disk remains low (<9 R g) throughout the transition from hard to soft state. This suggests that the changes observed in the accretion disk properties during the state transition are driven by variation in accretion rate, and not necessarily due to changes in the inner disk’s radius. We compare the aforementioned disk properties in two different outbursts with state transitions occurring at dissimilar luminosities and find identical evolutionary trends in the disk properties, with differences only seen in the corona’s kT e and τ. We also perform an analysis by employing a self-consistent Comptonized accretion disk model accounting for the scatter of disk photons by the corona, and measure the low inner disk truncation radius across the bright intermediate states, using the temperature-dependent values of the spectral hardening factor, thereby independently confirming our results from the reflection analysis.

Broadband X-ray observations of four gamma-ray narrow-line Seyfert 1 galaxies

Narrow-line Seyfert 1 galaxies (NLS1s) is one of the few classes of active galactic nuclei harboring powerful relativistic jets and detected in γ-rays. NLS1s are well-known X-ray sources. While in non-jetted sources the origin of this X-ray emission may be a hot corona surrounding the accretion disk, in jetted objects, especially beamed ones, the contribution of corona and relativistic jet is difficult to disentangle without a proper sampling of the hard X-ray emission. For this reason, we observed with NuSTAR the first four NLS1s detected at high energy γ-rays. These data, along with XMM-Newton and Swift/XRT observations, confirmed that X-rays originate both in the jet and in the accretion disk corona. Time variability in hard X-rays furthermore suggests that, as observed in flat-spectrum radio quasars, the dissipation region during flares could change its position from source to source, and it can be located both inside and outside the broad-line region. We find that jetted NLS1s, and other blazars as well, seem not to follow the classical fundamental plane of black hole (BH) activity, which therefore should be used as a BH mass estimator in blazars with extreme care only. Our results strengthen the idea according to which γ-NLS1s are smaller and younger version of flat-spectrum radio quasars, in which both a Seyfert and a blazar component co-exist.

Simultaneous Millimetre-wave and X-ray monitoring of the Seyfert galaxy NGC 7469

ABSTRACT We report on daily monitoring of the Seyfert galaxy ngc 7469, around 95 and 143 GHz, with the iram (Institut de Radioastronomie Millimetrique) 30- m radio telescope, and with the Swift X-ray and UV/optical telescopes, over an overlapping period of 45 d. The source was observed on 36 d with iram, and the flux density in both mm bands was on average ∼10 mJy, but varied by $\pm 50{{\ \rm per\ cent}}$, and by up to a factor of 2 between days. The present iram variability parameters are consistent with earlier monitoring, which had only 18 data points. The X-ray light curve of ngc 7469 over the same period spans a factor of 5 in flux with small uncertainties. Similar variability in the mm band and in the X-rays lends support to the notion of both sources originating in the same physical component of the active galactic nucleus (AGN), likely the accretion disc corona. Simultaneous monitoring in eight UV/optical bands shows much less variability than the mm and X-rays, implying this light originates from a different AGN component, likely the accretion disc itself. We use a tentative 14-d lag of the X-ray light curve with respect to the 95 GHz light curve to speculate on coronal implications. More precise mm-band measurements of a sample of X-ray-variable AGN are needed, preferably also on time-scales of less than a day where X-rays vary dramatically, in order to properly test the physical connection between the two bands.

First Hard X-Ray Observation of a Compact Symmetric Object: A Broadband X-Ray Study of a Radio Galaxy OQ+208 with NuSTAR and Chandra

Abstract Compact symmetric objects (CSOs) have been observed with Chandra and XMM-Newton to gain insights into the initial stages of a radio source evolution and to probe the black hole activity at the time of relativistic outflow formation. However, there have been no CSO observations to date at the hard X-ray energies (>10 keV), impeding our ability to robustly constrain the properties of the intrinsic X-ray emission and of the medium surrounding the young expanding jets. We present the first hard X-ray observation of a CSO performed with the Nuclear Spectroscopic Telescope Array (NuSTAR). Our target, OQ +208, is detected up to 30 keV, and thus we establish CSOs as a new class of the NuSTAR sources. We analyze the NuSTAR data jointly with our new Chandra and archival XMM-Newton data and find that a young (∼250 yr old) radio jet spanning the length of 10 pc coexists with cold obscuring matter, consistent with a dusty torus, with an equivalent hydrogen column density of N H = 1023–1024 cm−2. The primary X-ray emission is characterized by a photon index of Γ ∼ 1.45 and an intrinsic 0.5–30 keV luminosity of L ≃ 1043 erg s−1. The results of our spectral modeling and broad-line optical classification of the source suggest a porous structure of the obscuring torus. Alternatively, the source may belong to the class of optically unobscured/X-ray-obscured active galactic nucleus. The observed X-ray emission is too weak compared to that predicted by the expanding radio lobes model, leaving an accretion disk corona or jets as the possible origins of the X-ray emission from this young radio galaxy.

Hot, dense He iioutflows during the 2017 outburst of the X-ray transientSwift J1357.2−0933

AbstractTime-resolved SALT spectra of the short-period, dipping X-ray transient, Swift J1357.2−0933, during its 2017 outburst has revealed broad Balmer and He ii λ4686 absorption features, blueshifted by ∼600 km s−1. Remarkably these features are also variable on the ∼500 s dipping period, indicating their likely association with structure in the inner accretion disc. We interpret this as arising in a dense, hot (≳30 000 K) outflowing wind seen at very high inclination, and draw comparisons with other accretion disc corona sources. We argue against previous distance estimates of 1.5 kpc and favour a value ≳6 kpc, implying an X-ray luminosity LX ≳ 4 × 1036 erg s−1. Hence it is not a very faint X-ray transient. Our preliminary 1D Monte Carlo radiative transfer and photoionization calculations support this interpretation, as they imply a high intrinsic LX, a column density NH ≳ 1024 cm−2, and a low covering factor for the wind. Our study shows that Swift J1357.2−0933 is truly remarkable amongst the cohort of luminous, Galactic X-ray binaries, showing the first example of He ii λ4686 absorption, the first (and only) variable dip period and is possibly the first black hole ‘accretion disc corona’ candidate.

On High-energy Particles in Accretion Disk Coronae of Supermassive Black Holes: Implications for MeV Gamma-rays and High-energy Neutrinos from AGN Cores

Abstract Recent observations with ALMA have revealed evidence for nonthermal synchrotron emission from the core regions of two nearby Seyfert galaxies. This suggests that the coronae of accretion disks in active galactic nuclei (AGNs) can be conducive to the acceleration of nonthermal electrons, in addition to the hot, thermal electrons responsible for their X-ray emission through thermal Comptonization. Here, we investigate the mechanism of such particle acceleration, based on observationally inferred parameters for AGN disk coronae. One possibility to account for the observed nonthermal electrons is diffusive shock acceleration, as long as the gyrofactor η g does not exceed ∼106. These nonthermal electrons can generate gamma-rays via inverse Compton scattering of disk photons, which can appear in the MeV band, while those with energies above ∼100 MeV would be attenuated via internal γγ pair production. The integrated emission from all AGNs with thermal and nonthermal Comptonization can reproduce the observed cosmic background radiation in X-rays as well as gamma-rays up to ∼10 MeV. Furthermore, if protons are accelerated in the same conditions as electrons and η g ∼ 30, our observationally motivated model is also able to account for the diffuse neutrino flux at energies below 100–300 TeV. The next generation of MeV gamma-ray and neutrino facilities can test these expectations by searching for signals from bright, nearby Seyfert galaxies such as NGC 4151 and IC 4329A.

Modeling the X-Ray Power Spectra of AGN by Using Reprocessing Echoes from an Extended Corona

Abstract Characteristic signatures that X-ray reverberation from an extended corona can manifest in the observed PSD of AGN are investigated. The presence of two X-ray blobs illuminating an accretion disk can cause the interference between two reprocessing-echo components and produce distinct physical features in the PSD. The oscillatory structures (e.g., dips and humps) are seen, but, contrarily to the lamp-post case, the strongest dip is not always the one at the lowest frequency. Instead, we find that the frequency where the strongest dip is seen is associated with the lower-source height while the lowest frequency where the first dip appears is linked to the upper-source height. This is because the reverberation timescales increase with the source height. Accurate modeling of the PSD then helps put constraints on the lower and upper limits of the corona extent. Furthermore, the reverberation signatures are less pronounced with increasing number of sources that do not produce reflection (e.g., additional X-rays from fast, relativistic outflows). The amplitude of the oscillations also depends on the amount of dilution contributed by the X-ray sources, and thus encodes information about their relative brightness. Due to stronger dilutions, robust detection of these signatures with current observations will become even more difficult if the corona is extended. Future observations made by Athena will enable us to fit these characteristics with statistically significant details, and to reveal the nature of the disk–corona system.

Episodic Jets from Black Hole Accretion Disks

Abstract Several active galactic nuclei and microquasars are observed to eject plasmoids that move at relativistic speeds. We envisage the plasmoids as pre-existing current carrying magnetic flux ropes that were initially anchored in the accretion disk corona. The plasmoids are ejected outwards via a mechanism called the toroidal instability (TI). The TI, which was originally explored in the context of laboratory tokamak plasmas, has been very successful in explaining coronal mass ejections from the Sun. Our model predictions for plasmoid trajectories compare favorably with a representative set of multi-epoch observations of radio emitting knots from the radio galaxy 3C 120, which were preceded by dips in X-ray intensity.

Discovery of Redshifted He-like Iron Absorption Line from Luminous Accreting Neutron Star SMC X-1

Abstract We have analyzed X-ray data of SMC X-1 obtained with Suzaku and discovered the combination of an absorption line and an underlying broadened emission line centered at 6.4 keV in an observation performed on 2012 May 19. This absorption line is centered at with an absorption strength of , naturally interpreted as an He α resonance line of Fe at 6.7 keV that has a redshift of . Although Suzaku observed this system 10 times during 11 months in 2011–2012, the absorption feature has been seen only in a single observation when the neutron star (NS) was in a rising phase of the super-orbital modulation, which can be regarded as an egress from occultation by an extended accretion disk. We therefore attribute the line to a low density, highly ionized absorber in an accretion disk corona arising from the disk illuminated by the NS’s intense X-rays. This interpretation also agrees with a discussion on the photoionization degree and the line depth.

Wave Generation and Heat Flux Suppression in Astrophysical Plasma Systems

Abstract Heat flux suppression in collisionless plasmas for a large range of plasma β is explored using two-dimensional particle-in-cell simulations with a strong, sustained thermal gradient. We find that a transition takes place between whistler-dominated (high-β) and double-layer-dominated (low-β) heat flux suppression. Whistlers saturate at small amplitude in the low beta limit and are unable to effectively suppress the heat flux. Electrostatic double layers (DLs) suppress the heat flux to a mostly constant factor of the free-streaming value once this transition happens. The DL physics is an example of ion–electron coupling and occurs on a scale of roughly the electron Debye length. The scaling of ion heating associated with the various heat flux driven instabilities is explored over the full range of β explored. The range of plasma-βs studied in this work makes it relevant to the dynamics of a large variety of astrophysical plasmas, including the intracluster medium of galaxy clusters, hot accretion flows, stellar and accretion disk coronae, and the solar wind.

Simulating the Collapse of a Thick Accretion Disk due to a Type I X-Ray Burst from a Neutron Star

Abstract We use two-dimensional, general relativistic, viscous, radiation hydrodynamic simulations to study the impact of a Type I X-ray burst on a hot and geometrically thick accretion disk surrounding an unmagnetized, non-rotating neutron star. The disk is initially consistent with a system in its low/hard spectral state, and is subject to a burst that rises to a peak luminosity of 1038 erg s−1 in 2.05 s. At the peak of the burst, the temperature of the disk has dropped by more than three orders of magnitude and its scale height has gone down by more than one order of magnitude. The simulations show that these effects predominantly happen due to Compton cooling of the hot plasma, and clearly illustrate the potential cooling effects of bursts on accretion disk coronae. In addition, we demonstrate the presence of Poynting–Robertson drag, though it only enhances the mass accretion rate onto the neutron star by a factor of ∼3–4 compared to a simulation with no burst. Simulations such as these are important for building a general understanding of the response of an accretion disk to an intense X-ray impulse, which, in turn, will be crucial for deciphering burst spectra. Detailed analysis of such spectra offers the potential to measure neutron star radii, and hence constrain the neutron star equation of state, but only if the contributions coming from the impacted disk and its associated corona can be understood.

Visibility and Origin of Compact Interplanetary Radio Type IV Bursts

We have analyzed radio type IV bursts in the interplanetary (IP) space at decameter–hectometer (DH) wavelengths to determine their source origin and a reason for the observed directivity. We used radio dynamic spectra from the instruments on three different spacecraft, STEREO-A, Wind, and STEREO-B, which were located approximately 90 degrees apart from each other in 2011 – 2012, and thus gave a 360 degree view of the Sun. The radio data were compared to white-light and extreme ultraviolet (EUV) observations of flares, EUV waves, and coronal mass ejections (CMEs) in five solar events. We find that the reason that compact and intense DH type IV burst emission is observed from only one spacecraft at a time is the absorption of emission in one direction and that the emission is blocked by the solar disk and dense corona in the other direction. The geometry also makes it possible to observe metric type IV bursts in the low corona from a direction where the higher-located DH type IV emission is not detectable. In the absorbed direction we found streamers, and they were estimated to be the locations of type II bursts, caused by shocks at the CME flanks. The high-density plasma was therefore most probably formed by shock–streamer interaction. In some cases, the type II-emitting region was also capable of stopping later-accelerated electron beams, which were visible as type III bursts that ended near the type II burst lanes.

Signatures of the Disk–Jet Coupling in the Broad-line Radio Quasar 4C+74.26

Abstract We explore the disk–jet connection in the broad-line radio quasar 4C+74.26, utilizing the results of multiwavelength monitoring of the source. The target is unique in that its radiative output at radio wavelengths is dominated by a moderately beamed nuclear jet, at optical frequencies by the accretion disk, and in the hard X-ray range by the disk corona. Our analysis reveals a correlation (local and global significance of 96% and 98% respectively) between the optical and radio bands, with the disk lagging behind the jet by 250 ± 42 days. We discuss the possible explanation for this, speculating that the observed disk and the jet flux changes are generated by magnetic fluctuations originating within the innermost parts of a truncated disk, and that the lag is related to a delayed radiative response of the disk when compared with the propagation timescale of magnetic perturbations along a relativistic outflow. This scenario is supported by re-analysis of NuSTAR data, modeled in terms of a relativistic reflection from the disk illuminated by the coronal emission, which returns an inner disk radius . We discuss the global energetics in the system, arguing that while the accretion proceeds at the Eddington rate, with the accretion-related bolometric luminosity L bol ∼ 9 × 1046 erg s−1 ∼ 0.2L Edd, the jet total kinetic energy L j ∼ 4 × 1044 erg s−1, inferred from the dynamical modeling of the giant radio lobes in the source, constitutes only a small fraction of the available accretion power.

A NICER Spectrum of MAXI J1535–571: Near-maximal Black Hole Spin and Potential Disk Warping

Abstract We report on a Neutron star Interior Composition Explorer (NICER) observation of the Galactic X-ray binary and stellar-mass black hole candidate, MAXI J1535−571. The source was likely observed in an “intermediate” or “very high” state, with important contributions from both an accretion disk and hard X-ray corona. The 2.3–10 keV spectrum shows clear hallmarks of relativistic disk reflection. Fits with a suitable model strongly indicate a near-maximal spin parameter of and a disk that extends close to the innermost stable circular orbit, (1σ statistical errors). In addition to the relativistic spectrum from the innermost disk, a relatively narrow Fe K emission line is also required. The resolution of NICER reveals that the narrow line may be asymmetric, indicating a specific range of emission radii. Fits with a relativistic line model suggest an inner radius of for the putative second reflection geometry; full reflection models suggest that radii a few times larger are possible. The origin of the narrow line is uncertain, but a warp likely provides the most physically plausible explanation. We discuss our results in terms of the potential for NICER to reveal new features of the inner and intermediate accretion disk around black holes.

Heating of accretion-disk coronae and jets by general relativistic magnetohydrodynamic turbulence

Turbulence in an accretion disk launches Alfvén waves (AWs) that propagate away from the disk along magnetic-field lines. Because the Alfvén speed varies with distance from the disk, the AWs undergo partial non-WKB reflection, and counter-propagating AWs subsequently interact, causing AW energy to cascade to small scales and dissipate. To investigate this process, we introduce an Elsasser-like formulation of general relativistic magnetohydrodynamics (GRMHD) and develop the theory of general relativistic reduced MHD in an inhomogeneous medium. We then derive a set of equations for the mean-square AW amplitude $M_{+}$and turbulent heating rate $Q$under the assumption that, in the plasma rest frame, AWs propagating away from the disk are much more energetic than AWs propagating toward the disk. For the case in which the background flow is axisymmetric and time independent, we solve these equations analytically to determine$M_{+}$and $Q$as functions of position. We find that, for an idealized thin disk threaded by a large-scale poloidal magnetic field, the AW energy flux is${\sim}(\unicode[STIX]{x1D70C}_{\text{b}}/\unicode[STIX]{x1D70C}_{\text{d}})^{1/2}\unicode[STIX]{x1D6FD}_{\text{net,d}}^{-1/2}$times the disk’s radiative flux, where$\unicode[STIX]{x1D70C}_{\text{b}}$and$\unicode[STIX]{x1D70C}_{\text{d}}$are the mass densities at the coronal base and disk midplane, respectively, and$\unicode[STIX]{x1D6FD}_{\text{net,d}}$is the ratio (evaluated at the disk midplane) of plasma-plus-radiation pressure to the pressure of the average vertical magnetic field. This energy flux could have a significant impact on disk coronae and outflows. To lay the groundwork for future global simulations of turbulent disk coronae and jets, we derive a set of averaged GRMHD equations that account for reflection-driven AW turbulence using a sub-grid model.