Simple assumptions made regarding electron thermodynamics often limit the extent to which general relativistic magnetohydrodynamic (GRMHD) simulations can be applied to observations of low-luminosity accreting black holes. We present, implement, and test a model that self-consistently evolves an entropy equation for the electrons and takes into account the effects of spatially varying electron heating and relativistic anisotropic thermal conduction along magnetic field lines. We neglect the back-reaction of electron pressure on the dynamics of the accretion flow. Our model is appropriate for systems accreting at ≪ 10 −5 of the Eddington accretion rate, so radiative cooling by electrons can be neglected. It can be extended to higher accretion rates in the future by including electron cooling and proton-electron Coulomb collisions. We present a suite of tests showing that our method recovers the correct solution for electron heating under a range of circumstances, including strong shocks and driven turbulence. Our initial applications to axisymmetric simulations of accreting black holes show that (1) physically-motivated electron heating rates that depend on the local magnetic field strength yield electron temperature distributions significantly different from the constant electron to proton temperature ratios assumed in previous work, with higher electron temperatures concentrated in the coronal region between the disc and the jet; (2) electron thermal conduction significantly modifies the electron temperature in the inner regions of black hole accretion flows if the effective electron mean free path is larger than the local scale-height of the disc (at least for the initial conditions and magnetic field configurations we study). The methods developed in this work are important for producing more realistic predictions for the emission from accreting black holes such as Sagittarius A* and M87; these applications will be explored in future work.

We employ detailed numerical simulations to probe the mechanism of flow reversals in twodimensional turbulent convection. We show that the reversals occur via vortex reconnection of two attracting corner rolls having same sign of vorticity, thus leading to major restructuring of the flow. Large fluctuations in heat transport are observed during the reversal due to this flow reconfiguration. The flow configurations during the reversals have been analyzed quantitatively using large-scale modes. Using these tools, we also show why flow reversals occur for a restricted range of Rayleigh and Prandt numbers.PACS numbers: 47.55. 47.27.De Several experiments [1][2][3][4][5][6][7][8] and numerical simulations [8][9][10][11][12] on turbulent convection exhibit "flow reversals" in which the probes near the lateral walls of the container show random reversals (also see review articles [13]). These reversals have certain similarities with magnetic field reversals in dynamo and Kolmogorov flow [7]. Researchers typically study convection in a controlled setup called "Rayleigh-Bénard convection" in which a fluid confined between two plates is heated from below and cooled at the top. The two nondimensional numbers used to characterize the flow are the Rayleigh number (Ra), which is the ratio of the buoyancy term and the diffusive term, and the Prandtl number (Pr), which is the ratio of the kinematic viscosity and the thermal diffusivity. Cioni et al. [4] performed convection experiments on mercury, water, and helium gas in a cylindrical geometry and observed reversals for Ra > 10 8 . Sugiyama et al. [8] and Vasiliev and Frick [5] studied reversals in a rectangular box with water. No reversal was observed for a cubical box (aspect ratio 1), but a quasi two-dimensional box (aspect ratio ≤ 0.2) exhibits reversals for a band of Rayleigh and Prandtl numbers [8]. Surprisingly, a cubical box containing mercury shows reversals [7], indicating a strong role played by the Prandtl number and geometry in the reversal dynamics.Several theoretical models have been invoked to explain flow reversals. Benzi and Verzicco [9] and Sreenivasan et al. [14] used stochastic resonance, while Arajuo et al.[15] employed low-dimensional models with noise to explain reversals. Brown and Ahlers [3] and Mishra et al. [11] showed that in a cylindrical geometry, the flow reversals are induced by a rotation or cessation of large-scale flow structures. For two-dimensional box geometry, Sugiyama et al.[8] relate the flow reversal to the growth of the corner rolls due to the plume detachments from the boundary layers. Chandra and Verma [12] studied the reversals quantitatively by representing flow structures as Fourier modes and showed that during the reversals, the amplitude of the first Fourier mode (k x = 1, k y = 1) becomes very small, while the Fourier mode (k x = 2, k y = 2) gains strength. The growth of the secondary modes at the expense of the primary modes is akin to the cessation-led reversals reported by Brown and Ahlers [3] and Mishra et al. [11], ...

Based on direct numerical simulations and symmetry arguments, we show that the large-scale Fourier modes are useful tools to describe the flow structures and dynamics of flow reversals in Rayleigh-Bénard convection (RBC). We observe that during the reversals, the amplitude of one of the large-scale modes vanishes, while another mode rises sharply, very similar to the "cessation-led" reversals observed earlier in experiments and numerical simulations. We find anomalous fluctuations in the Nusselt number during the reversals. Using the structures of the RBC equations in the Fourier space, we deduce two symmetry transformations that leave the equations invariant. These symmetry transformations help us in identifying the reversing and non-reversing Fourier modes.PACS numbers: 47.55. 47.27.De, Many experiments [1][2][3][4][5][6][7] and numerical simulations [5,[8][9][10] on turbulent convection reveal that the velocity field of the system reverses randomly in time (also see review articles [11]). This phenomenon, known as "flow reversal", remains ill understood. This process gains practical importance due to its similarities with the magnetic field reversals in geodynamo and solar dynamo [12]. In this letter, we study the dynamics and symmetries of flow reversals in turbulent convection using the large-scale Fourier modes of the velocity and temperature fields.The experiments and simulations performed to explore the nature of flow reversals are typically for an idealized convective system called Rayleigh-Bénard convection (RBC) in which a fluid confined between two plates is heated from below and cooled at the top. Detailed measurements show that the first Fourier mode vanishes abruptly during some reversals [3,4]. These reversals are referred to as "cessation-led". Recently Sugiyama et al. [5] performed RBC experiments on water in a quasi two-dimensional box, and observed flow reversals with the flow profile dominated by a diagonal large-scale roll and two smaller secondary rolls at the corners. They attribute the flow reversals to the growth of the two smaller corner rolls as a result of plume detachments from the boundary layers.Several theoretical studies performed to understand reversals in RBC provide important clues. Broadly, these works involve either stochasticity (e.g., "stochastic resonance" [8, 13]), or low-dimensional models with noise [14,15]. Mishra et al.[10] studied the large-scale modes of RBC in a cylindrical geometry and showed that the dipolar mode decreases in amplitude and the quadrupolar mode increases during the cessation-led reversals. Regarding dynamo, low-dimensional models constructed using the large-scale modes and symmetry arguments reproduced dynamo reversals successfully [7,16].The theoretical models described above only focus on the large-scale modes. Here too, they provide limited information about these modes due to small number of measuring probes. In this letter we compute large-scale and intermediate-scale Fourier modes accurately using the complete flow profile. This helps us ...

Synchrotron emission and absorption determine the observational appearances of many astronomical systems. In this paper, we describe a numerical scheme for calculating synchrotron emissivities and absorptivities in all four Stokes parameters for arbitrary gyrotropic electron distribution functions, building on earlier work by Leung, Gammie, and Noble. We use this technique to evaluate the emissivities and the absorptivities for a thermal (Maxwell-Jüttner), isotropic power-law, and an isotropic kappa distribution function. The latter contains a powerlaw tail at high particle energies that smoothly merges with a thermal core at low energies, as is characteristic of observed particle spectra in collisionless plasmas. We provide fitting formulae and error bounds on the fitting formulae for use in codes that solve the radiative transfer equation. The numerical method and the fitting formulae are implemented in a compact C library called symphony. We find that the kappa distribution has a source function that is indistinguishable from a thermal spectrum at low frequency and transitions to the characteristic selfabsorbed synchrotron spectrum, n µ 5 2 , at high frequency; the linear polarization fraction for a thermal spectrum is near unity at high frequency; and all distributions produce O(10%) circular polarization at low frequency for lines of sight sufficiently close to the magnetic field vector.

Black holes accreting well below the Eddington rate are believed to have geometrically thick, optically thin, rotationally supported accretion discs in which the Coulomb mean free path is large compared to GM/c 2 . In such an environment, the disc evolution may differ significantly from ideal magnetohydrodynamic predictions. We present non-ideal global axisymmetric simulations of geometrically thick discs around a rotating black hole. The simulations are carried out using a new code grim, which evolves a covariant extended magnetohydrodynamics model derived by treating non-ideal effects as a perturbation of ideal magnetohydrodynamics. Non-ideal effects are modeled through heat conduction along magnetic field lines, and a difference between the pressure parallel and perpendicular to the field lines. The model relies on an effective collisionality in the disc from wave-particle scattering and velocity-space (mirror and firehose) instabilities. We find that the pressure anisotropy grows to match the magnetic pressure, at which point it saturates due to the mirror instability. The pressure anisotropy produces outward angular momentum transport with a magnitude comparable to that of MHD turbulence in the disc, and a significant increase in the temperature in the wall of the jet. We also find that, at least in our axisymmetric simulations, conduction has a small effect on the disc evolution because (1) the heat flux is constrained to be parallel to the field and the field is close to perpendicular to temperature gradients, and (2) the heat flux is choked by an increase in effective collisionality associated with the mirror instability.