A unified description of anti-dynamo conditions for incompressible flows

A general type of mathematical argument is described, which applies to all the cases in which dynamo maintenance of a steady magnetic field by motion in a uniform density is known to be impossible. Previous work has demonstrated that magnetic field decay is unavoidable under conditions of axisymmetry and in spherical or planar incompressible flows. These known results are encompassed by a calculation for flows described in terms of a generalized poloidal-toroidal representation of the magnetic field with respect to an arbitrary two dimensional surface. We show that when the velocity field is two dimensional, the dynamo growth, if any, that results, is linear in one of the projections of the field while the other projections remain constant. We also obtain criteria for the existence of and classification into two and three dimensional velocity results which are satisfied by a restricted set of geometries. In addition, we discuss the forms of spatial variation of the density and the resistivity that are allowed so that field decay still occurs for this set of geometries.Comment: 8 pages, Asian Journal of Physics, in pres

Stellar and gas dynamical model for tidal disruption events in a quiescent galaxy

A detailed model of the tidal disruption events (TDEs) has been constructed using stellar dynamical and gas dynamical inputs that include black hole (BH) mass $M_{\bullet}$, specific orbital energy $E$ and angular momentum $J$, star mass $M_{\star}$ and radius $R_{\star}$, and the pericenter of the star orbit $r_{p}(E,\hspace{1mm}J,\hspace{1mm}M_{\bullet})$. We solved the steady state Fokker--Planck equation using the standard loss cone theory for the galactic density profile $\rho (r) \propto r^{-\gamma}$ and stellar mass function $\xi(m)$ where $m=M_{\star}/M_{\odot}$ and obtained the feeding rate of stars to the BH integrated over the phase space as $\dot{N}_{t} \propto M_{\bullet}^\beta$, where $\beta= -0.3\pm 0.01$ for $M_{\bullet}>10^7 M_{\odot}$ and $\sim 6.8 \hspace{1mm} \times 10^{-5}$ Yr$^{-1}$ for $\gamma=0.7$. We use this to model the in-fall rate of the disrupted debris, $\dot{M}(E,\hspace{1mm}J,\hspace{1mm}m,\hspace{1mm}t)$, and discuss the conditions for the disk formation, finding that the accretion disk is almost always formed for the fiduciary range of the physical parameters. We also find the conditions under which the disk formed from the tidal debris of a given star with a super Eddington accretion phase. We have simulated the light curve profiles in the relevant optical g band and soft X-rays for both super and sub-Eddington accretion disks as a function of $\dot{M}(E,\hspace{1mm}J,\hspace{1mm}t)$. Using this, standard cosmological parameters, and mission instrument details, we predict the detectable TDE rates for various forthcoming surveys finally as a function of $\gamma$.Comment: 56 pages; 22 Figures, Accepted in the Astrophysical Journal, updated paper includes proof correction

Cosmic spin and mass evolution of black holes and its impact

We build an evolution model of the central black hole that depends on the processes of gas accretion, the capture of stars, mergers as well as electromagnetic torque. In case of gas accretion in the presence of cooling sources, the flow is momentum-driven, after which the black hole reaches a saturated mass; subsequently, it grows only by stellar capture and mergers. We model the evolution of the mass and spin with the initial seed mass and spin in $\Lambda$CDM cosmology. For stellar capture, we have assumed a power-law density profile for the stellar cusp in a framework of relativistic loss cone theory that include the effects of black hole spin, Carter's constant, loss cone angular momentum, and capture radius. Based on this, the predicted capture rates of $10^{-5}$--$10^{-6}$ yr$^{-1}$ are closer to the observed range. We have considered the merger activity to be effective for $z \lesssim 4$, and we self-consistently include the Blandford-Znajek torque. We calculate these effects on the black hole growth individually and in combination, for deriving the evolution. Before saturation, accretion dominates the black hole growth ($\sim 95\%$ of the final mass), and subsequently, stellar capture and mergers take over with roughly equal contribution. The simulations of the evolution of the $M_{\bullet} - \sigma$ relation using these effects are consistent with available observations. We run our model backward in time and retrodict the parameters at formation. Our model will provide useful inputs for building demographics of the black holes and in formation scenarios involving stellar capture.Comment: 56 pages, 27 figures; accepted for publication in the Astrophysical Journal; v2: updated references, typos fixed, editorial changes; v3: corrections during proof

Magnetohydrostatic flux tube model in the solar atmosphere

We construct two classes of the magnetohydrostatic equilibria of the axisymmetric flux tubes with twisted magnetic fields in the stratified solar atmosphere that span from the photosphere to the transition region. We built the models by incorporating specific forms of the gas pressure and poloidal current in the Grad-Shafranov equation. This model gives both closed and open field structure of the flux tube. The other open field model we construct is based on the self-similar formulation, where we have incorporated specific forms of the gas pressure, poloidal current and two different shape functions. We study the homology of the parameter space that is consistent with the solar atmosphere and find that the estimation of the magnetic structure inside the flux tubes is consistent with the observation and simulation results of the magnetic bright points.Comment: 2 pages, 1 figure; to appear in the proceedings of IAU Symposium 340: Long-term datasets for the understanding of solar and stellar magnetic cycle

Models of force-free spheres and applications to solar active regions

Here we present a systematic study of force-free field equation for simple axisymmetric configurations in spherical geometry. The condition of separability of solutions in radial and angular variables leads to two classes of solutions: linear and non-linear force-free fields. We have studied these linear solutions Chandrasekhar (1956) and extended the non-linear solutions given in Low \& Lou (1990) to the irreducible rational form $n= p/q$, which is allowed for all cases of odd $p$ and to cases of $q>p$ for even $p$. We have further calculated their energies and relative helicities for magnetic field configurations in finite and infinite shell geometries. We demonstrate here a method here to be used to fit observed magnetograms as well as to provide good exact input fields for testing other numerical codes used in reconstruction on the non-linear force-free fields.Comment: 7 pages, 2 figure

On Compressing U-net Using Knowledge Distillation

We study the use of knowledge distillation to compress the U-net architecture. We show that, while standard distillation is not sufficient to reliably train a compressed U-net, introducing other regularization methods, such as batch normalization and class re-weighting, in knowledge distillation significantly improves the training process. This allows us to compress a U-net by over 1000x, i.e., to 0.1% of its original number of parameters, at a negligible decrease in performance.Comment: 4 pages, 1 figur

A global galactic dynamo with a corona constrained by relative helicity

We present a model for a global axisymmetric turbulent dynamo operating in a galaxy with a corona which treats the supernovae (SNe) and magneto-rotational instability (MRI) driven turbulence parameters under a common formalism. The nonlinear quenching of the dynamo is alleviated by inclusion of small-scale advective and diffusive magnetic helicity fluxes, which allow the gauge invariant magnetic helicity to be transferred outside the disk and consequently build up a corona during the course of dynamo action. The time-dependent dynamo equations are expressed in a separable form and solved through an eigenvector expansion constructed using the steady-state solutions of the dynamo equation. The parametric evolution of the dynamo solution allows us to estimate the final structure of the global magnetic field and the saturated value of the turbulence parameter $\alpha_m$, even before solving the dynamical equations for evolution of magnetic fields in the disk and the corona, along with $\alpha$-quenching. We then solve these equations simultaneously to study the saturation of large-scale magnetic field, its dependence on the small-scale magnetic helicity fluxes and corresponding evolution of the force-free field in the corona. The quadrupolar large-scale magnetic field in the disk is found to reach equipartition strength within a timescale of 1 Gyr. The large-scale magnetic field in the corona obtained is much weaker in strength compared to the field inside the disk and has only a weak impact on the dynamo operation.Comment: 59 pages, 14 figures, 37 sub-figures, 2 tables and 7 appendices. Accepted for publication in Ap

Topological and statistical properties of nonlinear force-free fields

We use our semi-analytic solution of the nonlinear force-free field equation to construct three-dimensional magnetic fields that are applicable to the solar corona and study their statistical properties for estimating the degree of braiding exhibited by these fields. We present a new formula for calculating the winding number and compare it with the formula for the crossing number. The comparison is shown for a toy model of two helices and for realistic cases of nonlinear force-free fields; conceptually the formulae are nearly the same but the resulting distributions calculated for a given topology can be different. We also calculate linkages, which are useful topological quantities that are independent measures of the contribution of magnetic braiding to the total free energy and relative helicity of the field. Finally, we derive new analytical bounds for the free energy and relative helicity for the field configurations in terms of the linking number. These bounds will be of utility in estimating the braided energy available for nano-flares or for eruptions.Comment: 22 pages, 8 figures, 1 table, to appear in a special issue of Advances in Space Research on the "Dynamic Sun

Open and closed magnetic configurations of twisted flux tubes

We construct two classes of magnetohydrostatic (MHS) equilibria for an axisymmetric vertical flux tube spanning from the photosphere to the lower part of the transition region within a realistic stratified solar atmosphere subject to solar gravity. We assume a general quadratic expression of the magnetic flux function for the gas pressure and poloidal current and solve the Grad-Shafranov equation analytically. The solution is a combination of a homogeneous and a particular part where the former is separable by a Coulomb function in $r$ and exponential in $z$, while the particular part is an open configuration that has no $z$ dependence. We also present another open field solution by using a self-similar formulation with two different profile functions and incorporating stratified solar gravity to maintain the magnetohydrostatic equilibria, which is a modification of earlier self-similar models with a twist. We study the admitted parameter space that is consistent with the conditions in the solar atmosphere and derive magnetic and the thermodynamic structures inside the flux tube that are reasonably consistent with the photospheric magnetic bright points (MBPs) for both open and closed field Coulomb function and self-similar models as estimated from observations and simulations. The obtained open and closed field flux tube solutions can be used as the background conditions for the numerical simulations for the study of the wave propagation through the flux tubes. The solutions can also be used to construct realistic magnetic canopies.Comment: 34 pages; 21 figures with 29 sub-figures; 6 tables, Accepted in the Astrophysical Journa

Kinematics of and emission from helically orbiting blobs in a relativistic magnetized jet

We present a general relativistic (GR) model of jet variability in active galactic nuclei due to orbiting blobs in helical motion along a funnel or cone shaped magnetic surface anchored to the accretion disk near the black hole. Considering a radiation pressure driven flow in the inner region, we find that it stabilizes the flow, yielding Lorentz factors ranging between 1.1 and 7 at small radii for reasonable initial conditions. Assuming these as inputs, simulated light curves (LCs) for the funnel model include Doppler and gravitational shifts, aberration, light bending, and time delay. These LCs are studied for quasi-periodic oscillations (QPOs) and the power spectral density (PSD) shape and yield an increased amplitude ($\sim$ 12 %); a beamed portion and a systematic phase shift with respect to that from a previous special relativistic model. The results strongly justify implementing a realistic magnetic surface geometry in Schwarzschild geometry to describe effects on emission from orbital features in the jet close to the horizon radius. A power law shaped PSD with a typical slope of $-2$ and QPOs with timescales in the range of $(1.37 - 130.7)$ days consistent with optical variability in Blazars, emerges from the simulations for black hole masses $M_{\bullet} = (0.5 - 5) \times 10^8 M_{\odot}$ and initial Lorentz factors $\gamma_{jet,i} = 2 - 10$. The models presented here can be applied to explain radio, optical, and X-ray variability from a range of jetted sources including active galactic nuclei, X-ray binaries and neutron stars.Comment: 41 pages, 13 Figures (19 sub-figures), 2 Tables, 1 Appendix; minor corrections made; Accepted for publication in the Astrophysical Journa