# Posts Tagged numerical simulation

## Recent Postings from numerical simulation

### Numerical Simulation of Vertical Oscillations in an Axisymmetric Thick Accretion Flow around a Black Hole

We study time evolution of rotating, axisymmetric, two dimensional inviscid accretion flows around black holes using a grid based finite difference method. We do not use reflection symmetry on the equatorial plane in order to inspect if the disk along with the centrifugal barrier oscillated vertically. In the inviscid limit, we find that the CENtrifugal pressure supported BOundary Layer (CENBOL) is oscillating vertically, more so, when the specific angular momentum is higher. As a result, the rate of outflow produced from the CENBOL, also oscillates. Indeed, the outflow rates in the upper half and the lower half are found to be anti-correlated. We repeat the exercise for a series of specific angular momentum {\lambda} of the flow in order to demonstrate effects of the centrifugal force on this interesting behaviour. We find that, as predicted in theoretical models of disks in vertical equilibrium, the CENBOL is produced only when the centrifugal force is significant and more specifically, when {\lambda} > 1.5. Outflow rate itself is found to increase with {\lambda} as well and so is the oscillation amplitude. The cause of oscillation appears to be due to the interaction among the back flow from the centrifugal barrier, the outflowing winds and the inflow. For low angular momentum, the back flow as well as the oscillation are missing. To our knowledge, this is the first time that such an oscillating solution is found with an well-tested grid based finite difference code and such a solution could be yet another reason of why Quasi-Periodic Oscillations should be observed in black hole candidates which are accreting low angular momentum transonic flows.

### Extraction of Gravitational Waves in Numerical Relativity

A numerical-relativity calculation yields in general a solution of the Einstein equations including also a radiative part, which is in practice computed in a region of finite extent. Since gravitational radiation is properly defined only at null infinity and in an appropriate coordinate system, the accurate estimation of the emitted gravitational waves represents an old and non-trivial problem in numerical relativity. A number of methods have been developed over the years to "extract" the radiative part of the solution from a numerical simulation and these include: quadrupole formulas, gauge-invariant metric perturbations, Weyl scalars, and characteristic extraction. We review and discuss each method, in terms of both its theoretical background as well as its implementation. Finally, we provide a brief comparison of the various methods in terms of their inherent advantages and disadvantages.

### A supernova feedback implementation for the astrophysical simulation software Arepo [Replacement]

Supernova (SN) explosions play an important role in the development of galactic structures. The energy and momentum imparted on the interstellar medium (ISM) in so-called "supernova feedback" drives turbulence, heats the gas, enriches it with heavy elements, can lead to the formation of new stars or even suppress star formation by disrupting stellar nurseries. In the numerical simulation at the sub-galactic level, not including the energy and momentum of supernovas in the physical description of the problem can also lead to several problems that might partially be resolved by including a description of supernovas. In this thesis such an implementation is attempted for the combined numerical hydrodynamics and N-body simulation software Arepo (Springel, 2010) for the high density gas in the ISM only. This allows supernova driven turbulence in boxes of 400pc cubed to be studied. In a stochastic process a large amount of thermal energy is imparted on a number of neighbouring cells, mimicking the effect of a supernova explosions. We test this approach by modelling the explosion of a single supernova in a uniform density medium and comparing the evolution of the resulting supernova remnant to the theoretically-predicted behaviour. We also run a simulation with our feedback code and a fixed supernova rate derived from the Kennicutt-Schmidt relation (Kennicutt, 1998) for a duration of about 20 Myrs. We describe our method in detail in this text and discuss the properties of our implementation. vii

### A supernova feedback implementation for the astrophysical simulation software Arepo

Supernova (SN) explosions play an important role in the development of galactic structures. The energy and momentum imparted on the interstellar medium (ISM) in so called "supernova feedback" drives turbulence, heats the gas, enriches it with heavy elements, can lead to the formation of new stars or even suppress star formation by disrupting stellar nurseries. In the numerical simulation at the sub-galactic level, not including the energy and momentum of supernovas in the physical description of the problem can also lead to several problems that might partially be resolved by including a description of supernovas. In this thesis such an implementation is attempted for the combined numerical hydrodynamics and N-body simulation software Arepo (Springel, 2010). In a stochastic process a large amount of thermal energy is imparted on a number of neighbouring cells, mimicking the effect of a supernova explosions. We test this approach by modelling the explosion of a single supernova in a uniform density medium and comparing the evolution of the resulting supernova remnant to the theoretically-predicted behaviour. We also run a simulation with our feedback code and a fixed supernova rate derived from the Kennicutt-Schmidt relation (Kennicutt, 1998) for a duration of about 20 Myrs. We describe our method in detail in this text and discuss the properties of our implementation.

### Magneto-hydrodynamical Numerical simulation of wind production from black hole hot accretion flows at very large radii

Numerical simulations of black hole hot accretion flows have shown the existence of strong wind. Those works focus only on the region close to black hole thus it is unknown whether or where the wind production stops at large radii. To address this question, Bu et al. (2016) have performed hydrodynamic (HD) simulations by taking into account the gravitational potential of both the black hole and the nuclear star clusters. The latter is assumed to be $\propto \sigma^2 \ln(r)$, with $\sigma$ being the velocity dispersion of stars and $r$ be the distance from the center of the galaxy. It was found that when the gravity is dominated by nuclear stars, i.e., outside of radius $R_A\equiv GM_{\rm BH}/\sigma^2$, winds can no longer be produced. That work, however, neglects the magnetic field, which is believed to play a crucial dynamical role in the accretion and thus must be taken into account. In this paper, we revisit this problem by performing magneto-hydrodynamical (MHD) simulations. We confirm the result of Bu et al. (2016), namely wind can't be produced at the region of $R>R_A$. Our result, combined with the results of Yuan et al. (2015), indicates that the formula describing the mass flux of wind $\dot{M}_{\rm wind}=\dot{M}_{\rm BH}(r/20r_s)$ can only be applied to the region where the black hole potential is dominant. Here $\dot{M}_{\rm BH}$ is the mass accretion rate at the black hole horizon and the value of $R_A$ is similar to the Bondi radius.

### Integration of Particle-Gas Systems with Stiff Mutual Drag Interaction

Numerical simulation of numerous mm/cm-sized particles embedded in a gaseous disk has become an important tool in the study of planet formation and in understanding the dust distribution in observed protoplanetary disks. However, the mutual drag force between the gas and the particles can become so stiff, particularly because of small particles and/or strong local solid concentration, that an explicit integration of this system is computationally formidable. In this work, we consider the integration of the mutual drag force in a system of Eulerian gas and Lagrangian solid particles. Despite the entanglement between the gas and the particles under the particle-mesh construct, we are able to devise a numerical algorithm that effectively decomposes the globally coupled system of equations for the mutual drag force and makes it possible to integrate this system on a cell-by-cell basis, which considerably reduces the computational task required. We use an analytical solution for the temporal evolution of each cell to relieve the time-step constraint posed by the mutual drag force as well as to achieve the highest degree of accuracy. To validate our algorithm, we use an extensive suite of benchmarks with known solutions in one, two, and three dimensions, including the linear growth and the nonlinear saturation of the streaming instability. We demonstrate numerical convergence and satisfactory consistency in all cases. Our algorithm can for example be applied to model the evolution of the streaming instability with mm/cm-sized pebbles at high mass loading, which has important consequences for the formation scenarios of planetesimals.

### Integration of Particle-Gas Systems with Stiff Mutual Drag Interaction [Replacement]

Numerical simulation of numerous mm/cm-sized particles embedded in a gaseous disk has become an important tool in the study of planet formation and in understanding the dust distribution in observed protoplanetary disks. However, the mutual drag force between the gas and the particles can become so stiff, particularly because of small particles and/or strong local solid concentration, that an explicit integration of this system is computationally formidable. In this work, we consider the integration of the mutual drag force in a system of Eulerian gas and Lagrangian solid particles. Despite the entanglement between the gas and the particles under the particle-mesh construct, we are able to devise a numerical algorithm that effectively decomposes the globally coupled system of equations for the mutual drag force and makes it possible to integrate this system on a cell-by-cell basis, which considerably reduces the computational task required. We use an analytical solution for the temporal evolution of each cell to relieve the time-step constraint posed by the mutual drag force as well as to achieve the highest degree of accuracy. To validate our algorithm, we use an extensive suite of benchmarks with known solutions in one, two, and three dimensions, including the linear growth and the nonlinear saturation of the streaming instability. We demonstrate numerical convergence and satisfactory consistency in all cases. Our algorithm can for example be applied to model the evolution of the streaming instability with mm/cm-sized pebbles at high mass loading, which has important consequences for the formation scenarios of planetesimals.

### Integration of Particle-Gas Systems with Stiff Mutual Drag Interaction [Replacement]

Numerical simulation of numerous mm/cm-sized particles embedded in a gaseous disk has become an important tool in the study of planet formation and in understanding the dust distribution in observed protoplanetary disks. However, the mutual drag force between the gas and the particles can become so stiff, particularly because of small particles and/or strong local solid concentration, that an explicit integration of this system is computationally formidable. In this work, we consider the integration of the mutual drag force in a system of Eulerian gas and Lagrangian solid particles. Despite the entanglement between the gas and the particles under the particle-mesh construct, we are able to devise a numerical algorithm that effectively decomposes the globally coupled system of equations for the mutual drag force and makes it possible to integrate this system on a cell-by-cell basis, which considerably reduces the computational task required. We use an analytical solution for the temporal evolution of each cell to relieve the time-step constraint posed by the mutual drag force as well as to achieve the highest degree of accuracy. To validate our algorithm, we use an extensive suite of benchmarks with known solutions in one, two, and three dimensions, including the linear growth and the nonlinear saturation of the streaming instability. We demonstrate numerical convergence and satisfactory consistency in all cases. Our algorithm can for example be applied to model the evolution of the streaming instability with mm/cm-sized pebbles at high mass loading, which has important consequences for the formation scenarios of planetesimals.

### Integration of Particle-Gas Systems with Stiff Mutual Drag Interaction [Replacement]

Numerical simulation of numerous mm/cm-sized particles embedded in a gaseous disk has become an important tool in the study of planet formation and in understanding the dust distribution in observed protoplanetary disks. However, the mutual drag force between the gas and the particles can become so stiff, particularly because of small particles and/or strong local solid concentration, that an explicit integration of this system is computationally formidable. In this work, we consider the integration of the mutual drag force in a system of Eulerian gas and Lagrangian solid particles. Despite the entanglement between the gas and the particles under the particle-mesh construct, we are able to devise a numerical algorithm that effectively decomposes the globally coupled system of equations for the mutual drag force and makes it possible to integrate this system on a cell-by-cell basis, which considerably reduces the computational task required. We use an analytical solution for the temporal evolution of each cell to relieve the time-step constraint posed by the mutual drag force as well as to achieve the highest degree of accuracy. To validate our algorithm, we use an extensive suite of benchmarks with known solutions in one, two, and three dimensions, including the linear growth and the nonlinear saturation of the streaming instability. We demonstrate numerical convergence and satisfactory consistency in all cases. Our algorithm can for example be applied to model the evolution of the streaming instability with mm/cm-sized pebbles at high mass loading, which has important consequences for the formation scenarios of planetesimals.

### Numerical simulation and experimental study of PbWO4/EPDM and Bi2WO6/EPDM for the shielding of {\gamma}rays [Cross-Listing]

The MCNP5 code was employed to simulate the {\gamma}ray shielding capacity of tungstate composites. The experimental results were applied to verify the applicability of the Monte Carlo program. PbWO4 and Bi2WO6 were prepared and added into ethylene propylene diene monomer (EPDM) to obtain the composites, which were tested in the {\gamma}ray shielding. Both the theoretical simulation and experiments were carefully chosen and well designed. The results of the two methods were found to be highly consistent. In addition, the conditions during the numerical simulation were optimized and double-layer {\gamma}ray shielding systems were studied. It was found that the {\gamma}-ray shielding performance can be influenced not only by the material thickness ratio but also by the arrangement of the composites.

### Key Issues Review: Numerical studies of turbulence in stars

The numerical simulation of turbulence in stars has led to a rich set of possibilities regarding stellar pulsations, asteroseismology, thermonuclear yields, and formation of neutron stars and black holes. The breaking of symmetry by turbulent flow grows in amplitude as collapse is approached, which insures that the conditions at the onset of collapse are not spherical. This lack of spherical symmetry has important implications for the mechanism of explosion and ejected nucleosynthesis products. Numerical resolution of several different types of three--dimensional (3D) stellar simulations are compared; it is suggested that core collapse simulations may be under-resolved. New physical effects which appear in 3D are summarized. Connections between simulations of progenitor explosion and observations of supernova remnants (SNR) are discussed. Present treatment of boundaries, for mixing regions during He--burning, requires revision.

### Key Issues Review: Numerical studies of turbulence in stars [Cross-Listing]

The numerical simulation of turbulence in stars has led to a rich set of possibilities regarding stellar pulsations, asteroseismology, thermonuclear yields, and formation of neutron stars and black holes. The breaking of symmetry by turbulent flow grows in amplitude as collapse is approached, which insures that the conditions at the onset of collapse are not spherical. This lack of spherical symmetry has important implications for the mechanism of explosion and ejected nucleosynthesis products. Numerical resolution of several different types of three--dimensional (3D) stellar simulations are compared; it is suggested that core collapse simulations may be under-resolved. New physical effects which appear in 3D are summarized. Connections between simulations of progenitor explosion and observations of supernova remnants (SNR) are discussed. Present treatment of boundaries, for mixing regions during He--burning, requires revision.

### Numerical Simulation of Tidal Evolution of a Viscoelastic Body Modelled with a Mass-Spring Network [Replacement]

We use a damped mass-spring model within an N-body code to simulate the tidal evolution of the spin and orbit of a self-gravitating viscoelastic spherical body moving around a point-mass perturber. The damped mass-spring model represents a Kelvin-Voigt viscoelastic solid. We measure the tidal quality function (the dynamical Love number $\,k_2\,$ divided by the tidal quality factor $\,Q\,$) from the numerically computed tidal drift of the semimajor axis of the binary. The shape of $\,k_2/Q\,$, as a function of the principal tidal frequency, reproduces the kink shape predicted by Efroimsky (2012a; CeMDA 112$\,:\,$283) for the tidal response of near-spherical homogeneous viscoelastic rotators. We demonstrate that we can directly simulate the tidal evolution of spinning viscoelastic objects. In future, the mass-spring N-body model can be generalised to inhomogeneous and/or non-spherical bodies.

### Numerical Simulation of Tidal Evolution of a Viscoelastic Body Modeled with a Mass-Spring Network

We use a damped mass-spring model within an N-body code, to simulate the tidal evolution of the spin and orbit of a viscoelastic spherical body moving around a point-mass perturber. The damped spring-mass model represents a Kelvin-Voigt viscoelastic solid. We derive the tidal quality function (the dynamical Love number $\,k_2\,$ divided by the tidal quality factor $\,Q\,$) from the numerically computed tidal drift of the semimajor axis of the binary. The obtained shape of $\,k_2/Q\,$, as a function of the principal tidal frequency, reproduces the typical kink shape predicted by Efroimsky (2012a; CeMDA 112$\,:\,$283) for the tidal response of near-spherical homogeneous viscoelastic rotators. Our model demonstrates that we can directly simulate the tidal evolution of viscoelastic objects. This opens the possibility for investigating more complex situations, since the employed spring-mass N-body model can be generalised to inhomogeneous and/or non-spherical bodies.

### Simulating the Environment Around Planet-Hosting Stars - I. Coronal Structure

We present the results of a detailed numerical simulation of the circumstellar environment around three exoplanet-hosting stars. A state-of-the-art global magnetohydrodynamic (MHD) model is considered, including Alfv\'en wave dissipation as a self-consistent coronal heating mechanism. This paper contains the description of the numerical set-up, evaluation procedure, and the simulated coronal structure of each system (HD 1237, HD 22049 and HD 147513). The simulations are driven by surface magnetic field maps, recovered with the observational technique of Zeeman Doppler Imaging (ZDI). A detailed comparison of the simulations is performed, where two different implementations of this mapping routine are used to generate the surface field distributions. Quantitative and qualitative descriptions of the coronae of these systems are presented, including synthetic high-energy emission maps in the Extreme Ultra-Violet (EUV) and Soft X-rays (SXR) ranges. Using the simulation results, we are able to recover similar trends as in previous observational studies, including the relation between the magnetic flux and the coronal X-ray emission. Furthermore, for HD 1237 we estimate the rotational modulation of the high-energy emission due to the various coronal features developed in the simulation. We obtain variations, during a single stellar rotation cycle, up to 15\% for the EUV and SXR ranges. The results presented here will be used, in a follow-up paper, to self-consistently simulate the stellar winds and inner astrospheres of these systems.

### The numerical approach to quantum field theory in a non-commutative space [Cross-Listing]

Numerical simulation is an important non-perturbative tool to study quantum field theories defined in non-commutative spaces. In this contribution, a selection of results from Monte Carlo calculations for non-commutative models is presented, and their implications are reviewed. In addition, we also discuss how related numerical techniques have been recently applied in computer simulations of dimensionally reduced supersymmetric theories.

### The numerical approach to quantum field theory in a non-commutative space [Cross-Listing]

Numerical simulation is an important non-perturbative tool to study quantum field theories defined in non-commutative spaces. In this contribution, a selection of results from Monte Carlo calculations for non-commutative models is presented, and their implications are reviewed. In addition, we also discuss how related numerical techniques have been recently applied in computer simulations of dimensionally reduced supersymmetric theories.

### The numerical approach to quantum field theory in a non-commutative space

Numerical simulation is an important non-perturbative tool to study quantum field theories defined in non-commutative spaces. In this contribution, a selection of results from Monte Carlo calculations for non-commutative models is presented, and their implications are reviewed. In addition, we also discuss how related numerical techniques have been recently applied in computer simulations of dimensionally reduced supersymmetric theories.

### Numerical experiments on the detailed energy conversion and spectrum studies in a corona current sheet

In this paper, we study the energy conversion and spectra in a corona current sheet by 2.5-dimensional MHD numerical simulations. Numerical results show that many Petschek-like fine structures with slow-mode shocks mediated by plasmoid instabilities develop during the magnetic reconnection process. The termination shocks can also be formed above the primary magnetic island and at the head of secondary islands. These shocks play important roles in generating thermal energy in a corona current sheet. For a numerical simulation with initial conditions close to the solar corona environment, the ratio of the generated thermal energy to the total dissipated magnetic energy is around $1/5$ before secondary islands appear. After secondary islands appear, the generated thermal energy starts to increase sharply and this ratio can reach a value about $3/5$. In an environment with a relatively lower plasma density and plasma $\beta$, the plasma can be heated to a much higher temperature. After secondary islands appear, the one dimensional energy spectra along the current sheet do not behave as a simple power law and the spectrum index increases with the wave number. The average spectrum index for the magnetic energy spectrum along the current sheet is about $1.8$. The two dimensional spectra intuitively show that part of the high energy is cascaded to large $kx$ and $ky$ space after secondary islands appear. The plasmoid distribution function calculated from numerical simulations behaves as a power law closer to $f(\psi) \sim \psi^{-1}$ in the intermediate $\psi$ regime. By using $\eta_{eff} = v_{inflow}\cdot L$, the effective magnetic diffusivity is estimated about $10^{11}\sim10^{12}$~m$^2$\,s$^{-1}$.

### On improving analytical models of cosmic reionization for matching numerical simulation

The methods for studying the epoch of cosmic reionization vary from full radiative transfer simulations to purely analytical models. While numerical approaches are computationally expensive and are not suitable for generating many mock catalogs, analytical methods are based on assumptions and approximations. We explore the interconnection between both methods. First, we ask how the analytical framework of excursion set formalism can be used for statistical analysis of numerical simulations and visual representation of the morphology of ionization fronts. Second, we explore the methods of training the analytical model on a given numerical simulation. We present a new code which emerged from this study. Its main application is to match the analytical model with a numerical simulation. Then, it allows one to generate mock reionization catalogs with volumes exceeding the original simulation quickly and computationally inexpensively, meanwhile reproducing large scale statistical properties. These mock catalogs are particularly useful for CMB polarization and 21cm experiments, where large volumes are required to simulate the observed signal.

### Protostellar accretion traced with chemistry: Comparing synthetic C18O maps of embedded protostars to real observations [Replacement]

Context: Understanding how protostars accrete their mass is a central question of star formation. One aspect of this is trying to understand whether the time evolution of accretion rates in deeply embedded objects is best characterised by a smooth decline from early to late stages or by intermittent bursts of high accretion. Aims: We create synthetic observations of deeply embedded protostars in a large numerical simulation of a molecular cloud, which are compared directly to real observations. The goal is to compare episodic accretion events in the simulation to observations and to test the methodology used for analysing the observations. Methods: Simple freeze-out and sublimation chemistry is added to the simulation, and synthetic C$^{18}$O line cubes are created for a large number of simulated protostars. The spatial extent of C$^{18}$O is measured for the simulated protostars and compared directly to a sample of 16 deeply embedded protostars observed with the Submillimeter Array. If CO is distributed over a larger area than predicted based on the protostellar luminosity, it may indicate that the luminosity has been higher in the past and that CO is still in the process of refreezing. Results: Approximately 1% of the protostars in the simulation show extended C$^{18}$O emission, as opposed to approximately 50% in the observations, indicating that the magnitude and frequency of episodic accretion events in the simulation is too low relative to observations. The protostellar accretion rates in the simulation are primarily modulated by infall from the larger scales of the molecular cloud, and do not include any disk physics. The discrepancy between simulation and observations is taken as support for the necessity of disks, even in deeply embedded objects, to produce episodic accretion events of sufficient frequency and amplitude.

### Protostellar accretion traced with chemistry: Comparing synthetic C18O maps of embedded protostars to real observations

Context: Understanding how protostars accrete their mass is a central question of star formation. One aspect of this is to try and understand if the time evolution of accretion rates in deeply embedded objects is best characterised by a smooth decline from early to late stages, or by intermittent bursts of high accretion. Aims: We create synthetic observations of deeply embedded protostars in a large numerical simulation of a molecular cloud, which are compared directly to real observations. The goal is to compare episodic accretion events in the simulation to observations, and to test the methodology used for analysing the observations. Methods: Simple freeze-out and sublimation chemistry is added to the simulation, and synthetic C18O line cubes are created for a large number of simulated protostars. The spatial extent of C18O is measured for the simulated protostars, and compared directly to a sample of 16 deeply embedded protostars observed with the Submillimeter Array. If CO is distributed over a larger area than predicted based on the protostellar luminosity, it may indicate that the luminosity has been higher in the past, and that CO is still in the process of refreezing. Results: Approximately 1% of the protostars in the simulation show extended C18O emission, as opposed to approximately 50% in the observations, indicating that the magnitude and frequency of episodic accretion events in the simulation is too low relative to observations. The protostellar accretion rates in the simulation are primarily modulated by infall from the larger scales of the molecular cloud, and do not include any disk physics. The discrepancy between simulation and observations is taken as support for the necessity of disks, even in deeply embedded objects, to produce episodic accretion events of sufficient frequency and amplitude.

### The sparkling Universe: a scenario for cosmic void motions

We perform a statistical study of the global motion of cosmic voids using both a numerical simulation and observational data. We analyse their relation to large--scale mass flows and the physical effects that drive those motions. We analyse the bulk motions of voids, defined by the mean velocity of haloes in the surrounding shells in the numerical simulation, and by galaxies in the Sloan Digital Sky Survey Data Release 7. We find void mean bulk velocities close to 400 km/s, comparable to those of haloes (~ 500-600 km/s), depending on void size and the large--scale environment. Statistically, small voids move faster than large ones, and voids in relatively higher density environments have higher bulk velocities than those placed in large underdense regions. Also, we analyze the mean mass density around voids finding, as expected, large--scale overdensities (underdensities) along (opposite to) the void motion direction, suggesting that void motions respond to a pull--push mechanism. This contrasts with massive cluster motions who are mainly governed by the pull of the large-scale overdense regions. Our analysis of void pairwise velocities shows how their relative motions are generated by large--scale density fluctuations. In agreement with linear theory, voids embedded in low (high) density regions mutually recede (attract) each other, providing the general mechanism to understand the bimodal behavior of void motions. In order to compare the theoretical results and the observations we have inferred void motions in the SDSS using linear theory, finding that the estimated observational void motions are in statisticalagreement with the results of the simulation. Regarding large--scale flows, our results suggest a scenario of galaxies and galaxy systems flowing away from void centers with the additional, and morerelevant, contribution of the void bulk motion to the total velocity.

### Estimating SI violation in CMB due to non-circular beam and complex scan in minutes

Mild, unavoidable deviations from circular-symmetry of instrumental beams along with scan strategy can give rise to measurable Statistical Isotropy (SI) violation in Cosmic Microwave Background (CMB) experiments. If not accounted properly, this spurious signal can complicate the extraction of other SI violation signals (if any) in the data. However, estimation of this effect through exact numerical simulation is computationally intensive and time consuming. A generalized analytical formalism not only provides a quick way of estimating this signal, but also gives a detailed understanding connecting the leading beam anisotropy components to a measurable BipoSH characterisation of SI violation. In this paper, we provide an approximate generic analytical method for estimating the SI violation generated due to a non-circular (NC) beam and arbitrary scan strategy, in terms of the Bipolar Spherical Harmonic (BipoSH) spectra. Our analytical method can predict almost all the features introduced by a NC beam in a complex scan and thus reduces the need for extensive numerical simulation worth tens of thousands of CPU hours into minutes long calculations. As an illustrative example, we use WMAP beams and scanning strategy to demonstrate the easability, usability and efficiency of our method. We test all our analytical results against that from exact numerical simulations.

### Large-scale numerical simulations of star formation put to the test: Comparing synthetic images and actual observations for statistical samples of protostars [Replacement]

(abridged) Context: Both observations and simulations of embedded protostars have progressed rapidly in recent years. Bringing them together is an important step in advancing our knowledge about the earliest phases of star formation. Aims: To compare synthetic continuum images and SEDs, calculated from large-scale numerical simulations, to observational studies, thereby aiding in both the interpretation of the observations and in testing the fidelity of the simulations. Methods: The radiative transfer code RADMC-3D is used to create synthetic continuum images and SEDs of protostellar systems in a large numerical simulation of a molecular cloud. More than 13000 unique radiative transfer models are produced of a variety of different protostellar systems. Results: Over the course of 0.76 Myr the simulation forms more than 500 protostars, primarily within two sub-clusters. Synthetic SEDs are used to the calculate evolutionary tracers Tbol and Lsmm/Lbol. It is shown that, while the observed distributions of the tracers are well matched by the simulation, they generally do a poor job of tracking the protostellar ages. Disks form early in the simulation, with 40 % of the Class 0 protostars being encircled by one. The flux emission from the simulated disks is found to be, on average, a factor of 6 too low relative to real observations. The distribution of protostellar luminosities spans more than three order of magnitudes, similar to the observed distribution. Cores and protostars are found to be closely associated with one another, with the distance distribution between them being in excellent agreement with observations. Conclusions: The analysis and statistical comparison of synthetic observations to real ones is established as a powerful tool in the interpretation of observational results.

### Light Bridge in a Developing Active Region. II. Numerical Simulation of Flux Emergence and Light Bridge Formation

Light bridges, the bright structure dividing umbrae in sunspot regions, show various activity events. In Paper I, we reported on analysis of multi-wavelength observations of a light bridge in a developing active region (AR) and concluded that the activity events are caused by magnetic reconnection driven by magnetconvective evolution. The aim of this second paper is to investigate the detailed magnetic and velocity structures and the formation mechanism of light bridges. For this purpose, we analyze numerical simulation data from a radiative magnetohydrodynamics model of an emerging AR. We find that a weakly-magnetized plasma upflow in the near-surface layers of the convection zone is entrained between the emerging magnetic bundles that appear as pores at the solar surface. This convective upflow continuously transports horizontal fields to the surface layer and creates a light bridge structure. Due to the magnetic shear between the horizontal fields of the bridge and the vertical fields of the ambient pores, an elongated cusp-shaped current layer is formed above the bridge, which may be favorable for magnetic reconnection. The striking correspondence between the observational results of Paper I and the numerical results of this paper provides a consistent physical picture of light bridges. The dynamic activity phenomena occur as a natural result of the bridge formation and its convective nature, which has much in common with those of umbral dots and penumbral filaments.

### Formation of a condensate during charged collapse [Cross-Listing]

We observe a condensate forming in the interior of a black hole (BH) during numerical simulations of gravitational collapse of a massless charged (complex) scalar field. The magnitude of the scalar field in the interior tends to a non-zero constant; spontaneous breaking of gauge symmetry occurs and a condensate forms. This phenomena occurs in the presence of a BH without the standard symmetry breaking quartic potential; the breaking occurs via the dynamics of the system itself. We also observe that the scalar field in the interior rotates in the complex plane and show that it matches numerically the electric potential to within $1\%$. That a charged scalar condensate can form near the horizon of a black hole in the Abelian Higgs model without the standard symmetry breaking potential had previously been shown analytically in an explicit model involving a massive scalar field in an $AdS_4$ background. Our numerical simulation lends strong support to this finding, although in our case the scalar field is massless and the spacetime is asymptotically flat.

### Formation of a condensate during charged collapse [Replacement]

We observe a condensate forming in the interior of a black hole (BH) during numerical simulations of gravitational collapse of a massless charged (complex) scalar field. The magnitude of the scalar field in the interior tends to a non-zero constant; spontaneous breaking of gauge symmetry occurs and a condensate forms. This phenomena occurs in the presence of a BH without the standard symmetry breaking quartic potential; the breaking occurs via the dynamics of the system itself. We also observe that the scalar field in the interior rotates in the complex plane and show that it matches numerically the electric potential to within $1\%$. That a charged scalar condensate can form near the horizon of a black hole in the Abelian Higgs model without the standard symmetry breaking potential had previously been shown analytically in an explicit model involving a massive scalar field in an $AdS_4$ background. Our numerical simulation lends strong support to this finding, although in our case the scalar field is massless and the spacetime is asymptotically flat.

### Formation of a condensate during charged collapse [Replacement]

We observe a condensate forming in the interior of a black hole (BH) during numerical simulations of gravitational collapse of a massless charged (complex) scalar field. The magnitude of the scalar field in the interior tends to a non-zero constant; spontaneous breaking of gauge symmetry occurs and a condensate forms. This phenomena occurs in the presence of a BH without the standard symmetry breaking quartic potential; the breaking occurs via the dynamics of the system itself. We also observe that the scalar field in the interior rotates in the complex plane and show that it matches numerically the electric potential to within $1\%$. That a charged scalar condensate can form near the horizon of a black hole in the Abelian Higgs model without the standard symmetry breaking potential had previously been shown analytically in an explicit model involving a massive scalar field in an $AdS_4$ background. Our numerical simulation lends strong support to this finding, although in our case the scalar field is massless and the spacetime is asymptotically flat.

### Formation of a condensate during charged collapse

We observe a condensate forming in the interior of a black hole (BH) during numerical simulations of gravitational collapse of a massless charged (complex) scalar field. The magnitude of the scalar field in the interior tends to a non-zero constant; spontaneous breaking of gauge symmetry occurs and a condensate forms. This phenomena occurs in the presence of a BH without the standard symmetry breaking quartic potential; the breaking occurs via the dynamics of the system itself. We also observe that the scalar field in the interior rotates in the complex plane and show that it matches numerically the electric potential to within $1\%$. That a charged scalar condensate can form near the horizon of a black hole in the Abelian Higgs model without the standard symmetry breaking potential had previously been shown analytically in an explicit model involving a massive scalar field in an $AdS_4$ background. Our numerical simulation lends strong support to this finding, although in our case the scalar field is massless and the spacetime is asymptotically flat.

### Spin flips in generic black hole binaries [Cross-Listing]

We study the spin dynamics of individual black holes in a binary system. In particular we focus on the polar precession of spins and the possibility of a complete flip of spins with respect to the orbital plane. We perform a full numerical simulation that displays these characteristics. We evolve equal mass binary spinning black holes for $t=20,000M$ from an initial proper separation of $d=25M$ down to merger after 48.5 orbits. We compute the gravitational radiation from this system and compare it to 3.5 post-Newtonian generated waveforms finding close agreement. We then further use 3.5 post-Newtonian evolutions to show the extension of this spin {flip-flop} phenomenon to unequal mass binaries. We also provide analytic expressions to approximate the maximum {flip-flop} angle and frequency in terms of the binary spins and mass ratio parameters at a given orbital radius. Finally we discuss the effect this spin {flip-flop} would have on accreting matter and other potential observational effects.

### Spin flips in generic black hole binaries [Replacement]

We study the spin dynamics of individual black holes in a binary system. In particular we focus on the polar precession of spins and the possibility of a complete flip of spins with respect to the orbital plane. We perform a full numerical simulation that displays these characteristics. We evolve equal mass binary spinning black holes for $t=20,000M$ from an initial proper separation of $d=25M$ down to merger after 48.5 orbits. We compute the gravitational radiation from this system and compare it to 3.5 post-Newtonian generated waveforms finding close agreement. We then further use 3.5 post-Newtonian evolutions to show the extension of this spin {flip-flop} phenomenon to unequal mass binaries. We also provide analytic expressions to approximate the maximum {flip-flop} angle and frequency in terms of the binary spins and mass ratio parameters at a given orbital radius. Finally we discuss the effect this spin {flip-flop} would have on accreting matter and other potential observational effects.

### Spin flips in generic black hole binaries

We study the spin dynamics of individual black holes in a binary system. In particular we focus on the polar precession of spins and the possibility of a complete flip of spins with respect to the orbital plane. We perform a full numerical simulation that displays these characteristics. We evolve equal mass binary spinning black holes for $t=20,000M$ from an initial proper separation of $d=25M$ down to merger after 48.5 orbits. We compute the gravitational radiation from this system and compare it to 3.5 post-Newtonian generated waveforms finding close agreement. We then further use 3.5 post-Newtonian evolutions to show the extension of this spin {flip-flop} phenomenon to unequal mass binaries. We also provide analytic expressions to approximate the maximum {flip-flop} angle and frequency in terms of the binary spins and mass ratio parameters at a given orbital radius. Finally we discuss the effect this spin {flip-flop} would have on accreting matter and other potential observational effects.

### Spin flips in generic black hole binaries [Replacement]

We study the spin dynamics of individual black holes in a binary system. In particular we focus on the polar precession of spins and the possibility of a complete flip of spins with respect to the orbital plane. We perform a full numerical simulation that displays these characteristics. We evolve equal mass binary spinning black holes for $t=20,000M$ from an initial proper separation of $d=25M$ down to merger after 48.5 orbits. We compute the gravitational radiation from this system and compare it to 3.5 post-Newtonian generated waveforms finding close agreement. We then further use 3.5 post-Newtonian evolutions to show the extension of this spin {flip-flop} phenomenon to unequal mass binaries. We also provide analytic expressions to approximate the maximum {flip-flop} angle and frequency in terms of the binary spins and mass ratio parameters at a given orbital radius. Finally we discuss the effect this spin {flip-flop} would have on accreting matter and other potential observational effects.

### Visibility moments and power spectrum of turbulence velocity

Here we introduce moments of visibility function and discuss how those can be used to estimate the power spectrum of the turbulent velocity of external spiral galaxies. We perform numerical simulation to confirm the credibility of this method and found that for galaxies with lower inclination angles it works fine. This is the only method to estimate the power spectrum of the turbulent velocity fluctuation in the ISM of the external galaxies.

### Visibility moments and power spectrum of turbulence velocity [Replacement]

Here we introduce moments of visibility function and discuss how those can be used to estimate the power spectrum of the turbulent velocity of external spiral galaxies. We perform numerical simulation to confirm the credibility of this method and found that for galaxies with lower inclination angles it works fine. This is the only method to estimate the power spectrum of the turbulent velocity fluctuation in the ISM of the external galaxies.

### Solar wind turbulence from MHD to sub-ion scales: high-resolution hybrid simulations [Replacement]

We present results from a high-resolution and large-scale hybrid (fluid electrons and particle-in-cell protons) two-dimensional numerical simulation of decaying turbulence. Two distinct spectral regions (separated by a smooth break at proton scales) develop with clear power-law scaling, each one occupying about a decade in wave numbers. The simulation results exhibit simultaneously several properties of the observed solar wind fluctuations: spectral indices of the magnetic, kinetic, and residual energy spectra in the magneto-hydrodynamic (MHD) inertial range along with a flattening of the electric field spectrum, an increase in magnetic compressibility, and a strong coupling of the cascade with the density and the parallel component of the magnetic fluctuations at sub-proton scales. Our findings support the interpretation that in the solar wind large-scale MHD fluctuations naturally evolve beyond proton scales into a turbulent regime that is governed by the generalized Ohm's law.

### Solar wind turbulence from MHD to sub-ion scales: high-resolution hybrid simulations

We present results from a high-resolution and large-scale hybrid (fluid electrons and particle-in-cell protons) two-dimensional numerical simulation of decaying turbulence. Two distinct spectral regions (separated by a smooth break at proton scales) develop with clear power-law scaling, each one occupying about a decade in wave numbers. The simulation results exhibit simultaneously several properties of the observed solar wind fluctuations: spectral indices of the magnetic, kinetic, and residual energy spectra in the magneto-hydrodynamic (MHD) inertial range along with a flattening of the electric field spectrum, an increase in magnetic compressibility, and a strong coupling of the cascade with the density and the parallel component of the magnetic fluctuations at sub-proton scales. Our findings support the interpretation that in the solar wind large-scale MHD fluctuations naturally evolve beyond proton scales into a turbulent regime that is governed by the generalized Ohm's law.

### Vacuum high harmonic generation in the shock regime [Cross-Listing]

Electrodynamics becomes nonlinear and permits the self-interaction of fields when the quantised nature of vacuum states is taken into account. The effect on a plane probe pulse propagating through a stronger constant crossed background is calculated using numerical simulation and by analytically solving the corresponding wave equation. The electromagnetic shock resulting from vacuum high harmonic generation is investigated and a nonlinear shock parameter identified.

### Transport by meridional circulations in solar-type stars

Transport by meridional flows has significant consequences for stellar evolution, but is difficult to capture in global-scale numerical simulations because of the wide range of timescales involved. Stellar evolution models therefore usually adopt parameterizations for such transport based on idealized laminar or mean-field models. Unfortunately, recent attempts to model this transport in global simulations have produced results that are not consistent with any of these idealized models. In an effort to explain the discrepancies between global simulations and idealized models, we here use three-dimensional local Cartesian simulations of compressible convection to study the efficiency of transport by meridional flows below a convection zone in several parameter regimes of relevance to the Sun and solar-type stars. In these local simulations we are able to establish the correct ordering of dynamical timescales, although the separation of the timescales remains unrealistic. We find that, even though the generation of internal waves by convective overshoot produces a high degree of time dependence in the meridional flow field, the mean flow has the qualitative behavior predicted by laminar, "balanced" models. In particular, we observe a progressive deepening, or "burrowing", of the mean circulation if the local Eddington-Sweet timescale is shorter than the viscous diffusion timescale. Such burrowing is a robust prediction of laminar models in this parameter regime, but has never been observed in any previous numerical simulation. We argue that previous simulations therefore underestimate the transport by meridional flows.

### On the correction of conserved variables for numerical RMHD with staggered constrained transport

Despite the success of the combination of conservative schemes and staggered constrained transport algorithms in the last fifteen years, the accurate description of highly magnetized, relativistic flows with strong shocks represents still a challenge in numerical RMHD. The present paper focusses in the accuracy and robustness of several correction algorithms for the conserved variables, which has become a crucial ingredient in the numerical simulation of problems where the magnetic pressure dominates over the thermal pressure by more than two orders of magnitude. Two versions of non-relativistic and fully relativistic corrections have been tested and compared using a magnetized cylindrical explosion with high magnetization ($\ge 10^4$) as test. In the non-relativistic corrections, the total energy is corrected for the difference in the classical magnetic energy term between the average of the staggered fields and the conservative ones, before (CA1) and after (CA1') recovering the primitive variables. These corrections are unable to pass the test at any numerical resolution. The two relativistic approaches (CA2 and CA2'), correcting also the magnetic terms depending on the flow speed in both the momentum and the total energy, reveal as much more robust. These algorithms pass the test succesfully and with very small deviations of the energy conservation ($\le 10^{-4}$), and very low values of the total momentum ($\le 10^{-8}$). In particular, the algorithm CA2' (that corrects the conserved variables after recovering the primitive variables) passes the test at all resolutions. The numerical code used to run all the test cases is briefly described.

### Effects of Turbulent Viscosity on A Rotating Gas Ring Around A Black Hole: The Density Profile of Numerical Simulation

In this paper, we present the time evolution of a rotationally axisymmetric gas ring around a non rotating black hole using two dimensional grid-based hydrodynamic simulation. We show the way in which angular momentum transport is included in simulations of non-self-gravitating accretion of matter towards a black hole. We use the Shakura-Sunyaev {\alpha} viscosity prescription to estimate the turbulent viscosity. We investigate how a gas ring which is initially assumed to rotate with Keplerian angular velocity is accreted on to a back hole and hence forms accretion disc in the presence of turbulent viscosity. Furthermore, we also show that increase of the {\alpha} coefficient increases the rate of advection of matter towards the black hole. The density profile we obtain is in good quantitative agreement with that obtained from the analytical results. The dynamics of resulting angular momentum depends strongly on {\alpha}.

### Effects of Turbulent Viscosity on A Rotating Gas Ring Around A Black Hole: Results in Numerical Simulation [Replacement]

In this paper, we present the time evolution of a rotationally axisymmetric gas ring around a non rotating black hole using two dimensional grid-based hydrodynamic simulation. We show the way in which angular momentum transport is included in simulations of non-self-gravitating accretion of matter towards a black hole. We use the Shakura-Sunyaev {\alpha} viscosity prescription to estimate the turbulent viscosity for all major viscous stress tensors. We investigate how a gas ring which is initially assumed to rotate with Keplerian angular velocity is accreted on to a black hole and hence forms accretion disc in the presence of turbulent viscosity. We show that the centrifugal pressure supported sub-Keplerian flow with shocks forms when the ring starts to disperse with inclusion of relatively small amount of viscosity. But, if the viscosity is above the critical value, the shock disappears altogether and the whole disc becomes Kepleiran which is subsonic everywhere except in a region close to the horizon, where it supersonically enters to the black hole. We discovered a multiple valued Mach number solution and the corresponding density distributions that connects matter (a) from the initial Keplerian gas ring to a sub-Keplerian disc with shocks in presence of small amount of viscosity and (b) from the sub-Keplerian flow to a Keplerian disc in presence of huge amount of viscosity. We calculate the temporal variations of the magnitude of various time scales which ensure us about the stability of the flow.

### Numerical Simulation of Hot Accretion Flows (III): Revisiting wind properties using trajectory approach

Previous MHD simulations have shown that wind must exist in black hole hot accretion flows. In this paper, we continue our study by investigating the detailed properties of wind, such as mass flux and poloidal speed, and the mechanism of wind production. For this aim, we make use of a three dimensional GRMHD simulation of hot accretion flows around a Schwarzschild black hole. The simulation is designed so that the magnetic flux is not accumulated significantly around the black hole. To distinguish real wind from turbulent outflows, we track the trajectories of the virtual Largrangian particles from simulation data. We find two types of real outflows, i.e., a quasi-relativistic jet close to the axis and a sub-relativistic wind subtending a much larger solid angle. Most of the wind originates from the surface layer of the accretion flow. The poloidal wind speed almost remains constant once they are produced, but the flux-weighted wind speed roughly follows $v_{\rm p, wind}(r)\approx 0.25 v_k(r)$. The mass flux of jet is much lower but the speed is much higher, $v_{\rm p,jet}\sim (0.3-0.4) c$. Consequently, both the energy and momentum fluxes of the wind are much larger than those of the jet. We find that the wind is produced and accelerated primarily by the combination of centrifugal force and magnetic pressure gradient, while the jet is mainly accelerated by magnetic pressure gradient. Finally, we find that the wind production efficiency $\epsilon_{\rm wind}\equiv\dot{E}_{\rm wind}/\dot{M}_{\rm BH}c^2\sim 1/1000$, in good agreement with the value required from large-scale galaxy simulations with AGN feedback.

### Hemispheric Coupling: Comparing Dynamo Simulations and Observations

Numerical simulations that reproduce solar-like magnetic cycles can be used to generate long-term statistics. The variations in N-S hemispheric cycle synchronicity and amplitude produced in simulations has not been widely compared to observations. The observed limits on asymmetry show that hemispheric sunspot area production is no more than 20% asymmetric for cycles 12-23 and phase lags do not exceed 20% (2 yrs) of the total cycle period. Independent studies have found a long-term trend in phase values as one hemisphere leads the other for ~four cycles. Such persistence in phase is not indicative of a stochastic phenomenon. We compare the findings to results from a numerical simulation of solar convection recently produced with the EULAG-MHD model. This simulation spans 1600 yrs and generated 40 regular, sunspot-like cycles. While the simulated cycle length is too long and the toroidal bands remain at too high of latitudes, some solar-like aspects of hemispheric asymmetry are reproduced. The model reproduces the synchrony of polarity inversions and onset of cycle as the simulated phase lags do not exceed 20% of the cycle period. Simulated amplitude variations between the N and S hemispheres are larger than observed in the Sun. The simulations show one hemisphere persistently leads the other for several successive cycles, placing an upper bound on the efficiency of transequatorial magnetic coupling mechanisms. These include magnetic diffusion, cross-equatorial mixing within elongated convective rolls and transequatorial meridional flow cells. One or more of these processes may lead to magnetic flux cancellation whereby the oppositely directed fields come in close proximity and cancel each other across the magnetic equator late in the solar cycle. We discuss the discrepancies between model and observations and the constraints they pose on possible mechanisms of hemispheric coupling.

### Collective coordinate approximation to the scattering of solitons in modified NLS and sine-Gordon models

We use a collective coordinate approximation to model the scattering of two solitons in modified nonlinear Schr\"odinger and sine-Gordon systems. We find that the anomalies of the conservation laws of the charges as calculated using the collective coordinate approximation demonstrate the same dependence on the symmetry of the field configuration as that previously found analytically and using a full numerical simulation. This suggests that the collective coordinate approximation is a suitable method to investigate quasi-integrability in perturbed integrable models. We also discuss the general accuracy of this approximation by comparing our results with those of the full numerical simulations and find that the approximation is often remarkably accurate though less so when the models are a long way from the integrable case.

### The 3D MHD code GOEMHD3 for large-Reynolds-number astrophysical plasmas [Replacement]

The numerical simulation of turbulence and flows in almost ideal, large-Reynolds-number astrophysical plasmas motivates the implementation of almost conservative MHD computer codes. They should efficiently calculate, use highly parallelized schemes scaling well with large numbers of CPU cores, allows to obtain a high grid resolution over large simulation domains and which can easily be adapted to new computer architectures as well as to new initial and boundary conditions, allow modular extensions. The new massively parallel simulation code GOEMHD3 enables efficient and fast simulations of almost ideal, large-Reynolds-number astrophysical plasma flows, well resolved and on huge grids covering large domains. Its abilities are validated by major tests of ideal and weakly dissipative plasma phenomena. The high resolution ($2048^3$ grid points) simulation of a large part of the solar corona above an observed active region proved the excellent parallel scalability of the code using more than 30.000 processor cores.

### The 3D MHD code GOEMHD3 for large-Reynolds-number astrophysical plasmas

The numerical simulation of turbulence and flows in almost ideal, large-Reynolds-number astrophysical plasmas motivates the implementation of almost conservative MHD computer codes. They should efficiently calculate, use highly parallelized schemes scaling well with large numbers of CPU cores, allows to obtain a high grid resolution over large simulation domains and which can easily be adapted to new computer architectures as well as to new initial and boundary conditions, allow modular extensions. The new massively parallel simulation code GOEMHD3 enables efficient and fast simulations of almost ideal, large-Reynolds-number astrophysical plasma flows, well resolved and on huge grids covering large domains. Its abilities are validated by major tests of ideal and weakly dissipative plasma phenomena. The high resolution ($2048^3$ grid points) simulation of a large part of the solar corona above an observed active region proved the excellent parallel scalability of the code using more than 30.000 processor cores.

### Determination of the mass anomalous dimension for $N_f=12$ and $N_f=9$ SU($3$) gauge theories [Cross-Listing]

We show the numerical simulation result for the mass anomalous dimension of the SU($3$) gauge theory coupled to $N_f = 12$ fundamental fermions. We use two independent methods, namely the step scaling method and the hyperscaling method of the Dirac mode number, to determine the anomalous dimension in the vicinity of the infrared fixed point of the theory. We show the continuum extrapolations keeping the renormalized coupling constant as a reference in both analyses. Furthermore, some recent works seems to suggest the lower boundary of the conformal window of the SU($3$) gauge theory exists between $N_f=8$ and $10$. We also briefly report our new project, in which the numerical simulation of the SU($3$) gauge theory coupled to $N_f=9$ fundamental fermions has been performed.

### Three Einstein rings: explicit solution and numerical simulation

We investigated the effects of gravitational lensing for a system in which a lens is a point mass and a homogeneous disc with a central hole. In such system there is a variety of cases resulting in formation of one, two and three Einstein rings. We found an explicit solution and considered conditions for existence of the second Einstein ring arising on the disc. Numerical modelling of the images was made for various ratios of the central mass to the disc one and for various values of the disc surface density. We also analysed dependence of the magnification factor on a source position for such system. The result of our work can be used in search of astrophysical objects with a toroidal (ring) structure.