Spherical Wedge Research Articles

Formation of Polar Crown Filament Magnetic Fields by Supergranular Helicity Injection

To understand the magnetic fields of the polar crown filaments (PCFs) at high latitudes near polar regions of the Sun, we perform magnetofrictional numerical simulations on the long-term magnetic evolution of bipolar fields with roughly east–west polarity inversion lines (PILs) in a 3D spherical wedge domain near polar regions. The Coriolis-effect-induced vortical motions at the boundaries of several supergranular cells inject magnetic helicity from the photospheric boundary into the solar atmosphere. Supergranular-scale helicity injection, transfer, and condensation produce strongly sheared magnetic fields. Magnetic reconnections at footpoints of the sheared fields produce magnetic flux ropes (MFRs) with helicity signs consistent with the observed hemispheric helicity rule. The cross-sectional area of MFRs exhibits an uneven distribution, resembling a “foot-node-foot” periodic configuration. Experiments with different tilt directions of PILs indicate that the PCFs preferably form along PILs with the western end close to the polar region. The bending of PILs caused by supergranular flows, forming S-shape (Z-shape) PIL segments, promotes the formation of dextral (sinistral) MFRs. The realistic magnetic models we obtained can serve as starting points for the study of the plasma formation and eruption of PCFs.

Effects of the centrifugal force in stellar dynamo simulations

Context.The centrifugal force is often omitted from simulations of stellar convection either for numerical reasons or because it is assumed to be weak compared to the gravitational force. However, the centrifugal force might be an important factor in rapidly rotating stars, such as solar analogs, due to its Ω2scaling, where Ω is the rotation rate of the star.Aims.We study the effects of the centrifugal force in a set of 21 semi-global stellar dynamo simulations with varying rotation rates. Included in the set are three control runs aimed at distinguishing the effects of the centrifugal force from the nonlinear evolution of the solutions.Methods.We solved the 3D magnetohydrodynamic equations with the PENCILCODEin a solar-like convective zone in a spherical wedge setup with a 2πazimuthal extent. The rotation rate and the amplitude of the centrifugal force were varied. We decomposed the magnetic field into spherical harmonics and studied the migration of azimuthal dynamo waves (ADWs), the energy of different large-scale magnetic modes, and differential rotation.Results.In the regime with the lowest rotation rates, Ω = 5 − 10 Ω⊙, where Ω⊙is the rotation rate of the Sun, we see no marked changes in either the differential rotation or the magnetic field properties. For intermediate rotation, Ω = 20 − 25 Ω⊙, we identify an increase in the differential rotation as a function of centrifugal force. The axisymmetric magnetic energy tends to decrease with centrifugal force, while the non-axisymmetric one increases. The ADWs are also affected, especially in the propagation direction. In the most rapidly rotating set with Ω = 30 Ω⊙, these changes are more pronounced, and in one case the propagation direction of the ADW changes from prograde to retrograde. The control runs suggest that the results are a consequence of the centrifugal force and not due to the details of the initial conditions or the history of the run.Conclusions.We find that the differential rotation and properties of the ADWs only change as a function of the centrifugal force when rotation is rapid enough.

The geometry of the hot corona in MCG-05-23-16 constrained by X-ray polarimetry

ABSTRACT We report on the second observation of the radio-quiet active galactic nucleus MCG-05-23-16 performed with the Imaging X-ray Polarimetry Explorer (IXPE). The observation started on 2022 November 6 for a net observing time of 640 ks, and was partly simultaneous with NuSTAR (86 ks). After combining these data with those obtained in the first IXPE pointing on 2022 May (simultaneous with XMM–Newton and NuSTAR) we find a 2–8 keV polarization degree Π = 1.6 ± 0.7 (at 68 per cent confidence level), which corresponds to an upper limit Π = 3.2 per cent (at 99 per cent confidence level). We then compare the polarization results with Monte Carlo simulations obtained with the monk code, with which different coronal geometries have been explored (spherical lamppost, conical, slab, and wedge). Furthermore, the allowed range of inclination angles is found for each geometry. If the best-fitting inclination value from a spectroscopic analysis is considered, a cone-shaped corona along the disc axis is disfavoured.

Tempestuous life beyond R500: X-ray view on the Coma cluster with SRG/eROSITA

This is the second paper in a series of studies of the Coma cluster using the SRG/eROSITA X-ray data obtained during the calibration and performance verification phase of the mission. Here, we focus on the region adjacent to the radio source 1253+275 (radio relic, RR, hereafter). We show that the X-ray surface brightness exhibits its steepest gradient at ∼79′ (∼2.2 Mpc ≈ R200c), which is almost co-spatial to the outer edge of the RR. As in the case of several other relics, the Mach number of the shock derived from the X-ray surface brightness profile (MX ≈ 1.9) appears to be lower than needed to explain the slope of the integrated radio spectrum in the diffusive shock acceleration model (MR ≈ 3.5) if the magnetic field is uniform and the radiative losses are fast. However, the shock geometry is plausibly much more complicated than a spherical wedge centered on the cluster, given the non-trivial correlation between radio, X-ray, and SZ images. While the complicated shock geometry alone might cause a negative bias in MX, we speculate on a few other possibilities that may affect the MX − MR relation, including the shock substructure that might be modified by the presence of non-thermal filaments stretching across the shock and the propagation of relativistic electrons along the non-thermal filaments with a strong magnetic field. We also discuss the “history” of the radio galaxy NGC 4789, which is located ahead of the relic in the context of the Coma-NGC 4839 merger scenario.

Time-dependent boundary conditions for data-driven coronal global and spherical wedge-shaped models

ABSTRACT The development of an efficient and accurate method for boundary condition treatments is of fundamental importance to data-driven magnetohydrodynamic (MHD) modelling of the global solar corona and solar active region. Particularly, in a 3D spherical wedge-shaped volume, suitable to the numerical study of solar active region, the transverse terms calls for a delicate treatment at the computational domain’s edges and corners, and properly prescribed conditions for boundaries joining regions of different flow properties, so as to take account of the joint effect of incoming and outgoing waves. To provide a solution to the determination of boundary conditions, in this paper a systematic tactics is formulated for handling edges and corners and the prescribed conditions for inner/outer/edge/corner boundaries are proposed through the combination (CBC-ILW) of the time-dependent characteristic boundary conditions (CBCs) and the inverse Lax-Wendroff (ILW) procedure. First, a data-driven 3D MHD simulation has been carried out to study the dynamic evolution of the solar corona from 1Rs to 6.7Rs during the period between 2018 May 16 and August 6. The simulated results of the global coronal evolution provide a good comparison with observed coronal images during the period investigated. Then, the validity of 3D MHD-CBC-ILW is verified for a 3D spherical wedge model, by producing almost the same results as those taken out of the global model on a 3D spherical wedge-shaped volume.

Transition from anti-solar to solar-like differential rotation: Dependence on Prandtl number

Context. Late-type stars such as the Sun rotate differentially due to the interaction of turbulent convection and rotation. Aims. The aim of the study is to investigate the effects of the effective thermal Prandtl number, which is the ratio of kinematic viscosity to thermal diffusivity, on the transition from anti-solar (slow equator, fast poles) to solar-like (fast equator, slow poles) differential rotation. Methods. Three-dimensional hydrodynamic and magnetohydrodynamic simulations in semi-global spherical wedge geometry were used to model the convection zones of solar-like stars. Results. The overall convective velocity amplitude increases as the Prandtl number decreases, in accordance with earlier studies. The transition from anti-solar to solar-like differential rotation is insensitive to the Prandtl number for Prandtl numbers below unity, but for Prandtl numbers greater than unity, solar-like differential rotation becomes significantly harder to excite. Magnetic fields and more turbulent regimes with higher fluid and magnetic Reynolds numbers help to achieve solar-like differential rotation in near-transition cases where anti-solar rotation is found in more laminar simulations. Solar-like differential rotation occurs only in cases with radially outward turbulent angular momentum transport due to the Reynolds stress at the equator. The dominant contribution to this outward transport near the equator is due to prograde propagating thermal Rossby waves. Conclusions. The differential rotation is sensitive to the Prandtl number only for large Prandtl numbers in the parameter regime explored in this study. Magnetic fields have a greater effect on the differential rotation, although the inferred presence of a small-scale dynamo did not lead to drastically different results. The dominance of the thermal Rossby waves in the simulations is puzzling because they are not detected in the Sun. The current simulations are shown to be incompatible with the currently prevailing mean-field theory of differential rotation.

Experimental verification of analytical calculation approaches and FEM material models with the aim of determining friction of thermoplastics

The paper presents analytical approaches for calculating the effective contact area of the sphere-plane contact, which allow conclusions to be drawn about the coefficient of friction of thermoplastics with manageable effort. These approaches are verified experimentally utilizing friction and wear tests using the example of a steel sphere against PE-UHMW. The friction area of the sphere was varied using a self-constructed adjustment unit and a spherical wedge. With the help of parametric experiments regarding the angular position of the spherical wedge, a limiting contact angle was detected. This limiting angle allows the calculation of the deformative and adhesive friction. FE models are being developed for the simulative verification of the analytical approaches and further investigations of structures for friction reduction. These require specific material models as a basis for the representation of the stress-strain behavior in contact. The different FEM material models are calibrated and compared against each other based on experimental tests. The materials POM, PP, PMMA, and PE-UHMW were considered.

Experimental verification of analytical calculation approaches and FEM material models with the aim of determining friction of thermoplastics

The paper presents analytical approaches for calculating the effective contact area of the sphere-plane contact, which allow conclusions to be drawn about the coefficient of friction of thermoplastics with manageable effort. These approaches are verified experimentally utilizing friction and wear tests using the example of a steel sphere against PE-UHMW. The friction area of the sphere was varied using a self-constructed adjustment unit and a spherical wedge. With the help of parametric experiments regarding the angular position of the spherical wedge, a limiting contact angle was detected. This limiting angle allows the calculation of the deformative and adhesive friction. FE models are being developed for the simulative verification of the analytical approaches and further investigations of structures for friction reduction. These require specific material models as a basis for the representation of the stress-strain behavior in contact. The different FEM material models are calibrated and compared against each other based on experimental tests. The materials POM, PP, PMMA, and PE-UHMW were considered.

Spherical wedge billiard: From chaos to fractals and Talbot carpets.

We introduce the spherical wedge billiard, a dynamical system consisting of a particle moving along a geodesic on a closed non-Euclidean surface of a spherical wedge. We derive the analytic form of the corresponding Poincaré map and find very complex dynamics, ranging from completely chaotic to very regular, exhibiting fractal features. Further, we show that upon changing the billiard parameter, the fixed points of the Poincaré map merge in complex ways, which has origin in the spherical aberration of the billiard mapping. We also analyze in detail the regular regime when phase space diagram is closely related to Talbot carpets.

Mass transfer analysis of bioactive compounds in apple wedges impregnated with beetroot juice: A 3D modelling approach

The simultaneous mass transfer of bioactive compounds, water loss and solids gain occurring during osmotic dehydration of apple wedges with beetroot juice was investigated. Beetroot juice was adjusted to 50°Bx with sucrose and further used as osmotic medium at 30, 40 and 50 °C. Impregnation of bioactive compounds was described in terms of total antioxidant capacity (TAC), total phenolic compounds (TPC) and total betalains (betacyanins and betaxanthins) contents. An unsteady state 3D-diffusion model, where product is shaped as a prolate spheroidal wedge, was proposed for describing experimental data and for obtaining equilibrium points and diffusion coefficients of migrating species with and without considering the apple shrinkage. The effect of geometry mismatch on the estimation of diffusion coefficients when product is shaped as spherical wedges is also explored. Results indicated that after 480 min apple cuts impregnated with beetroot juice increased their TPC in about 7–20 mg GAE/100 g from 43 ± 2 mg GAE/100 g in fresh product. On the other hand, TAC was increased in about 45–123 mg Trolox/100 g from 57 ± 0.8 mg Trolox/100 g in fresh product for the same processing time, being betalains the main responsible for this change. Shrinkage had a larger effect on the estimation of diffusion coefficients than geometry mismatch under the explored experimental conditions. Diffusion coefficients for bioactive compounds corrected for apple shrinkage were in the range of 0.49–6.05 × 10−11 m2/s. The use of higher impregnation temperatures caused a significant increase in equilibrium points of TPC and TAC (p < 0.05).

Effects of a subadiabatic layer on convection and dynamos in spherical wedge simulations

ABSTRACTWe consider the effect of a subadiabatic layer at the base of the convection zone on convection itself and the associated large-scale dynamos in spherical wedge geometry. We use a heat conduction prescription based on the Kramers opacity law which allows the depth of the convection zone to dynamically adapt to changes in the physical characteristics such as rotation rate and magnetic fields. We find that the convective heat transport is strongly concentrated towards the equatorial and polar regions in the cases without a substantial radiative layer below the convection zone. The presence of a stable layer below the convection zone significantly reduces the anisotropy of radial enthalpy transport. Furthermore, the dynamo solutions are sensitive to subtle changes in the convection zone structure. We find that the kinetic helicity changes sign in the deeper parts of the convection zone at high latitudes in all runs. This region expands progressively towards the equator in runs with a thicker stably stratified layer.

Transition from axi- to nonaxisymmetric dynamo modes in spherical convection models of solar-like stars

Context. Both dynamo theory and observations of stellar large-scale magnetic fields suggest a change from nearly axisymmetric configurations at solar rotation rates to nonaxisymmetric configurations for rapid rotation. Aims. We seek to understand this transition using numerical simulations. Methods. We use three-dimensional simulations of turbulent magnetohydrodynamic convection in spherical shell wedges and considered rotation rates between 1 and 31 times the solar value. Results. We find a transition from axi- to nonaxisymmetric solutions at around 1.8 times the solar rotation rate. This transition coincides with a change in the rotation profile from antisolar- to solar-like differential rotation with a faster equator and slow poles. In the solar-like rotation regime, the field configuration consists of an axisymmetric oscillatory field accompanied by an m = 1 azimuthal mode (two active longitudes), which also shows temporal variability. At slow (rapid) rotation, the axisymmetric (nonaxisymmetric) mode dominates. The axisymmetric mode produces latitudinal dynamo waves with polarity reversals, while the nonaxisymmetric mode often exhibits a slow drift in the rotating reference frame and the strength of the active longitudes changes cyclically over time between the different hemispheres. In the majority of cases we find retrograde waves, while prograde waves are more often found from observations. Most of the obtained dynamo solutions exhibit cyclic variability either caused by latitudinal or azimuthal dynamo waves. In an activity-period diagram, the cycle lengths normalized by the rotation period form two different populations as a function of rotation rate or magnetic activity level. The slowly rotating axisymmetric population lies close to what in observations is called the inactive branch, where the stars are believed to have solar-like differential rotation, while the rapidly rotating models are close to the superactive branch with a declining cycle to rotation frequency ratio and an increasing rotation rate. Conclusions. We can successfully reproduce the transition from axi- to nonaxisymmetric dynamo solutions for high rotation rates, but high-resolution simulations are required to limit the effect of rotational quenching of convection at rotation rates above 20 times the solar value.

Convection-driven spherical shell dynamos at varying Prandtl numbers

(abidged) Context: Stellar convection zones are characterized by vigorous high-Reynolds number turbulence at low Prandtl numbers. Aims: We study the dynamo and differential rotation regimes at varying levels of viscous, thermal, and magnetic diffusion. Methods: We perform three-dimensional simulations of stratified fully compressible magnetohydrodynamic convection in rotating spherical wedges at various thermal and magnetic Prandtl numbers (from 0.25 to 2 and 5, respectively). Results: We find that the rotation profiles for high thermal diffusivity show a monotonically increasing angular velocity from the bottom of the convection zone to the top and from the poles toward the equator. For sufficiently rapid rotation, a region of negative radial shear develops at mid-latitudes as the thermal diffusivity is decreased. This coincides with a change in the dynamo mode from poleward propagating activity belts to equatorward propagating ones. Furthermore, the cyclic solutions disappear at the highest magnetic Reynolds numbers. The total magnetic energy increases with the magnetic Reynolds number in the range studied here ($5-151$), but the energies of the mean magnetic fields level off at high magnetic Reynolds numbers. The differential rotation is strongly affected by the magnetic fields and almost vanishes at the highest magnetic Reynolds numbers. In some of our most turbulent cases we find that two regimes are possible where either differential rotation is strong and mean magnetic fields relatively weak or vice versa. Conclusions: Our simulations indicate a strong non-linear feedback of magnetic fields on differential rotation, leading to qualitative changes in the behaviors of large-scale dynamos at high magnetic Reynolds numbers. Furthermore, we do not find indications of the simulations approaching an asymptotic regime where the results would be independent of diffusion coefficients.

Influence of a coronal envelope as a free boundary to global convective dynamo simulations

We explore the effects of an outer stably stratified coronal envelope on rotating turbulent convection, differential rotation, and large-scale dynamo action in spherical wedge models of the Sun. We solve the compressible magnetohydrodynamic equations in a two-layer model with unstable stratification below the surface, representing the convection zone, and a stably stratified coronal envelope above. The interface represents a free surface. We compare our model to models that have no coronal envelope. The presence of a coronal envelope is found to modify the Reynolds stress and the $\Lambda$ effect resulting in a weaker and non-cylindrical differential rotation. This is related to the reduced latitudinal temperature variations that are caused by and dependent on the angular velocity. Some simulations develop a near-surface shear layer that we can relate to a sign change in the meridional Reynolds stress term in the thermal wind balance equation. Furthermore, the presence of a free surface changes the magnetic field evolution since the toroidal field is concentrated closer to the surface. In all simulations, however, the migration direction of the mean magnetic field can be explained by the Parker--Yoshimura rule, which is consistent with earlier findings. A realistic treatment of the upper boundary in spherical dynamo simulations is crucial for the dynamics of the flow and magnetic field evolution.

Robustness of oscillatoryα2dynamos in spherical wedges

We study the connection between spherical wedge and full spherical shell geometries using simple mean-field $\alpha^2$ dynamos. We solve the equations for a one-dimensional time-dependent mean-field dynamo to examine the effects of varying the polar angle $\theta_0$ between the latitudinal boundaries and the poles in spherical coordinates. We investigate the effects of turbulent magnetic diffusivity and $\alpha$ effect profiles as well as different latitudinal boundary conditions to isolate parameter regimes where oscillatory solutions are found. Finally, we add shear along with a damping term mimicking radial gradients to study the resulting dynamo regimes. We find that the commonly used perfect conductor boundary condition leads to oscillatory $\alpha^2$ dynamo solutions only if the wedge boundary is at least one degree away from the poles. Other boundary conditions always produce stationary solutions. By varying the profile of the turbulent magnetic diffusivity alone, oscillatory solutions are achieved with models extending to the poles, but the magnetic field is strongly concentrated near the poles and the oscillation period is very long. By introducing radial shear and a damping term mimicking radial gradients, we again see oscillatory dynamos, and the direction of drift follows the Parker--Yoshimura rule. Oscillatory solutions in the weak shear regime are found only in the wedge case with $\theta_0 = 1^\circ$ and perfect conductor boundaries. A reduced $\alpha$ effect near the poles with a turbulent diffusivity concentrated toward the equator yields oscillatory dynamos with equatorward migration and reproduces best the solutions in spherical wedges.

Characterization of the compressive deformation behavior with strain rate effect of low-density polymeric foams

This study investigates the compressive deformation behavior of a low-density polymeric foam at different strain rates. The material tested has micron-sized pores with a closed cell structure. The porosity is about 94%. During a uni-axial compressive test, the macroscopic stress–strain curve indicates a plateau region during plastic deformation. Finite Element Method (FEM) simulation was carried out, in which the yield criterion considered both components of Mises stress and hydrostatic stress. By using the present FEM and experimental data, we established a computational model for the plastic deformation behavior of porous material. To verify our model, several indentation experiments with different indenters (spherical indentation and wedge indentation) were carried out to generate various tri-axial stress states. From the series of experiments and computations, we observed good agreement between the experimental data and that generated by the computational model. In addition, the strain rate effect is examined for a more reliable prediction of plastic deformation. Therefore, the present computational model can predict the plastic deformation behavior (including time-dependent properties) of porous material subjected to uni-axial compression and indentation loadings.

BIPOLAR MAGNETIC SPOTS FROM DYNAMOS IN STRATIFIED SPHERICAL SHELL TURBULENCE

Recent work by Mitra et al. (2014) has shown that in strongly stratified forced two-layer turbulence with helicity and corresponding large-scale dynamo action in the lower layer, and nonhelical turbulence in the upper, a magnetic field occurs in the upper layer in the form of sharply bounded bipolar magnetic spots. Here we extend this model to spherical wedge geometry covering the northern hemisphere up to 75° latitude and an azimuthal extent of 180°. The kinetic helicity and therefore also the large-scale magnetic field are strongest at low latitudes. For moderately strong stratification, several bipolar spots form that eventually fill the full longitudinal extent. At early times, the polarity of spots reflects the orientation of the underlying azimuthal field, as expected from Parker's Ω-shaped flux loops. At late times their tilt changes such that there is a radial field of opposite orientation at different latitudes separated by about 10°. Our model demonstrates the spontaneous formation of spots of sizes much larger than the pressure scale height. Their tendency to produce filling factors close to unity is argued to be reminiscent of highly active stars. We confirm that strong stratification and strong scale separation are essential ingredients behind magnetic spot formation, which appears to be associated with downflows at larger depths.

Optical schemes of spectrographs with a diffractive optical element in a converging beam

Optical schemes of spectrographs based on transmission concave holographic gratings working in converging beams are considered. General description of the design techniques are provided. Each of them is supported by a certain example with calculation and modeling results. In particular, it’s shown that combination of such element with a spherical wedge allows to create a spectrograph with correction of astigmatism and a variable-dispersion spectrograph.

ON THE CAUSE OF SOLAR-LIKE EQUATORWARD MIGRATION IN GLOBAL CONVECTIVE DYNAMO SIMULATIONS

We present results from four convectively-driven stellar dynamo simulations in spherical wedge geometry. All of these simulations produce cyclic and migrating mean magnetic fields. Through detailed comparisons we show that the migration direction can be explained by an $\alpha\Omega$ dynamo wave following the Parker--Yoshimura rule. We conclude that the equatorward migration in this and previous work is due to a positive (negative) $\alpha$ effect in the northern (southern) hemisphere and a negative radial gradient of $\Omega$ outside the inner tangent cylinder of these models. This idea is supported by a strong correlation between negative radial shear and toroidal field strength in the region of equatorward propagation.

Confirmation of bistable stellar differential rotation profiles

(abridged) Context: Solar-like differential rotation is characterized by a rapidly rotating equator and slower poles. However, theoretical models and numerical simulations can result in a slower equator and faster poles when the rotation is slow. Aims: We study the critical rotational influence under which differential rotation flips from solar-like (fast equator, slow poles) to an anti-solar one (slow equator, fast poles). We estimate the non-diffusive ($\Lambda$ effect) and diffusive (turbulent viscosity) contributions to the Reynolds stress. Methods: We present the results of three-dimensional numerical simulations of mildly turbulent convection in spherical wedge geometry. Here we apply a fully compressible setup which would suffer from a prohibitive time step constraint if the real solar luminosity was used. We regulate the convective velocities by varying the amount of heat transported by thermal conduction, turbulent diffusion, and resolved convection. Results: Increasing the efficiency of resolved convection leads to a reduction of the rotational influence on the flow and a sharp transition from solar-like to anti-solar differential rotation for Coriolis numbers around 1.3. We confirm the recent finding of a large-scale flow bistability: contrasted with running the models from an initial condition with unprescribed differential rotation, the initialization of the model with certain kind of rotation profile sustains the solution over a wider parameter range. Conclusions: Our results may have implications for real stars that start their lives as rapid rotators implying solar-like rotation in the early main-sequence evolution. As they slow down, they might be able to retain solar-like rotation for lower Coriolis numbers before switching to anti-solar rotation. This could partially explain the puzzling findings of anti-solar rotation profiles for models in the solar parameter regime.