Spiral Magnetic Field Research Articles

Spiral magnetic fields and their role on accretion dynamics in the circumnuclear disk of Sagittarius A*: Insight from λ = 850 μm polarization imaging

Abstract We showcase a study on the physical properties of the circumnuclear disk (CND) surrounding the supermassive black hole (SMBH) Sgr A* of the Galactic Center, emphasizing the role of magnetic field ($\boldsymbol {B}$ field) with $0.50\,$pc spatial resolution. Based on the sensitive $\lambda = 850\, \mu$m polarization data taken with the JCMT SCUBA2/POL2 (James Clerk Maxwell Telescope Submillimetre Common-User Bolometer Array 2), we analyzed ancillary datasets: CS $J = 2$–1 emission taken with ALMA (Atacama Large Millimeter/submillimeter Array), continuum emissions taken at $\lambda = 6\,$cm and at $\lambda = 37\, \mu$m taken with the VLA (Very Large Array) and SOFIA (the Stratospheric Observatory for Infrared Astronomy telescope). The $\boldsymbol {B}$ field within the CND exhibits a coherent spiral pattern. Applying the model described by Wardle and Königl (1990, ApJ, 362, 12; the WK model) to the observed $\boldsymbol {B}$ field pattern, it favors gas-pressure-dominant models without dismissing a gas-and-$\boldsymbol {B}$ field comparable model, leading us to estimate the $\boldsymbol {B}$-field strength in the ionized cavity around Sgr A* as $0.24^{+0.06}_{-0.04}\,$mG. Analysis based on the WK model further allows us to derive representative $\boldsymbol {B}$-field strengths for the radial, azimuthal, and vertical components as $(B_r, B_\phi , B_z) = (0.4 \pm 0.1, -0.7 \pm 0.2, 0.2 \pm 0.05)\,$mG, respectively. A key finding is that the $|B_\phi |$ component is dominant over $B_r$ and $B_z$ components, consistent with the spiral morphology, indicating that the CND’s $\boldsymbol {B}$-field is predominantly toroidal, possibly shaped by accretion dynamics. Considering the turbulent pressure, estimated plasma $\beta$ values indicate that the effective gas pressure should surpass the magnetic pressure. Assessing the CND of our MWG in the toroidal-and-vertical stability parameter space, we propose that such an “effective” magneto-rotational instability (MRI) may likely be active. The estimated maximum unstable wavelength, $\lambda _{\rm max} = 0.1 \pm 0.1\,$pc, is smaller than the CND’s scale height ($0.2 \pm 0.1\,$pc), which indicates the potential for the effective MRI intermittent cycles of $\sim 10^6\,$yr, which should profoundly affect the CND’s evolution, considering the estimated mass accretion rate of $10^{-2} M_{\odot }\,$yr$^{-1}$ to the SMBH.

The Coherent Magnetic Field of the Milky Way

We present a suite of models of the coherent magnetic field of the Galaxy based on new divergence-free parametric functions describing the global structure of the field. The model parameters are fit to the latest full-sky Faraday rotation measures (RMs) of extragalactic sources and polarized synchrotron intensity (PI) maps from the Wilkinson Microwave Anisotropy Probe and Planck. We employ multiple models for the density of thermal and cosmic-ray electrons in the Galaxy, needed to predict the sky maps of RMs and PI for a given Galactic magnetic field (GMF) model. The robustness of the inferred properties of the GMF is gauged by studying many combinations of parametric field models and electron density models. We determine the pitch angle of the local magnetic field (11° ± 1°), explore the evidence for a grand-design spiral coherent magnetic field (inconclusive), determine the strength of the toroidal and poloidal magnetic halo fields below and above the disk (magnitudes the same for both hemispheres within ≈10%), set constraints on the half-height of the cosmic-ray diffusion volume (≥2.9 kpc), investigate the compatibility of RM- and PI-derived magnetic field strengths (compatible under certain assumptions), and check if the toroidal halo field could be created by the shear of the poloidal halo field due to the differential rotation of the Galaxy (possibly). A set of eight models is identified to help quantify the present uncertainties in the coherent GMF spanning different functional forms, data products, and auxiliary input. We present the corresponding sky maps of rates for axion–photon conversion in the Galaxy and deflections of ultrahigh-energy cosmic rays.

Numerical modeling of time dependent Diffusive Shock Acceleration

Motivated by cosmic ray (CR) re-acceleration at a potential Galactic Wind Termination Shock (GWTS), we present a numerical model for time-dependent Diffusive Shock Acceleration (DSA). We use the stochastic differential equation solver (DiffusionSDE) of the cosmic ray propagation framework CRPropa3.2 with two modifications: An importance sampling module is introduced to improve statistics at high energies in order to keep the simulation time short. An adaptive time step is implemented in the DiffusionSDE module. This ensures to efficiently meet constraints on the time and diffusion step, which is crucial to obtain the correct shock spectra. The time evolution of the spectrum at a one-dimensional planar shock is verified against the solution obtained by the grid-based solver VLUGR3 for both energy-independent and energy-dependent diffusion. We show that the injection of pre-accelerated particles can lead to a broken power law spectrum in momentum if the incoming spectrum of CRs is harder than the re-accelerated spectrum. If the injected spectrum is steeper, the shock spectrum dominates at all energies. We finally apply the developed model to the GWTS by considering a spherically symmetric shock, a spiral Galactic magnetic field, and anisotropic diffusion. The time-dependent spectrum at the shock is modeled as a basis for further studies.

Tomographic Imaging of the Sagittarius Spiral Arm's Magnetic Field Structure

The Galactic global magnetic field is thought to play a vital role in shaping Galactic structures such as spiral arms and giant molecular clouds. However, our knowledge of magnetic field structures in the Galactic plane at different distances is limited, as measurements used to map the magnetic field are the integrated effect along the line of sight. In this study, we present the first ever tomographic imaging of magnetic field structures in a Galactic spiral arm. Using optical stellar polarimetry over a 17′×10′ field of view, we probe the Sagittarius spiral arm. Combining these data with stellar distances from the Gaia mission, we can isolate the contributions of five individual clouds along the line of sight by analyzing the polarimetry data as a function of distance. The observed clouds include a foreground cloud (d < 200 pc) and four clouds in the Sagittarius arm at 1.23, 1.47, 1.63, and 2.23 kpc. The column densities of these clouds range from 0.5 to 2.8 × 1021 cm−2. The magnetic fields associated with each cloud show smooth spatial distributions within their observed regions on scales smaller than 10 pc and display distinct orientations. The position angles projected on the plane of the sky, measured from the Galactic north to the east, for the clouds in increasing order of distance are 135°, 46°, 58°, 150°, and 40°, with uncertainties of a few degrees. Notably, these position angles deviate significantly from the direction parallel to the Galactic plane.

North–South Plasma Asymmetry Across Mercury's Near‐Tail Current Sheet

AbstractAmong nearly 300 near‐Mercury tail current sheet crossings performed by the MESSENGER spacecraft, we identified 37 traversals of an asymmetric current sheet, wherein the lobe densities on opposite sides differ by a factor of three or more. These asymmetric current sheet crossings primarily occur on the dawnside. A global magnetohydrodynamic (MHD) simulation was found to be in excellent agreement with the observations. The results suggest that the north–south density asymmetry is caused by solar wind entering via an upstream‐connected window in one hemisphere. Furthermore, the Parker spiral interplanetary magnetic field (IMF) controls the near‐tail density asymmetries, whereas Mercury's offset dipole magnetic field controls those in mid‐ or distant‐tail regions. We propose that hemispheric asymmetries in Mercury's magnetospheric convection occur under strong IMF conditions.

MATHEMATICAL MODEL OF PLASMA TRANSFER IN A HELICAL MAGNETIC FIELD

The paper presents the results of mathematical modeling of plasma transfer in a spiral magnetic field using new experimental data obtained at the SMOLA trap created at the Budker Institute of Nuclear Physics SB RAS. Plasma confinement in the trap is carried out by transmitting a pulse from a magnetic field with helical symmetry to a rotating plasma. New mathematical model is based on a stationary plasma transfer equation in an axially symmetric formulation. The distribution of the concentration of the substance obtained by numerical simulation confirmed the confinement effect obtained in the experiment. The dependences of the integral characteristics of the substance on the depth of corrugation of the magnetic field, diffusion and plasma potential are obtained.

Study on Stirring Effect of Spiral Magnetic Field in Continuous Casting of Round Blooms

Herein, for the continuous casting of round blooms with strand electromagnetic stirring, the effects of rotating magnetic field (RMF) and spiral magnetic field (SMF) on the electromagnetic, fluid flow, heat transfer, solidification, and macrostructure are investigated by numerical simulation and industrial experiments, respectively. The results show that the molten steel flows in a spiral shape along the casting direction with the RMF and SMF. Due to the influence of secondary flow caused by electromagnetic stirring (EMS), and the zone of high velocity of molten steel accounts for 13.7% of the whole stirring range with the RMF. The SMF has a strong stirring effect in the whole range of secondary flow, the zone of high velocity of molten steel accounts for 49.1% of the whole stirring range, and the liquid fraction of the SMF at the liquid core is also lower. The results of the industrial experiments prove that the SMF has a wider stirring depth than the RMF, and the center crack range is significantly reduced.

Interstellar Polarization Survey. III. Relation between Optical Polarization and Reddening in the General Interstellar Medium

Optical starlight can be partially polarized while propagating through the dusty, magnetized interstellar medium (ISM). The polarization efficiency describes the polarization intensity fraction per reddening unit, P V /E(B − V), related to the interstellar dust grains and magnetic field properties. The maximum value observed, [P V /E(B − V)]max, is thus achieved under optimal polarizing conditions of the ISM. Therefore, the analysis of polarization efficiency observations across the Galaxy contributes to the study of magnetic field topology, small-scale magnetic fluctuations, grain-alignment efficiency, and composition. Infrared observations from Planck satellite have set [P V /E(B − V)]max to 13% mag−1. However, recent optical polarization observations in Planck's highly polarized regions showed polarization efficiency values between 13.6% mag−1 and 18.2% mag−1 (depending on the extinction map used), indicating that [P V /E(B − V)]max is not well constrained yet. We used V-band polarimetry of the Interstellar Polarization Survey (consisting of ∼10,500 high-quality observations distributed in 34 fields of 0.°3 × 0.°3) to accurately estimate the polarization efficiency in the ISM. We estimated the upper limit of P V /E(B − V) with the weighted 99th percentile of the field. In five regions, the polarization efficiency upper limit is above 13% mag−1. Furthermore, we found [P V /E(B − V)]% mag−1 using diffuse intermediate-latitude (∣b∣ > 7.°5) regions with apparently strong regular Galactic magnetic field in the plane-of-sky. We studied the variations of P V /E(B − V) across the sky and tested toy models of polarization efficiency with Galactic longitude that showed some correspondence with a uniform spiral magnetic field.

An Analytical Model of Turbulence in Parker Spiral Geometry and Associated Magnetic Field Line Lengths

Understanding the magnetic connections from the Sun to interplanetary space is crucial for linking in situ particle observations with the solar source regions of the particles. A simple connection along the large-scale Parker spiral magnetic field is made complex by the turbulent random walk of field lines. In this paper, we present the first analytical model of heliospheric magnetic fields where the dominant 2D component of the turbulence is transverse to the Parker spiral. The 2D wave field is supplemented with a minor wave field component that has asymptotic slab geometry at small and large heliocentric distances. We show that turbulence spreads field lines from a small source region at the Sun to a 60° heliolongitudinal and heliolatitudinal range at 1 au, with a standard deviation of the angular spread of the field lines of 14°. Small source regions map to an intermittent range of longitudes and latitudes at 1 au, consistent with dropouts in solar energetic particle intensities. The lengths of the field lines are significantly extended from the nominal Parker spiral length of 1.17 au up to 1.6 au, with field lines from sources at and behind the west limb considerably longer than those closer to the solar disk center. We discuss the implications of our findings for understanding charged particle propagation and the importance of understanding the turbulence properties close to the Sun.

Solar Energetic Particle Acceleration at a Spherical Shock with the Shock Normal Angle Evolving in Space and Time

We present a 2D kinematic model to study the acceleration of solar energetic particles (SEPs) at a shock driven by a coronal mass ejection. The shock is assumed to be spherical about an origin that is offset from the center of the Sun. This leads to a spatial and temporal evolution of the angle between the magnetic field and the shock-normal direction (θ Bn ) as it propagates through the Parker spiral magnetic field from the lower corona to 1 au. We find that the high-energy SEP intensity varies significantly along the shock front due to the evolution of θ Bn . Generally, the west flank of the shock preferentially accelerates particles to high energies compared to the east flank and shock nose. This can be understood in terms of the rate of acceleration, which is higher at the west flank. Double power-law energy spectra are reproduced in our model as a consequence of the local acceleration and transport effects. These results will help us to better understand the evolution of SEP acceleration and provide new insights into large SEP events observed by multiple spacecraft, especially those close to the Sun, such as Parker Solar Probe and Solar Orbiter.

Dense Gas Formation via Collision-induced Magnetic Reconnection in a Disk Galaxy with a Bisymmetric Spiral Magnetic Field

Recently, a collision-induced magnetic reconnection (CMR) mechanism was proposed to explain a dense filament formation in the Orion A giant molecular cloud. A natural question is whether CMR works elsewhere in the Galaxy. As an initial attempt to answer the question, this paper investigates the triggering of CMR and the production of dense gas in a flat-rotating disk with a modified Bisymmetric spiral (BSS) magnetic field. Cloud−cloud collisions at field reversals in the disk are modeled with the Athena++ code. Under the condition that is representative of the warm neutral medium, the cloud−cloud collision successfully triggers CMR at different disk radii. However, dense gas formation is hindered by the dominating thermal pressure, unless a moderately stronger initial field ≳5 μG is present. The strong-field model, having a larger Lundquist number S L and lower plasma β, activates the plasmoid instability in the collision midplane, which is otherwise suppressed by the disk rotation. We speculate that CMR can be common if more clouds collide along field reversals. However, to witness the CMR process in numerical simulations, we need to significantly resolve the collision midplane with a spatial dynamic range ≳106. If Milky Way spiral arms indeed coincide with field reversals in BSS, it is possible that CMR creates or maintains dense gas in the arms. High-resolution, high-sensitivity Zeeman/Faraday rotation observations are crucial for finding CMR candidates that have helical fields.

On the properties of Alfvénic switchbacks in the expanding solar wind: Three-dimensional numerical simulations

Switchbacks—abrupt reversals of the magnetic field within the solar wind—have been ubiquitously observed by Parker Solar Probe (PSP). Their origin, whether from processes near the solar surface or within the solar wind itself, remains under debate and likely has key implications for solar wind heating and acceleration. Here, using three-dimensional expanding box simulations, we examine the properties of switchbacks arising from the evolution of outwards-propagating Alfvén waves in the expanding solar wind in detail. Our goal is to provide testable predictions that can be used to differentiate between properties arising from solar surface processes and those from the in situ evolution of Alfvén waves in switchback observations by PSP. We show how the inclusion of the Parker spiral causes magnetic field deflections within switchbacks to become asymmetric, preferentially deflecting in the plane of the Parker spiral and rotating in one direction toward the radial component of the mean field. The direction of the peak of the magnetic field distribution is also shown to be different from the mean field direction due to its highly skewed nature. Compressible properties of switchbacks are also explored, with magnetic-field-strength and density fluctuations being either correlated or anticorrelated depending on the value of β, agreeing with predictions from theory. We also measure dropouts in magnetic-field strength and density spikes at the boundaries of these synthetic switchbacks, both of which have been observed by PSP. The agreement of these properties with observations provides further support for the Alfvén wave model of switchbacks.

The Kinetic Expansion of Solar-wind Electrons: Transport Theory and Predictions for the Very Inner Heliosphere

Abstract We propose a transport theory for the kinetic evolution of solar-wind electrons in the heliosphere. We derive a gyro-averaged kinetic transport equation that accounts for the spherical expansion of the solar wind and the geometry of the Parker spiral magnetic field. To solve our three-dimensional kinetic equation, we develop a mathematical approach that combines the Crank–Nicolson scheme in velocity space and a finite-difference Euler scheme in configuration space. We initialize our model with isotropic electron distribution functions and calculate the kinetic expansion at heliocentric distances from 5 to 20 solar radii. In our kinetic model, the electrons evolve mainly through the combination of ballistic particle streaming, the magnetic mirror force, and the electric field. By applying fits to our numerical results, we quantify the parameters of the electron strahl and the core part of the electron velocity distributions. The strahl fit parameters show that the density of the electron strahl is around 7% of the total electron density at a distance of 20 solar radii, the strahl bulk velocity and strahl temperature parallel to the background magnetic field stay approximately constant beyond a distance of 15 solar radii, and β ∥s (i.e., the ratio of the strahl parallel thermal pressure to the magnetic pressure) is approximately constant with heliocentric distance at a value of about 0.02. We compare our results with data measured by the Parker Solar Probe. Furthermore, we provide theoretical evidence that the electron strahl is not scattered by the oblique fast-magnetosonic/whistler instability in the near-Sun environment.

Small-scale Magnetic Flux Ropes and Their Properties Based on In Situ Measurements from the Parker Solar Probe

Abstract We report small-scale magnetic flux ropes via the in situ measurements from the Parker Solar Probe during the first six encounters, and present additional analyses to supplement our prior work in Chen et al. These flux ropes are detected by the Grad–Shafranov-based algorithm, with their durations and scale sizes ranging from 10 s to ≲1 hr and from a few hundred kilometers to 10−3 au, respectively. They include both static structures and those with significant field-aligned plasma flows. Most structures tend to possess large cross helicity, while the residual energy is distributed over wide ranges. We find that these dynamic flux ropes mostly propagate in the antisunward direction relative to the background solar wind, with no preferential signs of magnetic helicity. The magnetic flux function follows a power law and is proportional to scale size. We also present case studies showing reconstructed two-dimensional (2D) configurations, which confirm that both the static and dynamic flux ropes have a common configuration of spiral magnetic field lines (also streamlines). Moreover, the existence of such events hints at interchange reconnection as a possible mechanism for generating flux rope-like structures near the Sun. Lastly, we summarize the major findings, and discuss the possible correlation between these flux rope-like structures and turbulence due to the process of local Alfvénic alignment.

Did the Event Horizon Telescope Detect the Base of the Submilliarsecond Tubular Jet in M87?

Abstract A high-sensitivity, 7 mm Very Long Baseline Array image of M 87 was previously analyzed in order to estimate the bulk flow jet velocity between 0.4 and 0.65 mas from the point of origin using the asymmetry between the well-characterized double-ridged counter-jet (unique to this image) and the double-ridged jet. We use this same image to estimate the cross-sectional area of this tubular stream. The velocity, acceleration, cross-sectional area, and flux density along this stream determine a unique, perfect magnetohydrodynamic jet solution that satisfies conservation of energy, angular momentum, and mass (a monotonic conversion of Poynting flux to kinetic energy flux along the jet). The solution is protonic and magnetically dominated. The bilateral jet transports ≈1.2 × 10−4 M ⊙ yr−1 and ≈1.1 × 1042 erg s−1, placing strong constraints on the central engine. A Keplerian disk source that also produces the Event Horizon Telescope (EHT) annulus of emission can supply the energy and mass if the vertical magnetic field at the equator is ∼1–3.5 G (depending on location). A Parker spiral magnetic field, characteristic of a wind or jet, is consistent with the observed EHT polarization pattern. Even though there is no image of the jet connecting with the annulus, it is argued that these circumstances are not coincidental and the polarized portion of the EHT emission is mainly jet emission in the top layers of the disk that is diluted by emission from an underlying turbulent disk. This is a contributing factor to the relatively low polarization levels that were detected.

Effect of solar wind density and velocity on the subsolar standoff distance of the Martian magnetic pileup boundary

Using a 3D multispecies magnetohydrodynamic model, we investigated the effect of the solar wind dynamic pressure (Pd) with different densities and velocities on the subsolar standoff distance (r0) of the Martian magnetic pileup boundary (MPB). We fixed the solar maximum condition, the strongest crustal field located in the dayside region, and the Parker spiral interplanetary magnetic field at Mars. We simulated 35 cases with a Pd range of 0.1494 to 7.323 nPa (solar wind number density n ∈ [1, 9] cm−3, and solar wind velocity V ∈ [−258, −1344] km s−1). The main results are as follows. (1) r0 decreases with increasing Pd according to the power-law relations. For the same Pd, a higher solar wind velocity (lower density) results in a larger r0 of the Martian MPB. (2) A higher solar wind density leads to a lower ratio of the compressed magnetic field strength to the crustal field strength and a larger plasma β under the same Pd. This indicates that the thermal pressure at the Martian MPB plays a significant role for the compressed magnetic field. Because the magnetic pileup process is stronger for a higher solar wind velocity, the magnetic pressure at the Martian MPB is increased. As a result, the thermal pressure decreases and r0 of the Martian MPB increases. (3) We present a new formula of r0 with the parameters of the solar wind dynamic pressure, number density, and velocity.

Constraining Suprathermal Electron Evolution in a Parker Spiral Field With Cassini Observations

AbstractSuprathermal electrons in the solar wind consist of the “halo,” present at all pitch angles, and the “strahl” which is a field‐aligned, beam‐like population. Examining the heliospheric evolution of strahl beams is key to understanding the in‐transit processing of solar wind suprathermal electrons, in particular, to identify electron scattering mechanisms and to establish the origin of the halo population. Not only does this have significant implications with regard to the kinetic processes occurring within the solar wind but also its thermodynamic evolution, as the suprathermal electrons carry the majority of the solar wind heat flux. In this investigation, an established model for suprathermal electron evolution in a Parker spiral interplanetary magnetic field is adapted from its original use. The model is constrained using solar wind strahl observed by the Cassini mission on its interplanetary journey to Saturn. The effects of large scale IMF geometry due to different solar wind velocities and application of different electron scattering factors are examined. It is found that slow solar wind speeds provide the closest match to the strahl width observations, both in terms of radial distance and electron energy trends, and that predominantly slower solar wind speeds were therefore likely observed by the Cassini mission en‐route to Saturn. It is necessary to include a strahl scattering factor which increases with electron energy in order to match observations, indicating that the strahl scattering mechanism must have an inherent energy dependence.

Time evolution of stream interaction region energetic particle spectra in the inner heliosphere

We analyze an energetic proton event associated with a stream interaction region (SIR) that was observed at Parker Solar Probe on day 320 of 2018 when the spacecraft was just 0.34 AU from the Sun. Using the Integrated Science Investigation of the Sun instrument suite, we perform a spectral analysis of the event and show how the observed spectra evolve over the course of the event. We find that the spectra from the first day of the event are much more consistent with local acceleration at a weak compression, while spectra from later on are more typical of SIR-related events in which particles accelerated at distant shocks dominate. After the first day, the spectra remain approximately constant, which indicates that the modulation of energetic particles during transit from the presumed source region is weaker than previously thought. We argue that these observations can be explained by a sub-Parker spiral magnetic field structure connecting the spacecraft to a source region in the SIR that is relatively close to the Sun. We further propose that acceleration at weak, pre-shock compressions likely plays an important role in observations of SIR-related events in the inner heliosphere and that future modelling of such events should consider acceleration all along the compression region, not just at the distant shock region.

The magnetic field across the molecular warped disk of Centaurus A

Magnetic fields are amplified as a consequence of galaxy formation and turbulence-driven dynamos. Galaxy mergers can potentially amplify the magnetic fields from their progenitors, making the magnetic fields dynamically important. However, the effect of mergers on magnetic fields is still poorly understood. We use thermal polarized emission observations to trace the magnetic fields in the molecular disk of the nearest radio active galaxy, Centaurus A, which is thought to be the remnant of a merger. Here, we detect that the magnetic field orientations in the plane of the sky are tightly following the $\sim3.0$ kpc-scale molecular warped disk. Our simple regular large-scale axisymmetric spiral magnetic field model can explain, to some extent, the averaged magnetic field orientations across the disk projected on the sky. Our observations also suggest the presence of small-scale turbulent fields, whose relative strength increases with velocity dispersion and column density. These results have strong implications for understanding the generation and role of magnetic fields in the formation of galaxies across cosmic time.

Magnetic convection in a nonuniformly rotating electrically conductive medium in an external spiral magnetic field

The research is devoted to the stability of convective flow in a nonuniformly rotating layer of an electrically conducting fluid in a spiral magnetic field. The stationary and oscillatory modes of magnetic convection are considered depending on the profile of the angular rotation velocity (Rossby number Ro) and on the profile of the external azimuthal magnetic field (magnetic Rossby number Rb). The non-linear dynamic system of Lorentz type equations is obtained by using the Galerkin method. Numerical analysis of these equations has shown the presence of chaotic behavior of convective flows. The criteria of the occurrence of chaotic movements are found. It depends on the parameters of convection: dimensionless numbers of Rayleigh Ra, Chandrasekhar Q, Taylor Ta, and external azimuthal magnetic field with the Rossby magnetic number Rb for Rayleigh (Ro = −1) and Kepler profiles of the angular rotation velocity of the medium.