Modelling the magnetic field and differential rotation effects for the vicinity of the base of convective zone in the Sun

In this work, some numerical solutions of magnetohydrodynamics equations are investigated in the presence of differential rotation with the use of previously developed algorithm. This algorithm includes the thin shell approximation and a special separation of variables which were used to obtain the radial and latitudinal variations of physical parameters in spherical coordinates. The magnetic field profile is chosen to produce comparable magnetic fluxes found in previous works. The sphericity and density shape parameters relevant to model is determined by using two different known differential rotation profiles. It is found that the shape of variations in physical parameters is strongly dependent to magnetic field profile and there is a considerable change in density with respect to reference model. It is as well shown that the spherical symmetric distributions of physical parameters are broken for the region of study.     

On the variations of some physical quantities above the sunspot bands near the equatorial plane

In this work, some solutions of magnetohydrodynamic (MHD) equations are searched in order to investigate some large scale physical quantities in the sunspot dominated latitudinal regions near the equatorial plane. Special separation of variables is used to obtain the radial and latitudinal changes in spherical coordinates. Present parametric analysis yields three important parameters which are the sphericity, density and radial components shape parameters in the latitudinal distributions of physical variables. In the region of interest there is a considerable change in physical quantities with respect to regions where sunspots do not appear.     

First order latitude effects in the solar wind

The Weber-Davis model of the solar wind is generalized to include the effects of latitude. The principal assumptions of perfect electrical conductivity, rotational symmetry, a polytropic relation between pressure and density, and a flow aligned magnetic field in a system rotating with the Sun, are retained. A flow aligned magnetic field in the rotating system may be expressed in terms of the flow velocity and density. Rotational symmetry fixes the longitudinal flow velocity V_φ in terms of the flow in the r?θ plane. Thus, the original three dimensional magnetohydrodynamic flow problem is reduced to a two dimensional hydrodynamic flow problem in the r?θ plane.There are three critical surfaces associated with the equations which supply conditions to determine three of six required boundary conditions. The specified boundary conditions at the base of the corona are the temperature, density, and magnitude of the magnetic field. The equations are then expanded about the radial, nonrotating Parker solution and an analytic solution is obtained for the resulting first order equations. The results show that for constant coronal boundary conditions there is a latitudinal flow toward the solar poles, as a result of magnetic stresses, which persists out to large distances for the Sun. Associated with this flow is a latitudinal component of the magnetic field. The radial flow parameters are, to within small first order differences, in agreement with those of the Parker and the Weber-Davis models of the solar wind.The equations are further generalized to permit first order latitudinal variations in the specified coronal boundary conditions. Results at 1 a.u. are presented for 5 per cent latitudinal differences between the equatorial and polar values. These results show that the solution at 1 a.u. is most sensitive to a latitudinal dependence in the boundary temperature and least sensitive to a latitudinal dependence in the magnetic field magnitude.A solution is then obtained for an approximate dipolar variation in the coronal magnetic field magnitude. This solution predicts that the latitudinal flow is initially toward the Equator due to magnetic channeling; however, this effect is rapidly overcome and the latitudinal flow at 1 a.u. is toward the pole and not significantly different from the solution for constant boundary conditions.     

Self‐similar evolutionary solutions for an accreting magneto‐fluid around a compact object with finite electrical conductivity

In this paper, we investigate the time evolution of an accreting magneto‐fluid with finite conductivity. For the case of a thin disk, the fluid equations along with Maxwell's equations are derived in a simplified, one‐dimensional model that neglects the latitudinal dependence of the flow. The finite electrical conductivity of the plasma is taken into account by Ohm's law; however, the shear viscous stress is neglected, as well as the self‐gravity of the disk. In order to solve the integrated equations that govern the dynamical behaviour of the magneto‐fluid, we have used a self‐similar solution. We introduce two dimensionless variables, S₀ and ε_ϱ, which represent the size of the electrical conductivity and the density behaviour with time, respectively. The effect of each of these on the structure of the disk is studied. While the pressure is obtained simply by solving an ordinary differential equation, the density, the magnetic field, the radial velocity, and the rotational velocity are presented analytically. The solutions show that the S0 and ε_ϱ parameters affect the radial thickness of the disk. Also, radial velocity and gas pressure are more sensitive to the electrical conductivity in the inner regions of disk. Moreover, the parameter ε_ϱ has a more significant effect on the physical quantities for small radii. (© 2015 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

P – ω Dynamo in Spherical Geometry

Based on the fundamental P – ω dynamo equation, using spherical polar coordinates, we carry out a study of turbulent plasma wave dynamo effect. For various rotation laws, different analytical solutions are derived. In the cases of no rotation and rigid rotation, the dynamo generates poloidal field only, while with differential rotation, regardless the differential rotation is radial or latitudinal, poloidal and toroidal fields are all generated. We may think that the solutions are the analytical forms of the magnetic field in a turbulent source region of celestial bodies. This revised version was published online in July 2006 with corrections to the Cover Date.     

On the effect of magnetic field on the low-l solar p-mode oscillations

One of the possible magnetic field effects on the stellar pulsations is known to be a splitting in the observed frequencies. Using this knowledge in the solar convection zone, there are two aims in this work Considering the Sun as an incompressible fluid, our first objective was to investigate the variation of the physical parameters in the 30% outermost convective solar layer, during a pulsation period. The second purpose was to calculate, by means of the spherical harmonics, the shifts on the low-l p-mode frequencies which could be caused by the presence of the magnetic field in the Sun. The first order perturbation approximation was used in order to calculate analytically the resulting frequency shifts and the small perturbations on the magnetic field, as well as the physical parameters, such as density, pressure and temperature, of a Standard Solar Model excluding both rotation and magnetic field (Christensen-Dalsgaard et al., 1996) in the unperturbed equilibrium case.     

The Relationship Between Kinematics and Spatial Structure of the Large-Scale Solar Magnetic Field

The rotation of large-scale solar magnetic fields has been investigated by analysing a 20-yr series of synoptic maps of the radial magnetic field. For this purpose, a specially adapted method of spectral analysis was used. We calculated rotation spectra of the magnetic field as functions of the rotation period, heliographic latitude, and longitudinal wave number, k. These spectra reveal the existence of a number of discrete, rigidly rotating components (modes) of the magnetic field, whose rotation periods lie in the wide range from 26.5 to 30.5 days. The significant spectral maxima lie in the (rotation period–latitude) plane close to the curve that represents the differential rotation of small-scale magnetic features. For the first harmonic of the magnetic field (k=1) the properties of the rotation spectra are consistent with those reported by Antonucci, Hoeksema, and Scherrer (1990). However, the distribution of the rigidly rotating modes over rotation period and their latitudinal structure change systematically with the harmonic number k. As k increases, the mean distance P in rotation period between the modes decreases, from 1.2 days for k=1 to 0.3–0.5 days for k=4. This decreasing period separation is accompanied by a decrease of the characteristic latitude separation between the mode maxima. The latitudinal and longitudinal discrete spatial scales of the non-axisymmetric magnetic field appear to be connected with each other, as well as with the temporal scale P.     

Continuous monitoring of nightside upper thermospheric mass densities in the martian southern hemisphere over 4 martian years using electron reflectometry

Details are presented of an improved technique to use atmospheric absorption of magnetically reflecting solar wind electrons to constrain neutral mass densities in the nightside martian upper thermosphere. The helical motion of electrons on converging magnetic field lines, through an extended neutral atmosphere, is modeled to enable prediction of loss cone pitch angle distributions measured by the Magnetometer/Electron Reflectometer (MAG/ER) experiment on Mars Global Surveyor at 400 km altitude. Over the small fraction of Mars' southern hemisphere (∼2.5%) where the permanent crustal magnetic fields are both open to the solar wind and sufficiently strong as to dominate the variable induced martian magnetotail field, spherical harmonic expansions of the crustal fields are used to prescribe the magnetic field along the electron's path, allowing least-squares fitting of measured loss cones, in order to solve for parameters describing the vertical neutral atmospheric mass density profile from 160 to 230 km. Results are presented of mass densities in the southern hemisphere at 2 a.m. LST at the mean altitude of greatest sensitivity, 180 km, continuously over four martian years. Seasonal variability in densities is largely explained by orbital and latitudinal changes in dayside insolation that impacts the nightside through the resulting thermospheric circulation. However, the physical processes behind repeatable rapid, late autumnal cooling at mid-latitudes and near-aphelion warming at equatorial latitudes is not fully clear. Southern winter polar warming is generally weak or nonexistent over several Mars years, in basic agreement with MGS and MRO accelerometer observations. The puzzling response of mid-latitude densities from 160° to 200° E to the 2001 global dust storm suggests unanticipated localized nightside upper thermospheric lateral and vertical circulation patterns may accompany such storms. The downturn of the 11-year cycle of solar EUV flux is likely responsible for lower aphelion densities in 2004 and 2006 (Mars years 27 and 28).     

Maximum mass of magnetic white dwarfs

We revisit the problem of the maximum masses of magnetized white dwarfs(WDs).The impact of a strong magnetic field on the structure equations is addressed.The pressures become anisotropic due to the presence of the magnetic field and split into parallel and perpendicular components.We first construct stable solutions of the Tolman-Oppenheimer-Volkoff equations for parallel pressures and find that physical solutions vanish for the perpendicular pressure when B(?) 10~(13) G.This fact establishes an upper bound for a magnetic field and the stability of the configurations in the(quasi) spherical approximation.Our findings also indicate that it is not possible to obtain stable magnetized WDs with super-Chandrasekhar masses because the values of the magnetic field needed for them are higher than this bound.To proceed into the anisotropic regime,we can apply results for structure equations appropriate for a cylindrical metric with anisotropic pressures that were derived in our previous work.From the solutions of the structure equations in cylindrical symmetry we have confirmed the same bound for B ~ 10~(13) G,since beyond this value no physical solutions are possible.Our tentative conclusion is that massive WDs with masses well beyond the Chandrasekhar limit do not constitute stable solutions and should not exist.     

A constructing method for a self-consistent three-dimensional solution of the magnetohydrostatic equations using full-disk magnetogram data

A technique is proposed for constructing self-consistent 3-D solutions satisfying the magnetohydrostatic (MHS) equations, and fitting observations along the line of sight of the magnetic field at the photosphere. The technique is a generalization of a potential-field extrapolation method (Rudenko, 2001) using full-disk magnetogram data. The solution of the problem under consideration is based on representing the magnetic field in terms of a scalar function, with its subsequent harmonic expansion in terms of the functional basic set of spherical functions that satisfies the specified boundary conditions. It is expected that a numerical realization of the proposed method will make possible a real-time modeling of the three-dimensional magnetic field, temperature, pressure and density distributions.     

The Observed Long- and Short-Term Phase Relation between the Toroidal and Poloidal Magnetic Fields in Cycle 23

The observed phase relations between the weak background solar magnetic (poloidal) field and strong magnetic field associated with sunspots (toroidal field) measured at different latitudes are presented. For measurements of the solar magnetic field (SMF) the low-resolution images obtained from Wilcox Solar Observatory are used and the sunspot magnetic field was taken from the Solar Feature Catalogues utilizing the SOHO/MDI full-disk magnetograms. The quasi-3D latitudinal distributions of sunspot areas and magnetic fields obtained for 30 latitudinal bands (15 in the northern hemisphere and 15 in the southern hemisphere) within fixed longitudinal strips are correlated with those of the background SMF. The sunspot areas in all latitudinal zones (averaged with a sliding one-year filter) reveal a strong positive correlation with the absolute SMF in the same zone appearing first with a zero time lag and repeating with a two- to three-year lag through the whole period of observations. The residuals of the sunspot areas averaged over one year and those over four years are also shown to have a well defined periodic structure visible in every two – three years close to one-quarter cycle with the maxima occurring at − 40° and + 40° and drifts during this period either toward the equator or the poles depending on the latitude of sunspot occurrence. This phase relation between poloidal and toroidal field throughout the whole cycle is discussed in association with both the symmetric and asymmetric components of the background SMF and relevant predictions by the solar dynamo models.     

Wave-trains in the solar wind

Applying an Alfvén-Wave-Extended-QRH-approximation and the method of characteristics, we solve the equations of motion for outwardly propagating Alfvén waves analytically for three different cases of an azimuthal dependence of the background solar wind, (a) for a pure fast-slow stream configuration, (b) for the situation where the high-speed stream originates from a diverging magnetic field region, and (c) for the case of (b) and an initially decreasing density configuration (‘coronal hole’). The reaction of these waves on the background state as well as mode-mode coupling effects are neglected. These three solar wind models are discussed shortly. For the superimposed Alfvén waves we find, on an average, that there is a strong azimuthal dependence of all relevant wave parameters which, correlated with the azimuthal distributions of the solar wind variables, leads to good agreements with observations. The signature of high-speed streams and these correlations could clearly indicate solar wind streams originating from ‘coronal holes’. Contrary to the purely radial dependent solar wind, where outwardly propagating Alfvén waves are exclusively refracted towards the radial direction, we now find a refraction nearly perpendicular to the direction of the interplanetary magnetic field in the compression region and closely towards the magnetic field direction down the trailing edge and in the low-speed regime.     

Analysis of the three-dimensional stellar velocity field using vector spherical functions

We apply vector spherical functions to problems of stellar kinematics. Using these functions allows all of the systematic components in the stellar velocity field to be revealed without being attached to a specific physical model. Comparison of the theoretical decomposition coefficients of the equations for a particular kinematical model with observational data can provide precise information about whether the model is compatible with the observations and can reveal systematic components that are not described by this model. The formalism of vector spherical functions is particularly well suited for analyzing the present and future (e.g., GAIA) catalogs containing all three velocity vector components: the propermotions in both coordinates and the radial velocity. We show that there are systematic components in the proper motions of Hipparcos stars that cannot be interpreted in terms of the linear Ogorodnikov-Milne model. The same result is also confirmed by an analysis of the radial velocities for these stars.     

The non-axisymmetrical configuration of a large-scale solar and stellar magnetic field

The non-axisymmetric and nonlinear solutions of the magnetostatic equations are given in three-dimensional space of spherical coordinates (r, θ, ?). These solutions are applied to the large-scale solar magnetic field. Their basic features are similar to a dipole field near the polar regions and the polarity reverses near the equator. These features agree with observations for the large-scale solar magnetic field. The solutions can also be applied to investigating the connection between the structure of the magnetic field and the density distribution of the corona. It is shown that the tops of the closed magnetic field associate with density enhancements. Similar results may apply to the large-scale configuration of the stellar field.     

The Magnetic Field at the Inner Boundary of the Heliosphere Around Solar Minimum

STEREO A and B observations of the radial magnetic field between 1 January 2007 and 31 October 2008 show significant evidence that in the heliosphere, the ambient radial magnetic field component with any dynamic effects removed is uniformly distributed. Based on this monopolar nature of the ambient heliospheric field we find that the surface beyond which the magnetic fields are in the monopolar configuration must be spherical, and this spherical surface can be defined as the inner boundary of the heliosphere that separates the monopole-dominated heliospheric magnetic field from the multipole-dominated coronal magnetic field. By using the radial variation of the coronal helmet streamers belts and the horizontal current – current sheet – source surface model we find that the spherical inner boundary of the heliosphere should be located around 14 solar radii near solar minimum phase.     

Coronal general magnetic field evolution as a new parameter of the solar cycle

The magnetic field lines of the corona associated with the solar-cycle surface general magnetic field are calculated by a potential-field approximation to study the solar-cycle evolution of the geometry of the coronal field. The surface field evolution used here is the radial field evolution, predicted by a model of the solar cycle driven by the dynamo action of the global convection, and justified observationally using Mount Wilson magnetic synoptic chart data. The evolution of the calculated coronal general field is now good for comparison with observations and shows the following. (i) The field of the polar and high-altitude corona has dipolar structure in almost all phases of the solar cycle except in a short time interval around maximum phase despite the quadrupolar structure of the general magnetic field at the surface; quadrupolar field forms loop-like structure in the lower corona. The almost-dipolar structure of the polar and high-latitude corona and the loop formation of the equatorial lower corona explain the appearance of the undisturbed minimum corona observed at eclipses. (ii) The polar field lines are directed almost radially at the minimum phase, which should be responsible for polar plumes. The field lines slowly open up to participate in the loop-like structure of the equatorial lower corona, and rapidly change their structure and polarity at the maximum phase, to resume the almost radial configuration slowly, (iii) During the rapidly changing maximum phase, the field lines do not penetrate deep into the interplanetary space resulting in the absence of polar plumes and the appearance of the circular corona- the maximum corona. In this phase, the coronal field should not be approximated by a dipole field. The surface field evolution which can explain such behaviors of the corona is characteristic of the solar-cycle process dominated by the latitudinal gradient of the differential rotation. If the radial gradient dominated in the subsurface process, the coronal evolution would look quite different and would show latitudinal propagation of enhancement of activity. Although nonaxisymmetric features should be superposed on the axisymmetric general field to express the real corona, the general field can be a basic coronal field in studying long-term interaction between the convection zone and the interstellar space especially in studying the magnetic braking of the solar rotation.     

Rayleigh-Taylor instability of a two-fluid layer under a general rotation field and a horizontal magnetic field

In this paper the Rayleigh-Taylor instability (RTI) of a two-fluid layer system under the simultaneous action of a general rotation field and a horizontal magnetic field is presented. An approximate and an exact solution of the eigenvalue equation are calculated. These solutions are important not only to understand more deeply the physical problem but also to determine the correct numerical solutions. Numerical calculations are done for an unstable density stratification in the cases of horizontal magnetic field parallel and perpendicular to the horizontal component of the angular velocity. For an adverse density stratification, it is shown that in comparison to previous works, the horizontal magnetic field creates new angular areas (of the angle of propagation of the perturbation) at which the perturbation is stable and propagates as traveling waves. It is also shown that the vertical component of the angular velocity has a destabilizing effect because it works to eliminate the stable angular areas.     

Large-Scale Magnetic Field of the Sun and Evolution of Sunspot Activity

The evolution of the large-scale magnetic field of the Sun has been studied using an algorithm of tomographic inversion. By analyzing line-of-sight magnetograms, we mapped the radial and toroidal components of the Sun??s large-scale magnetic field. The evolution of the radial and toroidal magnetic field components in the 11-year solar cycle has been studied in a time?Clatitude aspect. It is shown that the toroidal magnetic field of the Sun is causally related to sunspot activity; i.e., the sunspot formation zones drift in latitude and follow the toroidal magnetic fields. The results of our analysis support the idea that the high-latitude toroidal magnetic fields can serve as precursors of sunspot activity. The toroidal fields in the current cycle are anomalously weak and also show a barely noticeable equatorward drift. This behavior of the toroidal magnetic field suggests low activity levels in the current cycle and in the foreseeable future.     

A non-adiabatic analysis for axisymmetric pulsations of magnetic stars

This paper presents the results of a non-adiabatic analysis for axisymmetric non-radial pulsations including the effect of a dipole magnetic field. Convection is assumed to be suppressed in the stellar envelope, and the diffusion approximation is used to radiative transport. As in a previous adiabatic analysis, the eigenfunctions are expanded in a series of spherical harmonics. The analysis is applied to a  1.9-M_⊙  , main-sequence model  (log  T _eff= 3.913)  . The presence of a magnetic field always stabilizes low-order acoustic modes. All the low-order modes of the model that are excited by the κ-mechanism in the He  ii ionization zone in the absence of a magnetic field are found to be stabilized if the polar strength of the dipole magnetic field is larger than about 1 kG. For high-order p modes, on the other hand, distorted dipole and quadrupole modes excited by the κ-mechanism in the H ionization zone remain overstable, even in the presence of a strong magnetic field. It is found, however, that all the distorted radial high-order modes are stabilized by the effect of the magnetic field. Thus, our non-adiabatic analysis suggests that distorted dipole modes and distorted quadrupole modes are most likely excited in rapidly oscillating Ap stars. The latitudinal amplitude dependence is found to be in reasonable agreement with the observationally determined one for HR 3831. Finally, the expected amplitude of magnetic perturbations at the surface is found to be very small.     

Spherical Magnetic Vortex in an External Potential Field: A Dissipative Contraction

We consider the dissipative evolution of a spherical magnetic vortex with a force-free internal structure, located in a resistive medium and held in equilibrium by the potential external field. The magnetic field inside the sphere is force-free (the model of Chandrasekhar in Proc. Natl. Acad. Sci. 42, 1, 1956). Topologically, it is a set of magnetic toroids enclosed in spherical layers. A new exact MHD solution has been derived, describing a slow, uniform, radial compression of a magnetic spheroid under the pressure of an ambient field, when the plasma density and pressure are growing inside it. There is no dissipation in the potential field outside the sphere, but inside the sphere, where the current density can be high enough, the magnetic energy is continuously converted into heat. Joule dissipation lowers the magnetic pressure inside the sphere, which balances the pressure of the ambient field. This results in radial contraction of the magnetic sphere with a speed defined by the conductivity of the plasma and the characteristic spatial scale of the magnetic field inside the sphere. Formally, the sphere shrinks to zero within a finite time interval (magnetic collapse). The time of compression can be relatively small, within a day, even for a sphere with a radius of about 1 Mm, if the magnetic helicity trapped initially in the sphere (which is proportional to the number of magnetic toroids in the sphere) is quite large. The magnetic system is open along its axis of symmetry. On this axis, the magnetic and electric fields are strictly radial and sign-variable along the radius, so the plasma will be ejected along the axis of magnetic sphere outwards in both directions (as jets) at a rate much higher than the diffusive one, and the charged particles will be accelerated unevenly, in spurts, creating quasi-regular X-ray spikes. The applications of the solution to solar flares are discussed.