In this paper we study the dynamics of toroidal flux tubes emerging from the solar interior, through the photosphere and into the corona. Many previous theoretical studies of flux emergence use a twisted cylindrical tube in the solar interior as the initial condition. Important insights can be gained from this model, however, it does have shortcomings. The axis of the tube never fully emerges as dense plasma becomes trapped in magnetic dips and restrains its ascent. Also, since the entire tube is buoyant, the main photospheric footpoints (sunspots) continually drift apart. These problems make it difficult to produce a convincing sunspot pair. We aim to address these problems by considering a different initial condition, namely a toroidal flux tube. We perform numerical experiments and solve the 3D MHD equations. The dynamics are investigated through a range of initial field strengths and twists. The experiments demonstrate that the emergence of toroidal flux tubes is highly dynamic and exhibits a rich variety of behaviour. In answer to the aims, however, if the initial field strength is strong enough, the axis of the tube can fully emerge. Also, the sunspot pair does not continually drift apart. Instead, its maximum separation is the diameter of the original toroidal tube.
We numerically study the evolution of magnetic fields and fluid flows in a thin spherical shell. We take the initial field to be a latitudinally confined, predominantly toroidal flux tube. For purely toroidal, untwisted flux tubes, we recover previously known radial-shredding instabilities, and show further that in the nonlinear regime these instabilities can very effectively destroy the original field. For twisted flux tubes, including also a poloidal component, there are several possibilities, including the suppression of the radial-shredding instability, but also a more directly induced evo…
The nonaxisymmetric Tayler instability of toroidal magnetic fields due to axial electric currents is studied for conducting incompressible fluids between two coaxial cylinders without endplates. The inner cylinder is considered as so thin that even the limit of R_in \to 0 can be computed. The magnetic Prandtl number is varied over many orders of magnitudes but the azimuthal mode number of the perturbations is fixed to m=1. In the linear approximation the critical magnetic field amplitudes and the growth rates of the instability are determined for both resting and rotating cylinders. Without ro…
The buoyant transport of magnetic fields from the solar interior towards the surface plays an important role in the emergence of active regions, the formation of sunspots and the overall solar dynamo. Observations suggest that toroidal flux concentrations often referred to as "flux tubes", rise from their region of initiation likely in the solar tachocline towards the solar surface due to magnetic buoyancy. Many studies have assumed the existence of such magnetic structures and studied the buoyant rise of an isolated flux tube in a quiescent, field-free environment. Here, motivated by their fo…
Magnetohydrodynamic (MHD) waves are ubiquitous in the solar atmosphere. Alfven waves and magneto-sonic waves are particular classes of MHD waves. These wave modes are clearly different and have pure properties in uniform plasmas of infinite extent only. Due to plasma non-uniformity MHD waves have mixed properties and cannot be classified as pure Alfven or magneto-sonic waves. However, vorticity is a quantity unequivocally related to Alfven waves as compression is for magneto-sonic waves. Here, we investigate MHD waves superimposed on a one-dimensional non-uniform straight cylinder with constan…
Using two-dimensional magnetohydrodynamics calculations, we investigate a twist gen- eration mechanism on a magnetic flux tube at the base of the solar convection zone based on the idea of Choudhuri, 2003, Sol. Phys., 215, 31 in which a toroidal mag- netic field is wrapped by a surrounding mean poloidal field. During generation of the twist, the flux tube follows four phases. (1) It quickly splits into two parts with vortex motions rolling up the poloidal magnetic field. (2) Owing to the physical mechanism similar to that of the magneto-rotational instability, the rolled-up poloidal field is b…
We investigate dynamical coupling timescales of a neutron star's superfluid core, taking into account the interactions of quantized neutron vortices with quantized flux lines of the proton superconductor in addition to the previously considered scattering of the charged components against the spontaneous magnetization of the neutron vortex line. We compare the cases where vortex motion is constrained in different ways by the array of magnetic flux tubes associated with superconducting protons. This includes absolute pinning to and creep across a uniform array of flux lines. The effect of a tor…
A new method for reconstruction of coronal magnetic fields as force-free fields (FFFs) is presented. Our method employs poloidal and toroidal functions to describe divergence-free magnetic fields. This magnetic field representation naturally enables us to implement the boundary conditions at the photospheric boundary, i.e., the normal magnetic field and the normal current density there, in a straightforward manner. At the upper boundary of the corona, a source-surface condition can be employed, which accommodates magnetic flux imbalance at the bottom boundary. Although our iteration algorithm …
Analytic arguments have been advanced that the degree of differential rotation in a neutron star depends on whether the topology of the internal magnetic field is open or closed. To test this assertion, the ideal-magnetohydrodynamics solver PLUTO is employed to investigate numerically the flow of an incompressible, viscous fluid threaded by a magnetic field with open and closed topologies in a conducting, differentially rotating, spherical shell. Rigid body corotation with the outer sphere is enforced on the Alfvén time-scale, along magnetic field lines that connect the northern and southern h…
This article completes and extends a recent study of the Grad-Shafranov (GS) reconstruction in toroidal geometry, as applied to a two and a half dimensional configurations in space plasmas with rotational symmetry. A further application to the benchmark study of an analytic solution to the toroidal GS equation with added noise shows deviations in the reconstructed geometry of the flux rope configuration, characterized by the orientation of the rotation axis, the major radius, and the impact parameter. On the other hand, the physical properties of the flux rope, including the axial field streng…
The rise of flux-tubes with intense magnetic fields from the base of the convection zone to the solar surface has been substantiated as a probable mechanism for sunspot formation. The origin of flux-tubes of sufficient strength (~10^5 G) is however uncertain. This paper considers the instability of a large-scale toroidal magnetic field caused by the magnetic suppression of convective heat transport as a candidate for the flux tube forming mechanism. The consideration employs the analytical dependence of the eddy diffusion on the magnetic field supplied by mean-field magnetohydrodynamics. The i…
Active regions (ARs) often possess an observed net electric current in a single magnetic polarity. We show that such "non-neutralized" currents can arise from a geometric projection effect when a twisted flux tube obliquely intersects the photosphere. To this end, we emulate surface maps of an emerging AR by sampling horizontal slices of a semi-torus flux tube at various heights. Although the tube has no net toroidal current, its poloidal current, when projected along the vertical direction, amounts to a significant non-neutralized component on the surface. If the tube emerges only partially a…
We consider the effects of material flows on the dynamics of toroidal magnetic flux tubes located close to the base of the solar convection zone, initially within the overshoot region. The problem is to find the physical conditions in which magnetic flux can be stored for periods comparable to the dynamo amplification time, which is of the order of a few years. We carry out nonlinear numerical simulations to investigate the stability and dynamics of thin flux tubes subject to perpendicular and longitudinal flows. We compare the simulations with the results of simplified analytical approximatio…
We present the first 3D MHD study in spherical geometry of the non-linear dynamical evolution of magnetic flux tubes in a turbulent rotating convection zone. We study numerically the rise of magnetic toroidal flux ropes from the base of a modelled convection zone up to the top of our computational domain where bipolar patches are formed. We compare the dynamical behaviour of flux tubes in a fully convective shell possessing self-consistently generated mean flows such as meridional circulation and differential rotation, with reference calculations done in a quiet isentropic zone. We find that t…
We study how active-region-scale flux tubes rise buoyantly from the base of the convection zone to near the solar surface by embedding a thin flux tube model in a rotating spherical shell of solar-like turbulent convection. These toroidal flux tubes that we simulate range in magnetic field strength from 15 kG to 100 kG at initial latitudes of 1 degree to 40 degrees in both hemispheres. This article expands upon Weber, Fan, and Miesch (Astrophys. J., 741, 11, 2011) (Article 1) with the inclusion of tubes with magnetic flux of 10^20 Mx and 10^21 Mx, and more simulations of the previously investi…
We use a thin flux tube model in a rotating spherical shell of turbulent convective flows to study how active region scale flux tubes rise buoyantly from the bottom of the convection zone to near the solar surface. We investigate toroidal flux tubes at the base of the convection zone with field strengths ranging from 15 kG to 100 kG at initial latitudes ranging from 1 degree to 40 degrees with a total flux of 10^22 Mx. We find that the dynamic evolution of the flux tube changes from convection dominated to magnetic buoyancy dominated as the initial field strength increases from 15 kG to 100 kG…
We study the long term evolution of a set of 22 bipolar active regions (ARs) in which the main photospheric polarities are seen to rotate one around the other during several solar rotations. We first show that differential rotation is not at the origin of this large change in the tilt angle. A possible origin of this distortion is the nonlinear development of a kink-instability at the base of the convective zone; this would imply the formation of a non-planar flux tube which, while emerging across the photosphere, would show a rotation of its photospheric polarities as observed. A characterist…
A topological constraint on the dynamics of a magnetic field in a flux tube arises from the fixed point indices of its field line mapping. This can explain unexpected behaviour in recent resistive-magnetohydrodynamic simulations of magnetic relaxation. Here we present the theory for a general periodic flux tube, representing, for example, a toroidal confinement device or a solar coronal loop. We show how an ideal dynamics on the side boundary of the tube implies that the sum of indices over all interior fixed points is invariant. This constraint applies to any continuous evolution inside the t…
Recent studies of NOAA active region 10953, by Okamoto {\it et al.} ({\it Astrophys. J. Lett.} {\bf 673}, 215, 2008; {\it Astrophys. J.} {\bf 697}, 913, 2009), have interpreted photospheric observations of changing widths of the polarities and reversal of the horizontal magnetic field component as signatures of the emergence of a twisted flux tube within the active region and along its internal polarity inversion line (PIL). A filament is observed along the PIL and the active region is assumed to have an arcade structure. To investigate this scenario, MacTaggart and Hood ({\it Astrophys. J. Le…
In this paper we study the effects of a toroidal magnetic flux tube emerging into a magnetized corona, with an emphasis on large-scale eruptions. The orientation of the fields is such that the two flux systems are almost antiparallel when they meet. We follow the dynamic evolution of the system by solving the 3D MHD equations using a Lagrangian remap scheme. Multiple eruptions are found to occur. The physics of the trigger mechanisms are discussed and related to well-known eruption models.
Transverse oscillations of coronal loops are often observed and have been theoretically interpreted as kink magnetohydrodynamic (MHD) modes. Numerical simulations by Terradas et al. (2008, ApJ 687, L115) suggest that shear flows generated at the loop boundary during kink oscillations could give rise to a Kelvin-Helmholtz instability (KHI). Here, we investigate the linear stage of the KHI in a cylindrical magnetic flux tube in the presence of azimuthal shear motions. We consider the basic, linearized MHD equations in the beta = 0 approximation, and apply them to a straight and homogeneous cylin…
Magnetic field emerges at the surface of the Sun as sunspots and active regions. This process generates a poloidal magnetic field from a rising toroidal flux tube, it is a crucial but poorly understood aspect of the solar dynamo. The emergence of magnetic field is also important because it is a key driver of solar activity. We show that measurements of horizontal flows at the solar surface around emerging active regions, in combination with numerical simulations of solar magnetoconvection, can constrain the subsurface rise speed of emerging magnetic flux. The observed flows imply that the rise…
We study the emergence of a toroidal flux tube into the solar atmosphere and its interaction with a pre-existing field of an active region. We investigate the emission of jets as a result of repeated reconnection events between colliding magnetic fields. We perform 3D simulations by solving the time-dependent, resistive MHD equations in a highly stratified atmosphere. A small active region field is constructed by the emergence of a toroidal magnetic flux tube. A current structure is build up and reconnection sets in when new emerging flux comes into contact with the ambient field of the active…
The study of MHD waves is important both for understanding heating in the solar atmosphere and for solar atmospheric seismology. The analytical investigation of wave mode properties in a cylinder is of particular interest in this domain, as many atmospheric structures can be modeled as such in a first approximation. We use linearized ideal MHD to study quasimodes (global modes that are damped through resonant absorption) with a frequency in the cusp continuum, in a straight cylinder with a circular base and an inhomogeneous layer at its boundary which separates two homogeneous plasma regions i…
Elongated magnetic polarities are observed during the emergence phase of bipolar active regions (ARs). These extended features, called magnetic tongues, are interpreted as a consequence of the azimuthal component of the magnetic flux in the toroidal flux-tubes that form ARs. We develop a new systematic and user-independent method to identify AR tongues. Our method is based on determining and analyzing the evolution of the AR main polarity inversion line (PIL). The effect of the tongues is quantified by measuring the acute angle [ tau] between the orientation of the PIL and the direction orthog…
Long observational series for bipolar active regions (ARs) provide significant information about the mutual transformation of the poloidal and toroidal components of the global solar magnetic field. The direction of the toroidal field determines the polarity of leading sunspots in ARs in accordance with the Hale's polarity law. The vast majority of bipolar ARs obey this regularity, whereas a few percent of ARs have the opposite sense of polarity (anti-Hale ARs). However, the study of these ARs is hampered by their poor statistics. The data for five 11-year cycles (16-18 and 23,24) were combine…
We describe a 3D finite-difference spherical anelastic MHD (FSAM) code for modeling the subsonic dynamic processes in the solar convective envelope. A comparison of this code with the widely used global spectral anlastic MHD code, ASH (Anelastic Spherical Harmonics), shows that FSAM produces convective flows with statistical properties and mean flows similar to the ASH results. Using FSAM, we first simulate the rotating solar convection in a partial spherical shell domain and obtain a statistically steady, giant-cell convective flow with a solar-like differential rotation. We then insert buoya…
We have investigated the toroidal analog of ellipsoidal shells of matter, which are of great significance in Astrophysics. The exact formula for the gravitational potential $Ψ(R,Z)$ of a shell with a circular section at the pole of toroidal coordinates is first established. It depends on the mass of the shell, its main radius and axis-ratio $e$ (i.e. core-to-main radius ratio), and involves the product of the complete elliptic integrals of the first and second kinds. Next, we show that successive partial derivatives $\partial^{n +m} Ψ/\partial_{R^n} \partial_{Z^m}$ are also accessible by analy…
Recently Vishik anti-fast dynamo theorem, has been tested against non-stretching flux tubes [Phys Plasmas 15 (2008)]. In this paper, another anti-dynamo theorem, called Cowling's theorem, which states that axisymmetric magnetic fields cannot support dynamo action, is carefully tested against thick tubular and curved Riemannian untwisted flows, as well as thin flux tubes in diffusive and diffusionless media. In the non-diffusive media the Cowling's theorem is not violated in thin Riemann-flat untwisted flux tubes, where the Frenet curvature is negative. Nevertheless the diffusion action in the …
Context. Active regions (ARs) appear in the solar atmosphere as a consequence of the emergence of magnetic flux tubes. The presence of elongated magnetic polarities in line-of-sight (LOS) magnetograms indicates the existence of twist in the flux tubes forming them. These polarity elongations, called magnetic tongues, bias the measurement of AR characteristics obtained during their emergence phase (e.g. their tilt angle and magnetic flux, among others). In particular, obtaining a good estimation of the tilt angle evolution plays a key role in constraining flux-transport dynamo models. Aims. In …
Dynamics of slender magnetic flux tubes (MFT) in the accretion discs of T Tauri stars is investigated. We perform simulations taking into account buoyant, aerodynamic and turbulent drag forces, radiative heat exchange between MFT and ambient gas, magnetic field of the disc. The equations of MFT dynamics are solved using Runge-Kutta method of the fourth order. The simulations show that there are two regimes of MFT motion in absence of external magnetic field. In the region $r<0.2$ au, the MFT of radii $0.05 \leq a_0 \leq 0.16\,H$ ($H$ is the scale height of the disc) with initial plasma beta of…
