Formation of buoyant magnetic structures by a localized velocity shear

Motivated by considerations of the solar tachocline, we study the generation of strong buoyant magnetic structures by a sheared velocity field localized in a convectively stable background, using non-linear three-dimensional (3D) magnetohydrodynamic (MHD) simulations. The shear flow can spontaneously create strong tube-like toroidal (streamwise) magnetic structures from an imposed weak uniform poloidal (cross-stream) magnetic field. The structures are magnetically buoyant and therefore rise, and may evolve further to a rich variety of geometries, including kinked or arched shapes. The emergence process can repeat indefinitely with a characteristic period. These mechanisms may be relevant to the MHD processes in the solar tachocline and the creation and emergence of solar active regions.     

Oblique propagation and dissipation of Alfvén waves in coronal holes

We investigate the effect of viscosity and magnetic diffusivity on the oblique propagation and dissipation of Alfvén waves with respect to the normal outward direction, making use of MHD equations, density, temperature and magnetic field structure in coronal holes and underlying magnetic funnels. We find reduction in the damping length scale, group velocity and energy flux density as the propagation angle of Alfvén waves increases inside the coronal holes. For any propagation angle, the energy flux density and damping length scale also show a decrement in the source region of the solar wind (< 1.05 R_⊙) where these may be one of the primary energy sources, which can convert the inflow of the solar wind into the outflow. In the outer region (> 1.21 R_⊙), for any propagation angle, the energy flux density peaks match with the peaks of MgX 609.78 Å and 624.78 Å linewidths observed from the Coronal Diagnostic Spectrometer (CDS) on SOHO and the non-thermal velocity derived from these observations, justify the observed spectroscopic signature of the Alfvén wave dissipation.     

Diamagnetic pumping near the base of a stellar convection zone

The property of inhomogeneous turbulence in conducting fluids to expel large‐scale magnetic fields in the direction of decreasing turbulence intensity is shown as important for the magnetic field dynamics near the base of a stellar convection zone. The downward diamagnetic pumping confines a fossil internal magnetic field in the radiative core so that the field geometry is appropriate for formation of the solar tachocline. For the stars of solar age, the diamagnetic confinement is efficient only if the ratio of turbulent magnetic diffusivity η_T of the convection zone to the (microscopic or turbulent) diffusivity η_in of the radiative interior is η_T/η_in ≥ 10⁵. Confinement in younger stars requires larger η_T/η_in. The observation of persistent magnetic structures on young solar‐type stars can thus provide evidence for the nonexistence of tachoclines in stellar interiors and on the level of turbulence in radiative cores. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Latitudinal shear instability in the solar tachocline

In an attempt to explain the observed rotation profile in the solar radiative zone and the tachocline, Spiegel & Zahn proposed a model based on anisotropic turbulent angular momentum transport. Although very successful in reproducing some of the features of the solar tachocline, their model assumes without verification that the origin of the turbulence could be caused by latitudinal shear instability. This paper studies the weakly non-linear evolution of two-dimensional shear instability, in which the interaction between the global rotation profile and the Reynolds stresses can be described self-consistently. Provided that the initial rotation profile is sufficiently close to marginal stability (which is the case of the solar tachocline), the instability is shown to saturate and to relax to a marginally stable state, which differs very little from the observed rotation profile. It is therefore likely that the tachocline is in a state of marginal stability with respect to latitudinal shear instability, and shows that angular momentum transport in the tachocline is unlikely to be caused by shear-induced turbulence.     

The angular momentum transport in rotating turbulent convection

The angular momentum transport in rotating turbulent convection is simulated with the NIRVANA code for Taylor numbers up to 10⁶. The box consists of an unstable layer embedded in two stable overshoot layers. We find the expected anisotropies in the rotating anisotropic turbulence field: 〈u^′2_r〉 exceeds 〈u^′2_ϕ〉, and 〈u^′2_ϕ〉 exceeds 〈u^′2_θ〉. The resulting radial angular momentum transport is directed inwards and peaks in the middle of the convective layer. The resulting latitudinal angular momentum transport is equatorwards, peaks at the surface and is highly concentrated to the equatorial region. Apart from a factor of 2–3 the total amplitudes of the cross‐correlations are of the same order of magnitude. In the lower overshoot region (‘tachocline’) the cross‐correlations are negative. It is argued that the concentration of the latitudinal angular momentum transport towards the surface and towards the equator does not too strongly reduce its potential to produce rotation laws with accelerated equators. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Dynamics of the solar tachocline – I. An incompressible study

Gough & McIntyre have suggested that the dynamics of the solar tachocline are dominated by the advection–diffusion balance between the differential rotation, a large-scale primordial field and baroclinicly driven meridional motions. This paper presents the first part of a study of the tachocline, in which a model of the rotation profile below the convection zone is constructed along the lines suggested by Gough & McIntyre and solved numerically. In this first part, a reduced model of the tachocline is derived in which the effects of compressibility and energy transport on the system are neglected; the meridional motions are driven instead by Ekman–Hartmann pumping. Through this simplification, the interaction of the fluid flow and the magnetic field can be isolated and is studied through non-linear numerical analysis for various field strengths and diffusivities. It is shown that there exists only a narrow range of magnetic field strengths for which the system can achieve a nearly uniform rotation. The results are discussed with respect to observations and to the limitations of this initial approach. A following paper combines the effects of realistic baroclinic driving and stratification with a model that closely follows the lines of work of Gough & McIntyre.     

Interpreting Helioseismic Structure Inversion Results of Solar Active Regions

Helioseismic techniques such as ring-diagram analysis have often been used to determine the subsurface structural differences between solar active and quiet regions. Results obtained by inverting the frequency differences between the regions are usually interpreted as the sound-speed differences between them. These in turn are used as a measure of temperature and magnetic-field strength differences between the two regions. In this paper we first show that the “sound-speed” difference obtained from inversions is actually a combination of sound-speed difference and a magnetic component. Hence, the inversion result is not directly related to the thermal structure. Next, using solar models that include magnetic fields, we develop a formulation to use the inversion results to infer the differences in the magnetic and thermal structures between active and quiet regions. We then apply our technique to existing structure inversion results for different pairs of active and quiet regions. We find that the effect of magnetic fields is strongest in a shallow region above 0.985R _⊙ and that the strengths of magnetic-field effects at the surface and in the deeper (r<0.98R _⊙) layers are inversely related (i.e., the stronger the surface magnetic field the smaller the magnetic effects in the deeper layers, and vice versa). We also find that the magnetic effects in the deeper layers are the strongest in the quiet regions, consistent with the fact that these are basically regions with weakest magnetic fields at the surface. Because the quiet regions were selected to precede or follow their companion active regions, the results could have implications about the evolution of magnetic fields under active regions.     

MHD waves in the solar north polar coronal hole

The effects, hitherto not treated, of the temperature and the number density gradients, both in the parallel and the perpendicular direction to the magnetic field, of O VI ions, on the MHD wave propagation characteristics in the solar North Polar Coronal Hole are investigated. We investigate the magnetosonic wave propagation in a resistive MHD regime where only the thermal conduction is taken into account. Heat conduction across the magnetic field is treated in a non‐classical approach wherein the heat is assumed to be conducted by the plasma waves emitted by ions and absorbed at a distance from the source by other ions. Anisotropic temperature and the number density distributions of O VI ions revealed the chaotic nature of MHD standing wave, especially near the plume/interplume lane borders. Attenuation length scales of the fast mode is shown not to be smoothly varying function of the radial distance from the Sun (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Flux-dominated solar dynamo model with a thin shear layer

Flux-dominated solar dynamo models have demonstrated to reproduce the main features of the large scale solar magnetic cycle, however the use of a solar like differential rotation profile implies in the the formation of strong toroidal magnetic fields at high latitudes where they are not observed. In this work, we invoke the hypothesis of a thin-width tachocline in order to confine the high-latitude toroidal magnetic fields to a small area below the overshoot layer, thus avoiding its influence on a Babcock-Leighton type dynamo process. Our results favor a dynamo operating inside the convection zone with a tachocline that essentially works as a storage region when it coincides with the overshoot layer. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the possibility of a bimodal solar dynamo

A simple way to couple an interface dynamo model to a fast tachocline model is presented, under the assumption that the dynamo saturation is due to a quadratic process and that the effect of finite shear layer thickness on the dynamo wave frequency is analogous to the effect of finite water depth on surface gravity waves. The model contains one free parameter which is fixed by the requirement that a solution should reproduce the helioseismically determined thickness of the tachocline. In this case it is found that, in addition to this solution, another steady solution exists, characterized by a four times thicker tachocline and 4–5 times weaker magnetic fields. It is tempting to relate the existence of this second solution to the occurrence of grand minima in solar activity. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Statistical Assessment of Photospheric Magnetic Features in Imminent Solar Flare Predictions

In this study we use the ordinal logistic regression method to establish a prediction model, which estimates the probability for each solar active region to produce X-, M-, or C-class flares during the next 1-day time period. The three predictive parameters are (1) the total unsigned magnetic flux T _flux, which is a measure of an active region’s size, (2) the length of the strong-gradient neutral line L _gnl, which describes the global nonpotentiality of an active region, and (3) the total magnetic dissipation E _diss, which is another proxy of an active region’s nonpotentiality. These parameters are all derived from SOHO MDI magnetograms. The ordinal response variable is the different level of solar flare magnitude. By analyzing 174 active regions, L _gnl is proven to be the most powerful predictor, if only one predictor is chosen. Compared with the current prediction methods used by the Solar Monitor at the Solar Data Analysis Center (SDAC) and NOAA’s Space Weather Prediction Center (SWPC), the ordinal logistic model using L _gnl, T _flux, and E _diss as predictors demonstrated its automatic functionality, simplicity, and fairly high prediction accuracy. To our knowledge, this is the first time the ordinal logistic regression model has been used in solar physics to predict solar flares.     

Stability of the solar tachocline with magnetic fields

The stability of magnetic fields in the solar tachocline is investigated. We present stability limits for higher azimuthal wave numbers and results on the dependence of the stability on the location of toroidal magnetic fields in latitude. While the dependence of the wave number with the largest growth rate on the magnetic field strength and the magnetic Prandtl number is small, the dependence on the magnetic Reynolds number Rm indicates that lowest azimuthal modes are excited for very high Rm. Upon varying the latitudinal position of the magnetic field belts, we find slightly lower stability limits for high latitudes, and very large stability limits at latitudes below 10°, with little dependence on latitude in between. An increase of the maximum possible field was achieved by adding a poloidal field. The upper limit for the toroidal field which can be stored in the radiative tachocline is then 1000 G, compared to about 100 G for a purely toroidal field as was found in an earlier work. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Damped Oscillations of Coronal Loops

A mechanism of damped oscillations of a coronal loop is investigated. The loop is treated as a thin toroidal flux rope with two stationary photospheric footpoints, carrying both toroidal and poloidal currents. The forces and the flux-rope dynamics are described within the framework of ideal magnetohydrodynamics (MHD). The main features of the theory are the following: i) Oscillatory motions are determined by the Lorentz force that acts on curved current-carrying plasma structures and ii) damping is caused by drag that provides the momentum coupling between the flux rope and the ambient coronal plasma. The oscillation is restricted to the vertical plane of the flux rope. The initial equilibrium flux rope is set into oscillation by a pulse of upflow of the ambient plasma. The theory is applied to two events of oscillating loops observed by the Transition Region and Coronal Explorer (TRACE). It is shown that the Lorentz force and drag with a reasonable value of the coupling coefficient (c _d ) and without anomalous dissipation are able to accurately account for the observed damped oscillations. The analysis shows that the variations in the observed intensity can be explained by the minor radial expansion and contraction. For the two events, the values of the drag coefficient consistent with the observed damping times are in the range c _d ≈2 – 5, with specific values being dependent on parameters such as the loop density, ambient magnetic field, and the loop geometry. This range is consistent with a previous MHD simulation study and with values used to reproduce the observed trajectories of coronal mass ejections (CMEs).     

Steps for Building a Calibrated Flux-Transport Dynamo for the Sun

Given the complexity involved in a flux-transport-type dynamo driven by both Babcock – Leighton and tachocline α effects, we present here a step-by-step procedure for building a flux-transport dynamo model calibrated to the Sun as a guide for anyone who wishes to build this kind of model. We show that a plausible sequence of steps to reach a converged solution in such a dynamo consists of (i) numerical integration of a classical α – ω dynamo driven by a tachocline α effect, (ii) continued integration with inclusion of meridional circulation to convert the model into a flux-transport dynamo driven by only a tachocline α effect, (iii) final integration with inclusion of a Babcock – Leighton surface α effect, resulting in a flux-transport dynamo that can be calibrated to obtain a close fit of model output with solar observations.     

A New Look at Mode Conversion in a Stratified Isothermal Atmosphere

Recent numerical investigations of wave propagation near coronal magnetic null points (McLaughlin and Hood: Astron. Astrophys. 459, 641, 2006) have indicated how a fast MHD wave partially converts into a slow MHD wave as the disturbance passes from a low-β plasma to a high-β plasma. This is a complex process and a clear understanding of the conversion mechanism requires the detailed investigation of a simpler model. An investigation of mode conversion in a stratified, isothermal atmosphere with a uniform, vertical magnetic field is carried out, both numerically and analytically. In contrast to previous investigations of upward-propagating waves (Zhugzhda and Dzhalilov: Astron. Astrophys. 112, 16, 1982a; Cally: Astrophys. J. 548, 473, 2001), this paper studies the downward propagation of waves from a low-β to high-β environment. A simple expression for the amplitude of the transmitted wave is compared with the numerical solution.     

Solar coronal heating by magnetosonic waves

Solar coronal heating by magnetohydrodynamic (MHD) waves is investigated. ultraviolet (UV) and X-ray emission lines of the corona show non-thermal broadenings. The wave rms velocities inferred from these observations are of the order of 25–60 km s⁻¹ . Assuming that these values are not negligible, we solved MHD equations in a quasi-linear approximation, by retaining the lowest order non-linear term in rms velocity. Plasma density distribution in the solar corona is assumed to be inhomogeneous. This plasma is also assumed to be permeated by dipole-like magnetic loops. Wave propagation is considered along the magnetic field lines. As dissipative processes, only the viscosity and parallel (to the local magnetic field lines) heat conduction are assumed to be important. Two wave modes emerged from the solution of the dispersion relation. The fast mode magneto-acoustic wave, if originated from the coronal base can propagate upwards into the corona and dissipate its mechanical energy as heat. The damping length-scale of the fast mode is of the order of 500 km. The wave energy flux associated with these waves turned out to be of the order of 2.5×10⁵ ergs cm⁻² s⁻¹ which is high enough to replace the energy lost by thermal conduction to the transition region and by optically thin coronal emission. The fast magneto-acoustic waves prove to be a likely candidate to heat the solar corona. The slow mode is absent, in other words cannot propagate in the solar corona.     

The solar dynamo

The solar dynamo continues to pose a challenge to observers and theoreticians. Observations of the solar surface reveal a magnetic field with a complex, hierarchical structure consisting of widely different scales. Systematic features such as the solar cycle, the butterfly diagram, and Hale's polarity laws point to the existence of a deep-rooted large-scale magnetic field. At the other end of the scale are magnetic elements and small-scale mixed-polarity magnetic fields. In order to explain these phenomena, dynamo theory provides all the necessary ingredients including the effect, magnetic field amplification by differential rotation, magnetic pumping, turbulent diffusion, magnetic buoyancy, flux storage, stochastic variations and nonlinear dynamics. Due to advances in helioseismology, observations of stellar magnetic fields and computer capabilities, significant progress has been made in our understanding of these and other aspects such as the role of the tachocline, convective plumes and magnetic helicity conservation. However, remaining uncertainties about the nature of the deep-seated toroidal magnetic field and the effect, and the forbidding range of length scales of the magnetic field and the flow have thus far prevented the formulation of a coherent model for the solar dynamo. A preliminary evaluation of the various dynamo models that have been proposed seems to favor a buoyancy-driven or distributed scenario. The viewpoint proposed here is that progress in understanding the solar dynamo and explaining the observations can be achieved only through a combination of approaches including local numerical experiments and global mean-field modeling.Received: 5 May 2003, Published online: 15 July 2003     

A New Model for Propagating Parts of EIT Waves: A Current Shell in a CME

EIT waves are observed in EUV as bright fronts. Some of these bright fronts propagate across the solar disk. EIT waves are all associated with a flare and a CME and are commonly interpreted as fast-mode magnetosonic waves. Propagating EIT waves could also be the direct signature of the gradual opening of magnetic field lines during a CME. We quantitatively addressed this alternative interpretation. Using two independent 3D MHD codes, we performed nondimensional numerical simulations of a slowly rotating magnetic bipole, which progressively result in the formation of a twisted magnetic flux tube and its fast expansion, as during a CME. We analyse the origins, the development, and the observability in EUV of the narrow electric currents sheets that appear in the simulations. Both codes give similar results, which we confront with two well-known SOHO/EIT observations of propagating EIT waves (7 April and 12 May 1997), by scaling the vertical magnetic field components of the simulated bipole to the line of sight magnetic field observed by SOHO/MDI and the sign of helicity to the orientation of the soft X-ray sigmoids observed by Yohkoh/SXT. A large-scale and narrow current shell appears around the twisted flux tube in the dynamic phase of its expansion. This current shell is formed by the return currents of the system, which separate the twisted flux tube from the surrounding fields. It intensifies as the flux tube accelerates and it is co-spatial with weak plasma compression. The current density integrated over the altitude has the shape of an ellipse, which expands and rotates when viewed from above, reproducing the generic properties of propagating EIT waves. The timing, orientation, and location of bright and faint patches observed in the two EIT waves are remarkably well reproduced. We conjecture that propagating EIT waves are the observational signature of Joule heating in electric current shells, which separate expanding flux tubes from their surrounding fields during CMEs or plasma compression inside this current shell. We also conjecture that the bright edges of halo CMEs show the plasma compression in these current shells.     

Principal Components and Independent Component Analysis of Solar and Space Data

Principal components analysis (PCA) and independent component analysis (ICA) are used to identify global patterns in solar and space data. PCA seeks orthogonal modes of the two-point correlation matrix constructed from a data set. It permits the identification of structures that remain coherent and correlated or that recur throughout a time series. ICA seeks for maximally independent modes and takes into account all order correlations of the data. We apply PCA to the interplanetary magnetic field polarity near 1 AU and to the 3.25R _⊙ source-surface fields in the solar corona. The rotations of the two-sector structures of these systems vary together to high accuracy during the active interval of solar cycle 23. We then use PCA and ICA to hunt for preferred longitudes in northern hemisphere Carrington maps of magnetic fields.     

The Effect of Abnormal Granulation on Acoustic Wave Travel Times and Mode Frequencies

Observations indicate that in plage areas (i.e. in active regions outside sunspots) acoustic waves travel faster than in the quiet Sun, leading to shortened travel times and higher p-mode frequencies. Coupled with the 11-year variation of solar activity, this may also explain the solar cycle variation of oscillation frequencies. While it is clear that the ultimate cause of any difference between the quiet Sun and plage is the presence of magnetic fields of order 100 G in the latter, the mechanism by which the magnetic field exerts its influence has not yet been conclusively identified. One possible such mechanism is suggested by the observation that granular motions in plage areas tend to be slightly “abnormal”, dampened compared to the quiet Sun. In this paper we consider the effect that abnormal granulation observed in active regions should have on the propagation of acoustic waves. Any such effect is found to be limited to a shallow surface layer where sound waves propagate nearly vertically. The magnetically suppressed turbulence implies higher sound speeds, leading to shorter travel times. This time shift Δ τ is independent of the travel distance, while it shows a characteristic dependence on the assumed plage field strength. As a consequence of the variation of the acoustic cutoff with height, Δ τ is expected to be significantly higher for higher frequency waves within the observed regime of 3 – 5 mHz. The lower group velocity near the upper reflection point further leads to an increased envelope time shift, as compared to the phase shift. p-mode frequencies in plage areas are increased by a corresponding amount, Δ ν/ν=ν Δ τ. These characteristics of the time and frequency shifts are in accordance with observations. The calculated overall amplitudes of the time and frequency shifts are comparable to, but still significantly less than (by a factor of 2 to 5), those suggested by measurements.