Modelling photospheric magnetoconvection

The increasing power of computers makes it possible to model the non-linear interaction between magnetic fields and convection at the surfaces of solar-type stars in ever greater detail. We present the results of idealized numerical experiments on two-dimensional magnetoconvection in a fully compressible perfect gas. We first vary the aspect ratio λ of the computational box and show that the system runs through a sequence of convective patterns, and that it is only for a sufficiently wide box (λ ≥ 6) that the flow becomes insensitive to further increases in λ. Next, setting λ = 6, we decrease the field strength from a value strong enough to halt convection and find transitions to small-scale steady convection, next to spatially modulated oscillations (first periodic, then chaotic) and then to a new regime of flux separation, with regions of strong field (where convection is almost completely suppressed) separated by broad convective plumes. We also explore the effects of altering the boundary conditions and show that this sequence of transitions is robust. Finally, we relate these model calculations to recent high-resolution observations of solar magnetoconvection, in plage regions as well as in light bridges and the umbrae of sunspots.     

Converging and diverging convection around axisymmetric magnetic flux tubes

A numerical model of idealized sunspots and pores is presented, where axisymmetric cylindrical domains are used with aspect ratios (radius versus depth) up to 4. The model contains a compressible plasma with density and temperature gradients simulating the upper layer of the Sun's convection zone. Non-linear magnetohydrodynamic equations are solved numerically and time-dependent solutions are obtained where the magnetic field is pushed to the centre of the domain by convection cells. This central magnetic flux bundle is maintained by an inner convection cell, situated next to it and with a flow such that there is an inflow at the top of the numerical domain towards the flux bundle. For aspect ratio 4, a large inner cell persists in time, but for lower aspect ratios it becomes highly time dependent. For aspect ratios 2 and 3 this inner convection cell is smaller, tends to be situated towards the top of the domain next to the flux bundle, and appears and disappears with time. When it is gone, the neighbouring cell (with an opposite sense of rotation, i.e. outflow at the top) pulls the magnetic field away from the central axis. As this happens a new inner cell forms with an inflow which pushes the magnetic field towards the centre. This suggests that to maintain their form, both pores and sunspots need a neighbouring convection cell with inflow at the top towards the magnetic flux bundle. This convection cell does not have to be at the top of the convection zone and could be underneath the penumbral structure around sunspots. For an aspect ratio of 1, there is not enough space in the numerical domain for magnetic flux and convection to separate. In this case the solution oscillates between two steady states: two dominant convection cells threaded by magnetic field and one dominant cell that pushes magnetic flux towards the central axis.     

Spatially intermittent fields in photospheric magnetoconvection

Motivated by recent high-resolution observations of the solar surface, we investigate the problem of non-linear magnetoconvection in a three-dimensional compressible layer. We present results from a set of numerical simulations which model the situation in which there is a weak imposed magnetic field. This weak-field regime is characterized by vigorous granular convection and spatially intermittent magnetic field structures. When the imposed field is very weak, magnetic flux tends to accumulate at the edges of the convective cells, where it forms compact, almost 'point-like' structures which are reminiscent of those observed in the quiet Sun. If the imposed field is slightly stronger, there is a tendency for magnetic flux to become concentrated into 'ribbon-like' structures which are comparable to those observed in solar plages. The dependence of these simulations upon the strength of the imposed magnetic field is analysed in detail, and the concept of the fractal dimension is used to make a further, more quantitative comparison between these simulations and photospheric observations.     

Development of structure in pores and sunspots: flows around axisymmetric magnetic flux tubes

Flux elements, pores and sunspots form a family of magnetic features observed at the solar surface. As a first step towards developing a fully non-linear model of the structure of these features and of the dynamics of their interaction with solar convection, we conduct numerical experiments on idealized axisymmetric flux tubes in a compressible convecting atmosphere in cylindrical boxes of radius up to 8 times their depth. We find that the magnetic field strength of the flux tubes is roughly independent of both distance from the centre and the total flux content of the flux tube, but that the angle of inclination from the vertical of the field at the edge of the tube increases with flux content. In all our calculations, fluid motion converges on the flux tube at the surface. The results compare favourably with observations of pores; in contrast, large sunspots lie at the centre of an out-flowing moat cell. We conjecture that there is an inflow hidden beneath the penumbrae of large spots, and that this inflow is responsible for the remarkable longevity of such features.     

What is a flux tube? On the magnetic field topology of buoyant flux structures

We study the topology of field lines threading buoyant magnetic flux structures. The magnetic structures, visually resembling idealized magnetic flux tubes, are generated self-consistently by numerical simulation of the interaction of magnetic buoyancy and a localized velocity shear in a stably stratified atmosphere. Depending on the parameters, the system exhibits varying degrees of symmetry. By integrating along magnetic field lines and constructing return maps, we show that, depending on the type of underlying behaviour, the stages of the evolution, and therefore the degree of symmetry, the resulting magnetic structures can have field lines with one of three distinct topologies. When the x -translational and y -reflectional symmetries remain intact, magnetic field lines lie on surfaces but individual lines do not cover the surface. When the y symmetry is broken, magnetic field lines lie on surfaces and individual lines do cover the surface. When both x and y symmetries are broken, magnetic field lines wander chaotically over a large volume of the magnetically active region. We discuss how these results impact our simple ideas of a magnetic flux tube as an object with an inside and an outside, and introduce the concept of 'leaky' tubes.     

Magnetoconvection in an inclined magnetic field: linear and weakly non-linear models

Recent observations of sunspots have revealed a rich range of behaviour and a complicated magnetic field structure; magnetoconvection is the key physical process underlying these phenomena. Traditional studies of magnetoconvection have considered vertical, or sometimes horizontal, imposed fields. Tilted fields have received less attention, and yet these are crucial to sunspot dynamics, particularly in the penumbra where field lines are inclined at a variety of angles to the vertical. Tilting the field is also interesting from a purely theoretical viewpoint since it breaks many of the symmetries usually associated with convection problems. In this paper we study the linear stability of a layer permeated by an inclined magnetic field and go on to set up model equations in order to study the patterns formed in the weakly non-linear regime. Possible applications of the results to sunspots are discussed.     

Numerical simulations of rotating axisymmetric sunspots

A numerical model of axisymmetric convection in the presence of a vertical magnetic flux bundle and rotation about the axis is presented. The model contains a compressible plasma described by the non-linear MHD equations, with density and temperature gradients simulating the upper layer of the Sun's convection zone. The solutions exhibit a central magnetic flux tube in a cylindrical numerical domain, with convection cells forming collar flows around the tube. When the numerical domain is rotated with a constant angular velocity, the plasma forms a Rankine vortex, with the plasma rotating as a rigid body where the magnetic field is strong, as in the flux tube, while experiencing sheared azimuthal flow in the surrounding convection cells, forming a free vortex. As a result, the azimuthal velocity component has its maximum value close to the outer edge of the flux tube. The azimuthal flow inside the magnetic flux tube and the vortex flow is prograde relative to the rotating cylindrical reference frame. A retrograde flow appears at the outer wall. The most significant convection cell outside the flux tube is the location for the maximum value of the azimuthal magnetic field component. The azimuthal flow and magnetic structure are not generated spontaneously, but decay exponentially in the absence of any imposed rotation of the cylindrical domain.     

Magnetic flux participation in solar surface magnetism during solar cycle 24

This study aims at investigating surface magnetic flux participation among different types of magnetic features during solar cycle 24. State-of-the-art observations from SDO/HMI and Hinode/SOT are combined to form a unique database in the interval from April 2010 to October 2015. Unlike previous studies, the statistics presented in this paper are feature-detection-based. More than 20 million magnetic features with relatively large scale, such as sunspot/pore, enhanced and quiet networks, are automatically detected and categorized from HMI observations, and the internetwork features are identified from SOT/SP observations. The total flux from these magnetic features reaches 5.9×10²² Mx during solar minimum and2.4 × 10²³ Mx in solar maximum. Flux occupation from the sunspot/pore region is 29% in solar maximum.Enhanced and quiet networks contribute 18% and 21% flux during the solar minimum, and 50% and 9% flux in the solar maximum respectively. The internetwork field contributes over 55% of flux in the solar minimum, and its flux contribution exceeds that of sunspot/pore features in the solar maximum. During the solar active condition, the sunspot field increases its area but keeps constant flux density of about 150 G,while the enhanced network follows the sunspot number variation showing increasing flux density and area,but the quiet network displays decreasing area and somewhat increasing flux density of about 6%. The origin of the quiet network is not known exactly, but is suggestive of representing the interplay between mean-field and local dynamos. The source, magnitude and possible importance of ‘hidden flux' are discussed in some detail.     

Magnetic field inclination and atmospheric oscillations above solar active regions

Recent observational evidence for magnetic field direction effects on helioseismic signals in sunspot penumbrae is suggestive of magnetohydrodynamic (MHD) mode conversion occurring at lower levels. This possibility is explored using wave mechanical and ray theory in a model of the Sun's surface layers permeated by uniform inclined magnetic field. It is found that fast-to-slow conversion near the equipartition depth at which the sound and Alfvén speeds coincide can indeed greatly enhance the atmospheric acoustic signal at heights observed by Solar and Heliospheric Observatory/Michelson Doppler Imager and other helioseismic instruments, but that this effect depends crucially on the wave attack angle , i.e. the angle between the wavevector and the magnetic field at the conversion/transmission depth. A major consequence of this insight is that the magnetic field acts as a filter, preferentially allowing through acoustic signal from a narrow range of incident directions. This is potentially testable by observation.     

Modulation and symmetry changes in stellar dynamos

Stellar dynamos are governed by non-linear partial differential equations (PDEs) which admit solutions with dipole, quadrupole or mixed symmetry (i.e. with different parities). These PDEs possess periodic solutions that describe magnetic cycles, and numerical studies reveal two different types of modulation. For modulations of Type 1 there are parity changes without significant changes of amplitude, while for Type 2 there are amplitude changes without significant changes in parity. In stars like the Sun, cyclic magnetic activity is interrupted by grand minima that correspond to Type 2 modulation. Although the Sun's magnetic field has maintained dipole symmetry for almost 300 yr, there was a significant parity change at the end of the Maunder Minimum. We infer that the solar field may have flipped from dipole to quadrupole polarity (and back) after deep minima in the past and may do so again in the future. Other stars, with different masses or rotation rates, may exhibit cyclic activity with dipole, quadrupole or mixed parity. The origins of such behaviour can be understood by relating the PDE results to solutions of appropriate low-order systems of ordinary differential equations (ODEs). Type 1 modulation is reproduced in a fourth-order system while Type 2 modulation occurs in a third-order system. Here we construct a new sixth-order system that describes both types of modulation and clarifies the interactions between symmetry-breaking and modulation of activity. Solutions of these non-linear ODEs reproduce the qualitative behaviour found for the PDEs, including flipping of polarity after a prolonged grand minimum. Thus we can be confident that these patterns of behaviour are robust, and will apply to stars that are similar to the Sun.     

Convective intensification of magnetic fields in the quiet Sun

Kilogauss-strength magnetic fields are often observed in intergranular lanes at the photosphere in the quiet Sun. Such fields are stronger than the equipartition field B _e, corresponding to a magnetic energy density that matches the kinetic energy density of photospheric convection, and comparable with the field B _p that exerts a magnetic pressure equal to the ambient gas pressure. We present an idealized numerical model of three-dimensional compressible magnetoconvection at the photosphere, for a range of values of the magnetic Reynolds number. In the absence of a magnetic field, the convection is highly supercritical and characterized by a pattern of vigorous, time-dependent, 'granular' motions. When a weak magnetic field is imposed upon the convection, magnetic flux is swept into the convective downflows where it forms localized concentrations. Unless this process is significantly inhibited by magnetic diffusion, the resulting fields are often much greater than B _e and the high magnetic pressure in these flux elements leads to their being partially evacuated. Some of these flux elements contains ultraintense magnetic fields that are significantly greater than B _p. Such fields are contained by a combination of the thermal pressure of the gas and the dynamic pressure of the convective motion, and they are constantly evolving. These ultraintense fields develop owing to non-linear interactions between magnetic fields and convection; they cannot be explained in terms of 'convective collapse' within a thin flux tube that remains in overall pressure equilibrium with its surroundings.     

Dynamical effects in solar photospheric flux tubes

The interaction of an intense flux tube, extending vertically through the photosphere, with p-modes in the ambient medium is modelled by solving the time dependent MHD equations in the thin flux tube approximation. It is found that a resonant interaction can occur, which leads to the excitation of flux tube oscillations with large amplitudes. The resonance is not as sharp as in the case of an unstratified atmosphere, but is broadened by a factor proportional toH ^–2, whereH is the local pressure scale height. In addition, the inclusion of radiative transport leads to a decrease in the amplitude of the oscillations, but does not qualitatively change the nature of the interaction.     

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     

Magnetic and thermal phase shifts in the local helioseismology of sunspots

Phase perturbations due to inclined surface magnetic field of active region strength are calculated numerically in quiet Sun and simple sunspot models in order to estimate and compare the direct and indirect (thermal) effects of the fields on helioseismic waves. It is found that the largest direct effects occur in highly inclined field characteristic of penumbrae, and scale roughly linearly with magnetic field strength. The combined effects of sunspot magnetic and thermal anomalies typically yield negative travel-time perturbations in penumbrae. Travel-time shifts in umbrae depend on details of how the thermal and density structure differs from the quiet Sun. The combined shifts are generally not well approximated by the sum of the thermal and magnetic effects applied separately, except at low field strengths of around 1 kG or less, or if the thermal shift is small. A useful rule-of-thumb appears to be that travel-time perturbations in umbrae are predominantly thermal, whereas in penumbrae they are mostly magnetic.     

Dynamo action in simulations of penetrative solar convection with an imposed tachocline

We summarize new and continuing three-dimensional spherical shell simulations of dynamo action by convection allowed to penetrate downward into a tachocline of rotational shear. The inclusion of an imposed tachocline allows us to examine several processes believed to be essential in the operation of the global solar dynamo, including differential rotation, magnetic pumping, and the stretching and organization of fields within the tachocline. In the stably stratified core, our simulations reveal that strong axisymmetric magnetic fields (of ∼ 3000 G strength) can be built, and that those fields generally exhibit a striking antisymmetric parity, with fields in the northern hemisphere largely of opposite polarity to those in the southern hemisphere. In the convection zone above, fluctuating fields dominate over weaker mean fields. New calculations indicate that the tendency toward toroidal fields of antisymmetric parity is relatively insensitive to initial magnetic field configurations; they also reveal that on decade-long timescales, the magnetic fields can briefly enter (and subsequently emerge from) states of symmetric parity.We have not yet observed any overall reversals of the field polarity, nor systematic latitudinal propagation. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Helicity of solar active regions from a dynamo model

We calculate helicities of solar active regions based on the idea that poloidal flux lines get wrapped around a toroidal flux tube rising through the convection zone, thereby giving rise to the helicity. We use our solar dynamo model based on the Babcock-Leighton α-effect to study how helicity varies with latitude and time.     

Solar dynamo models with α ‐effect and turbulent pumping from local 3D convection calculations

Results from kinematic solar dynamo models employing α ‐effect and turbulent pumping from local convection calculations are presented. We estimate the magnitude of these effects to be around 2–3 m s^–1, having scaled the local quantities with the convective velocity at the bottom of the convection zone from a solar mixing‐length model. Rotation profile of the Sun as obtained from helioseismology is applied in the models; we also investigate the effects of the observed surface shear layer on the dynamo solutions. With these choices of the small‐ and large‐scale velocity fields, we obtain estimate of the ratio of the two induction effects, C _α /C _Ω ≈ 10^–3, which we keep fixed in all models. We also include a one‐cell meridional circulation pattern having a magnitude of 10–20 m s^–1 near the surface and 1–2 m s^–1 at the bottom of the convection zone. The model essentially represents a distributed turbulent dynamo, as the α ‐effect is nonzero throughout the convection zone, although it concentrates near the bottom of the convection zone obtaining a maximum around 30° of latitude. Turbulent pumping of the mean fields is predominantly down‐ and equatorward. The anisotropies in the turbulent diffusivity are neglected apart from the fact that the diffusivity is significantly reduced in the overshoot region. We find that, when all these effects are included in the model, it is possible to correctly reproduce many features of the solar activity cycle, namely the correct equatorward migration at low latitudes and the polar branch at high latitudes, and the observed negative sign of B _r B _ϕ . Although the activity clearly shifts towards the equator in comparison to previous models due to the combined action of the α ‐effect peaking at midlatitudes, meridional circulation and latitudinal pumping, most of the activity still occurs at too high latitudes (between 5° … 60°). Other problems include the relatively narrow parameter space within which the preferred solution is dipolar (A0), and the somewhat too short cycle lengths of the solar‐type solutions. The role of the surface shear layer is found to be important only in the case where the α ‐effect has an appreciable magnitude near the surface. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Spectropolarimetery of umbral fine structures from Hinode: evidence for magnetoconvection

We present spectropolarimetric analysis of umbral dots and a light bridge fragment that show dark lanes in G -band images. Umbral dots show upflow as well as associated positive Stokes V area asymmetry in their central parts. Larger umbral dots show downflow patches in their surrounding parts that are associated with negative Stokes V area asymmetry. Umbral dots show weaker magnetic field in central part and higher magnetic field in peripheral area. Umbral fine structures are much better visible in total circularly polarized light than in continuum intensity. Umbral dots show a temperature deficit above dark lanes. The magnetic field inclination shows a cusp structure above umbral dots and a light bridge fragment. We compare our observational findings with 3D magnetohydrodynamic simulations.