prior to surface appearance of sunspot flux tubes and magneto-thermal pulsation of the Sun

We found an evidence that the luminosity of the Sun systematically decreased about 20 days before sunspot surface appearance by analysing time-lag correlation of time derivatives of running mean time profiles of the data of the Active Cavity Radiometer Irradiance Monitor (ACRIM) I experiment on board of Solar Maximum Mission (SMM) and of the data of the daily sunspot number. This indicates that sunspot flux tube cooling and heat transport blocking by the flux tubes start to take place in the interior of the solar convection zone well before the sunspot surface appearance. From this finding and our previous finding that the luminosity of the Sun systematically increased and the blocked heat appeared on the surface about 50 days after the sunspot surface appearance, a new view of sunspot formation and dynamics and a new view of the luminosity modulation emerged. (i) Sunspots of a solar cycle are formed from clusters of flux tubes which can be seen in the running mean time profile of the sunspot number as a peak with duration on the order of 100 to 200 days. (ii) Heat flow is blocked by the cluster of sunspot flux tubes inside the convection zone to decrease the luminosity about 20 days before the surface emergence of the sunspot cluster. (iii) The blocked heat appears on the surface about 50 days after the surface emergence of the cluster of sunspot flux tubes to heat up the surface. This appears as a thermal pulse in the running mean time profile of the ACRIM dat in between the peaks of the sunspot running mean time profile. This process of heating the surface makes the temperature gradient less steep and weakens the buoyancy of sunspot flux tubes below the surface. (vi) The radiative cooling of the surface layer by the excess heat release steepens the temperature gradient so that the buoyancy of the sub-surface magnetic flux tubes becomes stronger to cause the next surge of emergence of a cluster of sunspots and other magnetic activities, which creates a peak in the time profile of the sunspot number. We call this peak a magnetic pulse of the Sun and the coupled process of alternating pulsed appearance of heat and sunspots the magneto-thermal pulsation of the Sun.     

Formation of polar starspots through meridional circulation

To explain the observed intermingling of polarities in the magnetic field distributions of rapidly rotating stars, surface magnetic flux transport models demand the presence of fast meridional flows.We combine simulations of the pre-eruptive and post-eruptive magnetic flux transport in cool stars to investigate the influence of a fast meridional circulation on the latitudinal eruption pattern of magnetic flux tubes and on the polar magnetic field properties. Magnetic flux tubes rising through the convection zone experience an enhanced latitude-dependent poleward deflection through meridional flows, which renders the wings of stellar butterfly diagrams convex. The larger amount of magnetic flux emerging at higher latitudes supports the intermingling of opposite polarities of polar magnetic fields and yields magnetic flux densities in the polar regions about 20% higher than in the case disregarding the pre-eruptive deflection. Taking the pre-eruptive evolution of magnetic flux into account therefore eases the need for the fast meridional flows predicted by previous investigations. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A coupled model of magnetic flux generation and transport in stars

We present a combined model for magnetic field generation and transport in cool stars with outer convection zones. The mean toroidal magnetic field, which is generated by a cyclic thin-layer α Ω dynamo at the bottom of the convection zone is taken to determine the emergence probability of magnetic flux tubes in the photosphere. Following the nonlinear rise of the unstable thin flux tubes, emergence latitudes and tilt angles of bipolar magnetic regions are determined. These quantities are put into a surface flux transport model, which simulates the surface evolution of magnetic flux under the effects of large-scale flows and turbulent diffusion. First results are discussed for the case of the Sun and for more rapidly rotating solar-type stars. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Spot signatures of a solar‐type dynamo in close binary stars

Magnetic activity signatures in the atmosphere of active stars can be used to place constrains on the underlying processes of flux transport and dynamo operation in its convective envelope. The ‘solar paradigm’ for magnetic activity suggests that the magnetic field is amplified and stored at the base of the convection zone. Once a critical field strength is exceeded, perturbations initiate the onset of instabilities and the growth of magnetic flux loops, which rise through the convection zone, emerge at the stellar surface, and eventually lead to the formation of starspots and active regions. In close binaries, the proximity of the companion star breaks the rotational symmetry. Although the magnitude of tidal distortions is rather small, non‐linear MHD simulations have nevertheless shown in the case of main‐sequence binary components that they can cause non‐uniform surface distributions of flux tube eruptions. The present work extends the investigation to post‐mainsequence components to explore the specific influence of the stellar structure on the surface pattern of erupting flux tubes. In contrast to the case of main‐sequence components, where the consistency between simulation results and observations supports the presumption of a solar‐like dynamo mechanism, the numerical results here do not recover the starspot properties frequently observed on evolved binary components. This aspect points out an insufficiency of the applied flux tube model and leads to the conclusion that additional flux transport and possibly amplification mechanisms have to be taken into account. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the appearance of magnetic flux in the solar photosphere

Ideas and models for the appearance of photospheric magnetic structure are confronted with observational data. Some findings are: The magnetic flux emerging in an active region consists of a bundle of flux tubes which were already concentrated before penetrating into the photosphere. A model of a rising bunch of flux tubes joining into a few strands at larger depths describes the coalescence of spots near the leading and following edges of the active region while more flux may surface near the center of the region. There is no observational evidence for appreciable helical twists in the flux bundles.Throughout the region's lifetime the magnetic elements move coherently, the whole magnetic structure rotates faster than the quiet photosphere. In active regions the convective flow at scales larger than the granulation is arrested by the magnetic structure. The long-lived supergranular cells around spots and in the enhanced network in turn determine the decay properties of spots and facular clusters. The modulation of the convective flow by the magnetic structure explains the slow dispersal of faculae.The hierarchy of magnetic elements (sunspots-pores-knots-facular clusters-facular points) may be explained by a set of magnetostatic flux tube models in the top of the convection zone. The underlying assumptions are that the heat flow along the magnetic field is reduced and that there is no heat exchange across the field except by radiation.A tentative model is proposed to account for the amplification, ascent and emergence of intense flux bundles. The assumptions are: (i) the field is concentrated in toroidal bundles by differential rotation, (ii) in the deep convection zone flux bundles are contained by the external turbulent pressure, and (iii) for field strengths up to the equipartition value efficient lateral heat exchange is possible. After a loop has surfaced radiative cooling and subsequent convective downflow reduce the temperature in the top of the flux tubes which then contract to field strengths well above the local equipartition value. There the heat flow is channelled along the field, which creates the conditions for the magnetostatic flux tube models without requiring a blocking of the heat flow somewhere within the tubes.The paper contains a brief review on the evolution of the magnetic field from the emergence in active regions up to the enigmatic disappearance, and a list of topics for further observational investigation.     

3-D non-linear evolution of a magnetic flux tube in a spherical shell: The isentropic case

We present recent 3-D MHD numerical simulations of the non-linear dynamical evolution of magnetic flux tubes in an adiabatically stratified convection zone in spherical geometry, using the anelastic spherical harmonic (ASH) code.We seek to understand the mechanism of emergence of strong toroidal fields from the base of the solar convection zone to the solar surface as active regions. We confirm the results obtained in cartesian geometry that flux tubes that are not twisted split into two counter vortices before reaching the top of the convection zone. Moreover, we find that twisted tubes undergo the poleward-slip instability due to an unbalanced magnetic curvature force which gives the tube a poleward motion both in the non-rotating and in the rotating case. This poleward drift is found to be more pronounced on tubes originally located at high latitudes. Finally, rotation is found to decrease the rise velocity of the flux tubes through the convection zone, especially when the tube is introduced at low latitudes. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Radiative heating and the buoyant rise of magnetic flux tubes in the solar interior

We study the effect of radiative heating on the evolution of thin magnetic flux tubes in the solar interior and on the eruption of magnetic flux loops to the surface. Magnetic flux tubes experience radiative heating because (1) the mean temperature gradient in the lower convection zone and the overshoot region deviates substantially from that of radiative equilibrium, and hence there is a non-zero divergence of radiative heat flux; and (2) the magnetic pressure of the flux tube causes a small change of the thermodynamic properties within the tube relative to the surrounding field-free fluid, resulting in an additional divergence of radiative heat flux. Our calculations show that the former constitutes the dominant source of radiative heating experienced by the flux tube.In the overshoot region, the radiative heating is found to cause a quasi-static rising of the toroidal flux tubes with an upward drift velocity 10^-3|| cm s^-1, where _e – _ad < 0 describes the subadiabaticity in the overshoot layer. The upward drift velocity does not depend sensitively on the field strength of the flux tubes. Thus in order to store toroidal flux tubes in the overshoot region for a period comparable to the length of the solar cycle, the magnitude of the subadiabaticity (< 0) in the overshoot region must be as large as 3 × 10^–4. We discuss the possibilities for increasing the magnitude of and for reducing the rate of radiative heating of the flux tubes in the overshoot region.Using numerical simulations we study the formation of -shaped emerging loops from toroidal flux tubes in the overshoot region as a result of radiative heating. The initial toroidal tube is assumed to be non-uniform in its thermodynamic properties along the tube and lies at varying depths beneath the base of the convection zone. The tube is initially in a state of neutral buoyancy with the internal density of the tube plasma equal to the local external density. We find from our numerical simulations that such a toroidal tube rises quasi-statically due to radiative heating. The top portion of the nonuniform tube first enters the convection zone and may be brought to an unstable configuration which eventually leads to the eruption of an anchored flux loop to the surface. Assuming reasonable initial parameters, our numerical calculations yield fairly short rise times (2–4 months) for the development of the emerging flux loops. This suggests that radiative heating is an effective way of causing the eruption of magnetic flux loops, leading to the formation of active regions at the surface.The National Solar Observatory is one of the National Optical Astronomy Observatories by the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the National Science Foundation.     

Heat flow near obstacles in the solar convection zone

Disturbances in the heat flow in the solar convection zone are calculated with a turbulent thermal diffusion coefficient based on a mixing length approximation. As a consequence of the radiative boundary condition at the surface and the strong increase of the diffusion coefficient with depth, the convection zone resembles a thermally superconducting shell enclosed between a thin surface layer and an interior core of low thermal conductivity. Thermal disturbances originating in the convection zone do not penetrate into the interior, and penetrate only weakly through the solar surface. A thermally isolating obstacle buried entirely in the convection zone casts a shadow of reduced temperature at the solar surface; the brightening surrounding this shadow is undetectable. The shadow is weak unless the object is located close to the surface (less than 2000 km). Assuming a sunspot to be an area of reduced thermal conductivity which extends a finite depth into the convection zone, the heat flow around this obstacle is calculated. The heat flux blocked below the spot (missing flux) spreads over a very extended area surrounding the spot. The brightening corresponding to this missing flux is undetectable if the reduction of the thermal conductivity extends to a depth greater than 1000 km. It is concluded that no effect other than a decrease of the convective efficiency is needed to explain the temperature change observed at the solar surface in and around a sunspot. The energy balance is calculated between magnetic flux tubes, oriented vertically in the solar surface, (magnetic elements in active regions and the quiet network) and their surroundings. Near the visible surface radiation enters the tube laterally from the surrounding convection zone. The heating effect of this influx is important for small tubes (less than a few arcseconds). Due to this influx tubes less than about 1 in diameter can appear as bright structures irrespective of the amount of heat conveyed along the tube itself. Through the lateral influx, small tubes such as are found in the quiet network act as little leaks in the solar surface through which an excess heat flux escapes from the convection zone.     

On the Connection Between Mean Field Dynamo Theory and Flux Tubes

Mean field dynamo theory deals with various mean quantities and does not directly throw any light on the question of existence of flux tubes. We can, however, draw important conclusions about flux tubes in the interior of the Sun by combining additional arguments with the insights gained from solar dynamo solutions. The polar magnetic field of the Sun is of order 10 G, whereas the toroidal magnetic field at the bottom of the convection zone has been estimated to be 100000 G. Simple order-of-magnitude estimates show that the shear in the tachocline is not sufficient to stretch a 10 G mean radial field into a 100000 G mean toroidal field. We argue that the polar field of the Sun must get concentrated into intermittent flux tubes before it is advected to the tachocline. We estimate the strengths and filling factors of these flux tubes. Stretching by shear in the tachocline is then expected to produce a highly intermittent magnetic configuration at the bottom of the convection zone. The meridional flow at the bottom of the convection zone should be able to carry this intermittent magnetic field equatorward, as suggested recently by Nandy and Choudhuri (2002). When a flux tube from the bottom of the convection zone rises to a region of pre-existing poloidal field at the surface, we point out that it picks up a twist in accordance with the observations of current helicities at the solar surface.     

Mechanism of cyclically polarity reversing solar magnetic cycle as a cosmic dynamo

We briefly describe historical development of the concept of solar dynamo mechanism that generates electric current and magnetic field by plasma flows inside the solar convection zone. The dynamo is the driver of the cyclically polarity reversing solar magnetic cycle. The reversal process can easily and visually be understood in terms of magnetic field line stretching and twisting and folding in three-dimensional space by plasma flows of differential rotation and global convection under influence of Coriolis force. This process gives rise to formation of a series of huge magnetic flux tubes that propagate along iso-rotation surfaces inside the convection zone. Each of these flux tubes produces one solar cycle. We discuss general characteristics of any plasma flows that can generate magnetic field and reverse the polarity of the magnetic field in a rotating body in the Universe. We also mention a list of problems which are currently being disputed concerning the solar dynamo mechanism together with observational evidences that are to be constraints as well as verifications of any solar cycle dynamo theories of short and long term behaviors of the Sun, particularly time variations of its magnetic field, plasma flows, and luminosity.     

The impact of meridional circulation on stellar butterfly diagrams and polar caps

Observations of rapidly rotating solar-like stars show a significant mixture of opposite-polarity magnetic fields within their polar regions. To explain these observations, models describing the surface transport of magnetic flux demand the presence of fast meridional flows. Here, we link subsurface and surface magnetic flux transport simulations to investigate (i) the impact of meridional circulations with peak velocities of  ≤125 m s⁻¹  on the latitudinal eruption pattern of magnetic flux tubes and (ii) the influence of the resulting butterfly diagrams on polar magnetic field properties. Prior to their eruption, magnetic flux tubes with low field strengths and initial cross-sections below  ∼300 km  experience an enhanced poleward deflection through meridional flows (assumed to be polewards at the top of the convection zone and equatorwards at the bottom). In particular, flux tubes which originate between low and intermediate latitudes within the convective overshoot region are strongly affected. This latitude-dependent poleward deflection of erupting magnetic flux renders the wings of stellar butterfly diagrams distinctively convex. The subsequent evolution of the surface magnetic field shows that the increased number of newly emerging bipoles at higher latitudes promotes the intermingling of opposite polarities of polar magnetic fields. The associated magnetic flux densities are about 20 per cent higher than in the case disregarding the pre-eruptive deflection, which eases the necessity for fast meridional flows predicted by previous investigations. In order to reproduce the observed polar field properties, the rate of the meridional circulation has to be of the order of 100 m s⁻¹, and the latitudinal range from which magnetic flux tubes originate at the base of the convective zone (≲50°) must be larger than in the solar case (≲35°).     

The rise of twisted magnetic flux tubes

Sunspots are caused by the eruption of magnetic flux tubes through the solar photosphere: current theories of the internal magnetic field of the Sun suggest that such tubes must rise relatively unscathed from the base of the convection zone. In order to understand how the structure of the magnetic field within a buoyant flux tube affects its stability as it rises, we have considered the quasi-two-dimensional rise of isolated magnetic flux tubes through an adiabatically stratified atmosphere. The magnetic field is initially helical; we have investigated a range of initial field configurations, varying the distribution and strength of the twist of the field.     

The three-dimensional breakup of a magnetic layer

Three-dimensional numerical simulations of the instability of a layer of magnetic field caused by magnetic buoyancy are carried out over a range of parameter values. The layer breaks up into a number of interlocking magnetic flux tubes that become increasingly three-dimensional, although strongly arched flux tubes are not observed. The introduction of background rotation has the principal effect of suppressing the instability. The α -effect, which measures the twist of the flux tubes induced by the rotation, is found to be positive (in the northern hemisphere) but small in magnitude.     

Stability of magnetic flux tubes in close binary stars

Photometric and Doppler imaging observations of active binaries indicate the existence of starspots at preferred longitudes (position angles with respect to the companion star). We investigate the stability of magnetic flux tubes in the convection zone of close, fast‐rotating binary stars and explore whether the observed preferred longitudes could be caused by tidal forces and the deformation of the active star. We assume a synchronized binary system with spin axes perpendicular to the orbital plane and a rotation period of a few days. The tidal force and the deviation from spherical structure are considered in lowest‐order perturbation theory. The magnetic field is in the form of toroidal magnetic flux rings, which are stored in mechanical equilibrium within the stably stratified overshoot region beneath the convection zone until the field has grown sufficiently strong for the undulatory instability to initiate the formation of rising loops. Frequencies and geometry of stable as well as growth rates of unstable eigenmodes are determined by linear stability analysis. Particular consideration is given to the question whether the effects of tidal forces and perturbations of the stellar structure can force a rising flux loop to enter the convection zone at specific longitudes.     

Fluid Dynamics and MHD of the Solar Convection Zone and Tachocline: Current Understanding and Unsolved Problems – (Invited Review)

We review recent progress and define unanswered scientific questions in five related topics: granulation- to supergranulation-scale convection and magnetic structures; global convection and circulation; the rise of magnetic flux tubes to the photosphere, and their injection into the base of the convection zone; tachocline fluid dynamics and MHD; and the solar dynamo. We close with a set of observational `targets' for helioseismologists to aim for.     

The emergence of magnetic flux

This paper first summarizes the morphology and dynamics of emerging flux regions and arch filament systems and then discusses detailed observations of a particular active region with emerging magnetic flux.The central part of the growing active region shows abnormal granulation and a weak magnetic field that, locally, is transverse. In the border zone, strong downward flows occur in the chromopshere and photosphere (small features with strong magnetic fields (faculae, pores) are formed here.) Near the leading and following edge, sunspots are formed by the coalescence of such small magnetic elements.The observational data are interpreted by means of a heuristic model of an emergent magnetic loop-shaped bundle consisting of many flux tubes. In this model we incorporate the theory of convective collapse and the buoyancy of flux tubes. The observed complexity in the structure and dynamics, including strong transverse fields and velocity shear, is attributed to the emergence of several flux regions within the active region at different orientations.     

Gas entry into non-spot magnetic tubes

Gas penetration into twisted magnetic tubes can occur by the inward diffusion of neutral atoms in the neighbourhood of the temperature minimum between photosphere and chromosphere, where the degree of ionization is low. Again, turbulent buffeting indents tubes in the convection zone and, in particular near the photosphere, provides a larger area where the overall diffusion rate may be enhanced. These processes do not contribute rapidly to the gas content of magnetic tubes, but diffusion near the temperature minimum may well be the source of the observed downflow in magnetic points.     

The effect of Kelvin-Helmholtz instability on rising flux tubes in the convection zone

If the solar dynamo operates at the bottom of the convection zone, then the magnetic flux created there has to rise to the surface. When the convection zone is regarded as passive, the rising flux is deflected by the Coriolis force to emerge at rather high latitudes, poleward of typical sunspot zones (Choudhuri and Gilman, 1987; Choudhuri, 1989). Choudhuri and D'Silva (1990) included the effects of convective turbulence on the rising flux through (a) giant cell drag and (b) momentum exchange by small-scale turbulence. The momentum exchange mechanism could enable flux tubes of radii not more than a few hundred km to emerge radially at low latitudes, but the giant cell drag mechanism required unrealistically small flux tube radii (a few meters for a reasonable giant cell upflow) to counteract the Coriolis force. We now include the additional effect of Kelvin-Helmholtz instability in a symmetrical flux ring caused by the azimuthal flow induced during its rise. The azimuthal flow crosses the threshold for the instability only if there is a giant cell upflow to drag the flux tubes appreciably. In the absence of such a drag, as in the case of a passive convection zone or in the case of momentum exchange by small-scale turbulence, the azimuthal velocity never becomes large enough to cause the instability, leaving the results of the previous calculations unaltered. The giant cell drag, aided by Kelvin-Helmholtz instability, however, becomes now a viable mechanism for curbing the Coriolis force - 10⁴ G flux tubes with radii of a few hundred km being dragged radially by upflows of 70 m s^-1.