An overview is given of the observational and the theoretical methods used to investigate solar magnetic fields. It includes an introduction to the Stokes parameters, their radiative transfer in the presence of a magnetic field, and empirical techniques used to measure various properties of solar magnetic features, such as the strength and direction of the magnetic field, magnetic flux, temperature, velocity, size and lifetime. The MHD equations are introduced and some of the most common simplifications used to describe solar magnetic features are outlined. The application of these techniques to small-scale magnetic features is surveyed. The results of empirical and theoretical investigations of small-scale solar magnetic features are reviewed. Current views on their magnetic structure, thermal stratification, velocity field, size, distribution and evolution are presented. Finally, some open questions concerning small-scale solar magnetic fields are listed.

Small-scale solar magnetic fields: An overview

From satellite observations the solar total irradiance is known to vary. Sunspot blocking, facular emission, and network emission are three identified causes for the variations. In this paper we examine several different solar indices measured over the past century that are potential proxy measures for the Sun's irradiance. These indices are (1) the equatorial solar rotation rate, (2) the sunspot structure, the decay rate of individual sunspots, and the number of sunspots without umbrae, and (3) the length and decay rate of the sunspot cycle. Each index can be used to develop a model for the Sun's total irradiance as seen at the Earth. Three solar indices allow the irradiance to be modeled back to the mid-1700s. The indices are (1) the length of the solar cycle, (2) the normalized decay rate of the solar cycle, and (3) the mean level of solar activity. All the indices are well correlated, and one possible explanation for their nearly simultaneous variations is changes in the Sun's convective energy transport. Although changes in the Sun's convective energy transport are outside the realm of normal stellar structure theory (e.g., mixing length theory), one can imagine variations arising from even the simplest view of sunspots as vertical tubes of magnetic flux, which would serve as rigid pillas affecting the energy flow patterns by ensuring larger-scale eddies. A composite solar irradiance model, based upon these proxies, is compared to the northern hemisphere temperature depatures for 1700-1992. Approximately 71% of the decadal variance in the last century can be modeled with these solar indices, although this analysis does not include anthropogenic or other variations which would affect the results. Over the entire three centuries, approx. 50% of the variance is modeled. Both this analysis and previous similar analyses have correlations of model solar irradiances and measured Earth surface temperatures that are significant at better than the 95% confidence level. To understand our present climate variations, we must place the anthropogenic variations in the context of natural variability from solar, volcanic, oceanic, and other sources.

A discussion of plausible solar irradiance variations, 1700-1992

The magnetic field of the Sun is the underlying cause of the many diverse phenomena combined under the heading of solar activity. Here we describe the magnetic field as it threads its way from the bottom of the convection zone, where it is built up by the solar dynamo, to the solar surface, where it manifests itself in the form of sunspots and faculae, and beyond into the outer solar atmosphere and, finally, into the heliosphere. On the way it transports energy from the surface and the subsurface layers into the solar corona, where it heats the gas and accelerates the solar wind.

https://iopscience.iop.org/article/10.1088/0034-4885/69/3/R02/pdf

The solar magnetic field

Results from a realistic simulation of three-dimensional radiative magnetoconvection in a strong background magnetic field corresponding to the conditions in sunspot umbrae are shown. The convective energy transport is dominated by narrow upflow plumes with adjacent downflows, which become almost field-free near the surface layers. The strong external magnetic field forces the plumes to assume a cusplike shape in their top parts, where the upflowing plasma loses its buoyancy. The resulting bright features in intensity images correspond well (in terms of brightness, size, and lifetime) to the observed umbral dots in the central parts of sunspot umbrae. Most of the simulated umbral dots have a horizontally elongated form with a central dark lane. Above the cusp, most plumes show narrow upflow jets, which are driven by the pressure of the piled-up plasma below. The large velocities and low field strengths in the plumes are effectively screened from spectroscopic observation because the surfaces of equal optical depth are locally elevated, so that spectral lines are largely formed above the cusp. Our simulations demonstrate that nearly field-free upflow plumes and umbral dots are a natural result of convection in a strong, initially monolithic magnetic field.

/pdf/magnetoconvection-in-a-sunspot-umbra-11uvpfk4en.pdf

Magnetoconvection in a sunspot umbra

Magnetic flux emergence from the solar convection zone into the overlying atmosphere is the driver of a diverse range of phenomena associated with solar activity. In this article, we introduce theoretical concepts central to the study of flux emergence and discuss how the inclusion of different physical effects (e.g., magnetic buoyancy, magnetoconvection, reconnection, magnetic twist, interaction with ambient field) in models impact the evolution of the emerging field and plasma.

/pdf/flux-emergence-theory-45pkofupwr.pdf

Flux Emergence (Theory)

/pdf/the-growth-and-decay-of-sunspots-3zb0hacvnd.pdf

The Growth and Decay of Sunspots

Context. The stability of sunspots is one of the long-standing unsolved puzzles in the field of solar magnetism and the solar cycle. The thermal and magnetic structure of the sunspot beneath the solar surface is not accessible through observations, thus processes in these regions that contribute to the decay of sunspots can only be studied through theoretical and numerical studies. Aims. We study the effects that destabilise and stabilise the flux tube of a simulated sunspot in the upper convection zone. The depth-varying effects of fluting instability, buoyancy forces, and timescales on the flux tube are analysed. Methods. We analysed a numerical simulation of a sunspot calculated with the MURaM code. The simulation domain has a lateral extension of more than 98 Mm× 98 Mm and extends almost 18 Mm below the solar surface. The analysed data set of 30 hours shows a stable sunspot at the solar surface. We studied the evolution of the flux tube at defined horizontal layers (1) by means of the relative change in perimeter and area, that is, its compactness; and (2) with a linear stability analysis. Results. The simulation shows a corrugation along the perimeter of the flux tube (sunspot) that proceeds fastest at a depth of about 8 Mm below the solar surface. Towards the surface and towards deeper layers, the decrease in compactness is damped. From the stability analysis, we find that above a depth of 2 Mm, the sunspot is stabilised by buoyancy forces. The spot is least stable at a depth of about 3 Mm because of the fluting instability. In deeper layers, the flux tube is marginally unstable. The stability of the sunspot at the surface affects the behaviour of the field lines in deeper layers by magnetic tension. Therefore the fluting instability is damped at depths of about 3 Mm, and the decrease in compactness is strongest at a depth of about 8 Mm. The more vertical orientation of the magnetic field and the longer convective timescale lead to slower evolution of the corrugation process in layers deeper than 10 Mm. Conclusions. The formation of large intrusions of field-free plasma below the surface destabilises the flux tube of the sunspot. This process is not visible at the surface, where the sunspot is stabilised by buoyancy forces. The onset of sunspot decay occurs in deeper layers, while the sunspot still appears stable in the photosphere. The intrusions eventually lead to the disruption and decay of the sunspot.

The stability of sunspots

A sunspot is distinguished from a pore by having a filamentary penumbra, corresponding to convective motions that carry energy into the spot from the surrounding field-free plasma. A simplified model of energy transport in sunspots is developed in order to model the transition from pores to spots as the magnetic flux is varied. The observed overlap between the radii of large pores and small spots implies that the filamentary convective mode sets in suddenly and rapidly, as in the idealized case where pore solutions lose stability at a bifurcation.

The abrupt development of penumbrae in sunspots

A simple model is used to study the interaction of isolated magnetic flux tubes with convection in the sun. Convective motion in granules and supergranules is represented by prescribed flows in three-dimensional cells with square cross sections and thin flux tubes mode under the action of magnetic buoyance, Lorentz curvature forces, and aerodynamic drag. Inflow at the base of a cell competes with outflow at its upper surface; small flux tubes tend to be swept to the cell boundaries while larger, more buoyant tubes are dragged to the axis of the cell. These results are compared with recent observations of small-scale granular and intergranular magnetic fields.

H. U. Schmidt

Papers

The Growth and Decay of Sunspots

The stability of sunspots

The abrupt development of penumbrae in sunspots

Buoyant magnetic flux tubes. II: Three-dimensional behaviour in granules and supergranules

Buoyant magnetic flux tubes in supergranules.