E. A. Kirichek

Sunspot as an isolated magnetic structure: Stability and oscillations

A simple energy model of a sunspot as a compact magnetic feature is described where the main energy contribution is provided by the coolest and most compressed part of the magnetic force tube of the spot at depths ranging from Wilson’s depression level (300–500 km) down to 2–3 thousand km. The equilibrium and stability conditions for such a system are analyzed using the variation principle, and oscillations of the system as a whole about the inferred equilibrium position are studied. The sunspot is shown to be stable in the magnetic field strength interval from 0.8–1 to 4–5 kG. The dependence of the eigenfrequency on magnetic field strength ω(B) is computed for the main oscillatory mode, where only the umbra of the sunspot takes part in oscillations, ω = ω1 (B). Lower subharmonics may appear in the case where penumbra too becomes involved in the oscillatory process: ω2 = ω1/2, ω3 = ω1/3. Theoretical curves agree well with the observational data obtained in Pulkovo using various independent methods: from temporal variations of sunspot magnetic field and from line-of-sight-velocity measurements. The periods of oscillations found range from 40 to 200 minutes.

The subphotospheric structure of a sunspot

Local helioseismology techniques yielding the temperature and flow-velocity distributions under a sunspot indicate an unambiguous sign for the horizontal gas-pressure difference between the spot and ambient medium at depths of 4 Mm and more. In the Parker sunspot model, the transverse equilibrium condition cannot be satisfied in these layers: a cluster of vertical, strongly compressed magnetic flux tubes in a plasma that is hotter than the ambient medium with flows that diverge sidewise cannot be in equilibrium. Equilibrium can be satisfied in the hot zone under the spot only if the magnetic flux tube expands sharply with depth, so that the mean magnetic-field strength decreases dramatically at depths exceeding 4 Mm. This corresponds to the “shallow” sunspot model that has been used to interpret long-period sunspot oscillations.

Magnetohydrostatics of a Vertical Flux Tube in the Solar Atmosphere: Coronal Loops, a Model of a Ring Flare Filament

The magnetohydrostatic theory of a twisted magnetic flux tube (rope) immersed in a realistic solar atmosphere is presented in a closed analytical form for the first time. General formulas that allow the equilibrium plasma density, pressure, and temperature distributions inside an axisymmetric vertical flux tube to be calculated from its magnetic structure, which is assumed to be known (fixed), have been derived. An analytical model of the external hydrostatic medium free of a magnetic field, the solar atmosphere, where the temperature profile of the semi-empirical tabulated Avrett-Loeser model is used, has been constructed. The distribution of plasma parameters in a twisted magnetic flux tube at small deviations of its internal magnetic structure from the force-free one has been calculated as an example of applying the general theoretical formulas. Since the tube cross section does not change with height, the constructed model can be applied to describe the vertical parts of coronal loops. It has been found that the plasma density in the magnetic flux tube rises when the field twisting exceeds the force-free level and falls with decreasing field twisting compared to the force-free level. This property of a twisted magnetic flux tube is of fundamental importance for justifying the mechanism of flare energy release in magnetic flux ropes. A model of a flare in a ring chromospheric configuration is considered.

Magnetic fields in a limb flare on July 19, 2012

The magnetic field intensity in the limb solar flare of July 19, 2012, was measured by the Zeeman bisector splitting in the Hα line. An average magnetic field is established to attain 200 G at a measurement error of ±100 G in the top of a radiant flare loop at a height of about 40 Mm. The confinement of a strong magnetic field in the high coronal arcade at a relatively weak external field of the corona (about 1–2 G) is considered using a model of a force-free magnetic flux rope with a fine magnetic structure at a scale of about 300 km.

Model of quiescent prominence with the helical magnetic field

A new exact analytical solution of the magnetohydrostatic problem describes the equilibrium of a solitary, dense-cool solar filament maintained against the gravity by magnetic force in hot solar corona at heights up to 20–40 Mm. The filament is assumed to be uniform along the axis (the translation symmetry). The magnetic field of the filament has the helical structure (magnetic flux rope) with a typical strength of a few Gauss in the region of minimal temperature (about 4000 K). The model can be applied to the quiescent prominence of both normal and inverse magnetic polarity.

Long-term oscillations of sunspots and a special class of artifacts in SOHO/MDI and SDO/HMI data

A specific type of artifacts (named as “p2p”), that originate due to displacement of the image of a moving object along the digital (pixel) matrix of receiver are analyzed in detail. The criteria of appearance and the influence of these artifacts on the study of long-term oscillations of sunspots are deduced. The obtained criteria suggest us methods for reduction or even elimination of these artifacts. It is shown that the use of integral parameters can be very effective against the “p2p” artifact distortions. The simultaneous observations of sunspot magnetic field and ultraviolet intensity of the umbra have given the same periods for the long-term oscillations. In this way the real physical nature of the oscillatory process, which is independent of the artifacts have been confirmed again. A number of examples considered here confirm the dependence between the periods of main mode of the sunspot magnetic field long-term oscillations and its strength. The dependence was derived earlier from both the observations and the theoretical model of the shallow sunspot. The anti-phase behavior of time variations of sunspot umbra area and magnetic field of the sunspot demonstrates that the umbra of sunspot moves in long-term oscillations as a whole: all its points oscillate with the same phase.

Free Oscillations of the Facula Node at the Stage of Slow Dissipation

A solar faculae having an appearance of quite long-lived magnetic nodes can perform (as well as sunspots, chromospheric filaments, coronal loops) free oscillations, i.e., they can oscillate about the stable equilibrium position as a single whole, changing quasi-periodically magnetic field averaged over the section with periods from 1 to 4 hours. Kolotkov et al. (2017) described the case in which the average magnetic field strength of the facula node considerably decreased during observations of SDO magnetograms (13 hours), and, at the same time, its oscillations acquired a specific character: the fundamental mode of free oscillations of the facula considerably increased in amplitude (by approximately two times), while the period of oscillations increased by three times. At the end of the process, the system dissipated. In this work, we present the exact solution of the equation of small-amplitude oscillations of the system with a time-variable rigidity, describing the oscillation behavior at which the elasticity of the system decreases with time, while the period and amplitude of oscillations grow.

Coronal Loop as an Element of the Potential Magnetic Arcade

A new analytical model of a fine singular 3D coronal loop is developed. The loop is a thin curved magnetic flux tube immersed in the potential magnetic arcade. The ambient corona is given by the hydrostatic model of a quiet solar atmosphere (Avrett and Loeser, 2008). The proposed 3D model of a fine coronal loop reproduces well the observed physical properties of coronal structures of this kind.

On the modeling of visible sunspot layers

One option for a stationary model of the asymmetric sunspot previously presented by the authors is considered. It is shown that the presence of sub-Alfvénic Evershed flows in the sunspot penumbra has almost no effect on the temperature distribution in the dense photospheric layers of the sunspot but significantly lowers the temperature of the chromosphere and the lower corona above the sunspot penumbra up to heights of 5–6 Mm.

Large convective cells in the sun: a theoretical model

The geometric parameters and relationships of the horizontal and vertical velocities in a giant convective cylindrical cell were analyzed based on a continuity equation with the given vertical distribution of unperturbed gas density in the convective zone of the Sun. It has been found that the lower edge of such a convective cell must be positioned at a depth of 20 Mm in the secondary helium ionization zone. In addition, under the assumption that the plasma conductivity in the convective zone is caused by small-scale turbulence, a new stationary solution was obtained for the problem of magnetic field diffusion in the pattern of slow conductive plasma flows in scales of a convective cell.