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There are two main properties of a shielded magnetic flux rope. The first is the net electric current through its cross-section should be zero, i.e. $I = 0$. The second is the existence of a non-zero pressure of external magnetic field ${{B_{ex}^2} {/ {\vphantom {{B_{ex}^2} {8\pi}}}} {8\pi }}$ in order to keep in balance the magnetic rope with the strong force-free inner structure in a rarefied solar atmosphere. The first condition requires the existence of a special cylindrical surface within the magnetic flux rope on which the current density j changes sign, so that the direction of the current at the rope’s periphery (return current) is opposite to the current at its axis (direct current). Numerical calculations have shown that, when the pressure ${{B_{ex}^2} {/ {\vphantom {{B_{ex}^2} {8\pi }}}} {8\pi }}$ drops to a certain limit, an indefinite increase in the electric current density j and the force-free parameter $\alpha $ takes place in the vicinity of this special surface resulting in a dissipative collapse of the system. Such a drop in ${{B_{ex}^2} {/ {\vphantom {{B_{ex}^2} {8\pi }}}} {8\pi }}$ may happen due to a multitude of reasons with the most obvious reason being the uprise of a magnetic flux rope into rarefied layers of the chromosphere or corona. Due to the dissipative collapse a violent energy release begins in a thin azimuthal shell at the periphery of the twisted magnetic flux tube. On the basis of these properties of the force-free magnetic flux rope, the problems of plasma instabilities excitation and coronal heating are discussed.

The paper presents a modified version of a previously published 3D model of the solar facular knot (Solov’ev and Kirichek, 2019), which successfully describes the main structural features of the solar facular formations. As in the main model, we use the solution (Schatzman, 1965) in the potential approximation as the basis of the magnetic structure of the facular element. The magnetic profile of the facula, as before, has the shape of a fountain, in which thin streams of plasma rise in the center of the node and flow on its periphery along magnetic field lines. This configuration is very similar to the observed patterns (Lites et al., 2004; Jafarzadeh et al., 2017; Pietarila et al., 2009). In the new modified facula model, we use a different method to introduce force corrections into the potential distribution of the magnetic field, which causes the appearance of additional terms in the formulas for pressure and density. In the new version of the model, a change in only geometric parameters leads to a sharp change in the temperature profiles of the facular knot: instead of a dark central dip in the center of the assembly, a bright, hot formation can occur. An observational example of such a large-scale transformation is found in the Solar Dynamics Observatory data.

Changes in the umbra brightness and in the magnetic field over time should occur on long-period sunspot oscillations in accordance with Birman’s idea of inhibition of circulation (overturning) convection by a strong vertical magnetic field in antiphase mode.When the sunspot as a whole shifts upward, its power flux tube expands with a fixed magnetic flux, the magnetic field decreases, and the umbra brightness should increase. Vice versa, when the magnetic flux tube of the sunspot is lowered, it is compressed, the field grows, and the umbra brightness should decrease. Until now, this oscillatory process, which is described by the theoretical model of a shallow sunspot, has remained unexplored. This paper attempts to prove the Birman effect (Biermann, 1941) on the example of long-period oscillations of the sunspot 2010/12/07-13 (NOAA 11 133), for which there are available data from both the Solar and Heliospheric Observatory (SOHO) and Solar Dynamics Observatory (SDO).

Two main features define a sunspot as an object: a strong magnetic field and a low umbra temperature. The same can be said about a facular knot, in the center of which there is a dark micropore, and the magnetic field is several hundred Gauss (which noticeably exceeds the level of equipartition of about 200–250 G), sometimes reaching 1000 or 1200 G. The temperature decrease in these activity elements is explained by the suppression of circulation (overturning) convection by a strong vertical magnetic field. The first to express this idea was Biermann (1941). Within the framework of this conception, a saturation effect should be expected: when the magnetic field completely suppresses convection, its further growth should not lead to a decrease in temperature. There is still no clear answer to the question of the saturation effect. We will try to answer it in the first part of the work according to the Solar Dynamics Observatory. We show that saturation occurs at B = 2300–2400 G. A further increase in the magnetic field of the spot no longer leads to a decrease in the brightness of its umbra. The relationship of the magnetic field and intensity is also manifested in long-period oscillations of the magnetic field of spots. In this case, the brightness of the spot umbra changes in antiphase with its magnetic field (Efremov et al., 2020) in accordance with the theoretical model of a shallow sun spot (Solov’ev and Kirichek, 2014, 2016). A similar effect was established by us in the second part of this work for facular knots: during slow facular oscillations with periods of more than 1 h, its brightness changes in antiphase with a magnetic field. The obtained results convincingly confirm the Birman hypothesis.

Based on the previously proposed steady-state 3D model of a solar facular knot, we have developed its modified version in which the solitary nature of the facular object is reflected: its magnetic field is sharply bounded in the radial direction. The facula temperature profile calculated for the steady state of such a magnetic structure is compared with the high-resolution filtergram taken in the photospheric TiO 7057 A line at Big Bear Solar Observatory on June 19, 2017, at the New Solar Telescope (with a 1.6-m aperture). The image shows two closely spaced very dark micropores with a diameter $${\sim}300$$ km each. The micropores are surrounded by a fan of thin, light, radially elongated fibrils resembling a sunspot penumbra. The theoretically calculated $$T$$ profile of the facular knot completely reproduces these morphological features of the object being modeled.

—Observations of facular areas of the Sun using the Helioseismic and Magnetic Imager device of the Solar Dynamics Observatory allow detecting long-period oscillations in the magnetic field of facular knots with periods ranging from 30 to 250 minutes. Statistically significant oscillation modes are selected by the method of time series decomposition in empirical modes. During the observation period, which is close to the knot’s lifetime, its parameters (magnetic field, area on the magnetogram, etc.) may change noticeably, although the facular formation retains its structural identity all this time. These changes also affect the effective rigidity of the system (response to external disturbances), which leads to a change in the nature of the oscillations themselves. Based on this, we consider the observed oscillation in the magnetic field of facular knots with periods from 1 to 4 hours as the proper small oscillations of a system whose effective rigidity changes over time. A simple analytical model of the process is presented. It is shown that all three specific modes of low-frequency oscillations detected in facular knots are well described by the proposed model.

A distinctive feature of chromospheric magnetic fields in the facular regions on the Sun is their very fine spatial structure. In accordance with the observations performed in Ca II K and H line (the level of the lower chromosphere, about 0.5–1 Mm over the photosphere), numerous “slender fibrils mapping the magnetic field” were recorded. It is noted that “…the loops are organized in canopy-like arches” (Jafarzadeh et al., 2017). In this paper we perform some modification of the facular model presented by Solov’ev and Kirichek (2019) as a steady magnetic “fountain” with many thin trickles of plasma flows. The radial-azimuthal temperature profiles of the facula are calculated. It is shown that the difference in the vertical and horizontal scales of the system does not improve the correspondence between the model parameters of the faculae and their observed characteristics.

According to a model of sunspot dissipation through a thin boundary layer between the sunspot magnetic flux tube and the surrounding medium, the rate of decrease in the sunspot area must slow as the spot size decreases until it reaches a critical threshold at which the linear law of sunspot dissipation gives way to a nonlinear one. This theoretically predicted effect is confirmed with SOHO/MDI data, which have a high spatial and temporal resolution in comparison with ground-based observations. This work confirms the dissipation slowdown effect in small sunspots. It is reported that the nonlinear dissipation phase derived from discrete receiver (charge-coupled device, CCD) data is somewhat flatter and more diffuse than that derived from continuous radiation receiver data (photographic plates).

Long-term variations of the magnetic field of solitary solar pores with periods in the range of 4–12 h are detected with the Helioseismic and Magnetic Imager (HMI) of the Solar Dynamics Observatory (SDO). These long-term oscillations of the pores obviously reflect the movements of these structures as whole magnetic formations around their equilibrium position. In this article, long-period oscillations of the solar pore are studied based on the SDO data for the first time. The results are interpreted with the solution obtained earlier for eigen oscillations of the system with time-varying stiffness.

Based on the measurements of their magnetic fields, solar prominences can be divided into two types: filaments with normal (N) or inverse (I) polarity. In an N-type prominence, the magnetic polarity is the same as in the underlying photosphere, while the opposite polarity occurs in an I-type prominence. This paper continues to develop the method to calculate the pressure, density, and temperature of any given magnetic structure of the configuration. We model the dense and cool prominence with a helical structure of magnetic field. The magnetic configuration under study is introduced in a hydrostatic solar atmosphere given by the model considered by Avrett and Loezer (2008). The presented analytical model describes both the normal-polarity prominences and prominences with inverse polarity.

A method for the theoretical calculation of large-scale magnetoplasma solar coronal structures with a rotational symmetry in the spherical coordinate system is presented. The method makes it possible to obtain analytical solutions for equilibrium spatial distributions of pressures, densities, and temperatures in any preset axisymmetrical magnetic configuration. The obtained solution is used to plot model polar coronal holes via the introduction of a small power correction in the force free distribution of the magnetic field for subpolar region. The thermodynamic parameters of the coronal hole represented in this model are close to the observed values.

Long-period oscillations in the brightness of the Sun as a star have been analyzed on the basis of measurements of the solar light reflected by planets when they come into the field of view of the Large Angle and Spectrometric Coronagraph (LASCO C3) onboard the Solar and Heliospheric Observatory (SOHO). We took sequences of FITS intensitygrams from the archive of the SOHO/LASCO C3 space observatory for the observational data; they were 1024 × 1024 pixels in size, and they were acquired with the 12-min interval. In 5-day continuous time series of the brightness of Mars and Jupiter, coinciding modes of 8–10 and 36–38 h were found. This coincidence apparently indicates their solar origin. In addition, for Mars, we obtained 25-day series of time variations of the reflected solar radiation, in which a mode with even a lower frequency of 8–10 days was detected.

The properties of the 12-h artifact in the data of the SDO/HMI instrument (Helioseismic and Magnetic Imager) caused by the nonzero radial velocity of the station relative to the Sun are investigated. The study has been carried out with respect to long-period oscillations of the magnetic field of sunspots for different station positions in the Earth’s orbit by the alternative spectral method of singular decomposition of the signal CaterPillarSSA. Features of artifact filtering, both in special positions of the station (at the points of aphelion and perihelion) and at arbitrarily selected orbital points, are considered. It is shown that the 12-h artifact mode can be completely filtered from the time series of the observed variable, not only at these two orbital points (because of the symmetry of the station’s radial velocity with respect to the zero mean here) but also at any others. It is shown that only a 12-h mode is physically justified, while the 24-h harmonic appears only as an artifact in the Fourier decomposition of the amplitude-modulated signal. It is emphasized that the values of the magnetic field measured with SDO/HMI are sensitive only to the station’s radial velocity absolute values with respect to the Sun and do not depend on its direction. It has been noted that the periods of sunspot oscillation as a whole obtained from SDO/HMI data after orbital artifact filtration fit well into the dependence diagram of the period of sunspot oscillations on the value of its magnetic field strength constructed earlier by SOHO/MDIdata.

The fundamental small-scale structures such as granules, faculae, micropores that are observed in the solar photosphere under high resolution are discussed. As a separate constituent of the fine structure, a continuous net of dark intergranular lanes is considered. The results due to image processing of micropores and facular knots obtained on modern adaptive optics telescopes are presented. For intergranular lanes and micropores, a steady-state magnetic diffusion model is proposed, in which the horizontal-vertical plasma flows converging to a intergranular lane (and to the body of a micropore) compensate for the dissipative spreading of the magnetic flux at a given scale. A theoretical estimate of the characteristic scales of these structures in the photosphere is obtained as 20–30 km for the thickness of dark intergranular lanes (and the diameter of the thinnest magnetic tube in the solar photosphere), 200–400 km for the diameter of micropores. A model of a facular knot with the darkening core on the axis, which physically represents a micropore, stabilizing the entire magnetic configuration over a time interval of up to 1 day, is briefly described.

For the first time, the ultra-low oscillation mode of the sunspot magnetic field strength has been detected with a high degree of confidence by ground-based observations of sunspots with the Global Oscillation Network Group (GONG) network of telescopes. Synchronous series of magnetograms derived from the GONG and Solar and Heliospheric Observatory/Michelson Doppler Imager (SOHO/MDI) have been processed. They were obtained on September 27–30, 2010, for the active region NOAA 11109 with a total duration of 80 h. The periods of magnetic field oscillations found by space data coincide with the periods defined with GONG. This confirms the physical reality of the oscillatory process. The power spectrum contains harmonics with periods of 26 h, 8–10 h, and 3–4 h.

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-Alfvenic 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.

A new exact analytical solution to the stationary problem of ideal magnetohydrodynamics is derived for an unipolar asymmetric sunspot immersed in a realistic solar atmosphere. The radial and vertical profiles of pressure, plasma density, and temperature in the visible layers of the sunspot are calculated. The reduction in plasma density in the magnetic funnel of the sunspot, corresponding to the Wilson depression, is also obtained. The magnetic structure of the sunspot is given analytically in a realistic way: a part of the magnetic flux of the sunspot approaches the surrounding photosphere at the outer edge of the penumbra. The magnetic field of the sunspot is not assumed to be axially symmetric. For the first time, the angular dependence of the physical variables in this model allows us to simulate not only a deviation from the circular shape of the sunspot, but also a fine filamentary structure of the sunspot penumbra. The Alfven Mach number (the ratio of the plasma speed to the Alfven speed) is zero at the center of the sunspot and rises slowly toward the periphery of the sunspot; this corresponds to the structure of the Evershed flow in the penumbra. The Evershed flow in our model is mainly concentrated in dark penumbral filaments, as is observed.

The global long term periodic oscillations of the Sun are investigated on the base of three independent sets of data: It is found that in the low-frequency spectrum of global solar oscillations, there are modes with periods of 8–10, 36–38 hours and the main stable prominent mode with a period of ∼9-days.

The magnetic structure of a typical sunspot is represented as a set of three interacting magnetic fluxes. The first one, F1, escapes upward from the sunspot umbra and closes through the corona on the sunspot of opposite polarity. The second, F2, forms the sunspot penumbra. It includes Evershed flows emerging from the sunspot, the heavy photospheric plasma of which cannot be pulled up to the corona. For this reason, the F2 flux at the outer edge of the penumbra should close on the photosphere adjacent to the sunspot, i.e., here the F2 field changes sign. The third flux, F3, is the external magnetic field flux and has the same polarity as F1. Thus, magnetic fields of different signs occur near the outer boundary of the sunspot penumbra. Magnetic reconnections will inevitably take place there. Smalland multiscale current sheets in which particles are accelerated are formed. The latter fill short and low magnetic loops that connect sunspots of different polarity in the bipolar group and form interspot radio sources at the tops of these loops.

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.

A number of fundamental questions as regards the physical nature of sunspots are formulated. In order to answer these questions, we apply the model of a round-shaped unipolar sunspot with a lower boundary consisting of cool plasma and with strong magnetic field at the depth of about 4 Mm beneath the photosphere, in accordance with the data of local helioseismology and with certain physically sound arguments (the shallow sunspot model). The magnetic configuration of a sunspot is assumed to be close to the observed one and similar to the magnetic field of a round solenoid of the appropriate size. The transverse (horizontal) and longitudinal (vertical) equilibria of a sunspot were calculated based on the thermodynamic approach and taking into account the magnetic, gravitational, and thermal energy of the spot and the pressure of the environment. The dependence of the magnetic field strength in the sunspot center, B 0, on the radius of the sunspot umbra a is derived theoretically for the first time in the history of sunspot studies. It shows that the magnetic field strength in small spots is about 700 Gauss (G) and then increases monotonically with a, tending asymptotically to a limit value of about 4000 G. This dependence, B 0(a) includes, as parameters, the gravity acceleration on the solar surface, the density of gas in the photosphere, and the ratio of the radius of the spot (including penumbra), a p, to the radius of its umbra a. It is shown that large-scale subsurface flows of gas in the sunspot vicinity, being the consequence but not the cause of sunspot formation, are too weak to contribute significantly to the pressure balance of the sunspot. Stability of the sunspot is provided by cooling of the sunspot plasma and decreasing of its gravitational energy due to the vertical redistribution of the gas density when the geometric Wilson depression of the sunspot is formed. The depth of a depression grows linearly with B 0, in contrast to the quadratic law for the magnetic energy. Therefore, the range of stable equilibria turns out to be limited: large spots, with radius a larger than some limit value (about 12–18 Mm, depending on the magnetic field configuration), are unstable. It explains the absence of very large spots on the Sun and the appearance of light bridges in big spots that divide the spot into a few parts. The sunspots with B 0≈2.6÷2.7 kilogauss (kG) and a≈5 Mm are most stable. For these spots, taken as a single magnetic structure, the period of their vertical eigen oscillations is minimal and amounts, according to the model, to 10–12 hours. It corresponds well to the period derived from the study of long-term oscillations of sunspots using SOHO/MDI data.

Aims. The main goal of this work is to analyze the structural and temporal evolution of small-scale magnetic structures (SSMSs) observed in the solar atmosphere, such as solitary faculae and pores, and reveal long quasi-periodic oscillations of these structures. Methods. The statistical method of regression analysis and the wavelet transform were used to obtain the periods of oscillations and dependences between the parameters of magnetic structures and periods of oscillations. Results. Long-period oscillations with periods in the interval of 18−260 min are found for the structurally stable phase of SSMSs at the level of the solar photosphere. These long-period oscillations were interpreted as natural oscillations of the structurally stable long-lived magnetic structures around their equilibrium position. These oscillations, which are of similar nature, are observed in the chromospheric bright formations associated with photospheric SSMSs. Dependences between the magnetic field and the continuum intensity of the facula elements were found. It is shown that the continuum intensity of a SSMS decreases when its magnetic field increases.

Long-period oscillations of the magnetic field and the line-of-sight velocity have been detected in sunspots based on the synchronous 90-h time series of magnetograms and dopplergrams obtained with the MDI(SOHO) device. The sunspot eigenmode (12–30 h), the periods of which nonlinearly depend on the magnetic field strength, predominates in the magnetic field oscillation spectrum. The mode, which is probably induced by motions of supergranulation cells (33 h), predominates in the sunspot line-of-sight velocity oscillation spectrum. A strong mode (33 h), which indicates that long-period quasi-oscillations of supergranulation exist, was also detected in the velocity power spectrum for a quiet photosphere, together with the known 5-min mode.

An analysis of oscillatory processes with periods not shorter than several tens of minutes in three isolated sunspots, which were observed during identical periods in the optical and radio bands, is illustrated. SDO/HMI magnetograms at an interval of 45 s and radio maps at a wavelength of 1.76 cm, obtained using a Nobeyama radioheliograph (NoRH), have been used. The time profiles, which were constructed based on the NoRH and SDO/HMI data, indicate that the oscillations of the radioemission correlate with those of the sunspot magnetic field. The wavelet spectra and cross-wavelet transform reveal common oscillation periods of 30–40, 70–100, and 150–200 min. The identical oscillation periods, found using fundamentally different methods from ground-based and space observations, confirm the solar nature of these oscillations, which can be interpreted as oscillations of a sunspot as a whole.

An analysis of sampled 1-s observational data on geomagnetic pulsations within the Pc3 range on the INTERMAGNET network of near-equatorial and low-latitude observatories spaced over longitude during the initial phase of a moderate magnetic storm (April 5–7, 2010) was carried out for the first time. The obtained results were compared with magnetic observations at the low-latitude Chambon-la-Foret (CLF) and subauroral Kerguelen (PAF) observatories, as well as with observations at six Australian observatories located at low and middle latitudes. Two time intervals were studied in detail: the sudden commencement (SC) of the storm and the onset of the great global substorm. In the first interval, maximal amplitudes of near-equatorial pulsations were observed in the near-noon sector; in the second interval, in the near-midnight sector. The dynamics of the spectral structure of Pc3 pulsations in the considered events was shown to be different in spite of the fact that in both cases an amplification of waves was observed in two close spectral bands of the Pc3 spectrum: ∼20–30 and ∼30–40 mHz. The considered Pc3 pulsations were characterized by very small azimuthal wavenumbers (0.5 and less). Possible generation mechanisms for the observed Pc3 pulsations are discussed.

As a result of processing long (up to 144 h) series of sunspot magnetograms obtained on the SOHO (Solar and Heliospheric Observatory) spacecraft with the MDI (Michelson Doppler Imager) instrument it is shown that the mode with a period of 800–1300 min is a limiting low-frequency oscillation mode of the magnetic field of a sunspot as a whole. Its period is essentially and nonlinearly depends on the sunspot magnetic field strength. In addition to this mode, higher harmonics are also revealed in the sunspot oscillation spectra in the bands 40–45, 60–80, 135–170, 220–250, and 480–520 min. The oscillation power in these bands monotonically and rapidly decreases with increasing frequency, which is characteristic for overtones arising due to the nonlinear nature of oscillations. The limiting oscillation mode stably exists in sunspots for 1.5–2 days, which coincides with the average lifetime of a supergranular cell. The mode with the period of 35–48 h observed in the power spectrum is not an eigen mode of sunspots, because its period is independent of its magnetic field strength. Probably, it occurs as a quasiperiod of an external exciting force caused by disturbances from supergranular cells surrounding the sunspot.

We have obtained new exact solutions describing the self-similar, uniform compression of twisted magnetic flux tubes (a magnetic flux rope) with a linear, force-free internal field with α = const in a resistive medium during the course of their dissipative evolution. Formally, the radius of the rope a decreases to zero over a finite time. This time turns out to be relatively short (several hours) for a rope with a radius of several hundred kilometers, if the helicity of the magnetic field in the rope is high: αa ≫ 1.

We applied special data-processing algorithms to the study of long-period oscillations of the magnetic-field strength and the line-of-sight velocity in sunspots. The oscillations were investigated with two independent groups of data. First, we used an eight-hour-long series of solar spectrograms, obtained with the solar telescope at the Pulkovo Observatory. We simultaneously measured Doppler shifts of six spectral lines, formed at different heights in the atmosphere. Second, we had a long time series of full-disk magnetograms (10 – 34 hour) from SOHO/MDI for the line-of-sight magnetic-field component. Both ground- and space-based observations revealed long-period modes of oscillations (40 – 45, 60 – 80, and 160 – 180 minutes) in the power spectrum of the sunspots and surrounding magnetic structures. With the SOHO/MDI data, one can study the longer periodicities. We obtained two new significant periods (> 3σ) in the power spectra of sunspots: around 250 and 480 minutes. The power of the oscillations in the lower frequencies is always higher than in the higher ones. The amplitude of the long-period magnetic-field modes shows magnitudes of about 200 – 250 G. The amplitude of the line-of-sight velocity periodicities is about 60 – 110 m s−1. The absence of low-frequency oscillations in the telluric line proves their solar nature. Moreover, the absence of low-frequency oscillations of the line-of-sight velocity in the quiet photosphere (free of magnetic elements) proves their direct connection to magnetic structures. Long-period modes of oscillation observed in magnetic elements surrounding the sunspot are spread over the meso-granulation scales (10″ – 12″), while the sunspot itself oscillates as a whole. The amplitude of the long-period mode of the line-of-sight velocity in a sunspot decreases rapidly with height: these oscillations are clearly visible in the spectral lines originating at heights of approximately 200 km and fade away in lines originating at 500 km. We found a new interesting property: the low-frequency oscillations of a sunspot are strongly reduced when there is a steady temporal trend (strengthening or weakening) of the sunspot’s magnetic field. Another important result is that the frequency of long-period oscillations evidently depends on the sunspot’s magnetic-field strength.

The new approach to the modeling of quiescent solar prominences is proposed. We solve the inverse magnetohydrostatic problem, when the pressure, density and temperature of plasma in the filament are calculated from the equilibrium equations using the given magnetic structure (magnetic flux function is proposed to be known). The new exact nonlinear solutions for dense (n ≈ (2−3) × 1011 cm−3) and cold (T ≈ (5−10) × 103 K) filaments, embedded in the plan, vertically stratified atmosphere (hot solar corona) free of magnetic field, are derived. The filaments are stretched along the horizontal axisy(the translational symmetry is assumed: ∂/∂y = 0) and located parallel to and above a photospheric, magnetic polarity reversal line. The magnetic field lines have a structure of magnetic flux rope with helical field lines in three-dimensional space; the strength of magnetic field falls rapidly with distance from a rope axis. No external longitudinal magnetic field is needed to equilibrate the prominence. The net electric current along the filament is equal to zero. The model of magnetic arcade with the deflection (sag) on the top, proposed by Pikelner (1971) as a basic form of normal prominence, is calculated also using the method proposed. It is shown that such magnetic arcade, having the magnetic field strength of few gauss only, can effectively maintain the equilibrium of cool dense filament at the heights about 50–60 Mm.

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.

Internal gravity waves excited in the overlying atmosphere by the proper vertical oscillations of an entire sunspot are studied. For simplicity, the oscillations of the sunspot are introduced into the model through the lower boundary conditions, by specifying oscillations in the vertical velocity. The characteristic radius of the oscillating region is assumed to equal the spot size and the frequency to correspond to the long-period natural oscillations of the spot. Results of numerical computations are presented. It is shown that internal gravity waves propagate nearly in the horizontal direction. Therefore, immediately above the spot, the wave energy drops exponentially with the height, in good agreement with the available observational data. The level where the amplitude and energy density of the wave are maximum rises slowly with distance from the spot.

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 = ω 1 (B). Lower subharmonics may appear in the case where penumbra too becomes involved in the oscillatory process: ω 2 = ω 1/2, ω 3 = ω 1/3. 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.

New observational data on long-period oscillations of the line-of-sight velocities detected via the Doppler shifts of spectral lines observed at various heights in and near sunspots are presented. The sunspots and nearby magnetic elements oscillate with periods ranging from 40 to 80 min. The oscillations in the line-of-sight velocities persist over the entire observation session (up to four hours). These results support theoretical models in which this phenomenon represents natural long-period oscillations (vertical-radial displacements) of entire magnetic elements (sunspots, pores, and magnetic knots) about some stable equilibrium positions.

Measurements and the interpretation of the time delay effect between long quasi-periodic oscillations of sunspot magnetic fields and nearby millimeter radio sources observed at 37 GHz were the main goals of this work. Ground-based radio telescope operated by Metsahovi Radio Observatory, Aalto University, Finland was used to obtain time series variations of radio intensity at 37 GHz frequency, as well as, the Helioseismic and Magnetic Imager instrument on-board the Solar Dynamics Observatory spacecraft was used to obtain the magnetic field time series variations. Lags (time delays) in the interval of 15–35 minutes were obtained by cross-correlation analysis of time series and by direct geometrical measurements of distances between the radio sources and nearby sunspots. These distances were in the interval of 11–24 Mm. Corresponding time delays were defined as the relation of these distances to the sound speed. Time delays obtained by two different independent methods turned to be very close. This fact confirms the interpretation of the phenomenon under the study as a process of propagation of disturbances from the slowly oscillating sunspot to the radio source with the sound speed.

N-Alkyl(aryl)-C-trifluoromethyl-C-chlorosulfinimides and styrenes react to give the 3+2 cycloaddition products, 5-chloro-5-trifluoromethyl-3-aryl-4-alkyl(aryl)-isothiazolidines 9a-h. The structure of the dehydrochlorination products derived from the compounds 9a-h is determined by the nature of the substituent near the nitrogen atom. The compounds having bulky tertiary-butyl group (9c,h) are transformed into the isothiazolines derivatives 10a,b, but compounds 9a,b,d-g form β-aminovinyl trifluoromethylthioketones 11a-f.

We analyzed unique observations of two flare events at frequencies of 93 and 140 GHz. The observations were carried out with an RT-7.5 radiotelescope at the Bauman State Technical University (Moscow) using the method of continuous active region tracking with spatial resolutions of 2.5 (at a frequency of 93 GHz) and 1.5 arc-minutes (at 140 GHz). The light curves of the bursts were analyzed and compared with the time profiles of soft and hard X-ray emission obtained by the GOES and RHESSI spacecraft. The radio delete this word flux density spectra were plotted. It was found that the radiation flux at a frequency of 140 GHz exceeded the flux at 93 GHz. This constitutes a new independent confirmation of the presence of a subterahertz flare component, the appearance of which may be associated with the thermal radiation of the hot plasma at the base of flare loops.

In this work we present the first results of study and comparison of the parameters of quasi-periodic long-term oscillations of microwave emission of large (>0.7 arcmin) sunspots as a result of simultaneous observations with two radioheliographs – NoRH (17 GHz) and Siberian Solar Radio Telescope (SSRT) (5.7 GHz) with 1 minute cadence. Radioheliographs have been working with quite large time overlap (about 5 hours) and have the high spatial resolution: 10 arcsec (NoRH) and 20 arcsec (SSRT). We have found that quasi-periodic long-term oscillations are surely observed at both frequencies with the periods in the range of 20–150 min. We detected common periods for common time of observations with two radioheliographs and interpret this as the consequence of the vertical-radial quasi-periodic displacements of sunspot as a whole structure.

The method of steepest descent, also known as the saddle-point method, is a natural development of Laplace's method applied to the asymptotic estimate of integrals of analytic functions. Mathematicians have often attributed the method of steepest descent to the physicist Peter Debye, who in 1909 worked it out in an asymptotic study of Bessel functions. Debye himself remarked that he had borrowed the idea of the method from an 1863 paper of Bernhard Riemann. The present article offers a detailed historical analysis of the creation of the method of steepest descent. We show that the method dates back to Cauchy and that, 25 years before Debye, the Russian mathematician Pavel Alexeevich Nekrasov had already used this technique and extended it to more general cases. Copyright 1997 Academic Press. La methode de la descente la plus rapide, que l'on appelle actuellement methode du col, represente un developpement naturel de la methode de Laplace pour l'estimation asymptotique des integrales des fonctions analytiques. La methode du col est generalement liee au nom du physicien Peter Debye, qui a utilise en 1909 les idees que Riemann a presentees dans son article de 1863 afin de donner une analyse asymptotique des fonctions de Bessel. Le present travail donne une etude bien detaillee de la creation de la methode du col. Tout d'abord nous montrons que la methode du col remonte aux travaux de Cauchy et ensuite que le mathematicien russe Pavel Alexeevich Nekrasov a applique cette methode un quart de siecle avant Debye et l'a etendue aux cas generaux. Copyright 1997 Academic Press. [FORMULA]

Electrochemical aspects of the functioning of exogenous quinone electron transfer mediators required for developing a photobioelectrode on the basis of photosynthetic reaction centres were studied. Methods for the chemical modification of the Pt and Pt-amalgamated electrodes with a naphthoquinone monolayer were developed and electrochemical reversibility for this immobilized quinone was observed. Moreover, it was found that modification of the Pt surface with cysteamine and especially its amalgamation resulted in an increase in electrochemical reversibility for ubiquinone solubilized in aqueous solution and serving as a diffusionally mobile electron transfer mediator. Adsorption of reaction centres as a monolayer at the electrode surface did not affect the electrochemical properties of the immobilized quinonoid underlayer significantly but limited considerably the electrochemical process for the solubilized electron transfer mediator. The photoelectric activity of bacterial reaction centres adsorbed at various electrode underlayers was studied and was shown to depend significantly on the underlayer type. The results obtained are discussed in connection with the electrochemical properties of quinone electron transfer mediators.

Stable immobilization of reaction centers from photosynthetic bacterium Rhodobacter sphaeroides R-26 has been achieved at modified platinum electrodes. Two types of modified electrodes were used for this purpose. The first one was obtained by cysteamine adsorption at platinum surface, the second one - by covalent binding of naphthoquinone via an amino group of cysteamine anchored by a SH group to the platinum surface. Reduced cytochrome c and ubiquinone-10 were used as an electron donor and diffusionally mobile mediator of electron transfer from the immobilized reaction centers to an electrode, respectively. The best photoelectric characteristics were obtained for the electrode coated with the reaction centers and having quinone covalently immobilized via a cysteamine underlayer. The increase of photoactivity during long-term rinsing of the electrodes in a solution of detergent, lauryldimethylamine- N-oxide, is surprising. This effect is enhanced if the rinsing is combined with electrode incubation in a solution containing reduced cytochrome c and ubiquinone-10. The methods developed for immobilization of reaction centers do not touch primary and secondary quinone binding sites and should permit changing of the primary and/or secondary quinones for optimizing electrode photoelectric properties.

To develop a photobioelectrode on the basis of bacterial photosynthetic reaction centres, the modification of Pt and Au electrodes by naphthoquinones as electron transfer mediators has been studied. Cysteamine and cystamine were used as bifunctional spacers. Thiol and disulphide groups of these molecules are anchors at the surface of the metallic electrodes, whereas the amino groups bind naphthoquinones covalently in the form of their amino derivatives. The stability and electrochemical reversibility (the rate constant of the redox process is ca. 10 s −1 ) are shown to be increased considerably as compared with similar electrodes where aminosilane is used as a bifunctional spacer. The immobilization processes of sulphur-containing amino compounds are found to be reversible. This allows a covalently bound quinone to be displaced easily by the excess of thiolic compound and then a new naphthoquinone is immobilized with another redox potential. It has been shown that the length of the spacer between the immobilized quinone and the metallic surface can be increased due to carbodiimide coupling of the carboxyl group of the amino acid-quinone derivative with the amino group of the immobilized thiolic compound.

To study in more detail the process of quinone immobilization at the electrodes via aminosilanes, model synthesis was carried out by us in solution

We present an investigation of oscillatory processes with periods in the range of several tens of minutes for some single sunspots of a new solar cycle, observed in 2010–2011 at the same time intervals in the optical and radio ranges. We used magnetograms from SDO/HMI with a cadence of 45 s, and radio images at a frequency of 17 GHz obtained with the Nobeyama Radioheliograph (NoRH). Radio images in intensity (Stokes parameter I ) and circular polarization (Stokes parameter V ) were synthesized with a cadence of ten seconds and ten-second averaging. Time profiles obtained with NoRH and SDO/HMI show a correlation between the radio emission of sunspots and a magnetic field. Wavelet spectra and cross-wavelet transforms give similar oscillation periods: 30–40 min, 60–70 min, 100–110 min, and 150–200 min. The same periods found by fundamentally different methods from ground-based and space observations confirm the solar nature of these oscillations. One of the possible interpretations of our results is that detected oscillations reflect eigen oscillations of a sunspot as a whole predicted by the shallow sunspot model.

This paper describes a method developed and used at the Main (Pulkovo) Observatory, Russian Academy of Sciences, for recording Doppler shifts, as well as the broadenings of spectral lines caused by their Zeeman splitting, from digital optical spectrograms of active regions of the sun. It is shown that, in fairly prolonged (more than 4h) continuous observations, long-period variations of the line-of-sight velocities and the magnetic field, with a characteristic period of about 80min, are detected in sunspots and in magnetic elements of the surrounding photosphere. These variations are interpreted as vertical or line-of-sight oscillations of the entire spot (or the magnetic element) as a whole, caused by perturbing action from the surrounding turbulent medium (the photosphere and the convective zone).