Aims. Propagation and energy transfer of torsional Alfvén waves in solar magnetic flux tubes of axial symmetry is studied.Methods. An analytical model of a solar magnetic flux tube of axial symmetry is developed by specifying a magnetic flux and deriving general analytical formulas for the equilibrium mass density and gas pressure. The main advantage of this model is that it can be easily adopted to any axisymmetric magnetic structure. The model is used to numerically simulate the propagation of nonlinear Alfvén waves in such 2D flux tubes of axial symmetry embedded in the solar atmosphere. The waves are excited by a localized pulse in the azimuthal component of velocity and launched at the top of the solar photosphere, and they propagate through the solar chromosphere, the transition region, and into the solar corona. Results. The results of our numerical simulations reveal a complex scenario of twisted magnetic field lines and flows associated with torsional Alfvén waves, as well as energy transfer to the magnetoacoustic waves that are triggered by the Alfvén waves and are akin to the vertical jet flows. Alfvén waves experience about 5% amplitude reflection at the transition region. Magnetic (velocity) field perturbations that experience attenuation (growth) with height agree with analytical findings. The kinetic energy of magnetoacoustic waves consists of 25% of the total energy of Alfvén waves. The energy transfer may lead to localized mass transport in the form of vertical jets, as well as to localized heating because slow magnetoacoustic waves are prone to dissipation in the inner corona.

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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 eighthour-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 lineof-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.

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We processed magnetograms that were obtained with the Michaelson Doppler Imager onboard the Solar and Heliospheric Observatory (SOHO/MDI). The results confirm the basic properties of long-period oscillations of sunspots that have previously been established and also reveal new properties. We show that the limiting (lowest) eigenmode of low-frequency oscillations of a sunspot as a whole is the mode with a period of 10 -12 up to 32 -35 hours (depending on the sunspot's magnetic-field strength). This mode is observed consistently throughout the observation period 5 -7 days, but its amplitude is subject to quasi-cyclic changes, which are separated by about 1.5 -2 days. As a result, the lower mode with periods of about 35 -48 hours appears in the power spectrum of sunspot oscillations. But this lowest mode is apparently not an eigenmode of a sunspot because its period does not depend on the magnetic field of the sunspot. Perhaps, the mode reflects the quasi-periodic sunspot perturbations caused by supergranulation cells that surround it. We also analyze SOHO/MDI artifacts, which may affect the low-frequency power spectra of sunspots.