Waves observed in the photosphere and chromosphere of sunspots show complex dynamics and spatial patterns. The interpretation of high-resolution sunspot wave observations requires modeling of three-dimensional non-linear wave propagation and mode transformation in the sunspot upper layers in realistic spot model atmospheres. Here we present the first results of such modeling. We have developed a 3D non-linear numerical code specially designed to calculate the response of magnetic structures in equilibrium to an arbitrary perturbation. The code solves the 3D nonlinear MHD equations for perturbations; it is stabilized by hyper-diffusivity terms and is fully parallelized. The robustness of the code is demonstrated by a number of standard tests. We analyze several simulations of a sunspot perturbed by pulses of different periods at subphotospheric level, from short periods, introduced for academic purposes, to longer and realistic periods of three and five minutes. We present a detailed description of the three-dimensional mode transformation in a non-trivial sunspotlike magnetic field configuration, including the conversion between fast and slow magneto-acoustic waves and the Alfvén wave, by calculation of the wave energy fluxes. Our main findings are the following: (1) the conversion from acoustic to the Alfvén mode is only observed if the the driving pulse is located out of the sunspot axis, but this conversion is energetically inefficient; (2) as a consequence of the cut-off effects and refraction of the fast magneto-acoustic mode, the energy of the evanescent waves with periods around 5 minutes remains almost completely below the level β = 1; (3) waves with frequencies above the cut-off propagate field-aligned to the chromosphere and their power becomes dominating over that of evanescent 5-minute oscillations, in agreement with observations.

We analyze the propagation of waves in sunspots from the photosphere to the chromosphere using time series of co-spatial Ca ii H intensity spectra (including its line blends) and polarimetric spectra of Si i λ 10827 and the He i λ 10830 multiplet. From the Doppler shifts of these lines we retrieve the variation of the velocity along the line-of-sight at several heights. Phase spectra are used to obtain the relation between the oscillatory signals. Our analysis reveals standing waves at frequencies lower than 4 mHz and a continuous propagation of waves at higher frequencies, which steepen into shocks in the chromosphere when approaching the formation height of the Ca ii H core. The observed nonlinearities are weaker in Ca ii H than in He i lines. Our analysis suggests that the Ca ii H core forms at a lower height than the He i λ 10830 line: a time delay of about 20 s is measured between the Doppler signal detected at both wavelengths. We fit a model of linear slow magnetoacoustic wave propagation in a stratified atmosphere with radiative losses according to Newton's cooling law to the phase spectra and derive the difference in the formation height of the spectral lines. We show that the linear model describes well the wave propagation up to the formation height of Ca ii H, where non-linearities start to become very important.

We present results of nonlinear, two-dimensional, numerical simulations of magneto-acoustic wave propagation in the photosphere and chromosphere of small-scale flux tubes with internal structure. Waves with realistic periods of three to five minutes are studied, after horizontal and vertical oscillatory perturbations are applied to the equilibrium model. Spurious reflections of shock waves from the upper boundary are minimized by a special boundary condition. This has allowed us to increase the duration of the simulations and to make it long enough to perform a statistical analysis of oscillations. The simulations show that deep horizontal motions of the flux tube generate a slow (magnetic) mode and a surface mode. These modes are efficiently transformed into a slow (acoustic) mode in the v A < c S atmosphere. The slow (acoustic) mode propagates vertically along the field lines, forms shocks, and remains always within the flux tube. It might effectively deposit the energy of the driver into the chromosphere. When the driver oscillates with a high frequency, above the cutoff, nonlinear wave propagation occurs with the same dominant driver period at all heights. At low frequencies, below the cutoff, the dominant period of oscillations changes with height from that of the driver in the photosphere to its first harmonic (half period) in the chromosphere. Depending on the period and on the type of the driver, different shock patterns are observed.Helioseismology, Asteroseismology, and MHD Connection

Context. In the solar atmosphere, the acoustic cutoff frequency is a local quantity which depends on the atmospheric height. It separates the low-frequency evanescent waves from the high-frequency propagating waves. Aims. We measure the cutoff frequency of slow magnetoacoustic waves at various heights of a sunspot umbra and compare the results with the estimations from several analytical formulae. Methods. We analyzed the oscillations in the umbra of a sunspot belonging to active region NOAA 12662 observed in the 10830 Å spectral region with the GREGOR Infrared Spectrograph and in the Fe i 5435 Å line with the GREGOR Fabry-Pérot Interferometer. Both instrumets are attached to the GREGOR telescope at the Observatorio del Teide, Tenerife, Spain. We have computed the phase and amplification spectra between the velocity measured from different pairs of lines that sample various heights of the solar atmosphere. The cutoff frequency and its height variation have been estimated from the inspection of the spectra. Results. At the deep umbral photosphere the cutoff frequency is around 5 mHz and it increases to 6 mHz at higher photospheric layers. At the chromosphere the cutoff is ∼ 3.1 mHz. The comparison of the observationally determined cutoff with the theoretically predicted values reveals an agreement in the general trend and a reasonable match at the chromosphere, but also significant quantitative differences at the photosphere. Conclusions. Our analyses show strong evidence of the variation of the cutoff frequency with height in a sunspot umbra, which is not fully accounted for by current analytical estimations. This result has implications for our understanding of wave propagation, the seismology of active regions, and the evaluation of heating mechanisms based on compressible waves.

Context. Active regions are the most prominent manifestations of solar magnetic fields; their generation and dissipation are fundamental problems in solar physics. Light bridges are commonly present during sunspot decay, but a comprehensive picture of their role in the removal of photospheric magnetic field is still missing. Aims. We study the three dimensional configuration of a sunspot and in particular its light bridge during one of the last stages of its decay. Methods. We present the magnetic and thermodynamical stratification inferred from full Stokes inversions of the photospheric Si i 10827 Å and Ca i 10839 Å lines obtained with the GREGOR Infrared Spectrograph of the GREGOR telescope at Observatorio del Teide, Tenerife, Spain. The analysis is complemented by a study of continuum images covering the disk passage of the active region, which are provided by the Helioseismic and Magnetic Imager on board the Solar Dynamics Observatory. Results. The sunspot shows a light bridge with penumbral continuum intensity that separates the central umbra from a smaller umbra. We find that in this region the magnetic field lines form a canopy with lower magnetic field strength in the inner part. The photospheric light bridge is dominated by gas pressure (high-β), as opposed to the surrounding umbra where the magnetic pressure is higher. A convective flow is observed in the light bridge. This flow is able to bend the magnetic field lines and to produce field reversals. The field lines close above the light bridge and become as vertical and strong as in the surrounding umbra. We conclude that it develops because of two highly magnetized regions which come closer during the sunspot evolution.

We aim at reproducing the height dependence of sunspot wave signatures obtained from spectropolarimetric observations through 3D MHD numerical simulations. A magneto-static sunspot model based on the properties of the observed sunspot is constructed and perturbed at the photosphere introducing the fluctuations measured with the Si i λ 10827Å line. The results of the simulations are compared with the oscillations observed simultaneously at different heights from the He i λ 10830Å line, the Ca ii H core and the Fe i blends in the wings of the Ca ii H line. The simulations show a remarkable agreement with the observations. They reproduce the velocity maps and power spectra at the formation heights of the observed lines, as well as the phase and amplification spectra between several pair of lines. We find that the stronger shocks at the chromosphere are accompanied with a delay between the observed signal and the simulated one at the corresponding height, indicating that shocks shift the formation height of the chromospheric lines to higher layers. Since the simulated wave propagation matches very well the properties of the observed one, we are able to use the numerical calculations to quantify the energy contribution of the magneto-acoustic waves to the chromospheric heating in sunspots. Our findings indicate that the energy supplied by these waves is too low to balance the chromospheric radiative losses. The energy contained at the formation height of the lowermost Si i λ 10827Å line in the form of slow magneto-acoustic waves is already insufficient to heat the higher layers, and the acoustic energy which reaches the chromosphere is around 3-9 times lower than the required amount of energy. The contribution of the magnetic energy is even lower.

The conversion of fast waves to the Alfvén mode in a realistic sunspot atmosphere is studied through threedimensional numerical simulations. An upward propagating fast acoustic wave is excited in the high-β region of the model. The new wave modes generated at the conversion layer are analyzed from the projections of the velocity and magnetic field in their characteristic directions, and the computation of their wave energy and fluxes. The analysis reveals that the maximum efficiency of the conversion to the slow mode is obtained for inclinations of 25• and low azimuths, while the Alfvén wave conversions peak at high inclinations and azimuths between 50• and 120• . Downward propagating Alfvén waves appear at the regions of the sunspot where the orientation of the magnetic field is in the direction opposite to the wave propagation, since at these locations the Alfvén wave couples better with the downgoing fast magnetic wave which is reflected due to the gradients of the Alfvén speed. The simulations show that the Alfvén energy at the chromosphere is comparable to the acoustic energy of the slow mode, being even higher at high inclined magnetic fields.

Spectropolarimetric temporal series from Fe i 6301.5 Å and Ca ii infrared triplet lines are obtained by applying the Stokes synthesis code NICOLE to a numerical simulation of wave propagation in a sunspot umbra from MANCHA code. The analysis of the phase difference between Doppler velocity and intensity core oscillations of the Fe i6301.5 Å line reveals that variations in the intensity are produced by opacity fluctuations rather than intrinsic temperature oscillations, except for frequencies between 5 and 6.5 mHz. On the other hand, the photospheric magnetic field retrieved from the weak field approximation provides the intrinsic magnetic field oscillations associated to wave propagation. Our results suggest that this is due to the low magnetic field gradient of our sunspot model. The Stokes parameters of the chromospheric Ca ii infrared triplet lines show striking variations as shock waves travel through the formation height of the lines, including emission self-reversals in the line core and highly abnormal Stokes profiles. Magnetic field oscillations inferred from the Ca ii infrared lines using the weak field approximation appear to be related with the magnetic field strength variation between the photosphere and the chromosphere.