9 Conclusion

There exists several observational reports of umbral oscillations in the literature and there have been several theoretical attempts to explain them. However, no generally accepted model exists for the understanding of these mechanical structures, their physical mechanisms and energy transport to the surroundings. In this paper we presented new solutions for magneto-atmospheric waves in an isothermal atmosphere with a vertical magnetic field in the presence of radiative heat exchange based on Newton's law of cooling. Radiation can radically alter the dynamical properties of wave modes in a fluid. This radiative heat exchange gives rise to a temporal decay of oscillations with a characteristic dimensionless decay time $\tilde \tau_{\rm D} = 1/~\Omega_{\rm I}$, where $\Omega_{\rm I}$ is the imaginary part of $\Omega$. Depending on the value of the radiative relaxation time $\tilde\tau_{\rm R}$, the modes are effectively damped by the radiative dissipation in as short a time as two oscillation periods; however, in the limits of very large or very small $\tilde\tau_{\rm R}$, corresponding to nearly adiabatic or nearly isothermal oscillations, the modes are essentially undamped. We would also like to point out the merits and demerits of using Newton's law to model heat exchange. At sufficiently low frequencies, the wavelength of a disturbance is so long, that it becomes optically thick (no matter how transparent the material is), and the Newtonian cooling approximation no longer holds. Conversely, at high frequencies the wavelength of a disturbance becomes so small that it is optically thin (no matter how opaque the material) and the Newtonian approximation holds good. Bünte & Bogdan (1994) have already pointed out that radiative effects on oscillations in photospheric and higher layers are clearly important. Radiative dissipation based upon Newton's cooling law is clearly an oversimplification of the problem; nevertheless it allows us to assess the effects of radiative damping on the modal structure. It also enables us to look at the full frequency spectrum and the interaction amongst various modes. Our treatment of the weak field limit has permitted an analysis of the $K - \Omega$ diagram in terms of asymptotic approximations; this has allowed us to understand the nature of the modes in a vertical magnetic field in the presence of radiative exchange. The insight so gained has proved useful in extending the computation to the moderate to strong field case. The transition region lines as observed by CDS on SoHO are capable of diagnosing, Alfven, slow and fast magnetoacoustic waves. The Alfvenic oscillations are essentially velocity oscillations and do not cause any density fluctuations. The compressional modes may however reveal themselves in the form of intensity oscillations through a variation in the emission measure. This fact, together with the oscillations in intensity, allows us to interpret the waves as slow magneto-acoustic in nature. We have computed the frequencies of the modes from the full MAG equation (see Eq. (11)) and found out that for our model atmosphere they correspond to the slow magneto-acoustic modes. The p₁ and p₂ mode frequencies fall very well within the observed range (compare Tables 2 and 4). Our observational results very much complement earlier results and provide additional input for the study of the characteristics of the wave modes. Our observations reveal that umbral oscillations are a localized phenomenon, where intensity and velocity both shows a clear peak around 6 mHz. In all the wavelet plots, we also notice a smaller peak in the global wavelet spectra and some power in the phase plot around 3 mHz, for part of the time sequence, which corresponds to the penumbra. In the theory part of this paper we have shown that the life time of the oscillations are dependent on the relaxation time scale and in some cases these oscillations could be damped within a few oscillations periods as well. Our observations also indicate that the oscillations seems to come in packets with life times of $\sim$10-20 min, which matches fairly well with the damping behavior of our MAG waves. We should also point out that the envelope of these packets do not show exponential decay, as one would expect from the theory, rather the intensity amplitude usually remain sinusoidal. An alternative explanation for the appearance of the packets could be due to the rotation of the sun under the slit than the actual length of the oscillations. We see oscillations for only 20 min as that may be the time necessary for a source of say, 2 arcsec wide in the 2 arcsec wide slit, to rotate out of the field of view, if the sun is rotating at say, 6 arcsec/hour. In general we find good agreement between the model and observations as far as the duration of oscillation and range of frequency is concerned.

Acknowledgements

DB expresses his gratitude to Profs. S. S. Hasan and Joergen Christensen-Dalsgaard for many valuable discussions which has enabled to develop the theory of the MAG waves. DB wishes to thank the FWO for a fellowship (G.0344.98). EOS is a member of the European PLATON Network. We would like to thank the CDS and EIT teams at Goddard Space Flight Center for their help in obtaining the present data. CDS and EIT are part of SoHO, the Solar and Heliospheric Observatory, which is a mission of international cooperation between ESA and NASA. Research at Armagh Observatory is grant-aided by the N. Ireland Dept. of Culture, Arts and Leisure. This work was supported by PPARC grant PPA/G/S/1999/00055. The original wavelet software was provided by C. Torrence and G. Compo, and is available at URL: http://paos.colorado.edu/research/wavelets/.