3.7 Strong perturbations: magnetic tubes and sunspots

The interaction of solar waves with photospheric magnetic fields has been studied extensively in recent years. Yet, applications in the context of local helioseismology have proved difficult. The source of this difficulty is that magnetic perturbations are not small near the solar surface (e.g., Crouch and Cally, 2004 ): the first Born approximation cannot be applied there. Only deeper inside the Sun, can magnetic effects be treated as small perturbations. In this paragraph, we review theoretical results regarding wave interaction with magnetic flux tubes and sunspots. Near the photosphere magnetic fields appear to be clumped into intense flux tubes with a typical field strength of order $1 kG$ and diameters of about $100 km$ . It is well known (see, e.g., Hollweg, 1990 ) that flux tubes support various modes of oscillation: the Alfvén modes (twisting motion), the sausage modes (propagating change in the cross-section of the tube), and the kink modes (bending of the tube). The interaction of acoustic waves with thin flux tubes, i.e., tubes with diameters much less than the wavelengths, has been studied extensively (see, e.g., Ryutova and Priest, 1993 ; Bogdan and Zweibel, 1985 ; Bogdan et al., 1996 ; Tirry, 2000 ). There is general agreement that scattering of acoustic waves by flux tubes contributes to the observed damping rates and frequency shifts (Rajaguru et al., 2001

; Komm et al., 2002 ).

Finsterle et al. (2004b ) have detected the interaction of high-frequency acoustic waves with the canopy magnetic field in the solar chromosphere. These exciting results were obtained by making observations of solar oscillations at different heights in the atmosphere (Finsterle et al., 2004a ). We note that there have been theoretical studies of the effect of the magnetic canopy on acoustic modes (see, e.g., Campbell and Roberts, 1989 ; Goldreich et al., 1991 ).

Thomas et al. (1982 ) first predicted that solar oscillations could be used to probe the internal structure of sunspots. Sunspots are known to absorb incident p-mode energy (Braun et al., 1987 ), introduce phase shifts between the incident and scattered waves (Braun et al., 1992 ; Duvall Jr et al., 1996 ; Lindsey and Braun, 2004 ), and cause mode mixing (Braun, 1995 ). Effects that have been suggested to cause phase shifts are the Wilson depression (Braun and Lindsey, 2000 ), flows (see, e.g., Duvall Jr et al., 1996 ; Kosovichev, 1996 ), inhomogeneous absorption (Woodard, 1997 ), temperature/density/wave-speed anomalies (see, e.g., Kosovichev, 1996 ; Brüggen and Spruit, 2000 ; Tong et al., 2003 ), and the direct effect of magnetic fields. Bogdan et al. (1998 ) and Cally et al. (2003 ) have produced models of the coupling of ambient p and f modes with various magnetic waves in the sunspot that explain some aspects of the data, and demonstrate that magneto-atmospheric waves can not be ignored in local helioseismology. Yet, according to Bogdan (2000 ), ‘ignorance triumphs over knowledge’. The main difficulties are nonlinear aspects of wave propagation, radiative transfer in magnetised plasmas, and the relationship between velocity measurements in sunspots and real fluid motions. The theoretical study of oscillations in sunspots is an entire field of research and is crucial for the interpretation of local helioseismic measurements in and around sunspots. We refer the reader to the reviews by Bogdan and Braun (1995 ) and Bogdan (2000 ) for further details and references.