Simulations of waves in sunspots

Magnetic field modifies the properties of waves in a complex way. Significant advances has been made recently in our understanding of the physics of waves in solar active regions with the help of analytical theories, numerical simulations, as well as hi-resolution observations. In this contribution we review the current ideas in the field, with the emphasis on theoretical models of waves in sunspots. 1. Motivation and questions Photospheres and chromospheres of sunspots are the regions where different physical agents come into play with almost equal weight. The restoring forces in the wave equation, such as magnetic Lorentz force, gas pressure gradient and buoyancy, are of the same order of magnitude, making arbitrary the division into pure wave modes and complicating theoretical models. A recent summary on the observational properties of sunspot oscillations together with some theoretical issues is presented by Bogdan & Judge (2006). The most recent observational results suggest that wave phenomena at different layers of the sunspots umbrae and penumbrae are related with each other (see e.g. Marco et al. 1996; Maltby et al. 1999, 2001; Brynildsen et al. 2000, 2002; Christopoulou et al. 2000, 2001; Rouppe van der Voort et al. 2003; Tziotziou et al. 2006; Bloomfield et al. 2007a). The interpretation of the observational material defines several groups of questions concerning the sunspot wave physics that can be summarized in the following way: • What drives the waves observed in sunspots? Are they externally driven by the quiet Sun p-modes? Are there sources of oscillations inside the umbra due to weak convection? • How can the oscillations observed at different sunspot layers be interpreted in terms of MHD waves? What are the relationships between photospheric and chromospheric oscillations? What causes the complex spatial pattern of oscillations such as chromospheric umbral flashes, penumbral waves, spatial coherency of waves over the umbra, etc.? • Why is the wave power suppressed in the umbral photosphere compared to the quiet Sun? Why is it enhanced in the chromosphere? • What mechanisms produce the change of the dominating frequency of waves in the umbra from 3 mHz in the photosphere to 5–6 mHz in the