Energy Loss of Solar $p$ Modes due to the excitation of Magnetic Sausage Tube Waves: Importance of Coupling the Upper Atmosphere

We consider damping and absorption of solar $p$ modes due to their energy loss to magnetic tube waves that can freely carry energy out of the acoustic cavity. The coupling of $p$ modes and sausage tube waves is studied in a model atmosphere composed of a polytropic interior above which lies an isothermal upper atmosphere. The sausage tube waves, excited by $p$ modes, propagate along a magnetic fibril which is assumed to be a vertically aligned, stratified, thin magnetic flux-tube. The deficit of $p$-mode energy is quantified through the damping rate, $\Gamma$ and absorption coefficient, $\alpha$. The variation of $\Gamma$ and $\alpha$ as a function of frequency and the tube's plasma properties is studied in detail. Previous similar studies have considered only a subphotospheric layer, modelled as a polytrope that has been truncated at the photosphere (Bogdan et al. (1996), Hindman & Jain 2008, Gascoyne et al. (2011)). Such studies have found that the resulting energy loss by the $p$ modes is very sensitive to the upper boundary condition, which because of the lack of a upper atmosphere have been imposed in a somewhat ad hoc manner. The model presented here avoids such problems by using an isothermal layer to model the overlying atmosphere (chromosphere), and consequently, allows us to analyse the propagation of $p$-mode driven sausage waves above the photosphere. In this paper we restrict our attention to frequencies below the acoustic cut-off frequency. We demonstrate the importance of coupling all waves (acoustic, magnetic) in the subsurface solar atmosphere with the overlying atmosphere in order to accurately model the interaction of solar $f$ and $p$ modes with sausage tube waves.