Flute oscillations of cooling coronal loops with variable cross-section

We consider fluting oscillations in a thin straight expanding magnetic flux tube in the presence of a background flow. The tube is divided into a core region that is wrapped in a thin transitional region, where the damping takes place. The method of multiple scales is used for the derivation of the system of governing equations. This system is applicable to study both standing and propagating waves. Furthermore, the system of equations is obtained for magnetic tubes with a sharp boundary. An adiabatic invariant is derived using the Wentzel-Kramer-Brillouin (WKB) method for a magnetic flux tube with slowly varying density, and the theoretical results are then used to investigate the effect of cooling on flute oscillations of a curved flux tube semi-circular in shape. We have analysed numerically the dependencies of the dimensionless amplitude for a range of values of the expansion factor and the ratio of internal to external plasma densities at an initial time. We find that the amplitude increases due to cooling and is higher for a higher expansion factor. Higher values of the wave number lead to localisation of the oscillation closer to the boundary. Finally, we show that the higher the value of the ratio of internal to external plasma densities, the higher the amplification of oscillation due to cooling. Therefore, we conclude that the wave number, density ratio, and the variation of tube expansion are all relevant parameters in the cooling process of an oscillating flux tube.