Fast magnetohydrodynamic oscillations of a coronal loop embedded in a potential coronal arcade

Context. Observations indicate variable widths exhibited by fan coronal loops and flare loops that tend to widen towards the apex. Short-period, quasi-periodic pulsations in solar flares are often interpreted in terms of the fast-sausage oscillations of flare loops and the collective vertical vibrations of arcade loops are attributed with the vertical kink mode. Both phenomena are used as a seismological tool to estimate the physical parameters in the corona. Aims. We performed an analytical study of fast sausage and kink oscillations in coronal loops, given the effects of loop curvature, expansion, and Alfvén speed variation. Methods. We modelled a coronal loop as a dense expanding curved magnetic slab embedded within a potential coronal arcade, using a zero- β plasma limit. We obtained the dispersion relation that governs fast waves in the model and studied it both numerically and analytically. Results. The effects of loop expansion and variable Alfvén speed reduce the cut-off frequency and increase the cut-off wavenumbers for fast sausage and kink waves. Moreover, the principal vertical kink mode has a cut-off and strongly attenuates in the leaky regime. The frequency increase is found to be minor for the global sausage mode both in the trapped and leaky regimes, with a frequency shift within a few percent. We found that in our model, where the Alfvén speed increases from the footpoints to the loop top, the spatial profile of the longitudinal fundamental is broadened and the antinodes of the first overtone are shifted towards the footpoints. Conclusions. Using the classical expression for the cut-off wavenumber of the global sausage mode in a straight waveguide results in an underestimation of the density contrast constraint in flare loops. Instead, the suggested formula accounting for variations in loop widths provides more accurate results. The frequency of the global sausage mode can be correctly determined with the straight slab model.