On the sausage magnetohydrodynamic waves in magnetic flux tubes: finite plasma beta and phase mixing

Over the past twenty years, there has been increasing evidence of the existence of sausage waves in the solar atmosphere. These observations make them useful tools in the context of atmospheric seismology. Here, we study sausage magnetohydrodynamic waves in a magnetic flux tube of non-zero plasma beta with a circular cross-section and a radially inhomogeneous plasma density. Solving numerically the equations of motion for an initial value problem, the spatio-temporal evolution of the velocity perturbations is obtained for different sets of parameters. We show that the ratio of the amplitudes of the longitudinal and radial perturbations is determined by the amount of plasma beta. Additionally, the longitudinal component of the velocity perturbation experiences phase-mixing within a layer surrounding the boundary of the flux tube with a rate depending on the amount of plasma beta. The results revealed that in the presence of a non-zero plasma beta, the flux tube exhibits oscillations in both the radial and longitudinal directions, characterized by a combination of two frequencies: one belonging to the slow continuum and the other to the Alfvén continuum. Also, the period of radial oscillation is obtained for different sets of parameters. The dependence of the period of the radial oscillation on the wavenumber confirms the results obtained in previous studies.