Higher Radial Harmonics of Sausage Oscillations in Coronal Loops

Impulsively excited sausage oscillations of a plasma cylinder with a smooth radial profile of Alfvén speed are analyzed with a numerical solution of the initial-value problem for a partial differential equation of the Klein-Gordon type, describing linear magnetoacoustic oscillations with a fixed axial wavelength and an azimuthal mode number. The range of analyzed ratios of Alfvén speeds outside and inside the cylinder is from 2 to 10. Both trapped and leaky regimes of the oscillations are considered. It is shown that even in the long-wavelength limit, i.e., for axial wavenumbers much smaller than the cutoff values, damping times of higher radial sausage harmonics could be significantly greater than the oscillation periods, i.e., several oscillation cycles could be present in the signal. The quality factors decrease with decfreasing ratios of Alfvén speeds outside and inside the cylinder. Oscillation periods of the second and third radial harmonics remain practically independent of the axial wavelength even when the wavelength is shorter than the radius of the cylinder. The ratios of oscillation periods of fundamental and higher radial and axial harmonics are found to be significantly different, up to a factor of two in the long-wavelength limit. It is concluded that higher radial harmonics could be responsible for the departure of observed sausage oscillation signals from a harmonic shape, especially during the first several cycles of the oscillation. Even in the absence of spatially resolved data, higher axial and radial harmonics can be distinguished from each other by the period ratios.