Fast sausage modes in magnetic tubes with continuous transverse profiles: effects of a finite plasma beta

TL;DR

This study derives a dispersion relation for fast sausage modes in magnetic tubes with continuous transverse profiles, analyzing how finite plasma beta influences mode properties and implications for solar flare QPPs.

Contribution

It provides a generic dispersion relation accounting for finite plasma beta and arbitrary transverse structuring, clarifying its weak influence on mode periods and the conditions affecting damping times.

Findings

Weak beta dependence of mode periods and critical wavenumber.

Damping times are sensitive to length-to-radius ratio and density profile.

Zero-beta theory suffices for trapped modes in QPP analysis.

Abstract

While standing fast sausage modes in flare loops are often invoked to interpret quasi-periodic pulsations (QPPs) in solar flares, it is unclear as to how they are influenced by the combined effects of a continuous transverse structuring and a finite internal plasma beta ($\beta_{\rm i}$). We derive a generic dispersion relation (DR) governing linear sausage waves in straight magnetic tubes for which plasma pressure is not negligible and the density and temperature inhomogeneities of essentially arbitrary form take place in a layer of arbitrary width. Focusing on fast modes, we find that $\beta_{\rm i}$ only weakly influences $k_{\rm c}$, the critical longitudinal wavenumber separating the leaky from trapped modes. Likewise, for both trapped and leaky modes, the periods $P$ in units of the transverse fast time depend only weakly on $\beta_{\rm i}$, which is compatible with the fact that the effective wavevectors of fast sausage modes are largely perpendicular to the background magnetic field. However, a weak $\beta_{\rm i}$ dependence of the damping times $\tau$ is seen only when the length-to-radius ratio $L/R$ is $\sim 50\%$ larger than some critical value $\pi/(k_{\rm c} R)$, which itself rather sensitively depends on the density contrast, profile steepness as well as on how the transverse structuring is described. In the context of QPPs, we conclude that the much simpler zero-beta theory can be employed for trapped modes, as long as one sees the deduced internal Alfv\'en speed as actually being the fast speed. In contrast, effects due to a finite beta in flare loops should be considered when leaky modes are exploited.