Standing Sausage Perturbations in Solar Coronal Loops with Diffuse Boundaries: An Initial Value Problem Perspective

Working in pressureless magnetohydrodynamics, we examine the consequences of some peculiar dispersive properties linear fast sausage modes (FSMs) one-dimensional cylindrical equilibria with a continuous radial density profile ($\rho_0(r)$). As recognized recently on solid mathematical grounds, cutoff axial wavenumbers may be absent for FSMs when $\rho_0(r)$ varies sufficiently slowly outside nominal cylinder. Trapped therefore exist arbitrary and contrasts, their phase speeds long-wavelength regime differing little from external Alfv$\acute{\rm e}$n speed. If these trapped indeed show up solutions to associated initial value problem (IVP), then have much better chance observed than expected classical theory, can invoked account considerably broader range periodicities practiced. However, fundamentals active region loops as an example, that this expectation is not seen our finite-difference IVP, reason which explored by superposing necessary eigenmodes re-solve IVP. At least parameters examine, eigenfunctions are characterized spatial extent well exceeding observationally reasonable perturbations, meaning negligible fraction energy mode receive. We conclude absence examined equilibrium does guarantee distinct temporal behavior.