Nonlinear damped standing slow waves in hot coronal magnetic loops

¹ Solar Physics and Space Plasma Research Centre (SP 2RC), University of Sheffield, Hicks Building, Hounsfield Road, Sheffield S3 7RH, UK
² Space Research Institute (IKI), Russian Academy of Sciences, 117997 Moscow, Russia
e-mail: m.s.ruderman@sheffield.ac.uk

Received: 27 January 2013
Accepted: 19 March 2013

Abstract

We study the nonlinear standing slow waves in coronal magnetic loops. We assume that the wave amplitude is sufficiently small and use it as a small parameter in the asymptotic expansions. In addition, we assume that the wave damping is sufficiently slow, which enables us to introduce the “slow” time related to the evolution of the wave amplitude and shape. We show that, in the leading-order approximation, the nonlinear standing wave is a superposition of two identical nonlinear waves propagating in the opposite directions. The two waves are governed by the Burgers equation. Using the Cole-Hopfe substitution, we obtain an analytical solution describing a standing wave that has the form of the linear fundamental eigenmode at the initial moment of time. This solution is used for the parametric study of nonlinear standing waves. In particular, the effect of nonlinearity on the oscillation damping time is investigated.