Damping of kink waves by mode coupling

I. Analytical treatment^⋆

¹ School of Mathematics and Statistics, University of St Andrews, St Andrews, KY16 9SS UK
² Solar Physics and Space Plasma Research Centre (SP2RC), University of Sheffield, Sheffield, S3 7RH, UK
³ Departament de Física, Universitat de les Illes Balears, 07122 Palma de Mallorca, Spain
e-mail: alan@mcs.st-andrews.ac.uk

Received: 24 October 2012
Accepted: 2 January 2013

Abstract

Aims. We investigate the spatial damping of propagating kink waves in an inhomogeneous plasma. In the limit of a thin tube surrounded by a thin transition layer, an analytical formulation for kink waves driven in from the bottom boundary of the corona is presented.

Methods. The spatial form for the damping of the kink mode was investigated using various analytical approximations. When the density ratio between the internal density and the external density is not too large, a simple differential-integral equation was used. Approximate analytical solutions to this equation are presented.

Results. For the first time, the form of the spatial damping of the kink mode is shown analytically to be Gaussian in nature near the driven boundary. For several wavelengths, the amplitude of the kink mode is proportional to (1 + exp(-z²/L_g²))/2, where L_g² = 16/ϵκ²k². Although the actual value of 16 in L_g depends on the particular form of the driver, this form is very general and its dependence on the other parameters does not change. For large distances, the damping profile appears to be roughly linear exponential decay. This is shown analytically by a series expansion when the inhomogeneous layer width is small enough.