Energy Propagation by Transverse Waves in Multiple Flux Tube Systems Using Filling Factors

Abstract

In the last few years, it has been found that transverse waves are present at all times in coronal loops or spicules. Their energy has been estimated with an expression derived for bulk Alfvén waves in homogeneous media, with correspondingly uniform wave energy density and flux. The kink mode, however, is localized in space with the energy density and flux dependent on the position in the cross-sectional plane. The more relevant quantities for the kink mode are the integrals of the energy density and flux over the cross-sectional plane. The present paper provides an approximation to the energy propagated by kink modes in an ensemble of flux tubes by means of combining the analysis of single flux tube kink oscillations with a filling factor for the tube cross-sectional area. This finally allows one to compare the expressions for energy flux of Alfvén waves with an ensemble of kink waves. We find that the correction factor for the energy in kink waves, compared to the bulk Alfvén waves, is between f and 2f , where f is the density filling factor of the ensemble of flux tubes.

Energy and energy flux in axisymmetric slow and fast waves

“…We thus integrate the energy over the tube cross-section, and thus discuss the energy per unit length along the magnetic field. To calculate the total energy we should also integrate over the z-direction from a height zero to a height L, where L is the length of the flux tube (see Van Doorsselaere et al 2014). Since the wave mode properties do not change with height this results in a simple multiplication with a factor L. Closed analytic expressions for KE , ME , IE , S , and T can be found when using Eqs.…”

“…The surface over which we averaged is a circular disk with radius d f . The idea behind this averaging over an area is linked to the paper by Van Doorsselaere et al (2014). In that paper MHD theory is used to write an expression for energy flux in multistrand coronal loops in terms of density filling factors.…”

“…In that paper MHD theory is used to write an expression for energy flux in multistrand coronal loops in terms of density filling factors. As explained in Van Doorsselaere et al (2014) the radius of a flux tube strand R can be linked with the influence radius of that flux tube strand d f using the density filling factor f , i.e.…”

“…The physical distance d is almost impossible to quantify directly since the transport of energy is difficult to identify, even using the highest-resolution solar observations. However, there are ways to infer this distance d. We turn to the paper by Van Doorsselaere et al (2014) and use the influence radius of a flux tube strand d f = R/ f to quantify the distance d. Using this influence radius d f in combination with Eq. (29) should result in finite values for the energy of the wave mode.…”

“…Goossens et al (2013a) used MHD theory to derive formulas that determine the energy of kink waves in the solar corona. Van Doorsselaere et al (2014) have expanded on this idea and used MHD theory to write an expression for energy flux in terms of density filling factors. A&A 578, A60 (2015) In this paper we focus on axisymmetric slow and fast sausage modes, which have recently been discovered in numerous structures, but have not been adequately modelled to estimate their energy content.…”