From thin-flux-tube approximation to two-mode approximation

Abstract

The well-known mathematical formalism for the reduction of the magnetohydrodynamic equations to an infinite set of coupled one-dimensional equations in the coefficients of the variable expansions in terms of the tube radius is explored. A way to truncate the set at an arbitrary order is shown. A comparison of the approximation with the exact solution for linear waves reveals that the approximation obtained is not restricted to thin flux tubes, contrary to what one would expect. The approximation is a two-mode approximation, and is valid for tubes of arbitrary diameter. The shortcomings of the approximation and the limits of its validity are discussed. The two-mode approximation can be applied to nonlinear waves in coronal loops and to chromospheric filaments in the solar atmosphere.