We give a short derivation of the basic equations, starting from Eqs. (3)–(5). The solutions are obtained in the three regions, inside the flux tube, outside the flux tube and in the transition layer.
Again using the thin tube limit, we have (A.3)Hence, at r = R + l/2 = R(1 + ϵ/2), we have (A.4)Since both ξr and P are continuous across the transition layer as ϵ → 0, we can state ξe = η + δξr, Pe = Pi + δP, where both δξr and δP tend to zero as ϵ → 0. Using (A.4), we have, correct to O(ϵ2), (A.5)Rearranging Eq. (A.5), the final equation for the propagating kink mode is (A.6)Note that the right hand side of (A.6) is of O(ϵ). It is the leading order expressions for δξr and δP that we now need to calculate and this is done from the transition layer solutions.
Integrating Eq. (3) across the thin transition layer, we have (A.7)Integrating (4) we have (A.8)Remembering that η is independent of r, is l times the average value of the operator ℒ and, for the linear density profile, the average is ℒk, thus, (A.9)since ℒkη = O(ϵ). Hence, for the linear density profile (A.10)
In this section we evaluate (A.12)Using the solution for ξϕ given by (26), the integral across the transition layer is made up of four terms. These are evaluated in turn.
Now the integral of the first term on the RHS of Eq. (26), due to the radial profile of the driving boundary condition of ξϕ, is For large kz, this is proportional to (kz)-1. The influence of the choice of boundary condition does becomes less important after several wavelengths.
For small values of kz, it is easier to start from the integral expression. Hence, the first two terms in the Taylor series are For small values of κ, term 2 can be shown to reduce to where Z = κkz/2, as above.
We can now bring together the expressions for all four terms to rewrite the kink mode equation, (A.11), as Expressing η as , our final equation is (A.14)where , , ℒ1 = d2/dz2 − 2ikd/dz and ℒe = −k2κ + ℒk. In Eq. (A.14), the operator, ℒ1, acting on the final terms in the curly brackets on the right hand side, results in terms that are small for κ ≪ 1. In fact, the terms remain small even for κ ≤ 1/2. Hence, we will neglect them and the comparison with the numerical results confirms this is a valid assumption (see Sect. 5).
Equation (A.14) is an inhomogeneous, integro-differential equation for , the slowly varying amplitude function that describes the damping of the kink mode.
© ESO, 2013