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Appendix A: Corrections to the eigenfunctions due to the density stratification

In the Appendix we estimate the corrections to the chosen eigenfunctions due to the density stratification. Analytical progress can be made in the small y/χ limit. Since χ is a value smaller than one, this condition would automatically mean that we work in the small y limit, provided χ is not becoming too small. We are interested only in the characteristics of fundamental mode of kink oscillations and its first harmonic. Following Eq. (3) with the boundary conditions v_r(L) = dv_r(0)/dz = 0 and v_r(0) = v_r(L) = 0 for the fundamental mode and first harmonic, we introduce a new variable so that

In the new notations the density inside the loop can be written as (A.1)where h is the loop height above the solar atmosphere (a similar equation can be written for the external density). Next, we are working in the approximation h/H_i = ϵ ≪ 1, so Eq. (3) becomes (A.2)where ξ = ρ_i(0)/ρ_e(0) > 1 is the density ratio and χ = H_e/H_i < 1, with H_e and H_i being the density scale heights inside and outside the loop. Let us we write v_r and Ω as (A.3)Substituting these expansions into Eq. (A.2) and collecting terms proportional to subsequent powers of ϵ we obtain These equations must be solved separately for the fundamental mode and its first harmonic taking into account the boundary conditions.

• For the fundamental mode

• For the first harmonic

Let us first calculate the correction to the fundamental mode. It is easy to show that the solution of Eq. (A.4) taking into account the above boundary condition becomes (A.6)and (A.7)It is important to note that the form of the above solution is exactly the same as the solution we employed for the eigenfunction, v_r. In the next order of approximation we obtain Eq. (A.5) which can be written as (A.8)This boundary value problem will permit solutions only if the right-hand side satisfies the compatibility condition that can be obtained after multiplying the left-hand side by the expression of and integrating with respect to the variable ζ between 0 and 1, or

After some straightforward calculus we can find that (A.9)As a result, Eq. (A.5) becomes (A.10)This differential equation will have the solution (A.11)Applying the boundary condition , we find the constant

In order to find the value of C₂, we use the property of orthogonality, i.e.

which result in

As a result, the first order correction to v_r corresponding to the fundamental mode is (A.12)In Fig. A.1 we plot the correction to the eigenfunction for ϵ = 0.1, χ = 0.9, and ξ = 10. Figure (A.1) shows that we can approximate v_r(z) by cos(πz/2L) since the first order correction brings changes of about 1(%), i.e. insignificant.

A.1. Corrections to the first harmonic

The same analysis can be repeated for the first harmonic, taking into account the right boundary conditions. After a straightforward calculation it is easy to show that The correction to the eigenfunction corresponding to the first harmonic has been plotted in Fig. A.2 for the same values as before. It is obvious that the changes introduced by stratification in the value of the eigenfunction are of the order of 2(%), i.e. negligably small.

The two figures show that the effect of density stratification becomes more important for higher harmonics. Given the very large values of χ we used for prominences, the approximations used in this Appendix will always be valid. Fo the graphical represention of corrections in Figs. A.1 and A.2 we used χ = 1. If we lower this value to, e.g. 0.7 the corrections would still be small since the maximum relative change in the eigenfunction describing the fundamental mode would be 1.1%, while for the first harmonic, this would increase to 3.5%.

The robustness of our analysis was checked using a full numerical investigation for arbitrary values of χ and y. A typical dependence of the P₁/P₂ period ratio with respect to L/πH_i for one value of χ is shown in Fig. A.3 where the solid line corresponds to the analytical and the dotted line represent the numerical results, in both cases the density is inhomogeneous with respect to the coordinate z. The loop is set into motion using a gaussian-shaped source and we use a full reflective boundary conditions at the two footpoints of the loop. After the oscillations are formed, we use the FFT procedure to obtain the values of periods. Our analysis shows that the differences between the results obtained using the variational method and a full numerical investigation are of the order of 7% but towards the large range of L/πH_i. Restricting ourself to realistic values, i.e. L/πH_i < 5, we see that the results obtained with the two methods coincide with great accuracy.

© ESO, 2012