ner_tags: O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula O O B-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O
tokens: first order in δ n The general solution of Eq. ( A.29 ) then reads where α n ( T 1 ) and β n ( T 1 ) are two functions to be determined with the first-order equation. Using Eq. ( A.31 ) in Eq. ( A.30 ), it is straightforward to show (e.g., via the method of variation of parameters) that, if non-null, the terms in brackets will induce terms proportional to τe ± iτ in the expression of , up to a factor on the order of α n and β n . This will make on the order of up to a timescale on the order of O(1/ δ n ) and, therefore, would break the perturbation ansatz in Eq. ( A.28 ). Therefore, to obtain a uniformly valid expression up to such timescales, the terms in brackets have to vanish. This condition leads to with A n and B n two arbitrary constants. As a consequence, the amplitude is ruled only by the coupling term on the right-hand side of Eq. ( A.30 ), which depends on the leading-order amplitudes with n ′ ≠ n . For n ′ ≠ n , it is obvious that the amplitudes follow the same zeroth-order and first-order equations as in Eqs. ( A.29 ) and ( A.30 ); however, because of the considered scaling, T k for k = 0 or 1 in these equations are replaced by with π = ( ω n ′ / ω n ). As a result, we get where A n ′ and B n ′ are two arbitrary constants. According to Eqs. ( A.30 )–( A.32 ) we thus find that is ruled at leading order by Using again the method of variation of parameters, we can show that the amplitude is equal to a series of terms as a function of n ′ that are ). At this point, we assume that the magnitude of slowly varies with n ′, so that it is for n ′∼ n ; this will be checked a posteriori (see the resulting velocity amplitudes in Fig. 2 ). Provided for n ∼ n ′ and otherwise, as justified in Appendix A.2.3 , the main contribution to results from the terms of the series associated with n ′ ∼ n ; as a result, with Δ ω n = | ω n − ω n − 1 |. Within the asymptotic limit, ω n /Δ ω n ∼ n ≫ 1 and thus in order of magnitude. This highlights afterward that, instead of δ n , it is more appropriate to scale the non-adiabatic perturbation of the mode amplitude by in such a way that the perturbation ansatz becomes and the new first-order perturbation . As a result, the relative error when considering up to a timescale on the order of T damp is on the order of . We check a posteriori at the end of Sect. 3.1 that in the considered asymptotic frequency range. Therefore, all the previous perturbation developments, fortunately, hold true and the coupling terms do not stand for secular terms. These terms thus have no effect on the two-timing solution provided by , and are just responsible for the relative error on the order of . In summary, provided that the mode amplitude slowly varies with the radial order, the solution of the homogeneous equation as a function of the variable t , which is uniformly valid up to a timescale on the order of O( T damp ), takes the form, according to Eqs. ( A.31 ) and ( A.32 ), of the adiabatic solution modulated by an exponential