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Appendix A: The azimuthal Fourier coefficients

The azimuthal Fourier coefficients are defined in Eq. (12). It is easy to verify that they satisfy the properties (A.1)(A.2)In this Appendix, we give only those Fourier coefficients that are of relevance to the problem at hand, namely those corresponding to K′ = Q′ = 0. Owing to the above-mentioned symmetry relations, there are only four independent coefficients, which are given by (A.3)If we chose the reference angle γ = 0 (see Landi Degl’Innocenti & Landolfi 2004), then from Eq. (A.6) of Frisch (2010), it is clear that are real and hence that are also real. Using Eq. (A.6) of Frisch (2010) in our Eq. (A.3), we obtain the explicit form for these coefficients (A.4)

Appendix B: The explicit form of the real components of

In this Appendix, we list the explicit analytic forms of the 25 non-zero real components of , defined in Eqs. (19), (20), and (25) of the text. For K = 2, we define (B.1)We can express all the 25 non-zero real components of in terms of as follows : (B.2)The above equations are valid for x < x_c in the case of a 1D cut-off assumption. For x > x_c, we have to set . In the case of approximation-II, the various appearing in the above equations should be taken inside the frequency integral appearing in Eq. (B.1), and Eqs. (106)–(113) of Bommier (1997b) have to be used.

The various appearing in the above set of equations are given by Eq. (6) in the case of the 1D cut-off assumption and Eqs. (A11)–(A18) of Anusha et al. (in press) in the case of the 2D frequency domains of Bommier (1997b). Below we give the explicit forms of appearing in those equations. We introduce the abbreviations, C_B = cosθ_B, S_B = sinθ_B, c₁ = cosχ_B, s₁ = sinχ_B, c₂ = cos2χ_B, and s₂ = sin2χ_B. In terms of the elements of the matrix given below (see also Appendix C in Frisch 2007), we have (B.3)where (B.4)

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