EDP Sciences
Free Access
Volume 581, September 2015
Article Number A118
Number of page(s) 22
Section Planets and planetary systems
DOI https://doi.org/10.1051/0004-6361/201526246
Published online 18 September 2015

Online material

Appendix A: Harmonic highest degree kc

thumbnail Fig. A.1

Rank of the highest harmonic degree kc as a function of E (with K = 10-4) and K (with E = 10-4) for different values of A (in logarithmic scales). Left: kcE. Right: kcK.

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The expression of the highest harmonic degree, given in Eq. (108), is plotted as a function of the Ekman number and of the thermal diffusivity for various regimes from A ≪ 1 to A ≫ 1. It corresponds to the upper bound of harmonic degree, which is sufficient to represent all the resonances of the spectrum.

Appendix B: Relative differences ηl and ηH

The results provided by analytical expressions of Eqs. (65) and (71) have been compared to those given by the complete formula of the energy dissipated by viscous friction (Eq. (38)). The colour maps of Fig. B.1 correspond to the computation of the width at mid-height l11 and height of the main resonance, which are summarized in Fig. 8. The relative differences ηl and ηH are calculated using the expressions (B.1)lana and coming from Eqs. (65) and (71), and lth and being computed with Eq. (38). The plots highlight the asymptotic domains, in dark blue, where the analytical formulae constitute a relevant approximation for the width and height of a resonance, and the critical transition zones where they cannot be applied. The light blue horizontal line corresponds to A ~ A11, the hyper-resonant case. The colour gradient in the regions near E ~ 10-2 and K ~ 10-2 indicates the values of A, E, and K for which the condition of the quasi-adiabatic approximation (Eq. (85)) is not satisfied. The same can be done with the energy dissipated by thermal diffusion.

thumbnail Fig. B.1

ηl and ηH as a function of A, E, and K in logarithm scales. White contours highlight critical zones where the quasi-adiabatic assumption () or the asymptotic condition (AA11) are not satisfied. Top left: ηl(E,A). Top right: ηl(K,A). Bottom left: ηH(E,A). Bottom right: ηH(K,A).

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© ESO, 2015

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