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Appendix A: Dependence on domain size

Above we have shown that we can recover many earlier results obtained in full spherical shells with wedges that span 150° in latitude and 90° in longitude. This gives at least a fourfold advantage in terms of computation time in comparison to a full shell. However, it is important to study the range within which we can still recover the same results as with larger wedges. In order to study this we perform two additional sets of runs that are listed in Table A.1. In Set E we vary the longitudinal extent from 22.5° to full 360°, with Δθ = 150° in all models. In Set F we keep the longitudinal extent fixed at Δφ = 90° and vary the latitudinal extent between 60° and 170°. As our base model we take Run A5 with fairly rapid rotation and complicated large-scale flows in the saturated state.

Figure A.1 shows the latitudinal profiles of the off-diagonal components of the Reynolds stress from the middle of the convectively unstable layer and the rotation profiles as functions of radius from three latitudes from Set E and Run A5. The Reynolds stresses are very similar in the latitude range  ± 45° in runs with Δφ = 90° or larger. Somewhat larger differences are seen near the latitudinal boundaries. Runs E1 and E2 with the smallest longitude extents show the same qualitative behaviour for stress components Q_rθ and Q_θφ but not for Q_rφ. The rotation profiles for Runs A5, E3, and E4 with Δφ = 90° − 270° are very similar. The most obvious deviations from the trend occur again for Runs E1 and E2 where the radial gradient of is negative at

the equator as opposed to the other runs where a positive gradient is found for r/R > 0.8. Surprisingly, Run E5 with a full 360° longitude extent also deviates from the trend seen in the intermediate φ-extents: the qualitative trend of is similar but the magnitude of the differential rotation is reduced. This is due to a non-axisymmetric m = 2 mode which is excited in this simulation. Large-scale hydrodynamical non-axisymmetries have been reported from rapidly rotating convection (e.g. Brown et al. 2008). However, it is not clear whether the non-axisymmetry in our Run E5 is due to the same mechanism because of the slower rotation.

Comparing simulations with different latitudinal extents (Fig. A.2), we find that domains confined between  ± 45° latitude still reproduce the essential features of the solutions. This is particularly clear for the Reynolds stresses which are very similar in the latitude range  ± 45° from the equator, with only Run F1 showing qualitatively different results in this range. There are also some differences at high latitudes between Runs A5 and F4. The rotation profiles are also very similar in the range  ± 30° with the exception of Run F1. Run A5 also shows a deviating profile at high latitudes.

These results suggest that a 90° longitude and 150° latitude extent is sufficient to capture the main features of the solutions at larger domains. The cost of this is that some features which are not of primary interest in the present study, such as the large-scale non-axisymmetric modes, are omitted.

	Fig. A.1 Off-diagonal Reynolds stresses from the middle of the convection zone (upper row), and as a function of radius at θ = 90° (lower row, left panel), θ = 60° (middle panel), and θ = 30° (right panel) for Runs E1–E5 and A5. Linestyles as indicated in the legend in the lower middle panel.
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	Fig. A.2 Same as Fig. A.1 but for Runs F1–F4 and A5. The left panel on the lower row shows from θ = 45°. Linestyles as indicated in the legend in the lower left panel
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© ESO, 2011