Solutions to the SHJ test case. Left panels are figures reproduced from Heng et al. (2011) using the finite-difference model (reproduced by permission of Oxford University Press), and the right panels are results from this work for the “shallow” case (see Table 1 for explanation). The top row shows the temperature field at σ = 0.675 and 346 days. The middle and bottom rows show the zonal mean plots for temperature and wind respectively (i.e. zonally and temporally, from 200–1200 days, averaged).

Results for the HD 209458b test case reproduced from Heng et al. (2011) (reproduced by permission of Oxford University Press). The top row shows the zonal mean plots (i.e. zonally and temporally, from 200–1200 days, averaged, using bar as the unit of pressure) of temperature (left) and zonal wind (right). The middle and bottom rows show the temperature (colour) and horizontal velocities (vectors) at pressures 213 (middle left), 21 600 (middle right), 4.69 × 10⁵ (bottom left) and 21.9 × 10⁵ Pa (bottom right) after 1200 days.

Same as in Fig. 4 but for our “shallow” case. The zonal mean plots present pressure in Pa (SI unit, where 1 bar = 1 × 10⁵ Pa).

Same as in Fig. 4 but for our “deep” case. The zonal mean plots present pressure in Pa (SI unit, where 1 bar = 1 × 10⁵ Pa).

Same as in Fig. 4 but for our “full” case. The zonal mean plots present pressure in Pa (SI unit, where 1 bar = 1 × 10⁵ Pa).

Insensitivity of the zonally and temporally averaged zonal wind (ms^-1) to the different modelling choices. The simulations in the left panels use the “shallow” and the right panels the “full” equation set. The top left panel shows a simulations where diffusion is applied to θ in addition to u and v. The top right panel shows a simulations with non-uniform vertical level placement to optimise sampling of the local scaleheight. The bottom left panel shows the results when the atmospheric height is decreased from H = 1.1 × 10⁷ m to H = 6.7 × 10⁶ m, and the bottom right panel when it is increased to 1.25 × 10⁷ m (although only the overlapping pressure domain of these simulations with that of the models in Heng et al. (2011), shown in Fig. 4, is displayed to aid comparison).

Horizontal velocity (vector arrows) and vertical velocity (colour scale) for the “full” case (see Table 1 for explanation) at 1 × 10⁵ Pa (1 bar) and after 100 (top left), 400 (top right), 800 (bottom left) and 1200 (bottom right) days. Although the colour scales differ, the contour lines are the same for all panels.

Vertical velocity, as a function of pressure, for the “shallow”, “deep” and “full” cases (see Table 1 for explanation) as the top, middle and bottom panels respectively. The left and right panels show vertical velocity as a function of longitude where a meridional average (performed in a point-wise fashion, i.e. ^∫ vdφ as opposed to ^∫ cosφvdφ, to emphasise differences in the vertical flow towards the polar regions) has been performed, and of latitude where a zonal average has been performed, respectively.