Axisymmetric steady solutions of the differential rotation as a function of radius at the latitude θ = π/4 in the Taylor–Proudman regime. Left panel: analytical solution Eq. (45) is plotted in black dashed lines and compared with the numerical simulations obtained for fixed P_r(N₀/Ω₀)² parameter (10⁻⁴) and contraction Reynolds number Re_c (10⁻²) with various Ekman numbers: 10⁻² (in grey), 10⁻³ (in cyan), 10⁻⁴ (in light blue), 10⁻⁵ (in dark blue), and 10⁻⁶ (in purple) respectively for runs 1.1–1.5 in Table 2. For E = 10⁻⁵ and E = 10⁻⁶, additional contraction Reynolds numbers are studied, namely 10⁻¹ (runs 2.1 and 3.1 in Table 2), 1 (runs 2.2 and 3.2 in Table 2), and 10 (runs 2.3 and 3.3 in Table 2). All curves are rescaled by . Right panel: differential rotation between the inner and outer spheres ΔΩ = Ω(r_i)−Ω(r₀) at the same latitude and normalised with the rotation rate taken at the outer sphere Ω₀, now plotted as a function of . The previous numerical solutions obtained for E = 10⁻⁵ (runs 1.4 and 2.1–2.3 in Table 2) are plotted in dark blue, while the simulations performed for E = 10⁻⁶ (runs 1.5 and 3.1–3.3 in Table 2) are plotted in purple.