Differential rotation normalised to the rotation rate of the outer sphere δΩ/Ω₀, rescaled with Re_c and plotted at the latitude θ = π/8 as a function of radius. In each case E = 10⁻⁴ and P_r(N₀/Ω₀)² = 10⁴. Left panel: analytical solution Eq. (56) is represented by dashed lines and numerical solutions (runs 1.1–1.4 in Table 7) as solid coloured lines for the linear regime Re_c = 10⁻². The colours green, blue, purple, and red correspond respectively to the values of the density contrasts between the inner and outer spheres ρ_i/ρ₀ = 1.75, 10.8, 20.9, and 60.2. Middle panel: numerical solutions are now represented for Re_c = 10⁻², 10⁻¹, 1, and 10 (red, purple, blue, and green, respectively) for ρ_i/ρ₀ = 20.9 (runs 1.3, 2.3, 3.3, and 4.3 in Table 7). Right panel: same, but when Re_c ≥ 1 the solution in dashed lines is numerically estimated by taking the full contraction term in Eq. (55) into account.