Stratification influence on the onset of shear instability in a spherical shell. Top: evolution of the critical Reynolds number Re_c (or equivalently the Rossby number Ro_c) as a function of stratification intensity (quantified by the Rayleigh number Ra). When Re > Re_c (red upward triangles), nonaxisymmetric modes are maintained in time. Conversely, when Re < Re_c (dark downward triangles), the kinetic energies of nonaxisymmetric modes exponentially decay with time. The insets highlight the pattern of unstable modes close to the threshold via color maps of the radial velocity in the equatorial plane. Simulations parameters: E = 10⁻⁵, Pr = 0.1. Bottom: critical Reynolds number Re_c for the onset of hydrodynamic instability with increasing stratification, for various thermal and rotation parameters.