Axisymmetric investigation of differential rotation in contracting stellar radiative zones

Context. Stars experience rapid contraction or expansion at different phases of their evolution. Modelling the angular momentum and chemical elements transport occurring during these phases remains an unsolved problem. Aims. We study a stellar radiative zone undergoing radial contraction and investigate the induced differential rotation and meridional circulation. Methods. We consider a rotating spherical layer crossed by an imposed radial velocity field that mimics the contraction and solve numerically the axisymmetric hydrodynamical equations in both the Boussinesq and anelastic approximations. An extensive parametric study is conducted to cover regimes of contraction, rotation, stable stratification and density stratification that are relevant for stars. Results. The differential rotation and the meridional circulation result from a competition between the contraction-driven inward transport of angular momentum and an outward transport dominated by either viscosity or an Eddington-Sweet type circulation, depending on the value of the $P_r \left( N_0 / \Omega_0 \right)^2 $ parameter, where $P_r$ is the Prandtl number, $N_0$ the Brunt-V\"ais\"ail\"a frequency and $\Omega_0$ the rotation rate. Taking the density stratification into account is important to study more realistic radial contraction fields but also because the resulting flow is less affected by unwanted effects of the boundary conditions. In these different regimes and for a weak differential rotation, we derive scaling laws that relate the amplitude of the differential rotation to the contraction timescale.