The Contribution of Sound Waves and Instabilities to the Penetration of the Solar Differential Rotation Below the Convection Zone

The response of a layer to a horizontal shear flow at its top the surface was studied numerically as an initial value problem. The geometry was Cartesian and the conservation equations were solved with the help of the Zeus-3D code. In the initial state, the pressure, p, and density, ρ, of the layer were assumed to be related by a polytropic equation of index 1.14, which best approximates the solar values in the region of interest. The values of p and ρ at the lower boundary of the layer, namely r=R _l=0.4 R _⊙, were taken to be the solar values. The upper boundary was chosen to be the base of the solar convection zone, r=R _c=0.7 R _⊙. The shear flow at the surface, v _φ(R _c), was proportional to the solar differential rotation, and acoustical oscillations were present in the layer. It is shown that if the initial state is stable, a dynamical coupling between sound waves and the shear flow transmits the surface flow to the inner regions of the layer, even in the absence of dissipation. The shear flow in the sublayer below the one at the surface is proportional to v _φ(R _c), to the time, and to the strength of the oscillations. The constant of proportionality is calculated from the numerical integrations, performed for times of the order of 100 hr. Extrapolation of these results to longer times shows that the surface shear flow is transmitted to the inner regions in a time of the order of of 30 000 years. If the initial state is unstable to the vertical shear, the region of maximum instability depends also on the horizontal shear, and is located away from the equator (where the vertical shear is maximum). As a consequence, the longitudinal flow below the surface shows two equidistant maxima across the equator, located at intermediate latitudes.