We present self-consistent numerical simulations of the Sun's convection zone and radiative interior using a two-dimensional model of the solar equatorial plane. The background reference state is a one-dimensional solar structure model. Turbulent convection in the outer convection zone continually excites gravity waves that propagate throughout the stable radiative interior and deposit their angular momentum. We find that angular velocity variations in the tachocline are driven by angular momentum transported by overshooting convective plumes rather than nonlinear interaction of waves. The mean flow in the tachocline is time dependent but not oscillatory in direction and not like a quasi-biennial oscillation (QBO). Since the forcing in this shallow region cannot be described by simple linear waves, it is unlikely that the interaction of such waves is responsible for the solar cycle or the 1.3 yr oscillation. However, in the deep radiative interior, the interaction of low-amplitude gravity waves, continually excited by the overshooting plumes, is responsible for the angular velocity deviations observed there, which do resemble a very low amplitude QBO. Near the center of the model Sun the angular velocity deviation is about 2 orders of magnitude greater than that in the bulk of the radiative region and reverses its direction (prograde to retrograde or vice versa) in the opposite sense of the angular velocity deviations that occur in the tachocline. Our simulations thus demonstrate how angular velocity variations in the solar core are linked to those in the tachocline, which themselves are driven by convective overshooting.