The effect of the external magnetic field on the spectral properties of one-electron non-uniform quantum ring with radially directed hills is analysed. The corresponding one-particle wave equation is separable in the adiabatic limit, when the layer thickness is essentially smaller than its lateral dimension. Our calculations show that the presence of a single axially directed hill as well as a rise of the central hole thickness produce a quenching of the Aharonov-Bohm (AB) oscillations of the lower energy levels and of the magnetic momentum. However, as the number of radially directed hills is increased, the system exhibits again oscillations, resulted from an enhancement of tunnelling circular currents.