The combined impact of rotation and magnetic fields on oscillations of stellar fluids is still not well known theoretically. It mixes Alfv´en and inertial waves. Neutron stars are a place where both effects may be at work. We aim to solve this problem in the context of the r-mode instability in neutron stars, as it appears when these modes are coupled to gravitational radiation.We consider a rotating spherical shell filled with a viscous fluid of infinite electrical conductivity and analyse propagation of model perturbations when a dipolar magnetic field is bathing the fluid layer. We perform an extensive numerical analysis and find that the m =2 r-mode oscillation is influenced by the magnetic field when the Lehnert number (the ratio of Alfv´en speed to rotation speed) exceeds a value proportional to the one-fourth power of the Ekman number (a non-dimensional measure of viscosity). This scaling is interpreted as the coincidence of the width of internal shear layers of inertial modes and the wavelength of the Alfv´en waves. Applied to the case of rotating magnetic neutron stars, we find that dipolar magnetic fields above 1014 G are necessary to perturb the r-mode instability.

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