Differential rotation and r-modes in magnetized neutron stars

Rezzolla et al. drew attention to the second-order secular drift associated with r-modes and claimed that it should lead to enhancement of the magnetic field and suppression of r-mode instability in magnetized neutron stars. We critically revise these results. We present a particular second-order r-mode solution with vanishing secular drift, thus refuting a widely believed statement that secular drift is an unavoidable feature of r-modes. This non-drifting solution is not affected by a magnetic field B, if B ≪ B crit ≈ 1017 (ν/600 Hz) G (ν is the spin frequency) and does not lead to secular evolution of the magnetic field. For a general second-order r-mode solution, the drift does not necessarily vanish. The solution can be presented as a superposition of two solutions: one describes the evolution of differential rotation of a non-oscillating star (secular drift; for an unmagnetized star it is an arbitrary stationary rotation stratified on cylinders; for a magnetized star differential rotation evolves on the Alfvén timescale and may lead to enhancement of the magnetic energy), and the other is the non-drifting r-mode solution mentioned above. This representation allows us to conclude that enhancement of the magnetic field energy is limited by the initial energy of differential rotation, which is much less than the total energy of the r-mode (by a factor ∝ α2, where α is the mode amplitude). Hence, enhancement of the magnetic field by drift cannot suppress the r-mode instability. The results can be generalized for any oscillation mode in any medium, if this mode has a non-drifting solution for B = 0.