We consider the magnetorotational instability ( MRI ) of a hydrodynamically stable Taylor-Couette flow with a helical external magnetic field in the inductionless approximation defined by a zero magnetic Prandtl number ( \textrm { Pm } = 0 ) .
This leads to a considerable simplification of the problem eventually containing only hydrodynamic variables .
First , we point out that the energy of any perturbation growing in the presence of magnetic field has to grow faster without the field .
This is a paradox because the base flow is stable without the magnetic while it is unstable in the presence of a helical magnetic field without being modified by the latter as it has been found recently by Hollerbach and Rüdiger [ Phys .
Rev .
Lett . 95 , 124501 ( 2005 ) ] .
We revisit this problem by using a Chebyshev collocation method to calculate the eigenvalue spectrum of the linearized problem .
In this way , we confirm that MRI with helical magnetic field indeed works in the inductionless limit where the destabilization effect appears as an effective shift of the Rayleigh line .
Second , we integrate the linearized equations in time to study the transient behavior of small amplitude perturbations , thus showing that the energy arguments are correct as well .
However , there is no real contradiction between both facts .
The linear stability theory predicts the asymptotic development of an arbitrary small-amplitude perturbation , while the energy stability theory yields the instant growth rate of any particular perturbation , but it does not account for the evolution of this perturbation .
Thus , although switching off the magnetic field instantly increases the energy growth rate , in the same time the critical perturbation ceases to be an eigenmode without the magnetic field .
Consequently , this perturbation is transformed with time and so looses its ability to extract energy from the base flow necessary for the growth .