  Recent numerical results of current-driven instabilities at low magnetic
Prandtl number and high Hartmann number support the possibility of a saturation
state characterized by helicity oscillations. We investigate the underlying
mechanism by analyzing this possibility using an higher-order Landau-Ginzburg
effective Lagrangian for the weakly non-linear amplitude dynamics, where the
magnetic and velocity perturbations are linearly dependent. We find that, if
the mirror symmetry between left- and right-handed modes is spontaneously
broken, it is impossible to achieve an oscillating helical state. We argue that
the result is likely to hold also adding higher-order terms and in the presence
of an explicit symmetry breaking. We conclude that an oscillating saturating
state for the Tayler instability is unlikely to depend on the interaction of
chiral modes.
