Near-resonant interactions of the magnetorotational instability in thin Keplerian discs

TL;DR

This paper derives equations describing weakly nonlinear near-resonant interactions in thin Keplerian discs, revealing complex dynamics including explosive growth of mode amplitudes, which can surpass strict-resonance cases.

Contribution

It introduces a novel set of coupled equations for near-resonant magnetorotational instability interactions, highlighting the explosive growth potential in such systems.

Findings

Unstable triads can grow explosively with finite-time blow-up.

Near-resonant interactions can be faster than strict-resonance cases.

Derived equations reveal complex dynamical behaviors.

Abstract

The equations that govern the weakly nonlinear near-resonant interaction of a parent magnetorotational instability with two linearly stable daughter modes in thin nonuniform magnetized Keplerian discs are derived. While the equations for the daughter modes resemble their first order classical uniform counterparts, the parent mode is governed by a second order forced Duffing equation. It is demonstrated that the solutions of those three coupled equations exhibit a wide and rich spectrum of dynamical behavior. In particular, it is shown that amplitudes of unstable triads may grow explosively with time, namely reaching infinite values in a finite time. Paradoxically, explosively unstable near- resonance triads may grow much faster than their strict-resonance counterparts.