On the magnetorotational instability and elastic buckling

TL;DR

This paper reveals a fundamental mathematical equivalence between the magnetorotational instability in astrophysics and elastic buckling in materials science, suggesting a universal underlying model.

Contribution

It demonstrates an asymptotic reduction showing the same dynamical equations govern both MRI and elastic buckling, indicating a potential universal class of such systems.

Findings

Derived a common wave equation for MRI and elastic buckling

Identified the system as a mean-field interacting Duffing oscillator

First proof of a strange attractor in a PDE context

Abstract

This paper demonstrates an equivalence between rotating magnetised shear flows and a stressed elastic beam. This results from finding the same form of dynamical equations after an asymptotic reduction of the axis-symmetric magnetorotational instability (MRI) under the assumption of almost-critical driving. The analysis considers the MRI dynamics in a non-dissipative near-equilibrium regime. Both the magnetic and elastic systems reduce to a simple one-dimensional wave equation with a nonlocal nonlinear feedback. Under transformation, the equation comprises a large number of mean-field interacting Duffing oscillators. This system was the first proven example of a strange attractor in a partial differential equation. Finding the same reduced equation in two natural applications suggests the model might result from other applications and could fall into a universal class based on symmetry.