Ideal incompressible axisymmetric MHD: Uncovering finite-time singularities

TL;DR

This paper provides numerical evidence for finite-time singularities in 3D axisymmetric ideal incompressible MHD equations, showing how initial field strength ratios influence the type and development of singularities.

Contribution

It uncovers the dependence of singularity formation on initial velocity and magnetic field ratios in ideal MHD, revealing different singularity types and their conditions.

Findings

Finite-time singularities occur depending on initial field strength ratio C.

Different singularity types (blow-up or cusp) are identified based on C.

Pressure evolution offers insights into singularity development.

Abstract

We provide compelling numerical evidence for the development of (potential) finite-time singularities in the three-dimensional (3D) axisymmetric, ideal, incompressible magnetohydrodynamic (IMHD) equations, in a wall-bounded cylindrical domain, starting from smooth initial data, for the velocity and magnetic fields. We demonstrate that the nature of the singularity depends crucially on the relative strength C of the velocity and magnetic fields at the time of initialisation: (i) if C < 1, then the swirl components, at the wall, evolve towards square profiles that lead to the intensification of shear at the meridional plane (r = 1, z = L/2) and the development of a finite-time singularity; (ii) if C = 1, there is no temporal evolution; (iii) if C > 1, then the swirl components, at the wall, evolve towards a cusp-type singularity. By examining the spatiotemporal evolution of the pressure, we obtain insights into the development of these singularities.