Pseudo-Spectrum of the Resistive Magneto-hydrodynamics Operator: Resolving the Resistive Alfven Paradox

TL;DR

This paper investigates the resistive Alfvén paradox in magnetohydrodynamics by analyzing the pseudospectrum of the RMHD operator, revealing that the ideal continuum is approximated by pseudospectra even at low resistivity, and characterizing the ill-conditioning of eigenmodes.

Contribution

It provides a rigorous analysis of the pseudospectrum of the RMHD operator, resolving the Alfvén paradox and quantifying the ill-conditioning of resistive eigenmodes.

Findings

The ε-pseudospectrum contains the Alfvén continuum for small resistivity.

Resistive eigenmodes are exponentially ill-conditioned with condition number proportional to exp(R_M^{1/2}).

The entire stable half-annulus of complex frequencies is resonant to order ε.

Abstract

The `Alfv\'en Paradox' is that as resistivity decreases, the discrete eigenmodes do not converge to the generalized eigenmodes of the ideal Alfv\'en continuum. To resolve the paradox, the $\epsilon$-pseudospectrum of the RMHD operator is considered. It is proven that for any $\epsilon$, the $\epsilon$- pseudospectrum contains the Alfv\'en continuum for sufficiently small resistivity. Formal $\epsilon-pseudoeigenmodes$ are constructed using the formal Wentzel-Kramers-Brillouin-Jeffreys solutions, and it is shown that the entire stable half-annulus of complex frequencies with $\rho{|\omega|^2}=|\bf{v} \cdot \bf{B}(x)|^2$ is resonant to order $\epsilon$, i.e.~belongs to the $\epsilon-pseudospectrum$. The resistive eigenmodes are exponentially ill-conditioned as a basis and the condition number is proportional to $\exp(R_M^{1\over 2})$, where $R_M$ is the magnetic Reynolds number.