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

Abstract

The `Alfvén Paradox' is that as resistivity decreases, the discrete eigenmodes do not converge to the generalized eigenmodes of the ideal Alfvén 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 Alfvén 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 $\rho |\omega {|}^2=|\mathbf{v}\cdot \mathbf{B}(\mathbf{x}){|}^2$ 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 $\exp (R_M^{\frac{1}{2}})$, where $R_M$ is the magnetic Reynolds number.