Pseudo-Spectrum of the Resistive Magneto-hydrodynamics Operator: Resolving the Resistive Alfven Paradox
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 $ε$-pseudospectrum of the RMHD operator is considered. It is proven that for any $ε$, the $ε$- pseudospectrum contains the Alfvén continuum for sufficiently small resistivity. Formal $ε-pseudoeigenmodes$ are constructed using the formal Wentzel-Kramers-Brillouin-Jeffreys solutions, and it is shown that the entire stable half-annulus of complex frequencies with $ρ{|ω|^2}=|\bf{v} \cdot \bf{B}(x)|^2$ is resonant to order $ε$, i.e.~belongs to the $ε-pseudospectrum$. 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.