The pseudospectrum of the resistive magnetohydrodynamics operator: Resolving the resistive Alfvén 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 resistive magnetohydrodynamic (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=‖k⋅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(R1/2M), where RM is the magnetic Reynolds number.