PSEUDO-SPECTRUM OF THE RESISTIVE MAGNETO-HYDRODYNAMICS OPERATOR: RESOLVING THE RESISTIVE ALFVÉN PARADOX D.Borba∗

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 -pseudospectrum of the RMHD operator is considered. It is proven that for any , the pseudospectrum contains the Alfvén continuum for sufficiently small resistivity. Formal − 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| = |k ·B (x)| is resonant to order , i.e. belongs to the − pseudospectrum. The resistive eigenmodes are exponentially ill-conditioned as a basis and the condition number is proportional to exp(R 1 2 M ), where RM is the magnetic Reynolds number.