THE PSEUDOSPECTRUM OF THE RESISTIVE MAGNETOHYDRODYNAMICS OPERATOR - RESOLVING THE RESISTIVE ALFVEN PARADOX

The ''Alfven paradox'' is that as resistivity decreases, the discrete eigenmodes do not converge to the generalized eigenmodes of the ideal Alfven continuum. To resolve the paradox, the epsilon-pseudospectrum of the resistive magnetohydrodynamic (RMHD) operator is considered. It is proven that for any epsilon, the epsilon-pseudospectrum contains the Alfven 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=\k.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)1/2), Where R(M) is the magnetic Reynolds number.