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(RM1/2), where RM is the magnetic Reynolds number. © 1994 American Institute of Physics.
ASJC Scopus subject areas
- Condensed Matter Physics