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On the spectrum of the magnetohydrodynamic mean-field \(\alpha^2\)-dynamo operator. (English) Zbl 1218.47008

The existence of magnetohydrodynamic mean-field \(\alpha^2\)-dynamos with spherically symmetric, isotropic helical turbulence function \(\alpha\) is related to a non-self-adjoint spectral problem for a coupled system of two singular second order ordinary differential equations. The authors establish global estimates for the eigenvalues of this system in terms of the turbulence function \(\alpha\) and its derivative \(\alpha^{\prime}\). They allow us to formulate an antidynamo theorem and a nonoscillation theorem. The conditions of these theorems, which again involve \(\alpha\) and \(\alpha^{\prime}\), must be violated in order to reach supercritical or oscillatory regimes.

MSC:

47A10 Spectrum, resolvent
76W05 Magnetohydrodynamics and electrohydrodynamics
47A55 Perturbation theory of linear operators
34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators
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