Small amplitude solutions of the dynamo problem. 2. The case of α2-dynamos

Abstract

Abstract The amplitudes of the magnetic field generated by various α2-dynamo models are investigated for the case when the magnetic Reynolds number, R, slightly exceeds the critical value, R(0) , at which dynamo action is marginally possible. The basic assumption is that the α-effect is modified by the presence of the magnetic field by an amount proportional to the local magnetic energy density, according to the quenching law α = α0 − α1 |B|2, where B is the magnetic field and α0 and α1 are positive constants. Solutions were obtained by expanding the field in powers of the amplitude, a, of B, and by solving the resulting perturbation equations up to third order in a with the help of the adjoint kinematic dynamo system. An expression for the growth rate, γ1, of an infinitesimal disturbance when R — R (0) is small was obtained, and the amplitude, a(0), of the final (stable) state was derived. Numerical integrations were performed for five models originally due to Roberts (1972) as well as for a model studie...