Effects due to induced azimuthal eddy currents in a self-exciting Faraday disk homopolar dynamo with a nonlinear series motor. I.: Two special cases

Abstract

The elucidation of the behaviour of physically realistic self-exciting Faraday-disk dynamos bears inter alia on attempts by theoretical geophysicists to interpret observations of geomagnetic polarity reversals. Hide [The nonlinear differential equations governing a hierarchy of self-exciting coupled Faraday-disk homopolar dynamos, Phys. Earth Planet. Interiors 103 (1997) 281–291; Nonlinear quenching of current fluctuations in a self-exciting homopolar dynamo, Nonlinear Processes in Geophysics 4 (1998) 201–205] has introduced a novel 4-mode set of nonlinear ordinary differential equations to describe such a dynamo in which a nonlinear electric motor is connected in series with the coil. The applied couple, α, driving the disk is steady and the Lorentz couple driving the motor is a quadratic function, x(1−ϵ)+ϵσx 2 , of the dynamo-generated current x, with 0≤ϵ≤1. When there are no additional biasing effects due to background magnetic fields etc., the behaviour of the dynamo is determined by eight independent non-negative control parameters. These include ρ, proportional to the resistance of the disk to azimuthal eddy currents, and β, an inverse measure of the moment of inertia of the armature of the motor. When β=0 (the case when the motor is absent and ϵ and σ are redundant) and ρ −1≠0 , the 4-mode dynamo equations reduce to the 3-mode Lorenz equations, which can behave chaotically [E. Knobloch, Chaos in the segmented disc dynamo, Phys. Lett. A 82 (1981) 439–440]. When β≠0 but ρ −1=0 , the 4-mode set of equations reduces to a 3-mode dynamo [R. Hide (1997), see above], which can also behave chaotically when ϵ=0 [R. Hide, A.C. Skeldon, D.J. Acheson, A study of two novel self-exciting single-disk homopolar dynamos: theory, Proc. R. Soc. Lond. A 452 (1996) 1369–1395] but not when ϵ=1 [R. Hide (1998), see above]. In the latter case, however, all persistent fluctuations are completely quenched [R. Hide (1998), see above]. In this paper we investigate two limiting cases of ϵ=0 and ϵ=1 in the 4-mode dynamo when azimuthal eddy currents are allowed to flow i.e. cases when ρ −1=0 ; in a companion paper [I.M. Moroz, R. Hide, Effects due to induced azimuthal eddy currents in the Faraday disk self-exciting homopolar dynamo with a nonlinear series motor: II The general case, 1999, submitted] we extend the present analysis to the general case of 0≤ϵ≤1. When ϵ=0, chaotic behaviour occurs even more extensively in parameter space in the presence of azimuthal eddy currents than in their absence. When ϵ=1, the quenching of chaotic and all other non-steady dynamo action is no longer complete, for aperiodic solutions are found within limited regions of parameter space where β is very small and α is very large.