Nonlinear feedback and coupling in self-exciting dynamos

Abstract

Systems of steadily-forced self-exciting Faraday-disk homopolar dynamos that incorporate series motors and the autonomous sets of nonlinear ordinary differential equations (ODEs) governing their behaviour are of intrinsic engineering and mathematical interest. And by providing instructive “low-dimensional” models of self-exciting fluid dynamos in stars and planets they elucidate nonlinear magnetohydrodynamic (MHD) feedback and coupling processes involving the spatial redistribution of kinetic energy by the action of Lorentz ponderomotive forces. The time (t)-dependence of the persistent dynamo current I(t) depends on the control parameters of the system, especially A/B defined by the relationship between I and Lorentz torque, T, that drives the armature of the series motor into rotation relative to the ambient magnetic field, M (amb.), namely T = AI + BI 2 where A and B depend on the design characteristics of the motor [Hide, R., Nonlinear quenching of current fluctuations in a self-exciting homopolar dynamo. Nonlinear Process. Geophys. 1997a, 4, 201–205; Hide, R., The nonlinear equations governing a hierarchy of self-exciting coupled Faraday-disk homopolar dynamos. Phys. Earth Planet. Interiors 1997b, 103, 281–291]. A is non-zero when “external” sources of M (amb.) are present, such as permanent magnets and/or other self-exciting dynamo “sub-units”, whereas B is non-zero when the motor includes a stationary “field winding” through which dynamo current is diverted. Numerical analysis of the governing ODEs using digital and (electronic circuit) analogue computers and guided by Hopf-bifurcation instability theory indicate (i) that when A is non-zero there are regions of multi-dimensional “control-parameter” space where I(t) exhibits large-amplitude chaotic fluctuations reminiscent of polarity reversals and excursions of the main geomagnetic field, and (ii) that such regions shrink in volume as A/B decreases in magnitude and they disappear altogether in the special (but physically realistic) case when A/B = 0, with the sign (direction) of I determined by the initial conditions and its magnitude by the other control parameters. This process whereby nonlinear Lorentz forces attenuate persistent fluctuations or even quench them altogether offers a theoretical basis for interpreting the highly irregular geomagnetic polarity timeseries in the palaeomagnetic record [Hide, R., Generic nonlinear processes in self-exciting dynamos and the long-term behaviour of the geomagnetic field, including polarity superchrons. Phil. Trans. Roy. Soc. Lond. A 2000, 358, 943–955]. In the absence of permanent magnets, the governing ODEs exhibit the same “magnetic symmetry” property characteristic of the governing partial differential equations (PDEs) of unbiased MHD-dynamos, namely that for every solution (u, B) there is a corresponding solution (u, −B) (where u is the instantaneous Eulerian flow velocity and B the magnetic field at a general point P).