Taylor state dynamos found by optimal control: axisymmetric examples

摘要: Earth’s magnetic field is generated in its fluid metallic core through motional induction a process termed the geodynamo. Fluid flow heavily influenced by combination of rapid rotation (Coriolis forces), Lorentz forces (from interaction electrical currents and fields) buoyancy; it believed that inertial force viscous are negligible. Direct approaches to this regime far beyond reach modern high-performance computing power, hence an alternative ‘reduced’ approach may be beneficial. Taylor (Proc. R. Soc. Lond. A, vol. 274 (1357), 1963, pp. 274–283) studied inertia-free viscosity-free model as asymptotic limit such rapidly rotating system. In theoretical limit, velocity organize themselves special manner, torque acting on every geostrophic cylinder zero, property referred Taylor’s constraint. Moreover, instantaneously uniquely determined buoyancy field. order find solutions mathematical system equations full sphere, we use methods optimal control ensure required conditions cylinders satisfied at all times, conventional time-stepping procedure implements constraints end each time step. A derivative-based used discover correct so always satisfied. We report new quantity, Taylicity, measures adherence constraint analysing squared torques, normalized energy field, over entire core. Neglecting buoyancy, solve sphere seek axisymmetric equations; invoke - -effects sidestep Cowling’s anti-dynamo theorem dynamo possesses non-trivial solutions. Our methodology draws fully spectral expansions for divergenceless vector fields. employ five Galerkin polynomial bases radius boundary honoured member basis set, whilst satisfying orthogonality relation defined terms energies. demonstrate via numerous examples there stable possess decreasing spectrum thus well-converged. Classic distributions invoked, well distributions. One -effect oscillatory rarely before seen. By comparing our state with one allows torsional oscillations develop decay, show equilibrium both configurations coincident. models, dominates ageostrophic flow. work corroborates some results previously reported Wu & Roberts (Geophys. Astrophys. Dyn., 109 (1), 2015, 84–110), presenting results; sets stage three-dimensional implementation where driven by, example, thermal convection.