A trio of simple optimized axisymmetric kinematic dynamos in a sphere

Abstract

Planetary magnetic fields are generated by the motion of conductive fluid in the planet's interior. Complex flows are not required for dynamo action; simple flows have been shown to act as efficient kinematic dynamos, whose physical characteristics are more straightforward to study. Recently, Chen et al . (2018, J. Fluid Mech. 839 , 1–32. ( doi:10.1017/jfm.2017.924 )) found the optimal, unconstrained kinematic dynamo in a sphere, which, despite being of theoretical importance, is of limited practical use. We extend their work by restricting the optimization to three simple two-mode axisymmetric flows based on the kinematic dynamos of Dudley & James (1989, Proc. R. Soc. Lond. A 425 , 407–429. ( doi:10.1098/rspa.1989.0112 )). Using a Lagrangian optimization, we find the smallest critical magnetic Reynolds number for each flow type, measured using an enstrophy-based norm. A Galerkin method is used, in which the spectral coefficients of the fluid flow and magnetic field are updated in order to maximize the final magnetic energy. We consider the t 0 1 s 0 1 , t 0 1 s 0 2 and t 0 2 s 0 2 flows and find enstrophy-based critical magnetic Reynolds numbers of 107.7, 142.4 and 125.5 (13.7, 19.6 and 16.4, respectively, with the energy-based definition). These are up to four times smaller than the original flows. These simple and efficient flows may be used as benchmarks in future studies.

Optimal kinematic dynamos in a sphere

“…A general question emerges naturally: what is the spatial structure of the optimal flow among all permissible solutions and what ingredients constitute its optimality? This has been formulated as a variational optimization problem in a periodic box [14], in a cube [15], in a unit sphere with general flows [16] and in a unit sphere with axisymmetric flows [17]. Much efficiency has been gained by the optimal solutions compared with previous working dynamos in the sense of the magnetic Reynolds number R m , which is a non-dimensional parameter measuring the ratio between magnetic induction and diffusion.…”

Kinematic dynamos in triaxial ellipsoids

“…Next, instead of imposing the velocity field, it would be worth searching for the optimal spatial structure of the flow (among all permissible polynomial fields) yielding the most efficient dynamo magnetic field. We could indeed extend previous dynamo variational algorithms in spheres [51,56,57] to the ellipsoid, to determine the minimum magnetic Reynolds number for dynamo action in ellipsoidal geometries. Simulating saturated dynamo fields in ellipsoids is finally a long-term endeavour, but it requires an efficient algorithm to be found for the nonlinear terms.…”

“…Moreover, they reduce to the modified Dudley-James flows in full spheres [32,35,36]. Solving the kinematic dynamo problem usually amounts to finding the critical value of the magnetic Reynolds number yielding σ = 0, which is often estimated using root-finding algorithms [35,36] or optimization methods [51,56,57]. Here, we directly compute the dynamo action of the flows in a large region of the parameter space to simplify the comparison between the different numerical methods.…”

Acoustic and inertial modes in planetary-like rotating ellipsoids

“…the dynamic mode decomposition), but physically based approaches can also be employed. For instance, insightful asymptotic models have been developed in spheres to study rapidly rotating convection [3] or dynamo magnetic fields [4,5]. More generally, the normal modes could be used to incorporate the key characteristics of the system [[6,7], for planetary flows].…”