Optimal kinematic dynamos in a sphere

Abstract

A variational optimization approach is used to optimize kinematic dynamos in a unit sphere and locate the enstrophy-based critical magnetic Reynolds number for dynamo action. The magnetic boundary condition is chosen to be either pseudo-vacuum or perfectly conducting. Spectra of the optimal flows corresponding to these two magnetic boundary conditions are identical since theory shows that they are relatable by reversing the flow field (Favier & Proctor 2013 Phys. Rev. E 88 , 031001 ( doi:10.1103/physreve.88.031001 )). A no-slip boundary for the flow field gives a critical magnetic Reynolds number of 62.06, while a free-slip boundary reduces this number to 57.07. Optimal solutions are found to possess certain rotation symmetries (or anti-symmetries) and optimal flows share certain common features. The flows localize in a small region near the sphere’s centre and spiral upwards with very large velocity and vorticity, so that they are locally nearly Beltrami. We also derive a new lower bound on the magnetic Reynolds number for dynamo action, which, for the case of enstrophy normalization, is five times larger than the previous best bound.

Kinematic dynamos in triaxial ellipsoids

“…2 in [48]). This is also significantly different from the situation explored in [51], where the optimized flows were almost identical for the two BCs and yielded nearly the same smallest critical value for Rm. Our results suggest instead that, for a given flow, the dynamo onset can be strongly affected by using PV BCs or PC BCs.…”

“…Note that the matrices in (3.5) are also found to be severely ill-conditioned for truncation degrees N≳20 in our implementation of the Galerkin method. This is likely owing to the lack of symmetries in the construction of the basis elements, which is unfortunately necessary to account for ellipsoidal geometries in Cartesian coordinates (contrary to previous numerical implementations in spheres [36,51], which can fully exploit separation of variables and the orthogonality of the spherical harmonics). Our numerical implementation in double-precision arithmetic is thus currently limited to the large-scale diffusive modes (figure 2 b ).…”

“…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.…”

“…Here, it is advantageous to construct the basis elements for the magnetic field using spectral decompositions that admit explicit polynomial expressions (e.g. as in spheres [35,36,51]). We first introduce suitable spectral decompositions in triaxial ellipsoids, and then describe the projection method.…”