On nonlinear transitions, minimal seeds and exact solutions for the geodynamo

TL;DR

This study develops and deploys an adjoint-based optimal control framework to identify minimal magnetic seeds that trigger subcritical dynamo action in a rotating spherical-shell geodynamo model. By coupling short-time transient optimisation with long-time Newton-hookstep convergence for nonlinear travelling waves, the authors map the dynamical landscape between hydrodynamic basins and magnetohydrodynamic attractors, revealing an edge state ($TW_2$) that governs transition pathways. They compare cost functionals, finding that optimising axisymmetric magnetic energy yields a robust minimal seed (OS1, OS2) that reliably triggers a dynamo, while seeds based on total magnetic energy can produce spurious local optima; OS2 represents a true minimal perturbation near the edge state. The results, anchored on Christensen’s benchmark with $(Ek, ilde{Ra},Pm,Pr,r_i/r_o)=(10^{-3},100,5,1,0.35)$, demonstrate how minimal seeds navigate toward an edge state and then to a dynamo attractor ($TW_1$), providing a systematic pathway to understanding geodynamo dynamics in parameter regimes that are computationally inaccessible. This framework sets the stage for exploring parameter variations (e.g., the distinguished limit of dormy2016) and assessing how subcritical seeds evolve with changing balances among Coriolis, pressure, and Lorentz forces in the geodynamo.

Abstract

Nearly fifty years ago, Roberts (1978) postulated that Earth's magnetic field, which is generated by turbulent motions of liquid metal in its outer core, likely results from a subcritical (finite-amplitude) dynamo instability characterised by a dominant balance between Coriolis, pressure and Lorentz forces. Here we numerically explore subcritical convective dynamo action in a spherical shell, using techniques from optimal control and dynamical systems theory to uncover the nonlinear dynamics of magnetic field generation. Through nonlinear optimisation, via direct-adjoint looping, we identify the minimal seed - the smallest magnetic field that attracts to a nonlinear dynamo solution. Additionally, using the Newton-hookstep algorithm, we converge stable and unstable travelling wave solutions to the governing equations. By combining these two techniques, complex nonlinear pathways between attracting states are revealed, providing insight into a potential subcritical origin of the geodynamo. This paper showcases these methods on the widely studied benchmark of Christensen et al. (2001), laying the foundations for future studies in more extreme and realistic parameter regimes. We show that the minimal seed reaches a nonlinear dynamo solution by first approaching an unstable travelling wave solution, which acts as an edge state separating a hydrodynamic solution from a magnetohydrodynamic one. Furthermore, by carefully examining the choice of cost functional, we establish a robust optimisation procedure that can systematically locate dynamo solutions on short time horizons with no prior knowledge of its structure.

Figures (8)

• Figure 1: Sketch of the numerical domain. The fluid domain $\mathcal{V}_s$ (in blue) is surrounded by two insulating, solid domains: the inner core $\mathcal{V}_i$ (in grey) and an outer mantle $\mathcal{V}_o$.
• Figure 2: Slices of the $m=4$ nonlinear dynamo state (TW1). The dashed line on the equatorial slice indicates where the meridional slices were taken.
• Figure 3: Magnetic energy (top) and kinetic energy (bottom) evolution with the rescaled benchmark with two different initial energies.
• Figure 4: Comparison of the magnetic energy (top) and kinetic energy (bottom) evolution with four different initial conditions with $M_0=344$. The initial conditions are the optimised seeds obtained by optimising the total magnetic energy with a random initial guess (solid blue line), optimising the total magnetic energy with an initial guess of the rescaled benchmark initial condition (RB) (dotted red line), and optimised seed obtained by optimising the energy in the $m=0$ part of the magnetic field starting from a random guess (dashed-dotted green line). We also show the timeseries obtained without optimisation, starting directly from the rescaled benchmark initial condition RB (dashed orange line). The plots on the left show the energy evolution in the optimisation window $t_\textrm{opt}=0.2$, and the plots on the right show the long-time evolution.
• Figure 5: Top: Slices of an example optimal seed (non-dynamo) obtained with the total magnetic energy cost functional, with $t_\textrm{opt}=0.2$ and $\textit{M}_0=344$. Middle: Slices of the optimal seed (dynamo) obtained with the axisymmetric magnetic energy cost functional with $t_\textrm{opt}=0.2$ and $\textit{M}_0=344$. Bottom: Slices of the optimal seed (dynamo) obtained with the axisymmetric magnetic energy cost functional with $t_\textrm{opt}=0.4$ and $\textit{M}_0=162$. Figures (a), (d) and (g) show meridional slices the radial component of the magnetic field, figures (b), (e) and (h) show meridional slices of the radial component of the current, and figures (c), (f) and (i) show equatorial slices of the latitudinal component of the magnetic field. The dashed lines on the equatorial slices indicates where the meridional slices were taken.