On nonlinear transitions, minimal seeds and exact solutions for 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.

22 pages, 8 figures

References in corpus (28)

• Dedalus: A Flexible Framework for Numerical Simulations with Spectral Methods
• Recurrent motions within plane Couette turbulence
• Turbulent geodynamo simulations: a leap towards Earth's core
• Turbulence transition and the edge of chaos in pipe flow
• Lower branch coherent states in shear flows: transition and control
• Spherical convective dynamos in the rapidly rotating asymptotic regime
• Minimal seeds for shear flow turbulence: using nonlinear transient growth to touch the edge of chaos
• An optimisation approach for analysing nonlinear stability with transition to turbulence in fluids as an exemplar
• Strong-Field Spherical Dynamos
• A self-sustaining nonlinear dynamo process in Keplerian shear flows
• Relative periodic orbits form the backbone of turbulent pipe flow
• Variational method for finding periodic orbits in a general flow
• The Openpipeflow Navier--Stokes Solver
• Force balance in numerical geodynamo simulations: a systematic study
• Relative periodic orbits in transitional pipe flow
• The critical layer in pipe flow at high Reynolds number
• Multiscale Mixing Efficiencies for Steady Sources
• Revealing the state space of turbulence using machine learning
• Searching turbulence for periodic orbits with dynamic mode decomposition
• Edge states as mediators of bypass transition in boundary-layer flows
• Variational methods for finding periodic orbits in the incompressible Navier-Stokes equations
• Optimization of the magnetic dynamo
• Systematic parameter study of dynamo bifurcations in geodynamo simulations
• Exact coherent structures in two-dimensional turbulence identified with convolutional autoencoders
• Optimal perturbations and transition energy thresholds in boundary layer shear flows
• Optimal mixing in two-dimensional stratified plane Poiseuille flow at finite Péclet and Richardson numbers
• Robust and efficient identification of optimal mixing perturbations using proxy multiscale measures
• Step-based checkpointing with high-level algorithmic differentiation