On spectral instability and kinematic dynamo action in three-dimensional incompressible flows
File(s)
Author(s)
Villringer, David
Type
Thesis or dissertation
Abstract
This thesis studies the kinematic dynamo problem, and related instabilities in magnetohydrody-namics. Its central question is whether a divergence-free velocity field can sustain exponential growth of a magnetic field, particularly as magnetic resistivity tends to zero. This phenomenon, known as the dynamo effect, is a classical explanation for magnetic fields in celestial bodies such as the Earth and the Sun, but comparatively few rigorous results are available. After introducing the kinematic dynamo equations, as well as multiple foundational results in their theory, we present several new rigorous constructions. We construct an explicit example of a fast dynamo on R^3, by considering spatially periodic velocity fields exhibiting the α-effect at fixed resistivity. Via rescaling and spatial localisation, these building blocks are glued together to produce fast dynamo action. Similarly, we also construct a subsequential fast dynamo on T^3, with growth along a subsequence of times.
We then study a pulsed-diffusion model on T^3, proving fast dynamo action under a time-periodic, Lipschitz-continuous velocity field. The argument is inherently perturbative in the magnetic resistivity, using anisotropic Banach spaces to employ techniques from the singular perturbation theory.
Finally, we study slow dynamo action in a class of helical velocity fields, known as the Ponomarenko dynamo. We establish the existence of growing modes with growth rate of order ε^(1/3) in the resistivity ε, for a broad class of velocity fields. The proof relies on resolvent analysis of the associated linear operators, yielding bounds on the associated spectral projectors. Using this, we then prove nonlinear instability of the full three-dimensional MHD equations around the Taylor–Couette flow and the trivial magnetic field.
Overall, this thesis provides new rigorous examples of slow and fast dynamo action, and develops tools for the study of instabilities in the singular limit of vanishing resistivity.
We then study a pulsed-diffusion model on T^3, proving fast dynamo action under a time-periodic, Lipschitz-continuous velocity field. The argument is inherently perturbative in the magnetic resistivity, using anisotropic Banach spaces to employ techniques from the singular perturbation theory.
Finally, we study slow dynamo action in a class of helical velocity fields, known as the Ponomarenko dynamo. We establish the existence of growing modes with growth rate of order ε^(1/3) in the resistivity ε, for a broad class of velocity fields. The proof relies on resolvent analysis of the associated linear operators, yielding bounds on the associated spectral projectors. Using this, we then prove nonlinear instability of the full three-dimensional MHD equations around the Taylor–Couette flow and the trivial magnetic field.
Overall, this thesis provides new rigorous examples of slow and fast dynamo action, and develops tools for the study of instabilities in the singular limit of vanishing resistivity.
Version
Open Access
Date Issued
2026-04-20
Date Awarded
2026-08-01
Copyright Statement
Attribution-NonCommercial 4.0 International Licence (CC BY-NC)
License URL
Advisor
Coti Zelati, Michele
Hairer, Martin
Sponsor
Imperial College London
Publisher Department
Department of Mathematics
Publisher Institution
Imperial College London
Qualification Level
Doctoral
Qualification Name
Doctor of Philosophy (PhD)