Helical waves in Earth's outer core

This thesis addresses the generation and dispersion of wave packets in Earth's outer core. The waves of interest feed off the buoyancy field, and are supported by strong rotation and/or a large-scale magnetic field, two key features of the dynamics in Earth's core. We aim to better understand the role of wave packets in shaping the turbulent convection, the process of magnetic induction, and maintaining Earth's magnetic field. Our numerical experiments focus on the emission of wave packets from localised sources, on the small length-scales of the convection, which stirs the fluid iron. The fluid flow in Earth's outer core is characterised by a small Rossby number, the ratio of nonlinear inertia and the Coriolis acceleration. However, due to computational constraints, many geodynamo simulations lie in a regime where the convective structures have a moderate Rossby number. These simulations, that over-estimate the influence of nonlinear inertia, undergo an abrupt transition from a columnar flow structure with a strongly dipolar magnetic field, to a state of disorganised flow accompanied by a multipolar field. This transition is commonly termed the \emph{dipolar-multipolar} transition, and the collapse of the dipole occurs when the \emph{`local'} Rossby number \citep[as introduced by][]{christensen2006} is greater than $0.1$. Separately, in the rotating turbulence literature, there is preliminary evidence of a similarly sharp transition from a columnar flow structure to incoherent turbulence when the Rossby number is $ Ro \sim 0.2-0.6$. We show, that when $Ro > Ro^{\text{crit}} \approx 0.4$, inertial wave packets are suppressed, and columnar flow structures break down. Furthermore, we highlight a relationship between the `local' Rossby number used to describe the dipolar-multipolar transition and our $Ro$, which places both transitions at approximately $Ro^{\text{crit}}$. Based on this evidence, we conjecture that the breakdown of columnar structures, followed by the dipole collapse, is caused by the suppression of inertial wave packets at the critical threshold. In the following series of simulations, we study the effects of an ambient magnetic field on the dispersion of inertial wave packets. In the presence of an large-scale field, inertial waves are modified into a spectrum of waves called \emph{magnetic-Coriolis} waves, which present in a variety of forms depending on the wave-frequency. We focus on the Earth-like regime of rapid rotation and a small Lehnert number, $Le$, the ratio of the Alfv\'en and inertial frequencies. Our simulations initiated with a single buoyant blob yield an excellent comparison to the diffusion-less analytical results of \cite{bardsley2016} at $Ro \rightarrow 0$ and $Le = 0.1$. We identify three wave-types, predicted by linear theory, based on the waves' group velocity, helicity characteristics, and magnetic to kinetic energy ratio. At Earth-like values of $Le$, we observe that magnetic-Coriolis wave packets distribute helicity in a way that is beneficial to planetary magnetic field generation. Furthermore, the emf induced by the wave packets is coherent for Earth-like values of $Le$, suggesting than an $\alpha$-effect associated with the waves has the potential to drive a \emph{helical wave dynamo} \citep{davidson2014}