The Dynamics of Geophysical and Astrophysical Turbulence

Abstract

Turbulence is ubiquitous within geophysical and astrophysical fluid flows. Its interaction with physical ingredients such as rotation and stratification gives rise to spectacular dynamics, including the layering of material properties, which in turn influence the transport and distribution of heat, momentum and tracers. We consider the effects of rotation and stratification individually in the study of two different problems of scientific interest, using idealised models which retain only the essential ingredients.

Our first problem investigates the role of rotation in geophysical flows. We consider a barotropic, stochastically-forced turbulent flow on a beta-plane, which is well known to exhibit the spontaneous formation and equilibration of persistent zonal jets. The equilibrated jets are not steady and the focus here is on their time variability, which is of interest both because of its relevance to the behaviour of naturally occurring jet streams and for the insights it provides into the dynamical mechanisms operating in these systems. We compare the behaviour of a nonlinear (NL) system to a quasilinear (QL) model in which eddy-eddy interactions are neglected. Both systems reveal a rich zoology of dynamics, nevertheless, key differences exist. The NL model admits the formation of systematically migrating jets, a phenomenon that has not been previously identified. Jets migrate north or south with a speed of translation that is a function of the Rhines scale and the frictional damping rate, occasionally changing their direction of migration. The QL model does not exhibit jet migration, but a generalised quasilinear (GQL) model, in which certain eddy-eddy interactions are systematically restored, does, demonstrating that long waves, generated by such interactions, play a key dynamical role. The importance of these waves, in addition to the role of random fluctuations, is affirmed using a statistical formulation in which the flow statistics are solved for directly.

Our second problem considers the interaction of a stable density stratification with a background velocity distribution, which can develop into stratified turbulence. Geophysical flows, in which the diffusivities of momentum and heat are commensurate, are often very strongly stratified, nevertheless, turbulence still occurs. Density layering is key to understanding the properties of this `layered anisotropic stratified turbulence' (LAST) regime. On the other hand, astrophysical flows are typically characterised by strong thermal diffusion, inhibiting the formation of density layers. This suggests that LAST dynamics cannot occur, raising the interesting question of whether analogous or fundamentally different regimes exist in the limit of strong thermal diffusion. This thesis addresses this question for the case of a vertically stratified, horizontally-forced Kolmogorov flow. Using linear stability theory, we show that three-dimensional perturbations of the horizontal shear are always unstable in the limit of strong stratification and strong thermal diffusion, causing the flow to develop vertical layers, and hence vertical shear, in the velocity field, thereby allowing vertical shear instabilities to develop. The subsequent nonlinear evolution and transition to turbulence is studied numerically using direct numerical simulations, where four distinct dynamical regimes emerge, depending upon the strength of the background stratification. By considering dominant balances in the governing equations, we derive scaling laws which explain the empirical observations.