## Linear and nonlinear dynamics in stratified shear flows

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Stably stratified shear flows, in which a less dense layer of fluid lies above and moves counter to a more dense layer below, are ubiquitous in geophysical fluid dynamics. These are often found to be unstable if the non-dimensional Richardson number Ri, quantifying the strength of stratification to shear, is sufficiently low. This is of particular importance in oceanography, where shear instabilities are conjectured to be important in the generation of turbulence in the deep ocean, an area of huge uncertainty in contemporary climate models. The Miles-Howard theorem tells us that for a steady, inviscid, parallel shear flow, if the local Richardson number is everywhere greater than one quarter, the flow is stable to infinitesimal perturbations. Though an important result, the strong restrictions in the applicability of this theorem mean care must be used when applying the criterion of Ri > 1/4 for stability. This thesis explores some of these limitations, beginning with an overview in chapter 1. Chapter 2 explores the infinitesimal restriction of the Miles-Howard theorem, by asking whether finite-amplitude perturbations could lead to significant nonlinear behaviour, in a so-called subcritical instability. It is found that while the classical Kelvin-Helmholtz instability does indeed exhibit subcriticality, nonlinear steady states are found only just above Ri = 1/4. Chapter 3 investigates in detail a hitherto unknown linear instability, which was discovered in chapter 2. Behaving similarly to the classic Holmboe instability, it exists for Ri > 1/4 when viscosity is introduced, and reveals new insights into the possible physical interpretations of stratified shear instability. Chapter 4 revisits the results of chapter 2 but considers two cases of the Prandtl number Pr, the ratio of diffusivity of the momentum to density. When Pr = 0.7, as is approximately the case for air, a simple supercritical instability is found. However, for Pr = 7, corresponding approximately to water, strong subcritical behaviour is observed, and it is demonstrated that finite-amplitude perturbations can trigger Kelvin-Helmholtz-like behaviour well above Ri = 1/4. Chapter 5 considers the time-varying, non-parallel flow of an oblique internal gravity wave incident on a shear layer. Using direct-adjoint looping, it is shown that the disturbances which maximise energy after a certain time, so-called linear optimal perturbations, can be convective-like rolls in the spanwise direction, rather than a shear instability, calling into question the relevance of the classical shear instabilities in oceanography. Chapter 6 concludes the thesis with a discussion of the implications of the results.

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Kerswell, R