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Abstract
A flow in which two fluids slide by one another or two layers of a single fluid flow at different speeds is called a shear flow. These flows are of interest in a broad range of physical systems, including atmospheric and astrophysical ones. Their potential to be unstable, with small perturbations growing by feeding off the flow's kinetic energy, motivates their study. Particularly in astrophysical systems, these instabilities often drive turbulence that dramatically modifies the rate at which heat, particles, and momentum are transported across the flow. This thesis studies the nature of these instabilities and the ensuing turbulence, investigating the canonical shear-flow instability, the Kelvin-Helmholtz (KH) instability, in three systems. Specifically, large-scale stable modes are explored, which transfer energy from perturbations back to the shear flow and tend to decay in time, in contrast to unstable modes which take energy away from the flow and tend to grow.
In systems where stable modes were investigated prior to this work, the same nonlinear interactions that transfer energy from large to small scales were shown to transfer energy from unstable to stable modes. Thus, they can potentially reach significant amplitudes despite their tendency to decay in the absence of nonlinear interactions. At large amplitudes, the energy sink they present can significantly modify the turbulent state, and accounting for them can significantly improve reduced models of turbulent transport. These prior studies all concerned plasma micro-instabilities or systems that were otherwise quasi-homogeneous. The KH instability is importantly macroscopic and inhomogeneous. Thus, this thesis presents a significant expansion to the set of systems in which stable modes have been investigated.
Three studies are described in this thesis. Each study concerns a different flow configuration and a different set of physical effects: a piecewise-linear, fixed shear layer in hydrodynamics, a reinforced, sinusoidal shear flow in gyrokinetics, and a smooth, freely-evolving shear layer in MHD. It is shown that stable modes are nonlinearly coupled to unstable modes in each system. When stable modes are not suppressed by added physical effects, the transfer of energy from unstable to stable modes plays an important role in saturation, and stable modes are expected or directly shown to be excited to significant amplitudes in the ensuing turbulence. The excitation of stable modes is linked to the transport of momentum against its gradient, an effect previously observed in experiments. It is shown that accounting for stable modes can significantly improve reduced models of momentum transport in regimes where they reach significant amplitude.
The major contributions of this work are twofold. First, nonlinear coupling to large-scale stable modes is shown to be a generic feature of KH-unstable shear flows. Stable modes are shown to provide a valuable interpretive framework for explaining how features of the turbulence change with system parameters (e.g.~understanding why adding a magnetic field in the direction of the shear flow enhances small-scale fluctuations), and they inform improvements to reduced models. Second, in carrying out this work, tools developed for studying stable modes in quasi-homogeneous systems have been extended to apply to inhomogeneous systems, with stable modes shown to be relevant in inhomogeneous systems for the first time. Thus, this work motivates investigations into stable modes more broadly in inhomogeneous systems and provides the necessary tools for these investigations.
Further details of each study are given in the abstracts of Chapters 2, 3, and 4.
