Couette flows are an invaluable tool in understanding some of the most important processes in astrophysical and geophysical fluid dynamics. When driven by rotation or electric currents, instability in such flows can be a key ingredient in a number of fundamental processes. For example, Taylor-Couette flow can give rise to the magnetorotational instabilities widely thought to be responsible for angular momentum transport in astrophysical disks.
In this thesis, we add to the understanding of instabilities in Couette flows by focussing on two specific applications; inductionless magnetorotational instability in Taylor-Couette flow, and narrow-gap spherical Couette flows.
After providing an overview of the numerical methods utilised throughout, we perform a linear stability analysis on inductionless magnetorotational instability, allowing for a fully generalised set-up which allows for every possible combination of imposed currents. A full exploration of the relevant parameter space is given.
We then introduce the generalised quasilinear approximation, which serves to restrict nonlinear interaction between modes. In doing so, we are able to ascertain which modal interactions are essential to the key flow dynamics. More importantly, this is formally equivalent to the cumulant expansions utilised in the growing field of direct statistical simulation. Therefore, we are able to posit the future usefulness of statistical simulations to general wall-bounded flows.
Finally, we utilise direct numerical simulation to probe the existence of axisymmetric pulse train solutions in narrow-gap spherical Couette flow, in which computations have been, until now, unable to utilise sufficiently narrow gap widths. As such, the only detailed prior solutions consist of asymptotic studies. We numerically confirm the existence of these pulse trains, and chart their initial bifurcations from the steady state solution. We also examine their magnetohydrodynamic equivalents, which have so far not been considered.
