Stefan G. Llewellyn Smith

Stability analysis is performed on surface quasigeostrophic systems subjected to a Kolmogorov-type "shear force" on the boundaries using linear and nonlinear approaches. For a SQG system of semi-infinite depth forced on the upper boundary, the most linearly unstable mode is 2.74 the energy injection length scale. This is contrary to two-dimensional fluid systems, where the linear instability is greatest for long waves. In the presence of damping, the most linearly unstable mode shifts toward shorter length scales. The nonlinear critical Reynolds number across different damping strengths is found to be qualitatively similar to that of Euler 2D systems. For an SQG system of finite thickness being forced on both boundaries, its behaviour approaches that of a semi-infinite SQG system at the large thickness limit. In the small thickness limit, the behaviour of a symmetrically forced fluid layer approaches that of a 2D system, while an antisymmetrically forced fluid layer is not susceptible to both linear and nonlinear instabilities. With ageostrophic effects, an SQG+ system of semi-infinite depth is much more prone to instabilities than an otherwise identical SQG system in the absence of damping due to the instability of long-wave modes. However, damping significantly suppresses such instabilities. With increasing damping, the most linearly unstable mode moves toward a smaller length scale. Contrary to the zero damping case, when the damping is sufficiently large, ageostrophic effects have a small but measurable stabilising effect.

Quasi-geostrophic flow is an asymptotic theory for flows in rotating systems that are in geostrophic balance to leading order. It is characterized by the conservation of (quasi-geostrophic) potential vorticity and weak vertical flows. Surface quasigeostrophy (SQG) is the special case when the flow is driven by temperature anomalies at a horizontal boundary. The next-order correction to QG, QG+, takes into account ageostrophic effects. We investigate point vortx dynamics in SQG+, building on the work of Weiss. The conservation laws for SQG point vortices that parallel the 2D Euler case no longer exist when ageostrophic effects are included. The trajectories of point vortices are obtained explicitly for the general two-vortex case in SQG and SQG+. For the three-vortex case, exact solutions are found for rigidly rotating and stationary equilibria consisting of regular polygons and collinear configurations. As in the 2D case, only certain collinear vortex configurations are rigid equilibria. Trajectories of passive tracers advected by point vortex systems are studied numerically, in particular their vertical excursions, which are non-zero because of ageostrophic effects. Surface trajectories can manifest local divergence even though the underlying fluid equations are incompressible.

A new transform-based approach is presented that can be used to solve mixed boundary value problems for Laplace’s equation in non-convex and other planar domains, specifically the so-called Lipschitz domains. This work complements Crowdy (2015, CMFT, 15, 655–687), where new transform-based techniques were developed for boundary value problems for Laplace’s equation in circular domains. The key ingredient of the present method is the exploitation of the properties of the Szegö kernel and its connection with the Cauchy kernel to obtain transform pairs for analytic functions in such domains. Several examples are solved in detail and are numerically implemented to illustrate the application of the new transform pairs.

The free fall of an ellipse in an infinite linearly-stratified fluid is investigated using a linear two-dimensional, Boussinesq, diffusionless, inviscid model. The oscillations of the ellipse decay because of radiation damping, but unlike the case of a circular cylinder, the ellipse can also rotate and move horizontally. The resulting equations are solved analytically for some simple cases for which there is little or no rotation. Motions with rotation are studied numerically using a spectral method to solve for the wave field in the fluid.

A new technique is presented that can be used to solve mixed boundary value problems for Laplace’s equation and the complex Helmholtz equation in bounded convex planar domains. This work is an extension of Crowdy (2015, CMFT, 15, 655–687) where new transform-based techniques were developed for boundary value problems for Laplace’s equation in circular domains. The key ingredient of the method is the analysis of the so-called global relation, which provides a coupling of integral transforms of the given boundary data and of the unknown boundary values. Three problems which involve mixed boundary conditions are solved in detail, as well as numerically implemented, to illustrate how to apply the new approach.

A theoretical analysis is presented of peristaltic pumping down a narrow conduit with permeable walls, motivated by the flushing action of lugworms and other marine organisms in sandy burrows. Flow in the conduit is dealt with using lubrication theory; the leakage into the surrounding medium is taken into account by exploiting slender-body theory to solve the associated Darcy problem. By adopting a model for the local force balance on the pumping surface, we bridge between the limits in which the pump operates with either fixed load or displacement. In the latter limit we characterize peristaltic waves with either fixed form or ones that partially collapse the conduit. We construct pump characteristics (the relation between the mean flux and net pressure drop) when the burrow wall is impermeable and pressures are fixed at each end, and compare the results with existing laboratory experiments performed on lugworms. We then consider how the peristaltic dynamics is changed when the wall is made permeable. Last, we consider pumping along an impermeable burrow into a leaky head shaft. The results reveal that the permeability of the conduit wall or end can greatly impact the direction and strength of the recirculating flow.