The dispersion relation is found for edge waves in a rotating stratified fluid over a constant sloping bottom. The dispersion relation is then extended to the case of arbitrary gentle bottom bathymetry. Superinertial trapped modes do not exist in the rigid-lid Boussinesq case. The effect of some of the approximations that have been made in this problem is discussed.

The large-time behaviour of a large class of solutions to the two-dimensional linear diffusion equation in situations with radial symmetry is governed by the function known as Ramanujan's integral. This is also true when the diffusion coefficient is complex, which corresponds to Schrödinger's equation. We examine the asymptotic expansion of Ramanujan's integral for large values of its argument over the whole complex plane by considering the analytic continuation of Ramanujan's integral to the left half-plane. The resulting expansions are compared to accurate numerical computations of the integral. The large-time behaviour derived from Ramanujan's integral of the solution to the diffusion equation outside a cylinder is not valid far from the domain boundary. A simple method based on matched asymptotic expansions is outlined to calculate the solution at large times and distances: the resulting form of the solution combines the inverse logarithmic decay in time typical of Ramanujan's integral with spatial dependence on the usual similarity variable for the diffusion equation.

The scattering of plane acoustic waves by a vortex in a two-dimensional superfluid is examined for small Mach number M. The solution is developed from a systematic expansion of the governing equations in three separate regions: an inner vortical region which scales as the healing length, an interaction region governed by irrotational hydrodynamics and an outer wave region. The solutions in the different regions are matched together. The leading-order scattered field occurs at O(M2δ) in the wave region, where δ is the small non-dimensional amplitude of the incoming acoustic wave. The far-field behaviour of the wave-region solution shows that a different form of the expansion is required in the forward scatter direction: this corresponds to the expression previously derived for the acoustic Magnus force.

The trajectory of a non-isolated monopole on the beta-plane is calculated as an asymptotic expansion in the ratio of the strength of the vortex to the beta-effect. The method of matched asymptotic expansions is used to solve the equations of motion in two regions of the flow: a near field where the beta-effect enters as a first-order forcing in relative vorticity, and a wave field in which the dominant balance is a linear one between the beta-effect and the rate of change of relative vorticity. The resulting trajectory is computed for Gaussian and Rankine vortices.

We investigate the scattering of a plane acoustic wave by an axisymmetric vortex in two dimensions. We consider vortices with localized vorticity, arbitrary circulation and small Mach number. The wavelength of the acoustic waves is assumed to be much longer than the scale of the vortex. This enables us to de ne two asymptotic regions: an inner, vortical region, and an outer, wave region. The solution is then developed in the two regions using matched asymptotic expansions, with the Mach number as the expansion parameter. The leading-order scattered wave eld consists of two components. One component arises from the interaction in the vortical region, and takes the form of a dipolar wave. The other component arises from the interaction in the wave region. For an incident wave with wavenumber k propagating in the positive X-direction, a steepest descents analysis shows that, in the far- eld limit, the leadingorder scattered eld takes the form i( − )eikX + 1 2 cos cot ( 1 2 )(2 =kR)1=2 ei(kR− =4), where is the usual polar angle. This expression is not valid in a parabolic region centred on the positive X-axis, where kR 2 = O(1). A di erent asymptotic solution is appropriate in this region. The two solutions match onto each other to give a leading-order scattering amplitude that is nite and single-valued everywhere, and that vanishes along the X-axis. The next term in the expansion in Mach number has a non-zero far- eld response along the X-axis.

The surface quasi-geostrophic (SQG) equations are a model for low-Rossby number geophysical flows in which the overall dynamics are governed by buoyancy evolution on the boundary. The model can be used to explore the transition from two-dimensional to three-dimensional mesoscale geophysical flows. We examine SQG vortices and the resulting flow to first order in Rossby number, $O(Ro)$. This requires solving an extension to the usual QG equation to compute the velocity corrections, and we demonstrate this mathematical procedure. As we show, it is simple to obtain the vertical velocity, but difficult to find the $O(Ro)$ horizontal corrections. Chaotic transport due to three SQG point vortices is studied with Poincar {e} sections and the Finite Time Braiding Exponent (FTBE). This chaotic transport is representative of the mixing in the flow, and these terms are used interchangeably in this work. Changes in transport from $O(Ro)$ vertical velocity terms are also examined, though without $O(Ro)$ horizontal velocities this is not a true solution to the governing equations. We then consider the SQG elliptic vortex solution developed by Dritschel (2011), in which all $O(Ro)$ velocities can be calculated. Results show that SQG point vortices exhibit greater mixing at the surface than classical point vortices. There appears to be a minimum FTBE near flow regime boundaries, and generally mixing is greater when the energy of the system is greater. There is also a critical depth below which the FTBE decreases sharply. Finally, including vertical velocity in the point vortex solution increases the observed mixing.

The motion of buoyant vortices is studied using reduced-order models. For a vortex filament, i.e. a curve in three-dimensional space along which vorticity is concentrated as a delta function, the evolution of the filament is calculated using a desingularized Biot–Savart integral. Buoyancy is added using a momentum balance argument while surface tension is also included. A set of equations that couples the transverse motion of the buoyant vortex filament and the axial flow within its core is derived. The new model is verified in an asymptotic limit by comparing it to the previous analytical solution for a thin vortex ring. In another approximation, axisymmetric contour dynamics is implemented to model a buoyant vortex ring of deforming core. A vortex patch is enclosed by a vortex sheet which emerges from baroclinic generation of vorticity. The evolution of the vortex sheet strength is derived from the Euler equation, and represents the effects of density, buoyancy and surface tension on the axisymmetric vortex ring. Numerical calculations for the integro-differential equations are carried out until the curvature singularity of vortex sheet evolution leads to a blowup. Finally, a linear stability analysis is performed for vortices in the presences of density and surface tension. The basic state solution is given as a perturbation series in a small parameter representing curvature of the vortex or a weak strain acting on it. With the small parameter in the basic state, instabilities are triggered by the resonances between two neutrally stable Kelvin waves for the unperturbed vortex. Those parametric instabilities are the curvature instability and the Moore–Saffman–Tsai–Widnall instability, corresponding to the parameter being curvature or a strain. The effects of density and surface tension on these instabilities are studied by calculating the growth rate and the instability bandwidth in various wavenumber regimes.

It is common knowledge that runners' ponytails will sway from side to side when running. This thesis treats the swaying phenomenon as a stability problem and discusses solution methods and results.

Geometrically nonlinear dynamical equations are derived for a flexible rod pointing vertically down and clamped at its base that is harmonically excited. Floquet theory is used to derive numerical methods to solve the problem. Results show the ponytail is always stable when it is unforced. When it is forced, it has a complex region of instability if it is treated as a flexible string, but that region becomes more limited if it is treated as a flexible rod. Adding different different types of damping can help dissipate the energy of the ponytail motion and further reduce the instability region, so that for small enough forcing the motion is stable.