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.

The daily cycle of heating and cooling from the rising and setting of the sun creates the potential for periodically-forced convection in the world's atmosphere and bodies of water. In this dissertation, we model a body of water experiencing daily heating from the sun and examine the onset of convection using linear and nonlinear stability analysis. We also find the first exact bound on a flow quantity in time-modulated convection. The setup considered consists of fluid between two surfaces that are infinite in the horizontal directions, with one of the surfaces heated in a time-varying manner. The most important parameters are the Rayleigh number, $Ra$, which captures the strength of the heating, and the nondimensional frequency of the time modulation, $\omega$.

Radiative heating from the sun is modeled as a source term with exponential decay in the vertical. When confined to a thin layer near the surface, the effects of radiative heating may be approximated using an imposed temperature or heat flux boundary condition, and this approach forms the main focus of the dissertation. Each method of heating has time dependence to capture the sun's daily cycle. For stability analysis, a sinusoidal time dependence is used, and for bounding, results are found for a generic profile.

The stability results show that while the critical Rayleigh numbers for both linear and nonlinear stability increase with increasing modulation frequency, the linear and nonlinear stability thresholds show very different behavior at high modulation frequencies. As expected, the nonlinear stability threshold is always below the linear stability threshold. For the linear stability threshold, the ratio $Ra \omega^{-3/2}$ or $Ra \omega^{-2}$ approaches a constant, with the power of the frequency depending on the type of heating. This is found to match the result from considering a semi-infinite domain. For the nonlinear stability threshold, the growth of the Rayleigh number with the frequency is approximately linear, and no semi-infinite domain result is found. Beyond stability, we find an exact bound on the temperature dissipation that grows like $\sqrt{Ra}$ and $\sqrt{\omega}$ in the large $Ra$ and $\omega$ limits, respectively.

The point vortex has been used as a simple model for flows with circulation, and has been desingularized into vortex patches and vortex sheets. In this work, we investigate the steady states of a combination of these two formulations, the Sadovskii vortex. The Sadovskii vortex is a uniform patch of vorticity surrounded by a vortex sheet. Numerical continuation is used to follow families of solutions. In the limiting cases of the vortex patch and vortex sheet cases, we confirm previous research, and in the vortex patch case show new solutions.

In the regime where both sources of circulation exist, we show the relationship between the vortex patch and vortex sheet solution families. The more complicated vortex patch solution families lead to the simpler vortex sheet solution family due to a splitting of the vortex patch families at bifurcation points in the presence of the vortex sheet. The more circular elliptical family remains attached to the family with a single pinch off, and this family extends all the way to the pure vortex sheet solutions. More elongated families below this one also split at bifurcation points, and these families do not exist in the vortex sheet regime.

In the presence of surface tension, vortex patch shapes are deformed where the background straining flow is into the vortices. This leads to oscillations on the boundary, similar to those found by Tanveer (1986) for bubbles in Hele-Shaw cells. The pinch off cusp for the patch vortex becomes desingularized in the presence of surface tension, and no longer touches. The effects of surface tension on the vortex sheet and Sadovskii vortex shapes are shown to be similar but smaller.

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.