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We analyze the double Hopf bifurcations which occur in two geophysical fluid dynamics models: (1) a two-layer quasigeostrophic potential vorticity model with forcing and (2) a mathematical model of the differentially heated rotating annulus experiment. The bifurcations occur at the transition between axisymmetric steady solutions and non-axisymmetric travelling waves. For both models, the results indicate that, close to the transition, there are regions in parameter space where there are multiple stable waves. Hysteresis of these waves is predicted. For each model, center manifold reduction and normal form theory are used to deduce the local behaviour of the full system of partial differential equations from a low-dimensional system of ordinary differential equations. In each case, it is not possible to compute the relevant eigenvalues and eigenfunctions analytically. Therefore, the linear part of the equations is discretized and the eigenvalues and eigenfunctions are approximated from the resulting matrix eigenvalue problem. However, the projection onto the center manifold and reduction to normal form can be done analytically. Thus, a combination of analytical and numerical methods are used to obtain numerical approximations of the normal form coefficients, from which the dynamics are deduced. The first model differs from those previously studied with bifurcation analysis since it supports a steady solution which varies nonlinearly with latitude. The results indicate that the forcing does not qualitatively change the behaviour. However, the form of the bifurcating solution is affected. The second model uses the Navier-Stokes equations in the Boussinesq approximation, in cylindrical geometry. In addition to the double Hopf bifurcation analysis, a detailed axisymmetric to non-axisymmetric transition curve is produced from the computed eigenvalues. A quantitative comparison with experimental data finds that the computed transition curve, critical wave numbers and drift rates of the bifurcating waves are reasonably accurate. This indicates that the analysis, as well as the approximations which are made, are valid.

Computational Fluid Dynamics (CFD) are used to investigate two speech phenomena. The first phenomenon is the English bilabial plosive /pa/. Simulations are compared with microphone recordings and high speed video recordings to study the penetration rate and strength of the jet associated with the plosive /pa/. It is found that the dynamics in the first 10ms of the plosive are critical to penetration rate, and the static simulation was not able to capture this effect. However, the simulation is able to replicate the penetration rate after the initial 10ms. The second speech phenomenon is the English fricative /sh/. Here, the goal is to simulate the sound created during /sh/ to understand the flow mechanisms involved with the creation of this sound and to investigate the simulation design required to predict the sound adequately. A variety of simulation methods are tested, and the results are compared with previously published experimental results. It is found that all Reynolds-Averaged Navier-Stokes (RANS) simulations give bad results, and 2D Large Eddy Simulations (LES) also have poor results. The 3D LES simulations show the most promise, but still do not produce a closely matching spectra. It is found that the acoustic analogy matches the direct measurements fairly well in 3D simulations. The studies of /pa/ and /sh/ are compared and contrasted with each other. From the findings of the studies, and using theoretical considerations, arguments are made concerning which CFD methods are appropriate for speech research. The two studies are also considered for their direct applications to the field and future research directions which might be followed.

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