In this thesis a number of issues related to oceanographic problems have been dealt with on the basis of applying asymptotic methods.  The first study focused on the tidal generation of internal waves, a process which is quantifed by the conversion rates. These have traditionally been calculated by using the WKB approximation. However, the systematic imprecision of this theory for the lowest modes as well as turbulence at the seabed level affect the results. To handle these anomalies we introduced another asymptotic technique, homogenization theory, which led to signifcant improvements, especially for the lowest modes.  The second study dealt with the dynamical aspects of a nonlinear oscillator which can be interpreted as a variant of the classical two-box models used in oceanography. The system is constituted by two connected vessels containing a fluid characterised by a nonlinear equation of state and a large volume differences between the vessels is prescribed. It is recognised that the system, when performing relaxation oscillations, exhibits almost-discontinuous jumps between the two branches of the slow manifold of the problem. The lowest-order analysis yielded reasonable correspondence with the numerical results.  The third study is an extension of the lowest-order approximation of the relaxation oscillations undertaken in the previous paper. A Mandelstam condition is imposed on the system by assuming that the total heat content of the system is conserved during the discontinuous jumps.  In the fourth study an asymptotic analysis is carried out to examine the oscillatory behaviour of the thermal oscillator. It is found that the analytically determined corrections to the zeroth-order analysis yield overall satisfying results even for comparatively large values of the vessel-volume ratio. / At the time of the doctoral defense, the following papers were unpublished and had a status as follows: Paper 3: Manuscript. Paper 4: Manuscript.

This thesis investigates the effects of the earth's rotation on internal waves from two perspectives of nonlinear internal wave theory: near-resonant triads and weakly nonlinear models. We apply perturbation theory (multiple scale analysis) to the governing equations of internal waves and develop a near-resonant internal wave triad theory. This theory explains a resonant-like phenomenon in the numerical results obtained from simulating internal waves generated by tide topography interaction. Furthermore, we find that the inclusion of the earth's rotation (nonzero $f$) in the numerical runs leads to a very special type of resonance: parametric subharmonic instability. Through using perturbation expansion to solve separable solutions to the governing equations of internal waves, we derive a new rotation modified KdV equation (RMKdV). Of particular interest, the dispersion relation of the new equation obeys the exact dispersion relation for internal waves for both small and moderate wavenumbers ($k$). Thus this new RMKdV is able to model wea kly nonlinear internal waves with various wavenumbers ($k$), better than the Ostrovsky equation which fails at describing waves of small $k$.

When wind blows over a stratified lake an interface tilt is often generated, and internal waves usually appear after the wind stops. Internal waves in lakes are studied in many literatures, but most assume a hydrostatic pressure balance. In this thesis we discuss the internal Poincare waves and Kelvin waves in a rotating, continuously stratified, flat-bottom, circular lake with fully nonlinear and non-hydrostatic effects. An analytical solution is derived for the linearized system and it provides initial conditions used in the MIT General Circulation Model (MITgcm). This model is chosen due to its non-hydrostatic capability. Both Poincare waves and Kelvin waves are considered. The analytical solution of the linear system is verified numerically when the wave amplitude is small. As the wave amplitude increases the waves become more nonlinear. Poincare waves steepen and generate solitary-like waves with shorter wavelengths, but most of the energy contained in these waves is transferred back and forth between the parent wave and the solitary-like waves. Kelvin waves, on the other hand, steepen and lose their energy to solitary-like waves. The appearance of the solitary-like waves is not absolutely clear and higher resolution is required to clear up the details of this process. This conclusion agrees with de la Fuente et al (2008) who discussed the internal waves in a two-layer model. Moreover, in the Kelvin waves case, unexpected small waves are generated at the side boundaries and travel inwards. The wave amplitude and wavelength of these spurious waves become smaller as the horizontal resolution increases. One possible reason to explain these waves is the use of square grids to approximate the circular lake.

This thesis investigates the effects of the earth's rotation on internal waves from two perspectives of nonlinear internal wave theory: near-resonant triads and weakly nonlinear models. We apply perturbation theory (multiple scale analysis) to the governing equations of internal waves and develop a near-resonant internal wave triad theory. This theory explains a resonant-like phenomenon in the numerical results obtained from simulating internal waves generated by tide topography interaction. Furthermore, we find that the inclusion of the earth's rotation (nonzero $f$) in the numerical runs leads to a very special type of resonance: parametric subharmonic instability. Through using perturbation expansion to solve separable solutions to the governing equations of internal waves, we derive a new rotation modified KdV equation (RMKdV). Of particular interest, the dispersion relation of the new equation obeys the exact dispersion relation for internal waves for both small and moderate wavenumbers ($k$). Thus this new RMKdV is able to model wea kly nonlinear internal waves with various wavenumbers ($k$), better than the Ostrovsky equation which fails at describing waves of small $k$.

When wind blows over a stratified lake an interface tilt is often generated, and internal waves usually appear after the wind stops. Internal waves in lakes are studied in many literatures, but most assume a hydrostatic pressure balance. In this thesis we discuss the internal Poincare waves and Kelvin waves in a rotating, continuously stratified, flat-bottom, circular lake with fully nonlinear and non-hydrostatic effects. An analytical solution is derived for the linearized system and it provides initial conditions used in the MIT General Circulation Model (MITgcm). This model is chosen due to its non-hydrostatic capability. Both Poincare waves and Kelvin waves are considered. The analytical solution of the linear system is verified numerically when the wave amplitude is small. As the wave amplitude increases the waves become more nonlinear. Poincare waves steepen and generate solitary-like waves with shorter wavelengths, but most of the energy contained in these waves is transferred back and forth between the parent wave and the solitary-like waves. Kelvin waves, on the other hand, steepen and lose their energy to solitary-like waves. The appearance of the solitary-like waves is not absolutely clear and higher resolution is required to clear up the details of this process. This conclusion agrees with de la Fuente et al (2008) who discussed the internal waves in a two-layer model. Moreover, in the Kelvin waves case, unexpected small waves are generated at the side boundaries and travel inwards. The wave amplitude and wavelength of these spurious waves become smaller as the horizontal resolution increases. One possible reason to explain these waves is the use of square grids to approximate the circular lake.

This work develops a nonlinear, three-dimensional spectral collocation method for the simulation of the incompressible Navier-Stokes equations for geophysical and environmental flows. These flows are often driven by the interaction of stratified fluid with topography, which is accurately accounted for in this model using a mapped coordinate system. The spectral collocation method used here evaluates derivatives with a Fourier trigonometric or Chebyshev polynomial expansion as appropriate, and it evaluates the nonlinear terms directly on a collocated grid. The coordinate mapping renders ineffective fast solution methods that rely on separation of variables, so to avoid prohibitively expensive matrix solves this work develops a low-order finite-difference preconditioner for the implicit solution steps. This finite-difference preconditioner is itself too expensive to apply directly, so it is solved pproximately with a geometric multigrid method, using semicoarsening and line relaxation to ensure convergence with locally anisotropic grids. The model is discretized in time with a third-order method developed to allow variable timesteps. This multi-step method explicitly evaluates advective terms and implicitly evaluates pressure and viscous terms. The model’s accuracy is demonstrated with several test cases: growth rates of Kelvin-Helmholtz billows, the interaction of a translating dipole with no-slip boundaries, and the generation of internal waves via topographic interaction. These test cases also illustrate the model’s use from a high-level programming perspective. Additionally, the results of several large-scale simulations are discussed: the three-dimensional dipole/wall interaction, the evolution of internal waves with shear instabilities, and the stability of the bottom boundary layer beneath internal waves. Finally, possible future developments are discussed to extend the model’s capabilities and optimize its performance within the limits of the underlying numerical algorithms.

Internal waves occur in the interface between two layers of fluids with density stratification. In order to better understand the characteristics of continuous internal waves, a series of experiments were conducted in a laboratory tank. The upper and lower layers are fresh water of 15 cm thick and salt water of 30 cm thick, respectively. The periods of internal waves are 2.5, 5.5 and 6.6 sec. A micro-ADV is used to measure velocity profiles. Wave profiles at the density interface and the free surface are monitored respectively by an ultrasonic and capacitance wave gauges. Our results indicate that particle velocities (u and w) above and below the density interface have opposite directions. The speed is peaked near the density interface and it becomes weaker further away from the interface. Empirical Mode Decomposition is used to remove noise from the observed particle velocities, and the period is consistent with those derived from the interface elevations. The observed particle velocities also compare favorably with the theoretical results. When internal waves propagate without the interference of a sloping bottom, the turbulence induced is rather insignificant. The turbulence is more significant only near the density interface. With the existence of a sloping bottom, the internal waves gradually shoal and deform, the crest becomes sharp and steep, finally the waves become unstable, break and overturn. In this study the effect of bottom slope and the steepness of internal waves on the reflectivity of incoming waves are investigated. The reflectivity is smaller with gentler slope, and it increases and reaches a constant value with steeper slopes. The observed energy dissipation rate£`is higher near the slope. Three methods were used to estimate the energy dissipation rate and shear stress; namely, the inertial dissipation, the TKE and auto-correlation method. The£` estimated from the auto-correlation method is larger than that from the other two methods, but their trend is similar. The energy dissipation rate is found to increase with a gentler sloping bottom.

Thesis (Ph. D.)--University of Washington, 2001. / Vita. Includes bibliographical references (p. 147-153).

No description available.

This work develops a nonlinear, three-dimensional spectral collocation method for the simulation of the incompressible Navier-Stokes equations for geophysical and environmental flows. These flows are often driven by the interaction of stratified fluid with topography, which is accurately accounted for in this model using a mapped coordinate system. The spectral collocation method used here evaluates derivatives with a Fourier trigonometric or Chebyshev polynomial expansion as appropriate, and it evaluates the nonlinear terms directly on a collocated grid. The coordinate mapping renders ineffective fast solution methods that rely on separation of variables, so to avoid prohibitively expensive matrix solves this work develops a low-order finite-difference preconditioner for the implicit solution steps. This finite-difference preconditioner is itself too expensive to apply directly, so it is solved pproximately with a geometric multigrid method, using semicoarsening and line relaxation to ensure convergence with locally anisotropic grids. The model is discretized in time with a third-order method developed to allow variable timesteps. This multi-step method explicitly evaluates advective terms and implicitly evaluates pressure and viscous terms. The model’s accuracy is demonstrated with several test cases: growth rates of Kelvin-Helmholtz billows, the interaction of a translating dipole with no-slip boundaries, and the generation of internal waves via topographic interaction. These test cases also illustrate the model’s use from a high-level programming perspective. Additionally, the results of several large-scale simulations are discussed: the three-dimensional dipole/wall interaction, the evolution of internal waves with shear instabilities, and the stability of the bottom boundary layer beneath internal waves. Finally, possible future developments are discussed to extend the model’s capabilities and optimize its performance within the limits of the underlying numerical algorithms.