UBC Theses and Dissertations

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UBC Theses and Dissertations

A numerical model of stratified circulation in a shallow-silled inlet Dunbar, Donald Stanley, 1953-


A numerical model has been developed for the study of stratified tidal circulation in Indian Arm - a representative inlet on the southern coast of British Columbia. Equations for horizontal velocity, salt conservation, continuity, density (calculated as a linear function of salinity), and the hydrostatic approximation govern the dynamics. All equations have been laterally integrated under the assumption of negligible cross-inlet variability. The model is time dependent and includes nonlinear advective terms, horizontal and vertical turbulent diffusion of salt and momentum, and variations in width and depth. Provisions for surface wind stress and a flux of fresh water are also included, although neither was utilized in this study. An explicit finite difference scheme centred in both time and space was used to solve for the horizontal and vertical velocity components, salinity, and surface elevation on a staggered rectangular grid. A backward Euler scheme was used to suppress the computational mode. Tests using a semi-implicit scheme to solve the finite difference equations over realistic topography led to numerical instabilities at modest values of the time step - in spite of the unconditional stability criteria - suggesting that linear stability analysis may give misleading results for strongly nonlinear systems. Surface elevations calculated from tidal harmonic analysis and salinity timeseries derived from linearly interpolated CTD casts were prescribed at the open boundary. Initial and boundary conditions based on observations in Burrard Inlet and Indian Arm during the winter of 1974-75 were used to study the inlet's response to tidal forcing and to simulate the deep-water renewal that occurred during this period. Coefficients for the horizontal turbulent diffusion of momentum and salt were set equal to 10⁶ cm² s⁻¹. Reducing this value by a factor of two was found to have little impact on the solution. A further reduction to 10³ cm² s⁻¹ led to numerical instabilities under conditions of dense water inflow. The side friction term in the momentum balance was tuned to match calculated and observed dissipation rates in Burrard Inlet; leading to good agreement between the observed and calculated barotropic tide. Contour plots of tidal amplitudes and phases for model currents and salinities revealed a standing wave pattern for the K₁ and M₂ internal tides in Indian Arm; thus allowing for the possibility of resonance. A comparison of model results with vertical amplitude and phase profiles from harmonic analysis of Cyclesonde current meter timeseries at two locations in Indian Arm was consistent with this result. A least-squares fit was made of the vertical modal structure in the model to the complex tidal amplitudes. This led to calculations of the kinetic energy contained in each of the modes along the model inlet for the M₂ and K₁ constituents. Most of the energy was found to be contained in the barotropic and first baroclinic modes, with the latter dominating in the deep basin, and the former dominating near the sill. Second mode energy was significant for the K₁ constituent at some locations in Indian Arm. There are clear indications in the model of barotropic tidal energy being radiated into the inlet basin via the internal tide. Simulations of the influx of dense water into Indian Arm yielded exchange rates that are consistent with observed values and suggest the possibility of fine-tuning the model coefficients to allow prediction of future overturning events.

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