UBC Theses and Dissertations
Laboratory, field and numerical investigations of Holmboe's instability Tedford, Edmund W.
The instabilities that occur at a sheared density interface are investigated in the laboratory, the Fraser River estuary and with Direct Numerical Simulations (DNS). In the laboratory, symmetric Holmboe instabilities are observed during steady, maximal two-layer exchange flow in a long channel of rectangular cross section. Internal hydraulic controls at each end of the channel isolate the subcritical region within the channel from disturbances in the reservoirs. Inside the channel, the instabilities form cusp-like waves that propagate in both directions. The phase speed of the instabilities is consistent with linear theory, and increases along the length of the channel as a result of the gradual acceleration of each layer. This acceleration causes the wavelength of any given instability to increase in the direction of flow. As the instabilities are elongated new instabilities form, and as a consequence, the average wavelength is almost constant along the length of the channel. In the Fraser River estuary, a detailed stability analysis is conducted based on the Taylor-Goldstein (TG) equation, and compared to direct observations in the estuary. We find that each set of instabilities observed coincides with an unstable mode predicted by the TG equation. Each of these instabilities occurs in a region where the gradient Richardson number is less than the critical value of 1/4. Both the TG predictions and echosoundings indicate the instabilities are concentrated either above or below the density interface. These ‘one-sided’ instabilities are closer in structure to the Holmboe instability than to the Kelvin-Helmholtz instability. Although the dominant source of mixing in the estuary appears to be caused by shear instability, there is also evidence of small-scale overturning due to boundary layer turbulence when the tide produces strong near-bed velocities. Many features of the numerical simulations are consistent with linear theory and the laboratory experiments. However, inherent differences be tween the DNS and the experiments are responsible for variations in the dominant wavenumber and amplitude of the wave field. The simulations exhibit a nonlinear ‘wave coarsening’ effect, whereby the energy is shifted to lower wavenumber in discrete jumps. This process is, in part, related to the occurrence of ejections of mixed fluid away from the density interface. In the case of the laboratory experiment, energy is transferred to lower wavenumber by the ‘stretching’ of the wave field by a gradually varying mean velocity. This stretching of the waves results in a reduction in amplitude compared to the DNS. The results of the comparison show the dependence of the nonlinear evolution of a Holmboe wave field on temporal and spatial variations of the mean flow.
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