An Experimental and Theoretical Investigation of Internal Wave Kinetic Energy Density in Variable Stratifications

An Experimental and Theoretical Investigation of Internal Wave Kinetic Energy Density in Variable Stratifications Allison Marie Lee Department of Mechanical Engineering, BYU Doctor of Philosophy Internal waves are generated in a fluid if the density increases continuously with depth. The variation in density with depth, or stratification, defines the natural frequency of the fluid N. Two common examples of stratified fluids are the ocean and atmosphere; internal waves are generated continuously in both mediums. Although there are many internal wave generation mechanisms, one common and frequently studied method is tidal flow over oceanic bathymetry. If the local natural frequency of the water near the topography is greater than the tidal frequency ω , internal waves will be generated by the tidal flow over the topography. If N < ω , only evanescent waves will be formed. Unlike internal waves, evanescent waves decay rapidly as they move vertically away from their generation site. As evanescent waves pass from an evanescent region (N < ω), through a turning depth (N = ω) and into a propagating region (N > ω), they become propagating internal waves. Because internal waves can propagate energy across large distances, they play an important role in oceanic mixing and the overall energy budget of the ocean. Knowing where these waves are formed from evanescent waves and their corresponding energy improves understanding of the impact on their surrounding area. Kinetic energy density of evanescent and internal waves formed from oscillatory flow over topography in evanescent regions is first estimated using synthetic schlieren experiments and a novel linear theory model. Experiments were performed with two Gaussian topographies in an exponential density profile. The linear theory model, which uses a set of equations that links the evanescent and propagating regions with the Airy function to overcome the discontinuity inherent with a turning depth, was compared to the experiments. Both methods showed that increasing Fr1, the strength of the evanescent region relative to the excitation frequency, causes the propagating kinetic energy to decrease. In addition, kinetic energy decreased with increasing distance between the topography and the turning depth. Because the model does not account for nonlinearities such as turbulence generation, it regularly overestimates propagating kinetic energy relative to the experiments. After comparing the model with synthetic schlieren experiments, it was used to estimate that 25% of the evanescent wave energy generated by an oceanic topography located at 15◦N, 130◦E can become propagating wave energy. The influence of topography shape and fluid density profile on kinetic energy density was also explored through a combination of experiments, a linear theory model, and numerical simulations. From numerical simulations, kinetic energy can be directly calculated with the velocity profile and indirectly with the density perturbation field, in the same manner as the synthetic schlieren experiments. Average propagating internal wave kinetic energy (KE∗ 2 ) as a function of Fr1D/H, which combines Fr1 with the relative distance between the topography and the turning depth D/H, was compared for all methods. KE∗ 2 decreases with increasing Fr1D/H for all methods. Also, far from the turning depth, the direct and indirect simulations indicate similar kinetic energy when in the propagating region, where a distance from the turning depth can be quantified based on N and ω . This work was expanded to include a medium Gaussian, steep Gaussian, sinusoidal, and complex topography with two layer linear, parabolic, cubic, and exponential density profiles to investigate the validity of assuming an average natural frequency in the evanescent region and the impact of the topographic slope on KE∗ 2 . A comparison of the density profiles indicated that using a two layer linear density profile has similar results compared to the other density profiles for estimating KE∗ 2 as a function of Fr1D/H. Also, KE ∗ 2 is non-negligible for Fr1D/H < 4. Increasing the maximum slope of a topography shape decreases the kinetic energy of the generated internal waves, though it was found that the energy is dependent upon the actual shape of the topography as well. Particle image velocimetry (PIV) experiments were performed and compared to synthetic schlieren (SS). While SS experiments generally resulted in an overestimate of kinetic energy relative to the PIV results, the trends from each experimental method matched well. It is recommended that SS be used in regions away from turning depths, but that either are valid in the evanescent and propagating regions. PIV methods should be used when results near the turning depth or the topography are desired.