Internal Wave Interactions with Time-Dependent Critical Levels Brian Casaday and J. C. Vanderhoff Department of Mechanical Engineering Brigham Young University, Provo, UT Internal Waves Internal gravity waves exist abundantly in our world in stably- stratified fluids, such as the ocean and atmosphere. When a finite amount of fluid from a stratified environment is removed from its equilibrium density location, buoyancy and momentum forces will compel the fluid to oscillate about the fluid’s equilibrium point. If the fluid is forced to oscillate at any frequency below the local buoyancy frequency, internal waves will be generated. Internal waves will propagate throughout the fluid in all three dimensions, moving vertically as well as horizontally. Internal Waves Internal gravity waves exist abundantly in our world in stably- stratified fluids, such as the ocean and atmosphere. When a finite amount of fluid from a stratified environment is removed from its equilibrium density location, buoyancy and momentum forces will compel the fluid to oscillate about the fluid’s equilibrium point. If the fluid is forced to oscillate at any frequency below the local buoyancy frequency, internal waves will be generated. Internal waves will propagate throughout the fluid in all three dimensions, moving vertically as well as horizontally. Why Ray Tracing? Ray Tracing is computationally cheap. Thousands of experiments can be simulated with the time and computational power required for a single experiment using other fully non-linear and three-dimensional methods. Internal waves were modeled using a linear ray-tracing code utilizing the WKB approximation that will be validated against realistic cases. Why Ray Tracing? Ray Tracing is computationally cheap. Thousands of experiments can be simulated with the time and computational power required for a single experiment using other fully non-linear and three-dimensional methods. Internal waves were modeled using a linear ray-tracing code utilizing the WKB approximation that will be validated against realistic cases. Waves Approaching a Mean Flow As Internal waves propagate through a changing background wind, the shear alters the properties of the internal waves, including wavenumber, frequency, and wave steepness. A critical level is a region where the background winds cause the relative frequency of the internal wave to go to zero. Critical levels are regions of instability where absorption or wave breaking is likely to occur. Waves Approaching a Mean Flow As Internal waves propagate through a changing background wind, the shear alters the properties of the internal waves, including wavenumber, frequency, and wave steepness. A critical level is a region where the background winds cause the relative frequency of the internal wave to go to zero. Critical levels are regions of instability where absorption or wave breaking is likely to occur. Inertial Waves Inertial waves are internal gravity waves that oscillate with a frequency close to the local Coriolis frequency, defined by the rotation of the earth. Inertial waves in the atmosphere tend to have horizontal wavelengths of hundreds to over a thousand kilometers, with vertical wavelengths of one to seven kilometers. Horizontal velocity amplitudes typically range from m/s. Inertial Waves Inertial waves are internal gravity waves that oscillate with a frequency close to the local Coriolis frequency, defined by the rotation of the earth. Inertial waves in the atmosphere tend to have horizontal wavelengths of hundreds to over a thousand kilometers, with vertical wavelengths of one to seven kilometers. Horizontal velocity amplitudes typically range from m/s. Waves Approaching a Moving Mean Flow Before testing the interaction of small waves with inertial waves, the small waves were tested against horizontal wind envelopes moving downward. Although not particularly prevalent in nature, this models the downward phase propagation of a single crest of an inertial wave, resulting in a time dependent critical level. Waves Approaching a Moving Mean Flow Before testing the interaction of small waves with inertial waves, the small waves were tested against horizontal wind envelopes moving downward. Although not particularly prevalent in nature, this models the downward phase propagation of a single crest of an inertial wave, resulting in a time dependent critical level. Conclusions Time-dependent critical levels affect small scale waves differently than stationary critical levels. Time-dependent critical levels decrease chances of wave instability. Wave instability, which may lead to breaking, arises from background shear and wave steepness, although one may dominate the approach to overturning. Detected inertial waves in the atmosphere typically do not produce low Richardson numbers (high shears) capable of instability. Future Work Validate ray-tracing results against experimental and non-linear simulations. Incorporate wave energy parameters into simulations. Calculate energy loss to dissipation versus loss to the mean flow during wave breaking. Conclusions Time-dependent critical levels affect small scale waves differently than stationary critical levels. Time-dependent critical levels decrease chances of wave instability. Wave instability, which may lead to breaking, arises from background shear and wave steepness, although one may dominate the approach to overturning. Detected inertial waves in the atmosphere typically do not produce low Richardson numbers (high shears) capable of instability. Future Work Validate ray-tracing results against experimental and non-linear simulations. Incorporate wave energy parameters into simulations. Calculate energy loss to dissipation versus loss to the mean flow during wave breaking. Waves Approaching an Atmospheric Inertial Wave Numerous tests were run as internal waves approached and interacted with an ideal inertial wave, with similar results as waves passed though a downward moving horizontal wind. Waves Approaching an Atmospheric Inertial Wave Numerous tests were run as internal waves approached and interacted with an ideal inertial wave, with similar results as waves passed though a downward moving horizontal wind. A ray-tracing program was used to simulate internal waves approaching a horizontal flow with a Gaussian envelope, a mean flow with a downward moving envelope, and a large-scale inertial wave. Left: Simulation showing a wave becoming unstable due to wave steepness. Right: Simulation showing a wave becoming unstable due to background shear f (Low Richardson Number) Simulation of internal waves approaching a moving mean wind. Unlike the stationary mean wind, the wave passes through the critical level without any singularities. Simulation showing a wave approaching a mean flow. Fluid flowing over topography is a common source of internal wave generation, such as winds flowing over Amsterdam Island as shown above. visibleearth.nasa.gov