Numerical Investigation of Internal Wave-Vortex InteractionsTyler D. Blackhurst and J.C. Vanderhoff

Department of Mechanical EngineeringBrigham Young University, Provo, Utah

ConclusionA comparison of two numerical simulations suggests that the initial location of the internal waves relative to the vortex dipole is a reason for the interaction differences, particularly for the center rays of each case. Both carry similarities to the results of Godoy-Diana, et al. (2006), which show that the center ray reaches a critical level. The center ray of the first simulation entered the dipole toward the rear and escaped before reaching a critical level. The center ray of the second simulation is absorbed by the dipole, but it is pulled in the horizontal by one of the vortices. Work is being done to better understand this phenomenon and the role played by the relative positioning and scaling of the internal waves and the vortex dipole.

The three-dimensional advantage of this numerical simulation is seen as the off-center rays can be analyzed. Their spreading acknowledges a transfer of energy among wavelengths. Also, thousands of rays from a multitude of positions can be easily simulated.

Further numerical work is being done with these and similar wave-vortex interactions. Chiefly, more exhaustive analyses of current and future results is required to understand the interactions. This includes, in addition to that mentioned above, analyses of wave amplitude and energy which will offer insight regarding breaking and turbulence.

ReferencesInternal wave image (2009). http://www.nasa.gov/topics/earth/features/turbulence.html

Godoy-Diana, R., Chomaz, J. & Donnadieu, C. (2006). Internal gravity waves in a dipolar wind: A wave-vortex interaction experiment in a stratified fluid. Journal of Fluid Mechanics, 548, 281-308.

AcknowledgementRocky Mountain NASA Space Grant Fellowship 2008-2010

The initial location along the length is different in the solution of Figures 6 and 7 than in the previous solution, placing the interaction nearer to the front of the dipole. The off-center rays are temporarily caught in the dipole, then escape and spread out. Because its slope is zero after the interaction, the center ray is absorbed by the dipole, having reached a critical level. However, in the horizontal it moves a few meters away from the centerline, only to return shortly thereafter. This phenomenon is unexpected and reoccurs periodically.

Figure 7 Spatial views of wave-vortex interaction shown in Figure 6.

Figure 6 Three-dimensional plot of rays. The center ray reaches a critical level.

Multiple differently-colored rays (representing internal waves) are symmetrically placed across the dipole axis in Figure 4. Each ray interacts independently in three dimensions with the vortex dipole. They have the

same initial location along the length and depth, as well as the same initial wavenumber vector.

Figure 5 plots three two-dimensional views of the interaction. The top plot gives a view similar to looking down into an experimental tank and the middle plot is a side view; in each, the dipole translates right to left. The bottom plot is a view from the end of a tank, with the vortex dipole translating into the plot. Interaction with the dipole causes the wavenumbers to change, changing also the angles of propagation. The off-center rays spread out from the dipole axis. The center ray, as expected, is equally influenced by each vortex and moves straight horizontally. Maintaining a non-zero vertical slope, the center ray was not absorbed by the dipole as would be expected had it reached a critical level.

Figure 5 Spatial views of wave-vortex interaction shown in Figure 4.

Figure 4 Three-dimensional plot of rays involved in wave-vortex interaction.

ResultsIntroductionInternal gravity waves, as seen in Figure 1, are inherent in the atmosphere and ocean due to their stable stratifications. As they propagate through

Figure 1 Internal waves in the atmosphere seen over the ocean (NASA.gov, 2009).

their medium, internal waves of various scales interact with diverse phenomena common in these fluids. The interaction of small-scale internal waves with a vortex dipole is of particular interest due to the rotation of the earth resulting in constant vortex generation. The speed and direction with which internal waves approach a vortex dipole can significantly affect the wave-vortex interaction, determining if theenergy of the internal waves will be absorbed, refracted, or unaffected by the dipole. Understanding the effects of such an interaction on internal waves plays an important role in accurately predicting weather patterns and potential turbulence, benefitting meteorology and aerospace.

MethodsTwo variations of internal waves interacting with a Lamb-Chaplygin vortex dipole are investigated through linear three-dimensional numerical simulations, governed by ray theory. Ray theory traces wave energy propagation. In addition, wave amplitudes are analyzed for potential wave breaking and turbulence.

Figure 2 Experimental setup of Godoy-Diana, et al. (2006)

Results are validated by an experiment presented by Godoy-Diana, Chomaz and Donnadieu (2006), the setup of which is shown in Figure 2. To form the vortex dipole, two flaps were closed at one end of a tank filled with a salt-water mixture. Internal waves were generated by the oscillation of circular cylinders.

The numerically-generated dipole vortex is given as a vector plot in Figure 3. It is shown as if one were looking down into the experimental tank, with the dipole translating from right to left.

Figure 3 Numerically-generated vortex dipole.