Experiments on rotating and reflecting gravity wave beams

The influence of rotation on the orientation of internal wave beams has been well documented. Disturbances from a point source generate a conical wave-beam structure whose angle to the horizontal θ is given by the dispersion relation

θ = sin^-1(ω²-4Ω²/N²-4Ω²)^1/2

where ω is the forcing frequency, N the buoyancy frequency and Ω the angular rotation rate of the system, that is oriented anti-parallel to gravity. In contrast to the non-rotating case, there are both lower (2Ω) and upper (N) cut-off frequencies, outside of which no disturbances can propagate. This dispersion relation has never previously been verified experimentally.

The experiment comprised an acrylic tank of diameter 1 meter mounted on a rotating table. Two stable salinity stratifications (N=1.06, 1.43 +/- 0.01s^-1) were established by filling the tank using the double bucket method. A sphere of radius 2.54 +/- 0.02cm was fixed to a thin rod and positioned at the center of the tank. The sphere was oscillated vertically with an amplitude of 1.50 +/- 0.01cm over a range of frequencies. Through the arrangement of a video camera, a mirror and a back lit pattern of dots the resulting conical wave beams were visualised using the ``synthetic schlieren'' method. The beam angles were objectively measured using the Radon transform. Agreement between experiment and theory is very good, and one can readily determine the existence of a lower cut off frequency, below which no beams propagate.

Recent theoretical and numerical investigations predict that localized nonlinear effects in the overlapping region of an incoming and reflected wavebeam radiate higher-harmonic beams. Specifically, a primary-harmonic wave beam propagating at angle θ to the horizontal and incident on a slope of angle α yields a reflected primary-harmonic wave beam, which propagates upslope or downslope depending on whether θ > α or θ < α. Nonlinearity induces higher harmonics in the overlapping region of the incoming and reflected primary harmonic beams, turning this region into a source of new beams, provided that the frequency of the induced higher harmonics is less than N. The higher harmonic beams propagate upslope or downslope, depending on whether sin^-1(nω/N) exceeds or deceeds α, n being the order of the harmonic.

Experiments were performed in a tank 120cm long, 66cm high and 20cm thick. An acrylic cylinder 2.54 +/- 0.02cm in radius and 20cm long was oscillated in a linear stratification with an amplitude of 0.5 +/- 0.01cm to generate internal wave beams emanating at an angle of 25^o. The arrangement was such that one beam was reflected from an inclined surface, and the reflections were observed using the synthetic schlieren set-up. The reflected first- and second-harmonic beams were readily observed for a number of different slope angles. In agreement with recent theory, no generated second-harmonic wavebeams were observed for large slope angles, as the energy in the radiated wave beams was too weak for experimental detection.