Theoretical & Applied Fluid Dynamics Laboratory
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Bragg Resonance with Regular Topography
Bragg Resonance in a 2-Layer Fluid
Real ocean density is not constant and varies with the depth (due to change in water temperature and salinity). In most oceans, density jumps from a lighter upper layer fluid density to a heavier lower layer fluid one over a surface called Thermocline. Waves – similar to surface waves – can propagate over Thermocline and are called internal (interfacial) waves. Internal waves, when they propagate over topography, may exchange energy (via nonlinear resonance) with surface waves. If bottom is periodically modulated, the phenomenon is called Bragg Resonance and very well understood for a homogeneous fluid. Bragg resonance of internal waves is important in explaining the generation of internal gravity waves particularly in littoral zones, dead water phenomenon (high resistance on ships in stratified waters), and has many applications in navigation and design of off-shore structures.
We have studied Bragg resonance of interfacial waves theoretically and numerically. Starting from perturbation analysis, and proceeding to higher order nonlinearities, closed-form solutions for the resonance condition and the growth rate of resonant waves are obtained. Part II of this work is devoted to the development of a very efficient spectral-based numerical scheme. Theoretical results are cross-validated with our direct simulation and effect of higher nonlinearity and complicated cases – which are beyond the capacity of analytical tools – are studied. It is shown for example, that by multiple resonances between a chosen set of waves, water surface may undergo chaotic oscillations. We also show that Bragg resonance in two-layer fluids can significantly contribute to the development of the ocean spectrum, and, is a potential mechanism for the mysterious phenomenon of generation of high-frequency interfacial waves in lakes and on continental shelves.
References:
– Alam, M.R., Liu, Y. and Yue, D.K.P., “Bragg resonance of waves in a two-layer fluid propagating over bottom ripples. Part I. Perturbation analysis.”, J. Fluid Mechanics, Volume 624 (2009), pages 191-224. [PDF]
– Alam, M.R., Liu, Y. and Yue, D.K.P., “Bragg resonance of waves in a two-layer fluid propagating over bottom ripples. Part II. Numerical simulation.”, J. Fluid Mechanics, Volume 624 (2009), pages 225-253. [PDF]
– Alam, M.R., Liu, Y. and Yue, D.K.P., “Chaotic Internal Wave Motion due to Multiple Bragg Resonances”, American Geophysical Union Fall Meeting. San Francisco, CA, 15-19 December 2008.