Mathematical Modelling of Nonlinear Ring Waves of Moderate Amplitude in Fluids

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Surface and internal waves in the oceans have a strong effect on offshore structures, underwater cables and submersibles. They also contribute to ocean mixing processes, which are important for climate. Internal waves generated in straits, river-sea interaction areas and by localised topographic features are nearly annular (ring-shaped) in form and often propagate over an underlying current. Our project aims to develop appropriate mathematical models and to use them to study the properties and behaviour of such and similar waves with an emphasis on waves of moderate amplitude.Recently, it was shown that despite the clashing geometries of the waves and the current, there exists a linear modal decomposition that is different from the known decomposition for plane waves [1]. This decomposition was used to describe analytically the distortion of the wavefronts of weakly-nonlinear surface and internal waves, and to systematically develop and use mathematical models in order to solve some applied problems [2, 3]. The proposed work will be devoted to developing new advanced models for the waves of moderate amplitude, and to using them in order to study the behaviour of both surface and internal waves for different types of stratification and currents, as well as developing our understanding of the two-dimensional aspects of stability of the ring waves. The research will employ a variety of analytical and numerical methods.References:[1] K.R. Khusnutdinova, X. Zhang, Long ring waves in a stratified fluid over a shear flow, J. Fluid Mech. 794 (2016) 17-44.[2] K.R. Khusnutdinova, X. Zhang, Nonlinear ring waves in a two-layer fluid, Physica D: Nonlinear Phenomena 333 (2016) 208-221.[3] K.R. Khusnutdinova, Long internal ring waves in a two-layer fluid with an upper-layer current, Russ. J. Earth Sci. 20 (2020) ES4006.