Mathematical Modelling of Nonlinear Ring Waves of Moderate Amplitude in Fluids

海洋中的表面波和内波对近海结构物、水下电缆和潜水器具有强烈的影响。它们还有助于海洋混合过程，这对气候很重要。在海峡、河海相互作用区和局部地形特征产生的内波几乎是环形的，通常在底层水流上传播。我们的项目旨在开发适当的数学模型，并使用它们来研究这种和类似波的性质和行为，重点是中等振幅的波。最近，研究表明，尽管波和流的几何形状相互冲突，但存在不同于已知平面波分解的线性模态分解[1]。这种分解被用来解析地描述弱非线性表面波和内波的波前畸变，并系统地开发和使用数学模型，以解决一些应用问题[2，3]。拟议的工作将致力于为中等振幅的波浪开发新的先进模型，并利用这些模型研究不同类型的分层和海流的表面波和内波的行为，以及加深我们对环形波稳定性的二维方面的理解。研究将采用各种分析和数值方法。参考文献：[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，具有上层流的两层流体中的长内环波，Russ. J. EarthSci. 20（2020）ES 4006。

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.