Nonlinear modeling of wave-topography interactions, shear instabilities and shear induced wave breaking using vortex method

Theoretical studies on linear shear instabilities as well as different kinds of wave interactions often use simple velocity and/or density profiles (e.g. constant, piecewise) for obtaining good qualitative and quantitative predictions of the initial disturbances. Furthermore, such simple profiles provide a minimal model to obtain a mechanistic understanding of otherwise elusive shear instabilities. In this work, we have extended this minimal paradigm into nonlinear domain using vortex method. Making use of unsteady Bernoulli's equation in presence of linear shear, and extending Birkhoff-Rott equation to multiple interfaces, we have devised a way to numerically simulate the interaction between multiple fully nonlinear waves that start from linear initial conditions. This initial value problem solving strategy is quite general, and has allowed us to simulate diverse problems that can be essentially reduced to the minimal system with interacting waves, e.g. spilling and plunging breakers, stratified shear instabilities (Holmboe, Taylor-Caulfield, stratified Rayleigh), jet flows, and even wave-topography interaction problem like Bragg resonance. Free-slip boundary being a vortex sheet, its effect can also be effectively captured using vortex method. We found that the minimal models capture key nonlinear features (e.g. wave breaking features like cusp formation and roll-ups) which are observed in experiments and/or extensive simulations with smooth, realistic profiles.