Strongly nonlinear internal waves in near two-layer stratifications: Generation, propagation and self-induced shear instabilities

This thesis consists of three parts. In the first part, periodic travelling-wave solutions for a strongly nonlinear asymptotic model of long internal wave propagation in an Euler incompressible two-fluid system are derived and extensively analyzed. The class of waves with a prescribed mean elevation, and zero-average momentum and volume flux is studied in detail. We found that the domain of existence of these periodic waves contains that of their Euler solution counterparts as a subset, and the agreement is good on the common domain. Among other findings, the model predicts the existence of periodic waves of substantially larger amplitudes than those of limiting solitary waves. This is relevant for modeling realistic oceanic internal waves, which often occur in wavetrains with multiple peaks.;The second part consists of optimizing a two-layer system to approximate a continuously stratified one. This work aims at extending the applicability of two-layer asymptotic models. The strategy is validated by comparing long solitary wave numerical solutions in continuous stratification against their two-layer asymptotic counterparts.;The third part is a numerical study of the shear instability induced by internal solitary waves in near two-layer stratifications. We emulate numerically the generation of solitary wave in the experiments of [19], by using a variable density Navier Stokes solver [1]. We validate the numerical code by comparison against strongly nonlinear model [7], optimally adjusted for finite-width pycnocline developed in the second part. While the general dynamical features reported in [19] emerge from the simulations, there are significant discrepancies, which seem resolvable only by further laboratory work. Good agreement is however obtained for the self-induced shear instability for large waves. To assess whether this is an intrinsic property of the wave or an effect of the generation technique, we study the evolution of stationary solutions of Euler equations, found with a variant of the algorithm in [51]. We determine local stability characteristics and construct an amplitude threshold for manifestation of the instability. We discuss the implication of locally unstable shear for the global stability properties of traveling wave solutions.