{'fraction_non_alphanumeric': 0.018626309662398137, 'fraction_numerical': 0.0, 'mean_word_length': 5.390334572490707, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Three-wave resonant interactions in unstable media

The paper considers the evolution of weakly nonlinear disturbances in linearly unstable stratified shear flows. We develop a generic Hamiltonian formulation for two-dimensional flows. The paper is focused on three-wave resonant interactions, which are always present in the stratified shear flows under consideration as the lowest-order nonlinear process. Two different types of shear flows are considered. The first one is the classical piecewise-linear model with constant density and vorticity in each layer. For such flows, linear instability is due to weak interaction of different modes. The second type is the Kelvin–Helmholtz model, consisting of two layers with different densities and velocities. Velocity shear is assumed to be weakly supercritical. We show that apart from the classical triplets consisting of stable waves, both flow types admit only triplets consisting of one weakly unstable and two neutrally stable waves, and we consider them in detail. Universal evolution equations for three resonantly interacting wave packets are derived for both cases. For the first flow type, the generic equations coincide with the system derived earlier for a particular case of resonant interactions between unstable and neutral baroclinic waves in a quasi-geostrophic two-layer model. The evolution equations for the Kelvin–Helmholtz model are new, and are studied numerically and analytically in detail. In particular, we demonstrate that resonant interaction with neutral waves can stabilize the growth of the linearly unstable wave. This mechanism is essentially different from the well-known nonlinear stabilization mechanism due to cubic nonlinearity.