Generation and propagation of mode-1 and mode-2 internal waves over bottom topography in a three-layer system
Phys. Rev. Fluids 11, 044804 – Published 27 April, 2026
DOI: https://doi.org/10.1103/9rz3-7p2k
Abstract
Hitherto, the generation and propagation of mode-2 oceanic internal waves and their mutual transformations with mode-1 waves remain inadequately understood, although their significance for localized energy balance and material transport has only recently been unravelled by observational data. To investigate this topic, the wave equations are advantageous not only for allowing transformations between modes 1 and 2 but also for enabling asymptotic and numerical analyses, among which the coupled Korteweg–de Vries (KdV) equations play an important role in the sparse literature. However, their accuracy has not been formally examined through comparisons with primitive equations or, loosely, other reduced models. To address these concerns, a fully dispersive and weakly nonlinear pseudodifferential equation system, or fully dispersive internal wave (FDIW) equations, for a three-layer fluid system is derived from the full Euler equations, which delineate wave amplitudes and velocity potentials at the two interfaces, accounting for bottom topography and background flow. Then regular perturbation analysis and weakly nonlinear analysis in the long-wave limit are conducted to obtain theoretical predictions of the wave amplitude ratio between two interfaces, six resonant conditions with sinusoidal bottom topography, and the associated reflection and transmission coefficients, as well as the phase speeds, vertical structures, and polarities of internal solitary waves. All these predictions are confirmed by direct numerical simulations of the FDIW equations, and a significant discrepancy with the coupled KdV equations indicates that the latter needs much care in practical usage. After that, the FDIW equations are used to investigate the evolution of initial linear waves past an uneven bottom, mimicking the propagation of internal tides in the ocean, and the generation of mode-1 and mode-2 nonlinear waves induced by a constant background current and barotropic tides passing over topography.
