Hitherto, the generation and propagation of mode-2 oceanic internal waves and their mutual transformations with mode-1 waves have yet to be adequately understood, albeit their significance on localised energy balance and material transport has just been unraveled by observational data. To investigate this topic, the wave equations are supposed to have the advantage of not only allowing for the transformations between mode-1 and mode-2 but also being feasible to conduct asymptotic and numerical analyses, among which the coupled Korteweg-de Vries (KdV) equations play an important role in the overall 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 pseudo-differential equation system, hereafter abbreviated as FDIW equations, in a three-layer fluid system is derived from the full Euler equations, which delineate wave amplitudes and velocity potentials along the two interfaces, taking into account the bottom topography and background flow. Then, regular perturbation analysis and weakly nonlinear analysis are conducted to obtain the theoretical predictions on 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 to mimic the propagation of internal tides in the ocean and the generation of mode-1 and mode-2 nonlinear waves induced by constant background current and barotropic tides passing over topography. 

internal waves,mode transformation