Properties of solitary waves propagating in a two-layer fluid are investigated comparing experiments and theory. In the experiments the velocity field induced by the waves, the propagation speed and the wave shape are quite accurately measured using particle tracking velocimetry (PTV) and image analysis. The experiments are calibrated with a layer of fresh water above a layer of brine. The depth of the brine is 4.13 times the depth of the fresh water. Theoretical results are given for this depth ratio, and, in addition, in a few examples for larger ratios, up to 100:1. The wave amplitudes in the experiments range from a small value up to almost maximal amplitude. The thickness of the pycnocline is in the range of approximately 0.13-0.26 times the depth of the thinner layer. Solitary waves are generated by releasing a volume of fresh water trapped behind a gate. By careful adjustment of the length and depth of the initial volume we always generate a single solitary wave, even for very large volumes. The experiments are very repeatable and the recording technique is very accurate. The error in the measured velocities non-dimensionalized by the linear long wave speed is less than about 7-8% in all cases. The experiments are compared with a fully nonlinear interface model and weakly nonlinear Korteweg-de Vries (KdV) theory. The fully nonlinear model compares excellently with the experiments for all quantities measured. This is true for the whole amplitude range, even for a pycnocline which is not very sharp. The KdV theory is relevant for small wave amplitude but exhibit a systematic deviation from the experiments and the fully nonlinear theory for wave amplitudes exceeding about 0.4 times the depth of the thinner layer. In the experiments with the largest waves, rolls develop behind the maximal displacement of the wave due to the Kelvin-Helmholtz instability. The recordings enable evaluation of the local Richardson number due to the flow in the pycnocline. We find that stability or instability of the flow occurs in approximate agreement with the theorem of Miles and Howard.

Solitary waves propagating horizontally in a stratified fluid are investigated. The fluid has a shallow layer with linear stratification and a deep layer with constant density. The investigation is both experimental and theoretical. Detailed measurements of the velocities induced by the waves are facilitated by particle tracking velocimetry (PTV) and particle image velocimetry (PIV). Particular attention is paid to the role of wave breaking which is observed in the experiments. Incipient breaking is found to take place for moderately large waves in the form of the generation of vortices in the leading part of the waves. The maximal induced fluid velocity close to the free surface is then about 80% of the wave speed, and the wave amplitude is about half of the depth of the stratified layer. Wave amplitude is defined as the maximal excursion of the stratified layer. The breaking increases in power with increasing wave amplitude. The magnitude of the induced fluid velocity in the large waves is found to be approximately bounded by the wave speed. The breaking introduces a broadening of the waves. In the experiments a maximal amplitude and speed of the waves are obtained. A theoretical fully nonlinear two-layer model is developed in parallel with the experiments. In this model the fluid motion is assumed to be steady in a frame of reference moving with the wave. The Brunt-Väisälä frequency is constant in the layer with linear stratification and zero in the other. A mathematical solution is obtained by means of integral equations. Experiments and theory show good agreement up to breaking. An approximately linear relationship between the wave speed and amplitude is found both in the theory and the experiments and also when wave breaking is observed in the latter. The upper bound of the fluid velocity and the broadening of the waves, observed in the experiments, are not predicted by the theory, however. There was always found to be excursion of the solitary waves into the layer with constant density, irrespective of the ratio between the depths of the layers.

Analytical and numerical results from recently developed strongly nonlinear asymptotic models are compared and validated with experimental observations of internal gravity waves and results from the numerical integrations of Euler equations for solitary waves at the interface of two-fluid systems. The focus of this investigation is on regimes where large amplitudes are attained, where the classical weakly nonlinear theories prove inadequate. Two asymptotically different regimes are examined in detail: shallow fluids, in which the typical wavelengths of the interface displacement are long with respect to the depths of both fluids, and deep fluids, where the wavelengths are comparable to, or less than, the depth of one of the two fluids. With the aim of illustrating the breakdown of the asymptotic assumptions, the transition from a shallow to a deep regime is examined through numerical computation of Euler system's solutions and by comparisons with solution to models.

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Periodic travelling wave solutions for a strongly nonlinear model of long internal wave propagation in a two-fluid system are derived and extensively analysed, with the aim of providing structure to the rich parametric space of existence of such waves for the parent Euler system. The waves propagate at the interface between two homogeneous-density incompressible fluids filling the two-dimensional domain between rigid planar boundaries. The class of waves with a prescribed mean elevation, chosen to coincide with the origin of the vertical (parallel to gravity) axis, and prescribed zero period-average momentum and volume-flux is studied in detail. The constraints are selected because of their physical interpretation in terms of possible processes of wave generation in wave-tanks, and give rise to a quadrature formula which is analysed in parameter space with a combination of numerical and analytical tools. The resulting model solutions are validated against those computed numerically from the parent Euler two-layer system with a boundary element method. The parametric domain of existence of model periodic waves is determined in closed form by curves in the amplitude–speed (A, c) parameter plane corresponding to infinite period limiting cases of fronts (conjugate states) and solitary waves. It is found that the existence domain of Euler solutions is a subset of that of the model. A third closed form relation between c and A indicates where the Euler solutions cease to exist within the model's domain, and this is related to appearance of ‘overhanging’ (multiple valued) wave profiles. The model existence domain is further partitioned in regions where the model is expected to provide accurate approximations to Euler solutions based on analytical estimates from the quadrature. The resulting predictions are found to be in good agreement with the numerical Euler solutions, as exhibited by several wave properties, including kinetic and potential energy, over a broad range of parameter values, extending to the limiting cases of critical depth ratio and extreme density ratios. In particular, when the period is sufficiently long, model solutions show that for a given supercritical speed waves of substantially larger amplitude than the limiting amplitude of solitary waves can exist, and are good approximations of the corresponding Euler solutions. This finding can be relevant for modelling field observations of oceanic internal waves, which often occur in wavetrains with multiple peaks.