A nonlinear model of the shoaling and refraction of interfacial solitary waves in the ocean. part I: Development of the model and investigations of the shoaling effect

Oceanic internal waves are frequently observed to be of large amplitude, and to exhibit significant wave front curvature, as seen in synthetic aperture radar images. The aim of this paper is to present a refraction model in two horizontal dimensions, including moderate amplitude effects, to model these common properties of internal waves. The model incorporates second-order nonlinearity and uses a ray method to allow arbitrary refraction. For these preliminary investigations with the model, a two-layer ocean is assumed with an upper-layer thickness of 50 m and a density difference across the layers of 0.5 kg m(-3). An important aspect affecting the refraction of nonlinear waves is shoaling, as the phase speed of a nonlinear wave depends on its amplitude. In Part I of this paper the development of the model is described, and the shoaling phenomenon is investigated in detail. The shoaling of interfacial internal solitary waves over both slowly varying and rapidly varying bathymetry is studied. A wedge-shaped bathymetry representing an idealized continental slope varying from 1000 to 140 m is considered. Comparisons are made against analytical results for the one-dimensional propagation of solitary waves over varying bathymetry, for both onshelf and offshelf propagation. In the case of onshelf propagation over slowly varying bathymetry, the predictions of the second-order extended Korteweg-de Vries (EKdV) model are that an initial growth in wave amplitude as the wave passes across the deeper part of the slope is compensated by a decay in amplitude as it approaches the shallow water. This effect is particularly pronounced for large amplitude internal waves, and is due to a theoretical limit on EKdV wave amplitude that is explained in detail. In contrast, predictions from a first-order KdV model indicate continuous growth in amplitude, sometimes to unrealistic values. The results agree with adiabatic theory provided the slope of the bathymetry is much weaker than the slope of the interface induced by the internal wave. For very rapid changes in bathymetry the first-order KdV model agrees with predictions of the fission of internal waves found by Djordevic and Redekopp. However, once again the EKdV model gives very different predictions, and in the cases considered no fission was observed, but sometimes a small amplitude dispersive wave train was produced behind the leading wave. Simulations of the offshelf propagation of internal waves suggested that in most cases the adiabatic theory of wave transformation was reversible, but in some cases, when the initial wave amplitude in 140-m depth was close to the theoretical limiting amplitude, the waves were so wide that only extremely weak bathymetric gradients satisfied the slowly varying criterion. These near-limiting waves tended to split as they moved off-shelf. The agreement of the model simulations with analytical results give confidence in applying the model to two horizontal dimensions, which is done in Part II of this paper.