On the two-layer high-level Green-Naghdi model in a general form

Abstract

The traditional high-level Green-Naghdi (HLGN) model, which uses the polynomial as the shape function to approximate the variation of the horizontal- and vertical-velocity components along the vertical direction for each-fluid layer, can accurately describe the large-amplitude internal waves in a two-layer system for the shallow configuration (h₂ / λ ≪ 1, h₁ / λ ≪ 1). However, for the cases of the deep configuration (h₂ / λ ≪ 1, h₁ / λ = O(1)), higher-order polynomial is needed to approximate the variation of the velocity components along the vertical direction for the lower-fluid layer. This, however, introduces additional unknowns, leading to a significant increase in computational time. This paper, for the first time, derives a general form of the HLGN model for a two-layer fluid system, where the general form of the shape function is used during the derivation. After obtaining the general form of the two-layer HLGN equations, corresponding solutions can be obtained by determining the reasonable shape function. Large-amplitude internal solitary waves in a deep configuration are studied by use of two different HLGN models. Comparison of the two HLGN models shows that the polynomial as the shape function for the upper-fluid layer and the production of exponential and polynomial as the shape function for the lower-fluid layer is a good choice. By comparing with Euler’s solutions and the laboratory measurements, the accuracy of the two-layer HLGN model is verified.