Transcritical flow of a stratified fluid: The forced extended Korteweg–de Vries model

Abstract

Transcritical, or resonant, flow of a stratified fluid over an obstacle is studied using a forced extended Korteweg–de Vries model. This model is particularly relevant for a two-layer fluid when the layer depths are near critical, but can also be useful in other similar circumstances. Both quadratic and cubic nonlinearities are present and they are balanced by third-order dispersion. We consider both possible signs for the cubic nonlinear term but emphasize the less-studied case when the cubic nonlinear term and the dispersion term have the same-signed coefficients. In this case, our numerical computations show that two kinds of solitary waves are found in certain parameter regimes. One kind is similar to those of the well-known forced Korteweg–de Vries model and occurs when the cubic nonlinear term is rather small, while the other kind is irregularly generated waves of variable amplitude, which may continually interact. To explain this phenomenon, we develop a hydraulic theory in which the dispersion term in the model is omitted. This theory can predict the occurrence of upstream and downstream undular bores, and these predictions are found to agree quite well with the numerical computations.