Transcritical flow of a stratified fluid over topography: analysis of the forced Gardner equation

TL;DR

This paper analytically investigates transcritical stratified fluid flow over topography using the forced Gardner equation, identifying conditions for stationary hydraulic transitions and describing complex wave structures with numerical validation.

Contribution

It introduces an analytical framework for transcritical stratified flow over topography using the forced Gardner equation, including diverse wave structures and parameter ranges.

Findings

Identification of parameter ranges supporting stationary transcritical flow

Description of various wave structures including solibores and undular bores

Numerical confirmation of analytical predictions

Abstract

Transcritical flow of a stratified fluid past a broad localised topographic obstacle is studied analytically in the framework of the forced extended Korteweg--de Vries (eKdV), or Gardner, equation. We consider both possible signs for the cubic nonlinear term in the Gardner equation corresponding to different fluid density stratification profiles. We identify the range of the input parameters: the oncoming flow speed (the Froude number) and the topographic amplitude, for which the obstacle supports a stationary localised hydraulic transition from the subcritical flow upstream to the supercritical flow downstream. Such a localised transcritical flow is resolved back into the equilibrium flow state away from the obstacle with the aid of unsteady coherent nonlinear wave structures propagating upstream and downstream. Along with the regular, cnoidal undular bores occurring in the analogous problem for the single-layer flow modeled by the forced KdV equation, the transcritical internal wave flows support a diverse family of upstream and downstream wave structures, including solibores, rarefaction waves, reversed and trigonometric undular bores, which we describe using the recent development of the nonlinear modulation theory for the (unforced) Gardner equation. The predictions of the developed analytic construction are confirmed by direct numerical simulations of the forced Gardner equation for a broad range of input parameters.