Resonantly generated internal waves in a contraction

Abstract

The near-resonant flow of a stratified fluid through a localized contraction is considered in the long-wavelength weakly nonlinear limit to investigate the transient development of nonlinear internal waves and whether these might lead to local steady hydraulic flows. It is shown that under these circumstances the response of the fluid will fall into one of three categories, the first governed by a forced Korteweg–de Vries equation and the latter two by a variable-coefficient form of this equation. The variable-coefficient equation is discussed using analytical approximations and numerical solutions when the forcing is of the same (positive) and of opposite (negative) polarity to that of free solitary waves in the fluid. For positive and negative forcing, strong and weak resonant regimes will occur near the critical point. In these resonant regimes for positive forcing the flow becomes locally steady within the contraction, while for negative forcing it remains unsteady within the contraction. The boundaries of these resonant regimes are identified in the limits of long and short contractions, and for a number of common stratifications.

Synthetic Aperture Radar observations of resonantly generated internal solitary waves at Race Point Channel (Cape Cod)

“…The long‐shore bathymetry between RP and the outer Cape Cod is nearly equivalent to a vertical wall (see Figures 1 and 2). Given the equally abrupt southern slopes of SB, and the relatively smooth flat bottom of RP Channel, it seems reasonable to explain the internal wave generation in the context of models of stratified flows through a channel with a lateral contraction [ Melville and Macomb , 1987; Clarke and Grimshaw , 1994; Baines , 1995]. The forcing in this case is different from that resulting of flow across a pronounced sill, as in the case of SB.…”

Steady transcritical flow over a hole: Parametric map of solutions of the forced Korteweg–de Vries equation

“…Here, A͑x , t͒ is the amplitude of the relevant linear long wave mode, where x , t are the spatial and time variables, respectively, c is the aforementioned linear long wave phase speed in the reference frame of the obstacle, while the coefficients r , s are determined by the background stratification, G 0 is a measure of the forcing magnitude, and f͑x͒ is a projection of the obstacle onto the relevant long wave mode. In the work of Grimshaw and Smyth 2 and Clarke and Grimshaw, 3 and in many other works on transcritical flow ͑see the references below͒, the forcing term was localized and the emphasis was on the transient development of upstream and downstream undular bores, emanating from a locally steady hydraulic flow over the obstacle.…”

Symmetries, solutions and conservation laws of a class of nonlinear dispersive wave equations

“…Therefore, equation (1.1) is called a KdV equation with forcing term or the forced equation. In recent years, several explicit asymptotic derivations for this generic model equation (1.1) have been studied in [2,3]. We should note that in the absence of the forcing term c 2 z x (x, t), the classical KdV equation is completely integrable [4,5,6,7] while the KdV equation with a forcing term is not known to be integrable.…”