Resonant Flow over Topography

Abstract

Weakly nonlinear long waves in shallow water are well-known to be described by the Korteweg-de Vries (KdV) equation. Indeed the KdV equation is canonical and also describes wave motion in stratified fluid, rotating fluids, plasmas and many other physical contexts. Here we describe the resonant forcing of these waves by flow past topography. This occurs when the linear long wave speed is nearly the same as the basic flow speed so that, in hydraulic terminology, the flow is nearly critical. The KdV equation is now modified by the inclusion of a forcing term, and becomes $$- ({{A}_{\tau }} + \Delta {{A}_{x}}) + 6A{{A}_{x}} + {{A}_{{xxx}}} + {{G}_{x}} = 0,$$ (1) , which we shall call the fKdV equation (forced Korteweg-de Vries equation). For the case of flow of a shallow fluid over topography, the free surface displacement is 2/3α 1/2 hA(X,τ), the bottom topography is 2/9αhG(X) the horizontal co-ordinate is ∈−1 hX and the time co-ordinate is 6∈−3(g/h)1/2 τ, while the basic flow speed is (gh)1/2(1 + 1/6α 1/2∆). Here h is the upstream depth of the fluid, and α and ∈ are two small parameters characterizing respectively the amplitude of the topography and the inverse of the horizontal length scale of the topography. Since the response is resonant, the amplitude of the response is proportional to α 1/2, and the derivation of (1) requires the KdV balance that α 1/2 = ∈2. The basic flow is exactly critical, or resonant, if ∆ = 0, but is supercritical (subcritical) if ∆ > 0(< 0). The fKdV equation (1) is generic and describes resonant flow over topography in a variety of other physical contexts. For instance, it also describes resonant flow of a stratified fluid over topography, for which case a detailed derivation is given by Grimshaw and Smyth (1986) (hereafter denoted by GS). In another application, Patoine and Wain (1982) and Malanotte-Rizzoli have shown that (1) describes the resonant generation of Rossby waves by topography. An analogous equation, the forced BDA equation describes resonant flow of a coastal current past a longshore topographic feature such as a headland (Grimshaw, 1987, Mitsudera and Grimshaw, 1990). In the classical case when (1) describes the resonant generation of water waves, either by a moving pressure distribution or by flow over topography, derivations and numerical solutions have been given by Akylas (1984), Cole (1985) and Lee et al (1989).KeywordsRossby WaveWater WavePhysical ContextStratify FluidCnoidal WaveThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.