Resonant generation of finite-amplitude waves by the flow of a uniformly stratified fluid over topography

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Abstract
The forced Korteweg-de Vries equation is now established as the canonical equation to describe resonant, or critical, flow over topography. However, when the fluid is uniformly and weakly stratified, this equation degenerates in that the quadratic nonlinear term is absent. This anomalous, but important, case requires an alternative theory which is the purpose of this paper. We derive a new evolution equation to describe this case which, while having some similarities to the forced Korteweg-de Vries equation, contains two important differences. First, a topography of amplitude α now produces a finite-amplitude response, whereas in the canonical forced Korteweg-de Vries equation, the response scales with α½. Secondly, the maximum amplitude the fluid flow response can achieve is limited by wave breaking, whose onset is characterized by an incipient flow reversal. Various numerical solutions of the new evolution equation are presented spanning a parameter space defined by a resonance detuning parameter, the topographic amplitude and a parameter measuring the strength of the stratification.

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