×

zbMATH — the first resource for mathematics

On the resonant generation of large-amplitude internal solitary and solitary-like waves. (English) Zbl 1137.76328
Summary: We discuss numerical simulations of the generation of large-amplitude solitary waves in a continuously stratified fluid by flow over isolated topography. We employ the fully nonlinear theory for internal solitary waves to classify the numerical results for mode-1 waves and compare with two classes of approximate theories, weakly nonlinear theory leading to the Korteweg–deVries and Gardner equations and conjugate flow theory which makes no approximation with respect to nonlinearity, but neglects dispersion entirely. We find that both weakly nonlinear theories have a limited range of applicability. In contrast, the conjugate flow theory predicts the nature of the limiting upstream propagating response (a dissipationless bore), successfully describes the bore’s vertical structure, and gives a value of the inflow speed, \(c_j\), above which no upstream propagating response is possible. The numerical experiments demonstrate the existence of a class of large-amplitude response structures that are generated and trapped over the topography when the inflow speed exceeds \(c_j\). While similar in structure to fully nonlinear solitary waves, these trapped disturbances can induce isopycnal displacements more than 100% larger than those induced by the limiting solitary wave while remaining laminar. We develop a theory to describe the vertical structure at the crest of these trapped disturbances and describe its range of validity. Finally, we turn to the generation of mode-2 solitary-like waves. Mode-2 waves cannot be truly solitary owing to the existence of a small mode-1 tail that radiates energy downstream from the wave. We demonstrate that, for stratifications dominated by a single pycnocline, mode-2 wave dissipation is dominated by wave breaking as opposed to mode-1 wave radiation. We propose a phenomenological criterion based on weakly nonlinear theory to test whether mode-2 wave generation is to be expected for a given stratification.

MSC:
76B55 Internal waves for incompressible inviscid fluids
76B25 Solitary waves for incompressible inviscid fluids
PDF BibTeX XML Cite
Full Text: DOI