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Resonant triad interactions of two-layer gravity waves in cylindrical basins

Matthew Durey*

Paul A. Milewski

  • School of Mathematics and Statistics, University of Glasgow, University Place, Glasgow G12 8QQ, United Kingdom

  • *Contact author: matthew.durey@glasgow.ac.uk
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Phys. Rev. Fluids 10, 124801 – Published 3 December, 2025

DOI: https://doi.org/10.1103/m657-9blf

Abstract

We present the results of a theoretical investigation into the existence and evolution of resonant triads of two-layer gravity waves confined to a cylinder of finite height. We consider a stable stratification, for which the lighter fluid lies above the heavier fluid, with the two fluids being separated by an interface and bounded above and below by rigid horizontal lids. We demonstrate that the lateral confinement can excite resonant three-wave interactions, resulting in pronounced interfacial sloshing and highly efficient energy exchange. Such interactions are absent in unbounded domains or rectangular cylinders and consist of a single vertical mode and a single interface. For a cylinder of arbitrary cross section, we prove necessary and sufficient conditions for three spatially correlated wave modes to resonate, with the existence of triads depending critically on the depth and density of each fluid. Upon deriving amplitude equations governing the long-time evolution of a resonant triad, we show that the triad evolution is generally periodic and determine conditions for which energy exchange is most propitious. We also identify a criticality condition exclusive to multilayer flows that describes the decoupling of the triad interaction. Finally, we discuss the implications of our findings on the formation of resonant internal wave interactions in geophysical basins.

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Phys. Rev. Fluids 10, 124801– Published 3 December, 2025
  • Received 10 July 2025
  • Accepted 24 October 2025
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