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Forward and inverse cascades by exact resonances in surface gravity waves

Zhou Zhang and Yulin Pan*

  • Department of Naval Architecture and Marine Engineering, University of Michigan, Ann Arbor, Michigan 48109, USA

  • *yulinpan@umich.edu
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Phys. Rev. E 106, 044213 – Published 24 October, 2022

DOI: https://doi.org/10.1103/PhysRevE.106.044213

Abstract

We study the energy transfer by exact resonances for deep-water surface gravity waves in a finite periodic spatial domain. Based on a kinematic model simulating the generation of active wave modes in a finite discrete wave number space SR, we examine the possibility of direct and inverse cascades. More specifically, we set an initially excited region which iteratively spreads energy to wave modes in SR through exact resonances. At each iteration, we first activate new modes from scale resonances (which generate modes with new lengths), then consider two bounding situations for angle resonances (which transfer energy at the same length scale): the lower bound where no angle resonance is included and the upper bound where all modes with the same length as any active mode are excited. Such a strategy is essential to enable the computation for a large domain SR with the maximum wave number R∼103. We show that for both direct and inverse cascades, the modal propagation to the boundaries of SR can be established when the initially excited region is sufficiently large; otherwise a frozen turbulence state occurs, with a sharp transition between the two regimes especially for the direct cascade. Through a study on the structure of resonant quartets, the mechanism associated with the sharp transition and the role of angular energy transfer in the cascades are elucidated.

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Phys. Rev. E 106, 044213– Published 24 October, 2022
  • Received 30 May 2022
  • Accepted 27 September 2022
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