Three-wave resonant interactions and zonal flows in two-dimensional Rossby-Haurwitz wave turbulence on a rotating sphere

Kiori Obuse and Michio Yamada
Phys. Rev. Fluids 4, 024601 – Published 1 February 2019

Abstract

This paper addresses three-wave resonant interactions of Rossby-Haurwitz waves in two-dimensional turbulence on a rotating sphere. Zonal modes are often omitted from the “resonant wave set” even when they satisfy the conditions for three-wave resonant interactions, as they do not transfer any energy to other modes in a resonant manner. However, the presence of zonal flows induces phase shifts in other modes, and it is not at all clear that their influence is negligible. Since it is expected that three-wave resonant interactions govern the entire dynamics of turbulence if the rotation rate of the sphere is sufficiently high, by analogy with the theorem regarding three-wave resonant interactions of Rossby waves on a β plane with sufficiently large β previously proven by Yamada and Yoneda [Physica D 245, 1 (2013)], an appropriate definition of the resonant wave set was determined by comparing the time evolution of several wave sets on a rapidly rotating sphere. It was found that zonal waves of the form Ylm=0exp(iωt) with odd l, where Ylm are the spherical harmonics, should be considered for inclusion in the resonant wave set to ensure that the dynamics of the resonant wave set determine the overall dynamics of the turbulence on a rapidly rotating sphere. Consequently, it is suggested that the minimal resonant wave set that must be considered in the discussion of the three-wave interaction of Rossby-Haurwitz waves is the set consisting of nonzonal resonant waves and zonal waves of the form Yl0exp(iωt) with odd l.

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  • Received 30 October 2017

DOI:https://doi.org/10.1103/PhysRevFluids.4.024601

©2019 American Physical Society

Physics Subject Headings (PhySH)

Fluid Dynamics

Authors & Affiliations

Kiori Obuse1 and Michio Yamada2

  • 1Graduate School of Environmental and Life Science, Okayama University, Okayama, 700-8530, Japan
  • 2Research Institute for Mathematical Sciences, Kyoto University, Kyoto, 606-8502, Japan

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Issue

Vol. 4, Iss. 2 — February 2019

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