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

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 beta plane with sufficiently large beta 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 Y-l(m=0) exp(i omega t) with odd l, where Y(l)(m )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 Y-l(0) exp(icot) with odd l.