Though highly impacting our lives, rotating turbulent flows are not well understood. These anisotropic three-dimensional fluctuating flows are governed by different nonlinear processes, each of which can be dominant in a different range of parameters. More than 20 years ago, Galtier used weak wave turbulence theory (WTT) to derive explicit predictions for the energy spectrum of rotating turbulence. The spectrum is an outcome of forward energy transfer by inertial waves, the linear modes of rotating fluid systems. This spectrum has not yet been observed in the presence of significant quasi-two-dimensional geostrophic modes. In this Letter, we show that the predicted WTT field spectra do exist in steady rotating turbulence, alongside with the more energetic quasi-two-dimensional turbulent field. By removing the 2D component from the steady state velocity field, we show that the residual three-dimensional field consists of inertial waves and agrees with WTT spectra predictions. Our analysis examines the dependence of the energy spectrum on all four relevant parameters and provides limits within which WTT predictions hold. These results provide a solid basis for new theoretical and experimental works focused on the coexistence of the quasi-2D field and the inertial waves field and on their interactions.

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• Nonlinear dynamics in fluids
• Rotating turbulence
• Turbulence
• Wave-turbulence interactions
• Weak turbulence