Multiscale phenomena in turbulent wind–wave flows at low Reynolds numbers

The low Reynolds number solution of the wind–wave interaction problem is found in Cimarelli et al. (2023 J. Fluid Mech. vol. 956, A13), to be characterised by a skewed pattern of small-elevation waves on the bottom of a turbulent wind where drag reduction is caused by a wave-induced Stokes sublayer. The inhomogeneous, anisotropic and multiscale phenomena at the basis of this interesting solution are analysed here by means of the generalised Kolmogorov equation. It is found that the large and coherent structures populating the wind are the result of an upward shift of the self-sustaining production mechanisms of turbulence and of intense reverse energy cascade phenomena. The upward shift of production and the intensification of the reverse cascade are recognised to be the result of a periodically distributed pumping of scale energy induced by the pressure field associated with the wave-induced Stokes sublayer. The low dissipative nature of the wind–wave interface region is also investigated and is found to be related to a layering effect generated by the simultaneous presence of wave-induced pressure fluctuations and of wind-induced velocity fluctuations that interact with each other in an incoherent manner. Finally, the theoretical framework provided by the generalised Kolmogorov equation is also used to rigorously define two relevant cross-over scales for the filtering formalism, the shear scale identifying the energy-containing motion and the split energy cascade scale identifying the cross-over between forward and backward cascades. Well- defined quantitative criteria for the definition of spatial resolution and for the selection of turbulence closures in coarse-grained approaches to the wind–wave problem are provided.