Quantifying air–water turbulence with moment field equations

Energy transfer in turbulent fluids is non-Gaussian. We quantify non-Gaussian energy transfer between the atmosphere and bodies of water using a turbulent diffusion operator coupled with temporally self-affine velocity distributions and a recursive integration method that produce multifractal measures. The measures serve as input to a system of moment field equations (derived from Navier–Stokes) that generate and track high-frequency gravity waves that propagate through the water surface (as a result of the air–water interactions). The dimension of the support of the air–water turbulence produced by our methods falls within the range of theory and observation, and correspondingly, hindcast statistical measures of the water-wave surface such as significant water-wave height and wave period are well correlated to observational buoy data. Further, our recursive integration method can be used by spectral resolving phase-averaged models to interpolate temporal wind data to smaller scales to capture the non-Gaussian behaviour of the air–water interaction.