We discuss some consequences of assuming that two different non-linear model equations, and real turbulence are nearly Gaussian. It is supposed when necessary that the process is driven and it is supposed that the processes have become statistically stationary. These problems are discussed from the viewpoint of the Wiener–Hermite expansion for non-linear, nearly Gaussian processes. Expected equilibria forms are related to corresponding expressions obtained from the zero-fourth-cumulant assumption. The spectrum for Burgers’ model and for incompressible fluid flow problems is found from this viewpoint to be E ∼ k−2. The kinematical properties leading to such spectra are discussed. It is noted, as has been remarked earlier, that this spectrum is characteristic of flows with near discontinuities. A conjecture is offered concerning how these discontinuities are related to Gaussianity.