Fluctuation effects on 3D Lagrangian mean and Eulerian mean fluid motion

We formulate equations for the slow time dynamics of fluid motion that self consistently account for the effects of the variability upon the mean. The time-average effects of the fluctuations introduce nonlinear dispersion that acts to spatially smooth the transport velocity of the mean flow relative to its circulation or momentum velocity, by the inversion of a Helmholtz operator whose length scale corresponds to the covariance of the fluctuations.