Eulerian and Lagrangian statistics in an exactly solvable turbulent shear model with a random background mean

We study the Lagrangian statistics of passively advected particles in an elementary velocity model for turbulent shear. The stochastic velocity model is exactly solvable and includes features that highlight the important differences between Lagrangian and Eulerian velocity statistics, which are not equal in the present context. A major element of the velocity model is the presence of a random, spatially uniform background mean, which is superimposed on a turbulent shear with a spectrum that typically follows a power law. We directly solve for the Eulerian and Lagrangian statistics and show how the sweeping motion of the background mean affects the Lagrangian velocity statistics with faster decaying correlations that oscillate more rapidly compared to the Eulerian velocity. This arises due to interaction of the cross-sweeps of the mean flow with the shear component, which determines Lagrangian tracer transport rates. We derive explicit expressions for the tracer dispersion that demonstrate how the dispersion rate depends on model parameters. We validate the predictions with numerical experiments in various test regimes that also highlight the behavior of Lagrangian particles in space. The proposed exactly solvable model serves as a test problem for Eulerian spectral recovery via Lagrangian data assimilation and parameter estimation methods.