Comparing the two-dimensional cascades of vorticity and a passive scalar

We compare two-dimensional vorticity and passive scalar cascades seen as a gradient enhancement process. Our criteria are based on conditional averages of the first and second Lagrangian derivatives of vorticity and passive scalar gradients in relation to the local flow geometry. In order to interpret these criteria, transient properties are derived for random vorticity and scalar fields, showing that the second-order Lagrangian derivatives of vorticity and passive scalar gradients may behave differently. Cascades obtained in numerical simulations of decaying and forced incompressible turbulence are analysed. First-order analysis reveals that the direct cascade in elliptic domains is more efficient than previously suspected. While several first-order diagnostics collapse to a single curve for vorticity and passive scalars, second-order diagnostics consistently show that the vorticity gradient exhibits faster temporal fluctuations than the passive scalar gradient, a property which we anticipate qualitatively in the study of random fields.