Small-Scale Isotropy and Ramp-Cliff Structures in Scalar Turbulence

Passive scalars advected by three-dimensional Navier-Stokes turbulence exhibit a fundamental anomaly in odd-order moments because of the characteristic ramp-cliff structures, violating small-scale isotropy. We use data from direct numerical simulations with grid resolution up to $8192^3$ at high P\'eclet numbers understand this as scalar diffusivity, $D$, diminishes, or Schmidt number, $Sc = \nu/D$, increases; here $\nu$ is kinematic viscosity fluid. The microscale Reynolds number varies 140 650 and $Sc$ 1 512. A simple model for structures shown characterize derivative statistics extremely well. It accurately captures how isotropy restored large-$Sc$ limit, additionally suggests slight correction Batchelor length scale relevant smallest field.