Steep cliffs and saturated exponents in three dimensional scalar turbulence.

The intermittency of a passive scalar advected by three-dimensional Navier-Stokes turbulence at a Taylor-scale Reynolds number of $650$ is studied using direct numerical simulations on a $4096^3$ grid; the Schmidt number is unity. By measuring scalar increment moments of high orders, while ensuring statistical convergence, we provide unambiguous evidence that the scaling exponents saturate to $1.2$ for moment order beyond about $12$, indicating that scalar intermittency is dominated by the most singular shock-like cliffs in the scalar field. We show that the fractal dimension of the spatial support of steep cliffs is about $1.8$, whose sum with the saturation exponent value of $1.2$ adds up to the space dimension of $3$, thus demonstrating a deep connection between the geometry and statistics in turbulent scalar mixing. The anomaly for the fourth and sixth order moments is comparable to that in the Kraichnan model for the roughness exponent of $2/3$.