Tracer density discontinuities in turbulent flows: simple model and scaling laws

Mixing in fully developed incompressible turbulent flows is known to lead to a cascade of discontinuity fronts of passive scalar fields. A one-dimensional (1D) variant of Baker's map is developed, capturing the main mechanism responsible for the emergence of these discontinuities. For this 1D model, expressions for the height-distribution function of the discontinuity fronts and structure function scaling exponents $\zeta_p$ are derived [for Kolmogorov turbulence, $\zeta_p=\frac 23\log_3(p+1)$]. These analytic findings are in a good agreement with both our 1D simulations, and the results of earlier numerical and experimental studies.