Inverse cascade and intermittency of passive scalar in one-dimensional smooth flow

Random advection of a Lagrangian tracer scalar field θ(t,x) by a one-dimensional, spatially smooth and short-correlated in time velocity field is considered. Scalar fluctuations are maintained by a source concentrated at the integral scale L. The statistical properties of both scalar differences and the dissipation field are analytically determined, exploiting the dynamical formulation of the model. The Gaussian statistics known to be present at small scales for incompressible velocity fields emerges here at large scales (x>>L). These scales are shown to be excited by an inverse cascade of θ² and the probability distribution function (PDF) of the corresponding scalar differences to approach the Gaussian form, as larger and larger scales are considered. Small-scale (x<<L) statistics is shown to be strongly non-Gaussian. A collapse of scaling exponents for scalar structure functions takes place: Moments of order p>=1 all scale linearly, independently of the order p. Smooth scaling x^p is found for -1<p<1. Tails of the scalar difference PDF are exponential, while at the center a cusped shape tends to develop when smaller and smaller ratios x/L are considered. The same tendency is present for the scalar gradient PDF with respect to the inverse of the Péclet number (the pumping-to-diffusion scale ratio). The tails of the latter PDF are, however, much more extended, decaying as a stretched exponential of exponent 2/3, smaller than unity. This slower decay is physically associated with the strong fluctuations of the dynamical dissipative scale.