Inverse cascade and intermittency of passive scalar in 1d smooth flow

Random advection of Lagrangian tracer scalar field $\theta (t,x)$ 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 Gaussianity known to be present at small scales for incompressible velocity fields emerges here at large scales ($x\gg L$). These scales are shown to be excited by an inverse cascade of $\theta ^{2}$ and the probability distribution function (PDF) of the corresponding scalar differences to approach the Gaussian form, as larger and larger scales are considered. Small scales ($x\ll L$) statistics is shown to be strongly non-Gaussian. Collapse of scaling exponents for scalar structure functions\thinspace takes place: moments of order $p\ge 1$ scale all linearly, independently of the order $p$. Smooth scaling $x^{p}$ is found for $-1<p<1$. Tails of scalar differences 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 scalar gradients PDF with respect to the inverse of the P\'{e}clet 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.