INVERSE CASCADE AND INTERMITTENCY OF PASSIVE SCALAR IN ONE-DIMENSIONAL SMOOTH FLOW

Random advection of a Lagrangian tracer scalar field theta(t,x) by a o ne-dimensional, spatially smooth and short-correlated in time velocity field is considered. Scalar fluctuations are maintained by a source c oncentrated at the integral scale L. The statistical properties of bot h scalar differences and the dissipation field are analytically determ ined, exploiting the dynamical formulation of the model. The Gaussian statistics known to be present at small scales for incompressible velo city fields emerges here at large scales (x much greater than L). Thes e scales are shown to be excited by an inverse cascade of theta(2) and the probability distribution function (PDF) of the corresponding scal ar differences to approach the Gaussian form, as larger and larger sca les are considered. Small-scale (x much less than L) statistics is sho wn to be strongly non-Gaussian. A collapse of scaling exponents for sc alar structure functions takes place: Moments of order p greater than or equal to 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 w hen smaller and smaller ratios x/L are considered. The same tendency i s present for the scalar gradient PDF with respect to the inverse of t he Peclet number (the pumping-to-diffusion scale ratio). The tails of the latter PDF are, however, much more extended, decaying as a stretch ed exponential of exponent 2/3, smaller than unity. This slower decay is physically associated with the strong fluctuations of the dynamical dissipative scale.