Random advection of a Lagrangian tracer scalar field [Formula Presented] 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 [Formula Presented] 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 [Formula Presented] These scales are shown to be excited by an inverse cascade of [Formula Presented] 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 [Formula Presented] statistics is shown to be strongly non-Gaussian. A collapse of scaling exponents for scalar structure functions takes place: Moments of order [Formula Presented] all scale linearly, independently of the order [Formula Presented] Smooth scaling [Formula Presented] is found for [Formula Presented] Tails of the scalar difference PDF are exponential, while at the center a cusped shape tends to develop when smaller and smaller ratios [Formula Presented] 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 [Formula Presented] smaller than unity. This slower decay is physically associated with the strong fluctuations of the dynamical dissipative scale.
|Number of pages||17|
|Journal||Physical Review E - Statistical, Nonlinear, and Soft Matter Physics|
|Publication status||Published - 1997|