Turbulent Decay of a Passive Scalar in Batchelor Limit: Exact Results from a Quantum-Mechanical Approach

We show that the decay of a passive scalar θ advected by a random incompressible flow with zero correlation time in Batchelor limit can be mapped exactly to a certain quantum-mechanical system with a finite number of degrees of freedom. The Schrödinger equation is derived and its solution is analyzed for the case when at the beginning the scalar has Gaussian statistics with correlation function of the form e−|x−y| 2 . Any equal-time correlation function of the scalar can be expressed via the solution to the Schrödinger equation in a closed algebraic form. We find that the scalar is intermittent during its decay and the average of |θ|α (assuming zero mean value of θ) falls as e−γαDt at large t, where D is a parameter of the flow, γα = α(6− α)/4 for 0 < α < 3, and γα = 9/4 for α ≥ 3, independent of α. Typeset using REVTEX †Email address: [email protected] ∗This work is supported in part by funds provided by the U.S. Department of Energy (D.O.E.) under cooperative research agreement #DF-FC02-94ER40818. The problem of advection of a passive scalar (e.g., temperature or density of a pollutant) by a random flow has attracted considerable interest recently [1–5]. In many cases, the steady state of the scalar exhibits the phenomenon of intermittency, i.e. violation of Gaussianity, which, while being simpler than the Navier–Stokes intermittency [6], might well be the key for the understanding of the latter. In this paper, we consider the problem of turbulent decay of a passive scalar, i.e. its advection without external injection, in the limit of very large correlation length of the velocity (the Batchelor limit.) We show that the problem can be reduced to the quantum mechanics of a system with a finite number of degrees of freedom, which allows for the exact computation of any equal-time correlation function of the scalar. For a particular, yet physically interesting, choice of the initial condition, the number of relevant degrees of freedom of the effective quantum theory is only three, and one can easily find all intermittency properties of the scalar during the decay. The method does not rely on any phenomenological model or uncontrollable approximation. The problem is to find statistical properties of a scalar θ satisfying the equation [7] ∂tθ + vi∂iθ = κ∆θ (1) where vi is a Gaussian random field which is white in time, 〈vi(t,x)vj(t ,y)〉 = δ(t− t)fij(r) (2) where r = |x − y|. In this paper we will consider only the case of three spatial dimensions. For incompressible flows, ∂ifij = 0. In the Batchelor limit [5,8], which corresponds to the viscous-convective range in real turbulence at high Prandtl numbers, r is much smaller than the correlation length of v, and fij(r) = V δij −D(2δijr 2 − rirj) (3)