Turbulent decay of a passive scalar in the Batchelor limit: Exact results from a quantum-mechanical approach

Abstract

We show that the decay of a passive scalar \ensuremath{\theta} advected by a random incompressible flow with zero correlation time in the Batchelor limit can be mapped exactly to a certain quantum-mechanical system with a finite number of degrees of freedom. The Schr\"odinger equation is derived and its solution is analyzed for the case where, at the beginning, the scalar has Gaussian statistics with correlation function of the form ${e}^{\ensuremath{-}|x\ensuremath{-}y{|}^{2}}.$ Any equal-time correlation function of the scalar can be expressed via the solution to the Schr\"odinger equation in a closed algebraic form. We find that the scalar is intermittent during its decay and the average of $|\ensuremath{\theta}{|}^{\ensuremath{\alpha}}$ (assuming zero mean value of \ensuremath{\theta}) falls as ${e}^{\ensuremath{-}{\ensuremath{\gamma}}_{\ensuremath{\alpha}}\mathrm{Dt}}$ at large $t,$ where $D$ is a parameter of the flow, ${\ensuremath{\gamma}}_{\ensuremath{\alpha}}=\frac{1}{4}\ensuremath{\alpha}(6\ensuremath{-}\ensuremath{\alpha})$ for $0<\ensuremath{\alpha}<3,$ and ${\ensuremath{\gamma}}_{\ensuremath{\alpha}}=\frac{9}{4}$ for $\ensuremath{\alpha}>~3,$ independent of \ensuremath{\alpha}.