Intermittency of passive-scalar decay: Strange eigenmodes in random shear flows

Abstract

The decay of the concentration of a passive scalar released in a spatially periodic shear flow with random time dependence is examined. Periodic boundary conditions are assumed, placing the problem in the strange-eigenmode regime where the concentration decay is exponential in the long-time limit. The focus is on the limit of small diffusivity κ⪡1 (large Péclet number), which is studied using a combination of asymptotic methods and numerical simulations. Two specific flows are considered: both have a sinusoidal velocity profile, but the random function of time is either (i) the amplitude of the sinusoid or (ii) its phase. The behavior of the passive scalar in each flow is very different. The decay rate (or Lyapunov exponent) λ, in particular, which characterizes the long-time decay in almost all flow realizations, scales as κ2∕3 in (i) and κ3∕8 in (ii). The temporal intermittency of the scalar decay, associated with fluctuations in the speed of decay, is examined in detail. It is quantified by comparing the decay rate λ with the decay rates γp of the ensemble-averaged pth moment of the concentration. The two flows exhibit some intermittency, with γp≠pλ. It is, however, much weaker for flow (i), where the γp and λ satisfy κ2∕3 power laws, than for flow (ii), where the γp are proportional to κ1∕2 and are therefore asymptotically smaller than λ. The results for flow (ii) highlight the possible difficulty in relating the behavior of the passive scalar in single flow realizations to predictions made for ensemble-averaged quantities such as concentration moments.