Effects of turbulent mixing on the nonequilibrium critical behaviour

Abstract

We study the effects of turbulent mixing on the critical behaviour of a nonequilibrium system near its second-order phase transition between the absorbing and fluctuating states. The model describes the spreading of an agent (e.g., infectious disease) in a reaction-diffusion system and belongs to the universality class of the directed bond percolation process, also known as the simple epidemic process, and is equivalent to the Reggeon field theory. The turbulent advecting velocity field is modelled by the Obukhov–Kraichnan's rapid-change ensemble: Gaussian statistics with the correlation function ⟨vv⟩ ∝ δ(t − t′)k−d−ξ, where k is the wave number, and 0 < ξ < 2 is a free parameter. Using the field theoretic renormalization group we show that, depending on the relation between the exponent ξ and the spatial dimension d, the system reveals different types of large-scale, long-time asymptotic behaviour, associated with four possible fixed points of the renormalization group equations. In addition to known regimes (ordinary diffusion, ordinary directed percolation process and passively advected scalar field), the existence of a new nonequilibrium universality class is established, and the corresponding critical dimensions are calculated to the first order of the double expansion in ξ and ε = 4 − d (one-loop approximation). It turns out, however, that the most realistic values ξ = 4/3 (Kolmogorov's fully developed turbulence) and d = 2 or 3 correspond to the case of a passive scalar field, when the nonlinearity of the Reggeon model is irrelevant, and the spreading of the agent is completely determined by the turbulent transfer.