Effects of turbulent mixing on critical behaviour in the presence of compressibility: renormalization group analysis of two models

Abstract

Abstract. Critical behaviour of two systems, subjected to the turbulent mixing, is studied by means of the field theoretic renormalization group. The first system, described by the equilibrium model A, corresponds to relaxational dynamics of a non-conserved order parameter. The second one is the strongly non-equilibrium reaction-diffusion system, known as Gribov process and equivalent to the Reggeon field theory. The turbulent mixing is modelled by the Kazantsev-Kraichnan "rapid-change" ensemble: time-decorrelated Gaussian velocity field with the power-like spectrumEffects of compressibility of the fluid are studied. It is shown that, depending on the relation between the exponent ξ and the spatial dimension d, the both systems exhibit four different types of critical behaviour, associated with four possible fixed points of the renormalization group equations. Three fixed points correspond to known regimes: Gaussian fixed point, original model without mixing and passively advected scalar field. The most interesting fourth point corresponds to a new type of critical behaviour, in which the nonlinearity and turbulent mixing are both relevant, and the critical exponents depend on d, ξ and the degree of compressibility. The critical exponents and regions of stability for all the regimes are calculated in the leading order of the double expansion in two parameters ξ and ε = 4 − d. For the both models, compressibility enhances the role of the nonlinear terms in the dynamical equations: the region in the ε-ξ plane, where the new nontrivial regime is stable, is getting much wider as the degree of compressibility increases. For the incompressible fluid, the most realistic values d = 3 and ξ = 4/3 (Kolmogorov turbulence) lie in the region of stability of the passive scalar regime. If the compressibility becomes strong enough, the crossover in the critical behaviour occurs, and these values of d and ξ fall into the region of stability of the new regime, where the advection and the nonlinearity are both important. In its turn, turbulent transfer becomes more efficient due to combined effects of the mixing and the nonlinear terms.

Directed-bond percolation subjected to synthetic compressible velocity fluctuations: Renormalization group approach

“…The nontrivial regimes FP I 5 and FP I 6 are realized only in the nonphysical region for large values of ε. This numerical result confirms our previous expectations [23,24]. It was pointed out [41] that genuine thermal fluctuations can change IR stability of the given universality class.…”

“…In dynamical models with MSR response fields [36], all the 1-irreducible diagrams without the fieldsψ vanish, and it is sufficient to consider functions with Nψ ≥ 1. As was shown in [24] the rapidity symmetry ψ(t) → −ψ(−t),ψ → −ψ(−t) requires also inequality N ψ ≥ 1 to hold. Using these considerations together with relation (10), possible UV divergent structures are expected only for the 1PI Green functions listed in Table 2.…”

Scaling behavior in interacting systems: joint effect of anisotropy and compressibility

“…we see that equality β a = 0 holds for a = 1/2, and is valid to all orders in the perturbation theory [77].…”

“…For a non-interacting scalar field ϕ a distinction between cases a) and b) is preserved during perturbation procedure. As it has been pointed out in [77] for interacting theories the situation differs on a fundamental level. In fact, both terms ∇·(vϕ) and (v·∇)ϕ have to be included.…”

Field theory approach in kinetic reaction: Role of random sources and sinks

“…Effective action (26) is the dynamical action of the Gribov process [6], also known as the Reggeon model. The effects of random drift in this case were analyzed in [7] using the Obukhov-Kraichnan compressible velocity field.…”