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

• Published in: Journal of physics. A, Mathematical and theoretical Vol. 43; no. 40; p. 405001
• Main Authors: Antonov, N V, Kapustin, A S
• Format: Journal Article
• Language: English; Russian
• Published: Bristol IOP Publishing 08.10.2010; IOP
• Subjects: Compressibility; Dynamical systems; Dynamics; Exponents; Mathematical models; Nonlinearity; Turbulence; Turbulent mixing
• ISSN: 1751-8121; 1751-8113; 1751-8121
• DOI: 10.1088/1751-8113/43/40/405001

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 the 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 spectrum {proportional to}k-d - Delta *x. Effects of compressibility of the fluid are studied. It is shown that, depending on the relation between the exponent Delta *x and the spatial dimension d, both the 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 both nonlinearity and turbulent mixing are relevant, and the critical exponents depend on d, Delta *x 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 Delta *x and Delta *e = 4 - d. For both models, compressibility enhances the role of the nonlinear terms in the dynamical equations: the region in the Delta *e-- Delta *x 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 Delta *x = 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 Delta *x fall into the region of stability of the new regime, where both advection and the nonlinearity are important. In its turn, turbulent transfer becomes more efficient due to combined effects of the mixing and the nonlinear terms.