Critical behaviour of a system, subjected to strongly anisotropic turbulent
mixing, is studied by means of the field theoretic renormalization group.
Specifically, relaxational stochastic dynamics of a non-conserved
multicomponent order parameter of the Ashkin-Teller-Potts model, coupled to a
random velocity field with prescribed statistics, is considered. The velocity
is taken Gaussian, white in time, with correlation function of the form
∝δ(t−t′)/∣k⊥∣d−1+ξ, where k⊥ is
the component of the wave vector, perpendicular to the distinguished direction
("direction of the flow") --- the d-dimensional generalization of the
ensemble introduced by Avellaneda and Majda [1990 {\it Commun. Math. Phys.}
{\bf 131} 381] within the context of passive scalar advection. This model can
describe a rich class of physical situations. It is shown that, depending on
the values of parameters that define self-interaction of the order parameter
and the relation between the exponent ξ and the space dimension d, the
system exhibits various types of large-scale scaling behaviour, associated with
different infrared attractive fixed points of the renormalization-group
equations. In addition to known asymptotic regimes (critical dynamics of the
Potts model and passively advected field without self-interaction), existence
of a new, non-equilibrium and strongly anisotropic, type of critical behaviour
(universality class) is established, and the corresponding critical dimensions
are calculated to the leading order of the double expansion in ξ and
ϵ=6−d (one-loop approximation). The scaling appears strongly
anisotropic in the sense that the critical dimensions related to the directions
parallel and perpendicular to the flow are essentially different.Comment: 21 page, LaTeX source, 7 eps figures. arXiv admin note: substantial
text overlap with arXiv:cond-mat/060701