RESUMO
Kraichnan's model of passive scalar advection in which the driving (Gaussian) velocity field has fast temporal decorrelation is studied as a case model for understanding the anomalous scaling behavior in the anisotropic sectors of turbulent fields. We show here that the solutions of the Kraichnan equation for the n-order correlation functions foliate into sectors that are classified by the irreducible representations of the SO(d) symmetry group. We find a discrete spectrum of universal anomalous exponents, with a different exponent characterizing the scaling behavior in every sector. Generically the correlation functions and structure functions appear as sums over all these contributions, with nonuniversal amplitudes that are determined by the anisotropic boundary conditions. The isotropic sector is always characterized by the smallest exponent, and therefore for sufficiently small scales local isotropy is always restored. The calculation of the anomalous exponents is done in two complementary ways. In the first they are obtained from the analysis of the correlation functions of gradient fields. The theory of these functions involves the control of logarithmic divergences that translate into anomalous scaling with the ratio of the inner and the outer scales appearing in the final result. In the second method we compute the exponents from the zero modes of the Kraichnan equation for the correlation functions of the scalar field itself. In this case the renormalization scale is the outer scale. The two approaches lead to the same scaling exponents for the same statistical objects, illuminating the relative role of the outer and inner scales as renormalization scales. In addition we derive exact fusion rules, which govern the small scale asymptotics of the correlation functions in all the sectors of the symmetry group and in all dimensions.
RESUMO
The statistical objects characterizing turbulence in real turbulent flows differ from those of the ideal homogeneous isotropic model. They contain contributions from various two- and three-dimensional aspects, and from the superposition of inhomogeneous and anisotropic contributions. We employ the recently introduced decomposition of statistical tensor objects into irreducible representations of the SO(3) symmetry group (characterized by j and m indices, where j=0ellipsisinfinity,-j
RESUMO
The main difficulty of statistical theories of fluid turbulence is the lack of an obvious small parameter. In this paper we show that the formerly established fusion rules can be employed to develop a theory in which Kolmogorov's statistics of 1941 (K41) acts as the zero order, or background statistics, and the anomalous corrections to the K41 scaling exponents zeta(n) of the nth-order structure functions can be computed analytically. The crux of the method consists of renormalizing a four-point interaction amplitude on the basis of the fusion rules. This amplitude includes a small dimensionless parameter, which is shown to be of the order of the anomaly of zeta(2), delta(2)=zeta(2)-2/3 approximately 0.03. Higher-order interaction amplitudes are shown to be even smaller. The corrections to K41 to O(delta(2)) result from standard logarithmically divergent ladder diagrams in which the four-point interaction acts as a "rung." The theory allows a calculation of the anomalous exponents zeta(n) in powers of the small parameter delta(2). The n dependence of the scaling exponents zeta(n) stems from pure combinatorics of the ladder diagrams. In this paper we calculate the exponents zeta(n) up to O(delta32). Previously derived bridge relations allow a calculation of the anomalous exponents of correlations of the dissipation field and of dynamical correlations in terms of the same parameter delta(2). The actual evaluation of the small parameter delta(2) from first principles requires additional developments that are outside the scope of this paper.
