Anomalous scaling in two models of passive scalar advection: Effects of anisotropy and compressibility - art. no. 036302

The problem of the effects of compressibility and large-scale anisotropy on anomalous scaling behavior is considered for two models describing passive advection of scalar density and tracer fields. The advecting velocity fiel d is Gaussian, delta correlated in time, and scales with a positive exponen t epsilon. Explicit inertial-range expressions for the scalar correlation f unctions are obtained; they are represented by superpositions of power laws with nonuniversal amplitudes and universal anomalous exponents (dependent only on epsilon and alpha, the compressibility parameter). The complete set of anomalous exponents for the pair correlation functions is found nonpert urbatively, in any space dimension d, using the zero-mode technique. For hi gher-order correlation functions, the anomalous exponents are calculated to O(epsilon (2)) using the renormalization group. As in the incompressible c ase, the exponents exhibit a hierarchy related to the degree of anisotropy: the leading contributions to the even correlation functions are given by t he exponents from the isotropic shell, in agreement with the idea of restor ed small-scale isotropy. As the degree of compressibility increases, the co rrections become closer to the leading terms. The small-scale anisotropy re veals itself in the odd ratios of correlation functions: the skewness facto r slowly decreases going down to small scales for the incompressible case, but starts to increase if alpha is large enough. The higher odd dimensionle ss ratios (hyperskewness, etc.) increase, thus signaling persistent small-s cale anisotropy; this effect becomes more pronounced for larger values of a lpha.