Anomalous scaling in two models of the passive scalar advection: Effects of anisotropy and compressibility

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 field is Gaussian, $\delta$-correlated in time, and scales with a positive exponent $\epsilon$. Explicit inertial-range expressions for the scalar correlation functions are obtained; they are represented by superpositions of power laws with nonuniversal amplitudes and universal (dependent only on $\epsilon$ and $\alpha$, the compressibility parameter) anomalous exponents. The complete set of anomalous exponents for the pair correlation functions is found nonperturbatively, in any space dimension d, using the zero-mode technique. For higher-order correlation functions, the anomalous exponents are calculated to $O(\epsilon^{2})$ using the renormalization group. Like in the incompressible case, the exponents exhibit a hierarchy related to the degree of anisotropy: the leading contributions to the even correlation functions are given by the exponents from the isotropic shell, in agreement with the idea of restored small-scale isotropy. As the degree of compressibility increases, the corrections become closer to the leading terms. The small-scale anisotropy reveals itself in the odd ratios of correlation functions: the skewness factor is slowly decreasing going down to small scales for the incompressible case, but becomes increasing if $\alpha$ is large enough. The higher odd dimensionless ratios (hyperskewness etc.) increase, thus signalling the persistent small-scale anisotropy; this effect becomes more pronounced for larger values of $\alpha$.