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The problem of the effects of compressibility and large-scale anisotropy onanomalous scaling behavior is considered for two models describing passiveadvection of scalar density and tracer fields. The advecting velocity field isGaussian, $\delta$-correlated in time, and scales with a positive exponent$\epsilon$. Explicit inertial-range expressions for the scalar correlationfunctions are obtained; they are represented by superpositions of power lawswith nonuniversal amplitudes and universal (dependent only on $\epsilon$ and$\alpha$, the compressibility parameter) anomalous exponents. The complete setof anomalous exponents for the pair correlation functions is foundnonperturbatively, in any space dimension d, using the zero-mode technique. Forhigher-order correlation functions, the anomalous exponents are calculated to$O(\epsilon^{2})$ using the renormalization group. Like in the incompressiblecase, the exponents exhibit a hierarchy related to the degree of anisotropy:the leading contributions to the even correlation functions are given by theexponents from the isotropic shell, in agreement with the idea of restoredsmall-scale isotropy. As the degree of compressibility increases, thecorrections become closer to the leading terms. The small-scale anisotropyreveals itself in the odd ratios of correlation functions: the skewness factoris slowly decreasing going down to small scales for the incompressible case,but becomes increasing if $\alpha$ is large enough. The higher odddimensionless ratios (hyperskewness etc.) increase, thus signalling thepersistent small-scale anisotropy; this effect becomes more pronounced forlarger values of $\alpha$.Comment: 9 pages, 3 figure

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