Anomalous scaling in two models of passive scalar advection: Effects of anisotropy and compressibility
Abstract
The problem of the effects of compressibility and largescale 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, δ correlated in time, and scales with a positive exponent ɛ. Explicit inertialrange expressions for the scalar correlation functions are obtained; they are represented by superpositions of power laws with nonuniversal amplitudes and universal anomalous exponents (dependent only on ɛ and α, the compressibility parameter). The complete set of anomalous exponents for the pair correlation functions is found nonperturbatively, in any space dimension d, using the zeromode technique. For higherorder correlation functions, the anomalous exponents are calculated to O(ɛ^{2}) using the renormalization group. As 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 smallscale isotropy. As the degree of compressibility increases, the corrections become closer to the leading terms. The smallscale anisotropy reveals itself in the odd ratios of correlation functions: the skewness factor slowly decreases going down to small scales for the incompressible case, but starts to increase if α is large enough. The higher odd dimensionless ratios (hyperskewness, etc.) increase, thus signaling persistent smallscale anisotropy; this effect becomes more pronounced for larger values of α.
 Publication:

Physical Review E
 Pub Date:
 March 2001
 DOI:
 10.1103/PhysRevE.63.036302
 arXiv:
 arXiv:nlin/0010029
 Bibcode:
 2001PhRvE..63c6302A
 Keywords:

 47.27.Te;
 05.10.Cc;
 Renormalization group methods;
 Nonlinear Sciences  Chaotic Dynamics
 EPrint:
 9 pages, 3 figures