Renormalization group analysis of models of advection of a vector admixture and a tracer field by a compressible turbulent flow

Using a quantum field theory renormalization group, we consider models of advection of a vector field and a tracer field by a compressible turbulent flow. Both advected fields are considered passive, i.e., they do not have a backward influence on the fluid dynamics. The velocity field is generated by the stochastic Navier–Stokes equation. We consider the model in the vicinity of the special space dimension $d=4$. Analysis of the model in the vicinity of this dimension allows constructing a double expansion in the parameters $y$ (related to the correlator of the random force for the velocity field) and $\varepsilon=4-d$. We show that in the framework of the one-loop approximation, the two models have similar scaling behavior, i.e., similar behavior of the correlation and structure functions in the inertial range. We calculate all critical dimensions, in particular, of tensor composite operators, in the leading order of the double expansion in $y$ and $\varepsilon$.