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The influence of helicity on the stability of scaling regimes, on theeffective diffusivity, and on the anomalous scaling of structure functions of apassive scalar advected by a Gaussian solenoidal velocity field with finitecorrelation time is investigated by the field theoretic renormalization groupand operator product expansion within two-loop approximation. The influence ofhelicity on the scaling regimes is discussed and shown in the plane ofexponents $\epsilon-\eta$, where $\epsilon$ characterizes the energy spectrumof the velocity field in the inertial range $E\propto k^{1-2\epsilon}$, and$\eta$ is related to the correlation time at the wave number $k$ which isscaled as $k^{-2+\eta}$. The restrictions given by nonzero helicity on theregions with stable fixed points which correspond to the scaling regimes areanalyzed in detail. The dependence of the effective diffusivity on the helicityparameter is discussed. The anomalous exponents of the structure functions ofthe passive scalar field which define their anomalous scaling are calculatedand it is shown that although the separate composite operators which definethem strongly depend on the helicity parameter the resulting two-loopcontributions to the critical dimensions of the structure functions areindependent of helicity. Details of calculations are shown.Comment: revtex4, 28 pages, 13 figures, final version published in Phys. Rev.

physical review. e, statistical, nonlinear and soft matter physics

Influence of helicity on anomalous scaling of a passive scalar advected by the turbulent velocity field with finite correlation time: Two-loop approximation