Helical turbulent Prandtl number in the $A$ model of passive advection:   Two loop approximation
  Using the field theoretic renormalization group technique in the two-loop approximation, turbulent Prandtl numbers are obtained in the general $A$ model of passive vector advected by fully developed turbulent velocity field with violation of spatial parity introduced via continuous parameter $\rho$ ranging from $\rho=0$ (no violation of spatial parity) to $|\rho|=1$ (maximum violation of spatial parity). In non-helical environments, we demonstrate that $A$ is restricted to $-1.723 \leq A \leq 2.800$ (rounded on the last presented digit) due to the constraints of two-loop calculations. When $\rho >0.749$ restrictions may be removed. Furthermore, three physically important cases $A \in \{-1, 0, 1\}$ are shown to lie deep within the allowed interval of $A$ for all values of $\rho$. For the model of linearized Navier-Stokes equations ($A = -1$) up to date unknown helical values of turbulent Prandtl number have been shown to equal $1$ regardless of parity violation. Furthermore, we have shown that interaction parameter $A$ exerts strong influence on advection diffusion processes in turbulent environments with broken spatial parity. In explicit, depending on actual value of $A$ turbulent Prandtl number may increase or decrease with $\rho$. By varying $A$ continuously we explain high stability of kinematic MHD model ($A=1$) against helical effects as a result of its closeness to $A = 0.912$ (rounded on the last presented digit) case where helical effects are completely suppressed. Contrary, for the physically important $A=0$ model we show that it lies deep within the interval of models where helical effects cause the turbulent Prandtl number to decrease with $|\rho|$. We thus identify internal structure of interactions given by parameter $A$, and not the vector character of the admixture itself to be the dominant factor influencing diffusion advection processes in the helical $A$ model. 
Michal Hnati\v{c} and Peter Zalom
cond-mat.stat-mech hep-th math-ph math.MP