Helical turbulent Prandtl number in the $A$ model of passive advection: Two loop approximation

TL;DR

This study uses a two-loop renormalization group approach to analyze how the turbulent Prandtl number varies with the model parameter A and spatial parity violation, revealing stability and sensitivity patterns.

Contribution

It provides the first detailed two-loop analysis of the helical turbulent Prandtl number in the A model of passive advection, highlighting the influence of the parameter A and parity violation.

Findings

A is restricted to -1.723 to 2.800 in non-helical environments.

For A = -1, the helical turbulent Prandtl number is always 1.

The turbulent Prandtl number can increase or decrease with parity violation depending on A.

Abstract

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.