An alternative formulation for exact scaling relations in hydrodynamic and magnetohydrodynamic turbulence

TL;DR

This paper introduces a new formulation for exact turbulence relations using two-point statistics, simplifying the expression of cascade rates and aiding the study of anisotropic space plasmas like the solar wind.

Contribution

It presents an alternative, simpler algebraic formulation for turbulence cascade relations involving mixed second-order structure functions and additional key vectors.

Findings

Cascade rate expressed via mixed second-order structure functions.

Key role of vectors like ${f u} imes {oldsymbol omega}$, ${f u} imes {f b}$, and ${f j} imes {f b}$.

Potential for faster statistical convergence in space plasma studies.

Abstract

We propose an alternative formulation for the exact relations in three-dimensional homogeneous turbulence using two-point statistics. Our finding is illustrated with incompressible hydrodynamic, standard and Hall magnetohydrodynamic turbulence. In this formulation, the cascade rate of an inviscid invariant of turbulence can be expressed simply in terms of mixed second-order structure functions. Besides the usual variables like the velocity ${\bf u}$, vorticity $\omega$, magnetic field ${\bf b}$ and the current ${\bf j}$, the vectors ${\bf u } \times {\boldsymbol \omega}$, ${\bf u} \times {\bf b}$ and ${\bf j} \times {\bf b}$ are also found to play a key role in the turbulent cascades. The current methodology offers a simple algebraic form which is specially interesting to study anisotropic space plasmas like the solar wind, with in principle a faster statistical convergence than the classical laws written in terms of third-order correlators.