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

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 $ω$, magnetic field ${\bf b}$ and the current ${\bf j}$, the vectors ${\bf u } \times {\boldsymbol ω}$, ${\bf u} \times {\bf b}$ 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.