General second-rank correlation tensors for homogeneous magnetohydrodynamic turbulence

The properties and structure of second-order (Cartesian) correlation tensors are derived for the general case of two solenoidal random vector fields. The theory is intended to describe homogeneous magnetohydrodynamic turbulence, with no assumed rotational or reflectional symmetries. Each correlation tensor can be written in terms of four scalar generating functions and the relationship of these functions to the potentials that generate the poloidal and toroidal components of the underlying vector fields is derived. The physical nature of the scalar functions is investigated and their true or pseudoscalar character is ascertained. In our general discussion we clarify several misleading statements dating back to Robertson's original paper in the field [Proc. Camb. Philos. Sec. 36, 209 (1940)]. It is also shown that using the one-dimensional correlation function, it is possible to obtain spectral information on the induced electric field in directions perpendicular to the measurement direction.