Magnetic second-moment tensor ($M_{ij}=⟨B_i{B}_j⟩$)

For a resolved magnetic field, the unaveraged tensor $B_i{B}_j$ describes magnetic tension; the isotropic magnetic pressure is treated separately in the magnetohydrodynamic total pressure. Its spatial or ensemble average need not have matrix rank one. Statistical isotropy in three dimensions gives $⟨B_i{B}_j⟩=⟨B^2⟩{\delta }_{ij}/3$, by taking the trace. In Alfvén velocity units this tensor has dimensions of squared velocity.

For constant magnetic diffusivity, apply the product rule to $M_{ij}=B_i{B}_j$ and the incompressible resistive induction equation. The identity ${\nabla }^2(B_i{B}_j)=B_i{\nabla }^2{B}_j+B_j{\nabla }^2{B}_i+2{\partial }_n{B}_i{\partial }_n{B}_j$ proves the displayed formula. Omitting the gradient-product term gives the wrong tensor evolution. Its average generally requires information beyond the averaged magnetic second-moment tensor.