Solenoidal isotropic spectral tensor (${\hat{C}}_{ij}=H(k)({\delta }_{ij}-k_i{k}_j/k^2)$)

For a statistically homogeneous, isotropic, reflection-symmetric solenoidal vector random field in three dimensions, the nonzero-wavevector spectral tensor is a scalar multiple of the transverse projector of a vector field. Isotropy gives $A(k){\delta }_{ij}+D(k)k_i{k}_j$, while zero divergence gives $A+D{k}^2=0$. Its trace is $2A$, giving the displayed normalization. Without reflection symmetry an additional antisymmetric, helical term can occur.