Scalar structure function ($S_C(r)=⟨[C(\mathbf{x}+\mathbf{r})-C(\mathbf{x}){]}^2⟩$)

For a homogeneous scalar, $S_C(r)=2[⟨C^2⟩-⟨C{C}^{\prime }⟩]$. Decorrelation at large separation gives saturation at twice the variance. If the scalar is mean-square differentiable and isotropic, $S_C(r)=r^2⟨|\nabla C{|}^2⟩/3+o(r^2)$ at small separation. Statistical isotropy and zero mean alone do not imply decorrelation at infinity.