Scalar structure function
= Scalar structure function
{title2=$S_C(r)=\langle[C(\boldsymbol x+\boldsymbol r)-C(\boldsymbol x)]^2\rangle$}
For a homogeneous scalar, $S_C(r)=2[\langle C^2\rangle-\langle CC'\rangle]$. Decorrelation at large separation gives saturation at twice the variance. If the scalar is mean-square differentiable and isotropic, $S_C(r)=r^2\langle|\nabla C|^2\rangle/3+o(r^2)$ at small separation. Statistical isotropy and zero mean alone do not imply decorrelation at infinity.