Long-range pressure correlations in turbulence ($⟨u_x^2{p}^{\prime }⟩=O(r^{-3})$)

The quadrupolar kernel of the pressure Poisson equation for incompressible flow allows mixed pressure-velocity correlations with an $r^{-3}$ tail. Its derivative in the evolution of a longitudinal triple correlation allows an $r^{-4}$ tail, potentially changing the Loitsyansky integral. The existence of a nonlocal kernel does not fix the tail coefficient or prevent cancellations. An isotropic mean pressure multipole can vanish while a mixed higher-order correlation remains nonzero.