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The concern is that the rapid-decay assumptions need not be preserved by the dynamics. In a constant-density incompressible flow, the divergence of the Navier-Stokes equation gives the pressure Poisson equation for incompressible flow, with $\pi =p/\rho$:For a localized eddy, inversion with the free-space Green function and two integrations by parts yieldAt large separation the differentiated kernel has leading form $(3n_i{n}_j-{\delta }_{ij})/r^3$. Thus an anisotropic stress moment of an individual eddy produces a quadrupolar fluid pressure of order $r^{-3}$, even if the velocity source is spatially localized. Correlating that fluid pressure with local $u_x^2$ allows $⟨u_x^2{p}^{\prime }⟩=O(r^{-3})$. Isotropy may cancel a mean multipole but does not automatically cancel such a mixed fourth-order statistic.

The triple-correlation evolution contains a pressure-gradient term of the form $-⟨u_x^2{\partial }_{x^{\prime }}{\pi }^{\prime }⟩$, so these long-range pressure correlations in turbulence can generate a longitudinal triple-correlation tail $K\sim a{r}^{-4}$. With $L=0$, part (ii) then gives $\dot{I}=8\pi u^{3}a$, which need not vanish. This is the substance of the objection to deducing exact conservation from initially short velocity correlations. Batchelor and Proudman's original long-range calculation concerned anisotropic turbulence; extending it to isotropic decay requires the appropriate higher-order correlations and allows cancellations. A permitted tail is not a proof that its coefficient stays appreciable in fully developed turbulence.

This possibility appealed to workers constructing heuristic turbulence closures. A closure that generated a changing low-wavenumber $k^4$ coefficient and a decay exponent different from $10/7$ could then be viewed as representing a pressure-mediated effect, rather than as automatically violating a universal invariant. The closure's far-field coefficient still needs physical validation; closure-generated nonconservation does not itself prove the true dynamics nonconserving.

For evidence available before this exam, the 2006 simulations of Ishida, Davidson and Kaneda found that $I$ approached an approximately constant value after a transient when the box was much larger than the integral scale of turbulence and the Reynolds number was large. The energy decay was correspondingly close to $u^2\sim t^{-10/7}$. The observed conclusion was approximate late-time conservation in that regime, indicating weak net long-range triple-correlation transfer, rather than exact conservation for every initial condition, box size or Reynolds number.