= Solution
Write $u^2=\langle u_x^2\rangle$ for the one-component <variance> and $Q_{LL}=u^2f(r)$. Isotropy gives $Q_{ij}=u^2[(f-g)n_in_j+g\delta_{ij}]$, and incompressibility gives $g=f+rf'/2$. Taking the trace yields
$$
C(r)=u^2[3f+rf']=\frac{u^2}{r^2}\frac{d}{dr}[r^3f(r)].
$$
Therefore, with a regular correlation at the origin,
$$
\boxed{L=4\pi u^2\lim_{r\to\infty}r^3f(r)}.
$$
For $L>0$, the <longitudinal velocity correlation> has the tail $u^2f(r)\sim L/(4\pi r^3)$. Its physical source is the velocity induced far from a localized eddy with nonzero <hydrodynamic impulse>. The <hydrodynamic Biot-Savart kernel> gives the dipolar far field
$$
\mathbf u_{\rm far}(\mathbf x)=\frac{3\mathbf n(\mathbf I\cdot\mathbf n)-\mathbf I}{4\pi r^3}+\text{higher multipoles}.
$$
The absence of a <vorticity> monopole and the surviving <impulse> dipole give the $r^{-3}$ power. Random eddy <impulses> thus produce long-range velocity correlations despite the <vorticity> itself being concentrated in localized structures. Incompressibility also explains how a distant <pressure> response can redistribute velocity without transporting local <vorticity> there.
There is no inconsistency with integrability of the trace correlation. At leading order, $g\sim-f/2$, so
$$
Q_{ij}(r)\sim\frac{L}{8\pi r^3}(3n_in_j-\delta_{ij}),\qquad
Q_{ii}(r)=0\quad\hbox{at this leading order}.
$$
The longitudinal and transverse dipolar tails cancel in the trace. The scalar <Saffman integral> can remain finite even though individual tensor components have long-range, direction-dependent tails.
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