Author’s summary: Approximately valid for large values of the time \(t\), a formal solution to the Hopf \(\Phi\) equation is obtained here as an asymptotic power series in \(t^{-1}\). This approximate solution is directly applicable to grid-generated isotropic homogeneous turbulence at large Reynolds numbers during the initial (inertial-force dominated) period of decay; thus, the solution accounts for the observed \(t^{-1}\) decay law and the fact that the longitudinal correlation function \(f\) is independent of \(t\). It is observed that the longitudinal correlation function measured by Frenkiel, Klebanoff, and Huang is consistent with the theoretical asymptotic behavior \(f = (\text{const})r^{-3}\) as \(r\to\infty\) and fitted by the expression \(f = [1+0.770(r/M)]^{-3}\), where \(M\) is the grid mesh length and the separation distance \(r\) is greater than the Taylor microscale \((10vt)^{1/2}\). Interestingly enough, this form for the longitudinal correlation function is shown to be derivable from a variational principle.