Kolmogorov decay law ($u^2\propto (t-t_{\ast }{)}^{-10/7},\ell \propto (t-t_{\ast }{)}^{2/7}$)

Assume a nonzero invariant Loitsyansky integral, self-preserving large-scale correlations, and high-Reynolds number freely decaying isotropic turbulence. Then $I\propto u^2{\ell }^5$ and $-d(u^2)/dt\propto u^3/\ell$ imply the displayed decay and growth laws. The assumptions on correlation shape and the dissipation coefficient are needed in addition to conservation of the integral.