= Solution
Let $\mathcal E=\langle|\mathbf u|^2\rangle/2$ be the turbulent <kinetic energy> per unit mass. In addition to $L>0$ and high <Reynolds number>, use the usual self-preserving large-eddy decay assumption: the normalized large-scale correlation has a fixed shape, so
$$
L=C_L\mathcal E\ell^3,\qquad \ell=\left(\frac{L}{C_L\mathcal E}\right)^{1/3},
$$
where $C_L>0$ is constant. For freely decaying high-Reynolds-number <turbulence>, the cascade <viscous dissipation> law is $\epsilon=C_\epsilon\mathcal E^{3/2}/\ell$. The kinetic-energy balance therefore gives
$$
\frac{d\mathcal E}{dt}=-C_\epsilon C_L^{1/3}L^{-1/3}\mathcal E^{11/6}.
$$
Put $A=C_\epsilon C_L^{1/3}L^{-1/3}$. Integrating,
$$
\boxed{\mathcal E(t)=\left[\mathcal E(t_0)^{-5/6}+\frac56A(t-t_0)\right]^{-6/5}},
\qquad \boxed{\mathcal E\propto(t-t_*)^{-6/5},\quad \ell\propto(t-t_*)^{2/5}}.
$$
The constant $t_*$ is a virtual time origin. This is the <Saffman decay law>. The energy decreases while the <integral scale of turbulence> grows so that the conserved large-volume <momentum> <variance> remains constant.
The exponent does not follow from the invariant alone: it also uses unforced decay, one evolving large-eddy length and an asymptotically constant cascade <viscous dissipation> coefficient. A changing normalized spectral shape or a confining boundary can invalidate that closure. Moreover $\mathrm{Re}\sim\mathcal E^{1/2}\ell/\nu\propto(t-t_*)^{-1/5}$, so at fixed <kinematic viscosity> the high-Reynolds-number decay regime eventually ends; the $-6/5$ law is not a claim about the ultimate viscous limit.
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