= Solution
The <zeroth law of turbulence> states that at fixed outer <velocity> $u$ and <integral scale of turbulence> $\ell$, the mean <viscous dissipation> per unit mass approaches a nonzero finite value as the <Reynolds number> $\mathrm{Re}=u\ell/\nu$ becomes large. In an established forward <energy cascade>, this gives $\epsilon\sim C_\epsilon u^3/\ell$, with $C_\epsilon$ of order one. The small-scale <velocity gradients> become large enough to offset decreasing <kinematic viscosity>; this is the <turbulent dissipation anomaly>.
At the <Kolmogorov microscales>, the local <Reynolds number> is of order one, $v\eta/\nu\sim1$, and the <viscous dissipation> is $\epsilon\sim\nu(v/\eta)^2$. Solving these balances gives
$$
\boxed{\eta=(\nu^3/\epsilon)^{1/4},\qquad v=(\nu\epsilon)^{1/4}}.
$$
Substituting the outer-scale estimate, and omitting order-one constants, gives
$$
\boxed{\eta\sim\ell\,\mathrm{Re}^{-3/4},\qquad v\sim u\,\mathrm{Re}^{-1/4}}.
$$
The associated <eddy turnover time> is $\eta/v=(\nu/\epsilon)^{1/2}$.
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