 Solution

A continuous non-turbulent edge should have vanishing $k$, $ϵ$, characteristic turbulent length $L$ and turnover frequency $\omega$. With $p,q>0$ and $\sigma <2$, the length in part (c) vanishes exactly when $\sigma <3/2$, while the frequency vanishes exactly when $\sigma >1$. Hence the requested formal consistency interval isAt the endpoints one of these characteristic quantities remains finite rather than vanishing. This is consistency of the ideal turbulent closure and its edge limits, not a claim of validity arbitrarily close to $x=0$ in a real fluid: ${\nu }_T\to 0$ eventually makes neglected molecular viscosity important. Nor does it establish the stronger local scale separation $L\ll x$; the displayed solution has $L/x\to \infty$, another reason to treat the exact edge as a formal model limit.