= Solution
A continuous non-turbulent edge should have vanishing $k$, $\epsilon$, 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 is
$$
\boxed{1<\sigma<\frac32,
\qquad\sigma_k<\sigma_\epsilon<1.5\sigma_k.}
$$
At 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: $\nu_T\to0$ 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.
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