= Critical-layer regularity of a neutral Hazel mode
For $0<k<1$, a <neutral mode of the equal-width Hazel model> behaves as $w\sim z^{1-k}$ near $z=0$. Its <derivative> diverges as $z^{-k}$, so it is not a classical continuously differentiable mode across the <critical level of an internal gravity wave>. The horizontal <velocity> is proportional to $w'/k$; its local <kinetic energy> is finite only for $k<1/2$, because $\int_0^\epsilon z^{-2k}\,dz$ converges precisely then. One possible neutral-mode convention is the boundary value from $U-c$ with $c_i\downarrow0$, which fixes the phase of the power on the negative-$z$ side. The <Miles–Howard theorem> concerns growing modes with nonreal $c$ and does not rule out such singular neutral limits.
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