= Wave-action criterion for critical-level overturning
{title2=$\cot^4\theta< Bk^3\widehat\omega/(N^3\sin^3\theta)$}
For upward <internal gravity waves>, let $B=c_{gz}N^2A_\zeta^2/\widehat\omega$ be constant. Combining $c_{gz}=(N/k)\sin\theta\cos^2\theta$ with the <monochromatic internal-wave overturning criterion> gives $\cot^4\theta<Bk^3\widehat\omega/(N^3\sin^3\theta)$. Near a <critical level> the onset balance is $\cot\theta\sim(Bk^3\widehat\omega/N^3)^{1/4}$. This is compatible with $\widehat\omega=N\cos\theta$; it fixes the overturning point rather than redefining the <dispersion relation>.
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