 Wave-action criterion for critical-level overturning

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