= Monochromatic internal-wave overturning criterion
{title2=$|mA_\zeta|>1$}
For a <plane internal gravity wave> with vertical displacement $\zeta$, linear density advection gives $\rho'=-\bar\rho_z\zeta$. In a constant stable gradient, $\rho_z=\bar\rho_z(1-\zeta_z)$, so static overturning begins when the maximum of $\zeta_z$ exceeds one. A monochromatic displacement has maximum $|mA_\zeta|$. The criterion predicts the failure of the small-amplitude field, rather than a valid continuation of linear theory beyond overturning.
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