= Potential-energy cost of homogenizing a linear stratification
{title2=$\Delta P=\rho_0\mathcal A N^2h^3/12$}
Homogenizing the top depth $h$ of a constant-gradient stable stratification over horizontal area $\mathcal A$ creates mean density $\rho_t+\rho_0N^2h/(2g)$. Comparing gravitational potential energies before and after gives $\Delta P=\rho_0\mathcal A N^2h^3/12$. The density jump at the base has <reduced gravity> $N^2h/2$. The cubic energy cost leads to a quadratic-in-depth coefficient multiplying the deepening rate.
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