Stable density stratification

A fluid has stable density stratification when a small vertical displacement produces a restoring buoyancy force. In a gravitational field this normally means that mass density increases downward.

Equating an inertial eddy turnover time to the inverse buoyancy frequency gives the scale separating buoyancy-affected motions from a smaller approximately isotropic cascade. It is distinct from the viscous Kolmogorov microscales. In very stable local surface-layer scaling it is proportional to the Obukhov length, up to closure constants.

Homogenizing the top depth $h$ of a constant-gradient stable stratification over horizontal area $\mathcal{A}$ creates mean density ${\rho }_t+{\rho }_0{N}^{2}h/(2g)$. Comparing gravitational potential energies before and after gives $\Delta P={\rho }_0\mathcal{A}{N}^2{h}^3/12$. The density jump at the base has reduced gravity $N^{2}h/2$. The cubic energy cost leads to a quadratic-in-depth coefficient multiplying the deepening rate.

A fluid has unstable density stratification when a vertical displacement amplifies itself. Denser fluid above lighter fluid can overturn by Rayleigh-Taylor instability or develop thermal convection when the density difference is thermal.