Density stratification

Density stratification is variation of a fluid's mass density with height. Its stable or unstable character depends on whether the buoyancy force restores or amplifies a displaced parcel.

The density gradient is the gradient of a spatial mass density field. With an upward vertical coordinate in the Boussinesq approximation, the squared buoyancy frequency is $N^2=-(g/{\rho }_{\ast }){\partial }_z{\rho }_0$. A negative vertical density gradient therefore gives stable density stratification.

A layer with a pronounced vertical change in salinity is a halocline. When relatively fresh surface water lies over saltier water, its contribution to density stratification inhibits vertical exchange. A cold Arctic halocline can separate sea ice from warmer Atlantic water; increased heat at depth does not imply an equally increased ocean heat flux at the ice unless the intervening exchange also changes.

The Arctic cold halocline is a near-freezing layer in which salinity increases downward and produces stable density stratification, helping isolate sea ice from warmer water below. It can be maintained through shelf-water input, cooling and limited winter convection of inflowing waters, followed by freshwater capping and advection. Its formation cannot be inferred from shelf area alone: water-mass volumes, salt budgets and residence times matter.

The ocean mixed layer is a near-surface layer with relatively weak vertical variations of temperature and mass density, maintained by mixing. Beneath it, stable density stratification can isolate colder water from recent surface heating. The mixed-layer depth is not necessarily the full local water depth.

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.