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The source supplies the kinematic buoyancy flux $B_0=F_0/{\rho }_0$. The unstratified line plume velocity scale is $B_0^{1/3}$. Comparing this with the stratification time scale $N_0^{-1}$ gives the stratified line-plume height scaleIf $z_s\gg H$, the initial plume is only weakly affected by the stable density stratification and reaches the ceiling much like an unstratified wall plume. If $z_s\sim H$, its buoyancy flux falls substantially during the rise; it approaches neutral buoyancy near the upper room, overshoots because of its momentum flux, and spreads as a horizontal buoyant intrusion. If $z_s\ll H$, the plume reaches neutral buoyancy low in the room and forms a low intrusion after a modest overshoot. In each sketch the plume widens by entrainment, while increasing $N_0$ lowers the neutral-buoyancy and overshoot heights.

The density anomaly of water makes mass density increase with temperature up to $T_m$ and decrease above $T_m$. Since temperature increases with depth in the stated layer, the density profile first increases to its maximum ${\rho }_m$ at the $T=T_m$ isotherm and then decreases toward the warm bottom.

The upper region has denser water below lighter water and hence stable density stratification. Below the density maximum, density decreases downward, so lighter water lies beneath denser water and gives unstable density stratification. The lower region therefore overturns by thermal convection, while the cold upper region remains a stagnant cap. This coexistence is penetrative convection.

For an inviscid Boussinesq approximation fluid in a nonrotating frame, write buoyancy as $\sigma$ and kinematic pressure as $p$. The governing equations areThey require density variations to be small compared with a constant reference density, while retaining those variations in buoyancy; the flow scale must be small compared with the background density scale height. Ideal flow additionally neglects viscosity and scalar diffusion.

Linearize aboutwhere $N$ is the buoyancy frequency, and defineFor two-dimensional disturbances, the linear equations areThe condition $N^2>0$ expresses stable density stratification.