What is atmospheric stratification? In the context of atmospheric stability, what are the meanings of stable or unstable stratification? What is stratification?
 A: Stratification is density variations causing the fluids to separate to different layers.
As shown in the YouTube video "Stratified Flow -lecture":

A: You could use the buoyancy frequency, $N$, as a measure of stratification and stability. $N$ is defined as:$$
N^2 = \frac{g}{\theta} \frac{\mathrm{d}\theta}{\mathrm{d}z}
\,,$$where $\theta$ is the potential temperature.
With some variations in answers here and there. The term $\frac{\mathrm{d}\theta}{\mathrm{d}z}$ is a measure of convective stability with negative values representing convective instability for which a buoyancy frequency does not exist as an air parcel will not oscillate but instead will ascend/descend in order to produce a stable stratification.
I perhaps wonder if the moist potential temperature should be used rather than the dry potential temperature when investigating moist atmospheric processes such as cyclogenesis.
A: Stratification means that density changes with height. If it is statically stable, the density decreases with height. However, it is so simple only for incompressible fluids. For example, for laboratory flows in water.
For compressible fluids, such as the atmosphere, but to some extent even the oceans,  we have to take into account, that when a fluid parcel moves into a region with different pressure, its temperature and density changes due to adiabatic expansion and compression. That's why potential temperature  and potential density are used. They express the temperature or density of various air parcels when adiabatically expanded/compressed the same reference pressure.
If the potential density decreases with height or the potential temperature rises with height, the atmospheric stratification is statically stable. If it is constant, it is statically neutral. If potential temperature decreases with height, the stratification is statically unstable.
However, again, it is so simple only when the gradient is the same everywhere. Otherwise one has to examine the complete potential temperature profiles and see whether there is any location, where air with higher potential temperature is located under air of lower potential temperature. That would indicate static instability for a larger layer (the nonlocal static stability).
