that's interesting because my understanding was that a lot of models used pressure as the vertical (height) coordinate and with tracking temperature at any given coordinate lets you know the density of the air at any point.
That's typical of models that use the anelastic approximation, where it's useful for a number of reasons to rewrite the equations replacing the true vertical with a vertically stratified variable. I've seen density, pressure and temperature used.
That's less of a different model and more a different way to rewrite the equations to make them easier to analyse or simulate.
We might be talking at slightly different angles here. There's a strong difference in the equations between compressibity of the fluid due to compression and changes in density due to temperature, chemical concentration, etc. The term compressibility usually refers to the first usage, and modelling it leads to sound waves in the system and has major implications for how the system is simulated, I mean it's an entirely different class of algorithms. The second, where density still changes but not due to compression, so no sound waves, that can be easily modelled without including full compressibility. This allows (generally simpler) incompressible models to still incorporate useful thermal physics where important, like in climate and weather. Also, the smaller the scale of the system the more compressibility matters so I wouldn't be surprised if compressibility starts to matter for e.g. Tornados. But I'm not certain on that...
That's less of a different model and more a different way to rewrite the equations to make them easier to analyse or simulate.
We might be talking at slightly different angles here. There's a strong difference in the equations between compressibity of the fluid due to compression and changes in density due to temperature, chemical concentration, etc. The term compressibility usually refers to the first usage, and modelling it leads to sound waves in the system and has major implications for how the system is simulated, I mean it's an entirely different class of algorithms. The second, where density still changes but not due to compression, so no sound waves, that can be easily modelled without including full compressibility. This allows (generally simpler) incompressible models to still incorporate useful thermal physics where important, like in climate and weather. Also, the smaller the scale of the system the more compressibility matters so I wouldn't be surprised if compressibility starts to matter for e.g. Tornados. But I'm not certain on that...