In meteorological terminology within dynamics, there are usually one of two unique assumptions made:
- Baroclinic- density is a function of both temperature and pressure : $\rho(x,y,z,t)=\frac{P(x,y,z,t)}{R_dT(x,y,z,t)}$
- Barotropic- density is a function of only pressure : $\rho(x,y,z,t)=\frac{P(x,y,z,t)}{R_dT_0}$
However, these simple, yet often used and powerful definitions miss something important- the effect of water vapor. To get around this, often meteorologists make virtual temperature, that is $$T_v(x,y,z,t)=T(x,y,z,t)[1+\epsilon q(x,y,z,t) ]$$ where $q$ is the mixing ratio and $\epsilon$ is the ratio of the molar mass of water vapor to the molar mass of dry air.
Incorporating this into the definition of baroclinic is easy: $\rho(x,y,z,t)=\frac{P(x,y,z,t)}{R_dT_v(x,y,z,t)}$
Incorporating this into a barotropic assumption however, is difficult. What would the approach be? I would consider this important conceptually for the tropics, where the barotropic assumption is most often made, but is mostly covered in water.
