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In meteorological terminology within dynamics, there are usually one of two unique assumptions made:

  1. Baroclinic- density is a function of both temperature and pressure : $\rho(x,y,z,t)=\frac{P(x,y,z,t)}{R_dT(x,y,z,t)}$
  2. 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.

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  • $\begingroup$ Wouldn't that remove the mathematical benefit of the barotropic assumption (that temperature doesn't vary over time/space)? What's more, systems without significant temperature gradients are physical realities... but not sure you can say the same about moisture. It's an interesting idea, though, look forward to see what more you/others have towards the thought. $\endgroup$ Feb 16, 2018 at 18:06
  • $\begingroup$ And feel like really you're in the realm of of $θ$ and $θ_e$ rather than $T$ and $T_V$ since T is always a function of height? $\endgroup$ Feb 16, 2018 at 18:08

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