# Given only pressure and potential equivalent temperature, can I determine other variables?

I am given a 3 box model of atmosphere where each layer is 300 mbar thick, and the potential equivalent temperature of each is given (from bottom to top - 350K, 325K, 375K). Am I able to determine the specific humidity of each layer with just this information? I am racking by brain, trying to make this work, but I feel I need more information.

In a single-fluid atmosphere without condensibles, the pressure is essentially proportional to $$P \sim \rho \, T$$ and potential temperature is essentially $$\theta \sim T P^{1-1/\gamma}$$, with the temperature $$T$$, and the adiabatic index of the gas $$\gamma$$.
So in this situation you could reconstruct density and temperature from $$P$$ and $$\theta$$, but if you add another fluid into the mix (which is your condensible), then I don't see how that should be possible.
• In other words, you can get to the same $θ_e$ on a skew-T starting at different temperatures... it could be dry air, where $θ = θ_e$... or it could be very moist air, such that $θ << θ_e$. So you can't use $θ_e$ to get either $θ$ or $r$ (moisture content), because it may be any contribution mix between them. – JeopardyTempest Oct 29 '18 at 4:23