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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.

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I think you cannot, might be wrong on that though.

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.

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  • $\begingroup$ 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. $\endgroup$ – JeopardyTempest Oct 29 '18 at 4:23
  • $\begingroup$ Is this from the same question as your other one? If so, I think you're probably on the wrong track here... see comment there, maybe you can get moving the right direction :) :) $\endgroup$ – JeopardyTempest Oct 29 '18 at 4:26

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