On moist potential temperatures and their ability to characterize differences in the properties of air parcels

A framework is introduced to compare moist `potential' temperatures. The equivalent potential temperature, $θ_e,$ the liquid water potential temperature, $θ_\ell,$ and the entropy potential temperature, $θ_s$, are all shown to be potential temperatures, in the sense that they measure the temperatures of certain reference state systems whose entropy is the same as that of the air-parcel. They only differ in the choice of reference state composition: $θ_\ell$ describes the temperature a condensate-free state, $θ_e$ a vapor-free state, and $θ_s$ a water-free state would require to have the same entropy as the given state. Although in this sense $θ_e,$ $θ_\ell,$ and $θ_s$ are all different flavors of the same thing, only $θ_\ell$ satisfies the stricter definition of a `potential temperature', as corresponding to a reference temperature accessible by an isentropic and closed transformation of a system in equilibrium; both $θ_e$ and $θ_\ell$ measure the `relative' enthalpy of an air parcel at their respective reference states; but only $θ_s$ measures air-parcel entropy. None mix linearly, but all do so approximately, and all reduce to the dry potential temperature, $θ$ in the limit as the water mass fraction goes to zero. As is well known, $θ$ does mix linearly and inherits all the favorable (entropic, enthalpic, and potential temperature) properties of its various -- but descriptively less rich -- moist counterparts. All, involve quite complex expressions, but admit relatively simple and useful approximations. Of the three moist `potential' temperatures, $θ_s$ is the least familiar, but the most well mixed in the broader tropics, a property that merits further study as a possible basis for constraining mixing processes.

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