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

TL;DR

This paper compares different moist potential temperatures, clarifying their thermodynamic meanings and properties, and highlights the unique features of the entropy potential temperature, $ heta_s$, especially its mixing behavior in tropical regions.

Contribution

It introduces a unified framework for understanding moist potential temperatures and clarifies the thermodynamic basis and differences among $ heta_e$, $ heta_ ext{l}$, and $ heta_s$.

Findings

$ heta_ ext{l}$ satisfies the strict definition of potential temperature.

$ heta_s$ measures air-parcel entropy and is well mixed in the tropics.

All moist potential temperatures reduce to dry potential temperature as water content vanishes.

Abstract

A framework is introduced to compare moist `potential' temperatures. The equivalent potential temperature, $\theta_e,$ the liquid water potential temperature, $\theta_\ell,$ and the entropy potential temperature, $\theta_s$, 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: $\theta_\ell$ describes the temperature a condensate-free state, $\theta_e$ a vapor-free state, and $\theta_s$ a water-free state would require to have the same entropy as the given state. Although in this sense $\theta_e,$ $\theta_\ell,$ and $\theta_s$ are all different flavors of the same thing, only $\theta_\ell$ 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 $\theta_e$ and $\theta_\ell$ measure the `relative' enthalpy of an air parcel at their respective reference states; but only $\theta_s$ measures air-parcel entropy. None mix linearly, but all do so approximately, and all reduce to the dry potential temperature, $\theta$ in the limit as the water mass fraction goes to zero. As is well known, $\theta$ 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, $\theta_s$ 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.