Reconciling the Discrepancies of Equivalent Potential Temperatures in Atmosphere: A General Pathway Rooted in Entropy Conservation

Abstract

Equivalent potential temperature is a widely used thermodynamic variable in atmospheric science. However, various formulations have been proposed in the literature over the last one and half‐century but their differences and connections are not straightforward to be understood. In this study, a general pathway is proposed to understand the differences and consistency between equivalent potential temperatures in the atmosphere. It is found that previous formulations of equivalent potential temperature can be explained in a mathematically and physically consistent way, and can be easily derived from the general pathway. In addition, new formulations of equivalent potential temperature under certain processes can be defined under the general pathway, without sophisticated mathematical derivation and inconsistent thermodynamic approximations. Our study provides insights to define new thermodynamic variables that can be applied to a wide range of physical conditions and advance our understanding of weather and climate systems.

Introduction

Equivalent potential temperature $$\left({\theta }_{e}\right)$$ is widely used in the atmospheric science to understand the weather and climate systems across a wide range of scales. The vertical gradient of equivalent potential temperature can be used as a proxy to measure the stability within clouds. Cloudy air is unstable if equivalent potential temperature decreases with height when total water is conserved. Due to the conserved property under reversible and adiabatic processes, equivalent potential temperature in the boundary layer can be compared to that in the upper troposphere to estimate the level of neutral buoyancy. It can also be treated as a tracer to study motions that are approximately adiabatic. For example, air masses in a synoptic weather system can be tracked backward along the surfaces of equivalent potential temperature to trace their origins. The contribution of mid-latitude eddies to the global mean circulation can be captured through averaging the flow along isentropes (Pauluis et al., 2008, 2010). At convective scales, isentropic analyses of vertical motions can be used to examine the mean features along trajectories with similar equivalent potential temperature and is a very useful tool to understand the energy cycle in systems with moist convection (Pauluis, 2016; Pauluis & Mrowiec, 2013). This has been successfully applied in the studies of tropical cyclones and multi-scale atmospheric overturning (Chen et al., 2018, 2020; Fang et al., 2017; Kamieniecki et al., 2018; Mrowiec et al., 2016). Some numerical models or physical parameterizations (e.g., Heus et al., 2010; Larson, 2017; Stevens et al., 2005) use liquid water potential temperature (a variant of equivalent potential temperature, see details in Section 2) as a prognostic variable in the thermodynamic equation. Without explicitly including the phase transition between water vapor and liquid water, this simplifies the parameterization of sub-grid scale turbulent fluxes for non-precipitating cloud-topped boundary layer and thereby can better capture the vertical mixing in the regime of shallow convection.

Despite wide applications in meteorology, there is no consensus of how equivalent potential temperatures should be formulated. In general, there are two pathways to derive equivalent potential temperature. The first one is to define the equivalent potential temperature as the temperature that the air parcel would attain if it is first lifted pseudo-adiabatically until all the water vapor has condensed, released its latent heat, and fallen out and then is compressed dry adiabatically to the standard pressure of 1,000 hPa (Holton & Hakim, 2013; Wallace & Hobbs, 2006). This is equivalent to integrate the differential equation of the first law of thermodynamics, essentially expressing adiabatic transformations, and define a temperature that is conserved in a closed system (Betts, 1973; Bezold, 1891; Bryan & Fritsch, 2004; Helmholtz, 1891; Rossby, 1932; Tripoli & Cotton, 1981). Another pathway is to essentially express the equivalent potential temperature as a function of total specific moist entropy of an air parcel (Eldred et al., 2022; Emanuel, 1994; Hauf & Höller, 1987; Marquet, 2011; Normand, 1921; Paluch, 1979; Pointin, 1984; Romps & Kuang, 2010), echoing the finding of Bauer (1908), who first clarified the relationships between the potential temperature and the dry entropy. Following the two pathways, various formulations of equivalent potential temperature have been proposed and are given different names. Table 1 gives a summary of these formulations along the history line (the empirical ones are not included here), starting from the simplest form of dry potential temperature with contribution from only dry air (Bezold, 1891; Helmholtz, 1891), to the ones with water vapor and liquid water (Betts, 1973; Emanuel, 1994; Normand, 1921; Paluch, 1979; Rossby, 1932), and to the most complicated ones with all three phases of water included (Bryan & Fritsch, 2004; Hauf & Höller, 1987; Marquet, 2011; Pauluis, 2016; Pointin, 1984; Romps & Kuang, 2010; Tripoli & Cotton, 1981).

Table 1 Various Formulations of Potential Temperature in the Literature and Our New Formulations (Last Four Rows)

Reference	Formulation
Helmholtz (1891)	$$\theta =T{\left(\frac{{p}_{r}}{p}\right)}^{\tfrac{{R}_{d}}{{c}_{pd}}}$$
Bezold (1891)
Normand (1921)	$${\theta }_{e-N21}=\theta \left[1+\frac{{L}_{lv}\left({T}_{r}\right){q}_{v}}{{q}_{d}{c}_{pd}T}\right]$$
Rossby (1932)	$${\theta }_{e-R32}=T{\left(\frac{{p}_{r}}{{p}_{d}}\right)}^{\tfrac{{R}_{d}}{{c}_{pd}}}\mathrm{exp}\left[\frac{{L}_{lv}\left({T}_{r}\right){q}_{v}}{{q}_{d}{c}_{pd}T}\right]$$
Betts (1973)	$${\theta }_{l-B73}=\theta \mathrm{exp}\left[-\frac{{L}_{lv}(T){q}_{l}}{{c}_{pd}T}\right]$$
Betts (1973)	$${\theta }_{e-B73}=\theta \mathrm{e}\mathrm{x}\mathrm{p}\left[\frac{{L}_{lv}(T){q}_{v}}{{c}_{pd}T}\right]$$
Paluch (1979)	$${\theta }_{q-P79}=T{\left(\frac{{p}_{r}}{{p}_{d}}\right)}^{\tfrac{{q}_{d}{R}_{d}}{\left({q}_{d}{c}_{pd}+{c}_{l}{q}_{t}\right)}}\mathrm{e}\mathrm{x}\mathrm{p}\left[\frac{{L}_{lv}(T){q}_{v}}{\left({q}_{d}{c}_{pd}+{c}_{l}{q}_{t}\right)T}\right]$$
Tripoli and Cotton (1981)	$${\theta }_{il-TC81}=\theta \mathrm{e}\mathrm{x}\mathrm{p}\left(-\frac{{L}_{lv}\left({T}_{r}\right){q}_{l}+{L}_{iv}\left({T}_{r}\right){q}_{i}}{{q}_{d}{c}_{pd}T}\right)$$
Tripoli and Cotton (1981)	$${\theta }_{\mathit{eiv-TC81}}=\theta \mathrm{e}\mathrm{x}\mathrm{p}\left(\frac{{L}_{lv}\left({T}_{r}\right){q}_{v}-{L}_{il}\left({T}_{r}\right){q}_{i}}{{q}_{d}{c}_{pd}T}\right)$$
Pointin (1984)	$${\theta }_{e-HH87},{\theta }_{q-P84}=$$
Hauf and Höller (1987)	$$T{\left(\frac{{p}_{r}}{{p}_{d}}\right)}^{\tfrac{{q}_{d}{R}_{d}}{\left({q}_{d}{c}_{pd}+{q}_{t}{c}_{l}\right)}}\mathrm{e}\mathrm{x}\mathrm{p}\left[\frac{{q}_{v}{L}_{lv}(T)-{q}_{i}{L}_{il}(T)}{\left({q}_{d}{c}_{pd}+{q}_{t}{c}_{l}\right)T}-\frac{{R}_{v}\left({q}_{v}\,\ln{\mathcal{H}}_{l}+{q}_{i}\,\ln\frac{{\mathcal{H}}_{l}}{{\mathcal{H}}_{i}}\right)}{{q}_{d}{c}_{pd}+{q}_{t}{c}_{l}}\right]$$
Emanuel (1994)	$${\theta }_{e-E94}=T{\left(\frac{{p}_{r}}{{p}_{d}}\right)}^{\tfrac{{q}_{d}{R}_{d}}{\left({q}_{d}{c}_{pd}+{c}_{l}{q}_{t}\right)}}{\left({\mathcal{H}}_{l}\right)}^{-\tfrac{{q}_{v}{R}_{v}}{\left({q}_{d}{c}_{pd}+{c}_{l}{q}_{t}\right)}}\mathrm{e}\mathrm{x}\mathrm{p}\left[\frac{{L}_{lv}{q}_{v}}{\left({q}_{d}{c}_{pd}+{c}_{l}{q}_{t}\right)T}\right]$$
Emanuel (1994)	$${\theta }_{l-E94}=T{\left(\frac{{p}_{r}}{p}\right)}^{\chi }{\left(1-\frac{{q}_{l}}{{q}_{d}{\epsilon}+{q}_{t}}\right)}^{\chi }{\left(1-\frac{{q}_{l}}{{q}_{t}}\right)}^{-\gamma }\mathrm{e}\mathrm{x}\mathrm{p}\left[-\frac{{q}_{l}{L}_{lv}(T)}{\left({q}_{d}{c}_{pd}+{q}_{t}{c}_{pv}\right)T}\right]$$,
$$\left(\chi =\frac{{q}_{d}{R}_{d}+{q}_{t}{R}_{v}}{{q}_{d}{c}_{pd}+{q}_{t}{c}_{pv}},\gamma =\frac{{q}_{t}{R}_{v}}{{q}_{d}{c}_{pd}+{q}_{t}{c}_{pv}},{\epsilon}=\frac{{R}_{d}}{{R}_{v}}\right)$$
Bryan and Fritsch (2004)	$${\theta }_{il\mathrm{4}\mathit{-}BF\mathrm{04}}=T{\left(\frac{{p}_{r}}{p}\right)}^{\chi }{\left(1-\frac{{q}_{l}+{q}_{i}}{{q}_{d}{\epsilon}+{q}_{t}}\right)}^{\chi }{\left(1-\frac{{q}_{l}+{q}_{i}}{{q}_{t}}\right)}^{-\gamma }\mathrm{exp}\left[-\frac{{q}_{l}{L}_{lv}(T)+{q}_{i}{L}_{iv}(T)}{\left({q}_{d}{c}_{pd}+{q}_{t}{c}_{pv}\right)T}\right]$$
Bryan and Fritsch (2004)	$${\theta }_{\mathit{il5-BF04}}=T{\left(\frac{{p}_{r}}{p}\right)}^{\chi }{\left(1-\frac{{q}_{l}+{q}_{i}}{{q}_{d}{\epsilon}+{q}_{t}}\right)}^{\chi }{\left(1-\frac{{q}_{l}+{q}_{i}}{{q}_{t}}\right)}^{-\gamma }\times $$
$$\mathrm{e}\mathrm{x}\mathrm{p}\left[-\frac{{q}_{l}{L}_{lv}(T)+{q}_{i}{L}_{iv}(T)}{\left({q}_{d}{c}_{pd}+{q}_{t}{c}_{pv}\right)T}+\frac{{R}_{v}\left({q}_{l}\,\ln{\mathcal{H}}_{l}+{q}_{i}\,\ln{\mathcal{H}}_{i}\right)}{{q}_{d}{c}_{pd}+{q}_{t}{c}_{pv}}\right]$$
Romps and Kuang (2010)	$${\theta }_{e-RK10}=T{\left(\frac{{p}_{r}}{{p}_{d}}\right)}^{\tfrac{{R}_{d}}{{c}_{pd}}}{\left(\frac{T}{{T}_{r}}\right)}^{\tfrac{{q}_{v}{c}_{pv}+{q}_{l}{c}_{l}+{q}_{i}{c}_{i}}{{q}_{d}{c}_{pd}}}{\left(\frac{{e}_{r}}{e}\right)}^{\tfrac{{q}_{v}{R}_{v}}{{q}_{d}{c}_{pd}}}\mathrm{e}\mathrm{x}\mathrm{p}\left[\frac{{q}_{v}{L}_{lv}\left({T}_{r}\right)-{q}_{i}{L}_{il}\left({T}_{r}\right)}{{q}_{d}{c}_{pd}{T}_{r}}\right]$$
Marquet (2011)	$${\theta }_{s-M11}=\theta \mathrm{e}\mathrm{x}\mathrm{p}\left({\Lambda }{q}_{t}\right)\mathrm{e}\mathrm{x}\mathrm{p}\left[-\frac{{q}_{l}{L}_{lv}(T)+{q}_{i}{L}_{iv}(T)}{{c}_{pd}T}+\frac{{R}_{v}\left({q}_{l}\,\ln{\mathcal{H}}_{l}+{q}_{i}\,\ln{\mathcal{H}}_{i}\right)}{{c}_{pd}}\right]\times $$
$${\left(\frac{T}{{T}_{r}}\right)}^{\lambda {q}_{t}}{\left(\frac{p}{{p}_{r}}\right)}^{-\kappa \delta {q}_{t}}{\left(\frac{{r}_{r}}{{r}_{v}}\right)}^{\gamma {q}_{t}}\frac{{\left(1+\eta {r}_{v}\right)}^{\kappa \left(1+\delta {q}_{t}\right)}}{{\left(1+\eta {r}_{r}\right)}^{\kappa \delta {q}_{t}}}$$
$${\Lambda }=\frac{{s}_{vr}-{s}_{dr}}{{c}_{pd}},\lambda =\frac{{c}_{pv}-{c}_{pd}}{{c}_{pd}},\kappa =\frac{{R}_{d}}{{c}_{pd}},\delta =\frac{{R}_{v}}{{R}_{d}}-1,\gamma =\frac{{R}_{v}}{{c}_{pd}},\eta =\frac{{R}_{v}}{{R}_{d}}$$
Pauluis (2016)	$${\theta }_{ei-P16}={T}_{r}{\left(\frac{T}{{T}_{r}}\right)}^{\tfrac{{q}_{d}{c}_{pd}+{q}_{i}{c}_{i}+\left({q}_{l}+{q}_{v}\right){c}_{l}}{{q}_{d}{c}_{pd}+{q}_{t}{c}_{i}}}{\left(\frac{{p}_{dr}}{{p}_{d}}\right)}^{\tfrac{{q}_{d}{R}_{d}}{{q}_{d}{c}_{pd}+{q}_{t}{c}_{i}}}\mathrm{e}\mathrm{x}\mathrm{p}\left[\frac{\left({q}_{v}+{q}_{l}\right)\frac{{L}_{il}\left({T}_{r}\right)}{{T}_{r}}+{q}_{v}\frac{{L}_{lv}(T)}{T}-{q}_{v}{R}_{v}\,\ln{\mathcal{H}}_{l}}{{q}_{d}{c}_{pd}+{q}_{t}{c}_{i}}\right]$$
Gu and Tan (2025)	$${\theta }_{\mathit{es1-GT25}}=T{\left(\frac{T}{{T}_{r}}\right)}^{\tfrac{{q}_{t}\left({c}_{pv}-{c}_{pd}\right)}{{c}_{pd}}}{\left(\frac{{p}_{dr}}{{p}_{d}}\right)}^{\tfrac{{q}_{d}{R}_{d}}{{c}_{pd}}}{\left[\frac{{e}_{sl}\left({T}_{r}\right)}{e(T)}\right]}^{\tfrac{{q}_{t}{R}_{v}}{{c}_{pd}}}\mathrm{e}\mathrm{x}\mathrm{p}\left[\frac{{q}_{t}\left({s}_{vr}-{s}_{dr}\right)}{{c}_{pd}}\right]\times $$
$$\mathrm{e}\mathrm{x}\mathrm{p}\left[-\frac{{q}_{l}{L}_{lv}(T)+{q}_{i}{L}_{iv}(T)}{{c}_{pd}T}+\frac{{R}_{v}\left({q}_{l}\,\ln{\mathcal{H}}_{l}+{q}_{i}\,\ln{\mathcal{H}}_{i}\right)}{{c}_{pd}}\right]$$
Gu and Tan (2025)	$${\theta }_{\mathit{es2-GT25}}=T{\left(\frac{T}{{T}_{r}}\right)}^{\tfrac{{q}_{t}\left({c}_{l}-{c}_{pd}\right)}{{c}_{pd}}}{\left(\frac{{p}_{dr}}{{p}_{d}}\right)}^{\tfrac{{q}_{d}{R}_{d}}{{c}_{pd}}}\mathrm{e}\mathrm{x}\mathrm{p}\left[\frac{{q}_{t}\left({s}_{lr}-{s}_{dr}\right)}{{c}_{pd}}\right]\times $$
$$\mathrm{e}\mathrm{x}\mathrm{p}\left[\frac{{q}_{v}{L}_{lv}(T)-{q}_{i}{L}_{il}(T)}{{c}_{pd}T}-\frac{{R}_{v}\left({q}_{v}\,\ln{\mathcal{H}}_{l}+{q}_{i}\,\ln\frac{{\mathcal{H}}_{l}}{{\mathcal{H}}_{i}}\right)}{{c}_{pd}}\right]$$
Gu and Tan (2025)	$${\theta }_{s-GT25}={T}_{r}\mathrm{e}\mathrm{x}\mathrm{p}\left(\frac{s-{s}_{dr}}{{c}_{pd}}\right)$$
Gu and Tan (2025)	$${\theta }_{e-GT25}=T{\left(\frac{{p}_{dr}}{{p}_{d}}\right)}^{\tfrac{{q}_{d}{R}_{d}}{{q}_{d}{c}_{pd}+{q}_{l}{c}_{l}+\left({q}_{v}+{q}_{i}\right){c}_{i}}}\mathrm{e}\mathrm{x}\mathrm{p}\left\{\frac{{q}_{v}{L}_{iv}(T)-{q}_{v}{R}_{v}T\,\ln{\mathcal{H}}_{i}}{\left[{q}_{d}{c}_{pd}+{q}_{l}{c}_{l}+\left({q}_{v}+{q}_{i}\right){c}_{i}\right]T}\right\}$$

However, the differences and connections between various forms of equivalent potential temperature have not been well understood yet. The differences are not only due to the different water species being considered, but also attributed to various assumptions adopted during the derivation (e.g., reversible, pseudo-adiabatic, dependence of heat capacity on water species, independence of latent heat release from temperature during phase change). Even though the same water species and similar assumptions are made, the differences between some formulations (Hauf & Höller, 1987; Marquet, 2011) can still be significant. These ambiguities leads to some confusion on which formulation is the best one that can incorporate the most comprehensive thermodynamic information when studying atmospheric motions. Pauluis et al. (2010) found that, in mid-latitudes, mean circulations in terms of total mass transport averaged on the surface of constant liquid water potential temperature (dry isentrope) and on the surface of constant equivalent potential temperature (moist isentrope) can differ in magnitudes of 1.5 and 3 times. Application of isentropic analysis to atmospheric convection requires equivalent potential temperature used for conditional averaging being able to effectively separate the updrafts and downdrafts, avoiding ambiguous definitions of isentropic streamfunctions and thus the quantitative analyses of energy cycles (Marquet, 2017; Marquet & Dauhut, 2018; Pauluis, 2018). The conservative nature of equivalent potential temperature during reversible phase changes and adiabatic processes makes it a good candidate as a potential prognostic thermodynamic variable to be used in the numerical models (Ooyama, 1990, 2001; Pointin, 1984; Tripoli & Cotton, 1981; Zeng et al., 2005, 2008), allowing larger computational time steps in long time simulations and at the same time improving the entropy balance of the system. Such formulations requires careful treatment of sources and sinks of moist entropy (Pressel et al., 2015; Raymond, 2013) and also consistent assumptions across dynamical cores and physical parameterizations, otherwise violating the laws of thermodynamics and resulting in unclosed global energy budget in a typical weather or climate model (Bowen & Thuburn, 2022a, 2022b; Eldred et al., 2022; Lauritzen et al., 2022; Ohno & Matsugishi, 2024; Staniforth & White, 2019; Thuburn, 2017). Given the essential role of equivalent potential temperature in moist thermodynamics of atmosphere, it is therefore of importance and intriguing to reconcile the diverse formulations in a physically and mathematically consistent pathway, which has not been proposed in the literature. This study will focus on addressing the following scientific questions:

• Is there a pathway that can unify our understanding on the various formulations of equivalent potential temperature in a general way that is mathematically and physically consistent, and meanwhile is relatively simple compared to previous studies?

• Are there any new formulations that have not been presented in previous studies, and how can they be defined without inappropriate approximations?

The rest of the paper is organized as follows. In Section 2, we propose a general pathway that combines the advantages of the two classical pathways to find equivalent potential temperatures. The new pathway invokes moist entropy conservation (Section 2.1) under certain adiabatic processes. Section 3 gives detailed mathematical derivations for three examples. Various formulations of equivalent potential temperature in previous studies can be easily covered. The physical interpretations of different adiabatic processes in Section 3 and the validity of entropy conservation are discussed in Section 4. Using this general method, a new formulation of equivalent potential temperature is proposed in Section 5, as well as the general idea of deriving other new equivalent potential temperatures. In Section 6, the physical difference of equivalent potential temperature and entropy potential temperature is explained, and a general form of the entropy potential temperature is introduced. Finally, the conclusion is given in Section 7.

A General Pathway

The General Principle

As described in Section 1, various formulations of equivalent potential temperatures in previous studies are defined following two pathways. The first is generally to find an invariant thermodynamic variable through integration of the first law of thermodynamics when considering certain adiabatic processes. The first law of thermodynamics can be written as, 1 $$du=Tds-pd\alpha +\sum\limits _{k=v,l,i}{g}_{k}d{q}_{k},$$ where $$u$$ is the internal energy, $$T$$ is the absolute temperature, $$p$$ is the air pressure, and $$\alpha $$ is the specific volume. $${g}_{v}$$, $${g}_{l}$$ and $${g}_{i}$$ are Gibbs free energy of water vapor, liquid water and ice water, respectively. $${q}_{v}$$, $${q}_{l}$$ and $${q}_{i}$$ are specific humidity of water vapor, liquid water and ice water, respectively. For adiabatic processes, we have 2 $$du=-pd\alpha ,$$

Subtracting Equation 2 from Equation 1, the governing equation of the total specific moist entropy can be deduced in the absence of external forcing, as follows. 3 $$\begin{array}{r}\hfill ds=-\sum\limits _{k=v,l,i}\frac{{g}_{k}}{T}d{q}_{k}=\frac{{g}_{v}}{T}d{q}_{t}+\frac{{g}_{v}-{g}_{l}}{T}d{q}_{l}+\frac{{g}_{v}-{g}_{i}}{T}d{q}_{i},\end{array}$$

The second identity uses the constraint of total water change $$d{q}_{t}=d{q}_{v}+d{q}_{l}+d{q}_{i}$$, where $${q}_{t}$$ is the total specific humidity of all water constituents. Equation 3 shows that the total moist entropy is conserved, either under the phase equilibrium $$\left({g}_{v}={g}_{l}={g}_{i}\right)$$ with no mass exchanges with outside $$\left(d{q}_{t}=0\right)$$, or no phase changes are allowed $$\left(d{q}_{k}=0\right)$$. Equation 3 suggests that the second pathway to define the equivalent potential temperature from the moist specific entropy, as was done in previous studies (Eldred et al., 2022; Emanuel, 1994; Hauf & Höller, 1987; Marquet, 2011; Normand, 1921; Paluch, 1979; Pointin, 1984; Romps & Kuang, 2010), is essentially the same as the first pathway. However, the second pathway only attempts to express the equivalent potential temperature as a function of moist entropy without explicitly considering possible physical processes involved, which might be important for practical applications. Therefore, it is necessary to synthesize the two previous approaches for a more generalized approach in deriving the equivalent potential temperature, which is critical to understand the differences and connections between various existing formulations of equivalent potential temperatures.

Here, we propose that entropy conservation under adiabatic processes can be used as a more physically intuitive and consistent pathway. The equivalent potential temperature can be derived by bringing the air parcel to the reference level (usually at triple point, $${T}_{r}=273.15$$ K, $${p}_{r}=1000$$ hPa) through putative physical processes under the assumption that the total moist specific entropy is conserved. In principle, the total specific moist entropy $$(s)$$ of an air parcel with a perfect mixture of perfect gases is the function of temperature $$T$$, air pressure $$p$$, the specific humidity of water vapor $${q}_{v}$$, liquid water $${q}_{l}$$ and ice water $${q}_{i}$$ so that it can be written as a function $$s\left(T,p,{q}_{v},{q}_{l},{q}_{i}\right)$$, which can be simplified as $$s\left(T,p,{q}_{t}\right)$$ if phase equilibrium and conservation of water mass are assumed $$\left({q}_{t}={q}_{v}+{q}_{l}+{q}_{i}\right)$$. If we denote the temperature that the parcel will acquire at reference pressure level $${p}_{r}$$ as the equivalent potential temperature $${\theta }_{e}$$, the specific humidity of water vapor, liquid water and ice water at reference level as $${q}_{v}^{\prime }$$, $${q}_{l}^{\prime }$$ and $${q}_{i}^{\prime }$$ respectively, then conservation of moist entropy implies, 4 $$s\left({\theta }_{e},{p}_{r},{q}_{v}^{\prime },{q}_{l}^{\prime },{q}_{i}^{\prime }\right)=s\left(T,p,{q}_{v},{q}_{l},{q}_{i}\right).$$ The RHS of Equation 4 gives the formulation of moist entropy for the moist air parcel. The exact formulation on the LHS of Equation 4 depends on how the parcel is moved to the reference pressure level. Equivalent potential temperature $${\theta }_{e}$$ can be derived from Equation 4 following different adiabatic processes. Our method is in general simple since we only need to consider how the parcel is moved to the reference level through adiabatic processes. This method can give all the formulations of equivalent potential temperature that appeared in previous studies. One should be aware that Equation 4 is strictly correct only when full phase equilibrium is applied during the adiabatic processes (see Section 4), in which case the total moist entropy on the RHS should be a function of ($$T$$, $$p$$, $${q}_{t}$$).

Expression of Moist Entropy

First, our approach needs an explicit expression of moist specific entropy of an air parcel with all three water species in the atmosphere. It can be generally written as the sum of specific humidity weighted specific entropies of dry air $$\left({s}_{d}\right)$$, water vapor $$\left({s}_{v}\right)$$, liquid water $$\left({s}_{l}\right)$$ and ice water $$\left({s}_{i}\right)$$ (Ambaum, 2020) as follows, 5 $$s={q}_{d}{s}_{d}+{q}_{v}{s}_{v}+{q}_{l}{s}_{l}+{q}_{i}{s}_{i},$$ where $$s$$ represents the total moist specific entropy (S1 in Table 2). Standard definitions of specific entropies for each constituent in the air parcel should be generally determined by the partial derivative of Gibbs potential with respect to their conjugate variable $$T$$. Eldred et al. (2022) presented a detailed procedure to derive the thermodynamics of a system with consistent thermodynamic assumptions. Following the definitions in Eldred et al. (2022), the specific entropy of dry air $$\left({s}_{d}\right)$$, water vapor $$\left({s}_{v}\right)$$, liquid water $$\left({s}_{l}\right)$$ and ice water $$\left({s}_{i}\right)$$ can be written as follows, 6 $${s}_{d}={s}_{dr}+{c}_{pd}\,\ln\frac{T}{{T}_{r}}-{R}_{d}\,\ln\frac{{p}_{d}}{{p}_{dr}},$$ 7 $${s}_{v}={s}_{vr}+{c}_{pv}\,\ln\frac{T}{{T}_{r}}-{R}_{v}\,\ln\frac{e(T)}{{e}_{sl}\left({T}_{r}\right)},$$ 8 $${s}_{l}={s}_{lr}+{c}_{l}\,\ln\frac{T}{{T}_{r}},$$ 9 $${s}_{i}={s}_{ir}+{c}_{i}\,\ln\frac{T}{{T}_{r}},$$ where $${s}_{dr}$$, $${s}_{vr}$$, $${s}_{lr}$$ and $${s}_{ir}$$ are reference entropies of dry air, water vapor, liquid water and ice water at reference state $${T}_{r}$$, which we choose at the triple point; $${c}_{pd}$$, $${c}_{pv}$$, $${c}_{l}$$ and $${c}_{i}$$ are specific heat capacity for dry air, water vapor, liquid water and ice water at constant pressure, respectively; $${R}_{d}$$ and $${R}_{v}$$ are the gas constants of dry air and water vapor, respectively; $${p}_{d}$$ and $$e(T)$$ are the dry air pressure and the water vapor pressure at temperature $$T$$, respectively; $${p}_{dr}$$ and $${e}_{sl}\left({T}_{r}\right)$$ are the dry air pressure and saturated water vapor pressure at reference state, respectively. Substituting Equations 6–9 into Equation 5 gives an expression of the moist specific entropy of the air parcel, as shown in Table 1 (S1).

Table 2 Transformations of Specific Humidity Weighted Total Moist Entropy Corresponding to Different Adiabatic Processes, Final State Composition, Phase Equilibrium Conditions (the Second Column) and Their Formulations (the Third Column)

Ref	Moist Entropy	Exact Expressions
S1	$${q}_{d}{s}_{d}+{q}_{v}{s}_{v}+{q}_{l}{s}_{l}+{q}_{i}{s}_{i}$$	$${q}_{d}{s}_{dr}+{q}_{v}{s}_{vr}+{q}_{l}{s}_{lr}+{q}_{i}{s}_{ir}$$
$${+}\left({q}_{d}{c}_{pd}+{q}_{v}{c}_{pv}+{q}_{l}{c}_{l}+{q}_{i}{c}_{i}\right)\ln\frac{T}{{T}_{r}}-$$
$${-}{q}_{d}{R}_{d}\,\ln\frac{{p}_{d}}{{p}_{dr}}-{q}_{v}{R}_{v}\,\ln\frac{e(T)}{{e}_{sl}\left({T}_{r}\right)}$$
S2	$${q}_{d}{s}_{d}+{q}_{t}{s}_{v}-{q}_{l}\left({s}_{v}-{s}_{l}\right)-{q}_{i}\left({s}_{v}-{s}_{i}\right)$$	$${q}_{d}{s}_{dr}+{q}_{t}{s}_{vr}+\left({q}_{d}{c}_{pd}+{q}_{t}{c}_{pv}\right)\ln\frac{T}{{T}_{r}}$$
$${-}{q}_{d}{R}_{d}\,\ln\frac{{p}_{d}}{{p}_{dr}}-{q}_{t}{R}_{v}\,\ln\frac{e(T)}{{e}_{sl}\left({T}_{r}\right)}$$
$${-}\frac{{q}_{l}{L}_{lv}(T)+{q}_{i}{L}_{iv}(T)}{T}+{R}_{v}\left({q}_{l}\,\ln{\mathcal{H}}_{l}+{q}_{i}\,\ln{\mathcal{H}}_{i}\right)$$
Step 1: Ice melts into liquid water;
Step 2: Liquid water vapourizes into water vapor;
Final state: dry air and all water vapor Full equilibrium
S3	$${q}_{d}{s}_{d}+{q}_{t}{s}_{l}+{q}_{v}\left({s}_{v}-{s}_{l}\right)-{q}_{i}\left({s}_{l}-{s}_{i}\right)$$	$${q}_{d}{s}_{dr}+{q}_{t}{s}_{lr}+\left({q}_{d}{c}_{pd}+{q}_{t}{c}_{l}\right)\ln\frac{T}{{T}_{r}}$$
$${-}{q}_{d}{R}_{d}\,\ln\frac{{p}_{d}}{{p}_{dr}}+\frac{{q}_{v}{L}_{lv}(T)-{q}_{i}{L}_{il}(T)}{T}$$
$${-}{R}_{v}\left({q}_{v}\,\ln{\mathcal{H}}_{l}+{q}_{i}\,\ln\frac{{\mathcal{H}}_{l}}{{\mathcal{H}}_{i}}\right)$$
Step 1: Ice melts into liquid water;
Step 2: water vapor condensates into liquid water;
Final state: dry air and all liquid water Full equilibrium
S4	$${q}_{d}{s}_{d}+{q}_{t}{s}_{i}+{q}_{v}\left({s}_{v}-{s}_{i}\right)+{q}_{l}\left({s}_{l}-{s}_{i}\right)$$	$${q}_{d}{s}_{dr}+{q}_{t}{s}_{ir}+\left({q}_{d}{c}_{pd}+{q}_{t}{c}_{i}\right)\ln\frac{T}{{T}_{r}}$$
$${-}{q}_{d}{R}_{d}\,\ln\frac{{p}_{d}}{{p}_{dr}}+\frac{{q}_{v}{L}_{iv}(T)+{q}_{l}{L}_{il}(T)}{T}$$
$${-}{R}_{v}\left({q}_{v}\,\ln{\mathcal{H}}_{i}-{q}_{l}\,\ln\frac{{\mathcal{H}}_{l}}{{\mathcal{H}}_{i}}\right)$$
Step 1: liquid water freezes into ice;
Step 2: water vapor deposits into ice;
Final state: dry air and all ice Full equilibrium
S5	$${q}_{d}{s}_{d}+{q}_{i}{s}_{i}+\left({q}_{l}+{q}_{v}\right){s}_{v}-{q}_{l}\left({s}_{v}-{s}_{l}\right)$$	$${q}_{d}{s}_{dr}+{q}_{i}{s}_{ir}+\left({q}_{l}+{q}_{v}\right){s}_{vr}$$
$${+}\left[{q}_{d}{c}_{pd}+{q}_{i}{c}_{i}+\left({q}_{l}+{q}_{v}\right){c}_{pv}\right]\ln\frac{T}{{T}_{r}}$$
$${-}{q}_{d}{R}_{d}\,\ln\frac{{p}_{d}}{{p}_{dr}}-\left({q}_{l}+{q}_{v}\right){R}_{v}\,\ln\frac{e(T)}{{e}_{sl}\left({T}_{r}\right)}$$
$${-}\frac{{q}_{l}{L}_{lv}(T)}{T}+{q}_{l}{R}_{v}\,\ln{\mathcal{H}}_{l}$$
Liquid water vapourizes into water vapor
Final state: dry air, ice and water vapor Full equilibrium
S6	$${q}_{d}{s}_{d}+{q}_{i}{s}_{i}+\left({q}_{l}+{q}_{v}\right){s}_{l}+{q}_{v}\left({s}_{v}-{s}_{l}\right)$$	$${q}_{d}{s}_{dr}+{q}_{i}{s}_{ir}+\left({q}_{l}+{q}_{v}\right){s}_{lr}$$
$${+}\left[{q}_{d}{c}_{pd}+{q}_{i}{c}_{i}+\left({q}_{l}+{q}_{v}\right){c}_{l}\right]\ln\frac{T}{{T}_{r}}$$
$${-}{q}_{d}{R}_{d}\,\ln\frac{{p}_{d}}{{p}_{dr}}+\frac{{q}_{v}{L}_{lv}(T)}{T}-{q}_{v}{R}_{v}\,\ln{\mathcal{H}}_{l}$$
Water vapor condensates into liquid water
Final state: dry air, ice and liquid water Full equilibrium
S7	$${q}_{d}{s}_{d}+{q}_{v}{s}_{v}+\left({q}_{l}+{q}_{i}\right){s}_{l}-{q}_{i}\left({s}_{l}-{s}_{i}\right)$$	$${q}_{d}{s}_{dr}+{q}_{v}{s}_{vr}+\left({q}_{l}+{q}_{i}\right){s}_{lr}$$
$${+}\left[{q}_{d}{c}_{pd}+{q}_{v}{c}_{pv}+\left({q}_{l}+{q}_{i}\right){c}_{l}\right]\ln\frac{T}{{T}_{r}}-{q}_{d}{R}_{d}\,\ln\frac{{p}_{d}}{{p}_{dr}}$$
$${-}{q}_{v}{R}_{v}\,\ln\frac{e(T)}{{e}_{sl}\left({T}_{r}\right)}-\frac{{q}_{i}{L}_{il}(T)}{T}+{q}_{i}{R}_{v}\,\ln\frac{{\mathcal{H}}_{i}}{{\mathcal{H}}_{l}}$$
Ice melts into liquid water
Final state: dry air, water vapor and liquid water Full equilibrium
S8	$${q}_{d}{s}_{d}+{q}_{v}{s}_{v}+\left({q}_{l}+{q}_{i}\right){s}_{i}+{q}_{l}\left({s}_{l}-{s}_{i}\right)$$	$${q}_{d}{s}_{dr}+{q}_{v}{s}_{vr}+\left({q}_{l}+{q}_{i}\right){s}_{ir}$$
$${+}\left[{q}_{d}{c}_{pd}+{q}_{v}{c}_{pv}+\left({q}_{l}+{q}_{i}\right){c}_{i}\right]\ln\frac{T}{{T}_{r}}-{q}_{d}{R}_{d}\,\ln\frac{{p}_{d}}{{p}_{dr}}$$
$${-}{q}_{v}{R}_{v}\,\ln\frac{e(T)}{{e}_{sl}\left({T}_{r}\right)}+\frac{{q}_{l}{L}_{il}(T)}{T}-{q}_{l}{R}_{v}\,\ln\frac{{\mathcal{H}}_{i}}{{\mathcal{H}}_{l}}$$
Liquid water freezes into ice
Final state: dry air, water vapor and ice Full equilibrium
S9	$${q}_{d}{s}_{d}+{q}_{l}{s}_{l}+\left({q}_{v}+{q}_{i}\right){s}_{i}+{q}_{v}\left({s}_{v}-{s}_{i}\right)$$	$${q}_{d}{s}_{dr}+{q}_{l}{s}_{lr}+\left({q}_{v}+{q}_{i}\right){s}_{ir}$$
$${+}\left[{q}_{d}{c}_{pd}+{q}_{l}{c}_{l}+\left({q}_{v}+{q}_{i}\right){c}_{i}\right]\ln\frac{T}{{T}_{r}}-{q}_{d}{R}_{d}\,\ln\frac{{p}_{d}}{{p}_{dr}}$$
$${+}\frac{{q}_{v}{L}_{iv}(T)}{T}-{q}_{v}{R}_{v}\,\ln{\mathcal{H}}_{i}$$
Water vapor deposits into ice
Final state: dry air, liquid water and ice Full equilibrium
S10	$${q}_{d}{s}_{d}+{q}_{l}{s}_{l}+\left({q}_{v}+{q}_{i}\right){s}_{v}-{q}_{i}\left({s}_{v}-{s}_{i}\right)$$	$${q}_{d}{s}_{dr}+{q}_{l}{s}_{lr}+\left({q}_{v}+{q}_{i}\right){s}_{vr}$$
$${+}\left[{q}_{d}{c}_{pd}+{q}_{l}{c}_{l}+\left({q}_{v}+{q}_{i}\right){c}_{pv}\right]\ln\frac{T}{{T}_{r}}-{q}_{d}{R}_{d}\,\ln\frac{{p}_{d}}{{p}_{dr}}$$
$${-}\left({q}_{v}+{q}_{i}\right){R}_{v}\,\ln\frac{e(T)}{{e}_{sl}\left({T}_{r}\right)}-\frac{{q}_{i}{L}_{iv}(T)}{T}+{q}_{i}{R}_{v}\,\ln{\mathcal{H}}_{i}$$
Ice sublimates into water vapor
Final state: dry air, liquid water and water vapor Full equilibrium

In addition, Equation 5 can also be rewritten in various forms. Table 2 illustrates all the possible transformations. Following are three examples, 10 $$s={q}_{d}{s}_{d}+{q}_{t}{s}_{v}-{q}_{l}\left({s}_{v}-{s}_{l}\right)-{q}_{i}\left({s}_{v}-{s}_{i}\right),$$ 11 $$s={q}_{d}{s}_{d}+{q}_{t}{s}_{l}+{q}_{v}\left({s}_{v}-{s}_{l}\right)-{q}_{i}\left({s}_{l}-{s}_{i}\right),$$ 12 $$s={q}_{d}{s}_{d}+{q}_{t}{s}_{i}+{q}_{v}\left({s}_{v}-{s}_{i}\right)+{q}_{l}\left({s}_{l}-{s}_{i}\right).$$

The entropy differences between different phases at air temperature $$T$$ are due to the latent heat absorption and the change of Gibbs free energy (or chemical affinity) if they do not occur at thermodynamic equilibrium state, 13 $${s}_{v}-{s}_{l}=\frac{{L}_{lv}(T)}{T}+\frac{{A}_{lv}(T)}{T},$$ 14 $${s}_{l}-{s}_{i}=\frac{{L}_{li}(T)}{T}+\frac{{A}_{il}(T)}{T},$$ 15 $${s}_{v}-{s}_{i}=\frac{{L}_{iv}(T)}{T}+\frac{{A}_{iv}(T)}{T},$$ where $${L}_{lv}$$, $${L}_{li}$$ and $${L}_{iv}$$ are latent heat of vapourization, melting and sublimation, respectively. They are functions of temperature $$T$$ and follow the Kirchhoff's law in thermodynamics. $${A}_{lv}$$, $${A}_{il}$$ and $${A}_{iv}$$ are the chemical affinities and are expressed as 16 $${A}_{lv}(T)={g}_{l}(T)-{g}_{v}(T),$$ 17 $${A}_{il}(T)={g}_{i}(T)-{g}_{l}(T),$$ 18 $${A}_{iv}(T)={g}_{i}(T)-{g}_{v}(T),$$ in which $${g}_{v}(T)$$, $${g}_{l}(T)$$ and $${g}_{i}(T)$$ are Gibbs free energies (or chemical potentials) of water vapor, liquid water and ice at temperature $$T$$, respectively. The affinity terms have the following relations with water vapor pressure and saturated water vapor pressures with respect to the plane surfaces of liquid water and ice 19 $$\frac{{A}_{lv}(T)}{T}=-{R}_{v}\,\ln{\mathcal{H}}_{l},$$ 20 $$\frac{{A}_{il}(T)}{T}={R}_{v}\,\ln\frac{{\mathcal{H}}_{l}}{{\mathcal{H}}_{i}},$$ 21 $$\frac{{A}_{iv}(T)}{T}=-{R}_{v}\,\ln{\mathcal{H}}_{i},$$ where $${\mathcal{H}}_{l}=\frac{e(T)}{{e}_{sl}(T)}$$, is the relative humidity at temperature $$T$$ with respect to the liquid water and $${\mathcal{H}}_{i}=\frac{e(T)}{{e}_{si}(T)}$$ is the relative humidity at temperature $$T$$ with respect to the ice water. $${e}_{si}(T)$$ is the saturated water vapor pressure with respect to ice water at temperature $$T$$.

Substituting Equations 13–15, and Equations 19–21 in the second column of Table 2 gives the other nine different explicit expressions of the total moist specific entropy. Despite the different expressions, all the formulations in Table 2 are mathematically equivalent, since they are from the same definition Equation 5. However, these formulations have different physical interpretations that are critical to derive their corresponding equivalent potential temperatures and will be explained in Section 4.

Mathematical Derivation of Equivalent Potential Temperature $$\left({\theta }_{e}\right)$$

Section 2.2 gives various formulations in the RHS of Equation 4. To derive the equivalent potential temperature, the exact form of the total moist specific entropy at the reference level, that is, the expression in the LHS of Equation 4 should be first defined. Recalling that the air parcel is brought to the reference level after a sequence of adiabatic processes, the key to determining the expression in the LHS of Equation 4 is to set up the details of the final state, that is, the constituents in the air parcel. There are generally two rules to consider,

• R1: To get an explicit expression for $${\theta }_{e}$$ by solving Equation 4, final state must not have any latent heat terms that are functions of temperature;

• R2: $${\theta }_{e}$$ should not contain any terms involving reference entropies, since the result of a sequence of adiabatic processes should not depend on the reference state. The reference entropies must be either eliminated by relating species entropy differences to latent heats, or ensuring that the factor multiplying any reference entropy is the same in the current and final state (e.g., $${q}_{d}$$ or $${q}_{t}$$) so that the contributions from the current and final states cancel each other.

Given the above rules, several examples are given to illustrate how most of the existing formulations of equivalent potential temperatures can be derived from our general pathway.

Example 1: Final State With Dry Air and Water Vapor

We start from the explicit expression of total moist specific entropy corresponding to Equation 10 (S2 in Table 2). Since the expression has two terms with reference entropies $$\left({s}_{dr}{q}_{d}+{s}_{vr}{q}_{t}\right)$$, the final state of the air parcel can be set up with only dry air and water vapor existing to fulfill the above rules. The specific humidity of water vapor equals the total specific humidity of all the water species at the initial state. That is to say, there is no precipitation falling from the parcel during the adiabatic processes. In such a configuration, the final state total entropy of the parcel can be written as $${q}_{d}{s}_{dr}+{q}_{t}{s}_{vr}+\left({q}_{d}{c}_{pd}+{q}_{t}{c}_{pv}\right)\ln\left({T}^{\prime }/{T}_{r}\right)-{q}_{d}{R}_{d}\,\ln\left({p}_{d}^{\prime }/{p}_{dr}\right)-{q}_{t}{R}_{v}\,\ln\left[{e}^{\prime }/{e}_{sl}\left({T}_{r}\right)\right]$$, where $${p}_{d}^{\prime }$$ and $${e}^{\prime }$$ are the dry air pressure and the water vapor pressure when the parcel is moving to the reference level, respectively. This leads to the following equation, 22 $\begin{align*}\hfill {q}_{d}{s}_{dr}+{q}_{t}{s}_{vr}+\left({q}_{d}{c}_{pd}+{q}_{t}{c}_{pv}\right)\ln\frac{{T}^{\prime }}{{T}_{r}}-{q}_{d}{R}_{d}\,\ln\frac{{p}_{d}^{\prime }}{{p}_{dr}}-{q}_{t}{R}_{v}\,\ln\frac{{e}^{\prime }}{{e}_{sl}\left({T}_{r}\right)}\\ \hfill ={q}_{d}{s}_{dr}+{q}_{t}{s}_{vr}+\left({q}_{d}{c}_{pd}+{q}_{t}{c}_{pv}\right)\ln\frac{T}{{T}_{r}}-{q}_{d}{R}_{d}\,\ln\frac{{p}_{d}}{{p}_{dr}}-{q}_{t}{R}_{v}\,\ln\frac{e(T)}{{e}_{sl}\left({T}_{r}\right)}\\ \hfill -\frac{{q}_{l}{L}_{lv}(T)+{q}_{i}{L}_{iv}(T)}{T}+{R}_{v}\left({q}_{l}\,\ln{\mathcal{H}}_{l}+{q}_{i}\,\ln{\mathcal{H}}_{i}\right).\end{align*}$

The last two terms in the RHS of Equation 22 vanishes when the initial parcel is under phase equilibrium. The idealized gas law can be written as, 23 $\begin{align*}\hfill p={p}_{d}+e={\rho }_{d}{R}_{d}T+{\rho }_{v}{R}_{v}T=\rho T\left({q}_{d}{R}_{d}+{q}_{v}{R}_{v}\right),\end{align*}$ where $${\rho }_{d},{\rho }_{v},\rho $$ are densities of dry air, water vapor and air parcel, respectively. Using Equation 23, we have the following relations, 24 $\begin{align*}\hfill e=\frac{{r}_{v}p}{{\epsilon}+{r}_{v}},{p}_{d}=\frac{{\epsilon}p}{{\epsilon}+{r}_{v}},\end{align*}$ where $${r}_{v}$$ is the mixing ratio of water vapor and $${\epsilon}={R}_{d}/{R}_{v}\approx 0.622$$. Substituting Equation 24 into Equation 22, we have 25 $\begin{align*}\hfill \left({q}_{d}{c}_{pd}+{q}_{t}{c}_{pv}\right)\ln\frac{{T}^{\prime }}{{T}_{r}}-\left({q}_{d}{R}_{d}+{q}_{t}{R}_{v}\right)\ln{p}^{\prime }+{q}_{d}{R}_{d}\,\ln\left(1+\frac{{r}_{t}}{{\epsilon}}\right)+{q}_{t}{R}_{v}\,\ln\left(1+\frac{{\epsilon}}{{r}_{t}}\right)\\ \hfill =\left({q}_{d}{c}_{pd}+{q}_{t}{c}_{pv}\right)\ln\frac{T}{{T}_{r}}-\left({q}_{d}{R}_{d}+{q}_{t}{R}_{v}\right)\ln\,p+{q}_{d}{R}_{d}\,\ln\left(1+\frac{{r}_{v}}{{\epsilon}}\right)+{q}_{t}{R}_{v}\,\ln\left(1+\frac{{\epsilon}}{{r}_{v}}\right)\\ \hfill -\frac{{q}_{l}{L}_{lv}(T)+{q}_{i}{L}_{iv}(T)}{T}+{R}_{v}\left({q}_{l}\,\ln{\mathcal{H}}_{l}+{q}_{i}\,\ln{\mathcal{H}}_{i}\right),\end{align*}$ where $${p}^{\prime }={p}_{d}^{\prime }+{e}^{\prime }$$ is the pressure of the air parcel during the process moving to the reference level; $${r}_{t}$$ is the total water mixing ratio. Setting the pressure $${p}^{\prime }$$ as the reference pressure $${p}_{r}$$, we can get the equivalent potential temperature $${\theta }_{e}$$ by solving Equation 25 26 $\begin{align*}\hfill {\theta }_{e}=T{\left(\frac{{p}_{r}}{p}\right)}^{\chi }{\left(\frac{{\epsilon}+{r}_{v}}{{\epsilon}+{r}_{t}}\right)}^{\chi }{\left(\frac{{r}_{v}}{{r}_{t}}\right)}^{-\gamma }\mathrm{e}\mathrm{x}\mathrm{p}\left[-\frac{{q}_{l}{L}_{lv}(T)+{q}_{i}{L}_{iv}(T)}{\left({q}_{d}{c}_{pd}+{q}_{t}{c}_{pv}\right)T}+\frac{{R}_{v}\left({q}_{l}\,\ln{\mathcal{H}}_{l}+{q}_{i}\,\ln{\mathcal{H}}_{i}\right)}{{q}_{d}{c}_{pd}+{q}_{t}{c}_{pv}}\right],\end{align*}$ where $$\chi =\left({q}_{d}{R}_{d}+{q}_{t}{R}_{v}\right)/\left({q}_{d}{c}_{pd}+{q}_{t}{c}_{pv}\right)$$, $$\gamma ={q}_{t}{R}_{v}/\left({q}_{d}{c}_{pd}+{q}_{t}{c}_{pv}\right)$$. This formulation is equivalent to the ice-liquid water potential temperature $${\theta }_{\mathit{il5-BF04}}$$ (Bryan & Fritsch, 2004). When the parcel is initially at thermodynamic equilibrium state, the exponential terms with relative humidity $$\ln{\mathcal{H}}_{l}$$ and $$\ln{\mathcal{H}}_{i}$$ become zero and the formulation becomes equivalent to another ice-liquid water potential temperature $${\theta }_{\mathit{il4-BF04}}$$ (Bryan & Fritsch, 2004). It reduces to the liquid water potential temperature $${\theta }_{l-E94}$$ (Emanuel, 1994) when the ice water and non-equilibrium term are neglected. If a parcel is initially at phase equilibrium ($$\ln{\mathcal{H}}_{l}$$ and $$\ln{\mathcal{H}}_{i}$$ vanish), the formulation Equation 26 can be simplified to the ice–liquid water potential temperature $${\theta }_{il-TC81}$$ (Tripoli & Cotton, 1981), when the water vapor has no impact on the specific heat capacity and the total pressure is approximately equal to the dry partial pressure (vapor pressure is neglected), as well as the dependence of latent heat during phase changes on the temperature is neglected. It should be noted that $${\theta }_{\mathit{il4-BF04}}$$, $${\theta }_{\mathit{il5-BF04}}$$, and $${\theta }_{il-TC81}$$ are derived with sophisticated mathematical transformations that are not easily to follow. Our method is simple and has a very clear physical picture.

Example 2: Final State With Dry Air and Liquid Water

In the second example, the RHS of Equation 4 uses the exact expression of Equation 11 (S3 in Table 2). In this case, to fulfill rules R1 and R2, the final state of the air parcel is chosen to contain only dry air and liquid water. Therefore, the expression of the total specific entropy of the air parcel is $${q}_{d}{s}_{dr}+{q}_{t}{s}_{lr}+\left({q}_{d}{c}_{pd}+{q}_{t}{c}_{l}\right)\ln\left({T}^{\prime }/{T}_{r}\right)-{q}_{d}{R}_{d}\,\ln\left({p}_{d}^{\prime }/{p}_{dr}\right)$$. Using the above expression on the LHS of Equation 4 results in the following equation, 27 $\begin{align*}\hfill {q}_{d}{s}_{dr}+{q}_{t}{s}_{lr}+\left({q}_{d}{c}_{pd}+{q}_{t}{c}_{l}\right)\ln\frac{{T}^{\prime }}{{T}_{r}}-{q}_{d}{R}_{d}\,\ln\frac{{p}_{d}^{\prime }}{{p}_{dr}}\\ \hfill ={q}_{d}{s}_{dr}+{q}_{t}{s}_{lr}+\left({q}_{d}{c}_{pd}+{q}_{t}{c}_{l}\right)\ln\frac{T}{{T}_{r}}-{q}_{d}{R}_{d}\,\ln\frac{{p}_{d}}{{p}_{dr}}+\\ \hfill \frac{{q}_{v}{L}_{lv}(T)-{q}_{i}{L}_{il}(T)}{T}-{R}_{v}\left({q}_{v}\,\ln{\mathcal{H}}_{l}+{q}_{i}\,\ln\frac{{\mathcal{H}}_{l}}{{\mathcal{H}}_{i}}\right).\end{align*}$

The equivalent potential temperature can be derived by solving the above equation through setting $${p}_{d}^{\prime }={p}_{r}$$, 28 $\begin{align*}\hfill {\theta }_{e}=T{\left(\frac{{p}_{r}}{{p}_{d}}\right)}^{\tfrac{{q}_{d}{R}_{d}}{\left({q}_{d}{c}_{pd}+{q}_{t}{c}_{l}\right)}}\mathrm{e}\mathrm{x}\mathrm{p}\left[\frac{{q}_{v}{L}_{lv}(T)-{q}_{i}{L}_{il}(T)}{\left({q}_{d}{c}_{pd}+{q}_{t}{c}_{l}\right)T}-\frac{{R}_{v}\left({q}_{v}\,\ln{\mathcal{H}}_{l}+{q}_{i}\,\ln\frac{{\mathcal{H}}_{l}}{{\mathcal{H}}_{i}}\right)}{{q}_{d}{c}_{pd}+{q}_{t}{c}_{l}}\right].\end{align*}$

This formulation is the same as the entropy potential temperature $${\theta }_{e-HH87}$$ (Hauf & Höller, 1987) and the wet equivalent potential temperature $${\theta }_{q-P84}$$ (Pointin, 1984). It can be related to the equivalent potential temperature $${\theta }_{e-RK10}$$ defined in Romps and Kuang (2010) by $${\theta }_{e}={\left({T}_{r}^{{r}_{t}{c}_{l}/{c}_{pd}}{\theta }_{e-RK10}\right)}^{{q}_{d}{c}_{pd}/\left({q}_{d}{c}_{pd}+{q}_{t}{c}_{l}\right)}$$ with a conserved quantity when $${q}_{t}$$ is constant. When the ice water is neglected, it reduces to the equivalent potential temperature $${\theta }_{e-E94}$$ (Emanuel, 1994). It also reduces to the wet potential temperature $${\theta }_{q-P79}$$ (Paluch, 1979) if there is no ice water and the parcel is initially at thermodynamic equilibrium ($$\ln{\mathcal{H}}_{l}$$ and $$\ln{\mathcal{H}}_{i}$$ vanish). When the impact of liquid water on the specific heat capacity is neglected and the initial state is at phase equilibrium, it can be simplified to the ice-vapor water potential temperature $${\theta }_{\mathit{eiv-TC81}}$$ in Tripoli and Cotton (1981) assuming the latent heat of condensation and melting takes the value at the triple point. Equation 26 further reduces to the formulation of $${\theta }_{e-R32}$$ (Rossby, 1932) when the parcel only has water vapor and is in thermodynamic equilibrium.

Example 3: Final State With Dry Air and Ice Water

The third example considers the final state with only dry air and ice water. The total specific entropy of the parcel at final state can be written as $${q}_{d}{s}_{dr}+{q}_{t}{s}_{ir}+\left({q}_{d}{c}_{pd}+{q}_{t}{c}_{i}\right)\ln\left({T}^{\prime }/{T}_{r}\right)-{q}_{d}{R}_{d}\,\ln\left({p}_{d}^{\prime }/{p}_{dr}\right)$$. In this case, the RHS of Equation 4 uses the exact expression of Equation 12 (S4 in Table 2). This gives the following equation, 29 $\begin{align*}\hfill {q}_{d}{s}_{dr}+{q}_{t}{s}_{ir}+\left({q}_{d}{c}_{pd}+{q}_{t}{c}_{i}\right)\ln\frac{{T}^{\prime }}{{T}_{r}}-{q}_{d}{R}_{d}\,\ln\frac{{p}_{d}^{\prime }}{{p}_{dr}}\\ \hfill ={q}_{d}{s}_{dr}+{q}_{t}{s}_{ir}+\left({q}_{d}{c}_{pd}+{q}_{t}{c}_{i}\right)\ln\frac{T}{{T}_{r}}-{q}_{d}{R}_{d}\,\ln\frac{{p}_{d}}{{p}_{dr}}\\ \hfill +\frac{{q}_{v}{L}_{iv}(T)+{q}_{l}{L}_{il}(T)}{T}-{R}_{v}\left({q}_{v}\,\ln{\mathcal{H}}_{i}-{q}_{l}\,\ln\frac{{\mathcal{H}}_{l}}{{\mathcal{H}}_{i}}\right).\end{align*}$

Setting $${p}_{d}^{\prime }={p}_{r}$$ and solving $$T$$ from Equation 29 give the equivalent potential temperature as, 30 $\begin{align*}\hfill {\theta }_{e}=T{\left(\frac{{p}_{r}}{{p}_{d}}\right)}^{\tfrac{{q}_{d}{R}_{d}}{\left({q}_{d}{c}_{pd}+{q}_{t}{c}_{i}\right)}}\mathrm{e}\mathrm{x}\mathrm{p}\left[\frac{{q}_{v}{L}_{iv}(T)+{q}_{l}{L}_{il}(T)}{\left({q}_{d}{c}_{pd}+{q}_{t}{c}_{i}\right)T}-\frac{{R}_{v}\left({q}_{v}\,\ln{\mathcal{H}}_{i}-{q}_{l}\,\ln\frac{{\mathcal{H}}_{l}}{{\mathcal{H}}_{i}}\right)}{{q}_{d}{c}_{pd}+{q}_{t}{c}_{i}}\right].\end{align*}$ To our knowledge, this formulation of equivalent potential temperature has not been published before. However, it is equivalent to the ice equivalent potential temperature in Pauluis (2016). This can be seen by using the entropy expression S6 in Table 2 on the RHS of Equation 4. This leads to the following equation, 31 $\begin{align*}\hfill {q}_{d}{s}_{dr}+{q}_{t}{s}_{ir}+\left({q}_{d}{c}_{pd}+{q}_{t}{c}_{i}\right)\ln\frac{{T}^{\prime }}{{T}_{r}}-{q}_{d}{R}_{d}\,\ln\frac{{p}_{d}^{\prime }}{{p}_{dr}}\\ \hfill ={q}_{d}{s}_{dr}+{q}_{i}{s}_{ir}+\left({q}_{l}+{q}_{v}\right){s}_{lr}+{q}_{v}\left(\frac{{L}_{lv}(T)}{T}-{R}_{v}\,\ln{\mathcal{H}}_{l}\right)\\ \hfill -{q}_{d}{R}_{d}\,\ln\frac{{p}_{d}}{{p}_{dr}}+\left[{q}_{d}{c}_{pd}+{q}_{i}{c}_{i}+\left({q}_{l}+{q}_{v}\right){c}_{l}\right]\ln\frac{T}{{T}_{r}}.\end{align*}$

Solving $${T}^{\prime }$$ when $${p}_{d}^{\prime }={p}_{r}$$ gives the same formulation as $${\theta }_{ei-P16}$$ (Pauluis, 2016), 32 $\begin{align*}\hfill {\theta }_{e}={T}_{r}{\left(\frac{T}{{T}_{r}}\right)}^{\tfrac{{q}_{d}{c}_{pd}+{q}_{i}{c}_{i}+\left({q}_{l}+{q}_{v}\right){c}_{l}}{{q}_{d}{c}_{pd}+{q}_{t}{c}_{i}}}{\left(\frac{{p}_{dr}}{{p}_{d}}\right)}^{\tfrac{{q}_{d}{R}_{d}}{{q}_{d}{c}_{pd}+{q}_{t}{c}_{i}}}\mathrm{e}\mathrm{x}\mathrm{p}\left[\frac{\left({q}_{v}+{q}_{l}\right)\frac{{L}_{il}\left({T}_{r}\right)}{{T}_{r}}+{q}_{v}\frac{{L}_{lv}(T)}{T}-{q}_{v}{R}_{v}\,\ln{\mathcal{H}}_{l}}{{q}_{d}{c}_{pd}+{q}_{t}{c}_{i}}\right].\end{align*}$ While the entropy expressions in the RHS of Equation 29 and Equation 31 are different, they are mathematically equivalent because both are different transformations of Equation 5, as shown in Section 2.2. Since the expressions in the LHS of Equation 29 and Equation 31 are the same due to the same final state, the equivalent potential temperatures Equations 30 and 32 should be identical despite their different formulations.

Physical Interpretation

The three examples in Sections 3.1–3.3 cover most of the equivalent potential temperatures in the literature listed in Table 1, except the entropy potential temperature (Marquet, 2011), which will be discussed in the next section. In general, no sophisticated mathematical derivations are used in deriving these equivalent potential temperatures, highlighting the advantage of our general pathway. However, it remains unclear how the final states in Sections 3.1–3.3 are achieved through adiabatic processes. In this section, we will give physical interpretations of adiabatic processes that lead to the different final states with their corresponding expressions of total moist specific entropy in the RHS of Equation 4.

We start from the first example in Section 3.1, in which the final state of the air parcel has dry air and water vapor. For a parcel with all three phases at the initial state, we can lower the air parcel adiabatically to first melt the ice water into the liquid water, all of which evaporates into water vapor afterward. The total specific entropy of the air parcel can be logically written as Equation 10 $$s={q}_{d}{s}_{d}+{q}_{t}{s}_{v}-{q}_{l}\left({s}_{v}-{s}_{l}\right)-{q}_{i}\left({s}_{v}-{s}_{i}\right),$$ in which the third and last term represent the transformation from liquid water to water vapor and from ice water to water vapor, respectively. The first two terms $${q}_{d}{s}_{d}+{q}_{t}{s}_{v}$$ represent that air contains dry air and water vapor after adiabatic processes. The parcel with dry air and water vapor is then brought to the reference level. If we assume that the adiabatic phase changes occur under phase equilibrium, the total specific entropy at the final state is equal to that at the initial state (Equation 25), and the equivalent potential temperature can be defined as the temperature of air parcel acquired at the reference level.

We can also perform a thought experiment by moving the air parcel adiabatically so that all the ice water melts into the liquid water first and all the water vapor condenses into the liquid water afterward. Then bringing the parcel adiabatically to the reference level leads to the final state in the example of Section 3.2. In the RHS of Equation 11, the term $${q}_{v}\left({s}_{v}-{s}_{l}\right)$$ represents the condensation of water vapor and $${-}{q}_{i}\left({s}_{l}-{s}_{i}\right)$$ denotes the melting of ice water. $${q}_{d}{s}_{d}+{q}_{t}{s}_{l}$$ means that the adiabatic processes leave dry air and liquid water in the air parcel.

In the third example of Section 3.3, a thought experiment can be performed by raising the parcel adiabatically, conserving the total water content, until all the water vapor and liquid water turn into the ice phase under full equilibrium. In this circumstance, the air parcel has dry air and only ice water. Hence the total specific entropy of the air parcel in the RHS of Equation 12 should have a term $${q}_{d}{s}_{d}+{q}_{t}{s}_{i}$$, since all liquid water and ice water have turned into ice. The other two terms $${q}_{l}\left({s}_{l}-{s}_{i}\right)$$ and $${q}_{v}\left({s}_{v}-{s}_{i}\right)$$ represent the adiabatic processes of freezing and depositing, respectively.

In the above examples, the final state of the air parcel has only one type of water phase, with another two phases transformed into the final remaining phase through a sequential of adiabatic processes. Other final states with two water phases are also possible. Thought experiments can be performed by assuming only one phase change (e.g., condensation/evaporation, freezing/melting, sublimation/deposition) during the adiabatic displacement, rather than the two phase change processes, as illustrated in the expressions of Equations 10–12. These lead to six other different formulations of moist specific entropy (S5–S10 in Table 2). Note that all formulations are mathematically equivalent but imply different adiabatic processes under full equilibrium. The details of the adiabatic processes corresponding to each formulation are explained in the second column of Table 2.

The formulations of specific moist entropy in previous studies are special cases in Table 2. For example, S2 and S3 are exactly the same as the formulations in Marquet (2011) and Hauf and Höller (1987), respectively. S1 reduces to the formulation used in Romps and Kuang (2010) if the reference entropies of dry air and liquid water are set to zero. S3 reduces to the formulation in Emanuel (1994) when ice water is excluded and the reference entropies of dry air and liquid water are also set to zero. S6 become the formulations in Pauluis (2016) when the reference entropies of dry air and ice water are set to zero, respectively. Most importantly, we find other formulations of S4, S5, S7, S8, S9, and S10 that do not exist in any previous studies. Although all formulations are mathematically identical when the reference entropies are considered, they imply different adiabatic processes and provide multiple options to define new formulations of equivalent potential temperature, as will be discussed in Section 5.

Strictly speaking, the hypothesized adiabatic processes corresponding to S2–S10 in Table 2 are highly idealized. There are several cases in which the total moist entropy is conserved. For example, adiabatic phase changes occur under full phase equilibrium $$\left({g}_{v}={g}_{l}={g}_{i}\right)$$, or adiabatic processes occur out of equilibrium but with no phase changes and exchanges with outside ($$d{q}_{k}=0$$ in Equation 3). However, not all the adiabatic processes can guarantee moist entropy conservation. Many processes (e.g., irreversible phase changes, diffusive mixing) can result in irreversible increase of total entropy (Pauluis & Held, 2002a, 2002b; Singh & O’Neill, 2022). Taking S4 or Equation 12 as an example, water vapor cannot fully turn into ice since equilibrium requires that the water vapor pressure should be saturated with respect to the ice, and hence implying that water vapor must exist under equilibrium even most of the water mass is in ice phase. The final state can only be achieved asymptotically in highly idealized process. Turning all the water vapor into ice in reality produces extra entropy due to the irreversible process and therefore cannot guarantee the conservation of total specific moist entropy. Similarly, other hypothesized adiabatic processes (S4, S6, S9) that eliminate water vapor in final state are irreversible and result in entropy production. Another possibility that can produce entropy is an initially unsaturated or super-saturated parcel. The parcel must first be brought to equilibrium and the entropy increases irreversibly before the subsequent hypothesized adiabatic processes occur. Furthermore, the air parcel may experience out-of-equilibrium when returning to the reference level after the phase changes are complete, because the saturation water vapor pressure changes during the process so that the phase equilibrium cannot be maintained. In this case, the exchange of water constituents with the environment (e.g., entrainment, precipitation) will result in entropy production. Hence, it is difficult to guarantee entropy conservation during the hypothesized adiabatic processes. These should be taken into account when using our general pathway for future applications.

New Formulations of Equivalent Potential Temperature

In addition to the three examples in Section 3, there are another six cases in which two water phases are present in the final states, as shown in Section 4. Following the general pathway, we are able to find some new formulations of equivalent potential temperature based on these cases that have not been discussed in previous studies. Here, we give an example to demonstrate how the new formulation can be derived. Other new formulations can be found following similar procedures.

We start from a final state with dry air, liquid water and ice water in the air parcel. This is related to the adiabatic processes corresponding to S9 in Table 2. To ensure that reference entropies cancel in the expression for $${\theta }_{e}$$, the specific humidity of liquid water in the final state should be the same as that in the initial state, and the specific humidity of ice water in the final state should equal the sum of those of ice water and water vapor in the initial state.

The total entropy of the parcel in the final state can be written as $${q}_{d}{s}_{dr}+{q}_{l}{s}_{lr}+\left({q}_{v}+{q}_{i}\right){s}_{ir}+\left[{q}_{d}{c}_{pd}+{q}_{l}{c}_{l}+\left({q}_{v}+{q}_{i}\right){c}_{i}\right]\ln\,\left({T}^{\prime }/{T}_{r}\right)-{q}_{d}{R}_{d}\,\ln\left({p}_{d}^{\prime }/{p}_{dr}\right)$$. Assuming full equilibrium during the adiabatic processes, its entropy should be the same as the total entropy in the initial state, that is, S9 in Table 2. This leads to the following equation, 33 $\begin{align*}\hfill {q}_{d}{s}_{dr}+{q}_{l}{s}_{lr}+\left({q}_{v}+{q}_{i}\right){s}_{ir}+\left[{q}_{d}{c}_{pd}+{q}_{l}{c}_{l}+\left({q}_{v}+{q}_{i}\right){c}_{i}\right]\ln\frac{{T}^{\prime }}{{T}_{r}}-{q}_{d}{R}_{d}\hspace*{.5em}\ln\frac{{p}_{d}^{\prime }}{{p}_{dr}}\\ \hfill ={q}_{d}{s}_{dr}+{q}_{l}{s}_{lr}+\left({q}_{v}+{q}_{i}\right){s}_{ir}+\left[{q}_{d}{c}_{pd}+{q}_{l}{c}_{l}+\left({q}_{v}+{q}_{i}\right){c}_{i}\right]\ln\frac{T}{{T}_{r}}-{q}_{d}{R}_{d}\hspace*{.5em}\ln\frac{{p}_{d}}{{p}_{dr}}\\ \hfill +{q}_{v}\left(\frac{{L}_{iv}(T)}{T}-{R}_{v}\hspace*{.5em}\ln{\mathcal{H}}_{i}\right).\end{align*}$

Setting $${p}_{d}^{\prime }={p}_{r}$$, the equivalent potential temperature of the parcel can be defined as, 34 $\begin{align*}\hfill {\theta }_{e-GT25}=T{\left(\frac{{p}_{r}}{{p}_{d}}\right)}^{\tfrac{{q}_{d}{R}_{d}}{\left[{q}_{d}{c}_{pd}+{q}_{l}{c}_{l}+\left({q}_{v}+{q}_{i}\right){c}_{i}\right]}}\mathrm{e}\mathrm{x}\mathrm{p}\left(\frac{{q}_{v}{L}_{iv}(T)-{q}_{v}{R}_{v}T\,\ln{\mathcal{H}}_{i}}{\left[{q}_{d}{c}_{pd}+{q}_{l}{c}_{l}+\left({q}_{v}+{q}_{i}\right){c}_{i}\right]T}\right).\end{align*}$ To our knowledge, this formulation of equivalent potential temperature has not been published before. Compared with other formulations, the new formulation follows an adiabatic process similar to the Wegener–Bergeron–Findeisen process, in which the ice water increases in the mixed-phase clouds in compensate with the consumption of water vapor (no conversion from liquid water to ice water) because the water vapor pressure falls between the saturation vapor pressure over water and the lower saturation vapor pressure over ice. Therefore, our result indicates a potential advantage of the new formulation to be applied for studying deep convection when there are mixed phase water and ice water in the upper-troposphere. However, one should keep in mind that entropy production will occur due to the thermodynamic non-equilibrium during the above adiabatic processes and hence the entropy conservation can only be achieved in highly idealized processes.

In general, we are able to find other formulations of equivalent potential temperatures that do not exist in the literature. Theoretically, there are 10 different processes that can bring the air parcel to the reference level, as illustrated in Table 2. The total specific moist entropy on the RHS of Equation 4 can also take 10 different forms as in Table 2. Hence, there should be 100 different equivalent potential temperatures, in terms of the exact formulations. However, since all the formulations of total specific entropy in Table 2 are mathematically equivalent, there are essentially 10 different formulations of equivalent potential temperature under adiabatic processes. The reader can define their own equivalent potential temperature following the general pathway with or without approximations for other specified adiabatic processes.

Entropy Potential Temperature

In Section 3, it is shown that three formulations of potential temperature in previous studies, that is, $${\theta }_{\mathit{il5-BF04}}$$ (Bryan & Fritsch, 2004), $${\theta }_{e-HH87}$$ (Hauf & Höller, 1987) and $${\theta }_{ei-P16}$$ (Pauluis, 2016) can be derived as the temperature of an air parcel has been moved to the reference level while conserving the moist entropy under certain adiabatic processes. Other formulations can be deduced with appropriate approximations. The results indicate that equivalent potential temperatures are the temperature measuring the moist entropy, echoing the findings of Hauf and Höller (1987) that various definitions of potential temperature can be unified by a single one which they call entropy potential temperature. However, Hauf and Höller (1987) only mathematically defined the potential temperature as the exponential of moist entropy without clearly stating the physical processes involved. This method can only lead to the formulation of $${\theta }_{e-HH87}$$ and its associated ones. Our pathway generally recovers other new formulations after Hauf and Höller (1987), because we do not only relate the equivalent potential temperatures with moist entropy, but also explicitly consider various possible adiabatic processes.

Nevertheless, the above equivalent potential temperatures can not exactly measure the moist entropy when the total water specific humidity $${q}_{t}$$ changes, since the moist entropy equals the logarithm of equivalent potential temperature times specific heat capacity with $${q}_{t}$$ involved (e.g., $$s=\left({c}_{pd}{q}_{d}+{c}_{l}{q}_{t}\right)\ln{\theta }_{e-HH87}+{q}_{d}{s}_{dr}+{q}_{t}{s}_{lr}$$). Marquet (2011) defined an entropy potential temperature $${\theta }_{s-M11}$$ (see Table 1) with contributions from the reference entropies of water vapor and dry air as an exponential term $$\mathrm{e}\mathrm{x}\mathrm{p}\left[{q}_{t}\left({s}_{vr}-{s}_{dr}\right)/{c}_{pd}\right]$$, which has been argued to be necessary to define a unique potential temperature that can be synonymous with the moist entropy (Marquet, 2011, 2016, 2017; Marquet & Dauhut, 2018). The key difference is that the equivalent potential temperatures in Section 3 is purely derived from a sequential adiabatic processes and thus the ultimate temperature of the air parcel achieved at the reference level does not depend on the reference state. In the following, we will show that the relevance of reference entropies in defining potential temperatures is due to the purpose of representing the total moist specific entropy with a pure dry air parcel, which is not a result of sequential process. In this case, the total specific entropy of a dry air is $${s}_{dr}+{c}_{pd}\hspace*{.5em}\ln\left({T}^{\prime }/{T}_{r}\right)-{R}_{d}\hspace*{.5em}\ln\left({p}_{d}^{\prime }/{p}_{r}\right)$$. We use the formulation S3 from Table 2 to represent the total specific moist entropy of an air parcel been moved so that all the liquid water evaporates and ice water sublimate into water vapor. Then equating the two gives 35 $\begin{align*}\hfill {s}_{dr}+{c}_{pd}\hspace*{.5em}\ln\frac{{T}^{\prime }}{{T}_{r}}-{R}_{d}\hspace*{.5em}\ln\frac{{p}_{d}^{\prime }}{{p}_{dr}}={q}_{d}{s}_{dr}+{q}_{t}{s}_{vr}+\left({q}_{d}{c}_{pd}+{q}_{t}{c}_{pv}\right)\ln\frac{T}{{T}_{r}}-{q}_{d}{R}_{d}\hspace*{.5em}\ln\frac{{p}_{d}}{{p}_{dr}}-{q}_{t}{R}_{v}\hspace*{.5em}\ln\frac{e(T)}{{e}_{sl}\left({T}_{r}\right)}\\ \hfill -\frac{{q}_{l}{L}_{lv}(T)+{q}_{i}{L}_{iv}(T)}{T}+{R}_{v}\left({q}_{l}\hspace*{.5em}\ln{\mathcal{H}}_{l}+{q}_{i}\hspace*{.5em}\ln{\mathcal{H}}_{i}\right),\end{align*}$ where $${s}_{vr}$$ is the reference entropy of water vapor. Such manipulation assumes that the entropy of the dry air parcel should be exactly the same as the total specific moist entropy of the moist air parcel. Although we can assume pseudo-adiabatic process in which the condensate drops out and the air parcel becomes purely dry, such process does not guarantee conservation of moist entropy. Hence Equation 35 actually does not imply meaningful physical process as the adiabatic processes in Sections 3 and 4. If we assume the dry air takes the total moist entropy of the air parcel in previous steps, we are able to get the temperature $${T}^{\prime }$$ of the parcel when it is moved to the reference pressure level as 36 $\begin{align*}\hfill {\theta }_{\mathit{es1-GT25}}=T{\left(\frac{T}{{T}_{r}}\right)}^{\tfrac{{q}_{t}\left({c}_{pv}-{c}_{pd}\right)}{{c}_{pd}}}{\left(\frac{{p}_{dr}}{{p}_{d}}\right)}^{\tfrac{{q}_{d}{R}_{d}}{{c}_{pd}}}{\left[\frac{{e}_{sl}\left({T}_{r}\right)}{e(T)}\right]}^{\tfrac{{q}_{t}{R}_{v}}{{c}_{pd}}}\mathrm{e}\mathrm{x}\mathrm{p}\left[\frac{{q}_{t}\left({s}_{vr}-{s}_{dr}\right)}{{c}_{pd}}\right]\times \\ \hfill \mathrm{e}\mathrm{x}\mathrm{p}\left[-\frac{{q}_{l}{L}_{lv}(T)+{q}_{i}{L}_{iv}(T)}{{c}_{pd}T}+\frac{{R}_{v}\left({q}_{l}\hspace*{.5em}\ln{\mathcal{H}}_{l}+{q}_{i}\hspace*{.5em}\ln{\mathcal{H}}_{i}\right)}{{c}_{pd}}\right],\end{align*}$ where $${\theta }_{\mathit{es1-GT25}}$$ is defined as the entropy potential temperature. The reference entropies of water vapor and dry air come into the formulation as $$\mathrm{e}\mathrm{x}\mathrm{p}\left[{q}_{t}\left({s}_{vr}-{s}_{dr}\right)/{c}_{pd}\right]$$, as in Marquet (2011). The total specific moist entropy can be written as a function of $${\theta }_{\mathit{es1-GT25}}$$, 37 $\begin{align*}\hfill s={s}_{dr}+{c}_{pd}\hspace*{.5em}\ln\frac{{\theta }_{\mathit{es1-GT25}}}{{T}_{r}}.\end{align*}$

The total specific moist entropy can also be rewritten using the entropy potential temperature $${\theta }_{s-M11}$$ defined in Marquet (2011) as 38 $$s=\left(1-{q}_{r}\right){s}_{dr}+{q}_{r}{s}_{vr}+{c}_{pd}\,\ln\frac{{\theta }_{s-M11}}{{\theta }_{sr}},$$ where $${q}_{r}$$ and $${\theta }_{sr}$$ are two constants. $${q}_{r}$$ is the specific humidity of saturated water vapor at reference state, and $${\theta }_{sr}$$ is defined as $${\theta }_{sr}\equiv {T}_{r}{\left(\frac{{p}_{0}}{{p}_{r}}\right)}^{\tfrac{{R}_{d}}{{c}_{pd}}}\mathrm{e}\mathrm{x}\mathrm{p}\left[\frac{{q}_{r}\left({s}_{vr}-{s}_{dr}\right)}{{c}_{pd}}\right]{\left(1+\frac{{R}_{v}}{{R}_{d}}{r}_{r}\right)}^{\tfrac{{R}_{d}}{{c}_{pd}}},$$ where $${p}_{0}$$ equals $${p}_{r}$$ and they are both reference pressure; $${r}_{r}$$ is the saturated mixing ratio of water vapor at reference state. Therefore, $${\theta }_{\mathit{es1-GT25}}$$ and $${\theta }_{s}$$ are almost identical except a constant, $${\theta }_{s-M11}={\theta }_{\mathit{es1-GT25}}{\left(1+{R}_{v}{r}_{r}/{R}_{d}\right)}^{{R}_{d}/{c}_{pd}}.$$ Following similar procedures, different formulations of entropy potential temperature can be derived by using other expressions of total specific moist entropy corresponding to other adiabatic processes in the RHS of Equation 35. For example, if the expression of S3 in Table 2 is used, we can get another formulation of entropy potential temperature as, 39 $\begin{align*}\hfill {\theta }_{\mathit{es2-GT25}}=T{\left(\frac{T}{{T}_{r}}\right)}^{\tfrac{{q}_{t}\left({c}_{l}-{c}_{pd}\right)}{{c}_{pd}}}{\left(\frac{{p}_{dr}}{{p}_{d}}\right)}^{\tfrac{{q}_{d}{R}_{d}}{{c}_{pd}}}\mathrm{e}\mathrm{x}\mathrm{p}\left[\frac{{q}_{t}\left({s}_{lr}-{s}_{dr}\right)}{{c}_{pd}}\right]\times \\ \hfill \mathrm{e}\mathrm{x}\mathrm{p}\left[\frac{{q}_{v}{L}_{lv}(T)-{q}_{i}{L}_{il}(T)}{{c}_{pd}T}-\frac{{R}_{v}\left({q}_{v}\,\ln{\mathcal{H}}_{l}+{q}_{i}\,\ln\frac{{\mathcal{H}}_{l}}{{\mathcal{H}}_{i}}\right)}{{c}_{pd}}\right].\end{align*}$

Despite the quite different expressions, $${\theta }_{\mathit{es1-GT25}}$$ and $${\theta }_{\mathit{es2-GT25}}$$ are essentially the same because both formulations satisfy the following relation as in Equation 37, 40 $\begin{align*}\hfill s={s}_{dr}+{c}_{pd}\,\ln\frac{{\theta }_{es}}{{T}_{r}}.\end{align*}$

Thus, the entropy potential temperature can be generally defined as, 41 $\begin{align*}\hfill {\theta }_{s-GT25}={T}_{r}\mathrm{e}\mathrm{x}\mathrm{p}\left(\frac{s-{s}_{dr}}{{c}_{pd}}\right).\end{align*}$

Unlike the adiabatic process in Section 3, we are always able to end up with the same entropy potential temperature whatever the formulation of total specific moist entropy is used on the RHS of Equation 37. However, the entropy potential temperature defined in this pathway is not a result of following a sequence adiabatic processes and thereby depends on the reference entropies.

Conclusions and Discussions

Equivalent potential temperature is widely used in weather and climate systems across a wide range of temporal and spatial scales. Various formulations of equivalent potential temperature are proposed in the literature but their differences are not easily understood. In this study, the two typical pathways are combined to form a more general and physically consistent pathway, the so called “moist entropy conservation” under certain adiabatic processes, to define the equivalent potential temperature. All existing formulations of equivalent potential temperature, except the empirical ones (e.g., Bolton, 1980; Davies-Jones, 2009), can be covered following this pathway. Table 3 summarizes how different equivalent potential temperatures in previous literature can be found under different adiabatic processes with appropriate approximations. Our study provides a general method to define other new formulations of equivalent potential temperature, depending on the exact adiabatic processes in Table 2, some of which have not been considered before. An example is provided to illustrate how a new formulation of equivalent potential temperatures can be defined under adiabatic processes, during which all the water vapor sublimates into ice water while the liquid water remains unchanged. Compared with previous pathways, our general pathway significantly simplifies the mathematical derivations and, meanwhile, provides a clear picture of physical processes without inappropriate approximations (e.g., condensates fall out of the air parcel).

Table 3 Various Formulations of Potential Temperature in the Literature and Our New Formulations (Last Three Rows)

Formulations	Derivation pathway and approximations
$${\theta }_{\mathit{il5-BF04}}$$	Three water phases included;
Adiabatic process corresponding to S2 in Table 2;
$${\theta }_{\mathit{il4-BF04}}$$	Same as $${\theta }_{\mathit{il5-BF04}}$$, phase equilibrium in initial state;
$${\theta }_{l-B73}$$	Approximation of $${\theta }_{\mathit{il5-BF04}}$$;
Only water vapor and liquid water;
No impact of water vapor on specific heat capacity;
Phase equilibrium in initial state;
$${\theta }_{il-TC81}$$	Approximation of $${\theta }_{\mathit{il5-BF04}}$$;
Three water phases included;
No impact of water vapor on specific heat capacity;
Phase equilibrium in initial state; total pressure equals dry air pressure;
Latent heat of vapourization and sublimation at triple point;
$${\theta }_{l-E94}$$	Approximation of $${\theta }_{\mathit{il5-BF04}}$$;
Only water vapor and liquid water, phase equilibrium in initial state;
$${\theta }_{e-HH87}$$	Three water phases included;
$${\theta }_{q-P84}$$	Adiabatic process corresponding to S3 in Table 2;
$${\theta }_{e-B73}$$	Approximation of $${\theta }_{e-HH87}$$;
Only water vapor and liquid water;
No impact of liquid water on specific heat capacity;
Phase equilibrium in initial state;
$${\theta }_{q-P79}$$	Approximation of $${\theta }_{e-HH87}$$;
Only water vapor and liquid water;
Phase equilibrium in initial state;
$${\theta }_{\mathit{eiv-TC81}}$$	Approximation of $${\theta }_{e-HH87}$$; three water phases included;
No impact of liquid water on specific heat capacity;
Phase equilibrium in initial state;
Latent heat of condensation and melting at triple point;
$${\theta }_{e-E94}$$	Approximation of $${\theta }_{e-HH87}$$;
Only water vapor and liquid water;
$${\theta }_{e-RK10}$$	Three water phases included;
Adiabatic process corresponding to S1 in Table 2;
Reference entropies of dry air and liquid water set to zero;
$${\theta }_{ei-P16}$$	Three water phases included;
Adiabatic process corresponding to S4 in Table 2;
$${\theta }_{s-M11}$$	Equating total specific entropy of the dry air parcel
With total specific moist entropy of the moist air parcel;
No approximations; depend on reference entropies;
$${\theta }_{e-GT25}$$	Three water phases included;
adiabatic process corresponding to S9 in Table 2;
$${\theta }_{\mathit{es1-GT25}}$$	No approximations; depend on reference entropies;
$${\theta }_{\mathit{es2-GT25}}$$	No approximations; depend on reference entropies;
$${\theta }_{s-GT25}$$	The general form of entropy potential temperature;
No approximations; depend on reference entropies;

We also show that the entropy potential temperature can be uniquely defined if we let the entropy of a dry air parcel equal the total specific moist entropy of the initially moist air parcel. Therefore, the entropy potential temperature is not purely derived from a sequence of physical processes while conserving moist entropy and thus leads to dependence on the reference entropies. Because the adiabatic processes in our pathway occur continuously, one advantage of the new method is that the equivalent potential temperatures do not depend on the reference entropies, which cannot be physically measured. One should also keep in mind that our general pathway to define the equivalent potential temperatures depends on the exact adiabatic processes leading to different final states. Some of these processes cannot strictly guarantee the conservation of moist entropy because of the lack of full phase equilibrium during phase changes or having mass exchanges of water constituents with the environment. Therefore, the degree to which different equivalent potential temperatures can be conserved depends on the adiabatic processes. For better thermodynamic analyses, one should choose an equivalent potential temperature defined from some specific adiabatic processes that conserve total moist entropy the most.

In the development of numerical models, there is an effort to use equivalent potential temperatures as prognostic variables in dynamical cores (Heus et al., 2010; Stevens et al., 2005) or in the physical parameterizations (Larson, 2017). Inconsistent thermodynamic approximations, for example, neglecting the heat capacity of condensate while there are liquid or ice water in the parcel; constant assumption of latent heat during phase change while the heat capacities of condensates are different; neglecting the non-equilibrium process while the parcel is sub-saturated, should be avoided such that laws of thermodynamics are respected and global energy budget can be improved (Bowen & Thuburn, 2022a, 2022b; Lauritzen et al., 2022; Ohno & Matsugishi, 2024; Thuburn, 2017). Our general pathway does not necessarily invoke any inappropriate approximations to define the equivalent potential temperatures and thus provides useful insights for choosing appropriate thermodynamic variables for model development.

The fact that the two typical pathways can be essentially combined to form a general pathway suggest that, compared with equivalent potential temperature, the moist entropy can be a better option as a prognostic thermodynamic variable used in the numerical models (Ooyama, 1990, 2001; Pressel et al., 2015; Zeng et al., 2005, 2008). This is because the total specific moist entropy is essentially the same under reversible processes, while on the contrary, the exact formulation of equivalent potential temperature is defined based on the adiabatic process that are difficult to conserve moist entropy, as discussed in Section 4. In addition, the total specific moist entropy is an extensive variable while the equivalent potential temperature is not. As a result, for adiabatic and reversible motions, both resolved and unresolved, the moist entropy can be mixed linearly and thus is appropriate for sub-grid scale models. Furthermore, the extensive property of moist entropy guarantees that the discretized thermodynamic equation obeys conservation law and hence can help improve the entropy balance in the numerical models.

Acknowledgments

We sincerely thank Professors Maarten Ambaum, Robert Plant, Jun-ichi Yano, Xin Qiu and Bowen Zhou for their helpful discussions. We are also grateful to the anonymous reviewers and Professor John Thuburn for their critical feedback, which significantly enhances the clarity and precision of the thermodynamic expressions and explanations, and strengthened the overall organization of this study. This study is supported by the National Natural Science Foundation of China under Grants 42192555 and the Fundamental Research Funds for the Central Universities 0207/14380225, 0207/14912209.

Notation

• $$\rho $$
• Density of moist air
• $$\alpha =\frac{1}{\rho }$$
• specific volume of moist air
• $${\rho }_{d}$$
• Density of dry air
• $${\rho }_{v}$$
• Density of water vapor
• $$T$$
• Air temperature
• $${T}_{r}$$
• Air temperature at reference state (273.15 K)
• $$p$$
• Air pressure
• $${p}_{0}$$
• Reference pressure (1,000 hPa)
• $${p}_{r}$$
• Reference pressure
• $${p}_{d}$$
• Partial pressure of dry air
• $${p}_{dr}$$
• Partial pressure of dry air at reference state
• $${R}_{d}$$
• Gas constant for dry air
• $${R}_{v}$$
• Gas constant for water vapor
• $${R}_{m}$$
• Gas constant for moist air
• $${\epsilon}$$
• $${=}\frac{{R}_{d}}{{R}_{v}}$$
• $${c}_{pd}$$
• Specific heat capacity for dry air at constant pressure
• $${c}_{pv}$$
• Specific heat capacity for water vapor at constant pressure
• $${c}_{l}$$
• Specific heat capacity for liquid water at constant volume
• $${c}_{i}$$
• Specific heat capacity for ice water at constant volume
• $${c}_{pm}$$
• Specific heat capacity for moist air
• $$e$$
• Partial pressure of water vapor
• $$e(T)$$
• Pressure of water vapor at temperature $$T$$
• $${e}_{r}$$
• Partial pressure of water vapor at reference state (611 Pa)
• $${e}_{sl}(T)$$
• Saturated water vapor pressure with respect to the liquid water surface at temperature $$T$$
• $${e}_{si}(T)$$
• Saturated water vapor pressure with respect to the surface of ice water at temperature $$T$$
• $${e}_{sl}\left({T}_{r}\right)$$
• Saturated water vapor pressure with respect to the liquid water surface at reference temperature $${T}_{r}$$
• $${q}_{d}$$
• Specific humidity of dry air
• $${q}_{v}$$
• Specific humidity of water vapor
• $${q}_{l}$$
• Specific humidity of liquid water
• $${q}_{i}$$
• Specific humidity of ice water
• $${q}_{t}$$
• Specific humidity of total water
• $${r}_{v}$$
• Mixing ratio of water vapor
• $${r}_{r}$$
• Saturated mixing ratio of water vapor at reference state
• $${r}_{t}$$
• Mixing ratio of total water
• $${q}_{r}$$
• Specific humidity of water at reference state $$\left(=\frac{{r}_{r}}{1+{r}_{r}}\right)$$
• $$s$$
• Total specific entropy of moist air
• $${s}_{d}$$
• Specific entropy of dry air
• $${s}_{v}$$
• Specific entropy of water vapor
• $${s}_{l}$$
• Specific entropy of liquid water
• $${s}_{i}$$
• Specific entropy of ice water
• $${s}_{dr}$$
• Specific entropy of dry air at reference state
• $${s}_{vr}$$
• Specific entropy of water vapor at reference state
• $${s}_{lr}$$
• Specific entropy of liquid water at reference state
• $${s}_{ir}$$
• Specific entropy of ice water at reference state
• $${L}_{lv}(T)$$
• Latent heat of vapourization
• $${L}_{il}(T)$$
• Latent heat of melting
• $${L}_{iv}(T)$$
• Latent heat of sublimation
• $${g}_{v}(T)$$
• Gibbs free energy of water vapor at temperature $$T$$
• $${g}_{l}(T)$$
• Gibbs free energy of liquid water at temperature $$T$$
• $${g}_{i}(T)$$
• Gibbs free energy of ice water at temperature $$T$$
• $${A}_{lv}(T)$$
• $${=}{g}_{l}(T)-{g}_{v}(T)$$
• $${A}_{il}(T)$$
• $${=}{g}_{i}(T)-{g}_{l}(T)$$
• $${A}_{iv}(T)$$
• $${=}{g}_{i}(T)-{g}_{v}(T)$$
• $${\mathcal{H}}_{l}$$
• Relative humidity with respect to the surface of liquid water at temperature $$T$$
• $${\mathcal{H}}_{i}$$
• Relative humidity with respect to the surface of ice water at temperature $$T$$
• $$\theta $$
• Dry potential temperature
• $${\theta }_{e-N21}$$
• Equivalent potential temperature (Normand, 1921)
• $${\theta }_{{e}_{R}32}$$
• Equivalent potential temperature (Rossby, 1932)
• $${\theta }_{l-B73}$$
• Liquid water potential temperature (Betts, 1973)
• $${\theta }_{e-B73}$$
• Equivalent potential temperature (Betts, 1973)
• $${\theta }_{q-P79}$$
• Wet equivalent potential temperature (Paluch, 1979)
• $${\theta }_{q-P84}$$
• Wet equivalent potential temperature (Pointin, 1984)
• $${\theta }_{il-TC81}$$
• Ice-liquid water potential temperature (Tripoli & Cotton, 1981)
• $${\theta }_{\mathit{eiv-TC81}}$$
• Ice-vapor water equivalent potential temperature (Tripoli & Cotton, 1981)
• $${\theta }_{e-HH87}$$
• Entropy potential temperature (Hauf & Höller, 1987)
• $${\theta }_{e-E94}$$
• Equivalent potential temperature (Emanuel, 1994)
• $${\theta }_{l-E94}$$
• Liquid water potential temperature (Emanuel, 1994)
• $${\theta }_{\mathit{il4-BF04}}$$
• Ice-liquid water potential temperature (Bryan & Fritsch, 2004)
• $${\theta }_{\mathit{il5-BF04}}$$
• Another ice-liquid water potential temperature (Bryan & Fritsch, 2004)
• $${\theta }_{e-RK10}$$
• Equivalent potential temperature (Romps & Kuang, 2010)
• $${\theta }_{s-M11}$$
• Entropy potential temperature (Marquet, 2011)
• $${\theta }_{ei-P16}$$
• Ice equivalent potential temperature (Pauluis, 2016)
• $${\theta }_{\mathit{es1-GT25}}$$
• Entropy potential temperature (Equation 36)
• $${\theta }_{\mathit{es2-GT25}}$$
• Entropy potential temperature (Equation 39)
• $${\theta }_{s-GT25}$$
• General form of entropy potential temperature (Equation 41)
• $${\theta }_{e-GT25}$$
• New equivalent potential temperature with (Equation 34)

Data Availability Statement

This is a theoretical study with no plotted figures and data set from observations or numerical simulations in the current version. Any possible further update with practical data set and numerical codes will be released on .

Erratum

The originally published version of this article contained typographical errors. In the last equation in Table 1, the furthest right parenthesis in the numerator of the exponential term has been removed. The thirteenth sentence of Section 3.1, beginning “Equation 26 further reduces to the formulation…” has been moved to the last sentence of the first paragraph of Section 3.2. This may be considered the authoritative version of record.

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