## 1. Introduction

It is possible to derive several different thermodynamical variables that are conserved during moist adiabatic processes. The most commonly used variable in research and operational forecasting is equivalent potential temperature, or *θ _{e}*. Conceptually,

*θ*is the potential temperature an air parcel would have if all the water vapor were condensed by lifting the parcel to zero pressure. (Typically, the ice phase is neglected, and any freezing of the condensed water at low temperatures is not considered; this assumption is also made herein.)

_{e}There are numerous mathematical formulations for *θ _{e}* in the atmospheric sciences literature. Most of them are approximate formulations of varying degrees of accuracy. In the next section, a review is provided of the two most accurate formulations, which are applicable to very different physical situations. One, originally developed by Iribarne and Godson (1973), is appropriate for nonprecipitating clouds. It has the nice property of being an exact formulation (under the physical processes considered). A second formulation, presented by Bolton (1980), is appropriate for clouds that are precipitating. However, it is an empirical formulation based on fits to numerically generated data; consequently, this formulation has been difficult to apply in certain theoretical studies.

The primary purpose of this article is to demonstrate that a reasonably accurate formulation for *θ _{e}* that is applicable to precipitating clouds can be derived with a small number of reasonable assumptions. The subsequent formulation has accuracy that is comparable to other approximate formulations, but has the distinct advantage of having consistent formulations for other thermodynamical variables (e.g., total moist entropy and enthalpy). The new formulation is reasonably accurate, inexpensive, adaptable, and attractive for theoretical studies, which is a combination of characteristics that may be unrivaled by all other formulations that have been presented previously.

## 2. Review

Before developing a mathematical formula for *θ _{e}*, several assumptions must be made about what processes are considered during condensation. For simplicity, the ice phase is typically neglected, and this assumption is also made herein. Another common assumption is that all phases of water shall be in equilibrium during condensation. Consequently, supersaturation is not permitted. Third, a critical assumption must be made about the fate of condensed water. This is a complex problem because liquid water can fall relative to air. Two cases represent the extreme ends of the spectrum: 1) all condensate moves with the air (and, thus, there is no precipitation), or 2) all condensate immediately falls relative to the air (and, thus, precipitation is instantaneous). These processes are considered separately in sections 2a and 2b.

### a. Reversible thermodynamics

*reversible moist adiabatic*process. It is probably most applicable to shallow clouds and cumulus congestus in which precipitation is negligible. The mathematics of reversible moist adiabatic processes are tractable, and can be found in several texts (e.g., Iribarne and Godson 1981; Emanuel 1994). In this case, the mathematical formulation of total moist entropy (hereinafter referred to as reversible moist entropy, or

*s*) is given by

^{r}*T*is absolute temperature;

*p*is the partial pressure of dry air;

_{d}*R*and

*R*are respectively the gas constants for dry air and water vapor;

_{υ}*c*is the specific heat at constant pressure for dry air;

_{p}*c*is the specific heat of liquid water;

_{l}*r*and

_{υ}*r*are respectively the mixing ratios of water vapor and total water;

_{t}*e*/

*e*is relative humidity, where

_{s}*e*is the partial pressure of water vapor and

*e*is the value of

_{s}*e*at saturated equilibrium at the same temperature; and

*L*is the latent heat of vaporization, which is a function of temperature according to Kirchoff’s Law:

_{υ}*dL*/

_{υ}*dT*=

*c*−

_{pυ}*c*, where

_{l}*c*is the specific heat of water vapor at constant pressure.

_{pυ}*θ*is simply a different measure of entropy. Hereinafter,

_{e}*θ*during reversible adiabatic conditions is referred to as

_{e}*reversible equivalent potential temperature*, or

*θ*

^{r}

_{e}. The variables

*s*and

^{r}*θ*

^{r}

_{e}are related as

*p*

_{0}is a reference pressure (typically 1000 hPa). From (1) and (2), it follows that

*θ*

^{r}

_{e}has an exact mathematical formulation:

Under strictly saturated conditions (i.e., *s ^{r}* was first presented by Iribarne and Godson (1973), and (3) with

*θ*

^{r}

_{e}is referred to as “wet equivalent potential temperature.” The terms with

### b. Pseudoadiabatic thermodynamics

On the other extreme, it can be assumed that all condensed water is immediately removed from a parcel. This is an irreversible process because, consequently, there is no water available for evaporation; thus, a warming (i.e., descending) parcel of air would follow dry adiabatic processes, not a saturated process, and thus a parcel could not be brought back to its original conditions. Following convention, this is referred to as a *pseudoadiabatic process*. Entropy and *θ _{e}* during this process are hereinafter referred to as pseudoadiabatic moist entropy

*s*and pseudoadiabatic equivalent potential temperature

^{p}*θ*

^{p}

_{e}. Because deep cumulus convection typically produces precipitation, these variables are probably most applicable to deep (of order 10 km) clouds. However, they will never be strictly conserved in such clouds because

*some*liquid water is inevitably lofted by convective updrafts.

*s*and

^{p}*θ*

^{p}

_{e}are more complicated than their counterparts from reversible thermodynamics because total water is not constant along a pseudoadiabat. Most derivations of

*s*begin by taking the differential of (1) and then neglect the term involving liquid water mixing ratio

^{p}*r*(because liquid water is immediately removed from air by fallout); this yields

_{l}*s*:

^{p}*T*

_{0}, such that the constants of integration are essentially zero. The term with the integral in (5) makes

*s*expensive to calculate. Furthermore, any attempts to create a mathematical definition of

^{p}*θ*

^{p}

_{e}from (5) would result in a similarly complex and expensive variable. Consequently, terms are usually omitted from these formulations, or approximations to certain terms are made, to circumvent the need for expensive calculations; however, these approximations introduce errors of various degrees. Hence, there are many different approximate mathematical formulations for

*θ*

^{p}

_{e}in the literature (e.g., Betts and Dugan 1973; Simpson 1978; Bolton 1980).

*T*is absolute temperature at the lifting condensation level. Bolton (1980) determined that (6) is accurate to within 0.3 K. Because of its superior accuracy relative to other approximate formulations, this has become a very popular mathematical formulation for

_{L}*θ*

^{p}

_{e}.

*s*was not presented by Bolton (1980). However, by analogy to the derivation in reversible conditions, a formulation for

^{p}*s*can be obtained by working backward from (6); that is

^{p}*θ*

^{p}

_{e}and

*s*, as defined by (6) and (7), are conserved well in pseudoadiabatic conditions.

^{p}There are two practical problems with using these formulations, however. First, Bolton (1980) derived (6) using certain formulations for *e _{s}*(

*T*) and other mathematical constants (

*R*,

*c*, etc.). Thus, (6) is not directly applicable to situations that use different formulations (for

_{p}*e*,

_{s}*R*,

*c*, etc.). For example, if (6) is used with output from a numerical cloud model that uses different formulations, then some error may be incurred.

_{p}The second problem is that these formulas are inconvenient for analytic studies that require moist thermodynamical expressions. As an example, Emanuel (1994, p. 124) used the mathematical formulation for entropy to derive a set of Maxwell relations. These equations can then be used in numerous ways, such as for the derivation of the moist adiabatic lapse rate. These Maxwell relations are also crucial for certain theoretical work, such as the analytical theory for the maximum possible intensity of tropical cyclones (Emanuel 1986, 1988). The complex form of (6) and (7) makes them impractical to use for such studies.

In summary, (6) and (7) may be very accurate and sufficient for most applications. However, their empirical derivation means they are not flexible (e.g., for use with different underlying thermodynamic formulations) and they are not tractable for analytical studies.

## 3. An approximation for pseudoadiabatic thermodynamics

*s*and

^{p}*θ*

^{p}

_{e}that are accurate, adaptable, and yet relatively simple so that they can be used for theoretical studies. To this end, a new derivation begins with a formula that governs entropy changes in pseudoadiabatic conditions (Emanuel 1994, p. 131):

*α*is the specific volume of dry air. This equation is derivable from (4) by using Kirchoff’s equation and the Clausius–Clapeyron equation. It is assumed that

_{d}*s*is constant under any processes considered herein (and thus

^{p}*ds*= 0). For simplicity, the simple case of condensation with pressure held constant (

^{p}*dp*= 0) is considered first. (In other words, the saturated enthalpy is considered first.) Additionally, the latent heat of vaporization is written relative to a constant value

*L*

_{0}, for reasons that will become clear later; that is,

*L*(

_{υ}*T*) ≡

*L*

_{0}+

*L**(

*T*). Under these assumptions, and after rearranging some terms, (8) can be expressed as

*L*

_{0}is

*O*(2.5 × 10

^{6}J kg

^{−1}), then under all atmospheric conditions both terms in parentheses (

*ϵ*

_{1}and

*ϵ*

_{2}) are of order 1. Thus, it would be reasonably accurate to set

*ϵ*

_{1}=

*ϵ*

_{2}= 1 to simplify calculations (e.g., Iribarne and Godson 1981, p. 142). In fact, making this assumption eliminates the expensive integral term in (5), which represents the contribution to entropy from water vapor. On the other hand, some error is incurred by making this assumption, although the error may be minimal for some applications. However, in deep moist convection total condensation is relatively large, and the error incurred by making such assumptions can become unacceptable [as shown, e.g., by Wilhelmson (1977) and Tripoli and Cotton (1981)].

Serendipitously, the value of *L*_{0} is unconstrained; that is, it can be set to *any* value (provided that *L** is redefined appropriately). Furthermore, changes in *L*_{0} imply changes in *ϵ*_{2}. It is possible, then, to choose a value for *L*_{0} such that *ϵ*_{2} ≈ *ϵ*_{1} across a broad range of environments. In other words, the error incurred by one assumption (i.e., neglecting the contribution to entropy by water vapor) can be compensated by another assumption (i.e., neglecting the temperature dependence of *L _{υ}*). Thus, the primary approximation changes from

*ϵ*

_{1}=

*ϵ*

_{2}= 1 to

*ϵ*

_{1}≈

*ϵ*

_{2}. Although several previous studies have used a constant value for

*L*(analogous to the value

_{υ}*L*

_{0}herein), it may be a novel idea to adjust

*L*

_{0}to compensate errors in approximate thermodynamic expressions.

To aid in the following discussion, Fig. 1 displays the variation in *L _{υ}* as a function of temperature (solid line). If

*L*

_{0}is chosen as a relatively large value (as shown in Fig. 1), then

*ϵ*

_{2}would be small at low temperatures; this situation is convenient, because

*ϵ*

_{1}is small at low temperatures (because

*r*is typically small). Thus,

_{υ}*ϵ*

_{1}≈

*ϵ*

_{2}, and both terms drop out of (9).

At relatively high temperatures, *r _{υ}* can be

*O*(0.02) and thus

*ϵ*

_{1}≈ 1.05. In the case shown in Fig. 1, at high temperatures

*ϵ*

_{2}≈ 0.95. So, at this temperature, these two variables are still of order 1, but the agreement is not quite as good. Nevertheless, it is still the case that

*ϵ*

_{1}≈

*ϵ*

_{2}, and thus both terms can be assumed to drop out of (9) with reasonable accuracy.

*L*

_{0}is adjusted to an optimal value to ensure minimal errors. This optimal value for

*L*

_{0}is determined numerically later in this article.

*T*, using the Clausius–Clapeyron equation, Kirchoff’s equation, and the equation of state, and finally integrating from very low temperatures, an approximate formulation for

*s*is obtained:

^{p}*s*is very similar to the formulation for

^{p}*s*[given by (1)], except the term with

^{r}*c*is omitted and

_{l}*L*is replaced by

_{υ}*L*

_{0}. In this case, the relationship between

*s*and

*θ*is given as

_{e}*θ*is

_{e}*s*, this formulation for

^{p}*θ*

^{p}

_{e}is very similar to its reversible counterpart [given by (3)], except that all terms with

*c*are omitted and

_{l}*L*is replaced by

_{υ}*L*

_{0}.

It is interesting to note that the same approximations are made in many numerical cloud models. That is, in many cloud models the terms involving *c _{l}* are omitted from the thermodynamic equation, and

*L*(

_{υ}*T*) is replaced with a constant value (although, the first of these two approximations is more common). The analysis herein suggests that these are adequate assumptions for heavily precipitating cases (i.e., cases wherein the pseudoadiabatic assumption is appropriate). The analysis herein further suggests that an optimal value for

*L*

_{0}can be chosen, such that

*ϵ*

_{1}≈

*ϵ*

_{2}for a broad range of atmospheric conditions. This optimal value is determined during numerical tests in a later section of this article.

## 4. Numerical evaluation

To evaluate the accuracy of the approximate equations derived in the previous section, a numerical code is used to generate moist adiabats under various conditions. The code was described in detail by Bryan and Fritsch (2004). They used it to evaluate several formulations of ice–liquid water potential temperature, but it is easily adapted to evaluate *θ _{e}*.

*L*(

_{υ}*T*) and

*e*(

_{s}*T*); herein, values and formulations are the same as those used by Bolton (1980).

The code works as follows. Values of temperature are obtained using (14) as an air parcel is displaced vertically. Herein, pressure increments of 1 hPa are used. The mixing ratio of total water is conserved during this displacement. If vapor pressure, or *e*, is less than saturation vapor pressure, or *e _{s}*, after a displacement, then

*r*is conserved (

_{υ}*dr*= 0). However, if

_{υ}*e*exceeds

*e*during a vertical displacement, then saturated equilibrium is assumed and water is condensed such that

_{s}*e*=

*e*. An iterative procedure is used to determine the temperature that satisfies both (14) and the condition

_{s}*e*=

*e*(

_{s}*T*).

To obtain an exact pseudoadiabatic lapse rate, (14) is used and condensate is immediately removed after the completion of each vertical displacement. This is standard practice to obtain a pseudoadiabatic lapse rate. A comparison of output from this code with results reported by Bolton (1980, in his Table 3) shows that the code used herein produces essentially identical results; the maximum difference is 0.05 K.

To test the accuracy of the approximate equations developed in the previous section, a separate set of integrations are conducted in which condensed water is carried aloft during vertical displacements (as in a reversible process), but (10) is used as the governing equation (assuming *ds ^{p}* = 0).

Results for a parcel starting at *p* = 1000 hPa and *T* = 30°C are shown in Fig. 2a, assuming *L*_{0} = 2.555 × 10^{6} J kg^{−1}. Two cases are shown in this figure. One case assumes that the initial parcel is saturated (

In the subsaturated case (*ϵ*_{1} (i.e., owing to the neglect of the contribution of water vapor to entropy). Because of this slight relative cooling, this parcel reaches saturation earlier than the parcel using the exact equation. Then, the parcel using the approximate equation follows a saturated pseudoadiabat while the parcel using the exact formulation cools further until saturation is achieved. The difference in saturation points is what accounts for the abrupt change in error at ∼850 hPa.

This test has been repeated for different values of the initial pressure, temperature, and relative humidity. In all cases, results from the approximate equation are good approximations to results from the exact pseudoadiabatic formulation. In fact, the cases shown in Fig. 2 produce the largest errors from all of these tests.

The sensitivity to the value chosen for *L*_{0} is illustrated in Fig. 3, assuming initial values *p* = 1000 hPa, *T* = 30°C, and *L*_{0} has an effect on the accuracy of the solution. For *L*_{0} = 2.50 × 10^{6} J kg^{−1} (the value valid at *T* = 0°C, which is a commonly used constant value of *L _{υ}* in numerical models), results show a cold bias that becomes unacceptably large (1.5 K) at low pressures.

## 5. Evaluation of equivalent potential temperature

In this section, the new formulation for *θ*^{p}_{e} (13) is compared with Bolton’s formulation (6). To this end, the numerical code is used to determine exact values of *θ*^{p}_{e} given specified values of *p*, *T*, and *θ*^{p}_{e} is the value of *θ* at the end of the integration (herein, when the parcel reaches 10 hPa) when using the exact pseudoadiabatic formulation.

Results are listed in Table 1. Also listed are *δ*_{1}, which is defined as the error when using Bolton’s formula, and *δ*_{2}, which is defined as the error when using the new formulation. In general, the new formulation produces larger errors than Bolton’s formulation. However, even the largest error (∼0.4 K), which occurs in very warm and moist environments, is acceptable for most applications.

As mentioned earlier, the value of *L*_{0} is unconstrained in the new approximate equations, but an optimal value should be chosen such that *ϵ*_{1} ≈ *ϵ*_{2} in most environments. To determine this optimal value, the tests shown in Table 1 were repeated with different values of *L*_{0} until the mean-square error of all cases listed in Table 1 was minimized. The value *L*_{0} = 2.555 × 10^{6} J kg^{−1} produced the best result and therefore has been used for all results shown herein.

## 6. Summary

New approximate mathematical formulas for pseudoadiabatic entropy, or *s ^{p}*, and equivalent potential temperature, or

*θ*

^{p}

_{e}, are derived herein. The primary assumptions are that the contribution to entropy from water vapor can be neglected (because

*ϵ*

_{1}≈ 1) and that the subsequent error can be compensated by using a constant value for the latent heat of vaporization (because

*ϵ*

_{2}≈ 1). These assumptions have been considered many times previously (e.g., Iribarne and Godson 1981). However, a further important conclusion that probably has not been presented previously is that

*L*

_{0}can be assigned a value that ensures that these two assumptions have compensating errors (i.e.,

*ϵ*

_{1}≈

*ϵ*

_{2}). Numerical evaluation herein has determined an optimal value of

*L*

_{0}= 2.555 × 10

^{6}J kg

^{−1}.

Further numerical evaluation herein demonstrates that the approximate thermodynamic equation in (10) can accurately reproduce the exact pseudoadiabatic process. Furthermore, consistent mathematical formulations are derived for pseudoadiabatic moist entropy, given by (11), and pseudoadiabatic equivalent potential temperature, given by (13). Most significant is that these formulations do not require expensive numerical integration (as do the exact equations for a pseudoadiabatic process), and were not derived using empirical fits to data [as was done by Bolton (1980)].

In tests with a numerical code, the new formulation of *θ*^{p}_{e} is found to have larger inherent errors when compared with the empirical formulation derived by Bolton (1980), although the errors are acceptable (<0.4 K) for most practical applications. Unlike Bolton’s formulation, the new formulation has a consistent mathematical definition for *s ^{p}* and a consistent first law of thermodynamics. Because these formulations are expressed in a convenient and simple physical form, they should be very useful for theoretical applications.

## Acknowledgments

The author thanks Richard Rotunno and Kerry Emanuel for their insightful reviews.

## REFERENCES

Betts, A. K., and F. J. Dugan, 1973: Empirical formula for saturation psuedoadiabats and saturation equivalent potential temperature.

,*J. Appl. Meteor.***12****,**731–732.Bolton, D., 1980: The computation of equivalent potential temperature.

,*Mon. Wea. Rev.***108****,**1046–1053.Bryan, G. H., and J. M. Fritsch, 2004: A reevaluation of ice-liquid water potential temperature.

,*Mon. Wea. Rev.***132****,**2421–2431.Emanuel, K. A., 1986: An air–sea interaction theory for tropical cyclones. Part I: Steady-state maintenance.

,*J. Atmos. Sci.***43****,**585–604.Emanuel, K. A., 1988: The maximum intensity of hurricanes.

,*J. Atmos. Sci.***45****,**1143–1155.Emanuel, K. A., 1994:

*Atmospheric Convection*. Oxford, 580 pp.Iribarne, J. V., and W. L. Godson, 1973:

*Atmospheric Thermodynamics*. 1st ed. D. Reidel, 222 pp.Iribarne, J. V., and W. L. Godson, 1981:

*Atmospheric Thermodynamics*. 2nd ed. D. Reidel, 259 pp.Paluch, I. R., 1979: The entrainment mechanism in Colorado cumuli.

,*J. Atmos. Sci.***36****,**2467–2478.Simpson, R. H., 1978: On the computation of equivalent potential temperature.

,*Mon. Wea. Rev.***106****,**124–130.Tripoli, G. J., and W. R. Cotton, 1981: The use of ice-liquid water potential temperature as a thermodynamic variable in deep atmospheric models.

,*Mon. Wea. Rev.***109****,**1094–1102.Wilhelmson, R. B., 1977: On the thermodynamic equation for deep convection.

,*Mon. Wea. Rev.***105****,**545–549.

Output from the numerical code. (a) Potential temperature along a pseudoadiabat using the exact formulation (solid) and using the approximate formulation (dashed). All cases begin at *p* = 1000 hPa and *T* = 30°C, but the curves on the right begin with

Citation: Monthly Weather Review 136, 12; 10.1175/2008MWR2593.1

Output from the numerical code. (a) Potential temperature along a pseudoadiabat using the exact formulation (solid) and using the approximate formulation (dashed). All cases begin at *p* = 1000 hPa and *T* = 30°C, but the curves on the right begin with

Citation: Monthly Weather Review 136, 12; 10.1175/2008MWR2593.1

Output from the numerical code. (a) Potential temperature along a pseudoadiabat using the exact formulation (solid) and using the approximate formulation (dashed). All cases begin at *p* = 1000 hPa and *T* = 30°C, but the curves on the right begin with

Citation: Monthly Weather Review 136, 12; 10.1175/2008MWR2593.1

Potential temperature errors using the approximate equation with different values for *L*_{0}.

Citation: Monthly Weather Review 136, 12; 10.1175/2008MWR2593.1

Potential temperature errors using the approximate equation with different values for *L*_{0}.

Citation: Monthly Weather Review 136, 12; 10.1175/2008MWR2593.1

Potential temperature errors using the approximate equation with different values for *L*_{0}.

Citation: Monthly Weather Review 136, 12; 10.1175/2008MWR2593.1

Results from numerical evaluation of different formulations for *θ _{e}^{p}*, where

*p*,

*T*, and

*θ*refers to the value retrieved from the numerical code;

_{e}^{p}*δ*

_{1}refers to the error of

*θ*using Bolton’s formulation (6); and

_{e}^{p}*δ*

_{2}refers to the error of

*θ*using the new formulation (13).

_{e}^{p}