Reappraising the appropriate calculation of a common meteorological quantity: Potential Temperature

The potential temperature is a widely used quantity in atmospheric science since it is conserved for air’s adiabatic changes of state. Its definition involves the specific heat capacity of dry air, which is traditionally assumed as constant. However, the literature provides different values of this allegedly constant parameter, which are reviewed and discussed in this study. Furthermore, we derive the potential temperature for a temperature-dependent parameterization of the specific heat capacity of dry air, thus providing a new reference potential temperature with a more rigorous basis. This new reference shows different 5 values and vertical gradients in the upper troposphere and the stratosphere compared to the potential temperature that assumes constant heat capacity. The application of the new reference potential temperature to the prediction of gravity wave breaking altitudes reveals that the predicted wave breaking height may depend on the definition of the potential temperature used.


Introduction
According to the book Thermodynamics of the Atmosphere by Alfred Wegener (1911), the first published use of the expression potential temperature in meteorology is credited to Wladimir Köppen (1888)  1 and Wilhelm von Bezold (1888), both following the conclusions of Hermann von Helmholtz (1888) (Kutzbach, 2016).Over 130 years ago, von Helmholtz perceived that within the atmosphere the heat exchange between air masses of different temperatures, which are relatively moved, is insufficiently explained by heat transfer due only to radiation and convection.He argued that wind phenomena (e.g., the trade winds), storm events, and the atmospheric circulation were more intense, of larger extent, and more persistent than observed if the air's heat exchange within the discontinuity region (the friction surface of the different air masses) was not mainly due to eddydriven mixing.On his way to analytically describe the heat exchange of different air masses within the atmosphere, in May of 1880, von Helmholtz introduced the air's immanent heat while its absolute temperature changes with changing pressure (von Helmholtz, 1888).In essence, von Helmholtz concluded that the temperature gained by a volume of dry air due to its adiabatic descent from a certain initial pressure level (p) to ground pressure (p 0 ) corresponds to the air's immanent heat.In November of the same year, in agreement with von Helmholtz and probably inspired by a presentation that was given in 1 In the publication year (1911) of Wegener's book, Köppen's daughter Else got engaged to Alfred Wegener (Reinke-Kunze, 2013) and they married in the year 1913 (Hallam, 1975).June by Köppen (1888), this property was renamed and reintroduced as the air's potential temperature (θ in the following) by von Bezold (1888) with the following definition for strictly adiabatic changes of state: where T and p are the absolute temperature and pressure, respectively, of an air parcel at a certain initial (pressure-) altitude level.The quantities θ and p 0 are corresponding values of the same air parcel's absolute temperature and pressure if the air was exposed to conditions at ground level.The dimensionless coefficient γ, nowadays called the isentropic exponent, was specified as 1.41 (von Bezold, 1888).
Moreover, in the same publication, von Bezold concluded that for moist air's adiabatic changes of state, its potential temperature remains unchanged as long as the change of state occurs within dry-adiabatic limits; and further, if there is condensation and precipitation, the potential temperature changes by a magnitude that is determined by the amount of water that falls out of the air parcel.From a modern perspective, it is clear that the air parcel is an isolated thermodynamic system, and adiabatic processes correspond to processes with conserved entropy (i.e., isentropic processes).The description of the immanent heat is then equivalent to the thermodynamic state function entropy, which corresponds to potential temperature of dry air in a one-to-one relationship.
In general, the potential temperature has the benefit of providing a practicable vertical coordinate (equivalent to the pressure level or the altitude above, e.g., sea level) to visualise and analyse the vertical distribution and variability of (measured) data related to any type of atmospheric parameter.Admittedly, the use of the potential temperature as a vertical coordinate is initially less intuitive than applying altitude or pressure coordinates.Indeed, the potential temperature bears a certain abstractness to describe an air parcel's state at a certain altitude level by its imaginary dry-adiabatic descent to ground conditions.However, one major advantage of using the potential temperature as a vertical coordinate is that the (measured) data are sortable with respect to the entropy state at which the atmospheric samples were taken.Thus, the comparison of repeated measurements of an atmospheric parameter on an equipotential surface (isentrope) or layer excludes any diabatic change of the air parcel's state due to an entropy-changing uplift or descent of the air mass.
Apart from characterising the isentropes, the vertical profiles of the potential temperature (θ as a function of height z) are used as the reference for evaluating the atmosphere's actual vertical temperature gradient, which allows characterising its static stability.Notably, von Bezold (1888) already proposed the potential temperature as an atmospheric stability criterion.
In its basic formulation, the potential temperature exclusively refers to the state of dry air, and thus the potential temperature characterises the atmosphere's static stability with respect to vertical displacements of a dry air parcel.In meteorology, the static stability parameter is expressed in terms of the (squared) Brunt-Väisälä frequency N in the form where g is the gravitational acceleration.The potential temperature twice enters the formulation of the stability parameter, as the denominator (θ −1 ) and as the vertical gradient ∂θ ∂z .In the research field of dynamical meteorology, the potential vorticity (PV) is often used.The PV is proportional to the scalar product of the atmosphere's vorticity (the air's local spinning motion) and its https://doi.org/10.5194/acp-2020-361Preprint.Discussion started: 4 May 2020 c Author(s) 2020.CC BY 4.0 License.stratification (the air's tendency to spread in layers of diminished exchange).More concretely, the PV is the scalar product of the absolute vorticity vector and the three-dimensional gradient of θ, i.e., not only the potential temperature's vertical gradient but also its partial derivatives on the horizontal plane add to the resulting PV, although, particularly at stratospheric altitudes, the vertical gradient constitutes the dominant contribution.For the analytical description of a fluid's motion within a rotational system, as is the atmosphere, the PV provides a quantity that varies exclusively due to diabatic processes.Occasionally, by means of the dynamical parameter PV, the tropopause height is defined (usually at 2 PV units, see, e.g., Gettelman et al., 2011) as is, e.g., the edge of a large-scale cyclone such as the polar winter vortex on specific θ levels (cf.Curtius et al., 2005).
While for a dry atmosphere (i.e., with little or no water vapour) the potential temperature is the correct conserved quantity (corresponding to entropy) for reversible processes, for an atmosphere containing water in two or more phases (vapour, liquid, and/or solid phases) energy transfers due to phase changes play a major role.Thus, the formulation of the potential temperature has to be extended (since entropy is still the right quantity for reversible processes, including phase changes).Starting from Gibbs' equation, some formulations are available, e.g., the entropy potential temperature defined by Hauf and Höller (1987) or more general versions as derived by Marquet (2011).In these formulations, phase changes and deviations from thermodynamic equilibrium are included.An approximation to these more general formulations is, e.g., the equivalent potential temperature, which includes latent heat release, assuming thermodynamic equilibrium (e.g., Emanuel, 1994).These formulations always rely on the assumption of reversible processes (i.e., conserved entropy).However, in the case of large hydrometeors, liquid or solid particles are removed due to gravitational acceleration, leading to an irreversible process.Sometimes for this situation a so-called pseudo adiabatic potential temperature is defined, assuming instantaneous removal of hydrometeors from the air parcel; usually, meaningful approximations to this quantity are given, since generally it cannot be derived from first principles.
In a strict sense, this is not a conserved quantity, since an irreversible process is considered.Equivalent potential temperature including phase changes for vapour and liquid water is often used for the determination of convective instabilities.The general formulation can be easily adapted for an ice equivalent potential temperature, i.e., for reversible processes in pure ice clouds (see, e.g., Spichtinger, 2014).Although the latent heat of sublimation is larger than the latent heat of vaporisation, the absolute mass content of water vapour decreases exponentially with decreasing temperature, leading to only small corrections due to phase changes in pure ice clouds.
At altitudes above the clouds' top, within the upper troposphere and across the tropopause, the air is substantially dried out compared to tropospheric in-cloud conditions.Therefore, above clouds and further aloft, e.g., within the stratosphere, the conventional dry-air potential temperature may suffice to provide a meaningful vertical coordinate.Moreover, the potential temperature is commonly used as a prognostic variable in numerical models for the formulations of the energy equation.
Thereby, very often both variants, the potential temperature as well as the equivalent potential temperature, are involved to account for dry air situations and cloud conditions.
In any case, the use of the potential temperature requires the following preconditions to be fulfilled: 1. θ should be based on a rigorous derivation to ensure its validity as a function of atmospheric altitude in order not to corrupt its character as a vertical coordinate that allows for appropriately comparing (measured) atmospheric parameters, and https://doi.org/10.5194/acp-2020-361Preprint.Discussion started: 4 May 2020 c Author(s) 2020.CC BY 4.0 License.
2. θ should approximate to the greatest possible extent the true entropy state of a probed air mass and should preferably account for the implied dependencies on atmospheric variables, even under the assumption that air behaves as an ideal gas, with the aim that the potential temperature behaves as a rational physical variable.Thus, still abiding by the ideal gas assumption, a re-assessment of the fundamental atmospheric quantity θ is suggested, which is based on the state-of-knowledge of air's thermodynamic properties, and this re-assessed θ is comprehensively examined concerning its ability to hold also for atmospheric conditions above the troposphere.
In principle, the concept of the potential temperature is transferable to all systems of thermally stratified fluids as is a planetary gas atmosphere or an ocean, to investigate heat fluxes (advection or diffusion) or the static stability of the fluid.In astrophysics, the potential temperature is used almost identically as in atmospheric sciences to describe dynamic processes and thermodynamic properties (e.g.static stability or vorticity) in the atmosphere of planets other than the Earth.Here, the same value p 0 = 1000 hPa, as applied to the Earth's atmosphere, is frequently used as a reference pressure for the atmosphere of other planets (Catling, 2015, Table 4), whereby the formulations of the specific heat capacity require adaptations to account for the individual gas composition of the respective planetary atmosphere.The Weather Research and Forecasting model (WRF) was extended to "planetWRF" to simulate the weather in the atmosphere of other planets.Here, the potential temperature is included in the prognostic model equations (Richardson et al., 2007), while it was pointed out by Li and Chen (2019) that this approach could suffer from not accounting for the temperature dependence of the isobaric specific heat capacity c p of the respective atmosphere's gas composition.The atmosphere of Jupiter's moon Titan, the only known moon with a substantial atmosphere, was comprehensively studied with frequent application of the potential temperature based on profile measurement of temperature and pressure in Titan's atmosphere by the Huygens-probe (Müller-Wodarg et al., 2014).Moreover, the potential temperature is a frequently used quantity in oceanography (e.g., McDougall et al., 2003;Feistel, 2008), while here the consideration of sea water's salinity and its impact on the specific heat capacity of sea water implies additional complexity.In particular, McDougall et al. (2003) suggests a re-assessment of the potential temperature as applied in oceanography to approximate the adiabatic lapse rate, thus this study bears certain parallels to the present investigation aiming at the reappraisal of the potential temperature for atmosphere-related purposes.These studies from other disciplines motivate the need for a re-assessment of the potential temperature for the atmospheric sciences.Thus, the approach provided herein proposes a modified calculation of the widely used quantity of the potential temperature by additionally accounting for the current state of knowledge concerning air's properties.
2 Derivation of the potential temperature for an ideal gas The Gibbs equation (see, e.g., Kondepudi and Prigogine, 1998) is a general thermodynamic relation to describe the state of a system with m components and reads as In the following, dry air is assumed to be the single component in the system.Expressing the Gibbs equation in its specific form (i.e., division by the total mass M a of dry air; note, lowercase letters indicate specific variables, e.g., h = H/M a , etc.) leads to Furthermore, approximating dry air as an ideal gas leads to the following simplifications: -The ideal gas law can be applied with the specific gas constant R a of dry air, which is where R is the molar gas constant in J mol −1 K −1 (Tiesinga et al., 2020;Newell et al., 2018) and M mol,a is the molar mass of dry air (Lemmon et al., 2000), composed of nitrogen N 2 , oxygen O 2 , and argon Ar.
-The specific enthalpy is given by with c p the specific heat capacity of dry air.
Based on these assumptions, the change of the specific entropy (within the fluid dry air) is given by For isentropic changes of state, i.e., ds = 0, equation (8) reduces to Note that the assumption of dry air being an ideal gas does not imply that in (9) the specific heat capacity c p is constant.While statistical mechanics excludes any pressure dependence in the ideal-gas heat capacity, the general derivation (cf.Appendix A) permits a temperature dependence of c p .However, usually the temperature dependence is neglected in atmospheric physics Treating c p as a constant, rearrangement of (9) leads to Integration of (10) over the range from ground-level pressure and temperature (p 0 , T 0 ) to the pressure and temperature at a specific height (p, T ) yields and, after another straightforward conversion, one arrives at With the definition θ cp = T 0 , equation ( 12) is transformed into the commonly used expression for determining the potential temperature for which the ground-level pressure p 0 is arbitrary but usually set to p 0 = 1000 hPa.This choice coincides with the definition of the World Meteorological Organisation (WMO, 1966) and the standard-state pressure (Tiesinga et al., 2020), but should not be confused with the standard atmosphere 101325 Pa (Tiesinga et al., 2020).In the following, θ cp denotes the potential temperature based on a constant c p and, when a specific value of c p is applied, the subscript c p in the potential temperature's notation is replaced by the corresponding c p value.
3 Examining the assumption of constant c p for dry air The general theory of thermodynamics, assuming dry air as an ideal gas, gives the expression for the constant specific heat capacity, which is based on the results of statistical mechanics and the equipartition theorem (e.g., Huang, 1987).In ( 14), the parameter f = f trans + f rot + f vib is equal to the total number of degrees of freedom of the gas molecules of which dry air consists.The individual contributions to f comprise the degrees of freedom of translation f trans , rotation f rot , and vibration f vib .Assuming further that dry air exclusively consists of the linear molecules N 2 and O 2 (implying f trans = 3 and f rot = 2, while the contribution of Ar remains disregarded) and additionally neglecting the vibrational degrees of freedom (f vib = 0), the general relation ( 14) reduces to Although the neglect of vibrational excitation, particularly at very low temperatures, seems plausible and appropriate, errors are already introduced by this assumption for the temperature range relevant in the atmosphere.
In atmospheric sciences, for the majority of computations that require the specific heat capacity of dry air, a constant value of c p may be appropriate.According to the WMO (1966), the recommended value for c p of dry air is 1005 J kg −1 K −1 and, furthermore (ibid.), it is defined that γ = cp cv = 7 5 = 1.4,cf.(1).This definition is consistent with the general thermodynamic theory together with all aforementioned additional assumptions and results in (15) as well.
Even assuming a universally valid constant c p , a single consistently used value of c p was not found.Instead, the specified values of c p vary among different textbooks and other sources.In Table 1, some of the available values of constant specific heat capacity for dry air are compiled, indicating a variability of c p that ranges from 994 J kg −1 K −1 to 1011 J kg −1 K −1 .
These different values of constant c p scatter within a small range (below ±1.1 %) around the WMO's recommendation 1005 J kg −1 K −1 , which may seem negligible if c p contributes only as a linear coefficient within an equation (e.g., in the expression of the correction factor ξ, cf.Weigel et al., 2016).Unfortunately, however, in the formulation of the potential temperature θ cp , cf. (13), the specific heat capacity c p does not contribute linearly but rather as the denominator in the exponent.
Thus, the variety of different c p values, although scattering within a small range, impacts the resulting θ cp significantly.To quantify this impact, a computation of θ cp by using (13) was based on the values of static pressure (p, cf. Figure 1a) and absolute temperature (T , cf. Figure 1b) corresponding to the US Standard Atmosphere (United States Committee on Extension to the Standard Atmosphere, 1976).
From the list of the different c p (cf.Table 1), two extreme values were selected, namely 994 J kg −1 K −1 (Wegener and Wegener, 1935) and 1011 J kg −1 K −1 (WMO, 1966), in order to initially illustrate the sensitivity of the resulting θ cp to variations in c p in the range of ∼ 1%, as referenced by literature.Specific distinctions will be discussed at a later stage, then mainly in relationship to the commonly used recommendation of the WMO (c p = 1005 J kg −1 K −1 , WMO, 1966).In Figure 2a, the individual profiles of θ cp are shown, and panel (b) exhibits the absolute difference ∆θ cp = θ 994 − θ 1011 , based on the c p values selected.Figure 2b shows the sensitive response of calculated θ cp to a small variability in c p .At an altitude of 5 km, the difference ∆θ cp already exceeds 1 K.  ; cf. also Table 1), and (b) the absolute difference ∆θc p = θ994 − θ1011 between the two resulting curves of θc p .
7 K, with increasing altitude up to 20 km.At 50 km, approximately where the stratopause is located, which is the chosen upper height limit for this investigation, the computed ∆θ cp reaches almost 75 K.
The impact of this sensitivity becomes important at altitudes of ∼ 10 km and above, thus, where the use of the potential 200 temperature becomes increasingly meaningful.Here, and in particular above the cloud tops, the small-scale and comparatively fast tropospheric dynamics (causing vertical transport and implying diabatic processes) become diminished, while further above, towards the stratosphere, an increasingly layered vertical structure of the atmosphere is taking over.
As discussed above, the potential temperature is remarkably sensitive to small variations (within the per-cent range) of air's specific heat capacity, as these variations affect the exponent of the equation for θ cp ; further proof of this, from the mathematical 205 perspective, is provided in Appendix B. The studies of Ooyama (1990Ooyama ( , 2001) ) document an interesting attempt to formulate, e.g., the energy balance equations for the moist atmosphere, wherein entropy replaces the more common formulation using the potential temperature.This substitution avoids the use of the potential temperature, which "is merely an exponential transform of the entropy expressed in units of temperature" (Ooyama, 2001), thus, within this equation, air's specific heat capacity is implied exclusively as a linear coefficient.Consequently, a parameterisation for the temperature dependence of the specific heat capacity (c p (T ), cf.Section 4) may be easily adopted.However, the crucial drawback of the entropy-based equations is that to gain a numerical model for, e.g., weather forecast purposes, the parameterisations of most of the physical processes within the atmosphere would require a reformulation.
It should be noted that not only do literature values of air's specific heat capacity c p vary, but also the values of the gas constant R a vary slightly due to different historical approximations for the molar gas constant R and for the composition of dry air.The variation of values for R a is typically only on the order of 10 −1 J kg −1 K −1 , whereas the variability in c p is on the order of a few J kg −1 K −1 (cf.Table 1).Therefore, within the exponent of the expression (13) for θ cp , the variability of c p has by far a stronger impact on the resulting θ cp value than the variability of R a .
However, accepting for a moment the WMO's definition (15) of c p (WMO, 1966), the variability of air's c p should naturally be constrained to certain limits.With the specific gas constant R a = 287.05J kg −1 K −1 (WMO, 1966), the WMO's definition leads to c p = 1004.675J kg −1 K −1 .In contrast, taking into account the uncertainty introduced in R a by the molar mass of dry air, cf.Equation ( 6), the resulting range for air's specific heat capacity is 1004.897J kg It may be surmised that the rounded value c p = 1005 J kg −1 K −1 as recommended by the WMO (1966) had the main goal to simplify certain calculations, which at the time may have been mostly done by hand.
4 Accounting for the temperature dependence of air's specific heat capacity Next, while retaining the ideal-gas assumption, we consider the dependence of air's c p on temperature over the atmospherically relevant range (180 K to 300 K).The temperature dependence of c p is, of course, not a new finding.Experimental approaches for determining the calorimetric properties of air and the temperature dependence of a fluid's specific heat capacity are described by Witkowski (1896), who investigated the change of the mean c p as a function of temperature intervals between room temperature (as a fixed reference) and various warmer and colder temperatures, for atmospheric pressures and slightly beyond.
Despite the potentially high uncertainty of the experimental results from these times, Witkowski (1896) already indicated that with decreasing temperature the experimentally determined c p values initially decline, then pass a minimum, and subsequently increase again at lower temperatures (T < 170 K).The description of refined experiments and ascertainable data of air's c p (T ) for temperatures below 293 K is summarised by Scheel and Heuse (1912), Jakob (1923), and Roebuck (1925, 1930), illustrating in comprehensive detail the experimental effort and providing the resulting data.The review by Awano (1936) compiled and compared the data of c p (T ) of dry air ("air containing neither carbon-dioxide nor steam", Awano, 1936) and he attestedat that time-the previously mentioned studies to constitute "the most reliable experiments".During the decades following these experiments, further insights were gained and landmarks were reached which are summarised in the comprehensive survey by Lemmon et al. (2000) of the progress of modern formulations for the thermodynamic properties of air and about the experiments the previous formulations were based on.
Dixon ( 2007) Awano (1936), 400 hPa Awano (1936), 200 hPa Awano (1936)  Figure 3 illustrates the range of suggested constant values for the specific heat capacity (see Table 1) together with the measurements that were made to obtain air's behaviour as a function of temperature and pressure.In the same figure, calculated values of c p (T ) of dry air are displayed resulting from the equation of state which was derived from experimental p, V , and T data by Vasserman et al. (1966), who provided an extensive review of previous experimental and theoretical works and of the state of knowledge at that time.In addition, Figure 3 exhibits two different parameterisations, by Lemmon et al. (2000) and by Dixon (2007), which account for the temperature dependence of the specific heat capacity c p (T ).Moreover, Figure 3 contains discrete values of dry air's c p (T ) extracted from the database REFPROP (Reference Fluid Thermodynamic and Transport Properties Database by NIST, the National Institute of Standards and Technology, Lemmon et al., 2018), which is based on parameterisations resulting from thermodynamic considerations discussed later.
The measurement data, as well as the parameterisations, clearly indicate a dependence of air's specific heat capacity on the temperature.At temperatures above 300 K, the data points by Jakob (1923) are surprisingly well captured by the parameterisations, while below 270 K the course of the parameterised and measured c p (T ) diverge significantly.Possible reasons for this include: the measurements of c p (T ) have a precision likely no better than 1 %, and there could be systematic errors, especially at low temperatures; the measured data reflect the true thermodynamic behaviour of the real gas, rather than that of an ideal gas.
However, it is immediately obvious from Figure 3 that a good agreement among (i) the experimentally determined c p (T ) data, (ii) a constant c p (e.g., 1005 J kg −1 K −1 ; WMO (1966)), and (iii) the parameterised c p (T ) is found only for a temperature interval ranging from 270 K to 290 K.For all air temperatures below 270 K, the constant value c p = 1005 J kg −1 K −1 fails to coincide with either the parameterised or the experimentally determined values of c p (T ).

The temperature dependence of the ideal-gas specific heat capacity
As already indicated by the data depicted in Figure 3, the specific heat capacity c p depends on the gas temperature.With regard to measured values, the lack of constancy may be due to real-gas effects or to a dependence of the ideal-gas heat capacity on temperature.In this section, we focus on the latter effect, denoting the ideal-gas isobaric specific heat capacity by c 0 p (T ), where the superscript 0 indicates the underlying ideal-gas assumption.For an individual gas, there is always a contribution from the three translational degrees of freedom, c 0 p,trans = 5 2 R i , where R i is the specific gas constant of the gas.If the molecule is assumed to be a rigid rotor, there is also a rotational contribution given by for linear (e.g., diatomic) molecules, 3 2 R i , for nonlinear molecules. (16) As mentioned previously, at finite temperatures molecules also have contributions to c 0 p (T ) from intramolecular vibrations (and, at high temperatures, excited electronic states).To arrive at a temperature-dependent parameterisation for the idealgas specific heat capacity of dry air, the compounds' individual contributions, considering all degrees of freedom, need to https://doi.org/10.5194/acp-2020-361Preprint.Discussion started: 4 May 2020 c Author(s) 2020.CC BY 4.0 License.be parameterised and then combined according to each compound's proportion in the mixture.For the following, dry air is considered a three-component mixture: the diatomic gases nitrogen (N 2 ) and oxygen (O 2 ) and the monatomic gas argon (Ar).
To determine the contribution of N 2 to c 0 p (T ), both Bücker et al. (2002) and Lemmon et al. (2000) use the ideal-gas heat capacity from the reference equation of state of Span et al. (2000) that compares well with the findings from other studies within an uncertainty ∆c 0 p of less than 0.02 %.For the contribution of O 2 , Lemmon et al. (2000) use the formulation given by Schmidt and Wagner (1985).Alternatively, Bücker et al. (2002) provide a slightly different formulation from the International Union of Pure and Applied Chemistry (IUPAC, Wagner and de Reuck, 1987), after refitting it to more recently obtained data, thereby achieving an overall uncertainty ∆c 0 p of less than ±0.015 % for O 2 (Bücker et al., 2002).However, the difference in the resulting specific heat capacity contribution by O 2 between the two approaches (Lemmon et al. (2000) or Bücker et al. (2002)) is comparatively small.
For a monoatomic gas such as Ar, vibrational and rotational contributions to the heat capacity do not exist, and Bücker et al. (2002) consider that argon's excited electronic states are relevant only at temperatures above 3500 K. Hence, the contribution of argon Ar to the specific heat capacity of air reduces to c 0 p = 5 2 R Ar .The approach by Bücker et al. (2002) additionally considers the contribution of further constituents of air, such as water, carbon monoxide, carbon dioxide, and sulfur dioxide.These authors provide an analytical expression for specific heat capacity, accounting for this more complex but proportionally invariant air composition which is specified to deviate from the used reference by ∆c 0 p ≤ ±0.015 % in the temperature range of 200 K ≤ T ≤ 3300 K.At atmospheric altitudes above the clouds' top, i.e., on average above ∼ 11 km, the air is assumed to have lost most of its water and is deemed as dry.Furthermore, for the following, trace gases that contribute to air's composition by molar fractions of less than that of Ar are neglected.

NIST's parameterisation of c
Besides a comprehensive survey of the available experimental data for the specific heat capacity of air, Lemmon et al. (2000) also provide state-of-the-art knowledge for other thermodynamic properties (isochoric heat capacity, speed of sound, vapour-liquid-equilibrium, etc.).Additionally, they give two approaches to derive air's thermodynamic properties, including the vapour-liquid equilibrium: 1. an empirical model-based equation of state for standard (dry) air considered as a pseudo-pure fluid, and 2. assembly of a mixture model from equations of state for each pure fluid.
Each approach allows calculating the thermodynamic properties, e.g., c p , of gas mixtures such as dry air, and both are realgas models with the ideal-gas behaviour as a boundary condition.The major difference between the models is that the first approach considers air as a pseudo-pure fluid while the second, more rigorous approach treats air as a mixture composed of N 2 , O 2 , and Ar, in molar fractions of 0.7812, 0.2096, and 0.0092, respectively, following Lemmon et al. (2000, their table 3).This of fluids over a wide range of temperatures and pressures.Lemmon et al. (2000) suggest that their mixture models allow calculation of the specific heat capacity of a gas mixture within an estimated uncertainty of 1 %.
Both the pseudo-pure fluid model and the mixture model of Lemmon et al. (2000) use the same expression for the ideal-gas heat capacity, which is rigorously given as a sum of the pure-component contributions: where x i denotes the molar fraction of species i, and C 0 p as well as the molar gas constant R are given in units of J mol −1 K −1 .Like Bücker et al. (2002), Lemmon et al. (2000) use the expression of Span et al. (2000) for the contribution of N 2 to the heat capacity and adopt C 0 p = 5 2 R for Ar.Together with the contribution by O 2 according to the formulation by Schmidt and Wagner (1985), the expression provided by Lemmon et al. (2000) for the ideal-gas heat capacity of dry air is with the scalar coefficients N i for dry air (ibid.), which is specified as valid for temperatures from 60 K to 2000 K.
The parameterisation (18) provides the isobaric specific heat capacity of dry air, considered as a mixture of ideal gases.This represents a more rigorous and accurate behaviour than assuming it to be a constant.The parameterisation from Dixon (2007) is not explicitly described to be based on particular assumptions or data sets.The author indicates his suggested parameterisation to hold within 0.1 % for temperatures between 200 K and 450 K.For elevated air temperatures, the deviation between the ideal-gas limit c 0 p (T ) (Lemmon et al., 2000) and Dixon's parameterisation substantially increases.This is most likely due to the chosen type of polynomial approximation (Dixon, 2007), which increasingly departs from the reference c 0 p (T ) for gas temperatures exceeding 450 K. Concerning the thermophysical properties of humid air, the study by Tsilingiris (2008) provides further insight.Its purpose was to evaluate the transport properties as a function of different levels of the relative humidity and as a function of temperature (from 273 K to 373 K) for the gas mixture of air with water vapour at a constant pressure (1013 hPa).The atmospherically relevant pressure range below 1013 hPa and temperatures smaller than 273 K were not considered.Although this study focused on providing a comprehensive account of moisture within air, mainly for technical purposes and engineering calculations, the possible usefulness of these findings to atmospheric investigations is also apparent.However, the impact of water vapour on the resulting gas mixture's c p (T ) is significantly larger (cf.Tsilingiris, 2008) than the uncertainty of dry air's c p (T ) that is discussed in the present work.Furthermore, the consideration of water vapour as a component of air requires very individual and case-specific computations of c p (T ) of moist air, as water vapour is among the most variable constituents of the atmosphere.
The effort required to produce an analytical formulation for gas properties which best reflects the true gas behaviour may indicate that for engineering purposes (pneumatic shock absorbers, engines' combustion efficiency, improvements of turbofan/prop propulsion, aerodynamics, material sciences, etc.), especially where pressures exceed atmospheric, the assumption of ideal-gas behaviour introduces excessive uncertainty.
5 The θ cp(T ) from the temperature-dependent specific heat capacity of air Previously introduced approaches for computing the specific heat capacity of dry air call for a brief discussion on how to use the obtained c p (T ) to derive the potential temperature.In the following, θ cp(T ) denotes the derived potential temperature that accounts for the temperature dependence of dry air's specific heat capacity.Furthermore, it should be noted that simply substituting any c p (T ) value into the conventionally used and defining equation ( 13) for θ cp (WMO, 1966) may appear seductive but definitely leads to results inconsistent with θ cp(T ) that is based on the reference parameterisation of dry air's c p (T ).Therefore, the thermodynamically consistent use of c p (T ) in the derivation of θ is described in the following.

Derivation of θ cp(T ) based on the temperature-dependent specific heat capacity of dry air
In the derivation of the potential temperature (cf.Section 2), we note that, until reaching the expression for isentropic changes of state (9), no specific assumption was made about the specific heat capacity.As soon as the temperature dependence of the specific heat capacity comes into play, the re-assessment of (9) leads to where θ cp(T ) is the desired potential temperature.
The rearrangement of ( 21) makes evident that the desired potential temperature is a zero of the function F (x), given by To arrive at the desired potential temperature θ cp(T ) for any given temperature and pressure, the equation 0 = F (x) must be solved for the variable x, which is the desired θ cp(T ) .Equation ( 22) has at most only one real zero, since its integrand is strictly positive which means F (x) is strictly monotonic.
In the following, the ideal-gas reference potential temperature θ ref is introduced, based on the formulation of the ideal-gas limit of dry air's specific heat capacity c 0 p (T ) in accordance with (18) as formulated by Lemmon et al. (2000).This reference potential temperature θ ref represents the zero of F (x) in ( 22), wherein c p (z) is to be replaced by c 0 p (T ).
It may be noted that further variants of a reference potential temperature are derivable by replacing c p (z) in ( 22) by any other expression of the specific heat capacity of air which may appear sufficiently accurate.The steps to compute or approximate the zero of the function ( 22), described in this study, are independent of the chosen heat capacity formulation.
Unfortunately, for a straightforward solution of the integral ( 22), the suggested parameterisation of c p is too complex and an analytically insolvable nonlinear equation 0 = F (x) could result.Thus, an approximation of the equation's desired zero is required.Newton's method (cf., e.g., Deuflhard, 2011) provides a standard approach to numerically approximate the zero of a nonlinear equation.Proceeding from an initial guess x 0 , Newton's method constructs a sequence {x k } k∈N defined by the recursion The constructed sequence {x k } k∈N converges to the equation's desired zero.For the herein described computations, the iteration is stopped as soon as the absolute difference |x k+1 − x k | of two consecutive iterations falls below 10 −8 K.
For the reference of air's specific heat capacity, c 0 p (T ), the integral (22) turns out not to be explicitly solvable.Therefore, with each iteration, the solution of the integral As a first guess x 0 for the Newton iteration, the conventional definition of θ cp based on a constant specific heat capacity (WMO, 1966) is inserted: In the course of Newton's method, the sequence {x k } k∈N will converge to the unique zero for any initial guess x 0 due to the monotonicity of F (x).However, the right choice of the initial guess x 0 substantially decreases the error of the first iteration x 1 , speeding convergence to the desired zero of the function F (x). Therefore, it may be comprehensible to use the conventional definition of θ cp as the first guess for the Newton iteration (23).
Solving the previously described root-finding problem by Newton's method over the comprehensive range of iteration steps (until the set requirement, i.e., |x k+1 − x k | < 10 −8 K, is fulfilled) finally leads to the reference potential temperature θ ref .This θ ref is based on the ideal-gas limit of dry air's specific heat capacity c 0 p (T ), which refers to the current thermodynamic state-ofknowledge and, thus, we use θ ref as our reference for the potential temperature in the following.For evaluating the results, the air temperature and pressure from the US Standard Atmosphere are used once more to set up the vertical profiles of the potential temperature.Figure 4a  Table 1), c p = 994 J kg −1 K −1 and c p = 1011 J kg −1 K −1 .Clearly, in particular at elevated altitudes, the courses of θ 994 and θ 1011 significantly deviate from the reference.To quantitatively evaluate the match between the different profiles, the relative difference of the four profiles, θ 994 , θ 1004 , θ 1005 and θ 1011 , with respect to the reference, i.e., ∆θ/θ ref = θ cp − θ ref /θ ref , is depicted in Figure 4b.The comparison impressively demonstrates that the θ cp profiles significantly depart from the reference by up to ∼ 250 K at 50 km altitude, corresponding to a relative difference of about 10 %.With both extremes of constant c p , the relative error level of 0.1 % is exceeded at altitudes below 5 km.While θ 994 continues to increasingly deviate from the reference, θ 1011 re-enters and crosses the 0.1 % relative error interval (grey-shaded area) at altitudes between ∼ 20 km and 22.5 km, before it reaches similar errors to the other θ cp profiles that are based on a constant c p .Notably, up to an altitude of 15 km, the reference potential temperature is comparably well matched by both the recommended θ 1005 (WMO, 1966) and θ 1004 (based on the frequently used alternative c p = 1004 J kg −1 K −1 , cf.Table 1).Until 15 km altitude, both constant c p values lead to errors of calculated θ cp which remain comparatively small within the 0.1 % relative error interval.However, above ∼ 17.5 km, both θ 1004 and θ 1005 exceed the 0.1 % relative error interval, and further aloft their relative error with respect to the reference θ ref increases rapidly.
In the context of numerical models of the atmosphere, the energy balance equation is occasionally formulated based on the potential temperature θ, thus θ constitutes a prognostic model variable.In such a case, the temperature T needs to be calculated from a given pair of values of pressure p and potential temperature θ.Using once more the defining equation ( 21), a zero of the function is to be computed.Since (25) corresponds to the function F defined in ( 22) with the exception of a negative sign, the identical approximation procedure as outlined above in this section for the calculation of (T, p) → θ may be applied mutatis mutandis to calculate the transformation (θ, p) → T .
In any case, a certain effort is required to implement the new formulation of the potential temperature in an atmospheric model, as this equation should be based on the implicit definition ( 21) and such a goal may be subject of future endeavours.

Approximations of the reference potential temperature
Of course, the previously described procedure to compute the potential temperature may appear to be anything but practical.
Indeed, due to the complications inherent with: the requirement to numerically solve the integral in the function F (x) and the need to use Newton's method for an iteration sequence to approach the zero of F (x), a convenient approach to re-assess the conventional definition of the potential temperature is not provided at all.This motivates the development of a more practical approximation of the reference potential temperature.To arrive at a practicable z dz becomes the first obstacle to a practical approximation.Therefore, a plausible initial step is to replace the integral by an expression that is easier to treat.This expression may be proposed as f (T ) − f (x), where the function f is defined as and which is recognisable as an approximated primitive of z , see Appendix C1.The choice of the functional form of f is motivated by the exact primitive of the integral in the case of a constant c p .
As previously discussed (cf.Section 5.1), the formulation of a new expression for the potential temperature based on the temperature-dependent specific heat capacity c p (T ) requires finding the zero of the equation 0 = F (x), where the function F (x) is defined in ( 22).Replacing the exact integral in ( 22) by the difference f (T ) − f (x) means that F (x) is substituted by the function Consequently, the resulting approximated reference potential temperature, i.e., the respective zero of the function F (x), is denoted as θ approx ref .
The difference between the approximation result and the reference, i.e., is then referred to as the basic error of the approximation.Note that the replacement of the function F by F only circumvents the integration in F ; the root-finding problem 0 = F (x) for the approximated reference potential temperature θ approx ref remains analytically not solvable.
Therefore, the second move towards a practical approximation procedure is to construct approximations θ (k) to the zero of F (x) by using Newton's method, see Appendix C2.Newton's method is an iterative procedure; the notation θ (k) refers to the k-th computed iterate.Hence, θ (k) constitutes an approximation to θ approx ref , and, in the limit k → ∞, the approximation error vanishes.Two formulations of Newton's method are distinguished in Appendix C2, i.e., the principal application of Newton's method, and its further derivative, called Householder's method.Both formulations require the stipulation of one of the iterates θ (k) as sufficient to obtain a result of appropriate accuracy.The higher the number of iterations, of course, the smaller is the error (28), whereas the basic error ( 27) remains unaffected by the number of iterations.Hence, in any case, the basic error ( 27) is to be accepted as at least implied in the final approximation, even though a well-chosen θ (k) could result in an approximation k) that is smaller than the basic error.
The various errors implied in the proposed approximation procedure combining for the approximation's total error, as well as accompanying details, are discussed in Appendix D. In brief, Figure 5a  temperature-pressure data pairs.Based on the parameters of the US Standard Atmosphere, the basic error inherent with the approximation remains below 1.25 K up to altitudes of 50 km.Thus, regarding the subsequent iteration process, a substantial improvement of the error compared to ∼ 1.5 K is not to be expected for the total error of approximating the reference potential temperature.
An error analysis exclusively based on the US Standard Atmosphere is constrained to specific combinations of the air's pressure and temperature, potentially suppressing latent errors that may emerge if certain fluctuations of the real atmosphere's temperature and pressure profiles are considered.Thus, the error analysis is extended to an atmospheric pressure (p) and temperature (T ) range, from 1000 hPa to 0.5 hPa and from 180 K to 300 K, such that the conditions within the entire troposphere and stratosphere, including the stratopause, are covered.Figure 5b illustrates the absolute basic error ( 27) for the extended ranges of pressure and temperature while Figure 5c illustrates the relative basic error The contours in Figures 5b and 5c mainly highlight two regions: at ∼ 100 hPa where ∆θ never rises above 0.75 K which corresponds to a maximum relative basic error of 0.15 %, and in a pressure range from ∼ 5 hPa to 1 hPa where a ∆θ of 1.25 K is never exceeded, corresponding to relative errors of at most 0.1 %.Note that the entire ∆θ scale ranges up to 3 K, which may only be reached at pressures below 0.8 hPa combined with temperatures above 280 K.
As previously discussed, the basic error is unavoidable and is to be accepted when applying the suggested substitution for the integral in the definition of the function F (x) in ( 22).However, as outlined in Appendix C2, the second iterate θ (2) of Newton's method (principal application), may thoroughly suffice for the final approximation to the reference potential temperature θ ref , as this iteration level already features an approximation error (28) which is negligibly small.Figure 6a illustrates the total relative error of the suggested approximation θ (2) with respect to the ultimate reference θ ref for the extended ranges of pressure and temperature.Indeed, the contour pattern in Figure 6a and the basic relative approximation error shown in Figure 5c are remarkably similar.Thus, the iteration process itself imparts only a minor contribution to the total error compared to the basic approximation error.
The total approximation error, which is is dominated by the unavoidable basic error (first bracket) and augmented by a negligible error inherent to the iteration (second bracket), also supporting the conclusion that the second iterate of Newton's method is an appropriate approximation procedure.Compute the quantities by using the previously obtained x0 and the coefficients (C4).
by using the previously obtained x1 and the coefficients (C4).

Set θ
(2) i = x2 as the final approximation to the reference potential temperature θ ref of the ideally behaving air.
Figure 7. Flowchart guiding through the process of computing the approximation θ (2) by using Newton's formulation (C5) until its second iteration, wherein T (in K) and p (in hPa) are the atmospheric air conditions in terms of temperature and pressure, respectively, and p0 is set to 1000 hPa (WMO, 1966).the real atmospheric conditions from those of the US Standard Atmosphere could increase the absolute error ∆θ from a few K to up to 10 K, corresponding to an increase of the relative error to 1 %.Further critical pressure levels are at ∼ 20 hPa and ∼ 5 hPa, where the error's magnitude increases to several tens and several hundreds of K, respectively.At a pressure of 0.5 hPa, an absolute error ∆θ of up to 500 K is reached, which corresponds to a relative error of 10 % or even more.
6 The potential temperature for air as a real gas To calculate real-gas effects on the potential temperature, we use the model embedded in REFPROP (Lemmon et al., 2018), a standard reference database from NIST.This model treats air as a mixture and employs state-of-the art reference equations of state for pure nitrogen (Span et al., 2000), oxygen (Schmidt and Wagner, 1985), and argon (Tegeler et al., 1999).The mixing rule and binary interaction parameters are taken from the GERG-2008 model (Kunz and Wagner, 2012).From its definition in terms of an isentropic process, the potential temperature θ real (T, p) is defined implicitly by where s is the specific entropy.Calculating θ real (T, p) is a two-step process.First, the specific entropy s is computed at temperature T and pressure p.Then, the temperature θ real is found that gives the same entropy s at the ground pressure p 0 .This is an iterative calculation, but it is accomplished automatically within the REFPROP software (Lemmon et al., 2018).
One caveat should be mentioned regarding the computed potential temperatures.The range of validity of the equations of state for the air components (Span et al., 2000;Schmidt and Wagner, 1985;Tegeler et al., 1999) extends only up to 2000 K.At very high altitudes, computed values of θ real exceed this limit.While all the equations extrapolate in a physically realistic way, their quantitative accuracy is less certain above 2000 K.This caveat also applies to the ideal-gas calculations; the correlations for c 0 p (T ) for N 2 and O 2 are extrapolations beyond 2000 K.However, since the same ideal-gas values are used in the real-gas calculations, any inaccuracy in c 0 p (T ) will cancel when evaluating the difference between ideal-gas and real-gas values of θ. Figure 8 illustrates the comparison between the real-gas potential temperature θ real and the ideal-gas reference potential temperature θ ref .Figure 8a shows the difference θ real − θ ref , once more along the p-T -profile of the US Standard Atmosphere.
Figure 8b accounts again for any p-T -combination of extended range but shows the relative difference instead.The difference between θ real and θ ref never exceeds 0.1 K for the absolute difference or 30 • 10 −5 = 0.03 % for the relative difference.As may be anticipated from the deviation of c 0 p shown in Figure 3 at low temperatures both from the experimentally determined values (which may be inaccurate) as well as from the REFPROP data, the real-gas effect on the specific heat capacity of dry air tends to increase towards the coldest gas temperatures.However, the difference between the real-and ideal-gas approaches results in essentially no substantial difference between the resulting θ's, neither at ground conditions (for any temperature at ∼ 1000 hPa) nor at very high altitudes (at pressures below ∼ 1 hPa).While the negligible difference between θ real and θ ref near ground levels is less surprising, the diminished difference at higher altitudes reflects that in this region the potential temperature reaches such high values that the difference between the real-gas and the ideal-gas specific heat capacity becomes insignificant.
Within the intermediate (stratospheric) region, the low pressures (and thus the low air densities) cause the ideal-gas assumption to be an accurate approximation even at low temperatures.In general, the degree to which a gas can be treated as an ideal gas is primarily a function of the (molar) density.For an ideal gas, the density is proportional to the quotient p T ; this is almost true also for real air.Hence, declining pressures together with rising temperatures both make the air's behaviour increasingly close to ideal.temperature and its vertical derivative, cf. ( 2), the resulting N 2 , a measure of atmospheric stability, is affected by the definition of θ.This may have implications for the investigation of upward propagating gravity waves, which are emitted from the upper troposphere or lower stratosphere by various processes, e.g., spontaneous imbalance (Plougonven and Zhang, 2014), flow over mountains (Palmer et al., 1986), or convection (Choi and Chun, 2011).Various properties of gravity waves directly depend on the vertical profile of the Brunt-Väisälä frequency N 2 .Specifically, the altitude of gravity wave breaking, if due to static instability, depends on N 2 .To explore the implication of the θ definition on gravity wave breaking, vertical profiles of temperature and horizontal wind speed are used as shown in panels (a) and (b) of Figure 9.These are typical for mid latitudes for the months June and December, respectively, and they have been taken from the Upper Atmosphere Research Satellite Reference Atmosphere Project (URAP) data (Swinbank and Ortland, 2003).
Note that these profiles extend up to 85 km, thus covering the entire stratosphere and most of the mesosphere, compared to the previously used vertical range reaching at most to 50 km (up to approximately stratopause level).Nevertheless, both the parameterised specific heat capacity of ideal-gas dry air (18) and the general derivation of the reference potential temperature (Section 5.1) are valid also at altitudes above 50 km.Consequently, the new reference potential temperature also remains valid up to mesospheric altitudes, even though the approximate reference potential temperature θ approx Following Lindzen (1981), static instability due to a gravity wave occurs whenever it can lead to an overturn of potential temperature, which is expressed as (e.g., Bölöni et al., 2016) Here m(z) is a gravity wave's vertical wave number at the altitude z, and B(z) denotes the vertically varying buoyancy amplitude of the same wave.Thus, the (minimum) gravity wave breaking altitude z b > z 0 is predicted as the lowermost altitude where the condition m(z To explore the implications of using the new reference potential temperature instead of the WMO-defined potential temperature on the predicted altitude of gravity wave breaking, a typical altitude of z 0 = 17.5 km is chosen as the initiation level of a gravity wave with horizontal wave number k(z 0 ) = 2π λx and vertical wave number m(z 0 ) = 2π λz .The initial buoyancy amplitude  with a scaling factor 0 < B fact < 1 defining the wave amplitude at the initiation level with respect to static instability.The dependence of m and B on altitude is then determined by the classic steady-state approach as outlined, e.g., by Bölöni et al. (2016).The selected parameter values for the scaling factor B fact are B fact ∈ {0.05, 0.06, 0.08, 0.1, 0.12, 0.14, 0.16, 0.18, 0.2, 0.22, 0.24, 0.25, 0.3, 0.5, 0.75, 0.9} , (34) while the selected horizontal wave lengths λ x at initiation height are Note that deviations of this scale are only found for the mid-latitude December vertical profiles of temperature and wind speed employed here, and hydrostatic gravity waves with initial horizontal wave lengths λ x ∈ {100 km, 50 km} and initial vertical wave lengths between approximately 1 km and 3 km.Significant differences of predicted wave breaking altitudes were most frequently observed with initiation height amplitude scaling factors B fact between 0.1 and 0.2, but larger values can also lead to significant differences, see Figure 9f.In essence, the improvement from the use of a more accurate potential temperature for predicting the altitude of gravity wave breaking is non-negligible, although not excessive.Nonetheless, these improvements may be of particular relevance for individual investigations, e.g., concerning the mesopause altitude, which involve specific vertical profiles at concrete atmospheric conditions and at locations other than the mid-latitudes.It needs to be emphasised, however, that predictions of the gravity wave breaking altitude are highly sensitive to variations along the vertical profiles of both the temperature and wind speed.Furthermore, the results of such predictions strongly depend on the chosen parameters at the gravity waves' initiation height.

Summary and Conclusions
Under the assumption that dry air is an ideal gas, a re-assessment of computing the potential temperature was introduced that accounts for the hitherto unconsidered temperature dependence of air's specific heat capacity.The new reference potential potential temperature was compared to a potential temperature θ real wherein the real-gas behaviour of dry air is considered.
In the range of temperatures from 180 K to 300 K and the range of pressures from 1000 hPa to 0.5 hPa, covering the atmospheric conditions of roughly the entire troposphere and stratosphere, the relative differences between θ ref and θ real are smaller than 0.03 % and may be considered negligible.Consequently, θ ref even provides a reasonable approximation to the potential temperature of the real gas.
The difference between the newly derived reference potential temperature θ ref and the conventionally determined potential temperature θ cp (with constant c p = 1005 J kg −1 K −1 , as recommended by the World Meteorological Organisation, WMO, 1966) increases with altitude, e.g., ∆θ ≥ 1 K at pressures p ≤ 60 hPa.
Derivation of a potential temperature that is consistent with thermodynamics and that accounts for the ideal-gas properties of dry air requires integration of Gibbs' equation and the subsequent solution of the resulting nonlinear equation.With a constant c p , both analytical steps are straightforward, resulting in the conventional expression (13) as suggested by WMO (1966).However, if instead the temperature dependence of air's specific heat capacity c p (T ) is considered, the integrals as well as the equations are not analytically solvable and, thus, the solution must be approximated.Both approximations were performed and described in detail.The integral was treated with the basic approximation and the solution of the nonlinear equation was approached by using the second iterate of Newton's method.As an alternative to Newton's classical method, a modified formulation of Householder's iteration method is provided, featuring accelerated convergence properties.One of the foremost implications of the re-assessed potential temperature's definition concerns the use of θ as a vertical coordinate for the sorting, grouping, and comparison of (measured) data, e.g., along or across isentropes.Thereby, the re-assessed potential temperature constitutes a more accurate consideration of the air's actual properties.This particularly concerns, e.g., the specific heat capacity which is conventionally assumed as constant and for which various values are given depending on the textbook consulted.
Significant errors and biases may arise if, for instance, the conventional derivation of θ (WMO, 1966) is used together with values for air's specific gas constant (R a ) or air's specific heat capacity (c p ), which better comply with the most recent state-of-knowledge.Moreover, the use of the standard pressure 1013.25 hPa instead of 1000 hPa as defined by WMO (1966) and consistently used herein as ground level pressure (p 0 ), may cause an additional deviation of the resulting θ.Thus, the re-assessment of θ's definition could largely diminish such errors and biases and improve the comparability of data.
Concerning investigations of the propagation of gravity waves within the upper atmosphere, one further implication was investigated that arises from using the re-assessed potential temperature instead of the conventional definition.For predictions concerning the altitude of a gravity wave's breaking, the atmosphere's static stability is analysed, which is a function of both the potential temperature θ and its vertical derivative ∂θ ∂z .Using the re-assessed reference potential temperature instead of its Moreover, the predictions concerning the presence of critical layers within the atmosphere may be impacted by using θ ref instead of the conventional θ.Of all studied cases, a limited number of predictions produced a vertical deviation on the order of 5 km.Of course, a comprehensive sensitivity study concerning these altitude predictions of gravity waves' breaking should be based on a larger variety of initiation parameters and vertical profiles of the temperature and wind fields from different geographical latitudes.However, such an investigation is beyond the scope of this study.The evaluation of the quantitative and/or qualitative significance of identified vertical shifts and deviations may be left to the reader.
On the one hand, such a re-assessment could take into account the current state of knowledge regarding the accuracy of thermodynamic variables and substance-related properties.On the other hand, this way the conceptional abstractness already inherent in θ is not further complicated by a misleading selection of parameters or reputed constants.There is no doubt that the conventional method is suitable for the description of most processes occurring within the troposphere.However, at stratospheric or even mesospheric altitudes, the neglect of the temperature dependence of the ideal-gas heat capacity in the convential definition increasingly distorts the resulting absolute values as well as the vertical course of the potential temperature.Ultimately, it seems obvious to profit from the computing capacities available today and from the known higher accuracy of physical variables and atmospheric parameters to carry out a reappraisal of the potential temperature, a useful (but not always consistently used) meteorological quantity.Alternatively, assuming the system's pressure as constant, its volume is variable with total derivative and, therefore, it results, defining the isobaric molar heat capacity C p .In general, this quantity depends on pressure as well as on temperature.However, if the gas is assumed as ideal, an important conclusion from the statistical description of an ideal gas is the fact that the internal energy U must be independent of the pressure (see, e.g., Fay, 1965).
Using this result, together with (A7) and the ideal gas law (A1), it follows In the previous computations, there is no restriction on the temperature dependence of the internal energy U (T ).Therefore, even under the assumption of ideal-gas behaviour, the specific heat capacity C p in (A8) is in general a function of temperature.As a further step, the function h(x) is numerically approximated, while c p (T ) in (C1) is replaced by the ideal-gas limit of air's specific heat capacity c 0 p (T ).The integration interval [T 1 , x] with T 1 ≤ x ≤ 2000 K is traversed in steps of at most 0.001 K while each step of the integration process is carefully approximated by using Simpson's rule.

(C4)
In Figure C1a the function h(x) is graphed, together with the approximation f (x), as well as the respective deviations h(x) − f (x) in Figure C1b.Evidently, the absolute error inherent to the approximations is comparatively small as, over the entire temperature range above 190 K, the approximation error never exceeds ±1 J kg −1 K −1 .Thus, the approximation error remains even smaller than the error caused by the scatter of given constant values of the specific heat capacity.Exclusively at temperatures below 190 K, the approximation error rapidly rises above 1 J kg −1 K −1 , bearing in mind that such absolute temperatures are only occasionally found in the atmosphere within a relatively narrow altitude interval at the cold point tropopause.
Moreover, the deviation of f (x) and h(x) from each other appears negligible as the profiles almost ideally coincide (cf. Figure C1a).

C2 Finalised approximation of the reference potential temperature
As discussed in Section 5.1, the new formulation for the potential temperature based on the temperature-dependent specific heat capacity c p (T ) requires solving the root-finding problem 0 = F (x), where the function F (x) is defined in (22).However, since F (x) contains an integral that complicates the root-finding process, this integral is substituted by the difference f (T ) − f (x), where f is given in Section C1.Therefore, F (x) is replaced by the function F (x) as defined in ( 26 The equation 0 = F (x) is still not analytically solvable, so Newton's method is once more required.Using again x 0 = θ 1005 as the initial guess, cf. ( 24), the iteration sequence for Newton's method is given by the recursion Instead of this standard formulation of Newton's method (C5), Householder's formulation may be used, which allows for reducing the computation time due to its accelerated convergence speed.For completeness, the required derivatives f , f in the recursion formulas (C5) and (C6) are 2 . (C7) The final step on the way to formulate a new expression for the potential temperature requires defining one of the iterates x k as appropriate enough for the approximations that result from applying the different methods: the standard of Newton's method (C5), simply referred to as Newton's method in the sequel, or -Householder's method (C6).
While the mathematical expressions in (C5) and (C6) are of increasing complexity, the convergence rate of the approximating sequence increases with rising mathematical complication.The preferred method is determined by the accuracy required, i.e., an elevated accuracy level is necessarily associated with elevated computational effort for the approximation method.A discussion of the approximation errors is found in Appendix D.
Table C1 collects values of the new reference potential temperature θ ref , together with the first two iterates θ (1) , θ (2) using Newton's method (C5) and the first iterate θ (1) Householder using Householder's method (C6) for five pairs of temperature and pressure along the US Standard Atmosphere, cf. Figure 1, which allows to verify a computation.The first height is chosen midway along the linearly decreasing temperature profile within the troposphere, while the other heights correspond to the kinks of the temperature profile.or (b) the second iterate θ (2) as the final approximation to the reference potential temperature θ ref .
range from ∼ 5 hPa to 1 hPa, featuring the same impact on ∆θ/θ ref of identical strength as the basic error.This result may not be surprising, since the second iteration step with both methods, Newton's and Householder's, was already proven to approach the approximation comparatively well, without worsening the total error level (cf. Figure D1b).
Consequently, concerning the required number of iterations and the method to use, the second iteration of Newton's method can be recommended to deliver appropriate results, with a relative error of less than 0.3 %, up to the stratopause level (∼ 50 km).
Householder's method features an accelerated convergence rate, and its use up to its first iterate θ (1) may be already appropriate for certain applications.According to the total error of Householder's method up to its first iterate θ (1) (Figure D2c), the resulting relative error remains below 7 % to a pressure level of ∼ 50 hPa and ∆θ stays below 0.3 % to pressures of ∼ 2 hPa.
Thus, Figure D2 may serve as guidance to decide how many iterations with one or the other method best meets the individual
https://doi.org/10.5194/acp-2020-361Preprint.Discussion started: 4 May 2020 c Author(s) 2020.CC BY 4.0 License.where T denotes the absolute temperature in K, S the entropy in J K −1 , H the enthalpy in J, V the volume in m 3 , µ k the chemical potential of component k in J kg −1 , M k the mass of component k in kg, and p the static pressure in Pa.Assuming no phase conversion or chemical reaction within the system, the mass of each component does not change, hence dM k = 0 for each component k.
https://doi.org/10.5194/acp-2020-361Preprint.Discussion started: 4 May 2020 c Author(s) 2020.CC BY 4.0 License.and, instead, c p is assumed as constant.Immediately below and in Section 3, the treatment of c p as a temperature-independent constant is discussed.The introduction of the temperature dependence then follows in Section 4.

Figure 1 .
Figure 1.Vertical profiles of (a) atmospheric pressure and (b) temperature as functions of height, corresponding to the US Standard Atmosphere.

Figure 2 .
Figure 2. Computed vertical course of the potential temperature θc p based on the two extremes of constant values for the specific heat capacity cp provided in the literature (panel (a); cf. also Table 1), and (b) the absolute difference ∆θc p = θ994 − θ1011 between the two https://doi.org/10.5194/acp-2020-361Preprint.Discussion started: 4 May 2020 c Author(s) 2020.CC BY 4.0 License.

Figure 3 .
Figure 3. Variety of suggested values for the specific heat capacity of air.Constant values of cp are displayed over the range documented in Table1(dashed lines).The parameterisations of air's cp(T ), accounting for its temperature dependence byLemmon et al. (2000, solid    magenta curve) and by Dixon (2007, solid cyan curve) are displayed.Discrete measurement and literature data at about 1000 hPa (i.e., as often specified, at about one atmosphere) are indicated by dots.In addition, the studies byAwano (1936) andVasserman et al. (1966) provide data at other atmospheric pressures, as indicated by squares, diamonds, and triangles.
fractional composition of dry air is assumed to be constant from ground level up to 80 km height (United States Committee on Extension to the Standard Atmosphere, 1976) and its fractional composition would have to be shifted significantly to cause a serious deviation of the resulting potential temperature.The contribution to the composition by carbon dioxide (CO 2 ) and of https://doi.org/10.5194/acp-2020-361Preprint.Discussion started: 4 May 2020 c Author(s) 2020.CC BY 4.0 License.any other trace species was assumed to be negligible.The validity of both approaches is specified for various states of dry air, from its solidification point (59.75 K) up to temperatures of 1000 K, and for pressures up to 100 MPa and even much further beyond the pressure range that is relevant for atmospheric investigations.Both the pseudo-pure fluid model and the mixture model are implemented in NIST's REFPROP database (cf.https://www.nist.gov/srd/refprop)for various physical properties 172111248 • 10 −9 , N 4 = −3.115413101• 10 −13 , https://doi.org/10.5194/acp-2020-361Preprint.Discussion started: 4 May 2020 c Author(s) 2020.CC BY 4.0 License.4.3 The parameterisation of c 0 p (T ) from an engineer's perspective ) https://doi.org/10.5194/acp-2020-361Preprint.Discussion started: 4 May 2020 c Author(s) 2020.CC BY 4.0 License.Integration of (20) from the basic state (p 0 , θ cp(T ) ) to any other state (p, T ) yields

zFigure 4 .
Figure 4. (a) Reference potential temperature θ ref together with the potential temperatures θ994 and θ1011 relying on constant cp values (e.g.994 and 1011 J kg −1 K −1 , cf.Table 1).(b) Relative differences (θ994 − θ ref ) /θ ref and (θ1011 − θ ref ) /θ ref between the reference potential temperature and potential temperatures relying on constant cp values.For comparison, also the relative difference (θ1005 − θ ref ) /θ ref is displayed, for which cp = 1005 J kg −1 K −1 corresponds to the WMO recommendation.All profiles are based on the values for temperature and pressure according to the US Standard Atmosphere.Note the linear axis-scaling inside and the logarithmic scaling outside of the greyshaded area in panel (b).
exhibits the resulting reference profile, i.e., θ ref .Additionally, for comparison with the reference, further potential temperature profiles θ cp are shown based on two extremes of given constant values of air's specific heat capacity (cf.https://doi.org/10.5194/acp-2020-361Preprint.Discussion started: 4 May 2020 c Author(s) 2020.CC BY 4.0 License.
https://doi.org/10.5194/acp-2020-361Preprint.Discussion started: 4 May 2020 c Author(s) 2020.CC BY 4.0 License.approximation procedure, the two principal steps in the suggested procedure are briefly outlined in the following, whereas the comprehensive details and intermediate derivation steps are found in Appendix C. Proceeding from the definition (22) of the function F (x), the computation of the integral

Figure 5 .
Figure 5. Absolute basic error ∆θ = θ ref − θ approx ref , cf. (27), from approximating the reference potential temperature along the US Standard Atmosphere (a) and for the extended pressure range 1000 hPa to 0.5 hPa and temperature range 180 K to 300 K (b).For orientation, the white solid line indicates the p-T -profile from the US Standard Atmosphere.The relative basic error |∆θ| /θ ref is shown in panel (c) for the extended pressure and temperature range.

Figure 7
Figure 7 presents step-wise instructions for the computation of the second iterate approximation to the reference potential temperature, and may serve as a guide to follow the numerous equations and intermediate analytical steps described throughout the derivations in Appendix C. For completeness, Figures 6b and 6cexhibit a final comparison by means of the logarithmic difference and the logarithmic exhibit a final comparison by means of the logarithmic difference and the logarithmic relative difference between the reference potential temperature θ ref and the conventional definition θ cp (WMO, 1966) based on a constant specific heat capacity c p = 1005 J kg −1 K −1 .Notably, over a wide altitude range within the troposphere (i.e., for atmospheric pressures greater than ∼ 100 hPa), the absolute error ∆θ = |θ 1005 − θ ref | remains below 1 K, cf.

Figure 8 .
Figure 8. Difference θ real − θ ref reflecting the deviation of the potential temperature θ real , based on the properties of air behaving as a real gas under variable temperature and pressure, from the herein derived potential temperature expression θ ref for the ideal-gas limit of the air's specific heat capacity c 0 p (T ).(a) Difference along the profile of the US Standard Atmosphere.(b) Relative difference in p-T -coordinates covering any combination of atmospherically relevant temperatures and pressures.

5257
Implications of the potential temperature on the prediction of gravity waves' breakingAs previously shown, the newly defined reference potential temperature θ ref deviates most from the WMO-defined potential temperature θ 1005 at high altitudes (cf.Figure6).More particularly, not only do the values from both θ definitions differ, but also their vertical derivatives, i.e., ∂θ ref ∂z and ∂θ1005 ∂z .As the Brunt-Väisälä frequency N 2 depends on both the potential https://doi.org/10.5194/acp-2020-361Preprint.Discussion started: 4 May 2020 c Author(s) 2020.CC BY 4.0 License.
ref may not match very well with θ ref for altitudes above 50 km.Based on the temperature profiles in Figure 9a and considering the hydrostatic assumption as fulfilled, the Bruntg = 9.81 m s −2 is the gravitational acceleration.The resulting vertical profiles of the Brunt-Väisälä frequencies are depicted in Figure 9c.Evidently, the values of N 2 ref and N 2 1005 deviate from each other and, thus, lead to different predictions of the atmosphere's actual stability.Notably, the difference N 2 ref − N 2 1005 increases with altitude as already implied by the increase of the difference θ ref − θ 1005 with altitude, shown in Figure 6b.

Figure 9 .
Figure 9. Vertical profiles of (a) temperature and (b) horizontal wind speed as typical for mid-latitudes in June and December, up to 85 km altitude.Resulting Brunt-Väisälä frequency N 2 (c), either based on θ ref (solid lines) or on θ1005 (dashed lines).Panels (d), (e), and (f): vertical profiles of N 2 ref and N 2 1005 for December (solid and dashed blue lines, respectively) with the modulus |m(z) • B(z)| (green lines), either based on θ ref (solid lines) or on θ1005 (dashed lines), cf.text for further details.The panels' titles document the individually chosen values of the parameters λx, λz and B fact .The thin grey horizontal lines indicate the altitude of the predicted wave breaking altitude, i.e.where |m(z) • B(z)| first coincides with N 2 (z) above the initiation height.
and the vertical wave length is varied between 100 m and 4000 m with 100 m increment.The aforementioned parameter values are used to compute the vertical profiles of |mB| and N 2 based on the mid-latitude December profiles which are displayed in panels (d), (e), and (f) in Figure9.The green lines illustrate the modulus |mB| of the product of the vertical wave number and the buoyancy amplitude, while the blue lines exhibit the altitude dependence of the Brunt-Väisälä frequency N 2 .The results shown as solid lines are based on the new reference potential temperature θ ref ; the results of the computations using θ 1005 are represented by dashed lines.The predicted altitude of wave breaking is indicated by the first crossover of |mB| and N 2 above the wave's initiation height, indicated by the thin grey horizontal lines.Apparently, the predicted altitudes of wave breaking differ by more than about 5 km, depending on the definition of θ used (cf.panels (d), (e), (f) of Figure9).
temperature θ ref was introduced, which is thermodynamically consistent and based on a state-of-the-art parameterisation of the ideal-gas specific heat capacity of dry air from the National Institute of Standards and Technology (NIST).This reference https://doi.org/10.5194/acp-2020-361Preprint.Discussion started: 4 May 2020 c Author(s) 2020.CC BY 4.0 License.
The suggested approximation steps to obtain a reference potential temperature have two main sources of error: the errorθ ref − θ approxref inherent in the integral's basic approximation and the error θ approx ref − θ (k) of the k-th Newton iterate.The latter error approaches zero as k → ∞, whereas the error resulting from the basic approximation remains well below 0.1 % (along the US Standard Atmosphere) for values of θ ref of up to ∼ 2000 K, hence up to stratopause altitudes.To keep this low error level also for θ ref > 2000 K, the approximation may require an extension by means of a higher-order polynomial.
https://doi.org/10.5194/acp-2020-361Preprint.Discussion started: 4 May 2020 c Author(s) 2020.CC BY 4.0 License.conventional definition can result in a shift of predicted altitudes where the wave breaking occurs.The analysed cases revealed the prediction's high sensitivity to variations in the initiation conditions and the vertical profiles of temperature and wind.
's volume is held constant, equation (A4) represents the definition of the constant-volume heat capacity C V in molar units, i.e., dQ = ∂U ∂T V,N dT = C V (p, T ) dT. (A5) https://doi.org/10.5194/acp-2020-361Preprint.Discussion started: 4 May 2020 c Author(s) 2020.CC BY 4.0 License.by an approximation f (x) that is defined asf (x) = b 0 + b 1 ln (x − b 2 ) + b 3 x + b 4 x 2 .(C3) Notably, if c p is constant, this function reduces to an exact primitive of the integrand cp z with b 3 = b 4 = 0.Moreover, in this case, the resulting root-finding problem 0 = F (x) is exactly solvable and finally leads to the known conventional definition (13) of the potential temperature.

Figure
Figure C1.(a) Numerically evaluated function h(x) together with its approximation f (x); (b) the absolute approximation error h(x)−f (x).

Table 1 .
The values of ∆θ cp reach approximately 2.5 K at 10 km altitude and rise further, above https://doi.org/10.5194/acp-2020-361Preprint.Discussion started: 4 May 2020 c Author(s) 2020.CC BY 4.0 License.Temperature-independent constant values given mainly in textbooks for the specific heat capacity cp of dry air. )