The potential temperature is a widely used quantity in atmospheric science since it is conserved for dry 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 parameterisation 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 values and vertical gradients, in particular in the stratosphere and above, compared to the potential temperature that assumes constant heat capacity. The application of the new reference potential temperature is discussed for computations of the Brunt–Väisälä frequency, Ertel's potential vorticity, diabatic heating rates, and for the vertical sorting of observational data.

According to the book

Wegener mentioned a talk given by Köppen in a footnote on p. 111. In the publication year (1911) of Wegener's book, Köppen's daughter Else got engaged to Alfred Wegener

Cf.

These early applications of entropy in meteorology are also documented in

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, comparing repeated measurements of an atmospheric parameter on an isentropic surface or layer excludes any diabatic change in the probed air mass.

Apart from characterising the isentropes, the vertical profiles of the potential temperature
(

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 the equation for the moist specific
entropy, as derived from the first law of
thermodynamics and the Gibbs equation, further
extensions of the dry-air potential temperature
have been developed

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 or the virtual potential temperature,
which includes water vapour, are commonly
used as prognostic variables in numerical models for the formulations
of the energy equation

In any case, the use of the potential temperature requires the following preconditions to be fulfilled:

In principle, the concept of the potential temperature is transferable to all systems of thermally
stratified fluids such as 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

Moreover, the potential temperature is a frequently used quantity in oceanography

The study is organised as follows. The derivation of the potential
temperature for an ideal gas with constant specific heat capacity

The Gibbs equation

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

The ideal-gas law

The specific enthalpy is given by

Based on these
assumptions, the change in the specific entropy
(within the fluid

Note that the assumption of dry air being an ideal gas does not imply that in
Eq. (

Treating

The general theory of thermodynamics, assuming dry air as an ideal gas, gives the expression

In atmospheric sciences, for the majority of computations that require the specific heat
capacity of dry air, a constant value of

Even assuming a universally valid constant

Synopsis of temperature-independent constant values given mainly in textbooks for the specific heat capacity

Vertical profiles of

Computed vertical course of the potential temperature

These different values of constant

Figure

As indicated above, the reason for this sensitivity
to small variations of air's specific heat capacity
is that it affects the exponent of the equation
for

It should be noted that not only do literature values of air's specific heat capacity

The value of

However, accepting for a moment the WMO's definition (

Next, while retaining the ideal-gas assumption, we consider the dependence of air's

Variety of suggested values for the specific heat capacity of air. Ranges of constant values for

Figure

The measurement data, as well as the
parameterisations, clearly indicate a dependence of air's specific heat capacity
on the temperature. At temperatures above

the measurements of

the measured data reflect the true thermodynamic behaviour of the real gas rather than that of an ideal gas.

As already indicated by the data depicted in Fig.

As mentioned previously, at finite temperatures molecules also have contributions to

To determine the contribution of

For the contribution of

For a monatomic gas such as

The approach by

Besides a comprehensive survey of the available experimental data
for the specific heat capacity of air,

an empirical model-based equation of state for standard (dry) air considered as a pseudo-pure fluid and

assembly of a mixture model from equations of state for each pure fluid.

Each approach allows calculating the thermodynamic properties, e.g.

Both the pseudo-pure fluid model and the mixture model of

Like

The parameterisation (

The parameterisation from

Concerning the thermophysical properties of humid air, the study by

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/turboprop propulsion, aerodynamics, material sciences, etc.), especially where pressures exceed atmospheric, the assumption of ideal-gas behaviour introduces excessive uncertainty.

Previously introduced approaches for computing the specific heat capacity of dry air call
for a brief discussion on how to use the obtained

In the derivation of the potential temperature
(cf. Sect.

The rearrangement of Eq. (

In the following, the ideal-gas reference potential temperature

It may be noted that further variants of a reference potential temperature
are derivable by replacing

Unfortunately, for a straightforward solution of the integral (

For the reference of air's specific heat capacity,

As a first guess

Solving the previously described root-finding problem by Newton's
method over the comprehensive range of iteration steps (until the set requirement,
i.e.

In the context of numerical models of the atmosphere, the energy balance equation is occasionally
formulated based on the potential temperature

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 Eq. (

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

the need to use Newton's method for an iteration sequence to
approach the zero of

Proceeding from the definition (

As previously discussed (cf. Sect.

The difference between the approximation result and the reference, i.e.

Therefore, the second move towards a practical approximation procedure
is to construct approximations

Absolute 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

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 (

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

The total approximation error, which is

Flowchart guiding through the process of computing the approximation

For completeness, Fig.

The use of the new reference potential temperature

Although it is beyond the scope of the present study to
provide a general derivation of an appropriate energy
equation based on

The calculation of both the reference potential temperature

To account for real-gas effects (that cause a behaviour other
than that of an ideal gas; cf. Sect.

One caveat should be mentioned regarding the computed potential temperatures.
The range of validity of the equations of state for the air components

Difference

Figure

As previously shown, the newly defined reference potential temperature

The formula for the (squared)
Brunt–Väisälä frequency

The Brunt–Väisälä frequency
is the oscillation frequency of an air parcel due to a
local density perturbation

Vertical profiles of

To illustrate the deviation of

For equations involving the potential
temperature, however, it should be
emphasised that the substitution of

Ertel's potential vorticity

Using the temperature profiles from Fig.

It is noteworthy, however, that the computations of both

For atmospheric investigations, e.g. in the region of the
upper troposphere and lower stratosphere (UT/LS), it is
common practice to set vertical profiles of atmospheric
parameters in relation to the potential temperature as
a vertical coordinate. This way, the increasingly isentropic
stratification of the atmosphere above the UT is taken into
account. The transport of an air mass along isentropic surfaces,
i.e. surfaces of constant potential temperature and entropy,
is to be regarded as adiabatic.
Hence, the air's composition and properties within the same
isentrope interval, regardless of the observation location, are
better comparable than they would be if based on other isopleths
(i.e. height or pressure coordinates). Investigations of air mass
compositions over time and from different
regions at the same

Diabatic heating rates refer to the rate of energy

Apart from the absolute heating rates for the change in absolute
temperature, the change in potential temperature due to a
diabatic heating rate

Taking the relation

In order to judge the magnitudes of the
relative differences (Eqs.

A standard diagnostic for the speed of the stratospheric
circulation is the time lag of the upward-propagating seasonal
signal in tropical stratospheric water vapour

Note that the determination of absolute temperatures

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 temperature

The difference between the newly derived reference potential temperature

Derivation of a potential temperature that is consistent with
thermodynamics and that accounts for the ideal-gas properties of
dry air requires the integration of Gibbs' equation and the subsequent
solution of the resulting nonlinear equation. With a constant

The suggested approximation steps to obtain a reference potential temperature
have two main sources of error: the error

One of the foremost implications of the re-assessed potential temperature's
definition concerns the use of

Significant errors and biases may arise if, for instance, the conventional derivation of

In addition to the vertical sorting of data, implications of
the new reference potential temperature were discussed for several
other applications in which the potential temperature is used.
On the one hand, results may appear
mostly unaffected by using

In contrast, examples were shown where the
computation of Ertel's potential vorticity and the
rate of change of potential temperature in response to
diabatic heating yields different results by
the use of

It should be emphasised that all these examples were based on assuming particular profiles of temperature and pressure together with other assumptions. Moreover, only a limited number of examples could be investigated, while the applications of potential temperature are numerous. Consequently, a well-founded, individual decision is required for each application of the potential temperature as to whether it is worth applying the more rigorous calculation in the particular context.

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

In the following, the derivation of the air's specific heat capacities

If the system's volume is held constant, Eq. (

Alternatively, assuming the system's pressure as constant, its volume is
variable with total derivative

Using this result, together with Eq. (

This section explores, from a mathematical perspective, the sensitivity of the potential temperature formulation (

This section summarises the detailed steps of approximating the function

Proceeding from the definition of a function

As a further step, the function

By solving a least-squares problem, the coefficients in Eq. (

Values of the new reference potential temperature

As discussed in Sect.

The equation

The final step on the way to formulate a new expression for the potential
temperature requires defining one of the iterates

the standard of Newton's method (

Householder's method (

While the mathematical expressions in Eqs. (

Table

The following aims at a comprehensive investigation of the errors inherent
with approximating the ultimate reference potential temperature

Total relative error along the US Standard Atmosphere arising from the iteration process by declaring

The analysis of the approximation error is initially based on the
pressure and temperature profiles of the US Standard Atmosphere.
Figure

It may be noted that Householder's method achieves a significantly lower error
level than Newton's method due to its accelerated rate of convergence.
Compared to the first iterate approximations,
computation up to the second iterate
(cf. Fig.

Relative error of the second iterates

As with the discussion of the basic error in Sect.

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

As discussed in
Sect.

The data used in this study are available from the corresponding author upon request (manuel.baumgartner@uni-mainz.de).

MB, RW, and PS conceived, designed, and carried out the main part of the research. UA contributed the URAP data and implications on breaking heights of gravity waves. AHH gave advice about the heat capacity and performed the calculations of real-gas potential temperatures. FP contributed the implications on diabatic heating. MB and RW wrote the manuscript with contributions and reviews from all authors.

The authors declare that they have no conflict of interest.

We thank Eric W. Lemmon for his advice on the equation of state of dry air,
Vera Bense for fruitful discussions on gravity wave breaking,
Gergely Bölöni for providing us the URAP vertical profiles in Sect.

This research has been supported by the Deutsche Forschungsgemeinschaft (grant no. 257899354, AC 71/12-2, and SP 1163/5-2) and the Bundesministerium für Bildung, Wissenschaft, Forschung und Technologie (grant no. 01LG1205A).

This paper was edited by Timothy Garrett and reviewed by Pascal Marquet and one anonymous referee.