Abstract

The relationship between dry static energy and potential temperature, cpθ = cpT + gz, is exact for an adiabatic temperature profile, and extremely close to exact for an isothermal profile. Even though it is extremely accurate, its use in atmospheres with nonadiabatic temperature profiles can lead to significant errors when comparing the entropies of isolated atmospheric layers. Use of the relation in this context leads to the incorrect conclusion that an adiabatic temperature profile has greater entropy than an isothermal profile for the same static energy. The relation fails in this application because of the extreme sensitivity of the column-integrated entropy to temperature.

1. Introduction

The specific, dry static energy (the sum of the specific enthalpy and geopotential) is defined as

 
formula

An often used approximation relates specific, dry static energy to potential temperature via

 
formula

Equation (2) is exact for an adiabatic temperature profile (Madden and Robitaille 1970; Betts 1974), and the subscript θ is included as a reminder of this. Equation (2) is often applied to atmospheres having nonadiabatic temperature profiles, and its accuracy is easily assessed by computing the dry static energy of an isothermal profile using both Eqs. (1) and (2) and comparing the results. For an isothermal profile (T = Tiso) the height in a hydrostatic atmosphere as a function of pressure is

 
formula

(R is the specific gas constant). Using this in Eq. (1) and comparing the result with Eq. (2) shows that Eq. (2) slightly overestimates the dry static energy, but the error is less than 1% in the lower troposphere (Fig. 1). Thus, Eq. (2) is very accurate even for nonadiabatic temperature profiles.

Fig. 1.

Static energy vs pressure for an isothermal atmosphere (280 K). The solid line is cpT + gz while the dashed line is cpθ.

Fig. 1.

Static energy vs pressure for an isothermal atmosphere (280 K). The solid line is cpT + gz while the dashed line is cpθ.

Despite the high accuracy of Eq. (2), especially for shallow atmospheric layers, it should not be used when comparing the column-integrated entropies of isolated atmospheric columns with various temperature profiles. This paper discusses why the approximation should not be used in this context, and shows that if Eq. (2) is used to calculate dry static energy in this application an incorrect conclusion that an adiabatic temperature profile has greater entropy than an isothermal profile is drawn.

2. Calculating the isothermal temperature of an isolated layer

In an isolated layer there are no fluxes of heat or mass in or out, so that the dry static energy of the column is constant. The integrated dry static energy of a hydrostatic column extending from the surface (p0) to some pressure p1 is

 
formula

An adiabatic profile has a temperature given by T = T0g z/cp and the height as a function of pressure is

 
formula

(κ equals R/cp) so the total dry static energy from Eq. (4) is

 
formula

For an isothermal profile, T = Tiso, and height is given by Eq. (3) and the total dry static energy of the column is

 
formula

Equating Eqs. (6) and (7) shows the isothermal temperature to be

 
formula

where Δ p = p0p1.

Using the approximation hθ = cpθ, the dry static energy of a hydrostatic column is

 
formula

For an adiabatic layer (θ constant) this gives

 
formula

where p* = 1000 hPa [note that if p0 is equal to p* then Eqs. (6) and (10) are identical] while for an isothermal layer it yields

 
formula

Equating Eqs. (10) and (11) gives an isothermal temperature of

 
formula

Equations (8) and (12) give virtually the same values for the isothermal temperature (Fig. 2), particularly for shallow layers. Equation (8) is the exact result, while Eq. (12) uses the approximation to static energy, and slightly underestimates the temperature. We might believe that two-tenths of a degree difference is not a significant source of error, but we shall see in the following section that even this small difference is enough to seriously underestimate the entropy of an air column and lead to the wrong conclusions about the equilibrium temperature profile of an isolated layer.

Fig. 2.

Calculated isothermal temperature vs layer thickness (T0 = 280 K). The solid line is from the exact formula [Eq. (8)] while the dashed line is from the approximation [Eq. (12)].

Fig. 2.

Calculated isothermal temperature vs layer thickness (T0 = 280 K). The solid line is from the exact formula [Eq. (8)] while the dashed line is from the approximation [Eq. (12)].

3. The entropy difference between an isothermal and an adiabatic profile

The entropy maximization principle states that in an isolated, unconstrained system the equilibrium position will have maximum entropy (Callen 1985). Gibbs (1928) applied the entropy maximization principle to an isolated, heterogeneous mass under the influence of gravity and proved that an isothermal profile has greater entropy than any other possible profile having the same static energy, and will therefore be the equilibrium temperature profile. (This should not be confused with the observation that a well-mixed boundary layer tends to have an adiabatic temperature profile. The boundary layer is hardly isolated, and has significant fluxes of heat across the bottom and top boundaries.) In this section we compare the entropies of adiabatic and isothermal profiles having the same static energy, and show that the isothermal temperature calculated from Eq. (12), which was derived using the approximation of Eq. (2), yields an incorrect result of an adiabatic profile having greater entropy than an isothermal profile.

The total entropy of a column of air is

 
formula

The entropy of an isothermal layer is

 
formula

while the entropy of an adiabatic layer is

 
formula

The difference in entropy between an isothermal profile and an adiabatic profile is found by subtracting Eq. (15) from (14), and is

 
formula

which integrates to

 
formula

A positive value of ΔS implies that the isothermal profile has greater entropy than the adiabatic profile, while a negative ΔS implies the opposite.

Figure 3 shows the relationship of ΔS with Tiso for a 200-hPa-thick layer extending from 1000 to 800 hPa. Also shown on the plot are the values of Tiso calculated using the exact expression [Eq. (8)] and the approximation [Eq. (12)]. The exact value is 271.66 K, while the approximate value is slightly smaller at 271.49 K. They differ by less than two-tenths of a degree, and for most applications we would think nothing of using the approximate value in place of the correct value. However, if we did so in this instance we would mistakenly calculate a negative ΔS, and incorrectly deduce that the adiabatic profile has greater entropy than an isothermal profile for the same static energy.

Fig. 3.

Difference in entropy between an isothermal column and an adiabatic column as a function of isothermal temperature. The top and bottom pressures are 800 and 1000 hPa, respectively. The adiabatic column has a potential temperature of 280 K. The dashed line (T = 271.66 K) is temperature from the exact formula [Eq. (8)]. The dot–dashed line (T = 271.49 K) is temperature from the approximation [Eq. (12)].

Fig. 3.

Difference in entropy between an isothermal column and an adiabatic column as a function of isothermal temperature. The top and bottom pressures are 800 and 1000 hPa, respectively. The adiabatic column has a potential temperature of 280 K. The dashed line (T = 271.66 K) is temperature from the exact formula [Eq. (8)]. The dot–dashed line (T = 271.49 K) is temperature from the approximation [Eq. (12)].

4. Conclusions

We have shown that a common approximation that dry static energy is directly proportional to potential temperature, though very accurate, should not be used when calculating and comparing the entropies of different temperature profiles in isolated atmospheric layers having constant static energy. This is because the total entropy of the air column is very sensitive to temperature, and errors in temperature of even a few tenths of a degree are large enough to cause serious errors in this application.

REFERENCES

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:
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Gibbs
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Footnotes

Corresponding author address: Dr. Alex DeCaria, Dept. of Earth Sciences, Millersville University, Millersville, PA 17551. Email: Alex.Decaria@millersville.edu