**4. Atmospheric temperature**

Before considering the role of Gibbs free energy in the atmosphere, we take a detailed look at a sample of evidence from aircraft flights, drawing on a selection of figures from Tuck [12–14, 17, 19] to examine what processes determine atmospheric temperature. The discussion starts with **Figures 1**–**4**, taken from reference [17], and questions and answers in [20].

**Figure 1** shows a temperature trace, and the plots for obtaining the Hurst exponent *H*1 and the intermittency *C*1 from it. It is a great circle flight in the lower stratosphere during Arctic summer. The value *H*1 = 0.56 ± 0.03 is observational confirmation of the theoretical value of 5/9 posited by the statistical multifractal equations of Lovejoy and Schertzer [7], in the case of temperature behaving like a passive scalar in the horizontal. We will see, however, that *T* does not always so behave in the vertical. Further evidence of statistical multifractal behaviour can be seen in **Figure 2**, where the probability distributions are shown for millions of points from ER-2 flights taken at 1 Hz – corresponding to 200 m horizontal resolution - during Arctic summer and winter conditions. The fat tails are characteristic; Gaussians are not seen. The Lévy exponent *α* expresses this, being on average 1.6 whereas a Gaussian has the value 2; its predicted range is 1.5 < *α* < 2, see [12, 13]. The implication of this is that the variance of *T* does not converge, although the mean does.

The evidence for correlations between ozone photodissociation rate with temperature and its intermittency is exemplified in **Figures 5** and **6**.

**Figure 3** shows the photodissociation rate *J*[O3] plotted against the intermittency of temperature *C*1(*T*) for flights of the ER-2 in the Arctic lower stratosphere for the spring–summer-autumn of 1997 and the winter of 2000. **Figure 4** shows *T* itself plotted against *C*1(*T*) for the same data. We examine in **Figures 5** and **6** evidence from a particular flight, 19970509 (yyyymmdd), which crossed the terminator in a region of low wind speeds, enabling flight in 'the same air mass' in sunlit and dark conditions. The separate sunlit and dark legs were not long enough for a separate statistical multifractal analysis. The flight was 35 days from summer solstice and had a higher sun than most of the flights on **Figures 3** and **4**. In **Figures 3** and **4**, the intermittency of temperature never drops to zero even when the ozone photodissociation rate does, although approaching it in the coldest and darkest points in winter; see the points in the lower left corners. The intermittency of temperature

#### **Figure 5.**

*Observations from the ER-2 on 19970509 (yyyymmdd), when 'racetrack' segments were flown either side of the terminator in a slow moving airmass at about 55 mbar. Black curve,* J*[O3]; O3, blue;* T*, red; green, east longitude. Note that temperature is cooler in the dark, while ozone does not change. Like wind speed and the tracer nitrous oxide in Figure 6, it is approximately symmetrical about the terminator, where* J*[O3]* ≈ *0.* T *has increased by about 0.4 K in two hours in the sunlit air.*

is highest in the sunniest and warmest points, clustered in the upper right corners of **Figures 3** and **4**. **Figures 5** and **6** can cast some light on the foregoing results, allowing a direct estimate of the radiative heating rate.

In **Figures 5** and **6**, the black trace defines *J*[O3] and hence goes to near zero at the terminator and beyond into night. Across the terminator, the low wind speed, ozone and tracer nitrous oxide, while varying about a mean in sunlight and dark, remain constant on average. That is not true of temperature, which is on average 0.4 K higher in sunlit conditions, 'in the same air'. A heating rate of 0.2 K/hour at 55 mbar can be calculated, and successfully checked by computation of the energy influx from radiation and the specific heat of air. The heating rate is consistent with the observation of *J*[O3].

In May, and September, the temperature in the sunlit part of the flight legs was on the warm side of the probability distribution compared to the dark side, again 'in the same air'. This result can be seen in **Figures 7** and **8**.

These considerations are also consistent with the concept of heating by unthermalized translationally hot oxygen atoms causing intermittency of temperature.

The behaviour of temperature in the vertical is not that of a well-mixed passive scalar ('tracer' in atmospheric usage). Its scaling is dominated by gravity [12, 13, 21–23]. Experimental evidence is shown in **Figures 9** and **10**.

#### **Figure 6.**

*Same flight as in Figure 5. Windspeed, light blue; nitrous oxide, brown. In a slow-moving air mass (no more than 3% movement during the flight), wind speed and tracer are approximately symmetrical about the terminator, with temperature in Figure 5 being the only variable showing asymmetry about* J*[O3]* ≈ *0.*

#### **Figure 7.**

*Probability distribution for temperature, normalised to unity, for the data in Figure 5. Population has moved from the more probable values on the dark side of the terminator to the sunlit side.*

*Scale Invariant Turbulence and Gibbs Free Energy in the Atmosphere DOI: http://dx.doi.org/10.5772/intechopen.95268*

#### **Figure 8.**

*Probability distribution for temperature, normalised to unity, either side of the terminator for three flights from (65o N,148o W) 19970911 (yyyymmdd), 19970914 (yyyymmdd) and 19970915 (yyyymmdd). As for 20040305 (yyyymmdd), the sunlit data have gained population from the dark data.*

#### **Figure 9.**

*Observations and* H*1 scaling for dropsonde descent at (42o 42′ 56" N, 170<sup>o</sup> 55′ 30" W), temperature, horizontal wind speed and relative humidity, 20040305.*

#### **Figure 10.**

*Composite variogram for all 885 useable dropsonde descents over the eastern Pacific Ocean, from 15o N to 60o N, like that in Figure 9 during winter storms 2004, 2005 and 2006. The scaling is excellent for all three of temperature, horizontal wind speed and relative humidity. See [23] for further discussion.*

**Figure 9** shows the vertical scaling of the temperature, horizontal wind speed and relative humidity for an individual dropsonde descent in early March 2004 [23]. The scaling exponent of temperature *H*1(*T*) is near unity; it does however have significant intermittency of ≈0.20 (not shown), demonstrating departures from hydrostatic equilibrium. The vertical scaling exponents *H*1(*s*) and *H*1(rh) of horizontal wind speed and relative humidity are significantly less than that of temperature, being not directly affected by gravity, unlike the total air density and hence temperature. **Figure 10** shows the grand average composite variogram of 885 dropsondes from the three years of Winter Storms 2004, 2005 and 2006. The vertical scaling of temperature, wind speed and relative humidity is further discussed in [23].
