**2. Atmospheric entropy and Gibbs free energy from scale invariance**

This account draws on those in Tuck [5, 12–14, 17, 19] and references therein. In order to maintain scale invariance, the same processes must be at work scale by scale over the range concerned, 40 metres to a great circle (40,000 km) in this case. That process is manifest as turbulence, driven by the thermodynamic need to minimise Gibbs free energy, which it does by dilution of energy density. **Figure 1** shows an example of an airborne temperature trace at 200 m resolution. Analyses for the scaling exponent *H* and the intermittency *C*1 are also shown, see below for definitions.

Satellite data have extended the aircraft data's range from an Earth radius to a circumference [16]. A view of the atmospheric circulation then emerges wherein organised flow, the general circulation, is driven by turbulence that enables the minimisation of Gibbs free energy. That free energy arises from the absorption of photons from the low entropy solar beam by ozone, some clouds, a little by water vapour and by the surface. The necessary dissipation is achieved by infrared radiation to space from water vapour, carbon dioxide, ozone, water dimers, nitrous oxide, methane and assorted halocarbons. Radiation of both solar and terrestrial photons is by gases whose fluctuating abundance is modulated by atmospheric turbulence. The radiative effect of aerosols and clouds, including absorption, emission and scattering is an important component in the radiative balance, and the abundances of both are also modulated by turbulence.

Why turbulence? Is a question addressed successively by Horace Lamb, Théodore von Kármán and Werner Heisenberg, all of whom anticipated a disappointment in or hope for divine enlightenment upon entry to heaven – possibly with a Shakespearean degree of irony. Richard Feynman described it as the last unsolved major problem in classical physics. Here the view is put forward that it is an emergent property of

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

#### **Figure 1.**

*Temperature on an approximately horizontal flight leg of 4700 seconds from (65<sup>o</sup> N,148<sup>o</sup> W) on a great circle to the NE, on 19970506 (yyyymmdd). The data are averaged to 5 Hz, or about 40 m horizontal resolution (top). Lower left, log–log plot to determine Hurst exponent* H*1; the value of 0.56 ± 0.03 confirms the theoretical value of 5/9 predicted by statistical multifractal theory (the flights combine both horizontal and vertical motion). Lower right, intermittency* C*1(*T*), which correlates with the mean* T *for the segment when the data for all suitable flight legs are so plotted.* C*1(*T*) is also correlated with the ozone photodissociation rate, see text and Figures 3 and 4.*

molecular populations in an asymmetric environment, following the molecular dynamics calculations of Alder & Wainwright [11], as taken up by Tuck [5, 12–14, 17]. In the atmosphere, it is driven by the Gibbs free energy arising from the entropy difference between the incoming organised beam of solar photons and the outgoing less organised flux of infrared photons over the whole 4π solid angle.

The statistical multifractal analysis is briefly outlined as follows.

It has been argued recently that *G* is computable from observations in a scale invariant medium, and shown to work [5, 14]:

$$G \equiv -\frac{K(q)}{q} \tag{1}$$

where *K*(*q*)/*q* is a scaling quantity related to partition function *f*, Boltzmann constant *k* and temperature *T* by:

$$T \equiv \frac{1}{kq} \tag{2}$$

$$f \equiv \exp\left\{-K(q)\right\} \tag{3}$$

The relationship of *K*(*q*) to the Hurst exponent *H* is given in Tuck [14] as.

$$H = H\_q + K(q) / q \tag{4}$$

where

$$H\_q = \frac{\zeta(q)}{q} \tag{5}$$


#### **Table 1.**

*Equivalence between statistical thermodynamic and scaling variables.*

and ζ(*q*) is the linear slope of a log–log plot of the first order structure function of the fluctuations of the observed variable versus a lag parameter covering the range of the variable. The Hurst exponent *H*, the intermittency *C*1 and the Lévy exponent *α* are the scaling exponents that comprise the statistical multifractal description of atmospheric variability. Their derivation can be found in [14], with their statistical thermodynamic equivalences in **Table 1**.

By applying scale invariance to the bulk medium, and separately to the microdroplet, the Δ*G* between the two may be obtained by difference. Application of high-resolution experimental and imaging techniques would then enable a comparison with the values obtained by quantum statistical methods applied to the reactant molecules [5]. The technique has been successfully applied to the air itself [14].

### **3. Some consequences of atmospheric statistical multifractality**

As observed, atmospheric variables display the fat-tailed probability distribution functions characteristic of statistical multifractality; the Gaussians associated with Einstein-Smoluchowski diffusion are conspicuous by their absence. An example for temperature is shown in **Figure 2**.

The results from Alder and Wainwright [11] combined with statistical multifractal analysis [5, 12, 13, 17] imply that the air is not at chemical equilibrium and consequently the gas constant *R* is not sufficient to describe the molecular behaviour of the atmosphere. All current atmospheric models employ the gas constant in this manner. An experiment is suggested to test this point: measure the populations of the rotational energy levels of the major constituents, N2 and O2 to see whether they are at equilibrium – populated according to the Boltzmann distribution – or not. Direct measurement of the probability distribution of air molecule velocities would be a significant advance and check on the theory. The temperature should be compared between that consistent with Eq. (2) and that necessary to account for the rotational and translational populations. It is an important point that the implied higher population than Boltzmann equilibrium means that the far wings of the water vapour and carbon dioxide will be stronger than at equilibrium because these wings are caused by the molecular collisions with the highest speeds. That is where there is significant influence in calculations of the atmospheric radiative transfer, because unlike many line centres, they are not self-absorbed. There is observational evidence supporting this view of atmospheric temperature. **Figure 3** shows a plot of the ozone photodissociation rate *J*[O3], against the intermittency of temperature, *C*1(*T*), for flights in the lower Arctic stratosphere in the summer of 1997 and in the winter of 2000.

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

#### **Figure 2.**

*Probability distribution functions (PDF) of temperature from ER-2 flights at 18–20 km in the Arctic April– September 1997 (top) and January–march 2000 (bottom). The millions of data points are averaged to 1 Hz, corresponding to approximately 200 m horizontal resolution. Note the long or fat tails; Gaussians are not seen.*

The correlation is consistent with the effects of translationally hot oxygen atoms from ozone photodissociation being unequilibrated and conveying this to the air molecules as a whole. **Figure 4** shows the average temperature against intermittency *C*1(*T*) for the flight leg, and the correlation persists.

The reason for this, and the theoretical explanation, is in Chapter 5.1 of reference [12] and Section 4.2 of [14]. Further discussion is given in the next section. There are no suitable observational data to apply this analysis in the troposphere.

Now we consider the effects of scale invariance and statistical multifractality on clouds and aerosols. This approach was pioneered by Schertzer and Lovejoy [8, 9] and described at length in the context of clouds and radiation in Lovejoy and Schertzer [7]. Both phenomena pose major problems for current general circulation models (GCMs) that are used to attempt predictions of the future evolutions of climate. Lovejoy [10] has demonstrated that such models do not predict climate, they predict macroweather, the fluctuations on scales from 10 days to 30 years. 10 days is the approximate time for air to circle the globe, and it shows the scaling expected, namely 23/9 dimensional statistical multifractality. On time scales longer than 30 years, the data show similar scaling. On the intermediate 10-day to 30-year scale, the observations scale differently, and constitute the macroweather scale, which is what the so-called climate models predict. Macroweather forecasting of course has potential utility, but it is not climate prediction.

#### **Figure 3.**

*For all suitable ER-2 flights April–September 1997 and January–march 2000.* J*[O3] on the ordinate,* C*1(*T*) on the abscissa. The ozone photodissociation rate is averaged over the flight leg, with vertical bars representing the standard deviation. The intermittency of temperature is derived from the slope of the curve for each flight leg, see Figure 1, bottom left. The horizontal bars are the standard error of the slope of the line fit.*

#### **Figure 4.**

*As for Figure 3, but with* T *averaged over the flight leg on the ordinate. The macroscopic temperature is proportional to the mean square velocity of the air molecules, supporting the suggestion that translationally hot photofragments from ozone photodissociation account for the correlations in Figures 3 and 4.*

Clouds are still a major uncertainty in any modelling, because they involve the physical chemistry of all three phases of water, the complex chemistry of gas-toparticle conversion, the role of aerosols of varying sizes in acting as condensation nuclei, particularly as regards the role of organic surfactants. All of these phenomena affect the transmission of radiation, both UV/visible and infrared. The altitude of the clouds is also critical, via their temperature and hence state. As an example, we note that the organic content of lower stratospheric aerosols plays a disproportionately large role in their radiative influence [18]. This whole area of aerosols, clouds and radiation needs examination by scale invariant techniques, from individual particles to cloud decks. The scale invariant Gibbs free energy is, we argue, an appropriate tool. Its effects are of course intimately related to atmospheric temperature and its maintenance via dissipation through intermolecular energy exchange and subsequent infrared radiation to space.
