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pair of subsystems we get:

j j j where dSE and dSj are the annual increments of entropy for Earth (in total) and for each of its subsystems, the sum Y.j dSSj is the annual export of entropy from Space, and the sum Xj diSj is the total annual production of entropy by subsystems. Then for the system "Earth" we have:

j j j j i.e. as one might follow, di Se = X di Sj; deSE = dSsE = X dSsj. (4.4)

In accordance with Eq. (4.1) the upper limit of the annual entropy production within all subsystems of the system "Earth" is equal to diSj = 1.8 X 1022 J/K.

Note that in spite of the validity of a condition of the Prigogine theorem about the minimum entropy production for the whole system, its application to each subsystem is not correct. Each of them is the non-linear thermodynamic system, in which the dependence of the exchange flows of energy and matter from each other may also be non-linear.

We shall calculate the entropy contained in each subsystem and entropy flows between the subsystems (Venevsky, 1991; Svirezhev, 2001a). For that we shall also use the ideas and methods from Morowitz's book (1968). These calculations are quite cumbersome and, therefore, we shall omit the details. But first let us note that the values of Sj in Eq. (4.2) have different orders of magnitude: SL » SA, SB, SP, SH, since the mass of lithosphere is more than six orders of magnitude higher than that of other subsystems.

So, for the subsystems we have:

Atmosphere. We use a polytropic model of the "static" atmosphere (Khrgian, 1983). Such a "static" atmosphere is a non-equilibrium thermodynamic system consisting of the isothermal layers with their temperatures (the temperature of the ground layer is 14°C). In accordance with Landau and Lifshitz (1995) the entropy of such a system is equal to the sum of entropies of each layer, which is the mixture of ideal gases N2, O2, CO2, argon and H2O vapour. Their specific entropies at temperature 25°C and a pressure of 1 atm are equal to sN2 = 192, sO2 = 205, sCO2 = 214, sAr = 155, sHzO = 154, correspondingly. They are expressed in J/K mole. The calculation of entropy for the mixture of gases, if their molar concentrations and specific entropy for each component are known, is a standard operation in thermodynamics (see, for instance, Landau and Lifshitz, 1995). Then the total entropy of the polytropic atmosphere (under the assumption of the constancy of its composition) is equal to SA < 3.5 X 1022 J/K. Since carbon dioxide is one of the "life-forming" gases, we can calculate the corresponding entropy separately: sA°2 < 1 X 1019 J/K biota. We assume that the subsystem "biota" is identified with the terrestrial phytomass and submitted into a thermostat with the temperature T = 14°C, i.e. with the temperature equal to the mean annual temperature of our planet. The standard (averaging) composition of biomass is the following: liquid H2O—44%, fixed H2O—6%, cellulose—37.5%, prOteins—8.4%, carbO-hydrates, lipids, etc.—4.1%. The total mass Of the dry matter is 1.84 X 1018 g. Since the specific mOlar entropies for each fraction are knOwn, then the total entropy is easily calculated: SB < 9.1 X 1018 J/K. By excluding the water entropy we immediately get the entropy Of dry biOmass equal tO SB < 2.8 X 1018 J/K. Estimating the value Of entropy by anOther methOd, MOrOwitz (1968) has SB < 2.7 X 1018 J/K, i.e. generally speaking, a very similar value.

The main uncertainty here is in the estimatiOn Of the fractiOn Of H2O in a living biOmass. The estimates vary in a very wide interval from 40 up tO 90% for different plant species. Apparently, the estimatiOn in 50% is understated and, therefore, the total entropy Of biOta is abOut 1019 J/K.

It is alsO necessary tO take intO accOunt that water in biOta can be in twO states: liquid and ice-like, fOr instance in crystal lattices Of cellulOse. AbsOlute mOlar entrOpy Of water in the latter state at 14°C is less than the same entropy for liquid water by < 3 J/KmOle. Therefore, increase Of the percent Of fixed water can decrease the total entropy Of biOta, but its Order Of magnitude is nOt higher than 1017.

So, the entropy Of biOta (biOsphere) is less by three Orders Of magnitude than the atmOsphere entrOpy and has the same Order as the CO2 entrOpy in the atmOsphere.

11.5. Thermodynamics model of the biosphere. 2. Annual increment of entropy in the biosphere

In Order tO estimate the annual increment Of entrOpy in the biOsphere (biOta) we need information abOut in- and OutflOws Of energy and matter. A lOt Of such data for 1970 was published by COstanza and Neil (1982), and we shall use them. Again we assume that the subsystem Of biOta is a thermOstat with T = 14°C.

The annual increment Of entrOpy dSB = diSB + dSsB + dSPB + dSHB + dSAB. (5.1)

Let us cOnsider each term Of Eq. (5.1) separately.

Calculating dSSB we assume that vegetatiOn uses Only direct sOlar energy. COstanza and Neil estimate this flOw as ESB = 1.04 X 1024 J/year; then dSsB = 4dEsB ^ < 2.403 X 1020 J/K year. (5.2)

3 Ts

The value Of dSPB is determined by flOws Of mineral elements frOm sOil intO plants and flOws Of dead Organic matter frOm biOta intO sOil. If the first flOws are neglected (it is pOssible since the mass percent Of mineral elements in the living biOmass is relatively lOw, as well as their specific entropies), then dSPB = — s218J^DOM, where the annual flOw Of dead Organic matter gDOM = 3.34 X 1017 g/year (living biOmass cOntains abOut 50% water), and s28O7 < 2.47 J/K g is the specific entropy Of living biOmass at 287 K. The first value is taken frOm COstanza and Neil, and the secOnd value was Obtained using the results Of SectiOn 11.4. In fact, SB < 9.1 X 1018 J/K, and the total mass Of the dry matter is

1.84 X 1018 g. Then = 9.1 X 1018/2 X 1.84 X 1018 < 2.47 J/Kg and dSPB = -ib87?DOM < "8.25 X 1017J/K year. (5.3)

The value of dSHB = ^Hrt'O ?h20 j where qH2O < 6.67 X 1019 g H2O/year is the mass of liquid water consumed annually by biota (Costanza and Neil, 1982) and sH80 < 3.89 X J/K g (Morowitz, 1968). Then 2

The value of dS^ consists of the following exchange flows of entropy:

1. Entropy flow caused by diffusion of CO2 through stomata into leaf, dSA°;

2. Entropy flow caused by diffusion of O2 through stomata into the atmosphere, d^Ag;

3. Entropy flow caused by evapotranspiration, d^Ag0.

Let us consider each of these flows separately. Analogously with dSHB we have:

### CO 287 O 287

d^AB2 = sC802qC02 and dSAj = "s0"'qO2j where qC02 and q02 are the annual amounts of carbon dioxide and oxygen consumed and released by vegetation in the process of photosynthesis, correspondingly: qC02 = 2.255 X 1017 g CO2/year and q02 = 1.64 X 1017 g O2/year. Accordingly with Morowitz, the values of corresponding specific entropies are equal to sC02 < 4.86 J/g K and s0827 < 6.41 J/g K. Then dSAO2 = sCO qCO2 < 1.096 X 1018 J/K year; 5)

The entropy flow dSAB0 is a jump of entropy caused by the phase transition "liquid water ! water vapour": dS^0 = qHAc^dQ/Tfl) where qH^b = 5.67 X 1019 g H20/year is the annual total mass of evapotranspirated water, dQ = -2514 J/g H20 is energy of the phase transition, and TB = 287 K is the temperature of biota. Then dSHO = qHAo(dQ/TB) < -4.97 X 1020 J/K. (5.6)

We shall summarise all these flows in accordance with their orders of magnitude. So, the sum of flows, having 20 decimal exponents, is dS20 = dSSB + dSHB + dSHB0 = (2.403 + 2.595 - 4.97)X1020 < 2.8X1018j (5.7)

i.e. the summation of the exchange flows of entropy related to the global energy and water cycles leads to the loss of two orders of magnitude. Note that the value of dS20 will be lower if a jump of entropy caused by the phase transition from fixed water to liquid in the process of evapotranspiration is taken into account, i.e. two phase transitions are considered. The entropy jump is equal to

287 BA 287 287

where p is the fraction of fixed water in the total evapotranspiration, and (f7^ - s^O) = -0.168 J/KgH20 is the difference of corresponding entropies. The value of p is usually estimated as 12%; then AS287 = -1.14X 1018J/K. Assuming that p = 30%; we get

AS287 = — 2.85X10 J/K. Therefore, with a very high probability we obtain that dS20 < 1 X 1018 J/K. Moreover, since the percent of water in biota is just a little understated, we can state that the value of dS20 has the order of magnitude lower than 18.

The summation of flows with 18 decimal exponents gives the value dS18 = dsAO2 + dSOB = (1.096 — 1.051) X 1018 < 4.5 X 1016 J/K year, i.e. the exchange flows of entropy related to CO2 and O2 are almost balanced, so that their sum is also reduced by two orders of magnitude.

In order to obtain the total entropy balance we have to estimate the internal production of entropy, diSB, by biota, which is mainly connected with chemical reactions of the formation of structural molecules of plant and animal biomass (the formation of proteins and cellulose). Organic compounds containing phosphorus take an active part in such a type of reaction, therefore dsA°j2, dSAB » diSB by virtue of relation of the orders of magnitude in the phosphorus and carbon biochemical cycles (Morowitz, 1968).

Finally, since the exchange entropy flow between the pedosphere and biota is dSPB = — 8.25 X 1017 J/K year, the annual entropy balance for the biosphere is equal to dSB < dS20 + dS18 + dSPB = 1 X 1018 + 4.5 X 1016 — 0.825 X 1018

So, with the accuracy of two orders of magnitude (entropy of the biosphere is about 1019 J/K) we can state that the entropy balance for the biosphere in 1970 was equal to zero, i.e. in spite of the anthropogenic impact the biosphere was in dynamic equilibrium. The balance is shown in Fig. 11.2.

The result is very important, since the hypothesis about a quasi-stationary state of the contemporary biosphere plays one of the main roles in globalistics (Svirezhev, 1997a, 1998a). Should we test this statement somehow? Generally speaking, numerous estimations show that zero energy balance of the biosphere is fulfilled with sufficient accuracy that testifies to the advantage of the stationary hypothesis, but this is not entirely sufficient. In order to test whether the thermodynamic condition is sufficient, the balance between the internal entropy production and its export into the environment has to be estimated. The estimation has shown that it is almost equal to zero. Therefore, the system is in dynamic equilibrium with its environment, i.e. the contemporary biosphere is in a quasi-stationary state.

11.6. Exergy of solar radiation: global scale

In Section 10.6 of Chapter 10, we calculated the intraseasonal exergy balance for several types of vegetation localised in different geographical sites (two forests, grass and agriculture). We shall consider here a global scale of the problem (see also Svirezhev et al., 2003). In particular, it is interesting to see what new characteristics are added to the overall portrait of our planet by the global map of total exergy.

In order to calculate the global spatial distribution of exergy we used the surface radiation budget (SRB) data set collected and processed by Rossow and Schiffer (1991)

Fig. 11.2. Annual entrOpy balance for the biOsphere (biOta) in the 1970s. SR—sOlar radiatiOn; CO2—carbOn diOxide (net); H2O—liquid phase Of water; DOM—dead Organic matter. All storages are in J/K, all flOws are in J/K year.

and Darnell et al. (1992, 1996). This data set was develOped On the 1° X 1° grid (360 X 180 pOints). It cOnsists Of mOnthly means cOvering the periOd July 1983-June 1991. It cOvers the spectral range Of 0.2-50 |xm, and is divided intO twO regiOns: the shOrt-wave (SW, 0.2-5.0) and the thermal lOng-wave (LW, 5.0-50) micrometers. BOth the dOwnward and net radiatiOn at the surface are given.

AmOng Others the fOllOwing parameters are given: all-sky dOwnward shOrt-wave flux (SWDWN, insOlatiOn), tOtal sky net shOrt-wave flux (SWNET, absOrbed), all-sky dOwnward lOng-wave flux (LWDWN), tOtal sky net lOng-wave flux (LWNET; all in W/m2). In Our nOtatiOns Ein = SWDWN + LWDWN, R = SWNET + LWNET, a = 1 — (SWNET + LWNET)/(SWDWN + LWDWN). As abOve, we again deal with twO spectral intervals, vs and Vl fOr shOrt- and lOng-wave radiatiOn, respectively, with the bOrder Of divisiOn at 5 |xm, and then each spectrum (incOming and OutgOing radiatiOn) is represented Only by twO numbers, which are the integrals in respect tO the cOrrespOnding interval. Other values used in the exergy calculatiOns are defined as Rs = jV R(v)dv = SWNET, Rl = ja R(v)dv = LWNET, as = EOut/Ef = 1 — (SWNET/SWDWN) and al = E°ut/E}n = 1 - (LWNET/LWDWN).

Determination of the SRB requires information concerning the surface conditions (temperature, reflectivity, emissivity), the overlying atmosphere (composition, transmis-sibility, temperature, etc.), and the top-of-the-atmosphere insolation. Extensive work has been done to validate the methods and results. Recent descriptions are given in Darnell et al. (1992), Gupta et al. (1992, 1993) and Whitlock et al. (1995). These studies include comparison with more detailed radiative transfer models, with the results of other SRB algorithms and with surface measurements.

We reduced the data to the 2° X 2° grid for easier handling. For our exergy analysis we used the data for 1990. The reason is that 1990 is often referred to in the climate change debate because it was decided to relate the dynamic of future CO2 emissions to 1990.

For calculation of the exergy spatial distribution we use formulas (6.4) and (6.5) (Chapter 10), and also data about the components of the radiation balance. The results are represented as a map of the annual global exergy (see Fig. 11.3a), where the data are averaged by the annual time interval. Note that since we have only the integral data for two spectral intervals, the exergy estimation gives the value rather lower than exact one.

In addition, a map of global vegetation (biomes map) was taken from the site of NASA's Earth observatory: http://earthobservatory.nasa.gov/Library/LandCover/ land_cover_3.html.

This map shows a breakdown of general vegetation types over large areas of Earth, and contains 14 categories (see Fig. 11.4b). To determine what type of vegetation covers a given region, NDVI data were analysed and periodically compared with ground test measurements of the mapped areas.

The spatial distribution of the annual total exergy is shown in Fig. 11.3a. One can see that the exergy reaches its maximal values in the "red-orange" domains 1-4, where the highest degree of transformation of incoming radiation occurs. What kind of specific properties do these domains possess? It is known that these are so-called "oceanic gyres", i.e. the regions of the World Ocean with maximal circulation (Peixoto and Oort, 1992). This circulation allows for a very efficient transformation of the incoming radiation. As radiation is highest in these areas, we see the global maximum of exergy here. A map of the energy balance would show similar results, but the exergy map manifests these domains much more distinctly. Regions 1 and 2 are the well-known "equatorial upwellings" (famous El Niño events happen in region 1), region 3 is the "Arab-Somali upwelling", region 4 is the "Darwin upwelling". In other words, exergy can be a good indicator for crucial regions of the ocean.

One of the possible interpretations of exergy is the maximal useful work which can be performed by the system in the course of movement towards thermodynamic equilibrium with the environment (see Chapter 5). The First Law of Thermodynamics states that SQ = dU + SA, i.e. heat SQ, absorbed by the system from its environment, is spent for the increase of internal energy of the system, dU, and the performance of work against external forces. In our case, the system is an active surface transforming incoming radiation, and SQ = Ein - Eout = R, SA = Ex, so that R = dU + Ex, or dU = R - Ex, (6.1)

where dU is the full differential of internal energy of the system. Therefore, U is a state

20 40 K 80 10D 120 140 100 100 200 220

20 40 K 80 10D 120 140 100 100 200 220

150° W 150° W 120* W W 60'W itf W 0* 50° E 60* £ 90* E 120° E 150° E 100° E

150° W 150° W 120* W W 60'W itf W 0* 50° E 60* £ 90* E 120° E 150° E 100° E

Fig. 11.3. Global maps (annual means, W/m2) of exergy (a) and internal energy increment AU (b).

Fig. 11.3. Global maps (annual means, W/m2) of exergy (a) and internal energy increment AU (b).

variable. Let us consider a 1-year interval. Then the values R = R and Ex = Ex have to be annual means of their intraseasonal values, but the annual change of internal energy is defined as AU = Uend — Ubeg where Ubeg and Uend are the values of U at the beginning and the end of the year. Since, in our case, all these processes are periodical with a 1-year period

Fig. 11.4. Continental maps of internal energy increment DU (annual mean, W/m ) (a) and vegetation (b).

then the values of the state variables at the beginning and the end of the year have to coincide, Uend = Ubeg. Therefore, AU = R - Ex = 0.

However, if we construct a global map of the annual increment of internal energy, A Uj we can see that the condition A U = 0 is far from being fulfilled everywhere. There are regiOns where the annual increment Of internal energy is pOsitive and very large (these are equatOrial regiOns Of the Oceans and cOntinents). There are alsO regiOns with large negative values Of increment (pOlar and sub-pOlar regiOns). FOrmally, this means that in all these regiOns the energy cOnservatiOn law is nOt fulfilled! HOwever, this cOntradictiOn is quickly resOlved if One recalls that Only the glObal value Of DU, DUGl = J"s DU(x, y)dxdy, where S is the whOle surface Of the glObe, has tO equal zerO. The real calculatiOns prOve this fact.

It is natural that in the prOcess Of integratiOn the regiOns with pOsitive and negative DU will cOmpensate each Other. A physical mechanism Of the cOmpensatiOn is a transitiOn Of sOme part Of internal energy (fOr instance, in the fOrm Of mass and energy transpOrt by aerial and Oceanic currents) frOm dOmains with pOsitive DU intO dOmains where DU < 0. Then, frOm the pOint Of view Of thermOdynamics, the isOline DU(x, y) = 0 and its neighbOurhOOd have tO be a relatively quiet zOne.

The cOrrespOnding map is shOwn in Fig. 11.3b, in which the isOline DU(x, y) = 0 and its vicinity are marked in green. These "green dOmains" separate the "red" and "yellOw" regiOns frOm each Other. Expert OpiniOn assumes that the "red" regiOns in the Ocean are characterised by a very high intensity Of kinetic mOvements, upwellings and a very active biOlOgical life. CarbOn diOxide is emitted frOm the Ocean intO the atmOsphere. The cOntinental "red" regiOns cOrrespOnd tO dOmains with the maximal biOlOgical prOductiOn. Maybe (but this is a hypOthesis) these regiOns are alsO CO2 sOurces. The "yellOw" regiOns are alsO very active, but the activity is Of a different sOrt. In the Ocean these regiOns absOrb carbOn diOxide frOm the atmOsphere and are the regiOns Of circular currents and dOwnwellings. The cOntinental "yellOw" regiOns are dOmains with lOw biOlOgical prOductiOn. It may be that these regiOns are CO2 sinks.

There is nOt enOugh data tO either validate Or disprOve this hypOthesis. The reasOning (nOt prOOf) is as fOllOws. It is knOwn that the trOpical Ocean releases CO2, while the pOlar Ocean absOrbs it. The reasOn is that CO2 sOlubility is inversely related tO the temperature. The internal energy increment DU, as distinguished frOm the radiatiOn balance, is a cOntinuOus functiOn: there are nO gaps On the bOundary "earth-Ocean". Therefore, it may be suggested that we have the same situation in terrestrial systems: tropical ecOsystems cOuld pOssibly release CO2, while pOlar Ones absOrb it. This fact is alsO cOnfirmed by vast reserves Of humus in taiga and brOad-leaved fOrests, and the absence Of humus in the tropics (cf. UlanOwicz and HannOn, 1987).

When we lOOk at the map Of cOntinental distribution Of DU (Fig. 11.4a), we get the visual impressiOn that the regiOns with pOsitive and high values fOr the increment Of internal energy cOrrespOnd tO areas with a highly develOped vegetatiOn cOver (like trOpical rain fOrests). FOr cOnfirmatiOn, it is sufficient tO cOmpare Fig. 2 with the vegetatiOn map represented in Fig. 11.4b. COntinuing the visual cOmparisOn between Other vegetation types and the cOrrespOnding gradatiOns Of the increment Of internal energy, we see that the glObal pattern formed by the spatial distribution Of DU (Fig. 11.4a) and the glObal vegetation pattern (Fig. 11.4b) are very similar, at least when we cOmpare these maps simply as integral picturesque Objects. All these are in gOOd agreement with the table Of "exergy utilisation and storage in a cOmparative set Of ecOsystems" (J0rgensen et al., 2000), accOrding tO which deserts utilise Only 2% Of exergy, while the cOntributiOn Of tropical and Old-growth deciduOus forests is 70-72%.

Unfortunately, we do not know any reliable statistical method which would allow us to estimate quantitatively the degree of similarity between the two colour patterns. For this reason, at this stage we restrict the analysis to the visual comparison.

Since the NPP of different types of vegetation (biomes) decreases in an almost monotonous way from the equator to the poles, and the character of decrease is similar for DU(x, y), then we can assume that there is a close correlation between these values, and the value of DU(x, y) would be a measure of NPP. In order to prove this, additional investigations are needed.

### 11.7. Exergy of the biosphere

It seems that Vinogradov (1959) was the first to compare the chemical compositions of living and non-living matter (in Earth's biota and crust, correspondingly). Later on many authors repeated these estimations, but their corrections were insignificant. Therefore, for the estimation of the chemical exergy of the biosphere we have used (Svirezhev, 1997b) the original Vinogradov data. Some of them are shown in Table 11.1.

If we assume that the Earth's crust (non-living matter) is a system in thermodynamic equilibrium, we can calculate the exergy of living matter, i.e. the exergy of the biosphere, where the non-living matter of the Earth's crust is considered as some reference state. In other words, we consider the biosphere as some chemical system (for instance, an "active membrane"), which either concentrates or disperses chemical elements in comparison with their basic concentrations in the Earth's crust.

Let si be the content of ith element in the biosphere (biota) in percent in respect to mass and s0 be the same in the crust, which is considered as a residence state. If mi is the atomic weight of ith element then its molar concentration ci will be equal to ci = (si/mi)M, where M is the total biomass of the biosphere. In accordance with the standard definition of chemical exergy (see formula (4.6), Chapter 5), the exergy of the biosphere will be n

i=1 mi where M and Mo are the total mass of the biosphere and the Earth's crust, correspondingly. Strictly speaking, we take into account only that part of the crust which interacts with

Table 11.1

Chemical composition of living (biota) and non-living (crust) matter (in % to weight)

Table 11.1

Chemical composition of living (biota) and non-living (crust) matter (in % to weight)

 Element