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Fig. 10.2. Annual mean values of the exergy efficiency coefficient and Kullback measure in dependence of the radiation efficiency coefficient for different sites. KF, Kiel beech forest; SF, South Germany pine forest; CF, Central Germany pine forest; HG, Hamburg grassland; KC, Kiel crop field; OC, Oklahoma crop field. Dotted lines, linear regression for Kullback's measure; solid lines, linear regressions for the exergy efficiency coefficient.

Oklahoma although the latter has a much higher input of energy. The forests in Central and South Germany also show very high values of exergy and exergy efficiency in relation to the sites with higher anthropogenic pressure.

Certainly, these values are insufficient for any satisfactory statistics. However, the tendencies are as follows (see Fig. 10.2). Firstly, all the points, both (hEx, Hr) and (K, Hr), can be arranged by lines of regression hex = aEx + bexhr and K = aK + bKhr into two groups corresponding to forest and grass-crop ecosystems.

The regression coefficients are equal to: aEx = 0.114; afK = 0.134; bEx = bfK = 0.4 for forests and aEx = 0.088; agK = 0.083; bEx = 0.38; bgK = 0.45 for grass and crops. Lines with almost the same gradients indicate that although both the exergy and the information increment depend on the radiation balance (the latter differs in different sites), the dependence is identical for different types of vegetation. This indicates the possible existence of some regulating mechanism that is common for all types of vegetation.

10.7. Simplified energy and entropy balances in the ecosystem

From the viewpoint of thermodynamics, any ecosystem is an open thermodynamic system. An ecosystem being in a "climax" state corresponds to a dynamic equilibrium when the entropy production within the system is balanced by the entropy outflow to its environment.

Let us consider one unit of the Earth's surface, which is occupied by some natural ecosystem (e.g. meadow, steppe, forest, etc.) and is maintained in a climax state. Since the main component of any terrestrial ecosystem is vegetation, we assume that the area unit is covered by a layer, including dense enough vegetation and the upper layer of soil with litter, where dead organic matter is decomposed. The natural periodicity in such a system is equal to 1 year, so that all processes are averaged by a 1-year interval.

Since the energy and matter exchanges between the system and its environment are almost completely determined by the first autotrophic level, i.e. vegetation, the vegetation layer is considered as the system, whereas its environment is the atmosphere and soil. We assume that all exchange flows of matter, and also energy and entropy, are vertical, i.e. we neglect all horizontal flows and exchange between ecosystems located at different spatial points.

In Section 10.3 we have already used the simplified equation of the annual energy balance for a vegetation layer (equality (3.1)): R = ywqw + qh + ycqc, where the item ywqw is the latent heat flux, the item ycqc = GPP, and the item qh is a sensible heat flux, i.e. a turbulent flux transporting heat from the layer surface into the atmosphere. Since in our case there is an additional income of heat from biomass oxidation (respiration and decomposition of dead organic matter), the left side of the balance has to be represented as R + qox (instead of R) where qox = Qmet + Qdec • Here Qmet is a metabolic heat, and Qdec is heat released in the process of decomposition. Finally, the equation of energy balance is written as

In such a form of representation, all items of the energy balance are arranged in two groups (in square brackets) and members of these groups differ from each other by the orders of magnitude. For instance, the characteristic energies of the processes of a new biomass formation and its decomposition (the second brackets) are lower by a few orders of magnitude than the radiation balance and the energies, which are typical for evapotranspiration and turbulent transfer (the first brackets). As a rule, the terms contained within the second bracket are usually omitted in the standard expression for energy balance: R = ywqw + qh (see, for instance, Budyko, 1977). This suggests that in this case a so-called "asymptotic splitting" could be used. Note that such a kind of method is widely used in the theory of climate: this is a so-called "quasi-geostrophic approximation" (Pedlosky, 1979). Thus, if we follow the logic of asymptotic splitting then instead of exact balance (7.1) we get the two asymptotic equalities:

R - Gwqw - qh < 0 and Qmet + Qdec - GPP < 0- (7:2)

We assume that the fulfilment of the first equality provides the existence of some "thermostat", which should be called the "environment". The thermostat maintains the constancy of temperature and pressure within the environment. Then the fulfilment of the second equality is determined by a consistency of the processes of production on the one hand, and metabolism of vegetation and also decomposition of dead organic matter in litter and soil on the other hand.

In accordance with Glansdorff and Prigogine (1971) the entropy production within the system is equal to d^/dt < qox/T, where T is the temperature and qox is the heat prOductiOn Of the system. As shOwn abOve, the tOtal heat prOductiOn is a result Of twO processes: metabOlism Or respiration (Qmet) and decOmpOsitiOn Of dead Organic matter (Qdec) . Since these processes can be cOnsidered as a burning Of a cOrrespOnding amOunt Of Organic matter, the values Of Qmet and Qdec can be alsO expressed in enthalpy units. Thus, dS 1

dt T

The value Of T is the mean air temperature in the given area averaged alOng either the entire year, Or the vegetatiOn periOd and the periOd when decOmpOsitiOn is pOssible. In the latter case, all Other values alsO have tO be averaged alOng this periOd. Since the secOnd equality Of Eq. (7.2) has tO hOld, then diS GPP

di T

At the dynamic equilibrium the internal entrOpy prOductiOn has tO be cOmpensated by the entrOpy expOrt frOm the system, sO that dS

The last equality is a quantitative formulation Of the sO-called "entropy pump" cOncept (Svirezhev, 1990, 1998b, 2000; Svirezhev and Svirejeva-HOpkins, 1998; SteinbOrn and Svirezhev, 2000). We assume that the entropy pump "sucks" entropy produced by the ecOsystem. As a result, natural ecOsystems dO nOt accumulate entrOpy, and as a result it can exist during a sufficiently lOng time period. The pOwer Of the entropy pump, ldeS/dtl, depends On the GPP Of the natural ecOsystem lOcated in situ and the lOcal temperature (see Eq. (7.5)). We assume that the lOcal climatic, hydrolOgical, sOil and Other environmental cOnditiOns are adjusted in such a way that Only a natural ecOsystem, which is typical Of these lOcal cOnditiOns, can exist and be in a steady state at this site. We assume that the local climatic, hydrological, soil and other environmental conditions are adjusted in such a way that only a natural ecosystem corresponding specifically to these local conditions can exist at this site and be here in a steady state (dynamic equilibrium).

All these statements cOnstitute the main cOntent Of the "entrOpy pump" cOncept.

Hence, there is nO "OverprOductiOn Of entrOpy" in natural ecOsystems, because the "entropy pump" sucks the entire entropy Out Of ecOsystems, by the same token preparing them for a new 1-year period. Maybe, the picture will be clearer if we cOnsider the process Of ecOsystem functiOning as a cyclic prOcess with 1-year natural periOdicity. At the initial pOint Of the cycle the ecOsystem is in thermOdynamic equilibrium with its envirOnment. Then, as a result Of the wOrk dOne by the environment On the system, it performs a forced transitiOn tO a new dynamic equilibrium. The transitiOn is accOmpanied by the creatiOn Of new biOmass, and the ecOsystem entrOpy decreases. After this the reversible spontaneous prOcess is started, and the system mOves tO the initial pOint prOducing the entrOpy in the cOurse Of this path. The processes that accOmpany the transition are metabOlism Of vegetatiOn and the decOmpOsitiOn Of dead Organic matter in litter and sOil. If the cyclic process is reversible, i.e. the cycle can be repeated infinitely, the total productiOn Of entropy by the system has tO be equal tO its decrease at the first stage. Substantively bOth these transitions take place simultaneously, so that the entropy is produced within the system, and at the same time it is "sucked out" by the environment. At the equilibrium state, the annual amounts of the entropy produced by the system and the decrease of entropy caused by the work of the environment on the system are equivalent.

Let us imagine that this balance was disturbed (this is a typical situation for agro-ecosystems). Under the impact of new energy and matter inflows, the system moves towards a new state, which differs from dynamic equilibrium of the natural ecosystem. As a rule, the entropy produced by the system along the reversible path to the initial point cannot be compensated by its decrease at the first stage of the cycle. We obtain a typical situation of entropy overproduction. The further fate of this overproduced entropy could be different. The entropy can be accumulated within the system. It degrades as a result, and after a while dies (the first fate). The second fate is that the entropy can be "sucked out" by the environment, and the equilibrium will be re-established. In turn, this can be realised in two ways: either by import of an additional low-entropy energy, which can be used for the system restoration, or, unfortunately, by environmental degradation.

10.8. Entropy overproduction as a criterion of the degradation of natural ecosystems under anthropogenic pressure

When we talk about a "reference ecosystem" we take into account a completely natural ecosystem, without any anthropogenic load impacts. To find such an ecosystem today in industrialised countries is almost impossible (except, maybe, in the territories of natural parks). Really all so-called natural ecosystems today are under anthropogenic pressure (stress, impact, pollution, etc.). All these stresses began to act relatively recently (in about the last 100-150 years) in comparison with characteristic relaxation times of the biosphere, so that we can assume with rather high probability that the mechanisms, responsible for the functioning of the "entropy pump", have not yet adapted to the new situation. On the other hand, plants, which are main components of the natural ecosystem, react to anthropogenic stress very quickly, as a rule, by reducing their productivity.

Let us assume that the considered area is influenced by anthropogenic pressure, i.e. the inflow of artificial energy (W) to the system takes place. We include in this notion ("the inflow of artificial energy") both the direct energy inflow (fossil fuels, electricity, etc.) and the inflow of chemical substances (pollution, fertilisers, etc.). The anthropogenic pressure can be described by the vector of direct energy inflow Wf = {Wf, W2,...} and the vector of anthropogenic chemical inflows q = {q1, q2,...} to the ecosystem. A state of the "anthopogenic" ecosystem can be described by the vector of concentrations of chemical substances C = {Q, C2,...}, and such a macroscopic variable as the mean gross primary production (GPP) of the ecosystem. The undisturbed state of the corresponding natural ecosystem in the absence of anthropogenic pressure is considered as a reference state and is denoted by C0 = {C0, C0,...} and GPP0.

We suppose that the first inflow is dissipated inside the system when transformed directly into heat and, moreover, modifies the plant productivity. The second inflow, changing the chemical state of the environment, also modifies the plant productivity. In other words, there is a link between the input variables Wf and the state variables C on the one hand, and the macroscopic variable GPP on the other hand. It is given by the function GPP = GPP(C, Wf). Obviously, if we deal with contamination that inhibits plants' productivity, then this function must be monotonously decreasing with respect to its arguments. On the contrary, if the anthropogenic inflows stimulate plants (as with fertilisers) then the function increases. The typical "dose-effect" curves belong to such a functional class.

By formalising the previous arguments we can represent the entropy production within this "disturbed" ecosystem as

dt T

where the scalar W is a convolution of the inflows Wf and q, i.e. the total anthropogenic inflow. Note that the convolution may also depend on C, since these internal concentrations are maintained by the inflows q. We shall demonstrate later how this is calculated.

In accordance with the "entropy pump" concept a certain part of the entropy produced is released at this point by the "entropy pump" with power ldeS/dtl = GPP0/T, so that the total entropy balance is

di T

We assume here that despite anthropogenic perturbation the disturbed ecosystem is tending to a steady state again. If we accept it, we must also assume that the transition from natural to anthropogenic ecosystem is performed sufficiently fast so that the "tuning" of the entropy pump does not change. Therefore, it is unique that the residual part of the entropy production given by Eq. (8.2) is compensated by the outflow of entropy to the environment. This compensation can occur only at the expense of environmental degradation (s > 0) resulting, for instance, from heat and chemical pollution, and a mechanical impact on the system.

We also assume that the relation of succession connects the "natural" and "anthropogenic" ecosystems. We would like to include a few words about the relation of succession. Let us assume that the anthropogenic pressure has been removed, and the succession from the anthropogenic ecosystem towards a natural one has been started. The next stage of this succession, according to our concept, would be a "natural" ecosystem. Really, if the anthropogenic pressure is weak, the "natural" ecosystem (as we understand it) will be of a "wild" ecosystem type, existing at this site.

It is necessary to note, if the anthropogenic ecosystem is an agro-ecosystem, surrounded by forest, a grass -shrubs ecosystem (not a forest) will be successionally close to its "natural" ecosystem. Finally, we can define a successionally close ecosystem (i.e. an ecosystem successionally close to an ecosystem under anthropogenic stress) as the first stage of succession of an "anthropogenic" ecosystem where the anthropogenic stress is removed.

What is the "dynamic" sense of "successional closeness", and why do we need the concept? The point is that in this approach we can compare only close steady states, their vicinities must be intersected significantly, and the time-scale of a quasi-stationary transition from a natural to anthropogenic ecosystem and vice versa must be small (in comparison to the temporal scale of succession).

Let us consider the non-local dynamics of the system. We stop the flux of artificial energy (i.e. the anthropogenic energy and chemical fluxes) into the ecosystem. As a result, if the ecosystem is not degraded, a succession will take place at the site, which tends towards the natural ecosystem type specific for the territory (grassland, steppe, etc.). This is a typical reversible situation. Under severe degradation a succession would also take place, but towards another type of ecosystem. This is quite natural, since the environmental conditions have been strongly perturbed (for instance, as a result of soil degradation). This is an irreversible situation. So, the "successional closeness" concept means that we remain in a framework of "reversible" thermodynamics. And if there is no input of artificial energy, the steady-state for a given site (locality) will be presented by the natural ecosystem, as the local characteristics of the "entropy pump" correspond exactly to the natural type of ecosystem.

Nevertheless, there is a small incorrectness. When we discussed a successionally closed system above, we assumed implicitly that at any stage of the succession the system is in a dynamic equilibrium. Since a succession is a transition process between two steady states, this statement is incorrect. However, so far as we can suggest that the temporal scale of ecological succession is much greater than the same for anthropogenic processes, we can consider succession as a thermodynamically quasi-stationary process (simultaneously, we remain inside the model of equilibrium thermodynamics). Nevertheless, if we want to construct a thermodynamic model of succession, we should drop the hypothesis of quasi-stationary transition.

Bearing in mind the meaning of s, it is obvious that the s value can be used as the criterion for environmental degradation or as the "entropy fee" which has to be paid by society (actually suffering from the degradation of environment) for modern industrial technologies.

Of course, there is another way to balance the entropy production within the system. For instance, we can introduce an artificial energy and soil reclamation, pollution control (or, generally, ecological technologies). Using the entropy calculation we can estimate the necessary investments (in energy units).

10.9. Energy and chemical loads or how to convolute the vector data

Let us bear in mind that formula (8.2) describing the "degradative" part of the entropy in a disturbed ecosystem also represents the convolution W as a sum of two items: W = Wf + Wch: If the first item Wf, which is a convolution of the vector Wf, can be associated with the direct inflows of such a type of artificial energy as electricity, fossil fuels, etc. (energy load) then the second item Wch is associated with the inflows of chemical elements that maintain molar concentrations C within the system (chemical load).

Since all direct inflows are measured in common energy units, the convolution of Wf is defined simply: Wf = Y.i Wf: The problem is how to calculate Wch (Svirezhev, 1998b).

Let the system, which is described only by the chemical concentrations as state variables, move from an initial state C0 towards a state with C that is maintained by the pressure of the chemical load. We assume that the basic concentrations C0 correspond to a natural ("wild") ecosystem. We assume as well that if the chemical inflows q were stopped, then the system would evolve to its natural state. In this case the maintenance of concentration C within the system means that the work against chemical potentials is performed. As a result the entropy is produced inside the system with the rate dS = R X Ai,ioq = R X lnCQ/Cq (9.1)

i i where A^ are the affinities. Then

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