where Dw = (D^D™)/ (D1 + ADw^'), L = l1 / Ll2. Then the rate of energy dissipation by the vegetation layer of thickness l2 and unique area will be equal to

In order to illustrate these formulas we shall use a case study, namely a beech forest near Kiel in the North Germany (Steinborn, 2000). All data are for the year 1991. The first group of data contains information about climatic conditions of the given location: the mean atmospheric temperature during vegetation period is T = 16 °C, the temperature of leaves is Ti = 18 °C, the number of so-called "shining hours" is equal to 1646 h < 6 X 106 s, the concentrations of water vapour Cf(T) = 1.4 X 10~2 g H2O/gair and Cw = 1.16 X 10~2gH20/ g air (relative humidity is 83%). The next group of data provides information about the architecture of the vegetation layer: its thickness is l2 = 10 m = 103 cm, the thickness of leaf is 10"2 cm and the leaf area index is L = 4. Finally, the general information follows: the coefficients of molecular and turbulent diffusion are Dw = 0.25 cm2 s and d12) = 104 cm2/s, the density of dry air at 20°C is p = 1.2 X 10 3 g air/cm3 and the heat capacity cp = 1 J/K g air.

By substituting all these values into Eq. (4.6) we get C < 0.12 X 10-4 W/cm3, and, in accordance with Eq. (4.7), T(AiS/dt) = 0.12 X 10-4 X 103 = 0.012 W/cm2. Bearing in mind that the mean radiation balance for summer months is equal to 0.0126 W/cm2, the coincidence is fantastic! We do not know the reason for such a coincidence: either casual or, maybe, causal. At any rate, it is necessary to test the relation using other experimental data.

In fact, all energy absorbed by the vegetation layer of a beech forest is spent on its maintenance, including the production of new biomass, where the share of spent energy is so small that it is impossible to extract it from the total expenditure. This is a value of the next order of magnitude in comparison with the radiation balance. Nevertheless, it becomes possible when using results from Section 10.2.

Indeed, it was shown that the production of 1 kg C of biomass is accompanied by transpiration of 879 l of water. Since this biomass is equivalent to 4.2 X 107 J, the production of 1 J is accompanied by transpiration of 0.021 g H2O. The flux of evaporated water was estimated in the formula (2.3):

where Dw = 0.9 X 104 cm2/s. By substituting the corresponding values into the expression we get qw = 0.257 X 10-4 g H2O/cm2 s: Since this "pump" is working in the course of "shining time", the total amount of evaporated water is equal to (0.257 X 10-4 g H2O/cm2 s)(6 X 106 s) = 1.542 X 102 g H2O/cm2 that corresponds to 7.34 X 103 J/cm2. If we compare this value with the gross annual production of our beech forest in 1990, 7.064 kJ/cm2, we again obtain a very curious coincidence.

Finally, if we assume that all these coincidences are not casual but rather reflect some deep causal links then the equality "radiation balance < energy dissipation" can be used for estimation of the annual production of vegetation. Indeed, the formula for dissipation can be represented as

l2 Cw l2 H

where h is the relative humidity. Here we omit the terms describing the heat flows—they are small in comparison with the latent heat flow, which is determined by the transpiration and subsequent transportation of water vapour into the atmosphere. Since the annual gross production

where ST is the "shining time" in seconds; then, by combining Eqs. (4.8) and (4.9), and taking into account that Diss < R in W/cm2 we get

10.5. Vegetation as an active surface: the solar energy degradation and the entropy of solar energy

In the process of any detailed experimental study of the ecosystem several different measurements are carried out; in particular, the data about incoming and outgoing radiations are collected. Naturally, the following idea arises: to attempt to use such values as energy, entropy and exergy as macroscopic characteristics of the ecosystem and its most important component as vegetation. The problem is how to calculate them using the observed data. Several authors recently developed different approaches to the problem (Ulanowicz and Hannon, 1987; Aoki, 1987; Schneider and Kay, 1994; Svirezhev and Steinborn, 2001).

Vegetation is regarded as an active surface interacting with solar radiation and transforming it. Absorption, reflection and emission of radiation are consequences of this interaction, resulting in a new composition of spectrum of outgoing radiation. From a macroscopic point of view this is equivalent to the change of energy and information. It is natural to assume that the difference between these macroscopic variables calculated for incoming and outgoing radiation is defined not only by the properties of these radiation fluxes but also by the properties (state) of the active surface (vegetation), and, as a consequence, by the properties of an ecosystem. A natural idea comes to mind: to use the data of a continuous measurement of the components of radiation balance for the calculation of energy, entropy and exergy. A continuous measurement of the components of the radiation balance for different types of vegetation has become an almost standard procedure giving us information about the transformation of incoming solar radiation by vegetation. These values are used to calculate the dynamics of the radiation balance, i.e. dynamics of the integrated value, which deals with the energy conservation law. However, for a complete description of vegetation as a "thermodynamic machine", the principles responsible for the entropy concept should be implemented in addition to the First Law, namely the Second Law, the Prigogine theorem and the exergy concept (J0rgensen, 1992c).

If we look at the first two principles we see that they can be called "classic" ones, because they use the classic concept of entropy. But as soon as we try to consider the vegetation as some thermodynamic machine and to estimate its efficiency we immediately encounter the following problems: What is a working body of the machine? What is the temperature of the working body? For instance, Ulanowicz and Hannon are considering the Bose-Einstein gas as the working body. What is the temperature of this gas? You can answer that the temperature of incoming photons is equal to the temperature of the Sun (5700 K), but let us remember Gibbs has said that all material processes on our planet are taking place at its temperature. Thus, in all thermodynamic calculations we have to set the Earth's temperature. The first point of view is represented in the work of Essex (1984) and Peixoto and Oort (1992). The second point of view was formulated in the classic Gibbs works; nevertheless, the discussion is continuing.

Another approach differing from the classic one was suggested by Schneider and Kay (1994). According to them a complex structure develops spontaneously if a system is maintained far from thermodynamic equilibrium by a gradient of energy (like solar radiation). The more complex is the structure of a system, the more effectively it "dissipates the gradient". This idea—defined as the "non-equilibrium principle"—is a modern account of Schrodinger's (1944) "order from disorder" premise. Comparing ecosystems that receive the same amount of solar radiation and have the same soil properties and water balance, the most highly developed ecosystem would reradiate its energy on the lowest exergy level, i.e. the ecosystem would display the coldest black body temperature (Kutsch et al., 2001).

This can be explained with the radiation balance, R = Em - Eout, which is the net radiation flux, or in other words, the net radiation transformed into non-radiative energy at the surface (soil, water, vegetation, etc.). If the totals of incoming and outgoing radiation are represented as sums of long- and short-wave components, then Eln = E™ + E™ and Eout = E°ut + Esout where Eln, Efut, Ef1 and Esout are these components, respectively. Usually it is assumed that Ein < Ef and Eout < Efut + asEin where as (0 # as # 1) is the so-called short-wave albedo. Then R < (1 - as)Eln - E[out.

Schneider and Kay (1994) propose the ratio of radiation balance and net incoming solar (short-wave) radiation Rn/K* = 1 - (L*/K*) (here the notations used are: Rn = R = K* - L*, L* = E°ut and K* = ( 1 - as)Eln) as a functional measure of the system's ability to dissipate the incoming energy. The higher this ratio, and the lower the surface temperature of the system, the higher is its ability to dissipate the radiative gradient. As an argument they give the following results of observations for different surfaces. The surface temperature T is determined as T = (L*/es)1/4 where e is the emissivity, and s the Stefan-Boltzmann constant.

If, as Schneider and Kay suggested, we consider ecosystems as energy degraders, then of course all these data are very interesting: they show that the quarry degrades 62% of the net incoming radiation, K*, into energy in the form of molecular motion, Rn, while the 400-year-old forest degrades 91% (Table 10.1). If the more developed ecosystem degrades more energy, then both the ecosystem temperature and the ratio Rn/K are good indicators of ecosystem integrity. But there are "underwater rocks" in this concept: it is not clear what the term "degradation of energy" means in this case. Formally speaking, if the quarry degrades a lower quantity of energy then it is more organised. On the other hand, if we consider the surface as a heat machine then the expression Rn/K* may be interpreted as its efficiency. Then what is the meaning of the 91% efficiency of the mature fire forest? If, as Schneider and Kay suggested, the ratio Rn/K* points to the degree of exergy degradation,

Table 10.1

Radiative estimates from a thermal infrared multispectral scanner for different ecosystem types in the H.J. Andrews Experimental Forest, Oregon (USA)

Quarry

Clear-cut Douglas fir Natural forest plantation

400-year-old Douglas fir forest

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