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Data from Luvall and Holbo (1989).

then the interpretation of this value is not quite understandable. By introducing the different information content of incoming and outgoing radiation into our calculation we shall suggest below another concept of exergy which seems more appropriate. Moreover, we can investigate not only the amount of absorbed energy but also its quality, as done in Section 5.6.

Ulanowicz and Hannon (1987) agree with this point of view assuming that "living systems generate more entropy because they are more effective in utilising energy than are the ambient physical systems". Within the framework of classic thermodynamics they suggest a quantitative measure for the entropy calculation of different wavelength intervals. Similar efforts have been made by Press (1976) and Aoki (1987, 1995, 1998), who stress the principal difference between the entropy of scattered and direct solar radiation.

10.6. Vegetation as an active surface: exergy of solar radiation

In this section we apply the method which was described in Section 5.6 and allows us to calculate the exergy of solar radiation interacting with such an active surface as the vegetation (see also Svirezhev and Steinborn, 2001; Svirezhev et al., 2003). But before this, we shall say a few words about standard integral characteristics, describing the interaction widely used in actinometric measurements. Naturally we might be partially repetitive.

Incoming solar radiation is usually described by its energy spectrum, or the density distribution of energy, Eln(v; x, y, t) where v [ [v^, vmax] is the spectral frequency of radiation, the interval V : [ vmin, vmax] includes all possible values of spectral frequencies, t is time, and (x, y) are the coordinates of some point on the globe. At this point the incoming radiation interacts with a surface, and the result of this interaction is the transformation F of incoming radiation into the outgoing radiation Eout( v; x, y), so that at the given point and

■ F(x,y,t) t time Eln(v) ) E (v). It is obvious that the properties of the transformation F are connected with the characteristics of the reflecting and transforming surface, which could be an ocean surface, vegetation cover, a surface of ice, desert, etc. Since the inflow Eln(v) and the outflow Eout(v) are known and the transformation F is unknown, the surface transforming incoming solar radiation can be considered as a "black box". By analysing different relations between incoming and outgoing radiation we get some information about the properties of the "black box". Note that the outflow Eout(v) contains two components: the reflected radiation and radiation which is passed over the surface, so that Eout(v) = E™'(v) + E°ut(v). In reality, the second component is not registered and, as some estimations show, its value is significantly lower than the first one. That is why we shall assume that Eout(v) < Erout(v).

The simplest form of transformation operator is a shift operator R(v):

Here and later on we shall omit the notations x, y, and t by assuming that all the operations are performed at a given point (x, y) and time t. The convolution R = J v R(v)dv plays a very important role in meteorology, and is called the radiation (or energy) balance (see above). Continuous measurement of the radiation balance for the Earth's surface and its components, such as incoming and outgoing short- and long-wave solar radiation, constitute a standard meteorological procedure. Since short-term variations of these values can be considered to be merely a noise, it is possible to apply averaging for monthly and annual intervals. In this way, maps of the annual and monthly radiation balance were obtained (see, for instance, Budyko, 1963).

The second simple form of transformation operator is a contracting operator a(v)

where a(v) is the so-called spectral albedo. Its convolution a = (1/V)J"n [Eout(v)/ Em(v)]dvis the mean (or integral) albedo. This value is also a very important parameter in meteorology. Note that the value A as defined above is mathematically correct, but unfortunately, in meteorology a different definition of albedo is used (we shall call it the standard albedo): A = Eout/Ein, where Ein = J"VEin(v)dvand Eout = JVEout(v)dvare the total energies of incoming and outgoing radiation, respectively. This definition is also certainly correct, but the point is that it is yet another definition of the mean. In fact, by applying to the convolution of Eq. (6.2) the mean value theorem we get ¡VEout(v)dv = ¡v a(v)Em(v)dv = a $v Em(v)dv, where a is a "mean" value a(ff) for U [ V. It is often assumed that the interval V contains only short-wave radiation (see Section 10.5). Thus, these values are usually shown as the albedo of different surfaces. In the standard BATS scheme (Matthews, 1983, 1984) the interval V is divided into two sub-intervals, Vs and Vl (Vs U Vl = V, Vs > Vl = 0) for short- and long-wave radiation, respectively. The border of division is 0.7 ^m. As a result, the two standard albedo, as = Eout/E™ and al = Eout/Ein, are derived, where the energy spectra are integrated with respect to corresponding intervals. In standard actinometric measurements the entire spectral interval is divided into the following sub-intervals: Vs: 0.3-2.8 Mm and Vl: 2.8-100 Mm. It is natural that with the higher number of spectral intervals the accuracy of our calculation will increase.

Both types of above-considered operators are linear, and other linear operators do not exist. However, we think that the real transformation operator is more complex and nonlinear. We suggest using the exergy of solar radiation as such an operator (Section 5.6), which is a Kullback measure of the information increment. We recall the basic definition of the exergy of solar radiation in the context of measurements of the components of the radiation balance.

The spectral exergy (or the exergy flow, but as a rule we shall omit the word "flow") is equal to

Eout v

Ex(v) = Eout(v)ln -¿V + R(v) = Ein(v)a(v)ln[a(v)] + R(v). (6.3)

The total exergy, Ex, is a convolution of the spectral exergy: Ex = Ex(v)dv. If the standard Kullback measure of the increment of information is equal to K = JVpout(v)ln[pout(v)/pin(v)]dv, where pin(v) = Ein(v)/Ein and pout(v) = Eout(v)/Eout is used, then the expression for the total exergy is rewritten as (compare with Eq. (6.3)

Ex = —Out(v)ln —^ d v + R = —OutK + —Out ln(—Out/—in) + R J v —m(v)

= -»[1 + a(K + In a - 1)] = R 1 + a(K + ln * ~ 1) . (6.4)

The value hEx = Ex/—ln = 1 + a(K + ln a — 1) can be cOnsidered as a specific increment Of informatiOn, ex, per unit Of incOming energy. It is interesting that the tOtal exergy is expressed by a nOn-linear cOmbinatiOn Of such typical meteOrOlOgical parameters as the standard albedO a = —Out/—ln and the radiatiOn balance R, and alsO the new variable K (Kullback's information measure).

Since Only twO spectral intervals are used, the value Of K in Eq. (6.4) is

AlsO, in view Of the fact that instead Of the integral with respect tO the full spectrum we have Only the integral data fOr twO spectral intervals, the fOrmula fOr exergy calculatiOn will be apprOximate. If One lOOks carefully at the fOrmula, it can be seen that the errOr appears Only when we replace the value Of K, Or mOre cOrrectly the factOr:

It is easy to show that the factor F is represented as F = Jn Eln(v)a(v)lna(v)dv — Ema in a. Using the so-called Jensen inequality (Beckenbach and Bellman, 1961) we can write:

Em(v)a(v)ln a(v)dv $ E™as ln as and Ein(v)a(v)ln a(v)dv $ ^lnal ln al.

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