where Dw2) is the integral diffusion coefficient, and l2 a characteristic length of the path at the second stage. At this stage diffusion can almost be considered as a turbulent one, and its coefficient depends mainly on the intensity of turbulent exchange. The mean height of vegetation layer, dv, can be considered as the characteristic length l2, so that l2 = dv.
Since the fluxes at the first and second stages flow through different areas (LAs at the first, As at the second stage) and the continuity condition has to be fulfilled, q^LAs = q^As, or qff = Lq^ = qw (L is the leaf area index, for instance, L < 4-5 for oak forest). The latter is a flux of water vapour through the boundary of vegetation layer with single area (As = 1). At the first stage of our study we assume that the exchange of heat, water vapour and gases is performed only through the boundary between the vegetation layer and the atmosphere; the exchange between vegetation and soil is neglected (although it exists). The continuity equation allows us to exclude the concentration C from Eqs. (2.1) and (2.2). As a result we get
where the generalised coefficient of diffusion is equal to
If we consider two extreme cases, then formula (2.4) is represented in two different forms:
The first case is when a1 « 1, i.e. A » Dw^/D^. Then Dw/l2 < D11)/l1. Therefore, as follows from Eq. (2.3)
and if humidity at the external side of the leaf is close to the air humidity, then qw < q*1). This means that the total transpiration is mainly determined by the diffusive transport of water vapour within leaves, i.e. the leaf is a "bottleneck" for transpiration. Since the internal diffusion is determined in general by molecular mechanisms, we can set that D^ is the coefficient of the molecular diffusion of water vapour in air, D^ < 0.25 cm2/s. The effective area of diffusion depends on the total area of open stomata, which in turn is a function of stomata control. The latter can significantly modify the value of the transpiration rate.
Dw2) is the coefficient of turbulent diffusion in the surface layer, usually estimated (by the order of magnitude) as D® < 104 cm2/s. If there is no wind or the vegetation layer is weakly ventilated, this value is less than 104 cm2/s by 1-2 orders of magnitude.
We see that the coefficient of turbulent diffusion is higher by four orders of magnitude than the coefficient of molecular diffusion. Since at standard conditions
D^/D™ < 0.25 X 10—4, the necessary condition of the first extreme case is the realisation of the inequality l1 >> 0.25 X 10~4Ll2 < 10~4l2. In other words, the vegetation layer has to be thin and consists of plants, which have thick pulpy leaves with small number of stomata. For instance, if l2 < 100 cm then the thickness of a leaf l1 >> 10~2 cm.
The necessary condition of the second extreme case is a2 << 1, i.e. A << D^/D®. In this case Dw < D®, i.e. the turbulent diffusion plays the main role in the process of the subsequent transportation of water from the external side of the leaf into the surrounding air. The case could be realised if l1 << 10~4l2, i.e. vegetation consists of high trees with thin leaves, which have a lot of stomata. Note that the mean thickness of an oak leaf is about 2 X 10"2 cm, so that if l2 < 10 m = 103 cm then l1/l2 ~ 10~5.
The heat transport from the interior of leaves into the atmosphere is realised by means of molecular diffusion at the first stage with coefficient Dh1) called the thermal diffusivity of air, and the turbulent diffusion in the second stage. Since Dh1) < 0.23 cm /s at 25°C, it is natural to assume that Dh1) < D*^, and also Dh2) < D®. In this case the diffusion coefficient Dh is the same as before; the characteristic lengths are also the same. Then the turbulent heat flux is equal to
where cp = 1 J/g K is the air heat capacity, Ti and T are the temperatures of the leaf interior and the atmosphere air, respectively. The flux of water vapour connected with evapotranspiration is often called the "latent heat flux". In this case the value qw is multiplied by the value of the specific enthalpy of water vapour, so that the latent heat flux is expressed in J/cm2 s. The turbulent heat flux is also called the "sensible heat flux".
The entire transport path of carbon dioxide from the atmosphere to the vegetation layer, then inside the leaf until it reaches chlorophyll cells, can also be divided by two stages, as we were already doing above with water vapour. By analogy, the flux of CO2 from the atmosphere into the leaf is equal to qc = pD(C - Cc), (2.7)
where Cc and Cf are the concentrations of CO2 (measured in grams of CO2 per gram of dry air) in the atmosphere outside the vegetation and within the leaf at the level of parenchyma. The generalised coefficient of diffusion is
where dÇ' is the effective coefficient of CO2 diffusion through stomata and intercellular space, when the gas moves from the external side of a leaf to the parenchyma, Dc2' is the coefficient of turbulent diffusion, and A = l1/Ll2. If with respect to D® we can set DP = D*2', then for Dc1' we again assume that the transport of CO2 is determined by its molecular diffusion in air along this path. Therefore, Dc1' < 0.61DW < 0.15 cm2/s.
The second gaseous flux is a flux of oxygen from the interior of the leaf to the exterior of the vegetation layer. By analogy with the case of carbon dioxide we have
where Cox and Cfx are the concentrations of O2 (measured in grams of O2 per gram of dry air) in the atmosphere outside the vegetation and within the leaf at the level of parenchyma. The generalised coefficient of diffusion
Dox + ADox where Do^1 is the effective coefficient of oxygen diffusion through stomata and intercellular space when the gas moves from the external side of a leaf to the parenchyma, Do2 is the coefficient of turbulent diffusion, and A = l1/Ll2. Naturally then Dox) = DW). Since the coefficient of molecular diffusion of oxygen in air D\1 < 0.8DW1) < 0.2 cm2/s,
It would be interesting to estimate the relations between these fluxes for vegetation in reality. These estimations help us to answer many questions. For instance, how many grams of water have to evaporate for a plant to assimilate 1 g of carbon from the atmosphere? Or, another question is: how many joules of heat are released directly in the process of assimilation of the atmosphere carbon?
Let us estimate the ratio qw/qc, which is expressed as grams of water per gram of carbon dioxide. Using Eqs. (2.3), (2.4), (2.7) and (2.8) we get qw _ Dw Cw — C ,2 12
The ratio of diffusion coefficient can be represented as
Dc dc1)dc2)(dw1) + adw2)) 1 + Ak ' l213j where k = duP/d!^ < 4 X 10—4. Interestingly, even this ratio is contained within the narrow enough interval of 1 -1.64, whereas the value of k and A can vary within rather broad limits. If assuming l1 ~ 10—2 cm, l2 ~ 1m and L = 4, then Ak ~ 1 and Dw/Dc ~ 1 :32: Now this estimation is quite satisfactory for the given ratio.
It is known (Budyko, 1977) that on average the difference Cc — Cf constitutes about 10% of the CO2 concentration in the atmosphere, so that Cc — Cf < 0.1Cc, i.e. plants are normally using only a small part of the potential carbon dioxide flux. If we now assume that at temperate latitudes in summer the mean daily temperature is 20°C and the relative humidity is 50%, then C7 - Cw < 1.8 X 10~2 - 0.6 X 10~2 = 1.2 X 10^2 g H2O/g dry air. By taking into account
Cc = 0.66 X 10-3 g CO2/g air (it corresponds to 330 ppm) substituting all these values into Eq. (2.12) we finally get qw/qc < 240 g H2O/g CO2.
It follows from this that a plant has to evaporate 240 l of water in order to assimilate 1 kg of carbon dioxide, or in order to assimilate 1 kg of carbon it is necessary to evaporate 879 l of water.
In order to answer the second question we have to calculate the rate qh/qc. Using formulas (2.6)-(2.8) we get
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