Eddy diffusion coefficients vertical

We are using an equivalent vertical-mean eddy diffusion coefficient because we are computing a vertical mean concentration. 

If the lake is stratified, vertical transport is commonly the time-limiting step for complete mixing. This was the reason for applying the two-box model to the case of PCE in Greifensee (Illustrative Example 21.5). Now we go one step further. We consider a vertical water column of mean depth h with a constant vertical eddy diffusion coefficient Ez. The flux Fa/VJ of PCE escaping to the atmosphere is given by Eq. 20-la ... 

Mechanisms and rates of transport of nuclear test debris in the upper and lower atmosphere are considered. For the lower thermosphere vertical eddy diffusion coefficients of 3-6 X 106 cm.2 sec. 1 are estimated from twilight lithium enhancement observations. Radiochemical evidence for samples from 23 to 37 km. altitude at 31° N indicate pole-ward mean motion in this layer. Large increases in stratospheric debris in the southern hemisphere in 1963 and 1964 are attributed to debris from Soviet tests, transported via the mesosphere and the Antarctic stratosphere. Most of the carbon-14 remains behind in the Arctic stratosphere. 210Bi/ 210Pb ratios indicate aerosol residence times of only a few days at tropospheric levels and only several weeks in the lower stratosphere. Implications for the inventory and distribution of radioactive fallout are discussed. 

Above 120 km. the atmosphere is essentially in diffusive equilibrium (34). For the altitude interval between 80 and 120 km. Colegrove et al. (6) estimated an average vertical eddy diffusion coefficient of 4 X 106 cm.2 sec."1 by a careful evaluation of the 0/02 concentration ratio profile over this altitude interval. Using their eddy diffusion coefficient and following the procedure of Kellogg (22) it is estimated that molecular... 

The pattern of eddy mixing in the mesosphere apparently may be highly nonisotropic, similar to that indicated for the lower thermosphere. The upper limit of vertical eddy diffusion coefficients for the mesosphere... 

The second stage realizes a two-step procedure that re-calculates the ozone concentration over the whole space S = (tp, A, z) (, A)e l 0atmospheric boundary layer (zH 70 km), whose consideration is important in estimating the state of the regional ozonosphere. These two steps correspond to the vertical and horizontal constituents of atmospheric motion. This division is made for convenience, so that the user of the expert system can choose a synoptic scenario. According to the available estimates (Karol, 2000 Kraabol et al., 2000 Meijer and Velthoven, 1997), the processes involved in vertical mixing prevail in the dynamics of ozone concentration. It is here that, due to uncertain estimates of Dz, there are serious errors in model calculations. Therefore the units CCAB, MFDO, and MPTO (see Table 4.9) provide the user with the principal possibility to choose various approximations of the vertical profile of the eddy diffusion coefficient (Dz). 

A simple, quantitative, steady-state diffusion model (36) demonstrates the importance of physical processes in shaping the vertical distribution of phytoplankton. This model uses values of the eddy diffusion coefficient K from the theoretical model of James (35), which reproduces accurately the annual cycle of vertical temperature structure for this area of the Celtic Sea. The submodels for photosynthetic production, light, and grazing can be varied to any of the established models nutrient luxury or nutrient limitation of growth can be included. The model reproduces the main features of the UOR observations in the Celtic Sea and English Channel. 

Clark (55) reviewed the laboratory data and concluded that there was no homogeneous gas phase process capable of accounting for the CO and O2 mixing ratio limits at any altitude. More recently, McElroy and McConnell (93) have concluded that vertical transport with high eddy diffusion coefficients may explain the mixing ratios in the Mars upper atmosphere. The same mechanism cannot, however, account for the lower atmosphere. 

Many of the questions about the origin of the suboxic zone and the redox reaction zones would be easier to answer if we could calculate vertical fluxes. Unfortunately, neither the mechanism nor the rate of vertical transport are well understood. Estimates of the vertical advection velocity (w) and eddy diffusion coefficients (K.) are available in the literature (e.g., 5, 32, 39, 40), but they are probably not realistic, considering the importance of horizontal ventilation discussed earlier. 

The distribution of a number of dissolved species (02, C-14, Ra-226, salinity) in the Central Pacific water column, at depths between 1 and 4 km, has been shown (11) to be consistent with a steady-state model of the water column in which the concentration-depth profiles are stationary and the concentrations at the boundaries 1 and 4 km are stipulated at their present values. The physical model of the water column is based on two transport mechanisms vertical eddy diffusion (eddy diffusion coefficient K — 1.3 cm2 sec"1) and upwelling of deep water (advection velocity U = 1.4 X 10 5 cm sec"1, or approximately 1 cm per day) (11). 

In these expressions, Gq and G represent additional dissipative terms to be added to the thermodynamic equation (3.69). Kh represents the vertical eddy diffusion coefficient for heat, which is related to the vertical eddy diffusion coefficient for momentum and tracers (Eq. 3.78) by the Prandtl number Pr ... 

As indicated, the flux may be expressed either in units of molecules/m2 s or in units of kg/m2 s. Here, p and n are the density and number density of air, respectively, and K is called the eddy diffusion coefficient. This quantity must be treated as a tensor because atmospheric diffusion is highly anisotropic due to gravitational constraints on the vertical motion and large-scale variations in the turbulence field. Eddy diffusivity is a property of the flowing medium and not specific to the tracer. Contrary to molecular diffusion, the gradient is applied to the mixing ratio and not to number density, and the eddy diffusion coefficient is independent of the type of trace substance considered. In fact, aerosol particles and trace gases are expected to disperse with similar velocities. 

Table 1-7 summarizes vertical eddy diffusion coefficients for the troposphere and the lower stratosphere as derived from various observations, mainly of tracers. For the troposphere the Kz data of Davidson et al. (1966) are the least trustworthy, because they are based on tracers originating in the stratosphere. The best values probably are those of Bolin and Bischof (1970), derived from the seasonal variation of C02. Above the tropopause... 

Table 1-7. Values for the Eddy Diffusion Coefficient Kz (Vertical Transport) Derived Mainly from Tracer Observations...

Fig. 1-10. One-dimensional vertical eddy diffusion coefficient Kz derived from trace gas observations in the stratosphere (1) from nitrous oxide, (2,3) from methane. (1) Schmeltekopf el al. (1977), (2) Wofsy and McElroy (1973), (3) Hunten (1975).

Fig. 5.4 Oxygen microgradient (data points) at the sediment-water interface compared to the ratio, E/D (logarithmic scale), between the vertical eddy diffusion coefficient, E, and the molecular diffusion coefficient, D. Oxygen concentration was constant in the overflowing seawater. It decreased linearly within the diffusive boundary layer (DEL), and penetrated only 0.7 mm into the sediment. The DEL had a diickness of 0.45 mm. Its effective thickness, 8 is defined by the intersection between the linear DEL gradient and die constant bulk water concentration. The diffusive boundary layer occurs where E becomes smaller than D, i.e. where E/D = 1 (arrow). Data from Aarhus Eay, Denmark, at 15 m water depth during fall 1990 (Gundersen et al. 1995).

It turns out that turbulent diffusion can be described with Fick s laws of diffusion that were introduced in the previous section, except that the molecular diffusion coefficient is to be replaced by an eddy or turbulent diffusivity E. In contrast to molecular diffusivities, eddy dififusivities are dependent only on the phase motion and are thus identical for the transport of different chemicals and even for the transport of heat. What part of the movement of a turbulent fluid is considered to contribute to mean advective motion and what is random fluctuation (and therefore interpreted as turbulent diffusion) depends on the spatial and temporal scale of the system under investigation. This implies that eddy diffusion coefficients are scale dependent, increasing with system size. Eddy diffusivities in the ocean and atmosphere are typically anisotropic, having much large values in the horizontal than in the vertical dimension. One reason is that the horizontal extension of these spheres is much larger than their vertical extension, which is limited to approximately 10 km. The density stratification of large water bodies further limits turbulence in the vertical dimension, as does a temperature inversion in the atmosphere. Eddy diffusivities in water bodies and the atmosphere can be empirically determined with the help of tracer compounds. These are naturally occurring or deliberately released compounds with well-estabhshed sources and sinks. Their concentrations are easily measured so that their dispersion can be observed readily. 

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