Vertical turbulent diffusivity

Fig. 7. Vertical turbulent diffusivity profile corresponding to Eq. (9.11). From Lamb and Duran (1977).

Vertical temperature profiles in Greifensee, a lake near Zurich (Switzerland) with a surface area of 8.6 km2 and a maximum depth of 32 m, show a distinct thermocline during summer and autumn (see figure). Imboden and Emerson (1978) determined the coefficient of vertical turbulent diffusion, Ez, to lie between 0.01 and 0.04 cm2s 1 during this time of the year. 

You have worked hard to study the internal dynamics of tetrachloroethene (PCE) and to calculate vertical turbulent diffusion coefficients in lakes. A friend of yours is more interested in the process of air-water exchange. One day, she sees some of your PCE data lying on your desk. She is very happy with the table below and... 

Turbulent Exchange Model Reynolds Splitting Model Vertical Turbulent Diffusion... 

Illustrative Example 22.2 Vertical Turbulent Diffusion Coefficient in a Lake... 

Nci exchange. The top 2 m of the lake are well mixed. Vertical turbulent diffusivity... 

Fig. 22.6 Comparison of vertical temperature profiles measured at consecutive times t, and t,+l can be used to determine the vertical turbulent diffusivity Ez. From Im-boden et al. (1979).

Owing to vertical (turbulent) diffusion, heat is transported from regions of warm water to adjacent colder layers. Mathematically this appears as a heat flux against the vertical temperature gradient (remember Fick s first law, Eq. 18-6). Thus, at a later time, ti+u we expect to find warmer water between z and zB. The change of the heat content with time A is ... 

Figure 2 The correlation between stability frequency, N, and vertical turbulent diffusivity, Ez, according to Eq. 22-32 yields q = 0.5.

Radioactive or stable isotopes of noble gases are also used to determine vertical turbulent diffusion in natural water bodies. For instance, the decay of tritium (3H)— either produced by cosmic rays in the atmosphere or introduced into the hydrosphere by anthropogenic sources—causes the natural stable isotope ratio of helium, 3He/ 4He, to increase. Only if water contacts the atmosphere can the helium ratio be set back to its atmospheric equilibrium value. Thus the combined measurement of the 3H-concentration and the 3He/4He ratio yields information on the so-called water age, that is, the time since the analyzed water was last exposed to the atmosphere (Aeschbach-Hertig et al., 1996). The vertical distribution of water age in lakes and oceans allows us to quantify vertical mixing. 

Another procedure is based on the measurement of the radioactive isotope radon-222 (half-life 3.8 days), the decay product of natural radium-226. At the bottom of lakes and oceans, radon diffuses from the sediment to the overlying water where it is transported upward by turbulence. Broecker (1965) was among the first to use the vertical profile of 222Rn in the deep sea to determine vertical turbulent diffusivity in the ocean. 

Figure 22.8 Vertical profile of dissolved excess radon-222 activity (i.e., the radon-222 activity exceeding the activity of its parent nucleus radium-226) in the bottom waters of Greifensee (Switzerland) serves to compute vertical turbulent diffusivity Ez. Activity units are decay per minute per liter (dpm L ). Data from Imboden and Emerson (1978).

Which assumptions are necessary for determining vertical turbulent diffusion coefficients from repeated vertical temperature measurements made at a single location in the middle of a lake ... 

Explain the relationship between vertical turbulent diffusivity in surface waters and vertical stratification of the water column. 

P 22.3 Determine Vertical Turbulent Diffusivity in a Lake from Measurements of Tetrachloroethene (PCE)... 

In a lake (maximum depth 20 m) two vertical profiles of tetrachloroethene were measured at a time interval of one month (see table below). Calculate the vertical turbulent diffusivity, E at 8, 12, and 16 m depth. For simplicity assume that the cross-section of the lake, A(z), is independent of depth z. (Note that the same data were used in Problem 20.5 to calculate the air-water exchange rate of PCE.)... 

In Illustrative Example 19.4, the dissolution of a non-aqueous-phase liquid (NAPL) into groundwater was discussed. Here we consider a similar (although somewhat hypothetical) case. Assume that a mixture of chlorinated solvents totally covers the flat bottom of a small pond (maximum depth zmax = 4 m, surface area Asurface = 104 m2) forming a dense non-aqueous-phase liquid (DNAPL). The DNAPL is contaminated by benzene which dissolves into the water column and is vertically transported by turbulent diffusion. The pond is horizontally well mixed. The vertical turbulent diffusion coefficient is , = 0.1 cm2s l and approximately constant over the whole water column. 

P 22.6 Radon Profiles and Vertical Turbulent Diffusivity at the Bottom of the Ocean... 

First, recall that the nondimensional Damkohler number, Da (Eq. 22-11 b), allows us to decide whether advection is relevant relative to the influence of diffusion and reaction. As summarized in Fig. 22.3, if Da 1, advection can be neglected (in vertical models this is often the case). Second, if advection is not relevant, we can decide whether mixing by diffusion is fast enough to eliminate all spatial concentration differences that may result from various reaction processes in the system (see the case of photolysis of phenanthrene in a lake sketched in Fig. 21.2). To this end, the relevant expression is L (kr / Ez)1 2, where L is the vertical extension of the system, Ez the vertical turbulent diffusivity, and A, the first-order reaction rate constant (Eq. 22-13). If this number is much smaller than 1, that is, if... 

D = vertical turbulent diffusion coefficient R = production rate... 

Estimates of the vertical turbulent diffusivity in the Baltic Sea have been performed mainly by three different methods. 

The vertical turbulent diffusivity is parameterized in numerical models by... 

Imboden and Schwarzenbach (1985) have illustrated how the mass-balance equation is a means of accounting for chemical and biological reactions that produce or consume a chemical within a test volume, and for transport processes dial import or export the chemical across the boundaries. Each process acting on a chemical can be characterized by an environmental first-order rate constant, expressed in units of time-1. Transport mechanisms include water renewal by nvers, horizontal and vertical turbulent diffusion, advection by lake particles, and settling of particles (Imboden and Schwarzenbach, 1985). Chemical reaction i ales and reaction half-lives for a wide variety of reactions have been summarized by I loffmann (1981), Pankow and Morgan(1981), Morgan and Stone(1985),and Santsehi (1988). 

Schematic of a vertical turbulent diffusion flame in crossflow.

Quantification of vertical exchange rates and vertical turbulent diffusivities... 

The two box models discussed above can be extended to a multibox model. In the limit of infinitely small boxes the multibox model corresponds to the continuous model of Jassby and Powell (1975). Using Equation (40), the vertical turbulent diffusivity as function of depth can be obtained from x if t is at steady state and can be treated as ideal tracer with source strength of lyr/yr ... 

Peelers F, Kipfer R, Hofer M, Imboden DM, Domysheva VM (2000b) Vertical turbulent diffusion and upwelling in Lake Baikal estimated by inverse modeling of transient tracers. J Geophys Res 105 14283 and 3451-3464... 

FIGURE 19.6 Relationship between characteristic times of vertical turbulent diffusion tj) and a number of atmospheric reactions tc) for a layer of thickness Az = 10 m and different thermal stability classes (Kramm et al., 1993). For example, the HNO3—NH3—NH4NO3 equilibrium (reaction 2) has a reaction timescale comparable to that of turbulent diffusion under unstable and neutral conditions. 

Matyukhin, V. J., and 0. N. Prokofyev. 1966. Experimental determination of the coefficient of vertical turbulent diffusion in water for settling particles. Soviet Hydrol. (Am. Geophys.Union), No 3. 

Vertical turbulent diffusion rates are usually sufficiently fast that vertical homogeneity can be assumed. A typical vertical dispersion coefficient is 100 cm /s or 0.01 m /s thus a characteristic time for vertical mixing in a 1 m deep river is about 100 s or a few minutes. If the water residence time in the reach is 2000 s (e.g., a 1000 m reach with a velocity of 0.5 m/s) vertical mixing is essentially complete and vertical homogeneity can be assumed. 

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