Vertical turbulence

Chen, C, J., and W. Rodi. Vertical Turbulent Buoyant Jets . 4 Review of Experimental Data. Pergamon Press, New York. 

Mizuchina, T. 1982. An experimental study of vertical turbulent jet with negative buoyancy. Vi drrne- u. Stoffuhertragung, vol. 16, no. 1. 

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

Figure 18.11 Diffusion distance, L, vs. diffusion time, t, for typical diffusivities calculated from the Einstein-Smoluchowski relation L = (2Dt)m, Eq. 18-8. The following diffusivities, D, are used (values in cm2s ) He in solid KC1 at 25°C KT10 molecular in water 1 O 5 molecular in air KT1 vertical (turbulent) in ocean 10° vertical (turbulent) in atmosphere 105 horizontal (turbulent) in ocean 106 to 108. Values adapted from Lerman (1979).

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.

Figure 22.7 Vertical turbulent diffu-sivity Ez versus square of stability frequency V2 in two Swiss lakes (see Eq. 22-32). (a) For Umersee (maximum depth 196 m), a basin of Lake Lucerne, the data refer to 10-100 m depth and indicate shear-produced turbulence. (b) For Zugersee (maximum depth 198 m) the values are calculated for an extreme storm of about two days duration. The data refer to the depth interval between 10 and 70 m they show a mixture between turbulence production by local shear and large-scale motion. (Fromlmboden and Wuest., 1995.)...

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... 

The increase in vertical turbulence intensity caused by cooling tower plumes can be estimated for each temperature gradient and increment of distance from the tower. This can be represented by well-known turbulence parameters developed for Gaussian plume models ... 

In [48], it was shown that, in order to simulate the cold intermediate layer of the Black Sea (see [3]), one should take into account the dependence of the vertical turbulent mixing coefficient on the density stratification of the waters. In this case, the optimal coefficients in the well-known formula by Munk-Anderson for the Black Sea occurred to be an order of magnitude lower than those for the World Ocean. 

In addition to the results presented above, we should also note the studies of the climatic BSGC [56] based on the basic Russian prognostic model [57]. The distinctive features of [56] were related to the dependence of the coefficients of horizontal turbulence on lateral velocity shears and to the specifying of the monthly climatic temperature and salinity field at the surface [29] instead of the heat and moisture fluxes. Despite the relatively coarse horizontal calculation grid (about 22 km), this allowed the authors to reproduce [56] a relatively distinct MRC jet and the known NSAEs off the Turkish and Caucasian coasts and off the Danube River mouth. The results of the tuning in [56] of the Munk-Anderson s formula for the coefficient of the vertical turbulent exchange from the point of view of reproduction of the actual CIL were used in [53,54]. 

The T,S structure of the Black Sea waters presented in Fig. 3 is caused by the weak turbulent diffusion below the UML, which is characterized by a diffusivity of about 10 5 m2 s 1 [6,7,11-13], which is one to two orders of magnitude lower than the values usual in the open ocean. The reason for the weak vertical turbulent exchange in the Black Sea is the great differences in the densities of the primary water masses (the freshwater mass and that of the Sea of Marmara). 

The principal features of this structure are related to the very weak vertical turbulent exchange of the T,S properties between, on the one hand, the freshened surface and the cold intermediate water masses and, on the other hand, the significantly more saline deep water mass. 

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