Shear diffusion model

The growth rate of the patch size a2(t) determines the dilution of a chemical into the environment and thus the drop of the maximum concentration of the chemical. Therefore, the power law relating a2(t) and t is of great practical importance, for instance, to predict the behavior of a pollutant cloud in the environment. As it turns out, the specific power law which follows from Okubo s theory, Eq. 22-42, greatly exaggerates the effect of dilution compared to observations made in the field. The shear diffusion model which will be discussed next gives a more realistic picture. 

Figure 22.10 Fit of measured horizontal cloud size, o2, to the shear diffusion model by Carter and Okuto (1965), Eq. 22-44, of several tracer experiments in different basins of Lake Lucerne (Switzerland) and in Lake Neuchatel (Switzerland) (different symbols). From Peeters et al. (1996).

As pointed out by Peeters et al. (1996), based on their own experiments and on the reinterpretation of published field data, the adequate model to describe horizontal diffusion in lakes and oceans is the shear diffusion model by Carter and Okubo (1965). The model is described in Box 22.4. The most important consequence of this model is that the 4/3 law and the equivalent t3-power law for c2(t) expressed by Eq. 22-42 are replaced by an equation which corresponds to a continuous increase of the exponent m from 1 to 2 (Box 22.4, Eq.l) ... 

The meaning and typical sizes of the coefficients Ax and A2 are discussed in Box 22.4. From Eq. 22-44 we note that for small times t, c2(t) grows as f, whereas for large times it grows as i2. The critical time, crit, defined in Eq. 3 of Box 22.4 separates the two regimes. Figure 22.10 shows a2(() curves from different experiments conducted in Swiss lakes. In Illustrative Example 22.3 the shear diffusion model is applied to the case of an accident in which a pollutant is added to the thermocline of a lake. 

As shown by Peeters et al. (1996), horizontal diffusion experiments in lakes and oceans can be best described with the shear-diffusion model of Carter and Okubo (1965). The model yields the following relation between cloud size, ct2, and time, t ... 

Equation (1) contains two parameters and A2. In the framework of the shear diffusion model they can be identified with real physical quantities ... 

Typical sizes of Ah A2, and cnt are given in the table below. The values are taken from Peeters et al. (1996) these authors successfully reanalyzed several published tracer experiments in terms of the shear-diffusion model. An example is shown in Fig. 22.10. 

It is this flux which dilutes concentration patches moving along the streamlines. At this point, the connection to the shear diffusion model of horizontal mixing becomes clearer. An example of how dispersion is causing the dilution of a pollutant patch with its environment has been discussed in Illustrative Example 22.3 others will follow in Chapters 24 and 25. 

Advanced Topic P22.5 Shear Diffusion Model and Apparent Diffusivity... 

Protein Adsorption and Desorption Rates and Kinetics. The TIRF flow cell was designed to investigate protein adsorption under well-defined hydrodynamic conditions. Therefore, the adsorption process in this apparatus can be described by a mathematical convection-diffusion model (17). The rate of protein adsorption is determined by both transport of protein to the surface and intrinsic kinetics of adsorption at the surface. In general, where transport and kinetics are comparable, the model must be solved numerically to yield protein adsorption kinetics. The solution can be simplified in two limiting cases 1) In the kinetic limit, the initial rate of protein adsorption is equal to the intrinsic kinetic adsorption rate. 2) In the transport limit, the initial protein adsorption rate, as predicted by Ldveque s analysis (23), is proportional to the wall shear rate raised to the 1/3 power. In the transport-limited adsorption case, intrinsic protein adsorption kinetics are unobservable. 

Debye s original rokational diffusion model corresponds to ki = 2Dy where is the rotational diffusion coefficient which for nis model of the molecule as a sphere of volume V reorienting in a medium of shear viscosity Q ia s k T/6qV giving the Debye relaxation time... 

Molecular dynamics calculations are more time-consuming than Monte Carlo calculations. This is because energy derivatives must be computed and used to solve the equations of motion. Molecular dynamics simulations are capable of yielding all the same properties as are obtained from Monte Carlo calculations. The advantage of molecular dynamics is that it is capable of modeling time-dependent properties, which can not be computed with Monte Carlo simulations. This is how diffusion coefficients must be computed. It is also possible to use shearing boundaries in order to obtain a viscosity. Molec-... 

Trajectory models require spatiaUy and temporaUy resolved wind fields, mixing-height fields, deposition parameters, and data on the spatial distribution of emissions. Lagrangian trajectory models assume that vertical wind shear and horizontal diffusion are negligible. Other limitations of trajectory and Eulerian models have been discussed (30). 

Approaches used to model ozone formation include box, gradient transfer, and trajectoty methods. Another method, the particle-in-cell method, advects centers of mass (that have a specific mass assigned) with an effective velocity that includes both transport and dispersion over each time step. Chemistry is calculated using the total mass within each grid cell at the end of each time step. This method has the advantage of avoiding both the numerical diffusion of some gradient transfer methods and the distortion due to wind shear of some trajectory methods. 

For dynamical studies of diffusion, conformational and transport behavior under shear stress, or kinetics of relaxation, one resorts to dynamic models [54,58,65] in which the topological connectivity of the chains is maintained during the simulation. 

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