Vertical gradient model

Reaction rates of nonconservative chemicals in marine sediments can be estimated from porewater concentration profiles using a mathematical model similar to the onedimensional advection-diffusion model for the water column presented in Section 4.3.4. As with the water column, horizontal concentration gradients are assumed to be negligible as compared to the vertical gradients. In contrast to the water column, solute transport in the pore waters is controlled by molecular diffusion and advection, with the effects of turbulent mixing being negligible. 

Density layer oq = 16.10-16.15 kgm 3. This layer constitutes the lower part of the redox zone. The onset of hydrogen sulfide occurs just below the depths of maximum particulate manganese and iron. The reduction of Mn(III) and Mn(IV) by sulfide is very intensive [63,75] and model estimates [88] suggest these reactions can balance the hydrogen sulfide flux from below. A deeper phosphate maximum occurs about 5-10 m below the appearance of hydrogen sulfide. The vertical gradient of hydrogen sulfide increases at this depth (Fig. 2). 

Even if rate measurements in sediments are made using whole core incubations, e.g., when the inhibitor is a gas, it is still difficult to obtain a depth distribution of the rate (usually, an areal rate is obtained). A sophisticated measurement and model based system that avoids direct rate measurements has been used to overcome this problem. Microelectrodes, which have very high vertical resolution, are used to measure the fine scale distribution of oxygen and NOs" in freshwater sediments. By assuming that the observed vertical gradients represent a steady state condition, reaction-diffusion models can then be used to estimate the rates of nitrification, denitrification and aerobic respiration and to compute the location of the rate processes in relation to the chemical profiles (e.g., Binnerup et ai, 1992 Jensen et ai, 1994 Meyer et ai, 2001 Rysgaard et ai, 1994). Recent advances and details of the microelectrode approach can be found in the Chapter by Joye and Anderson (this volume). 

Most current models of the ocean reproduce the major features of oceanic carbon the vertical gradient in DIG, the seasonal and latitudinal patterns of pco in surface waters, and the interannual variability in pco observed during El Ninos (Prentice et al., 2001). However, ocean models do not capture the spatial distribution of at depth (Orr et al., 2001), and they do not show an interhemispheric transport of carbon that is suggested from atmospheric CO2 measurements (Stephens et al., 1998). The models also have a tight biological couphng between carbon and nutrients, which seems not to have existed in the past and may not exist in the future. The issue is addressed below in Section 8.10.4.2.2. 

Ocean process tracers and idealized models have also been used extensively to study the ventilation of the main thermocline. The main thermocline includes the upper 1 km of the tropical to subpolar ocean where the temperature and potential density vertical gradients are particularly steep. Thermocline... 

Nine boimdary conditions of the 3D SMART model are relevant for the concerned sea-atmosphere simulation. The horizontal open boundary condition 0 = 3c/3n =o with n J. 5/2 defines the gradient at the boundary to be zero for both water and air. This suppresses dispersion but permits advection through the open boundary. The no-flux boundary condition 0 = uc — Ddcdn g assures that neither dispersive nor advective flux passes through a solid boundary. At the sea and air boundary layer, dispersion is replaced with the Henry coefficient equilibrium assumption c , = Cat/h. This requires that natural gas is simulated in both, sea and atmosphere. The vertical open boundary condition for the atmosphere extrapolates the vertical gradient to the other side of the... 

In order to have an intuitive idea of how a more efficient solution can be achieved, let us consider a trivial bidimensional example in which the PES for is identical but displaced with respect to that of So (notice that this model is treated in detail in Chapter 8, where it is defined as the vertical gradient, VG). We know that the solution of the problem can simply be written as a product of two time-dependent wavepackets, which are coherent states, if the two PESs are harmonic. Hence... 

An important breakthrough was achieved in the late 1980s when the vertical gradient freeze (VGF) technique was developed [3, 4]. In the following period it was possible to improve the crystal-growth conditions continuously with respect to decreased thermal stress, especially by a rigorous application of modeling and process-simulation tools [5, 6]. 

Table 10.1 Types of calculation for the excited electronic state (ES) needed for the simulation of the vibrationally resolved electronic spectra with the vertical gradient (VG), adiabatic shift (AS), vertical Elessian (VH), and adiabatic Elessian (AEI) models.

Between V 9 and the concentration of solids gradually becomes more uniform in the vertical direction. This transition has been modeled by several authors as a concentration gradient where turbulent diffusion balances gravitational settling. See, for example, Karabelas (AJChE]., 23, 426 34 [1977]). 

The simple model given above does not take account of the facts that industrial refractories are poly crystalline, usually non-uniform in composition, and operate in temperature gradients, both horizontal and vertical. Changes in the coiTosion of multicomponent refractories will also occur when there is a change in the nature of tire phase in contact with the conoding liquid for example in Ca0-Mg0-Al203-Cf203 refractories which contain several phases. 

Two-zone models are especially useful for stratification and zoning strategies because of the typical vertical accumulation of heat, contaminants, or water vapor within these strategies. The level of the boundary between the lower and the upper zone is usually determined on the level of the highest temperature or/and concentration gradient. 

Most room models contain only one zone air node, thus assuming perfect mixing of the zone air and a homogenous temperature distribution in the space. Spatial temperature variations, such as vertical temperature gradients, are not considered. For specific applications such as displacement ventilation or atria, models with several zone air nodes in the vertical direction have been developed. ... 

We then consider a model problem of bouyancy recirculation in a closed cavity, using the same mesh of the previous example. Here the vertical boundaries are held at fixed temperatures, the left hotter than the right, while the horizontal boundaries are left unconstrained. A linear temperature gradient is thus set up between the left and right boundaries. The cooler and denser fluid at the right will tend to move down and displace the warm fluid at the left, setting up a clockwise circulation as seen in the streamline contour plot of Figure 6. 

Gradient diffusion was assumed in the species-mass-conservation model of Shir and Shieh. Integration was carried out in the space between the ground and the mixing height with zero fluxes assumed at each boundary. A first-order decay of sulfur dioxide was the only chemical reaction, and it was suggested that this reaction is important only under low wind speed. Finite-difference numerical solutions for sulfur dioxide in the St. Louis, Missouri, area were obtained with a second-order central finite-difference scheme for horizontal terms and the Crank-Nicolson technique for the vertical-diffusion terms. The three-dimensional grid had 16,800 points on a 30 x 40 x 14 mesh. 

In the past, various resin flow models have been proposed [2,15-19], Two main approaches to predicting resin flow behavior in laminates have been suggested in the literature thus far. In the first case, Kardos et al. [2], Loos and Springer [15], Williams et al. [16], and Gutowski [17] assume that a pressure gradient develops in the laminate both in the vertical and horizontal directions. These approaches describe the resin flow in the laminate in terms of Darcy s Law for flow in porous media, which requires knowledge of the fiber network permeability and resin viscosity. Fiber network permeability is a function of fiber diameter, the porosity or void ratio of the porous medium, and the shape factor of the fibers. Viscosity of the resin is essentially a function of the extent of reaction and temperature. The second major approach is that of Lindt et al. [18] who use lubrication theory approximations to calculate the components of squeezing flow created by compaction of the plies. The first approach predicts consolidation of the plies from the top (bleeder surface) down, but the second assumes a plane of symmetry at the horizontal midplane of the laminate. Experimental evidence thus far [19] seems to support the Darcy s Law approach. 

Lakes and oceans are often vertically stratified. That is, two or more fairly homogeneous water layers are separated by zones of strong concentration and density gradients. In Chapter 21, two- and multibox models will be developed to describe the distribution of chemicals in such systems. In these models, volume fluxes, Qex, are introduced to describe the exchange of water and solutes between adjacent boxes (Fig. 19.5). Qex has the same dimension as, for instance, the discharge of a river, [L3TT ]. The net mass flux, LFnet, from box 1 into box 2 is given by ... 

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