Models depth averaging

Fig. 1.30. Model-derived depth-averaged surface currents in (a) October, (b) December, (c) April, and (d) August. The depth average extends from sea surface to 830 m or bottom, whichever is shallower (Lee and Chao, 2003) (With permission from Elsevier s Copyright Clearance Center)...

The measured data and the results of 2D depth averaged model longitudinal changes of depth and depth-averaged velocity are compared in figure 6. The calculation was conducted... 

Some 2-D numerical models applicable to spills in rivers have been developed. A very promising model has been developed by Holly (43) to compute both steady-state and time-dependent depth-averaged mixing of a conservative, neutrally buoyant pollutant in a steady but nonuniform channel flow of arbitrary cross section. The main limitations here lie in the steady river flow and conservative substance. A similar model has been presented by Harden and Shen (34) and may have some computational advantages. Section III,B,4 notes the distances required to achieve 1-D conditions, indicating that the most profitable area for numerical modeling of diffusion processes may be the 2-D case. 

For most of the tests and accidents, raw data on debris density are presented as debris mass density in kg per m. For the China Lake test and the Steingletscher accident, an in-depth investigation of this relationship was performed. The surveyed data showed that for 1 kg of rock mass, 0.8 to 1.4 pieces of hazardous debris result (see also Fig. 26.17). For the development of the new model, an average value of one piece of hazardous debris per one kg of rock mass was finally assumed. 

Several different partial differential equations are available for implementation in the computer models with varjdng degree of success. The classic depth averaged shallow water equation originaUy used by Leendertse has been used by many to solve for the water surface elevation and velocity field. The classic Laplace s equation or Helmholtz equation was used by many to solve for the velocity potential. 

Two-dimensional (2D) models represent a second tier of spatial complexity with respect to sediment transport models. Models in this class have become more prevalent during the last couple of decades due to advancements in computer hardware and software capabilities. Two-dimensional models typically solve the depth-averaged flow continuity and Navier-Stokes equations with respect to hydrodynamic behavior and mass balance equations with respect to sediment transport. Computational methods employed in 2D models include finite difference, finite element, and finite volume. Examples of 2D models include Environmental Eluid Dynamics Code (EFDC), SEDZLJ, SEDZL, USTARS, MIKE21, and Delft 2D. 

Three-dimensional (3D) models represent the highest tier of spatial and process complexity and solve the flow continuity equation and the Navier-Stokes equations for conservation of mass and momentum in three-dimensional space. Three-dimensional models are favored over depth-averaged models for water body systems where density stratification occurs or hydraulic structures significantly impact hydrodynamic behavior. Examples of 3D coupled hydrodynamic/sediment transport models include EEDC, ECOMSED, MIKE-3, RMA-10, and Delft 3D. 

The nonlinear probability function developed by Partheniades (1992) is also an available option in EFDC. This probability of deposition function is based on the concept that under quiescent conditions all solids settling near the bed will reach and adhere to the bed, while at some threshold flow-induced bed surface stress (ted) none of the solids settling near the bed will reach and adhere to the bed. In the absence of site-specific data, this critical deposition stress is generally treated as a calibration parameter with a wide range of reported values from laboratory and field observations of 0.6 to 11 dyn/cm (TetraTech, 2007c). The near-bed solids concentration (5d) is also an estimated value, computed in EFDC based on either the bottom layer concentration (in 3D model applications) or depth-averaged water colunm concentration (in ID and 2D models). 

Model is insensitive to expected range of changes in summertime water depth in the tidal reach. Predicted DO values differ from standard profile by an average of 1 percent saturation. 

The Henry s law constant value of 2.Ox 10 atm-m /mol at 20°C suggests that trichloroethylene partitions rapidly to the atmosphere from surface water. The major route of removal of trichloroethylene from water is volatilization (EPA 1985c). Laboratory studies have demonstrated that trichloroethylene volatilizes rapidly from water (Chodola et al. 1989 Dilling 1977 Okouchi 1986 Roberts and Dandliker 1983). Dilling et al. (1975) reported the experimental half-life with respect to volatilization of 1 mg/L trichloroethylene from water to be an average of 21 minutes at approximately 25 °C in an open container. Although volatilization is rapid, actual volatilization rates are dependent upon temperature, water movement and depth, associated air movement, and other factors. A mathematical model based on Pick s diffusion law has been developed to describe trichloroethylene volatilization from quiescent water, and the rate constant was found to be inversely proportional to the square of the water depth (Peng et al. 1994). 

In the higher resolution (GR15) the representation of continental shelves is much better than in the coarse resolution (T42), both in terms of area, as well as in terms of water depth. T42 resolves only 51% of the total shelf area found in ET0P02. Most of the shelves resolved by the model are deeper than the ones in ET0P02, whith an average deviation of 36 %. The continental shelves in the Mediterrainian Sea, in Central Africa and the Amazon Continental Shelf are not represented in T42. 

Glaciochemical horizons are intervals of core with substantially higher or lower than average concentrations of certain chemical constituents. If a historical event of known age can be correlated with the event horizon in the core, the assigned age of that interval can be used to confirm the depth-age relationship which has been determined from seasonal variations or other dating methods. In addition, in deep ice where annual layers are too thin to count seasons reliably and dating is only possible by model calculations [15,30], these horizons provide check points for calculated ages. 

From the modelling results for bilayers composed of unsaturated lipids one can begin to speculate about the various roles unsaturated lipids play in biomembranes. One very well-known effect is that unsaturated bonds suppress the gel-to-liquid phase transition temperature. Unsaturated lipids also modulate the lateral mobility of molecules in the membrane matrix. The results discussed above suggest that in biomembranes the average interpenetration depth of lipid tails into opposite monolayers can be tuned by using unsaturated lipids. Rabinovich and co-workers have shown that the end-to-end distance of multiple unsaturated acyl chains was significantly less sensitive to the temperature than that of saturated acyls. They suggested from this that unsaturated... 

In Section 2.2, the Reynolds-averaged Navier-Stokes (RANS) equations were derived. The resulting transport equations and unclosed terms are summarized in Table 2.4. In this section, the most widely used closures are reviewed. However, due to the large number of models that have been proposed, no attempt at completeness will be made. The reader interested in further background information and an in-depth discussion of the advantages and limitations of RANS turbulence models can consult any number of textbooks and review papers devoted to the topic. In this section, we will follow most closely the presentation by Pope (2000). 

Srinivasan etal.,64 in a phenomenological development, split the etch rate into thermal and photochemical components and used zeroth-order kinetics to calculate the thermal contribution to the etch rate. An averaged time-independent temperature that is proportional to the incident fluence was used to determine the kinetic rate constant. The photochemical component of the etch rate was modeled using, as previously discussed, a Beer s law relationship. The etch depth per pulse is expressed, according to this model, in the form... 

In this model, the rate of river runoff (uriver) expressed as the depth of a layer of water produced by spreading the annual river-water input across the entire surfece area of the ocean. The annual amount of river water entering the ocean is 47,000 km /y (Figure 2.1). Assuming that the average area of the ocean is equal to that at the sea surfece (3.6 x 10 cm ), the river input represents the annual addition of a layer of water approximately 10 cm deep, making y ver = lOcm/y. 

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