Advective-diffusion equation, sediment

If particle mixing is assumed to be analogous to diffusion with sediment accumulation and radionuclide decay, the steady-state profile for excess activity of a nonexchangeable radionuclide is defined by an advective-diffusion equation. 

After making these adjustments for diffusion in sediments, the mass balance and vertical concentration patterns of nonconservative solutes in saturated sediments can be described by the following one-dimensional advective-diffusive general diagenetic equation (GDE) (Berner, 1980 Aller, 2001 Jprgensen and Boudreau, 2001) ... 

The mass balance and vertical concentration patterns of nonconservative solutes in saturated sediments can be described by the one-dimensional advective-diffusive general diagenetic equation (GDE). 

The standard method of estimating chemical migration in a cap is via a transient advection-diffusion model as described by Palermo et al. [1]. This model is applied to the chemical isolation layer of a cap, which is the cap thickness after removing components for porewater expression via consolidation of underlying sediment, consolidation of the cap, and bioturbation of the upper cap layers. Normally, an analytical solution to the mass conservation equation, assuming that the cap is semi-infinite, is employed in such an... 

An equation for the mixing of a tracer associated with the solid phase in a sediment profile is derived from the general diagenic equation (Berner, 1980). For a radioactive tracer with steady-state conditions and where bed porosity changes with depth can be considered negligible, the defining equation is the familiar equation of advective-diffusive transport ... 

The MTC representing the combined diffusive and advective (i.e., sedimentation rate) process kb must be estimated using Equations 13.6 and 13.7 by accounting for the Peclet number. 

The concentration of small ions in the atmosphere is determined by 1) the rate of ion-pair production by the cosmic rays and radioactive decay due to natural radioactive substances, 2) recombination with negative ions, 3) attachment to condensation nuclei, 4) precipitation scavenging, and 5) transport processes including convection, advection, eddy diffusion, sedimentation, and ion migration under the influence of electric fields. A detailed differential equation for the concentration of short-lived Rn-222 daughter ions including these terms as well as those pertaining to the rate of formation of the... 

The model (Fig. 23.6) consists of three compartments, (a) the surface mixed water layer (SMWL) or epilimnion, (b) the remaining open water column (OP), and (c) the surface mixed sediment layer (SMSL). SMWL and OP are assumed to be completely mixed their mass balance equations correspond to the expressions derived in Box 23.1, although the different terms are not necessarily linear. The open water column is modeled as a spatially continuous system described by a diffusion/advection/ reaction... 

The corresponding dynamic equations of the open water column are constructed from Eq. 22-6. They are completed by the sediment-water boundary flux derived in Eq. 23-38. We assume that the net vertical advection of water is zero. Thus, the vertical water movement is incorporated in the turbulent diffusivity, Ez. The assumption implies that if chemicals are directly introduced at depth z (term 1), they would not be accompanied by significant quantities of water. Typically, such inputs are due to sewage outlets (treated or untreated) into the lake. We get ... 

Because pore-water distributions over only the upper few decimeters of sediment wUl be considered here, advection can be ignored as relatively unimportant compared to diffusion and reaction (Lerman, 1975). Equation... 

If there are no or very few irrigated burrows present in the sediment, lateral diffusion is not significant and the r dependence of Eq. (6.12) can be ignored. In that case, the equation becomes the more traditional onedimensional transport-reaction equation used to model pore-water solute profiles where advection is relatively unimportant (Berner, 1971 1980 Lerman, 1979). Both the cylindrical microenvironment model and the onedimensional Cartesian coordinate model will be used here to quantify the Mn distributions at NWC and DEEP. 

Of the 41 listed in Table 4.1 the 16 most common mass transport processes representing the air, water, and soil and sediment media appear in Table 4.2. The media of prime concern often dictate the most convenient phase concentration used in the flux equation. For example, water quality models usually have Cw as the state variable and therefore the flux expression must have the appropriate MTC group based on Cw and these appear in the center column of Table 4.2. Aquatic bed sediment models usually have Cs, the chemical loading on the bed solids, as the state variable. The MTC groups in the right eolumn are used. All the MTC groups in Table 4.2 contain a basic transport parameter that reflects molecule, element, or particle mobility. Both diffusive and advective types appear in the table. These are termed the individual phase MTCs with SI units of m/s. Examples of each type in Table 4.2 include for water solute transport and Vg for sediment particle deposition (i.e., setting). 

Each flux has units of mg/m s. The individual processes flux equations may contain both diffusive and advective fluxes. Mimic the examples presented in Section 4.4.2 above. Flux expressions with concentration gradients and effective diffusion coefficients are normal for the soil and sediment side of these interfaces. For the air and water side of interfaces, concentration difference flux equations are normally used for the diffusive processes. Table... 

Estimating the MTC for the bank water exchange process. Details concerning the theory were presented above in Sections 11.3.2 and 11.3.3. The chemical flux concept in equation form, such as Equation 11.2, requires that advection be connected to other on-going in-bed transport processes such as diffusion. Advection in a chemodynamic context cannot be considered a stand-alone process. The molecular diffusion transport process is used in Equation 11.5 however, it can be generalized to accommodate any diffusive-type process and the appropriate MTC. As such the appropriate sediment-side MTC is... 

For Pe = 1, the sedimentation transport process is equal in direction and magnitude to the diffusion driven transport process and the term in brackets in Equation 13.6= 1.58 so the chemical flux into the bed MTC is 58% larger that represented by Equation 13.7. At Pe = 1.6, the term in brackets = 2.00 and the advective sedimentation rate is 2 x the purely diffusive rate. At higher values of Pe, sedimentation increasingly dominates the numerical value of the MTC at Pe = 9, it is 90% of the MT process in the bed. Under this limiting condition... 

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