Advection particle

There is a connection between the Lagrangian representation based on advected particles and the Eulerian representation using concentration fields. As in the case of pure advection the solution of the advection-diffusion equation can be given in terms of trajectories of fluid elements. Equation (2.6) can be generalized for the diffusive case using the Feynman-Kac formula (see e.g. Durrett (1996)) as... 

The advection velocity variability generates additional advective particle displacements , relative to the mean particle displacement v. Analogous to Brownian motion, the effect of a large number of independent... 

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. 

The physical transport of particles in a river occurs by two primary modes bedload and suspended load. Bedload consists of material moved along the bed of the river by the tractive force exerted by flowing water. Bedload may roll or hop along the bottom, and individual particles may remain stationary for long periods of time between episodes of movement. Suspended load consists of material suspended within the flow and that is consequently advected by flowing water. Rivers and streams are naturally turbulent, and if the upward component of turbulence is sufficient to overcome the settling velocity of a particle, then it will tend to remain in suspension because the particles become resuspended before they can settle to the bottom of the flow. Suspended load consists of the finest particles transported by a river, and in general is composed of clay- and silt-sized... 

The two prime mechanisms of carbon transport within the ocean are downward biogenic detrital rain from the photic zone to the deeper oceans and advection by ocean currents of dissolved carbon species. The detrital rain creates inhomogeneities of nutrients illustrated by the characteristic alkalinity profiles (Fig. 11-9). The amount of carbon leaving the photic zone as sinking particles should not be interpreted as the net primary production of the surface oceans since most of the organic carbon is recycled... 

Figure 7. Sediment eontains derived both seavenged from the water eolumn during particle settling and contained in solid material. Ra produced in the sediments is highly soluble in pore waters and diffuses into the overlying water or is advected across the sediment-water interface by discharging groundwater. Rn is produced within the water column from dissolved Ra and within the underlying sediments.

POC/ Th (mol C/dpm is the ratio on sinking particles and is the decay constant of " Th (0.029 d ). This approach makes no assumptions about residence times, although it implicitly assumes that sinking biogenic particles are the principal carriers of " Th atoms, that the POC/ Th ratio on sinking particles can be measured, that steady state applies and that horizontal and vertical transport of " Th via advection of water are negligible. 

Such OGCM modeling also suggests the importance of ice-cover in controlling the amount of Thxs advection (Henderson et al. 1999a). Low particle fluxes beneath sea-ice may lead to low scavenging rates in these areas, particularly where ice cover is permanent. In these areas, °Thxs may be advected to the edge of the ice sheet where it is... 

The use of ( Paxs/ Thxs) to assess past circulation rates only works well where the residence time of deep waters is low and advection therefore dominates over the removal of Pa by particle scavenging (Yu et al. 2001a). The Atlantic Ocean fits these... 

The RTD quantifies the number of fluid particles which spend different durations in a reactor and is dependent upon the distribution of axial velocities and the reactor length [3]. The impact of advection field structures such as vortices on the molecular transit time in a reactor are manifest in the RTD [6, 33], MRM measurement of the propagator of the motion provides the velocity probability distribution over the experimental observation time A. The residence time is a primary means of characterizing the mixing in reactor flow systems and is provided directly by the propagator if the velocity distribution is invariant with respect to the observation time. In this case an exact relationship between the propagator and the RTD, N(t), exists... 

Interphase Material Transfer. In some cases there is unidirectional bulk transfer of material and associated chemical between compartments (e.g. sediment deposition or atmospheric particle fallout) in which case the rate is given by an expression similar to that for advection in which Gg (m3/h) is the rate of transfer of the material namely... 

It is desirable to calculate new bulk phase Z values for the four primary media which include the contribution of dispersed phases within each medium as described by Mackay and Paterson (1991) and as listed earlier. The air is now treated as an air-aerosol mixture, water as water plus suspended particles and fish, soil as solids, air and water, and sediment as solids and porewater. The Z values thus differ from the Level I and Level II pure phase values. The necessity of introducing this complication arises from the fact that much of the intermedia transport of the chemicals occurs in association with the movement of chemical in these dispersed phases. To accommodate this change the same volumes of the soil solids and sediment solids are retained, but the total phase volumes are increased. These Level III volumes are also given in Table 1.5.2. The reaction and advection D values employ the generally smaller bulk phase Z values but the same residence times thus the G values are increased and the D values are generally larger. 

In this model, two level-set functions (d, p) are defined to represent the droplet interface (d) and the moving particle surface (p), respectively. The free surface of the droplet is taken as the zero in the droplet level-set function 0> and the advection equation (Eq. (3)) of the droplet level-set function (droplet surface. The particle level-set function (4>p) is defined as the signed distance from any given point x in the Eulerian system to the particle surface ... 

The data impose IJZ=0.2. The concentration profiles have been drawn for IJZ=0.1, 0.5, and 10 (Figure 8.24). Also drawn are the fluxes of dissolved species i for the same values of the parameters, which makes it possible to estimate the flux carried by sinking particles. For instance, a quick graphic examination reveals that, for IJz = 0.5, the flux of species i reaching the bottom Z with the rain of particles is approximately —0.1—(—0.7) or 60 percent of the dissolved flux advected at the base. 

As continuing sedimentation increases the depth of a sedimentary layer relative to the seafloor, the overlying pressure increases because of the increased weight of the additional particles. The increased pressure leads to particle compaction if the pore waters can escape upward. Under these conditions, sedimentation generates an upward advective flow of pore water. This flow has the potential to transport solutes. 

---