Advection

Advection is the process by which material contained in a flowing fluid is transported by bulk motion of the fluid. An important example is blood flow, which delivers oxygen and nutrients to the tissues of the body. Maintaining blood flow is essential to maintaining life in higher organisms. 

In advection the mass flux is driven by a continuous velocity field. Given a velocity field v and concentration field c, the mass flux density is T, = ctv. Substitution of this expression into Equation (3.45) yields 

Equation (3.47) is known as the advection equation. For one-dimensional fluid flow the advection equation reduces to 

One-dimensional advection will be used in Chapter 8 as a component in models of cellular biochemical systems that are coupled to blood-tissue solute exchange. 

Advection of aqueous solution in permeable rocks is an important mass transport mechanism as well as reaction and diffusion. 

The dominant transport process in a moving fluid is advection that consists of the rearrangement of fluid elements in space as a result 

The left hand side is the Lagrangian derivative of C, that is the rate of change of the concentration along a path following a fluid element. This allows to write the time evolution of the concentration C in a fluid element as  

This equation also describes the motion of very small suspended particles with negligible inertia that take on the local velocity of the medium instantaneously. The ensemble of solutions corresponding to all the initial positions within the domain of the flow ro V defines a mapping between initial and final positions of each fluid element 

The Lagrangian map gives a full description of the flow that is equivalent to the so called Eulerian description based on specifying 

From the Lagrangian map, the velocity field can be easily recovered by differentiation with respect to time. The inverse procedure, obtaining the Lagrangian map from the velocity field, requires integration of (2.4) and it is usually not possible doing it other than numerically. In any case, one can express the formal solution of equation (2.2) in terms of the solution of (2.3) and the inverse of the Lagrangian map 1 as 

Following the release of a toxicant into an environmental compartment, transport processes will determine its spatial and temporal distribution in the environment. The transport medium (or fluid) is usually either air or water, while the toxicant may be in dissolved, gaseous, condensed, or particulate phases. We can categorize physical transport as either advection or diffusion. 

Homogeneous Advection. The homogeneous advective transport rate (N, g/h) is simply described in mathematical terms by the product of the chemical concentration in the advecting medium (C, g/m3) and the flow rate of the medium (G, m3/h)  

For example, if the flow of water out of a lake is 1000 m3/h and the concentration of the toxicant is 1 xg/m3, then the toxicant is being advected from the lake at a rate of 1000 p,g/h (or 1 mg/h). The emission rates for many toxicant sources can be calculated in the same way. 

In surface waters advective currents often dominate the transport of toxicants, and they can be estimated from hydrodynamic models or current measurements. In many cases advective flow can be approximated by the volume of water exchanged per unit 

Advective air and water currents are much smaller in soil systems but still influence the movement of chemicals that reside in soil. Advection of water in the saturated zone is usually solved numerically from hydrodynamic models. Advection of air and water in the unsaturated zone is complicated by the heterogeneity of these soil systems. Models are usually developed for specific soil property classes, and measurements of these soil properties are made at a specific site to determine which soil-model layers to link together. 

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Nguen, N. and Reynen, J., 1984. A space-time least-squares finite element scheme for advection-diffusion equations. Cornput. Methods Appl Mech. Eng. 42, 331- 342. 

Donea, J. and Quartapelle, L., 1992. An introduction to finite element methods for transient advection problems. Comput. Methods Appl Meek Eng. 95, 169-203. 

The quantity k is related to the intensity of the turbulent fluctuations in the three directions, k = 0.5 u u. Equation 41 is derived from the Navier-Stokes equations and relates the rate of change of k to the advective transport by the mean motion, turbulent transport by diffusion, generation by interaction of turbulent stresses and mean velocity gradients, and destmction by the dissipation S. One-equation models retain an algebraic length scale, which is dependent only on local parameters. The Kohnogorov-Prandtl model (21) is a one-dimensional model in which the eddy viscosity is given by... 

Lagrangian trajectory models can be viewed as foUowing a column of air as it is advected in the air basin at the local wind velocity. Simultaneously, the model describes the vertical diffusion of poUutants, deposition, and emissions into the air parcel as shown in Eigure 4. The underlying equation being solved is a simplification of equation 5 ... 

Fig. 7. Comparison of various transport schemes for advecting a cone-shaped puff in a rotating windfield after one complete rotation (a), the exact solution (b), obtained by an accurate numerical technique (c), the effect of numerical diffusion where the peak height of the cone has been severely tmncated and (d), where the predicted concentration field is very bumpy, showing the effects of artificial dispersion. In the case of (d), spurious waves are...

Regardless of the source, the resultant oil slicks are essentially surface phenomena that are affected by several transportation and transformation processes. With respect to transportation, the principal agent for the movement of slicks is the wind, but length scales are important. Whereas small (i.e. relative to the slick size) weather systems, such as thunderstorms, tend to disperse the slick, cyclonic systems can move the slick essentially intact. Advection of a slick is also affected by waves and currents. To a more limited extent, diffusion can also act to transport the oil. 

Aeration of the hypolimnion (lower, colder layer of water in a stratified lake) without disruption of stratification has been used in deep lakes. This has the advantage of not increasing the temperature of the hypolimnion and prevents the advection of nutrient-rich water into the epilimnion (upper, warmer layer of water in a stratified lake). Oxygen injection is preferred in order to prevent the build up of nitrogen super-saturation which is toxic to fish. "... 

Large Deformation Wave Codes quantity advected... 

N. Ashgriz and J.Y. Poo, FLAIR Flux Line-Segment Model for Advection and Interface Reconstruction, J. Comput. Phys. 93 (1991). 

Errors in advection may completely overshadow diffusion. The amplification of random errors with each succeeding step causes numerical instability (or distortion). Higher-order differencing techniques are used to avoid this instability, but they may result in sharp gradients, which may cause negative concentrations to appear in the computations. Many of the numerical instability (distortion) problems can be overcome with a second-moment scheme (9) which advects the moments of the distributions instead of the pollutants alone. Six numerical techniques were investigated (10), including the second-moment scheme three were found that limited numerical distortion the second-moment, the cubic spline, and the chapeau function. 

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. 

Long, P. E., and Pepper, D. W., A comparison of six numerical schemes for calculating the advection of atmospheric pollution, in "Proceedings of the Third Symposium on Atmospheric Turbulence, Diffusion and Air Quality." American Meteorological Societv, Boston, 1976, pp. 181-186. 

At times when the surface pressure gradient is weak, resulting in light winds in the atmosphere s lowest layers, and there is a closed high-preSsure system aloft, there is potential for the buildup of air pollutant concentrations. This is especially true if the system is slow-moving so that light winds remain in the same vicinity for several days. With light winds there will be little dilution of pollutants at the source and not much advection of the polluted air away from source areas. 

The Gaussian Plume Model is the most well-known and simplest scheme to estimate atmospheric dispersion. This is a mathematical model which has been formulated on the assumption that horizontal advection is balanced by vertical and transverse turbulent diffusion and terms arising from creation of depletion of species i by various internal sources or sinks. In the wind-oriented coordinate system, the conservation of species mass equation takes the following form ... 

Multiple pathways are a major concern since depostion of PIC would have occurred. Specific soil conditions determine attenuation rates of penta PIC leachate. Once penta reaches the water table, other transport and fate processes become important. Penta exists in two forms ionized and non-ionized. The ionized form is soluble in water, while the non-ionized form is not. The ratio of the two forms in water is dependent on the pH of the aquifer. In alkaline environments penta PIC tend to be more soluble and more susceptible to advective transport and biological decay. Half-lives of penta leachate in groundwater have been estimated ranging from 27 days to 58 years. 

Its equations account for advection, Coriolis effects, turbulent heat, momentum, moisture iran.sport, and viscosity. The system treats diurnaliy varying winds such as the land-sea breeze and... 

Heat, described as power, as temperature on the described surface, or as advection of heat along with substance flowing out of the production process. 

The first two terms on the right side of Equafion (40) describe the contributions from transport by advection and by turbulent flux, respectively. The separation of the motion flux into advection and turbulent flux is somewhat arbitrary depending upon the circumstances the averaging time can be anything from a few minutes to a year or even more. 

An estimate of the advective fluxes (processes 1, 2, and 3) requires knowledge of the concentration of the species in solutions and in the solid... 

Time scales of transport can also be applied to situations when no well-defined reservoirs can be defined. If the dominant transport process is advection by mean flow or sedimentation by gravity, the time scale characterizing the transport between two places is simply tadv = L/V where L is the distance and V the transport velocity. Given a t)q)ical wind speed of 20 m/s in the mid-latitude tropospheric westerlies, the time of transport around the globe would be about 2 weeks. 

Hundreds to thousands of km - the synoptic scale, in which motions are those of whole weather systems. Advection is the dominant transport process. 

The motion of substances on the synoptic scale is often assumed to be pure advection. The flux through a unit area perpendicular to the wind is simply the product of wind velocity and concentration. If F is flux, V the velocity, and c concentration. 

The motions on the largest spatial scales amount to the aggregate of the world s synoptic weather systems, often called the general circulation. Both with respect to substances that have atmospheric lifetimes of a day or more and with regard to the advection of water, it is useful to depict the nature of this general circulation. The mean circulation is described to some extent in terms of the Hadley and Ferrell cells shown in Fig. 7-4. They describe a coupled circulation... 

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