Atmospheric diffusion equation boundary conditions

Before we proceed to solve (18.56), a few comments about the boundary conditions are useful. When the x diffusion term is dropped in the atmospheric diffusion equation, the equation becomes first-order in x, and the natural point for the single boundary condition on x is at x = 0. Since the source is also at x = 0 we have an option of whether to place the source on the RHS of the equation, as in (18.56), or in the x = 0 boundary condition. If we follow the latter course, then the x = 0 boundary condition is obtained by equating material fluxes across the plane at x = 0. The result is... 

We saw that by assuming constant eddy diffusivities Kxx. Kyy, and K77, the solution of the atmospheric diffusion equation has a Gaussian form. Thus it should be possible to obtain (18.88) or (18.89) as a solution of an appropriate form of the atmospheric diffusion equation. More importantly, because of the ease in specifying different physical situations in the boundary conditions for the atmospheric diffusion equation, we want to include those situations that we were unable to handle easily in Section 18.9.1, namely, the existence of an inversion layer at height H and partial absorption at the surface. Readers not concerned with the details of this solution may skip directly to Section 18.9.3. 

The advantage of the deposition velocity representation is that all the complexities of the dry deposition process are bundled in a single parameter, vd. The disadvantage is that, because vd contains a variety of physical and chemical processes, it may be difficult to specify properly. The flux F is assumed to be constant up to the reference height at which C is specified. Equation (19.1) can be readily adapted in atmospheric models to account for dry deposition and is usually incorporated as a surface boundary condition to the atmospheric diffusion equation. 

The atmospheric diffusion equation in three dimensions requires horizontal boundary conditions, two for each of the x. y, and z directions. The only exception are global-scale models simulating the whole Earth s atmosphere. One usually specifies the concentrations at the horizontal boundaries of the modeling domain as a function of time ... 

Dispersion Models Based on Inert Pollutants. Atmospheric spreading of inert gaseous contaminant that is not absorbed at the ground has been described by the various Gaussian plume formulas. Many of the equations for concentration estimates originated with the work of Sutton (3). Subsequent applications of the formulas for point and line sources state the Gaussian plume as an assumption, but it has been rigorously shown to be an approximate solution to the transport equation with a constant diffusion coefficient and with certain boundary conditions (4). These restrictive conditions occur only for certain special situations in the atmosphere thus, these approximate solutions must be applied carefully. 

In Equation 12.3, D is normally assumed to be independent of both penetrant concentration and polymer relaxations at low concentration, and x is the thickness of the membrane. This is especially true for gases such as oxygen (O2), carbon dioxide (CO2) at atmospheric pressure, and some organic compounds. Many theories have been proposed and many models have been developed to describe diffusion in polymers a detailed description of these models can be found elsewhere [7]. The diffusion processes through the membrane can generally be considered unidirectional and perpendicular to the flat surface, and solutions to the diffusion equations are obtained from the boundary conditions where the Henry s or Langmuir-Henry s law is applied. 

The quasilaminar sublayer resistance / b describes the excess resistance for the transfer of matter from the atmosphere to the surfaces of the vegetation, that is, the difference between the resistance for matter and the resistance for momentum. It is primarily associated with molecular diffusion through quasi laminar boundary layers. Several parameterizations for Rb have been developed, but that employed by Brook et al. (1999), which like Equations 7.3 and 7.6 is valid for conditions of neutral atmospheric stability, is particularly easy to apply ... 

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