Perturbed fluxes, calculation

The model-calculated partitioning of the human-induced perturbation fluxes In the global coastal margin of (a) nitrogen and (b) phosphorus for the period from 1850 to the present (2000) and projected to 2035 under the business-as-usual scenario, In units of lO mol/y and Mtons/y. The anthropogenic sources are plotted on the (-) side and the resulting accumulations and enhanced export fluxes are plotted on the +) side. Source-. From Mackenzie, F. T, et al. (2002). Chemical Geology 190, 13-32. 

The formulations presented in this section are for the calculation of perturbed fluxes. Analogous formulations can be derived and applied for the calculation of perturbed adjoints, kernels for the integral transport equations (see Section IV), generalized functions (see Section V), and other distribution-function perturbations. The presentation is restricted to exposition of the general formulation of the methods, without considering the technical details of the solution. 

If the perturbation in the neutron flux is restricted to the small region P, it is neither necessary nor economical to solve the perturbed flux equation of the flux-difference equations over the entire volume (P -t- U) of the reactor. In the following we outline the approaches used for applying these equations for local calculations, and describe some applications for these calculations. 

Following FerrelK, the second term in Equation 2 can be expressed as a Green-Kubo integral over a flux-flux correlation function. The transport is due to a velocity perturbation caused by two driving forces, the Brownian force and frictional force. The transport coefficient due to the segment-segment interaction can be calculated from the Kubo formula(9 ... 

The pressure momentum is conserved together with the momentum flux in a detonation wave of the structure calculated by us, as the sum p+ (D — u)2 is conserved in a shock wave maintained by the motion of a piston where each of the terms is separately conserved. The stability of such a shock wave with respect to small perturbations is beyond doubt. It is clear that when the chemical reaction ceases, detonation will decay, but the shock wave will also change, entering a medium with different properties. 

The starting point for this calculation is the general relation between the Laplace transforms of the concentration perturbation c, and the time-varying flux/(t)... 

Usually, for pristine watersheds, biologic and exchange reactions are assumed to be at steady state and the weathering flux is the difference between the watershed output and the precipitation input. In perturbed watersheds, the calculations become more complex due to agricultural inputs or changes in the exchangeable ions due to watershed acidification. 

Like the flume method, fluxes derived from chamber experiments do not integrate all marsh processes, and only measure changes in nutrient concentration caused by diffusion or uptake during over bank tides the technique may also perturb the system. It is also inappropriate to use this technique to estimate annual nurient fluxes, via the marsh surface, if the calculated fluxes are based on constant inundation of the marsh surface (as were those of Scudlark Church, 1989) and not the periodic submergence which actually occurs. It would be relatively easy to correct for this error, if surface water level had been continually monitored at the site, by calculating the periods of inundation from the tidal records and details of the marsh topography. However, it should be noted this extrapolation is another source of uncertainty. 

It was found that the sand-mud transition zone could be adequately represented by allowing the sedimentary processes to proceed at constant rates. When this was done, the calculated sand flux agrees well with the sand flux that has been measured over the sand-wave field in the eastern Sound. Since the tides and the estuarine circulation control the sand fluxes, the resulting distribution of sand approaches a steady state very quickly. It is likely that the sand-mud transition was established soon after the Sound became an arm of the sea and has persisted unchanged to the present day. As a corollary to this hypothesis, it would seem that Mattituck Sill has been accreting over the lifetime of the Sound. Small variations in the sand content preserved in short cores are probably due to the perturbation of the sand distribution by a series of winter storms rather than individual storm events. 

Exact calculation of reactivity and of the effects of perturbation on other integral parameters has to take into account the effect of the perturbation on the flux (or adjoint) distribution. Several methods allowing for the flux perturbation have been recently developed, or are already routinely in use. Three of these methods are reviewed in this work the direet calculation of the flux perturbation (Section III) integral transport theory methods (Section IV), and generalized perturbation theory (Section V, B). 

Perturbation theory was originally developed and traditionally applied for calculating the effect of perturbation on integral parameters. A recent development is the application of perturbation techniques for calculating the effect of perturbation on the flux and adjoint distributions. These techniques are being incorporated into new methods developed for the solution of... 

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