Analytic solutions of reaction diffusion equations and implications for the concept of an air parcel

Abstract

In the atmosphere, hydroxyl radical concentrations can be estimated by considering the relative change in the concentration of two hydrocarbons of differing reactivity. This approach is based on three assumptions: (1) that background concentrations of the two hydrocarbon are zero; (2) that transport processes will influence all hydrocarbon concentrations equally; that is, hydrocarbon changes can be separated into the product of a chemical term and a transport term; and (3) that hydrocarbons have the same spatial and temporal emission pattern. In this paper, analytical solutions to a steady state reaction diffusion equation are derived. The general solutions to this problem are nonseparable, with the degree of nonseparability defined by a single parameter that is a simple function of the system's intrinsic timescales. When this parameter is evaluated for diffusivities typical of the boundary layer (∼102 m2/s) and hydrocarbon reactivities that are sufficiently slow to be practically useful, it can be readily shown that for all practical purposes, separability can be assumed. This separability influences the spatial distribution of loss but not the net global loss. Thus even under nonseparable conditions, although the apparent local loss rate may be considerably less than the actual kinetic loss rate implied by hydrocarbon reactivity, when the apparent local loss rates are integrated to deduce a global loss rate, there will be no underestimate in the global loss rate, since the chemical loss rate is a linear function of hydrocarbon concentrations. It is conjectured that when much higher dispersion rates common in photochemical transport models (∼105–106 m2/s) are invoked, local photochemical balances may be perturbed when chemical loss rates are either spatially inhomogeneous or influenced by reactant concentrations. Thus in highly diffusive models the influence of highly reactive chemical species may be extended further from the source regions than is realistic, even though the globally averaged loss rates would still be consistent with the magnitude of the globally averaged sink.