Solutions to the 3D Transport Equation and 1D Diffusion Equation for Passive Tracers in the Atmospheric Boundary Layer and Their Applications

A solution to the 3D transport equation for passive tracers in the atmospheric boundary layer (ABL), formulated in terms of Green's function (GF), is derived to show the connection between the concentration and surface fluxes of passive tracers through GF. Analytical solutions to the 1D vertical diffusion equation are derived to reveal the nonlinear dependence of the concentration and flux on the diffusivity, time, and height, and are employed to examine the impact of the diffusivity on the diurnal variations of CO2 in the ABL. The properties of transport operator H and their implications in inverse modeling are discussed. It is found that H has a significant contribution to the rectifier effect in the diurnal variations of CO2. Since H is the integral of GF in time, the narrow distribution of GF in time justifies the reduction of the size of H in inverse modeling. The exponential decay of GF with height suggests that the estimated surface fluxes in inverse modeling are more sensitive to the observations in the lower ABL. The solutions and first mean value theorem are employed to discuss the uncertainties associated with the concentration-mean surface flux equation used to link the concentrations and mean surface flux. Both analytical and numerical results show that the equation can introduce big errors, particularly when surface flux is sign indefinite. Numerical results show that the conclusions about the evolution properties of passive tracers based on the analytical solutions also hold in the cases with a more complicated diffusion coefficient and time-varying ABL height.