The dispersion of contaminants in the Planetary Boundary Layer(PBL) by turbulent flows is a basic environmental problem. A great number of works have recently been carried out to study the dispersion of airborne pollutants in the Convective and Stable Boundary Layers(CBL and SBL). However less attention has been paid to the dispersion in the Residual Layer(RL), the remnant of the daytime CBL. In the RL, the dispersion of contaminants mainly occurs due to the decay of energy-containing eddies in the CBL.

This study describes the derivation of an eddy diffusivity for convective decaying turbulence in the residual layer. The approach is based on the budget equation for turbulent kinetic energy(TKE) where buoyant and shear terms are disregard and turbulence is assumed to be homogeneous and isotropic. In such equation the inertial energy transfer term is closed using a model suggested by Pao on the basis of dimensional considerations. In doing this, we obtain a budget equation for TKE and an analytical solution for the 3-D Energy Density Spectrum(EDS). Furthermore by the isotropy condition, the time decaying vertical velocity variance is calculated and compared with LES data.

The comparison is pretty good until t_*=tz_i/w_*~1. For larger times, the vertical velocity variance calculated from our model decays with the time according to a power law with the exponent -2. Employing an expression suggested by Hanna, described in terms of the spectral wave-number maximum, two vertical decaying eddy diffusivities are derived. One utilizing the decaying vertical velocity variance calculed by LES and the other calculated from our model. Both eddy diffusivities show a good agreement until t_*~1. As a consequence of the exponent -2 obtained for the decaying vertical velocity variance calculated from LES, for larger times the LES eddy diffusivity decays more strongly. Finally, the eddy diffusivities are fitted by empirical curves. These algebric fittings can be used in atmospheric diffusion models to parameterize the dispersion in a decaying convective turbulence in the residual layer.