Some aspects of simulating large scale atmospheric mixing

Abstract

The classical picture of one-dimensional diffusion in a homogeneous fluid results in a symmetrical distribution of the conservative property around the peak concentration while the latter remains fixed in space. This paper examines the asymmetry and movement of the peak resulting from mixing in a non-homogeneous fluid. A one-dimensional model comprising 21 boxes permits the diffusive flux across the wall of each box equal to the coefficient of eddy diffusion times the air density multiplied by the gradient of (mixing ratio) concentration. The end boxes are half the volume of the others and there is no flux across the ends of the model. The divergence of this flux creates the change in concentration in each box. With constant diffusion coefficients and density, the peak (mixing ratio) concentration initially in the center of the model remains fixed and symmetry exists for all times as theory predicts. Firstly, placing the point source near either edge produces asymmetry due to reflection of the Gaussian curve, also in accord with theory. The peak concentration actually shifts with time, its movement depending on the magnitude of the eddy diffusion coefficient and proximity to the edge. Secondly, the peak concentration initially in the middle of the model moves upward for the case of a constant diffusion coefficient but with density decreasing as in the standard atmosphere. For example, for a coefficient of 104 cm2 sec?1. The rate of rise increases with an increasing coefficient and vice versa. When the model contains a strong diffusion regime, or a troposphere, to about 9 km and a transition to smaller coefficients above 17 km, the stratosphere, then the rate of rise of the peak concentration increases (rising from 20 to 40 km in 24 rather than 30 months for a stratospheric coefficient of 104 cm2 sec?1). The removal of tracer material from below also increases the ascent rate of the peak concentration. Thirdly, models with constant density but variable coefficients of diffusion produce considerable asymmetry and some drift of the peak concentration with altitude. Finally, the one-dimensional model may be adapted to a spherical earth with constant horizontal air density. Material added at the equator diffuses symmetrically into both hemispheres as expected. But injections at other latitudes produce asymmetries such that, for example, the peak concentration shifts from 45° to the pole in 30 months if the coefficient of horizontal diffusion is 109 cm2 sec?1 An attempt will be made to explain the asymmetries and try to distinguish between possible real atmospheric effects and those due to the model simulation. The artificial substitution of a three dimensional world by a one dimensional model must be recognized.DOI: 10.1111/j.2153-3490.1966.tb00247.x