Estimation of a length scale to use with the quench level approximation for obtaining chemical abundances

The "quench level" approximation for estimating the observed abundance of chemically reacting species in the presence of convective dynamics states that the chemical reaction is quenched at the level where the time scales for the chemical reaction, t(chem), and for convective dynamics, t(dyn), are equal. The dynamical time constant, t(dyn), can be computed using t(dyn), L-2/K-eddy, where L is a length scale and K-eddy is the vertical eddy diffusion coefficient. Usually K-eddy is left as a free parameter, and lacking any better information, L is taken to be the pressure scale height, H. Here it is shown that an effective length scale L-eff, can be estimated from the e-folding length scales of t(chem) and t(dyn) and the pressure dependence of the equilibrium value of the chemical abundance. The improved accuracy of using L = L-eff instead of L = H is demonstrated by comparing the results of three different mathematical "models" for convective dynamics. The three models are the quench level approximation, a second computation that uses diffusive mixing to describe convection, and a third computation that explicitly integrates the equations of motion and chemical equilibration. Each of the three models uses a different fundamental quantity in describing the strength or efficiency of mixing that depends on the length scale L in a different way. When L = H is used, the three models give three different results, but when L = L-eff is used, results from the three models are virtually identical. The effective length scale, L-eff, is different for different chemical reactions and can be far from a scale height. As an example, for the isotropic enrichment of (D/H) in CH4 over (D/H) in H-2, the effective length scale is L-eff = 0.14H at a typical value of K-eddy = 10(8) cm(2) sec(-1). Using L = 0.14H leads to an enrichment factor of 1.17 on Jupiter compared to 1.21 when L = H is used. (C) 1998 Academic Press.