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,tchem, and for convective dynamics,tdyn, are equal. The dynamical time constant,tdyn, can be computed usingtdyn=L2/Keddy, whereLis a length scale andKeddyis the vertical eddy diffusion coefficient. UsuallyKeddyis left as a free parameter, and lacking any better information,Lis taken to be the pressure scale height,H. Here it is shown that an effective length scaleLeff, can be estimated from thee-folding length scales oftchemandtdynand the pressure dependence of the equilibrium value of the chemical abundance. The improved accuracy of usingL=Leffinstead ofL=His 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 scaleLin a different way. WhenL=His used, the three models give three different results, but whenL=Leffis used, results from the three models are virtually identical. The effective length scale,Leff, 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 CH4over (D/H) in H2, the effective length scale isLeff= 0.14Hat a typical value ofKeddy= 108cm2sec−1. UsingL= 0.14Hleads to an enrichment factor of 1.17 on Jupiter compared to 1.21 whenL=His used.

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