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Nonlinear Advection Algorithms Applied to Interrelated Tracers: Errors and Implications for Modeling Aerosol–Cloud Interactions

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  • 1 Pacific Northwest National Laboratory, Richland, Washington
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Abstract

Monotonicity constraints and gradient-preserving flux corrections employed by many advection algorithms used in atmospheric models make these algorithms nonlinear. Consequently, any relations among model variables transported separately are not necessarily preserved in such models. These errors cannot be revealed by traditional algorithm testing based on advection of a single tracer. New types of tests are developed and conducted to evaluate the monotonicity of a sum of several number mixing ratios advected independently of each other—as is the case, for example, in models using bin or sectional representations of aerosol or cloud particle size distributions. The tests show that when three tracers with an initially constant sum are advected separately in one-dimensional constant velocity flow, local errors in their sum can be on the order of 10%. When cloudlike interactions are allowed among the tracers in the idealized “cloud base” test, errors in the sum of three mixing ratios can reach 30%. Several approaches to eliminate the error are suggested, all based on advecting the sum as a separate variable and then using it to normalize the sum of the individual tracers’ mixing ratios or fluxes. A simple scalar normalization ensures the monotonicity of the total number mixing ratio and positive definiteness of the variables, but the monotonicity of individual tracers is no longer maintained. More involved flux normalization procedures are developed for the flux-based advection algorithms to maintain the monotonicity for individual scalars and their sum.

Corresponding author address: Mikhail Ovtchinnikov, Pacific Northwest National Laboratory, P.O. Box 999, MSIN: K9-24, Richland, WA 99352. Email: mikhail@pnl.gov

Abstract

Monotonicity constraints and gradient-preserving flux corrections employed by many advection algorithms used in atmospheric models make these algorithms nonlinear. Consequently, any relations among model variables transported separately are not necessarily preserved in such models. These errors cannot be revealed by traditional algorithm testing based on advection of a single tracer. New types of tests are developed and conducted to evaluate the monotonicity of a sum of several number mixing ratios advected independently of each other—as is the case, for example, in models using bin or sectional representations of aerosol or cloud particle size distributions. The tests show that when three tracers with an initially constant sum are advected separately in one-dimensional constant velocity flow, local errors in their sum can be on the order of 10%. When cloudlike interactions are allowed among the tracers in the idealized “cloud base” test, errors in the sum of three mixing ratios can reach 30%. Several approaches to eliminate the error are suggested, all based on advecting the sum as a separate variable and then using it to normalize the sum of the individual tracers’ mixing ratios or fluxes. A simple scalar normalization ensures the monotonicity of the total number mixing ratio and positive definiteness of the variables, but the monotonicity of individual tracers is no longer maintained. More involved flux normalization procedures are developed for the flux-based advection algorithms to maintain the monotonicity for individual scalars and their sum.

Corresponding author address: Mikhail Ovtchinnikov, Pacific Northwest National Laboratory, P.O. Box 999, MSIN: K9-24, Richland, WA 99352. Email: mikhail@pnl.gov

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