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

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the total number mixing ratio and positive definiteness of the variables, but the monotonicity of individual tracers is no longer maintained.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.

Evaluating advection/transport schemes using interrelated tracers, scatter plots and numerical mixing diagnostics

“…The total chlorine thus provides an important constraint on the chemistry. Similar arguments can be made for aerosol-cloud interactions (Ovtchinnikov and Easter, 2009) where important physical properties are derived from several tracers. The issue that may arise in this context is that, when models transport individual tracers with a shape-preserving scheme, there is in general no guarantee that the sum of the tracers (or any other function that is a function of more than two tracers) is shape-preserving, and therefore the sum may contain unphysical values.…”

“…For the first tracer χ we choose initial condition φ (sc) with (λ 1 , θ 1 ) = (3π/4, 0), (λ 2 , θ 2 ) = (5π/4, 0), c = 1/3 and g = 1/10 (Figure 4(a)). Following Ovtchinnikov and Easter (2009), the initial condition for the second tracer ξ is displaced with respect to χ , in this case 10…”

“…As alluded to above, tracers are usually transported separately although they may be related through functional relations. Truncation errors may perturb preexisting functional relations between tracers (Thuburn and McIntyre, 1997, hereafter referred to as TM97) and cause unphysical values in derived quantities (Ovtchinnikov and Easter, 2009). For pairs of tracers that are linearly related, a linear transport scheme will preserve that linear relation but, for higher than firstorder schemes, will not be shape-preserving.…”

“…As far as the authors are aware, the only exceptions to the widely used single tracer testing methodology in atmospheric transport scheme development are the onedimensional tests proposed by TM97 and Ovtchinnikov and Easter (2009). Both use a constant wind (so the analytical solution is simply the translation of the initial condition) and consider two tracers that are related nonlinearly (TM97) and three tracers that add up to a constant (Ovtchinnikov and Easter, 2009).…”

“…Both use a constant wind (so the analytical solution is simply the translation of the initial condition) and consider two tracers that are related nonlinearly (TM97) and three tracers that add up to a constant (Ovtchinnikov and Easter, 2009). None of these idealized multi-tracer tests have been extended to two dimensions or been applied in highly deformational/divergent flows, and the numerical mixing has not been systematically categorized or quantified.…”