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Nonlinear Advection Algorithms Applied to Interrelated Tracers: Errors and Implications for Modeling Aerosol–Cloud Interactions MIKHAIL OVTCHINNIKOV AND RICHARD C. EASTER Pacific Northwest National Laboratory, Richland, Washington (Manuscript received 29 April 2008, in final form 15 August 2008) 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the total number mixing ratio and positive definiteness of the variables, but the monotonicity of individual tracers is no longer maintained.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.
1. Introduction Advection or transport of trace substances and conserved properties by fluid is one of the most important processes to be represented in spatially gridded atmospheric models of many scales from boundary layer models to global climate models. Many numerical schemes are available (see, e.g., the review by Rood 1987). With models striving to maintain sharp spatial gradients on a grid whose resolution is severely limited by computational constraints, first-order schemes are unacceptably diffusive. Higher-order schemes are more accurate but produce undesirable oscillations near sharp spatial gradients in advected fields often encountered in atmospheric modeling (e.g., fronts and/or cloud boundaries). Corrections and constraints must therefore be applied to preserve monotonicity or, at least, the positivedefiniteness of variables, such as gas and particle mixing
Corresponding author address: Mikhail Ovtchinnikov, Pacific Northwest National Laboratory, P.O. Box 999, MSIN: K9-24, Richland, WA 99352. E-mail:
[email protected] DOI: 10.1175/2008MWR2626.1 Ó 2009 American Meteorological Society
ratios, for which negative values are nonphysical. Some schemes also apply local flux corrections, which depend on spatial distribution of advected variables (e.g., Chlond 1994; Walcek 2000, hereinafter W00). Although these modifications provide the algorithms with desired properties such as monotonicity and improved gradient preservation, they also make these algorithms nonlinear. Consequently, any relations among interrelated tracers advected separately are not necessarily preserved. This presents a serious problem for models in which variables derived from several tracers represent important properties. In this study, we focus on evaluating and correcting errors introduced when conventional advection algorithms are applied to a sectional or bin representation of particle size distributions in which particle spectra are represented discretely using a number of size categories. Specifically, we are concerned with maintaining monotonicity of a total number (or mass) mixing ratio integrated over all size categories. It will be shown that improving accuracy of advection for individual tracers can compete against preservation of this integrated property. Although there might be no universally optimal
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OVTCHINNIKOV AND EASTER TABLE 1. Advection algorithms.
Identifier B92 P86 SG90 W00
Description
Reference
Monotone flux limitation in the area-preserving fluxform advection algorithm Conservation of second-order moments The multidimensional positive definite advection transport algorithm—nonoscillatory option Minor flux adjustment near mixing ratio extremes
solution to this dilemma, the rationale here is that for studying cloud–aerosol interaction the compromise will often be weighted toward maintaining spatial distribution of total particle number (or mass) mixing ratio at the expense of perturbing spatial distributions of the mixing ratio for individual bins. This study was motivated by our own experiences in implementing and applying a detailed bin representation of aerosol particles and cloud droplets similar to Bott (1997) in a threedimensional cloud-resolving model (Ovtchinnikov and Ghan 2005) using the W00 advection algorithm. Initial simulations of marine stratus clouds produced nonphysical spatial variations of total particle number comparable to variations expected from physical aerosol processing in these clouds. Although interrelated tracers are transported separately in many models, the effect of advection algorithms on tracer correlations has rarely been addressed. In a recent study, Wright (2007) showed that when aerosol size distribution is represented by a number of moments treated as independent tracers during transport, the application of typical advection algorithms often results in the production of ‘‘invalid moment sets’’ for which no underlying size distribution could exist. He described a number of possible solutions to deal with the phenomenon, including vector transport, advection of surrogate quantities, and replacement of invalid moment sets. Addressing the same problem, McGraw (2007) introduced a nonnegative least squares solution that eliminates the need to arbitrarily choose the lead moment as in vector transport but requires inversion of matrix of dimension m 3 n, where m is the number of tracers (in his case moments) and n is the number of neighboring grid cells contributing to the solution. The applicability of these approaches designed for moment representations to bin representations of size distribution has not been investigated. Also, with m exceeding several tens or even a few hundreds as in many bin models, the approach could become prohibitively expensive. In this study, we first develop tests to isolate and quantify errors in the total number mixing ratio of several tracers transported separately and suggest pos-
Bott (1992) Prather (1986) Smolarkiewicz and Grabowski (1990) Walcek (2000)
sible remedies for the problem. These tests are then applied to a number of advection algorithms commonly used in atmospheric modeling. The rest of the paper is organized as follows: Section 2 describes the approach including schemes to be tested and normalization procedures to be applied; formulations of the tests and their results are given in section 3; and finally section 4 summarizes the most important findings of the study and discusses their implications for studies of cloud– aerosol interaction.
2. Approach a. Advection algorithms Atmospheric models employ a variety of advection algorithms. In this paper, we test four commonly used algorithms listed in Table 1. The actual FORTRAN code for the monotone flux limitation in the areapreserving flux-form advection algorithm is taken from Easter (1993), with modifications for monotonicity as in Bott (1992, hereinafter B92); it is hereinafter called the B92 algorithm. Note that for the test problems considered here that have nondivergent flow, the Easter (1993) algorithm becomes identical to that of B92. In this study, we use the fourth-order version of the algorithm. The code for the multidimensional positive definite advection transport algorithm (MPDATA) with nonoscillatory option as described by Smolarkiewicz and Grabowski (1990, hereinafter SG90) is taken from a large-eddy simulation (LES) model by Khairoutdinov and Kogan (1999). The code for the method of minor flux adjustment near mixing ratio extremes (hereafter, the W00 algorithm) is an adaptation of the original code (W00) by Ovtchinnikov and Ghan (2005). Finally, the code for the algorithm based on conservation of secondorder moments described in Prather (1986, hereinafter P86) has been provided by its author with an additional flux-limiting procedure that insures monotonicity (M. Prather 2008, personal communication). The performance of these and other algorithms in a variety of single-tracer advection tests has been examined extensively (e.g., W00 and references therein). In
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this study, the above four algorithms are subjected to a number of multicomponent tests. These tests, described in detail in section 3, show that none of the algorithms locally preserve the sum (and hence the monotonicity of the sum) of three mixing ratios when all the tracers are advected independently. In the rest of this section, we describe procedures that can be used to force monotonicity of the total mixing ratio, which is equivalent to conserving the total mixing ratio in the tests used here. These procedures are based on introducing the total mixing ratio as another variable to be advected separately from and in addition to individual mixing ratios and using that tracer to constrain the solution for other species.
b. Scalar normalization We describe normalization procedures in the context of the W00 algorithm, and we use Walcek’s (2000) notation, but with an additional index (m) added to differentiate between multiple tracer species. Thus, Qti, m and Qt1Dt i, m are the mixing ratios of trace species m (m 5 1, . . ., N) in grid cell i at times t and t 1 Dt, respectively. A new variable representing the sum of the mixing ratios to be conserved is defined as N
Qti, N11
5
Qti, m ,
(1)
m51
where Qti, N11 is recomputed at each time step before the advection routine is called and then advected separately from other individual species. Following advection, mixing ratios for individual tracers at a grid cell are adjusted to match the total in that grid cell. The adjustment can take different forms, the most intuitive being to make the correction for each tracer proportional to this tracer’s relative contribution to the total mixing ratio: 0 t1Dt @ t1Dt (Qt1Dt i, m )adj 5 Qi, m 1 QN11
3
Qt1Dt i, m N
k51
Qt1Dt i, k
c. Flux normalization The flux normalization procedure is an extension of the monotonicity constraints applied in B92 and W00. Its purpose is to enforce monotonicity on the sum of multiple trace-species mixing ratios. One-dimensional monotonic schemes generally march downstream through the grid and adjust outflow fluxes, where needed, so that mixing ratios at t 1 Dt are locally monotonic. With the proposed flux normalization, the outflow fluxes of individual tracers are adjusted simultaneously so that 1) the sum of the individual fluxes matches the outflow flux of the separately advected sum of individual tracers, and 2) the monotonicity of the individual tracers is maintained. This first constraint guarantees that the sum of the individually advected tracers will match the separately advected sum of individual tracers. The first modification to the W00 algorithm is that calculations of outflow-fluid mixing ratios (Qf,i1½,m), fluxes, and Qt1Dt i, m are done simultaneously for m 5 1, . . . , N 1 1. The algorithm basically marches downstream through the grid (or through portions of the grid where u stays positive or negative). At each grid cell i, after the W00 monotonicity constraints are applied [W00’s (8), (9a), and (9b)], a new final step is added. Consider the case of u $ 0. The individual Qf,i1½,m are now adjusted so that their sum will equal to Qf,i1½,N11. Define N
A5
A Qt1Dt i, k
k51
5 Qt1Dt i, m
additional computational burden for implementing this approach in a model includes performing the summaN tion m51 Qi, m twice per time step, applying an advection algorithm to one extra tracer, Qi, N11 , and computing adjustment (2), which amount to four operations per tracer per grid interval.
1
N
Qt1Dt i, N11 N
k51
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(2)
Qt1Dt i, k
By design, this normalization maintains the total number concentration of aerosol particles and preserves the positive definiteness of individual scalars. The method is simple and requires no changes to the advection code. As such it can be applied in almost any atmospheric model with minimal modifications. The
Qf , i11/2, m m51
and consider the case of A , Qf,i1½,N11, so that the Qf,i1½,m must be increased. Define dm5 0 if Qf,i1½,m 5 Qfmax,i1½,m, and dm 5 1 otherwise. Here, Qfmax,i1½,m is the maximum value of Qf,i1½,m that will produce monotonic values of Qt1Dt i, m ; it comes from W00’s (9a) with Qti,1mDt set to Qt1Dt min , i, m . The Qf,i1½,m are adjusted using N
B5
dm Qf , i11/2, m
m51
Cm 5 Qf , i11/2, m [11 dm (Qf , i11/2, N11 A)/B] Qf adj, i11/2, m 5 min (Qf max , i11/2, m , Cm ).
ð3Þ
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TABLE 2. Normalization procedures. Label
Normalization type
Equation
Comment
S Fa Fb Fc
Scalar Flux Flux Flux
(2) (3) (4) (3) then (4)
One-step adjustment Iterated as needed One-step adjustment Two-step adjustment
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monotonicity-constrained limit. The Qf,i1½,m are therefore adjusted using N
B5
Qf max , i11/2, m
m51
D 5 Qf , i11/2, N11 A Qf adj, i11/2, m 5 Qf , i11/2, m
If none of the Qfadj,i1½,m for which dm 5 1 attain their respective maximum values, then SQfadj,i1½,m will equal Qf,i1½,N11,. The Qfadj,i1½,m are now used to recalculate t1Dt the Qt1Dt i, m [W00’s (3)], and the sum of these Qi, m will t1Dt equal Qi, N11 . When one or more of the Qfadj,i1½,m attain their maximum values, it is possible to iterate on the adjustment step, starting each iteration with the latest Qfadj,i1½,m and updating the dm. The iterating will eventually be successful because it can be shown [using W00’s (7a), (7b), (8), (9a), and 9b)] that SQfmax,i1½,m $ Qfmax,i1½,N11 $ Qf,i1½,N11. For the case of A . Qf,i1½,N11, so that the Qf,i1½,m must be decreased, the algorithm follows the above but with ‘‘max’’ and ‘‘min’’ interchanged throughout. For the case of u , 0, the algorithm is similar except that in grid cell i the individual Qf,i2½,m are adjusted so that their sum will equal to Qf,i2½,N11. The flux normalization iterative procedure using (3) is labeled ‘‘Fa’’ in Table 2. We note that there are potentially many ways to adjust the individual Qf,i1½,m to sum to Qf,i1½,N11. The above approach attempts to apply the same relative adjustment to each Qf,i1½,m, which is reasonable from a physical standpoint. In a cloud model, for example, applying the same absolute adjustment to cloud-dropsized bins and rain-drop-sized bins (whose number mixing ratios are much different in magnitude) would seem undesirable. If the number of tracers N is large, this approach could require many iterations and become prohibitively expensive. There are several options for dealing with this. One option is to limit the maximum number of iterations to a small value (e.g., # 3) and on the final iteration (if it is needed) set all the dm 5 1 and set Qfadj,i1½,m 5 Cm. In this case, some of the Qfadj,i1½,m may exceed their maximum values, and some of the Qt1Dt i, m may be nonmonotonic. A second option might be to apply the scalar normalization after some fixed number of iterations. Note that in both these approaches monotonicity for individual scalars would be relaxed to achieve monotonicity for the sum. A third option is again to complete a small number of iterations and then make the final correction to the flux proportional to the difference between the flux and its
1(Qf max , i11/2, m Qf , i11/2, m ) D/(B A) . (4) With this approach, no iterations are necessary because Qfadj,i1½,m # Qfmax,i1½,m. This is because Qf , i11/2, N11 # Qf max , i11/2, N11 # Qf max , i11/2, m , so that Qf , i11/2, N11 2 A # B 2 A, and therefore D/(B 2 A) # 1. Note that (4) will always produce a monotonic solution for all tracers and their sum and can be applied directly, without using iterations (3) at all. This solution is labeled ‘‘Fb’’ in Table 2. Because large corrections could be applied to fluxes that are small but far from their monotonic limits, the physical meaning of this correction is not obvious, and it may be less accurate than (3). In our tests with three tracers, we found that applying (4) after just one iteration of (3) produces a solution very close to the one obtained by iterating (3) to the end. The third flux normalization procedure (labeled ‘‘Fc’’ in Table 2) tested in this study is therefore just that: one iteration (3) followed by adjustment (4). Note that applying normalization procedures discussed in this section requires the solution for the advection problem to be written in the form of flux convergence–divergence. This is true for the W00 and B92 algorithms but is not the case for the SG90 and P86 schemes. Consequently, the flux normalization procedures are only tested with W00 and B92. Furthermore, the W00 and B92 algorithms have been modified to simultaneously calculate advection of all tracers, including the newly added sum of individual tracers. Although the development of a new multicomponent advection algorithm is outside the scope of this paper, several modifications to the W00 algorithm have been tested in this study. The original W00 algorithm employs local Courant number-dependent sharpening factors to limit numerical diffusion and improve gradient preservation near extremes [W00’s (10)]. These adjustments are made to each advected tracer individually. At the suggestion of one of the reviewers, we also tested
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applying the same sharpening factor to all tracers whenever it is applied to any one of them. We found the results to be extremely case sensitive because this approach reduces the error in the sum of top-hat distributions for Courant numbers ,0.5 but has an opposite effect for Courant numbers .0.5 and for the irregular shape test, which is arguably more realistic. Because this option does not provide a robust improvement in the performance it is not discussed further in this paper. Another aspect of the W00 algorithm is that monotonicity is achieved in two steps: W00’s (6a) and (6b), which improve monotonicity, followed by his (8), (9a), and (9b), which ensure monotonicity. As suggested by one reviewer, the flux normalization procedure could be applied to the Qf,i1½,m after both of these steps. This modification does produce some small reductions in RMS errors for most but not all tracer distributions in the tests of section 3a, but it increases computational cost.
3. Multicomponent advection tests results a. One-dimensional advection tests The simplest imaginable multicomponent test involves advecting two tracers (Q1 and Q2) whose spatial distributions are initially constrained by a constant sum. For example, Q1 has a top-hat (square-wave) shape with a constant positive background value, and Q2 (x) 5 C 2 Q1 (x), where C is a constant that exceeds the maximum of the initial Q1. With the B92, P86, and SG90 algorithms, the sum of the two tracers is maintained during advection, indicating that the nonlinear adjustments to two ‘‘mirror image’’ tracers are equal in magnitude but opposite in sign. The W00 algorithm needed a slight modification from its original formulation. This algorithm determines local extreme and applies corrections to the fluxes in grid cells surrounding these extremes. We find that spurious extreme and nonextreme sequences generated in W00 by the round-off errors in the areas of otherwise uniform mixing ratios resulted in errors in the total sum for the two components that were larger by an order of magnitude compared to other algorithms. The problem is resolved by allowing some tolerance in determining the extremes. Specifically, we define a local extreme at cell i if Qi $ [max(Qi21,Qi11) – e] or Qi # [min(Qi21,Qi11) 1 e], where Qi is a mixing ratio of a tracer Q at a cell i and e is a prescribed tolerance factor much smaller than the meaningful advected mixing ratios. The original W00 formulation corresponds to e 50 whereas in this study we use e 5 1026 (in conjunction with nonzero Qi ; 1–10). With this modification to W00, all four algorithms perform equally well in the two-tracer test, but the special symmetry of this twotracer test makes it inadequate for our purposes.
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The tests discussed in the remainder of this section consist of advecting three tracers having an initially constant sum. Initial distributions of the two tracers (Q1 and Q2) are prescribed as either a top-hat (Fig. 1a) or irregular (Fig. 1b) signal, similar to SG90. For Q2, the disturbance is slightly shifted and increased in magnitude relative to Q1. In both cases, the distribution of the third tracer Q3 is computed as a residual from the constant sum. The tracers are advected over 50 grid cells in a 100-gridcell domain grid using a constant velocity and density flow. Figure 1 shows the results of the tests for all four advection algorithms for a Courant number of 0.1. Here, we focus on the behavior of the sum the three mixing ratios. The top panels in Fig. 1 show that the errors in the sum are significant and reach 10% and 5% for the tests with top-hat and irregular shapes, respectively. The performance of the algorithms is quantified by computing root-mean-square errors (RMSEs) shown in Fig. 2 as functions of the Courant number. These RMSEs are computed over the computational domain comprised of 100 grid intervals. Tests are made with Courant numbers equal to 50/M, where M is the number of time steps needed to move 50 grid cells. Extending W00’s results to a nearly continuous range of Courant numbers, Fig. 2 illustrates that the errors are only weakly sensitive to the Courant number and that the W00 scheme delivers consistently better scores for individual tracers. The origin for the unusually noisy behavior of the RMSE for the W00 algorithm is not clear, but this may not be important in atmospheric models in which the Courant number varies both spatially and temporally. With respect to the goals of this study, the most important results of the tests are concentrated in the top panels of Fig. 2, which show that the better performance of a scheme for individual tracers does not guarantee a better performance for the their sum. Indeed, the W00 scheme, which provides the smallest RMSE for most of the individual tracers, gives one of the largest RMSEs for their sum. The P86 algorithm results in the smallest RMSE for the sum, particularly for the top-hat distributions, while generating slightly higher RMSEs for several individual tracers than W00. Better performance of the P86 algorithm comes at a cost because this scheme requires higher memory and computational resources than the other three. Additional moments needed by the P86 scheme increase the number of variables that need to be stored by a factor of 3 for onedimensional advection and factors of 6 and 10 for twoand three-dimensional problems, respectively (P86; Rotman et al. 2001). The amount of additional computations also varies depending on the application, but
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FIG. 1. Numerical solutions for (a) top-hat and (b) irregular shape distributions of three tracers’ advection of 50 grid cells at a Courant number of 0.1 using the W00, B92, SG90, and P86 algorithms . Tops of panels show the sum of the three mixing ratios, with deviations from horizontal lines representing errors due to violation of monotonicity of the sum. The exact solution is indicated by a thick gray line.
for a two-dimensional rotation test the algorithm was shown to be about 2 times slower than the W00 or B92 algorithms (W00). It should be noted that in real atmospheric models Courant numbers are usually low, rarely exceeding 0.2. It is also true, however, that sharp gradients in tracer mixing ratios can be correlated with areas of elevated Courant numbers. One example of this is cloud updrafts within which sharp reductions in cloud condensation nuclei (CCN) concentration and increases in droplet concentration are formed by active droplet nucleation. Figure 3 shows the effect of the normalization procedures listed in Table 2 on the solution using the W00 and B92 schemes at a Courant number of 0.1. (Differences from applying Fa and Fc normalizations are not discernible in this and other figures in the paper and therefore these two solutions are represented by a single line.) All scalar and flux normalization procedures completely eliminate errors in the sum, but the adjustments they introduce to the original solutions for individual tracers are quite different. As mentioned earlier, the scalar normalization may result in nonmonotonic behavior, which is most clearly evident in Q1 and Q2 overshoots for the top-hat distributions in the W00 solution (Fig. 3, top left). The flux normalizations preserve monotonicity for individual tracers but result in greater numerical diffusion. For the top-hat distributions, the
increased diffusion smoothes the gradients (Figs. 3a,b), whereas for the wavy pattern it introduces a noticeable phase shift downwind (Figs. 3c,d). Of the three flux normalization corrections, Fb introduces the largest distortions to the advected patterns. The effects of the normalizations are qualitatively the same in W00 and B92, but the differences among the original and normalized solutions are much smaller for B92 (Figs. 3b,d). To quantify the effects of normalization procedures on the solution we again compute RMSE, which is shown in Fig. 4 as a function of a Courant number. Naturally, normalization procedures eliminate RMS errors for the sum, but in many cases they increase errors for individual tracers. The Fb normalization produces the largest RMSE in most cases, particularly at low Courant numbers. The W00 scheme, which originally had the lowest RMSE, experiences the largest error increase (Figs. 4a,e). In contrast, for B92 the normalizationinduced changes in RMSE are very small (Figs. 4b,f). The RMSEs for the Fa and Fc normalizations in this test are exactly the same as for B92 and differ on average by less than 3% for W00; these results are therefore shown as a single line in Fig. 4. The scalar normalization S generally results in small RMSE increases for individual tracers. One exception is the advection of top-hat shapes with the W00 scheme, in which the overshoots and undershoots seen in Fig. 3a are large enough to contribute significantly
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FIG. 2. RMSE as a function of Courant number for the W00, B92, SG90, and P86 algorithms (without normalization) for (a) top-hat and (b) irregular shape tests.
to the RMSE for Q1 and Q2 (Fig. 4a). Note that only scalar normalization is applied to the SG90 and P86 algorithms as discussed in section 2c. Overall, the changes in RMSE for various normalization procedures are comparable to or smaller than the differences among various algorithms shown in Fig. 2.
b. ‘‘Cloud base’’ tests In clouds and especially at cloud boundaries, we find not only sharp gradients but also transformations among different species. At cloud base, for example, gradients in CCN and droplet concentrations are, in fact, created through such transformations. Indeed, activation of CCN causes particles to move from the aerosol to the small droplet (SD) category, and subsequent droplet growth continue to shift particles to larger size bins. This process is different from a simple advection of various gradients with the flow discussed earlier in the sense that here the boundary (or gradient in tracer’s mixing ratio) is fixed in space while the air continuously flows through it. In this section, we describe a test that in very basic terms mimics the following processes near cloud base: advection of CCN and droplets, activation of CCN, and growth of droplets.
The minimum number of tracers needed to describe the above processes is three. We therefore introduce QCCN, QSD, and QLD to represent the number mixing ratios for CCN, small droplets, and large droplets (LDs), respectively. We consider a one-dimensional updraft with a constant velocity W 5 0.4 m s21 resolved on a spatial grid with uniform spacing of Dz 5 20 m using a time step of Dt 5 5 s, which provides a Courant number of 0.1 for the flow. The initial condition is a cloud-free domain (i.e., no small or large droplets) with a normalized CCN number mixing ratio of 1. We therefore have QCCN,0 5 1; QSD,0 5 QLD,0 5 0. The microphysical transformations are specified by ›QCCN /›t 5 R1 QCCN ›QSD /›t 5 R1 QCCN R2 QSD ›QLD /›t 5 R2 QSD ,
(5)
where R1 5 1/t 1 when z . zb and R1 5 0 elsewhere, and zb 5 0 m (bottom of grid cell 11) is the cloud base height. Similarly R2 5 1/t 2 when z . zc, R2 5 0 elsewhere, and zc 5 20 m (bottom of grid cell 12). We use t 1 5 t2 5 25 s, so that the e-folding times for CCNto-SD and SD-to-LD conversions both correspond to vertical transport distances of 10 m. Because the
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FIG. 3. Effects of various normalization procedures on the (left) W00 and (right) B92 solutions for the (top) top-hat and (bottom) irregular distributions of mixing ratios at Courant number of 0.1. Nonnormalized solutions are shown by thin solid lines.
conversion distance is less than the grid spacing, one expects that the advection algorithms will have difficulty in simulating the tracer distributions. This is not an extreme test, however, because the grid spacing and
conversion times are representative of those in 3D LES simulations used to study cloud–aerosol interactions. Simulations are performed using a process splitting technique, so at each time step advection (with or without
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FIG. 4. RMSE as a function of Courant number for the W00, B92, SG90, and P86 algorithms with and without normalization for (a)–(d) top-hat and (e)–(h) irregular shape tests. RMSEs for the nonnormalized solutions are shown by gray lines.
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FIG. 6. RMSE as a function of Courant number for the cloudlike test for the W00, B92, SG90, and P86 algorithms.
FIG. 5. Solutions for the cloudlike tests using the W00, SG90, B92, and P86 algorithms (without normalization). Altitude is shown relative to the cloud base height, which is marked by a horizontal dashed line. The analytical steady-state solution is represented by the thick gray line. See text for details.
normalization) and microphysical transformations are computed sequentially. The microphysical transformations [(5)] are integrated in a manner that conserves total number, using small substeps for high accuracy. Figure 5 shows profiles of the three tracers and their sum after a 5-min simulation using the four advection algorithms listed in Table 1. Above cloud base, QCCN begins to move to the QSD category, mimicking CCN activation and formation of small droplets. At the first in-cloud layer at 10 m, the CCN number mixing ratio is depleted to less than a half of the original value, with the rest of CCN activated by the third in-cloud layer at 50 m. The SD mixing ratio reaches its maximum at the first in-cloud layer (10 m) and declines higher up because the loss rate of these droplets to SD-to-LD transfer exceeds the generation rate via CCN-to-SD. The latter is proportional to the CCN mixing ratio and declines sharply as the CCN mixing ratio decreases. Note that QLD increases above the 10-m level and approaches unity above 50 m as all small droplets grow into that category. The most striking result of the test is in the sporadic behavior of the sum of the three tracers (Fig. 5) above the cloud base. This sum is initially constant and should remain constant throughout the simulation because all three tracers are transported with the same updraft velocity (no sedimentation is considered in this test), and microphysical transformations do not alter the total. Deviation of the total from unity is an artifact of the advection algorithm. The total mixing ratio is overestimated by 20% at the first cloudy grid cell and has
errors on the order of 5% over a layer that is 150 m thick. Note that because at heights above 70 m all microphysical transformations have been already completed, the resulting errors are limited to LD; this LD disturbance continues to be simply advected with the flow. An analytical steady-state solution for this test can be calculated by integrating the resulting ordinary differential equations with respect to height. This solution is also shown in Fig. 5. Comparison of simulation results with the analytical solution illustrates that large errors in the total just above the cloud base are associated primarily with the peak values of QSD, which are significantly overestimated by the W00, B92, and SG90 algorithms and slightly underestimated by the P86 algorithm. As with the advection tests in section 3a, the P86 solution is the most accurate, with the maximum error in the sum of the three mixing ratios being less than 5%. The Courant number dependency of the RMSE for the four algorithms in this test is shown in Fig. 6. There is a general trend of diminishing RMSE with increasing Courant number. As noted before, most models typically operate at Courant numbers below 0.2, and P86 outperforms the other algorithms in this low end of the studied range. Figure 7 shows changes in vertical distribution of QCCN, QSD, and QLD resulting from applying various normalization procedures after computing the tracer’s advection using W00. The sum of the three is conserved in this test. The correction of the sum occurs primarily because of the reduction in the peak QSD value. The normalized QSD in the first grid cell above cloud base is reduced from 0.91 to 0.63, which closely matches the analytical solution. The differences among the different normalizations are minimal. Applying normalizations to other advection algorithms produces very similar results (not shown).
4. Summary and discussion The advection of tracers in a flow of near-incompressible fluid is an important process that drives many phenomena
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gradients are nearly stationary in space. This test mimics advection across a cloud base, where the gradients are formed and maintained through microphysical transformations among aerosol and cloud particles while the location of the gradients in these fields remains nearly fixed in space. These tests are applied to four different advection algorithms. The results of the tests can be summarized as follows: d
d
FIG. 7. Effect of scalar and flux normalizations on the results of the cloudlike test using the W00 algorithm. The nonnormalized solution is shown for reference (thick gray line).
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: d
in the atmosphere. Accurate representation of this process in numerical models is notoriously difficult. Even for a relatively narrow class of atmospheric models, such as finite-difference-based mesoscale and cloud-scale models, there is no single, universally best algorithm. Consequently, the choice of a scheme to be used in a given model or study is often made based on scheme’s theoretical accuracy and performance in simple tests and its computational cost. It has been known that the nonlinearity of advection algorithms may distort certain relations among species advected separately. However, because the vast majority of traditional testing has been conducted on idealized distributions of a single tracer, and because errors due to advection are difficult to isolate in three-dimensional atmospheric models with complex physics and/or chemistry, the magnitude of the problem has not been sufficiently quantified. In this study, we evaluate the errors in the sum of several tracers’ mixing ratios advected separately and develop corrective procedures applicable to a multidimensional model with sectional (or bin) representation of aerosol and cloud particle size distributions. Two types of tests are developed and conducted in this study. The first represents simultaneous transport of several correlated spatial distributions and is used to evaluate errors from moving coincident boundaries in tracer fields with opposite gradients. This test mimics advection of a cloud boundary in which the spatial gradients in the CCN and cloud droplet mixing ratios are, in general, reversed. The second group of tests represents a flow passing through a region where tracer
When three species, initially with a constant sum, are advected separately in one-dimensional constant velocity flow, local errors in the sum can be on the order of 10%. When cloud-like interactions are allowed among the species in the idealized cloud base test, errors in the total sum of three mixing ratios can reach up to 30%.
d
A simple scalar normalization procedure maintains monotonicity of the sum of individual tracers and preserves 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.
The problem is considered here from the point of view of preserving monotonicity of the total concentration of aerosol and cloud particles, but it is relevant to any model that relies on preserving certain relations among independently advected variables (multimoment aerosol/cloud modules, sectional/bin microphysics, multicomponent chemistry, etc.) The problem will likely become more acute as more and more variables and physical processes are included in atmospheric models. When applied to modeling aerosol–cloud interaction, the advection schemes considered in this paper are likely to produce artificial redistribution of cloud and aerosol particles that may taint the model signals of aerosol effects on cloud properties and cloud processing of aerosols. This study provides an illustration of advection error’s magnitude and offers procedures to preserve monotonicity of the sum of a group of tracers. The proposed procedures perform well in the tests described in this paper, but the study is not without limitations. The most obvious one is that the tests presented here are one dimensional and based on constant velocity flow. A similar setup occurs in a treatment of particle sedimentation in
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which fall velocities of particles in any size bin can be nearly constant or vary little with height, although particles in different bins can have substantially different fall velocities. Most flows of interest, however, are multidimensional and more complicated in structure. There is an indication that errors in two- and three-dimensional flows exceed those in one-dimensional flow by a factor of 2 or more (M. Prather 2007, personal communication). Additional studies are needed to quantify these errors in multidimensional models and determine how they may affect simulated cloud–aerosol interactions. The tests were conducted and the corrective procedures were designed for preservation of monotonicity of the sum of mixing ratios. In models using a singlemoment sectional (bin) representation for the treatment of particle size distributions, this is equivalent to conserving the monotonicity of total number or mass mixing ratio and therefore is a reasonable physically based constraint. In models using a double-moment bin representation (number and mass for each bin), the procedures could be applied to the number tracers and mass tracers separately, thus preserving monotonicity both for total number and for mass. An extension of the proposed normalization approach to preservation of the monotonicity of any positive linear combination (i.e., a linear combination with positive coefficients) of individually transported tracers is trivial. In models that use multimoment treatments (with three or more moments) of particle size distributions, interrelations between individual moments that need preservation may not be reduced to their linear combination. In that case, the proposed normalization procedures would not be applicable and another approach should be used (e.g., McGraw 2007). All normalization procedures discussed in this paper ensure the monotonicity of the total mixing ratio during advection by altering the mixing ratios of individual tracers. In making a decision about which procedure (if any) to apply in a specific model, it is important to understand how the individual tracers are changed. Both scalar and flux normalizations can distort distributions of individual tracers and often lead to comparable increases in RMSE in their solutions (Fig. 4). There is, however, a principal difference between scalar and flux normalizations. The flux normalizations only redistribute the tracers’ mixing ratio in space. As shown in section 3, this can lead to increased numerical diffusion, which smoothes spatial gradients but does not violate domain-integral properties of individual tracers. In the case of scalar normalization, the monotonicity of total mixing ratio is forced by adjusting the mixing ratios for individual tracers in each grid cell independently. The required redistribution therefore occurs
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among different tracers in a given grid cell, as opposed to flux normalization, which is based on spatial redistribution of a mixing ratio among neighboring cells. Consequently, if tracers represent number mixing ratios, this scalar normalization approach preserves the total number mixing ratio but may lead to nonconservation of a total mass mixing ratio in the domain. (Conversely, if the transported species represent mass mixing ratios, then forcing conservation of total mass may violate conservation of total number mixing ratio.) In a cloud model in which the cloud droplet spectrum is represented by number mixing ratios in several size categories, redistribution of particles among these size bins after the advection may change the liquid water content in a grid cell. If the model’s thermodynamics is based on variables conserved during condensation/evaporation, such as total water mixing ratio and liquid water potential temperature (e.g., Khairoutdinov and Kogan 1999; Ovtchinnikov and Ghan 2005), then the domain water and energy conservation is not affected and normalization results in small changes to a few diagnosed variables (e.g., the water vapor mixing ratio and absolute temperature). If, on the other hand, the water vapor mixing ratio and/or absolute temperature are among the prognostic model variables, then the water and/or energy balance in a the domain could be affected and must be monitored carefully. Finally, it is instructive to look at additional computations needed to perform the proposed normalizations. Both scalar and flux normalizations require transporting one additional tracer (the sum of the individual tracers). The relative increase in computations for this transport is inversely proportional to the number of tracers contributing to the total. When a model carries a large number of tracers (e.g., several dozens of particle size bins), the additional computational burden will be small (on the order of few percent). This paper exposes for the first time errors that nonlinear advection algorithms may induce in the sum of multiple tracers. These errors are quantified here in a number of highly idealized one-dimensional tests; additional studies are needed to address their effects on simulated aerosol–cloud interactions in realistic multidimensional models. These studies should also put these advection errors in the context of other uncertainties in such models. Acknowledgments. The authors thank Michael Prather for providing the code of his advection algorithm and Weiguo Wang for help with the initial algorithm testing. This research was supported by the Pacific Northwest National Laboratory (PNNL) Directed Research and Development (LDRD) program as part of the Aerosol
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Climate Initiative. PNNL is operated by Battelle for the U.S. Department of Energy. Comments by Jerome Fast and two anonymous reviewers helped to improve the manuscript. REFERENCES Bott, A., 1992: Monotone flux limitation in the area-preserving fluxform advection algorithm. Mon. Wea. Rev., 120, 2592–2602. ——, A numerical model of the cloud-topped planetary boundary layer: Impact of aerosol particles on the radiative forcing of stratiform clouds. Quart. J. Roy. Meteor. Soc., 123, 631–656. Chlond, A., 1994: Locally modified version of Bott’s advection scheme. Mon. Wea. Rev., 122, 111–125. Easter, R. C., 1993: Two modified versions of Bott’s positive-definite numerical advection scheme. Mon. Wea. Rev., 121, 297–304. Khairoutdinov, M. F., and Y. L. Kogan, 1999: A large eddy simulation model with explicit microphysics: Validation against aircraft observations of a stratocumulus-topped boundary layer. J. Atmos. Sci., 56, 2115–2131. McGraw, R., 2007: Numerical advection of correlated tracers: Preserving particle size/composition moment sequences dur-
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ing transport of aerosol mixtures. J. Phys.: Conf. Series, 78, 012045, doi:10.1088/1742-6596/78/1/012045. Ovtchinnikov, M., and S. J. Ghan, 2005: Parallel simulations of aerosol influence on clouds using cloud-resolving and singlecolumn models. J. Geophys. Res., 110, D15S10, doi:10.1029/ 2004JD005088. Prather, M. J., 1986: Numerical advection by conservation of second-order moments. J. Geophys. Res., 91, 6671–6681. Rood, R. B., 1987: Numerical advection algorithms and their role in atmospheric transport and chemistry models. Rev. Geophys., 25, 71–100. Rotman, D. A., and Coauthors, 2001: Global Modeling Initiative assessment model: Model description, integration, and testing of the transport shell. J. Geophys. Res., 106, 1669–1691. Smolarkiewicz, P. K., and W. W. Grabowski, 1990: The multidimensional positive definite advection transport algorithm: Nonoscillatory option. J. Comput. Phys., 86, 355–375. Walcek, C. J., 2000: Minor flux adjustment near mixing ratio extremes for simplified yet highly accurate monotonic calculation of tracer advection. J. Geophys. Res., 105, 9335–9348. Wright, D. L., 2007: Numerical advection of moments of the particle size distribution in Eulerian models. J. Aerosol Sci., 38, 352–369.
