1. Introduction
Solving for the advective terms in the fluid conservation equations is an essential part of atmospheric models, and numerous discretization methods have been designed to this end (e.g., see review in Rood 1987). However, errors using numerical advection schemes are unavoidable in nearly all circumstances. Depending on the particular application, several properties may be of primary importance when considering advection schemes, including mass conservation, small diffusion and dispersion, shape preservation, and positive definiteness, among others (Rasch and Williamson 1990; Lauritzen et al. 2011; Thuburn 2008). The representation of advection near sharp gradients, such as those related to clouds boundaries in atmospheric models, is especially challenging. Near those regions, diffusive, loworder schemes tend to smooth the fields out substantially, while dispersive, higherorder schemes typically lead to spurious oscillations and unphysical negative values of the advected quantities. These issues are commonly addressed via the application of higherorder schemes with flux limiters (e.g., the positivedefinite and shapepreserving flux limiters described by Skamarock and Weisman 2009; Wang et al. 2009).
Besides an adequate description of individual fields, preserving functional relationships between advected quantities is also necessary for the representation of physical processes in general. Linear and semilinear advection schemes preserve linear relationships between two tracers, but tend to distort linear relationships involving more than two advected variables and any nonlinear relationships (Thuburn and McIntyre 1997). In numerical models, advectioninduced deviations from preexisting relationships between tracers may mimic the role of mixing in real flows or produce unrealistic effects. Assessing the extent to which these errors impact simulations of realistic flows is an important step in order to facilitate analysis of the representation of physical processes in models. Nevertheless, convergence metrics traditionally employed to evaluate the performance of advection schemes are not necessarily useful to this end. Lauritzen and Thuburn (2012) proposed different metrics that infer and quantify the physical realizability of the numerical mixing based on the type of deviations from the tracers’ initial relationships. These metrics can be readily applied to tracer advection in deformational/divergent flows in two or three dimensions. For example, using a suite of twodimensional linear transport tests on the sphere, Lauritzen et al. (2012) and Lauritzen et al. (2014) showed that these mixing metrics revealed large differences among multiple advection algorithms regarding their ability to preserve functional relationships. An implementation of these metrics in idealized threedimensional tests on the sphere is described in Kent et al. (2014).
The preservation of tracer interrelationships has been assessed mostly in the context of the advection of reactive gas species (e.g., Thuburn and McIntyre 1997) and aerosols (e.g., McGraw 2007; Wright 2007). However, preserving relationships between hydrometeor properties in atmospheric models, such as cloud droplet mass and number mixing ratios, is essential for the description of microphysical processes that feed back to the atmospheric state (e.g., Morrison et al. 2016). A classic example of an important nonlinear scalar relationship in cloud models is the supersaturation, which is typically diagnosed from separately advected fields of temperature and moisture (e.g., Grabowski 1989; Grabowski and Smolarkiewicz 1990; Stevens et al. 1996; Grabowski and Morrison 2008; Hoffmann 2016). Grabowski (1989) showed that errors in the supersaturation appear even when temperature and moisture are advected with nonoscillatory schemes, owing to incompatibilities in the advection of temperature and moisture (the relationship between the variables not being conserved), exacerbated by the highly nonlinear condensation/evaporation source term. In addition, nonlinear relationships between advected variables are usually employed to represent conversion rates in cloud microphysics schemes, such as Khairoutdinov and Kogan’s (2000) bulk autoconversion parameterization:
Several approaches have been proposed to mitigate deviations in nonlinear relationships between advected variables. For instance, Grabowski and Morrison (2008) suggested an alternative approach for improving the representation of a shapepreserved supersaturation field, based on advecting the absolute supersaturation (difference between the water vapor and saturation vapor mixing ratios) using a nonoscillatory scheme and subsequently adjusting the temperature and moisture fields to maintain consistency with the supersaturation field via cloud condensation/evaporation. Morrison et al. (2016) showed that remapping the set of prognostic advected variables so that key diagnostic quantities can be expressed as simple ratios of the advected quantities can mitigate errors in the diagnostic quantities. This idea was applied to model the shape parameter of gamma hydrometeor particle size distributions in the threemoment bulk microphysics schemes of Paukert et al. (2019) and Milbrandt et al. (2021). An essentially similar strategy could be applied to other quantities that depend nonlinearly on advected prognostic variables, but, in cases with many relevant nonlinear relationships, choosing the right set of advective quantities may not be straightforward. More general strategies to improve the synchronicity of the advection of primary quantities have also been proposed. For example, Grabowski and Smolarkiewicz (1990) suggested a method that mitigates the errors in the derived supersaturation by synchronously applying a fluxcorrected transport approach to the coupled system of thermodynamics variables affected by condensation/evaporation. A generalization of this approach was later presented by Schär and Smolarkiewicz (1996) for the advection of arbitrary coupled quantities. Other efforts to improve the compatibility of advected quantities based on synchronized advection approaches have been reported, for instance, by VanderHeyden and Kashiwa (1998), Wright (2007), McGraw (2007), and Risi et al. (2010).
Given the general occurrence of sharp gradients and a highly deformational multidimensional flow field, representing tracer transport in turbulent convective clouds constitutes a particularly difficult test for advection algorithms. Studies have shown that the representation of macrophysical properties of cumulus clouds in largeeddy simulation (LES) models can be strongly influenced by the treatment of advection (e.g., Matheou et al. 2011). Based on an axisymmetric, anelastic model, Costa et al. (2000) demonstrated that changing the order of accuracy of the advection scheme can impact the width of the droplet size distribution throughout a convective cloud. To our knowledge, there has not been a comprehensive study addressing the preservation of tracer interrelationships in simulations of 3D turbulent flow associated with convective clouds. Our study is intended to address this gap. The following primary questions are addressed: 1) How does advection impact linear and nonlinear tracer interrelationships, including sums of three tracers, in LES of a turbulent convective cloud? 2) How does the order of the advection scheme impact these tracer interrelationships? Several tracers following different initial relationships are considered, mimicking potential effects of advection artifacts on cloud hydrometeor and aerosol properties. We also investigate the effects of including the scalar normalization procedure proposed by Ovtchinnikov and Easter (2009) in order to preserve the sum of three advected tracers. Thus, we extend the work of Ovtchinnikov and Easter (2009), who used a modeling framework consisting of constant onedimensional flow, to realistic 3D turbulent cloud conditions. We also assess how applying the normalization procedure impacts other linear and nonlinear tracer relationships. The metrics suggested by Lauritzen and Thuburn (2012), designed to evaluate the physical realizability of deviations from the initial tracer relationships induced by advection schemes in highly deformational, multidimensional flows, are employed here. In addition to the analysis of the metrics for instantaneous model outputs, as in Lauritzen and Thuburn (2012), we also assess their evolution with time after tracer initialization. For further quantification, we include an analysis of the probability density functions of the tracers and their sum.
2. Methods
a. Model description
Our simulations were performed using the 3D, compressible, nonhydrostatic Cloud Model 1 (CM1; Bryan and Fritsch 2002; Bryan 2017) using grid spacings (horizontal and vertical) of 50, 100, and 200 m. For temporal integration, an explicit Klemp–Wilhelmson timesplitting scheme (Klemp and Wilhelmson 1978) was employed, using a thirdorder Runge–Kutta method for the slowmode terms (Wicker and Skamarock 2002). Advection in CM1 discretized equations is represented by a fluxform term plus a divergence term [Eqs. (6) and (7) in Bryan and Fritsch 2002]. The advection of the velocity variables followed a fifthorder finitedifference spatial discretization (Wicker and Skamarock 2002) at all Runge–Kutta steps. Advection of scalars used the fifthorder scheme of Wicker and Skamarock (2002) at the first two Runge–Kutta steps and a weighted essentially nonoscillatory (WENO) scheme at the third Runge–Kutta step, as is standard in CM1.^{1} For the advected thermodynamic variables related to pressure, temperature, water vapor and hydrometeors, a seventhorder WENO scheme was employed (Balsara and Shu 2000). The order of accuracy of the WENO scheme employed for the advection of the inert passive tracers varied from third to seventh (WENO3, WENO5, and WENO7). One set of tests included the scalar normalization suggested by Ovtchinnikov and Easter (2009), which scales each individual tracer to locally preserve the sum of three tracers after advection (MRWENO3N, MRWENO5N, and MRWENO7N). A positivedefinite flux limiter (similar to Skamarock and Weisman 2009) was applied to all scalar variables, including the inert passive tracers, at the last Runge–Kutta step. These numerical tests are summarized in Table 1.
Nomenclature and characteristics of the numerical tests.
To facilitate the analysis and interpretation, inert passive tracers were initialized at t = 100 min after the cloud developed and reached 6 km altitude. Although passive tracers by definition do not feedback on the flow dynamics, differences in the order of the operations in simulations employing different WENO schemes for the tracers’ advection induced tiny changes in the results of the numerical integrations (i.e., as a result of the nonassociative property of floatingpoint operations). The rapid growth of such perturbations in turbulent medium led to some flow differences between simulations in which only the order of accuracy of the tracer advection was varied. However, since the convective updraft was already well developed at t = 100 min, changes caused by varying the order of the advection scheme for the tracers had little impact on the main characteristics of the flow in the remaining 50 min of the simulations (not shown). To improve robustness, for each configuration listed in Table 1, three different flow realizations were run by modifying the initial conditions. Note that the modifications introduced in the initial conditions were identical across all configurations in Table 1. Initial conditions were varied by applying different sets of random perturbations to the low level potential temperature and moisture fields.
Subgridscale turbulent mixing was represented in the model by solving the subgrid turbulence kinetic energy equation similar to Deardorff (1980). However, the effects of subgridscale turbulent mixing on the tracers were neglected, in order to isolate the effects of the resolved advection on them.
As initial conditions, we used vertical profiles of potential temperature and water vapor mixing ratio from an atmospheric sounding launched at 1730 UTC 11 September 2014 from Manacapuru, Brazil (Holdridge et al. 2014), as part of the Observations and Modeling of the Green Ocean Amazon (GoAmazon2014/5) Experiment (Martin et al. 2016), similar to Hernández Pardo et al. (2020) (Fig. 1 therein). Random perturbations ranging over ±0.01 K and ±0.025 g kg^{−1} were applied to the initial fields of potential temperature and moisture, respectively, below the 21st model level to help spin up 3D turbulence. As noted above, the three different flow realizations used different random number seeds for these perturbations. Initial zerowind conditions were assumed.
Cloud microphysical processes were represented with the bulk scheme of Morrison et al. (2009), predicting mass and number mixing ratios of rain drops, cloud ice, snow and hail, and mass mixing ratio of cloud droplets. A cloud droplet number concentration of 250 cm^{−3} was assumed.
b. Tracer initialization
Here we are specifically interested in how numerical artifacts implicit in advection calculations in LES of convective clouds affect relationships among tracers, particularly relevant to cloud hydrometeor and aerosol properties. To this end, we focus on the impact of advection on the sum of three tracers, and nonlinear relationships between tracers, defining two specific types of inert passive tracers: “environmental” tracers and “incloud” tracers. These tracer categories are used as a simple analogy to aerosol and cloud quantities, respectively. We denote each tracer mixing ratio as
All tracer mass mixing ratios (kg kg^{−1}) were initialized at t = 100 min. At this time, the cloud depth exceeded 6 km, exhibiting strong updrafts (up to ∼28 m s^{−1}) and a fast cloud growth rate. Our tracer configuration was inspired by the 2D tests from Lauritzen and Thuburn (2012). For each class (i.e., environmental or incloud), we defined the following:

three tracers that added up to a constant, two of them being uniform within partially overlapping areas and zero, otherwise:${\text{tr}}_{{\chi}_{1}}=\{\begin{array}{ll}1,\hfill & z\phantom{\rule{.25em}{0ex}}\le \phantom{\rule{.25em}{0ex}}5\phantom{\rule{.25em}{0ex}}\text{km}\hfill \\ 0,\hfill & z>5\phantom{\rule{.25em}{0ex}}\text{km}\hfill \end{array},$${\text{tr}}_{{\chi}_{2}}=\{\begin{array}{ll}1,\hfill & z\ge 3\phantom{\rule{.25em}{0ex}}\text{km}\hfill \\ 0,\hfill & z<3\phantom{\rule{.25em}{0ex}}\text{km}\hfill \end{array},$${\text{tr}}_{{\chi}_{3}}=3{\text{tr}}_{{\chi}_{1}}{\text{tr}}_{{\chi}_{2}},$where z is height in kilometers;

three tracers that added up to a constant, with different gradients with respect to height and a quadratic relationship between two of them:${\text{tr}}_{{\chi}_{4}}=\text{max}\{\text{min}[0.2(z2)+1,1],0\},$${\text{tr}}_{{\chi}_{5}}={({\text{tr}}_{{\chi}_{4}})}^{2},$${\text{tr}}_{{\chi}_{6}}=3{\text{tr}}_{{\chi}_{4}}{\text{tr}}_{{\chi}_{5}};$

an additional tracer (initialized inside cloud only) to illustrate the conservation of linear relationships during advection:${\text{t}{\text{r}}^{\prime}}_{{c}_{4}}=3{\text{tr}}_{{c}_{4}}.$
3. Results
In a real flow, “mixing” refers to the effect of molecular diffusion, whereby mass is transported through particle random motion establishing a net flux opposing the concentration gradient, in contrast to bulk transport, or “advection.” Hence, “turbulent mixing” refers to the combined effect of molecular diffusion (mixing), and advection in a turbulent environment (“stirring”). The time scale of turbulent mixing is typically much shorter than the time scale of the mixing associated with pure molecular diffusion. This is a consequence of the deformation of the scalar field resulting from advection in a turbulent flow, which increases the scalar’s gradients and surface area, thus accelerating mixing via molecular diffusion.
LES models resolve the largest scales within the turbulence spectrum (i.e., energycontaining eddies and largest eddies in the inertial subrange), while the effects of smallerscale turbulence (i.e., smaller eddies in the inertial subrange and in the dissipative subrange) and molecular diffusion are parameterized. Therefore, the representation of turbulent mixing in LES models constitutes a combination of resolved and unresolvedscale processes. Because the effects of subgridscale turbulent mixing (i.e., advection from unresolved smallscale turbulent eddies, plus mixing via molecular diffusion) on the tracers were not included in the model, any turbulentmixinglike effects on the tracers must have occurred from a combination of resolvedscale advection (turbulent stirring) and associated numerical artifacts. The terms “numerical mixing” here refer to both mixinglike effects and other related but unphysical effects (i.e., effects that do not resemble realistic mixing). The impacts of numerical mixing on the individual tracers and their interrelationships will be detailed in the remainder of this section.
a. Analysis of individual tracers
Figure 1 illustrates the cloudedge contour (i.e., q_{c} = 10^{−6} kg kg^{−1}) and the distribution of the tracers, at t = 100 and 105 min, for one realization of the MRWENO7 case. The irregular and asymmetric shape of the cloud evidences the existence of turbulent eddies at scales smaller than the main updraft. The turbulent flow characteristics are further illustrated by vertical velocity w energy spectra at t = 105 min, for levels between the base and the top of the cloud (i.e., 2 km ≤ z ≤ 8 km) for simulations using 50, 100, and 200m grid spacings (Fig. 2). A nearly −5/3 slope of the energy spectra at scales smaller than the scale of the updraft (∼3–4 km) suggests that the simulations were able to reproduce the upper portion of the turbulence inertial subrange, consistent with previous LES studies of deep convection (e.g., Bryan et al. 2003; Lebo and Morrison 2015). This is evident for all three grid spacings tested. As expected, the higher the model resolution was, the smaller the wavelengths the model was able to resolve within the inertial subrange. However, at wavelengths smaller than ∼6Δx, steeper slopes (i.e., < −5/3, dotted contours in Fig. 2) indicate the effects of dissipation from numerical artifacts (Bryan et al. 2003, and references therein).^{2}
Vertical velocity energy spectra in the direction of the x axis, averaged over 2 km ≤ z ≤ 8 km and 5 km ≤ y ≤ 15 km, at t = 105 min in a single flow realization of the LRWENO7, MRWENO7, and HRWENO7 cases. The dotted section of each spectrum corresponds to wavelengths smaller than 6Δx. The black dashed line represents a −5/3 slope.
Citation: Monthly Weather Review 150, 10; 10.1175/MWRD220025.1
Vertical velocity energy spectra in the direction of the x axis, averaged over 2 km ≤ z ≤ 8 km and 5 km ≤ y ≤ 15 km, at t = 105 min in a single flow realization of the LRWENO7, MRWENO7, and HRWENO7 cases. The dotted section of each spectrum corresponds to wavelengths smaller than 6Δx. The black dashed line represents a −5/3 slope.
Citation: Monthly Weather Review 150, 10; 10.1175/MWRD220025.1
Vertical velocity energy spectra in the direction of the x axis, averaged over 2 km ≤ z ≤ 8 km and 5 km ≤ y ≤ 15 km, at t = 105 min in a single flow realization of the LRWENO7, MRWENO7, and HRWENO7 cases. The dotted section of each spectrum corresponds to wavelengths smaller than 6Δx. The black dashed line represents a −5/3 slope.
Citation: Monthly Weather Review 150, 10; 10.1175/MWRD220025.1
Substantial deformation of the tracers’ fields occurred in the simulated turbulent flow. A distinctive stripe of subcloudbase values of
Analysis of probability density functions (PDFs) of the tracers demonstrates the overall impact of advection on individual tracers’ mixing ratios. Note that the evolution of the tracers’ PDFs here reflects the balance between numerical diffusion and dispersion for each test. Since numerical diffusion tends to smooth out the tracers’ fields, its direct effect is a decrease in the width of the spectrum of values of each tracer. For instance, if enough time were considered, a purely diffusive advection scheme would ultimately produce uniform tracer fields (i.e., monodisperse tracer PDFs). In contrast, by producing oscillations, numerical dispersion has the effect of broadening the tracers’ PDFs. Since the initial PDF (t = 100 min) for each tracer was the same in all tests for a given flow realization, comparing the evolution of the PDFs across different cases is key for understanding the effects of numerical diffusion and dispersion. Faster narrowing of the PDFs indicate greater numerical diffusion, while broadening and “overshooting” of PDFs above the initial maximum tracer value indicate numerical dispersion errors.
The PDF associated with
Probability density function for
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Probability density function for
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Probability density function for
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Interestingly, the response to varying the model grid spacing was nonmonotonic, with broader PDFs in the 200m (LR) compared to the 100m (MR) tests (Fig. 3d), but narrower PDFs in the 100m compared to the 50m (HR) tests (Fig. 3e). Such behavior may have been caused by differences in the flow characteristics across the various grid spacing tests, especially since no subgridscale parameterization was applied to represent the effects of mixing by unresolved eddies on the tracers. For instance, at 200m grid spacing, where the turbulent flow was only marginally resolved (Fig. 2), turbulent mixing associated with the resolvedscale flow was likely weaker than at 100 or 50m gridspacing. We speculate that a large increase in turbulence as the model grid spacing was decreased from 200 to 100 m may have been enough to compensate the impact on the tracer PDF from a fundamental decrease in numerical diffusion with decreased grid spacing (i.e., the improved ability to retain sharp gradients using a smaller grid spacing). On the other hand, from 100 to 50m grid spacing, a smaller increase in resolved scale turbulence may have been unable to compensate for decreased numerical diffusion as the grid spacing was decreased. This is consistent with changes in the resolved w kinetic energy (E_{w}) integrated over wavenumber (κ) in the inertial subrange as the model grid spacing was decreased. Theoretically, given a E_{w}(κ) ∼ κ^{−5/3} relationship, the rate of increase of
b. Relationships between tracers
As mentioned above, the transport of tracers within a real fluid occurs via molecular diffusion and advection. A property of advection is that it preserves preexisting relationships between tracers’ mixing ratios, while mixing from molecular diffusion produces deviations from initial nonlinear relationships (Thuburn and McIntyre 1997). Although there is no physically based source of mixing in models without a representation of molecular diffusion, either explicit or parameterized, some degree of artificial mixinglike effects are unavoidably introduced by advection schemes (i.e., numerical mixing). Besides introducing deviations from nonlinear tracer interrelationships that mimic real mixing, numerical mixing can also disrupt tracer relationships in unphysical ways. In this section, we assess the impact of numerical mixing on the relationships between idealized inert passive tracers in the simulations.
To facilitate understanding of what would be expected from numerical mixing, let us first consider the impact of mixing on the relationships between tracers in real flows. As a result of the mixing of two parcels via molecular diffusion, a tracer mixing ratio would tend to be homogeneously distributed in the total volume of the combined parcels, and its value would correspond to a weighted average of the original mixing ratios in the parcels. In a scatterplot where the axes correspond to the mixing ratios of different tracers with identical diffusivities undergoing molecular diffusion simultaneously (following Lauritzen and Thuburn (2012), this type of scatterplot will be hereinafter called a “mixing diagram”), mixing would manifest as a translation of points into the region delimited by the convex envelope of the initial distribution (Penney et al. 2020). This is a constraint for the effects of numerical mixing to be physically realizable (Thuburn and McIntyre 1997; Lauritzen and Thuburn 2012). As in Lauritzen and Thuburn (2012), we refer to numerical mixing events satisfying this condition as “realistic mixing.” Numerical mixing events for which the resulting points in the corresponding mixing diagram are located outside of the convex envelope are classified as “unrealistic mixing” (or “unmixing”). Points with unmixing that exceeds the initial range of values on any of the axes are referred to here as “overshooting” events, in contrast to “rangepreserving” unmixing.
1) Linear relationships between two tracers
For the case in which two tracers are initially linearly correlated, the convex envelope of the original distribution in the mixing diagram would be the shortest line segment containing all points. Thus, all points deviating from the initial linear relationship classify as unmixing. Although preexisting linear relationships can be theoretically preserved by linear and semilinear advection schemes (Lin and Rood 1996), they can be disrupted by some flux limiters, such as the positivedefinite filter considered here (Lauritzen et al. 2015). Moreover, even for algorithms that are able to preserve linear correlations under exact arithmetic, the propagation of roundoff errors during finiteprecision calculations can also disrupt such relationships.
Figure 4 illustrates the impact of the advection schemes on linearly related tracers
Mixing diagrams for
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Mixing diagrams for
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Mixing diagrams for
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According to the reasoning above, all numerical mixing events producing deviations from the relationship defined by
Figure 5 shows that, although the number of rangepreserving unmixing events (i.e., the number grid points where the relationship between the tracers classifies as rangepreserving unmixing) increased with time, the intensity of the unmixing decreased on average. Presumably this behavior was associated with the decrease of the tracers’ gradients over time from numerical diffusion. Moreover, both
(a)–(e) Rangepreserving unmixing metric for
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(a)–(e) Rangepreserving unmixing metric for
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(a)–(e) Rangepreserving unmixing metric for
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2) Nonlinear relationships between two tracers
Nonlinear relationships between tracers are expected to be altered by any advection scheme as a result of truncation errors (Thuburn and McIntyre 1997). In this case, the physical realizability of the resulting joint distribution of the tracers can be visually inspected by means of mixing diagrams for each pair of tracers. Figure 6 shows examples of the mixing diagrams resulting from simulations with the different WENO advection orders, for single realizations each of the LR, MR and HR resolution cases. This figure refers to two of the tracers that initially followed a quadratic relationship:
Mixing diagrams for
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Mixing diagrams for
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Mixing diagrams for
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Mixing diagrams for
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Mixing diagrams for
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Mixing diagrams for
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Following Lauritzen and Thuburn (2012), we calculated
Figure 8 shows
Realistic mixing, rangepreserving unmixing, and overshooting metrics for (a)–(f)
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Realistic mixing, rangepreserving unmixing, and overshooting metrics for (a)–(f)
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Realistic mixing, rangepreserving unmixing, and overshooting metrics for (a)–(f)
Citation: Monthly Weather Review 150, 10; 10.1175/MWRD220025.1
In contrast to
A plausible explanation for the differences between the trends of
Different degrees of scatter among the panels in Fig. 6 suggest that, for each grid spacing tested, the intensity of numerical mixing in the model was significantly affected by the order of accuracy of the WENO schemes. Figures 8a, 8d, 8g, and 8j show that, although the differences among ensemble averages using different advection orders were smaller than the spread of the three different realizations for a given advection order, the order of the WENO scheme was positively correlated with
3) Sum of three tracers
In section 3b(1), we analyzed the conservation of the simplest case of linear relationships, i.e., the linear relationship between two variables. In this section we assess the conservation of the sum of three variables, a multivariate linear relationship. The convex envelope of a set of points initially following a linear relationship in three dimensions is the smallest section of the plane containing all points in the original distribution. Points deviating from the corresponding plane section (defined here by
The difference between summing three tracers advected individually,
Difference between advecting the sum of the tracers’ mixing ratios and calculating their sum after advecting individual tracers, at the y = 11km cross section, t = 105 min, in a single flow realization of the MRWENO7 case. The black continuous contours represent the q_{c} = 10^{−6} kg kg^{−1} isoline.
Citation: Monthly Weather Review 150, 10; 10.1175/MWRD220025.1
Difference between advecting the sum of the tracers’ mixing ratios and calculating their sum after advecting individual tracers, at the y = 11km cross section, t = 105 min, in a single flow realization of the MRWENO7 case. The black continuous contours represent the q_{c} = 10^{−6} kg kg^{−1} isoline.
Citation: Monthly Weather Review 150, 10; 10.1175/MWRD220025.1
Difference between advecting the sum of the tracers’ mixing ratios and calculating their sum after advecting individual tracers, at the y = 11km cross section, t = 105 min, in a single flow realization of the MRWENO7 case. The black continuous contours represent the q_{c} = 10^{−6} kg kg^{−1} isoline.
Citation: Monthly Weather Review 150, 10; 10.1175/MWRD220025.1
(a)–(h) Probability density functions of
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(a)–(h) Probability density functions of
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(a)–(h) Probability density functions of
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Figure 10 confirms the comparison above based on visual inspection of Fig. 9. That is, errors in the sum were more frequent for the incloud tracers than for the environmental tracers, and slightly more frequent for
(a)–(h) The 95th percentile of
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(a)–(h) The 95th percentile of
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(a)–(h) The 95th percentile of
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c. Scalar normalization
Some errors in the sum still persisted when the normalization procedure was applied (MRN tests, Fig. 10), presumably owing to machine roundoff errors. However, the sum of the tracers was significantly improved in MRN compared to MR tests. Figures 10d and 10h show that applying the normalization in these simulations reduced the frequency of
Keep in mind that this normalization procedure preserves monotonicity of the sum of the tracers at the expense of losing monotonicity of the individual tracers (Ovtchinnikov and Easter 2009). To explore this aspect, we analyzed the effect of the normalization procedure on the relationships between the pairs of tracers discussed in sections 3b(1) and 3b(2), as well as on individual tracers’ PDFs. Figure 3b shows that the effect of the normalization on the PDF of
For the nonlinear relationships, the impact of including the normalization procedure on the numerical mixing metrics (
Overall, the impact of the normalization on the relationship between any tracer pairs depends on the initial relationship, the position of the points in the mixing diagram, the correction factor C [Eq. (10)], and if one or both tracers in the relationship are included in the sum to be preserved (i.e., which tracers are affected by the normalization). To illustrate this in more depth, let us analyze different scenarios.
First, consider the sum S = tr_{i} + tr_{j} + tr_{k}, where the terms represent individual tracers’ mixing ratios, and an additional tracer tr_{l} given by tr_{l} = f(tr_{k}) initially. This situation includes the linear relationship between
In the second scenario, consider the same sum S = tr_{i} + tr_{j} + tr_{k}, combined with an initial relationship between tr_{i} and tr_{j}, tr_{j} = g(tr_{i}). In this case, the normalization is applied to both tracers in the relationship pair. For the tracers analyzed here, the combination of the quadratic relationship between
4. Conclusions and discussion
This study sheds light on the impact of numerical advection schemes on relationships between advected quantities in LES models. We analyzed simulations of an idealized, isolated convective cloud, using weighted essentially nonoscillatory (WENO) advection schemes with different orders of accuracy as well as different grid spacings and several inert passive tracers. The main findings are as follows:

Numerical mixing associated with sharp gradients of the transported quantities in a highly deformational flow led to significant deviations of the tracers’ initial interrelationships for the nonlinearly related tracers and for the sum of three tracers. For example, absolute deviations from the initial nonlinear (quadratic) relationships reached ∼0.38 (equivalent to a ∼161% error). Five minutes after the tracers were initialized in the standard tests, 5% of the grid points inside and near the cloud (i.e., points with
${\text{tr}}_{{c}_{1}}+{\text{tr}}_{{c}_{2}}+{\text{tr}}_{{c}_{3}}>{10}^{6}$ ) had relative errors in the sum of three tracers between ∼1% and 16%, depending on the particular case and the tracer triplet considered. Absolute deviations from the initial linear relationships of up to ∼0.19 (equivalent to a ∼9% error) occurred in the standard cases (i.e., without the scalar normalization procedure). However, overall errors in the linear relationships were very small, with the average of the shortest Euclidean distance from each point to the initial relationship (for tracer pairs in the mixing diagrams) generally several orders of magnitude smaller than the tracer values themselves and deviations in the nonlinear tracer relationships. 
A considerable fraction of the deviations from the initial nonlinear relationships between pairs of tracers classified as unrealistic mixing, particularly for the environmental tracer, including rangepreserving unmixing and overshooting events. For the environmental tracers following a quadratic relationship, the number of unrealmixing events represented on average ∼13% of the total number of grid points exhibiting deviations from the initial relationship inside and near the cloud (not considering the tests that included the scalar normalization procedure).

The intensity of realistic and unrealistic (including rangepreserving unmixing and overshooting) mixing events associated with tracer advection, quantified as the average of the shortest Euclidean distance from points of tracer pairs to the initial functional relationship in the mixing diagrams, as well as the errors in the sum of three tracers, increased with the order of accuracy of the advection scheme. This suggests that the smoothing effect of lowerorder (i.e., more diffusive) schemes opposed the development of inconsistencies in the transport of interrelated tracers. Increased intensity of numerical mixing may seem counterintuitive given reduced numerical diffusion with higher advection order. However, the higherorder schemes maintained a wider range of tracer values because of reduced numerical diffusion, leading to a greater average of the shortest Euclidean distance of points to the initial relationship.

The scalar normalization suggested by Ovtchinnikov and Easter (2009) greatly reduced errors in the sum of three tracers. For instance, 5 min after the tracers were initialized, the 95th percentile of the relative error in the sum decreased by more than ∼88% between the standard seventhorder WENO test with 100m grid spacing and analogous tests applying the normalization procedure, considering grid points inside and near the cloud. When this normalization was applied, deviations from the initial nonlinear relationships remained comparable to the standard model setup without tracer normalization. Relatively larger deviations from the linear relationship were induced by the normalization, compared to the standard model setup, but they were still much smaller than the deviations occurring in the nonlinear relationships and generally very small relative to the tracer values themselves.

The above results were consistent across different model grid spacings, varying from 50 to 200 m (horizontal and vertical), despite the structural differences of the resolved flow as the grid spacing was modified. Robustness was further demonstrated by slightly modifying the initial lowlevel potential temperature and moisture fields to generate threemember ensembles for each combination of grid spacing and advection scheme tested. The main results above were consistent across the ensembles.
It is worth noting that, even if the frequency of occurrence of significant deviations from tracers’ initial relationships was low relative to the total number of grid points within and near the cloud, the subsequent misrepresentation of aerosol or cloud microphysical processes in such instances may impact the development of the cloud as a whole, especially owing to nonlinear behavior and the turbulent nature of the flow.
Deviations from nonlinear relationships introduced by the advection scheme can be considered realistic as long as they resemble the effect of molecular diffusion in real flows. In practice this means that resulting points in a mixing diagram need to be contained within the convex envelope of the initial distribution (Thuburn and McIntyre 1997; Penney et al. 2020). However, physical consistency also requires the intensity of such realistic numerical mixing to be representative of natural conditions. Misrepresenting the degree of realistic mixing can have a negative impact on simulations by inducing an erroneous representation of physical processes and properties such as those mentioned in the introduction. Since inferring the true solution for natural conditions of the flow is not straightforward, evaluating the occurrence of realistic mixing is an extremely difficult task. One option could be to use highresolution simulations as a reference, but, as discussed here, changes in the structure of the flow across different resolutions within the LES subrange should be carefully considered. Unrealistic numerical mixing (both rangepreserving unmixing and overshooting), on the other hand, should always be avoided in order to prevent unphysical results.
Ovtchinnikov and Easter (2009) tested normalization methods to preserve the tracer sum in a simple onedimensional constant flow model, and we extended this to the case of a 3D turbulent cloud. One of the main conclusions of our study is that the simple tracer normalization method can (nearly) preserve sums while not noticeably increasing deviations in nonlinear relationships compared to the standard case without normalization. While deviations in linear relationships were increased with the normalization (when one tracer in the relationship was normalized and the other was not), errors remained small relative to the nonlinear relationships and the tracer values themselves. Moreover, if the normalization factor was applied to both tracers in a linear relationship this would preserve the relationship to within machine precision. In general, these results suggest that the normalization approach may be useful for many applications with little drawback. This is especially true in cases where the sum of the tracers is a key relationship to be preserved, such as for the set of variables representing particle size distributions in bin microphysics schemes. As Ovtchinnikov and Easter (2009) pointed out, care should be taken when adjusting the loworder moments of a binned distribution, since this could affect higherorder moments that are important for processes such as collision–coalescence.
Overall, our results indicate that the higher the order of the advection scheme, the worse its performance in preserving both linear and nonlinear tracer relationships (including sums of three tracers). Hence, we emphasize that “highorder” does not necessarily imply high accuracy in terms of important physically motivated metrics that depend on interrelated advected quantities. Nonetheless, it would be useful to investigate the extent to which these issues are associated with the schemes themselves or hidden in (sometimes ad hoc) limiters such as the positivedefinite limiter employed here. On the other hand, the excessive diffusion generated by lowerorder schemes, besides smoothing out fields of individual tracers generally, may lead to undesirable behavior when considering multiple scalars. For instance, artificial mixing induced by numerical diffusion at the cloud interface may result in increased supersaturation associated with the decrease of N_{c} by dilution (e.g., Hernández Pardo et al. 2020) and oscillations in supersaturation owing to evaporation (Stevens et al. 1996; Hoffmann 2016). Therefore, the tradeoff between retaining more detail in the spatial fields and larger deviations of the tracer relationships using higherorder advection schemes should be considered carefully depending on the particular application. Ultimately, efforts in the development of advection schemes should consider the reduction of unrealistic mixing, not just overshooting but also rangepreserving unmixing, as an important goal.
Finally, the impact of advection on tracer interrelationships decreased with time overall in these tests. This was associated with the decrease in the tracers’ gradients over time from numerical diffusion combined with the absence of tracer sources or sinks. With sources/sinks, deviations in tracer relationships might be important over the full duration of the simulation, and the magnitude of the deviations might be greater than that estimated here (Lauritzen et al. 2015). Future studies on the impact of advection schemes on tracer relationships, including sources/sinks in the conservation equations to emulate realistic aerosol and cloud microphysical processes, would be useful.
Applying WENO at all Runge–Kutta steps in additional tests (not shown) had a negligible impact on the analyses presented here.
The 6Δx threshold considered in Bryan et al. (2003) corresponds to a fifthorder advection scheme. For the seventhorder advection employed here the actual threshold below which flow features are damped may be different, but investigating this issue in detail is beyond the scope of this study.
Acknowledgments.
We thank George Bryan for hosting and maintaining CM1, and for the technical support regarding the characteristics of the WENO schemes implemented in this model. We also thank Kamal Kant Chandrakar for the implementation of the cumulus congestus case study in CM1. Lianet Hernández Pardo and Mira Pöhlker appreciate financial support from the Max Planck Society (MPG). This publication includes data analysis and visualizations created with NCL (NCAR 2018). We would like to acknowledge highperformance computing support from Cheyenne (doi:10.5065/D6RX99HX) provided by NCAR’s Computational and Information Systems Laboratory. This material is based upon work supported by the National Center for Atmospheric Research, which is a major facility sponsored by the National Science Foundation under Cooperative Agreement 1852977.
Data availability statement.
We employed the CM1 model version 19.8 (https://www2.mmm.ucar.edu/people/bryan/cm1/). The atmospheric sounding employed to initialize the model was obtained from the Atmospheric Radiation Measurement (ARM) user facility (Holdridge et al. 2014).
REFERENCES
Balsara, D. S., and C.W. Shu, 2000: Monotonicity preserving weighted essentially nonoscillatory schemes with increasingly high order of accuracy. J. Comput. Phys., 160, 405–452, https://doi.org/10.1006/jcph.2000.6443.
Borges, R., M. Carmona, B. Costa, and W. S. Don, 2008: An improved weighted essentially nonoscillatory scheme for hyperbolic conservation laws. J. Comput. Phys., 227, 3191–3211, https://doi.org/10.1016/j.jcp.2007.11.038.
Bryan, G. H., 2017: The governing equations for CM1. Tech. Rep., National Center for Atmospheric Research, 24 pp.
Bryan, G. H., and J. M. Fritsch, 2002: A benchmark simulation for moist nonhydrostatic numerical models. Mon. Wea. Rev., 130, 2917–2928, https://doi.org/10.1175/15200493(2002)130<2917:ABSFMN>2.0.CO;2.
Bryan, G. H., J. C. Wyngaard, and J. M. Fritsch, 2003: Resolution requirements for the simulation of deep moist convection. Mon. Wea. Rev., 131, 2394–2416, https://doi.org/10.1175/15200493(2003)131<2394:RRFTSO>2.0.CO;2.
Costa, A. A., G. P. Almeida, and A. J. C. Sampaio, 2000: A binmicrophysics cloud model with highorder, positivedefinite advection. Atmos. Res., 55, 225–255, https://doi.org/10.1016/S01698095(00)000661.
Deardorff, J. W., 1980: Stratocumuluscapped mixed layers derived from a threedimensional model. Bound.Layer Meteor., 18, 495–527, https://doi.org/10.1007/BF00119502.
Grabowski, W. W., 1989: Numerical experiments on the dynamics of the cloud–environment interface: Small cumulus in a shearfree environment. J. Atmos. Sci., 46, 3513–3541, https://doi.org/10.1175/15200469(1989)046<3513:NEOTDO>2.0.CO;2.
Grabowski, W. W., and P. K. Smolarkiewicz, 1990: Monotone finitedifference approximations to the advectioncondensation problem. Mon. Wea. Rev., 118, 2082–2098, https://doi.org/10.1175/15200493(1990)118<2082:MFDATT>2.0.CO;2.
Grabowski, W. W., and H. Morrison, 2008: Toward the mitigation of spurious cloudedge supersaturation in cloud models. Mon. Wea. Rev., 136, 1224–1234, https://doi.org/10.1175/2007MWR2283.1.
Hernández Pardo, L., H. Morrison, L. A. T. Machado, J. Y. Harrington, and Z. J. Lebo, 2020: Drop size distribution broadening mechanisms in a bin microphysics Eulerian model. J. Atmos. Sci., 77, 3249–3273, https://doi.org/10.1175/JASD200099.1.
Hoffmann, F., 2016: The effect of spurious cloud edge supersaturations in Lagrangian cloud models: An analytical and numerical study. Mon. Wea. Rev., 144, 107–118, https://doi.org/10.1175/MWRD150234.1.
Holdridge, D., J. Kyrouac, and E. Keeler, 2014: BalloonBorne Sounding System (SONDEWNPN). Atmospheric Radiation Measurement (ARM), accessed 5 April 2019, https://doi.org/10.5439/1021460.
Jiang, G.S., and C.W. Shu, 1996: Efficient implementation of weighted ENO schemes. J. Comput. Phys., 126, 202–228, https://doi.org/10.1006/jcph.1996.0130.
Kent, J., P. A. Ullrich, and C. Jablonowski, 2014: Dynamical core model intercomparison project: Tracer transport test cases. Quart. J. Roy. Meteor. Soc., 140, 1279–1293, https://doi.org/10.1002/qj.2208.
Khairoutdinov, M., and Y. Kogan, 2000: A new cloud physics parameterization in a largeeddy simulation model of marine stratocumulus. Mon. Wea. Rev., 128, 229–243, https://doi.org/10.1175/15200493(2000)128<0229:ANCPPI>2.0.CO;2.
Klemp, J. B., and R. B. Wilhelmson, 1978: The simulation of threedimensional convective storm dynamics. J. Atmos. Sci., 35, 1070–1096, https://doi.org/10.1175/15200469(1978)035<1070:TSOTDC>2.0.CO;2.
LasherTrapp, S. G., W. A. Cooper, and A. M. Blyth, 2005: Broadening of droplet size distributions from entrainment and mixing in a cumulus cloud. Quart. J. Roy. Meteor. Soc., 131, 195–220, https://doi.org/10.1256/qj.03.199.
Lauritzen, P. H., and J. Thuburn, 2012: Evaluating advection/transport schemes using interrelated tracers, scatter plots and numerical mixing diagnostics. Quart. J. Roy. Meteor. Soc., 138, 906–918, https://doi.org/10.1002/qj.986.
Lauritzen, P. H., P. A. Ullrich, and R. D. Nair, 2011: Atmospheric transport schemes: Desirable properties and a semiLagrangian view on finitevolume discretizations. Numerical Techniques for Global Atmospheric Models, P. Lauritzen et al., Eds., Springer, 185–251.
Lauritzen, P. H., W. C. Skamarock, M. J. Prather, and M. A. Taylor, 2012: A standard test case suite for twodimensional linear transport on the sphere. Geosci. Model Dev., 5, 887–901, https://doi.org/10.5194/gmd58872012.
Lauritzen, P. H., and Coauthors, 2014: A standard test case suite for twodimensional linear transport on the sphere: Results from a collection of stateoftheart schemes. Geosci. Model Dev., 7, 105–145, https://doi.org/10.5194/gmd71052014.
Lauritzen, P. H., A. J. Conley, J.F. Lamarque, F. Vitt, and M. A. Taylor, 2015: The terminator “toy” chemistry test: A simple tool to assess errors in transport schemes. Geosci. Model Dev., 8, 1299–1313, https://doi.org/10.5194/gmd812992015.
Lebo, Z. J., and H. Morrison, 2015: Effects of horizontal and vertical grid spacing on mixing in simulated squall lines and implications for convective strength and structure. Mon. Wea. Rev., 143, 4355–4375, https://doi.org/10.1175/MWRD150154.1.
Lin, S.J., and R. B. Rood, 1996: Multidimensional fluxform semiLagrangian transport schemes. Mon. Wea. Rev., 124, 2046–2070, https://doi.org/10.1175/15200493(1996)124<2046:MFFSLT>2.0.CO;2.
Martin, S. T., and Coauthors, 2016: Introduction: Observations and modeling of the green ocean amazon (GoAmazon2014/5). Atmos. Chem. Phys., 16, 4785–4797, https://doi.org/10.5194/acp1647852016.
Matheou, G., D. Chung, L. Nuijens, B. Stevens, and J. Teixeira, 2011: On the fidelity of largeeddy simulation of shallow precipitating cumulus convection. Mon. Wea. Rev., 139, 2918–2939, https://doi.org/10.1175/2011MWR3599.1.
McGraw, R., 2007: Numerical advection of correlated tracers: Preserving particle size/composition moment sequences during transport of aerosol mixtures. J. Phys.: Conf. Ser., 78, 012045, https://doi.org/10.1088/17426596/78/1/012045.
Milbrandt, J. A., H. Morrison, D. T. Dawson, and M. Paukert, 2021: A triplemoment representation of ice in the predicted particle properties (P3) microphysics scheme. J. Atmos. Sci., 78, 439–458, https://doi.org/10.1175/JASD200084.1.
Morrison, H., G. Thompson, and V. Tatarskii, 2009: Impact of cloud microphysics on the development of trailing stratiform precipitation in a simulated squall line: Comparison of oneand twomoment schemes. Mon. Wea. Rev., 137, 991–1007, https://doi.org/10.1175/2008MWR2556.1.
Morrison, H., A. A. Jensen, J. Y. Harrington, and J. A. Milbrandt, 2016: Advection of coupled hydrometeor quantities in bulk cloud microphysics schemes. Mon. Wea. Rev., 144, 2809–2829, https://doi.org/10.1175/MWRD150368.1.
Morrison, H., M. Witte, G. H. Bryan, J. Y. Harrington, and Z. J. Lebo, 2018: Broadening of modeled cloud droplet spectra using bin microphysics in an Eulerian spatial domain. J. Atmos. Sci., 75, 4005–4030, https://doi.org/10.1175/JASD180055.1.
Moser, D. H., and S. LasherTrapp, 2017: The influence of successive thermals on entrainment and dilution in a simulated cumulus congestus. J. Atmos. Sci., 74 375–392, https://doi.org/10.1175/JASD160144.1.
NCAR, 2018: The NCAR Command Language Version 6.5.0. UCAR/NCAR/CISL/TDD, accessed 2 March 2022, https://doi.org/10.5065/D6WD3XH5.
Ovtchinnikov, M., and R. C. Easter, 2009: Nonlinear advection algorithms applied to interrelated tracers: Errors and implications for modeling aerosol–cloud interactions. Mon. Wea. Rev., 137, 632–644, https://doi.org/10.1175/2008MWR2626.1.
Passarelli, R. E., 1978: An approximate analytical model of the vapor deposition and aggregation growth of snowflakes. J. Atmos. Sci., 35, 118–124, https://doi.org/10.1175/15200469(1978)035<0118:AAAMOT>2.0.CO;2.
Paukert, M., J. Fan, P. J. Rasch, H. Morrison, J. A. Milbrandt, J. Shpund, and A. Khain, 2019: Threemoment representation of rain in a bulk microphysics model. J. Adv. Model. Earth Syst., 11, 257–277, https://doi.org/10.1029/2018MS001512.
Penney, J., Y. Morel, P. Haynes, F. Auclair, and C. Nguyen, 2020: Diapycnal mixing of passive tracers by Kelvin–Helmholtz instabilities. J. Fluid Mech., 900, A26, https://doi.org/10.1017/jfm.2020.483.
Rasch, P. J., and D. L. Williamson, 1990: Computational aspects of moisture transport in global models of the atmosphere. Quart. J. Roy. Meteor. Soc., 116, 1071–1090, https://doi.org/10.1002/qj.49711649504.
Risi, C., S. Bony, F. Vimeux, and J. Jouzel, 2010: Waterstable isotopes in the LMDZ4 general circulation model: Model evaluation for presentday and past climates and applications to climatic interpretations of tropical isotopic records. J. Geophys. Res., 115, D12118, https://doi.org/10.1029/2009JD013255.
Robertson, I., S. J. Sherwin, and J. M. R. Graham, 2004: Comparison of wall boundary conditions for numerical viscous free surface flow simulation. J. Fluids Struct., 19, 525–542, https://doi.org/10.1016/j.jfluidstructs.2004.02.007.
Rood, R. B., 1987: Numerical advection algorithms and their role in atmospheric transport and chemistry models. Rev. Geophys., 25, 71–100, https://doi.org/10.1029/RG025i001p00071.
Schär, C., and P. K. Smolarkiewicz, 1996: A synchronous and iterative fluxcorrection formalism for coupled transport equations. J. Comput. Phys., 128, 101–120, https://doi.org/10.1006/jcph.1996.0198.
Shima, S.I., and W. W. Grabowski, 2020: International Cloud Modeling Workshop 2021—Isolated cumulus congestus based on SCMS campaign: Comparison between Eulerian bin and Lagrangian particlebased microphysics. The 10th Int. Cloud Modeling Workshop, Pune, India, Indian Institute of Tropical Meteorology (IITM), https://iccp2020.tropmet.res.in/CloudModelingWorkshop2020.
Skamarock, W. C., and M. L. Weisman, 2009: The impact of positivedefinite moisture transport on NWP precipitation forecasts. Mon. Wea. Rev., 137, 488–494, https://doi.org/10.1175/2008MWR2583.1.
Stevens, B., R. L. Walko, W. R. Cotton, and G. Feingold, 1996: The spurious production of cloudedge supersaturations by Eulerian models. Mon. Wea. Rev., 124, 1034–1041, https://doi.org/10.1175/15200493(1996)124<1034:TSPOCE>2.0.CO;2.
Thuburn, J., 2008: Some conservation issues for the dynamical cores of NWP and climate models. J. Comput. Phys., 227, 3715–3730, https://doi.org/10.1016/j.jcp.2006.08.016.
Thuburn, J., and M. E. McIntyre, 1997: Numerical advection schemes, crossisentropic random walks, and correlations between chemical species. J. Geophys. Res., 102, 6775–6797, https://doi.org/10.1029/96JD03514.
Tost, H., P. Jöckel, A. Kerkweg, R. Sander, and J. Lelieveld, 2006: Technical note: A new comprehensive scavenging submodel for global atmospheric chemistry modelling. Atmos. Chem. Phys., 6, 565–574, https://doi.org/10.5194/acp65652006.
VanderHeyden, W. B., and B. A. Kashiwa, 1998: Compatible fluxes for van Leer advection. J. Comput. Phys., 146, 1–28, https://doi.org/10.1006/jcph.1998.6070.
Vehkamäki, H., M. Kulmala, I. Napari, K. E. J. Lehtinen, C. Timmreck, M. Noppel, and A. Laaksonen, 2002: An improved parameterization for sulfuric acid–water nucleation rates for tropospheric and stratospheric conditions. J. Geophys. Res., 107, 4622, https://doi.org/10.1029/2002JD002184.
Vignati, E., J. Wilson, and P. Stier, 2004: M7: An efficient sizeresolved aerosol microphysics module for largescale aerosol transport models. J. Geophys. Res., 109, D22202, https://doi.org/10.1029/2003JD004485.
Wang, H., W. C. Skamarock, and G. Feingold, 2009: Evaluation of scalar advection schemes in the Advanced Research WRF Model using largeeddy simulations of aerosol–cloud interactions. Mon. Wea. Rev., 137, 2547–2558, https://doi.org/10.1175/2009MWR2820.1.
Wicker, L. J., and W. C. Skamarock, 2002: Timesplitting methods for elastic models using forward time schemes. Mon. Wea. Rev., 130, 2088–2097, https://doi.org/10.1175/15200493(2002)130<2088:TSMFEM>2.0.CO;2.
Wright, D. L., 2007: Numerical advection of moments of the particle size distribution in Eulerian models. J. Aerosol Sci., 38, 352–369, https://doi.org/10.1016/j.jaerosci.2006.11.011.