Impact of Advection Schemes on Tracer Interrelationships in Large-Eddy Simulations of Deep Convection

Lianet Hernández Pardo aMultiphase Chemistry Department, Max Planck Institute for Chemistry, Mainz, Germany

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Hugh Morrison bMesoscale and Microscale Meteorology Laboratory, National Center for Atmospheric Research, Boulder, Colorado

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Peter H. Lauritzen cClimate and Global Dynamics Laboratory, National Center for Atmospheric Research, Boulder, Colorado

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Mira Pöhlker aMultiphase Chemistry Department, Max Planck Institute for Chemistry, Mainz, Germany
dExperimental Aerosol and Cloud Microphysics Department, Leibniz Institute for Tropospheric Research, Leipzig, Germany
eLeipzig Institute for Meteorology, Faculty of Physics and Earth Sciences, University of Leipzig, Leipzig, Germany

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Abstract

This study investigates the preservation of tracer interrelationships during advection in large-eddy simulations of an idealized deep convective cloud, which is particularly relevant to chemistry, aerosol, and cloud microphysics models. Employing the Cloud Model 1, advection is represented using third-, fifth-, and seventh-order weighted essentially non-oscillatory schemes. As a simplified analogy for cloud hydrometeors and aerosols, several inert passive tracers following linear and nonlinear relationships are initialized after the cloud reaches ∼6-km depth. Numerical mixing in the simulated turbulent convective clouds leads to significant deviations from the initial nonlinear relationships between tracers. In these simulations, a considerable fraction of the grid points where the tracers’ nonlinear relationships are altered from advection are classified as unrealistic (e.g., ∼13% for the environmental tracers on average), including errors from range-preserving unmixing and overshooting. Errors in the sum of three tracers are also relatively large, ranging between ∼1% and 16% for 5% of the grid points in and near the cloud. The magnitude of unrealistic mixing and errors in the sum of three tracers generally increase with the order of accuracy of the advection scheme. These results are consistent across model grid spacings ranging from 50 to 200 m, and across three different flow realizations for each combination of grid spacing and advection scheme tested. Tests employing a previously proposed scalar normalization procedure show substantially reduced errors in the sum of three tracers with a relatively small negative impact on other tracer relationships. This analysis, therefore, suggests efficacy of the normalization procedure when applied to turbulent three-dimensional cloud simulations.

Significance Statement

In nature, transporting several quantities through bulk motions of a fluid does not affect preexisting relationships between them. However, this is not always accomplished in numerical models of the atmosphere, because of intrinsic limitations in the transport algorithms employed. We aim to investigate how these errors behave in 3D realistic simulations of a cumulus cloud, where the turbulent flow constitutes a particular challenge. We show that relationships between quantities are significantly and frequently perturbed during bulk transport in the model. Moreover, our results suggest that increasing complexity of the bulk-transport algorithms (in a way that is conventionally employed for improving the representation of individual quantities) tends to worsen the representation of relationships between two or three quantities.

© 2022 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Hernández Pardo’s current affiliation: Institute for Atmospheric and Environmental Sciences, Goethe University Frankfurt, Frankfurt, Germany.

Corresponding author: Lianet Hernández Pardo, hernandezpardo@iau.uni-frankfurt.de

Abstract

This study investigates the preservation of tracer interrelationships during advection in large-eddy simulations of an idealized deep convective cloud, which is particularly relevant to chemistry, aerosol, and cloud microphysics models. Employing the Cloud Model 1, advection is represented using third-, fifth-, and seventh-order weighted essentially non-oscillatory schemes. As a simplified analogy for cloud hydrometeors and aerosols, several inert passive tracers following linear and nonlinear relationships are initialized after the cloud reaches ∼6-km depth. Numerical mixing in the simulated turbulent convective clouds leads to significant deviations from the initial nonlinear relationships between tracers. In these simulations, a considerable fraction of the grid points where the tracers’ nonlinear relationships are altered from advection are classified as unrealistic (e.g., ∼13% for the environmental tracers on average), including errors from range-preserving unmixing and overshooting. Errors in the sum of three tracers are also relatively large, ranging between ∼1% and 16% for 5% of the grid points in and near the cloud. The magnitude of unrealistic mixing and errors in the sum of three tracers generally increase with the order of accuracy of the advection scheme. These results are consistent across model grid spacings ranging from 50 to 200 m, and across three different flow realizations for each combination of grid spacing and advection scheme tested. Tests employing a previously proposed scalar normalization procedure show substantially reduced errors in the sum of three tracers with a relatively small negative impact on other tracer relationships. This analysis, therefore, suggests efficacy of the normalization procedure when applied to turbulent three-dimensional cloud simulations.

Significance Statement

In nature, transporting several quantities through bulk motions of a fluid does not affect preexisting relationships between them. However, this is not always accomplished in numerical models of the atmosphere, because of intrinsic limitations in the transport algorithms employed. We aim to investigate how these errors behave in 3D realistic simulations of a cumulus cloud, where the turbulent flow constitutes a particular challenge. We show that relationships between quantities are significantly and frequently perturbed during bulk transport in the model. Moreover, our results suggest that increasing complexity of the bulk-transport algorithms (in a way that is conventionally employed for improving the representation of individual quantities) tends to worsen the representation of relationships between two or three quantities.

© 2022 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Hernández Pardo’s current affiliation: Institute for Atmospheric and Environmental Sciences, Goethe University Frankfurt, Frankfurt, Germany.

Corresponding author: Lianet Hernández Pardo, hernandezpardo@iau.uni-frankfurt.de

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, low-order schemes tend to smooth the fields out substantially, while dispersive, higher-order schemes typically lead to spurious oscillations and unphysical negative values of the advected quantities. These issues are commonly addressed via the application of higher-order schemes with flux limiters (e.g., the positive-definite and shape-preserving 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 semi-linear 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, advection-induced 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 two-dimensional 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 three-dimensional 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 non-oscillatory 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: qautocqc2.47Nc1.79, and Passarelli’s (1978) bulk snow-aggregation rate: Naggqs(2+bs)/3Ns(4bs)/3; where qc, Nc, qs and Ns are the mass and number mixing ratios for cloud droplets and snow, respectively, and bs is a constant. Nonlinear relationships are a key for many aspects of aerosol microphysical modeling as well. Similar to microphysical processes involving cloud droplet, rain, and ice particles, many processes related to aerosol formation and growth are typically parameterized involving nonlinear relationships between advected variables. Some examples are: the parameterization of new particle formation from sulfuric acid and water described by Vehkamäki et al. (2002), which depends nonlinearly on the temperature and relative humidity; aerosol coagulation in Vignati et al. (2004), represented by a nonlinear function of the total number and mean size of each aerosol mode; and the nucleation- and impaction-scavenging parameterization in Tost et al. (2006), which depends on nonlinear relationships between the properties of aerosol, cloud droplet and precipitation particles. In the context of bin or sectional cloud and aerosol microphysical models, it is particularly important to conserve the sum of several variables, as the bulk properties of the particles are derived from tens or hundreds of variables representing small sections of their size distributions. For instance, Ovtchinnikov and Easter (2009) showed that the separate advection of three tracers in a one-dimensional constant-velocity flow led up to 10% error locally in their sum, and 30% when cloud-like interactions were allowed.

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 shape-preserved supersaturation field, based on advecting the absolute supersaturation (difference between the water vapor and saturation vapor mixing ratios) using a non-oscillatory 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 three-moment 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 flux-corrected 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 large-eddy 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 one-dimensional 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 time-splitting scheme (Klemp and Wilhelmson 1978) was employed, using a third-order Runge–Kutta method for the slow-mode terms (Wicker and Skamarock 2002). Advection in CM1 discretized equations is represented by a flux-form term plus a divergence term [Eqs. (6) and (7) in Bryan and Fritsch 2002]. The advection of the velocity variables followed a fifth-order finite-difference spatial discretization (Wicker and Skamarock 2002) at all Runge–Kutta steps. Advection of scalars used the fifth-order scheme of Wicker and Skamarock (2002) at the first two Runge–Kutta steps and a weighted essentially non-oscillatory (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 seventh-order 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 (MR-WENO3-N, MR-WENO5-N, and MR-WENO7-N). A positive-definite 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.

Table 1

Nomenclature and characteristics of the numerical tests.

Table 1

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 non-associative property of floating-point 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.

Subgrid-scale turbulent mixing was represented in the model by solving the subgrid turbulence kinetic energy equation similar to Deardorff (1980). However, the effects of subgrid-scale 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 zero-wind conditions were assumed.

Fig. 1.
Fig. 1.

Mixing ratios of the various tracers (as labeled in the plots) along the y = 11-km cross section, at (left) t = 100 min (i.e., tracer initialization time) and (right) t = 105 min, for a single flow realization of the MR-WENO7 case. The black continuous contours represent the qc = 10−6 kg kg−1 isoline defining the cloud boundary.

Citation: Monthly Weather Review 150, 10; 10.1175/MWR-D-22-0025.1

The simulation time was 2.5 h, with time steps of 1, 3, and 6 s for the 50-, 100-, and 200-m grid-spacing tests (HR, MR and LR in Table 1), respectively. In all cases, the horizontal domain was 21.6 km × 21.6 km, with periodic lateral boundary conditions. Semi-slip and free-slip (e.g., Robertson et al. 2004) boundary conditions were assumed at the bottom and top of the 13-km vertical domain, respectively, with a 2-km depth damping layer above 11-km height. Surface fluxes were specified following the cumulus congestus case from the 10th International Cloud Modeling Workshop (Shima and Grabowski 2020), similar to Lasher-Trapp et al. (2005) and Moser and Lasher-Trapp (2017) among other studies. During the first hour of the simulations, constant, uniform, density-normalized (i.e., divided by air density) fluxes J of sensible heat (∼0.1 J g−1 m s−1) and water vapor (4 × 10−5 g g−1 m s−1) were assumed at the surface. After 1 h, Gaussian-shaped surface fluxes were employed to force convection, given by
J=Joexp[(xxc)2+(yyc)217002],
where (x; y) is the horizontal position within the model domain (in m) and (xc; yc) represent the coordinates (m) of the center of the domain. Jo represents the maximum density-normalized surface flux at the center of the Gaussian (∼0.9 J g−1 m s−1 for the sensible heat and 3.6 × 10−4 g g−1 m s−1 for the water vapor).

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 “in-cloud” tracers. These tracer categories are used as a simple analogy to aerosol and cloud quantities, respectively. We denote each tracer mixing ratio as trχi, with χ ∈ {e, c} (i.e., environmental or in-cloud, respectively) and i ∈ {1, 2, … , 6} indicating the tracer index. A threshold cloud droplet mixing ratio qc > 10−6 kg kg−1 was used to define in-cloud points for the tracer initialization.

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 in-cloud), we defined the following:

  • three tracers that added up to a constant, two of them being uniform within partially overlapping areas and zero, otherwise:
    trχ1={1,z5km0,z>5km,
    trχ2={1,z3km0,z<3km,
    trχ3=3trχ1trχ2,
    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:
    trχ4=max{min[0.2(z2)+1,1],0},
    trχ5=(trχ4)2,
    trχ6=3trχ4trχ5;
  • an additional tracer (initialized inside cloud only) to illustrate the conservation of linear relationships during advection:
    t rc4=3trc4.

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., energy-containing eddies and largest eddies in the inertial subrange), while the effects of smaller-scale 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 unresolved-scale processes. Because the effects of subgrid-scale turbulent mixing (i.e., advection from unresolved small-scale turbulent eddies, plus mixing via molecular diffusion) on the tracers were not included in the model, any turbulent-mixing-like effects on the tracers must have occurred from a combination of resolved-scale advection (turbulent stirring) and associated numerical artifacts. The terms “numerical mixing” here refer to both mixing-like 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 cloud-edge contour (i.e., qc = 10−6 kg kg−1) and the distribution of the tracers, at t = 100 and 105 min, for one realization of the MR-WENO7 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 200-m 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

Fig. 2.
Fig. 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 LR-WENO7, MR-WENO7, and HR-WENO7 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/MWR-D-22-0025.1

Substantial deformation of the tracers’ fields occurred in the simulated turbulent flow. A distinctive stripe of subcloud-base values of tre4 and tre5 (i.e., ∼1 kg kg−1, Figs. 1c4,c5), extending up to z ∼ 5 km at x ∼ 10 km, at t = 105 min, suggests that ascent within the updraft core at lower levels just above cloud base was nearly undilute. In contrast, strong dilution of the in-cloud tracers occurring at and nearby the cloud top and lateral edges (e.g., Figs. 1b4–b6) and the mixing of diluted environmental tracers into the cloud above z > ∼4 km in Fig. 1a5 indicate the role of numerical mixing.

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 trc4 is shown in Fig. 3. Only points at which trc4>106 and trc1+trc2+trc3>106 (i.e., inside or near the cloud) are considered. In general, the PDFs varied slightly among the ensemble of different flow realizations (generated by different initial potential temperature and moisture perturbations), as indicated by the shaded areas in Figs. 3a–c. In the specific case of trc4, numerical diffusion manifested as a progressive narrowing of the PDFs toward the smallest values over time (owing to the predominance of trc4=0 kgkg1 values for the domain as a whole at the tracer initialization time). As expected, the largest values of trc4 occurred more frequently at a given simulation time as the order of the advection increased, consistent with higher-order schemes being more accurate and producing less numerical diffusion than low-order schemes (e.g., 1D sensitivity tests shown in Morrison et al. 2018). In contrast, dispersion errors evidently increased using the highest-order scheme, seen by the overshooting (trc4>1) at t = 105 min in the WENO7 tests. These trends were consistent across the various model resolutions tested as seen by comparing different panels in Fig. 3, for all the tracers employed here (not shown).

Fig. 3.
Fig. 3.

Probability density function for trc4 at different times in the simulations listed in Table 1. The lines and shaded areas represent average and minimum/maximum values, respectively, from the three-member ensembles encompassing different flow realizations. Note that t = 100 min is the tracer’s initialization time. Since the tracer fields are almost identical at t = 100 min in tests with different advection orders, for each flow realization and model grid spacing tested, only data from the WENO7 case are shown.

Citation: Monthly Weather Review 150, 10; 10.1175/MWR-D-22-0025.1

Interestingly, the response to varying the model grid spacing was non-monotonic, with broader PDFs in the 200-m (LR) compared to the 100-m (MR) tests (Fig. 3d), but narrower PDFs in the 100-m compared to the 50-m (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 subgrid-scale parameterization was applied to represent the effects of mixing by unresolved eddies on the tracers. For instance, at 200-m grid spacing, where the turbulent flow was only marginally resolved (Fig. 2), turbulent mixing associated with the resolved-scale flow was likely weaker than at 100- or 50-m grid-spacing. 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 50-m 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 (Ew) integrated over wavenumber (κ) in the inertial subrange as the model grid spacing was decreased. Theoretically, given a Ew(κ) ∼ κ−5/3 relationship, the rate of increase of abEw(κ)dκ as b is increased (corresponding to an increase in the largest resolved wavenumber as the model grid spacing is decreased) is a decreasing function of b, where a,bR and 0 < a < b. Based on the spectra in Fig. 2, the integral of Ew(κ) from the minimum wavenumber in the inertial subrange (assumed to be the value of κ for which Ew is maximum) to κ = (6Δx)−1 increased ∼31% from LR-WENO7 to MR-WENO7, whereas from MR-WENO7 to HR-WENO7 it increased only ∼11%. While interesting, further analyzing these compensating mechanisms and sensitivities to model resolution is beyond the scope of this study. Rather, hereafter we discuss the impact of varying the advection order at different resolutions mainly to evaluate the robustness of the results.

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 mixing-like 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 “range-preserving” 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 semi-linear advection schemes (Lin and Rood 1996), they can be disrupted by some flux limiters, such as the positive-definite filter considered here (Lauritzen et al. 2015). Moreover, even for algorithms that are able to preserve linear correlations under exact arithmetic, the propagation of round-off errors during finite-precision calculations can also disrupt such relationships.

Figure 4 illustrates the impact of the advection schemes on linearly related tracers trc4 and t rc4 for a single realization of each case in Table 1. For now, let us focus on the LR, MR and HR cases (i.e., first three rows in Fig. 4). Analysis of the joint PDFs shows that linear relationships were generally conserved for all advection schemes tested, although higher-order WENO schemes tended to produce small deviations from the initial relationship that were more evident in the MR-WENO7 case. The largest absolute deviation from the linear relationship, considering points with 0trc41and0t rc43 in all simulations (excluding MR-N tests), was ∼0.19, representing a ∼9% error.

Fig. 4.
Fig. 4.

Mixing diagrams for trc4vst rc4, at t = 105 min in a single flow realization of the cases listed in Table 1. Only points at which trc1+trc2+trc3>106 (i.e., inside or near the cloud) are considered. Individual points are represented by blue dots. Locations with PDF values larger than 0.2 are indicated by a color scale.

Citation: Monthly Weather Review 150, 10; 10.1175/MWR-D-22-0025.1

According to the reasoning above, all numerical mixing events producing deviations from the relationship defined by t rc4=3trc4and0trc41 are considered unmixing events. Among the unmixing occurrences, points where 0trc41and0t rc43, classify as range-preserving unmixing, and those with trc4>1ort rc4>3 classify as overshooting. We quantified deviations from the initial linear relationship based on the average of the shortest Euclidean distance from each point to the line defined by t rc4=3trc4(0trc41), for points with 0trc41and0t rc43(L¯u, quantifying range-preserving unmixing) and points with trc4>1ort rc4>3(L¯o, associated with overshooting). To reduce the impact of rounding errors related to single-precision calculations, we considered only points with deviations from t rc4=3trc4and(0trc41) larger than 10−6, for both range-preserving unmixing and overshooting. To include only points inside or near the cloud, where most transport occurred, we used a threshold of trc1+trc2+trc3>106.

Figure 5 shows that, although the number of range-preserving unmixing events (i.e., the number grid points where the relationship between the tracers classifies as range-preserving 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 L¯u and the number of range-preserving unmixing points increased with the order of the advection scheme, suggesting that the smoothing effect of numerical diffusion opposed the development of unmixing errors. This result is consistent across the various model grid spacings tested. Nonetheless, values of L¯u of order 10−4 or smaller for the LR, MR and HR cases indicate that deviations from the linear relationship were very small overall. L¯o behaved similar to L¯u in terms of the response to different advection orders, but decreased faster with time so that no overshooting occurred after t = 105 min in any of the simulations (not shown).

Fig. 5.
Fig. 5.

(a)–(e) Range-preserving unmixing metric for trc4vst rc4 as a function of time in the simulations listed in Table 1. The lines and shaded areas represent average and minimum/maximum values, respectively, from the three-member ensembles encompassing different flow realizations. The color scale in (a)–(d) represents the number of points undergoing range-preserving unmixing. The number of points in LR and HR cases was multiplied by 8 and 8−1, respectively, to account for differences in the total number of points within the model domain between these cases and the MR cases.

Citation: Monthly Weather Review 150, 10; 10.1175/MWR-D-22-0025.1

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: trc4 versus trc5. The mixing diagrams for tre4 versus tre5 and trc4 versus trc5, for different times in a single realization of the MR-WENO7 case, are shown in Fig. 7. As Thuburn and McIntyre (1997) and Lauritzen and Thuburn (2012) pointed out (see Fig. 1 in the latter), deviations from the initial relationship between these tracers (i.e., the quadratic curve with 0trχ41) resemble realistic mixing if contained within the “hull” delimited by the trχ5=trχ42andtrχ5=trχ4 curves, with 0trχ41. This constitutes the convex envelope of the initial distribution. Note that, since the initial relationship does not change curvature in the interval 0trχ41, the line trχ5=trχ4(for0trχ41) represents the uttermost relationship between these tracers, arising from numerical mixing, that resembles real mixing. That is, it contains linear combinations of the extreme values of the initial set of points [see section 3b(1)]. Numerical mixing producing points outside of the hull classify as unmixing (range-preserving unmixing if 0trχ41and0trχ51, or overshooting if trχ4>1ortrχ5>1). Considering all of the simulations (except those applying the scalar normalization of Ovtchinnikov and Easter 2009), the maximum absolute deviation from the initial nonlinear relationships was ∼0.38 (corresponding to a ∼161% error).

Fig. 6.
Fig. 6.

Mixing diagrams for trc4vstrc5, at t = 105 min in single flow realizations of the cases listed in Table 1. Only points at which trc1+trc2+trc3>106 (i.e., inside or near the cloud) are considered. Individual points are represented by blue dots. Locations with PDF values larger than 0.4 are indicated by a color scale. The black lines indicate the region corresponding to realistic mixing.

Citation: Monthly Weather Review 150, 10; 10.1175/MWR-D-22-0025.1

Fig. 7.
Fig. 7.

Mixing diagrams for tre4vstre5andtrc4vstrc5, at different times in a single flow realization of the MR-WENO7 case. Only points at which trc1+trc2+trc3>106 (i.e., inside or near the cloud) are considered. Individual points are represented by blue dots. Locations with PDF values larger than 0.4 are indicated by a color scale. The black lines indicate the region corresponding to realistic mixing.

Citation: Monthly Weather Review 150, 10; 10.1175/MWR-D-22-0025.1

Following Lauritzen and Thuburn (2012), we calculated L¯r,L¯u, and L¯o as the average of the shortest Euclidean distance from each point to the curve defined by the initial quadratic relation trχ5=trχ42and0trχ41, separately for each region of the mixing diagrams (i.e., realistic mixing, range-preserving unmixing, and overshooting, respectively). Similar to the analysis of the linear tracer relationship in section 3b(1), only points with deviations from the trχ5=trχ42 curve (for 0trχ41) larger than 10−6 were considered. The same margin of error was allowed when classifying points into the realistic mixing, range-preserving unmixing and overshooting categories. We used a threshold of trc1+trc2+trc3>106 to focus on points inside or near the cloud. We emphasize that L¯r,L¯u, and L¯o are not unique measures of the intensity of realistic mixing, range-preserving unmixing and overshooting occurring in the simulations. Since they constitute averages over variable-size samples (i.e., the number of realistic mixing, range-preserving unmixing and overshooting events vary across simulations, even across tests with the same grid spacing), these metrics are supplemented by considering the number of occurrences of each type of event as well. On average, the number of range-preserving unmixing and overshooting events represented ∼13% and ∼1% of the occurrences of deviations from the quadratic relationship for the environmental tracers and the in-cloud tracers, respectively, considering all the simulations (except those applying the scalar normalization of Ovtchinnikov and Easter 2009).

Figure 8 shows L¯r,L¯u,andL¯o as a function of time as well as the number of occurrences of realistic mixing, range-preserving unmixing, and overshooting for the tracer pairs trχ4trχ5, for each advection order tested. Similar to the linear tracer relationship [section 3b(1)], while the number of occurrences of range-preserving unmixing increased, L¯uandL¯o decreased with time overall. Again, this appears to be associated with the decrease of the tracers’ gradients with time owing to numerical diffusion. Irregularities in the time series of L¯o for tre4tre5 (Figs. 8c,f) are likely associated with a lower confidence of the mean compared to L¯u, since there were many fewer points where overshooting occurred.

Fig. 8.
Fig. 8.

Realistic mixing, range-preserving unmixing, and overshooting metrics for (a)–(f) tre4vstre5 and (g)–(l) trc4vstrc5 as a function of time in the simulations listed in Table 1. The lines and shaded areas represent average and minimum/maximum values, respectively, from three-member ensembles encompassing different flow realizations. The color scale in (a)–(c) and (g)–(i) represents the number of points undergoing realistic mixing, range-preserving unmixing, or overshooting.

Citation: Monthly Weather Review 150, 10; 10.1175/MWR-D-22-0025.1

In contrast to L¯u,L¯r first increased, reached its maximum sometime after the tracer initialization, and then decreased monotonically toward the end of the simulations. This may be explained by changes in the range of values across which numerical mixing occurred over time. For example, Fig. 7 shows that right after tracer initialization the increasing number of realistic mixing events progressively filled out the hull, increasing L¯r. However, as the points in the mixing diagrams concentrated toward the left corner of the hull over time – that is, as tracer values decreased overall from numerical diffusion – the average distance to the quadratic curve began to decrease. This phase of decreasing-L¯r started earlier for the in-cloud tracers trc4trc5, which were diluted faster than the environmental tracers tre4tre5 (110 min versus 125–130 min).

A plausible explanation for the differences between the trends of L¯r and L¯u is that, besides depending on the tracers’ gradients and ranges of values, the occurrence of unmixing is also determined by the curvature of the tracer relationship. Points undergoing realistic mixing move closer to a linear relationship after which intense unmixing events are less likely to occur later on [see section 3b(1)]. Therefore, the occurrence of realistic mixing may itself limit the subsequent occurrence of unmixing, favoring a decrease of L¯u and L¯o with time in these tests.

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 L¯r overall, for each grid spacing tested. This suggests the role of numerical diffusion in reducing the range of tracer values on which the advection schemes operate, which in turn reduced L¯r. Thus, the higher-order advection schemes that produced less numerical diffusion generated greater L¯r values. L¯u for tre4tre5 also increased as the advection order increased, across all the model resolutions tested (Figs. 8b,e). In contrast, however, L¯u for trc4trc5 was negatively correlated with the advection scheme order (Figs. 8h,k). This may be related to the relatively small number of range-preserving unmixing points for trc4trc5, up to two orders of magnitude less than that for tre4tre5, indicating a lower confidence of the average for the latter. Moreover, the number of range-preserving unmixing points for trc4trc5 differed with the advection order; it increased by about a factor of 2 between the WENO3 and WENO7 cases (Fig. 8h). This is in contrast with the number of points undergoing range-preserving unmixing for tre4tre5 which was similar across the simulations at a given time (Fig. 8b). Large differences are also seen for the number of points classified as overshooting among the cases. Overall, caution should be exercised when directly comparing metrics derived from a significantly different number of points.

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 trχi+trχi+1+trχi+2, i ∈ {1, 4}, χ ∈ {e, c}, and the initial ranges of values of each tracer) classify as unmixing. Note that the definition of linear and semi-linear advection schemes is based on the conservation of linear relationships between only two variables. Despite constituting a linear function according to the general mathematical definition, conservation of the sum of three or more variables is not implied by linear or semi-linear schemes (Thuburn and McIntyre 1997). This is especially relevant for models with bin-microphysics and/or a sectional treatment of aerosols, where tens to hundreds of variables constituting a discretized particle size distribution are advected individually.

The difference between summing three tracers advected individually, F(trχi)+F(trχi+1)+F(trχi+2), versus first calculating the sum at the time the tracers were initialized and then advecting the sum, F(trχi+trχi+1+trχi+2), for i ∈ {1, 4} and χ ∈ {e, c}, at t = 105 min, is illustrated in Fig. 9. Errors in the sum of the advected tracers relative to the advected sum (hereinafter called “errors in the sum”) were more frequent inside the cloud, where resolved-scale turbulent flow produced strong gradients of the tracers. In addition, errors in the sum were also more frequent for the in-cloud tracers (trc1,trc2,trc3, and trc4,trc5,trc6) compared to the environmental tracers (tre1,tre2,tre3, and tre4,tre5,tre6), consistent with the existence of stronger gradients of the former in the upper half of the cloud (Fig. 1), where subcloud-scale turbulent-like eddies predominated. Likewise, the occurrence of sharper gradients at the edges of the cloud for tre2 compared to tre5 (Figs. 1a5,c5) likely led to more frequent errors in the sum for tre1,tre2,tre3 compared to that for tre4,tre5,tre6.

Fig. 9.
Fig. 9.

Difference between advecting the sum of the tracers’ mixing ratios and calculating their sum after advecting individual tracers, at the y = 11-km cross section, t = 105 min, in a single flow realization of the MR-WENO7 case. The black continuous contours represent the qc = 10−6 kg kg−1 isoline.

Citation: Monthly Weather Review 150, 10; 10.1175/MWR-D-22-0025.1

Figure 10 shows PDFs of errors in the sum, normalized by the advected sum of the tracers’ mixing ratios,
ΔSχi​​F(trχi+trχi+1+trχi+2)[F(trχi)+F(trχi+1)+F(trχi+2)]F(trχi+trχi+1+trχi+2),
for i ∈ {1, 4} and χ ∈ {e, c}. Only points for which F(trχi+trχi+1+trχi+2)>106 were considered. An additional threshold of F(trc1+trc2+trc3)>106 was considered, to restrict the analysis to the cloudy points and immediately adjacent areas.
Fig. 10.
Fig. 10.

(a)–(h) Probability density functions of ΔSχi, i ∈ {1, 4} and χ ∈ {e, c}, at t = 105 min in the simulations listed in Table 1. The lines and shaded areas represent average and minimum/maximum values, respectively, from the three-member ensembles encompassing different flow realizations [minimum/maximum ranges are not shown in (b), (e), (f), and (h) for clarity].

Citation: Monthly Weather Review 150, 10; 10.1175/MWR-D-22-0025.1

Figure 10 confirms the comparison above based on visual inspection of Fig. 9. That is, errors in the sum were more frequent for the in-cloud tracers than for the environmental tracers, and slightly more frequent for tre1,tre2,tre3 than for tre4,tre5,tre6. For each model grid spacing tested, increasing the order of accuracy of the WENO scheme exacerbated the frequency of occurrence of relatively large |ΔSχi| values. To further illustrate this result, we calculated the 95th percentile of |ΔSχi| as a function of time in each simulation. Figure 11 shows a positive correlation between the 95th percentile of |ΔSχi| and the order of accuracy of WENO at all times in the simulations, and all grid spacings tested. Relative errors in the sum ranged between ∼1% and ∼16% for 5% of the grid points inside and near the cloud at t = 105 min. Nevertheless, these errors tended to decrease with time, likely associated with the decrease of the tracers’ gradients through numerical diffusion.

Fig. 11.
Fig. 11.

(a)–(h) The 95th percentile of |ΔSχi|, i ∈ {1, 4} and χ ∈ {e, c}, as a function of time for the simulations listed in Table 1. The lines and shaded areas represent average and minimum/maximum values, respectively, from the three-member ensembles encompassing different flow realizations [minimum/maximum ranges are not shown in (b), (e), (f), and (h) for clarity].

Citation: Monthly Weather Review 150, 10; 10.1175/MWR-D-22-0025.1

c. Scalar normalization

We tested one of the normalization procedures suggested by Ovtchinnikov and Easter (2009), to force conservation of the tracer sum in the model at each time step, in the 100-m grid spacing simulations. Note that Ovtchinnikov and Easter (2009) also proposed a more complicated flux adjustment technique that preserves the sum while maintaining monotonicity of individual tracers (not tested here), but it increases numerical diffusion and is computationally more expensive. In the simpler approach tested here, to adjust the sum of the tracers after advection, at each grid point, the individual tracers’ mixing ratios were corrected according to their relative contribution to the total tracer sum:
(trχk)adj=trχkC,whereC=F(trχi+trχi+1+trχi+2)*F(trχi)+F(trχi+1)+F(trχi+2),
χ ∈ {e, c}, i ∈ {1, 4}, and k ∈ {i, i + 1, i + 2}. The asterisk (*) in the numerator is used to indicate that in this case the sum of the tracers is calculated immediately before advection at each time step. Note that this normalization does not affect conservation of individual tracer mixing ratios across the domain, provided they are conserved during advection calculations, to within machine precision.

Some errors in the sum still persisted when the normalization procedure was applied (MR-N tests, Fig. 10), presumably owing to machine round-off errors. However, the sum of the tracers was significantly improved in MR-N compared to MR tests. Figures 10d and 10h show that applying the normalization in these simulations reduced the frequency of |ΔSc1|=|ΔSc4|=1 by approximately one order of magnitude compared to the standard model configuration without tracer normalization. Greater improvements were obtained for the environmental tracers. For instance, the 95th percentiles of |ΔSc1|and|ΔSc4| decreased by ∼88% and ∼92%, respectively, from the MR-WENO7 to MR-WENO7-N tests, at t = 105 min (Figs. 11d,h). In turn, the 95th percentiles of |ΔSe1|and|ΔSe4| decreased by ∼95% at t = 105 min between the MR-WENO7 and MR-WENO7-N tests (Figs. 11b,f).

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 trc4 was negligible, only barely noticeable for the largest values of trc4 at t = 105 min in the WENO7 case. Minor impact of the normalization on individual tracers’ PDFs is likely associated with the relatively low frequency of occurrence of the largest errors in the sum, compared to the total number of grid points considered (i.e., points inside and near the cloud), However, deviations from the linear relationship between trc4andt rc4 were much more evident in the MR-N tests (Figs. 4j,k,l) compared to the standard simulations, leading to an order of magnitude increase in L¯u (Figs. 5b,d,e). This increase in L¯u occurred for all advection orders tested, though the relative increase was greater for the lower order schemes (WENO5 and particularly WENO3). At any rate, when the normalization procedure was applied, values of L¯u for the linear relationship tested here were still much smaller than the values of L¯u for the nonlinear relationships in the standard tests (Fig. 8). Moreover, deviations from the linear relationship remained generally very small relative to the tracer values themselves in the mixing diagrams (Fig. 4).

For the nonlinear relationships, the impact of including the normalization procedure on the numerical mixing metrics (L¯r,L¯u,andL¯o) was rather minor (Fig. 8). Although the normalization procedure may have affected the position of individual points in the mixing diagrams corresponding to the nonlinear relationships analyzed here, the net result was basically a redistribution of the realistic mixing, range-preserving unmixing, and overshooting events, but without expanding the range over which the points were spread. This is consistent with the tracers initially following a quadratic relationship (i.e., trχ4andtrχ5) also being two of the three tracers whose sum is evaluated. Deviations from the nonlinear relationships between trχ4andtrχ5 likely imply errors in the sum of trχ4,trχ5,andtrχ6 as well. Applying a correction factor to preserve the latter would generally be expected to modulate the former, though evidently not in a way that leads to substantial changes in the mixing diagrams for trχ4andtrχ5 here.

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 = tri + trj + trk, where the terms represent individual tracers’ mixing ratios, and an additional tracer trl given by trl = f(trk) initially. This situation includes the linear relationship between trc4andt rc5 [Eq. (8)] together with the sum trc4+trc5+trc6 analyzed here. In this case, the normalization would only affect one of the tracers in the relationship pair (i.e., trk), and the result would be a translation of each point in the trk-vs-trl mixing diagram along the trk axis. Whether the translation is up or down the trk axis would depend on if C > 1 or C < 1 at that particular point. For linear and semi-linear advection schemes, any linear tracer relationship f will be preserved during the advection step (to within machine precision). Thus, applying the normalization to trk would tend to move some points away from f in the mixing diagram. This is consistent with an order magnitude increase in mean Lu for the linear relationship between trc4andt rc4 from applying normalization to trc4 [as part of the {trχ4,trχ5,trχ6} triplet]. However, as we noted, errors in the linear relationship in simulations applying normalization remained small relative to the tracer values and deviations in the nonlinear tracer relationships here.

In the second scenario, consider the same sum S = tri + trj + trk, combined with an initial relationship between tri and trj, trj = g(tri). 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 trχ4andtrχ5 [Eq. (6)] and the sum trχ4+trχ5+trχ6 fits into this situation. Normalization has no impact if g is a linear relationship, which is easily understood by noting that in this situation the normalization scales both tri and trj by C at each point. If g is nonlinear, then numerical mixing will cause deviations from the initial relationship during the advection step, leading to a “filling out” of the hull in the tri-vs-trj mixing diagram as detailed in section 3b(2). Applying normalization in this case would move each point in the mixing diagram diagonally along a straight line intersecting that point and the origin. For the nonlinear quadratic relationship between trχ4andtrχ5 analyzed here, the distances of most points to the hull edge along these lines, relative to the tracer values, are much larger than C. This implies that most of these points stay within the hull after normalization and hence remain within the region of realistic mixing. Furthermore, the spreading of points into the hull during advection would likely lead to an overestimation of the sum (because for points in the hull trip+trjp>trip+trip2), hence C < 1 for the normalization, bringing the points toward the origin but still inside the hull. This reasoning is consistent with the small effect of the normalization on the mixing metrics for the quadratic relationship discussed earlier in this section.

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 non-oscillatory (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 trc1+trc2+trc3>106) 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 range-preserving unmixing and overshooting events. For the environmental tracers following a quadratic relationship, the number of unreal-mixing 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 range-preserving 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 lower-order (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 higher-order 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 seventh-order WENO test with 100-m 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 low-level potential temperature and moisture fields to generate three-member 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 high-resolution 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 range-preserving 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 one-dimensional 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 low-order moments of a binned distribution, since this could affect higher-order 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 “high-order” 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 positive-definite limiter employed here. On the other hand, the excessive diffusion generated by lower-order 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 Nc by dilution (e.g., Hernández Pardo et al. 2020) and oscillations in supersaturation owing to evaporation (Stevens et al. 1996; Hoffmann 2016). Therefore, the trade-off between retaining more detail in the spatial fields and larger deviations of the tracer relationships using higher-order 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 range-preserving 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.

1

Applying WENO at all Runge–Kutta steps in additional tests (not shown) had a negligible impact on the analyses presented here.

2

The 6Δx threshold considered in Bryan et al. (2003) corresponds to a fifth-order advection scheme. For the seventh-order 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 high-performance 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).

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Save
  • Balsara, D. S., and C.-W. Shu, 2000: Monotonicity preserving weighted essentially non-oscillatory schemes with increasingly high order of accuracy. J. Comput. Phys., 160, 405452, https://doi.org/10.1006/jcph.2000.6443.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Borges, R., M. Carmona, B. Costa, and W. S. Don, 2008: An improved weighted essentially non-oscillatory scheme for hyperbolic conservation laws. J. Comput. Phys., 227, 31913211, https://doi.org/10.1016/j.jcp.2007.11.038.

    • Search Google Scholar
    • Export Citation
  • 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, 29172928, https://doi.org/10.1175/1520-0493(2002)130<2917:ABSFMN>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Bryan, G. H., J. C. Wyngaard, and J. M. Fritsch, 2003: Resolution requirements for the simulation of deep moist convection. Mon. Wea. Rev., 131, 23942416, https://doi.org/10.1175/1520-0493(2003)131<2394:RRFTSO>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Costa, A. A., G. P. Almeida, and A. J. C. Sampaio, 2000: A bin-microphysics cloud model with high-order, positive-definite advection. Atmos. Res., 55, 225255, https://doi.org/10.1016/S0169-8095(00)00066-1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Deardorff, J. W., 1980: Stratocumulus-capped mixed layers derived from a three-dimensional model. Bound.-Layer Meteor., 18, 495527, https://doi.org/10.1007/BF00119502.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Grabowski, W. W., 1989: Numerical experiments on the dynamics of the cloud–environment interface: Small cumulus in a shear-free environment. J. Atmos. Sci., 46, 35133541, https://doi.org/10.1175/1520-0469(1989)046<3513:NEOTDO>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Grabowski, W. W., and P. K. Smolarkiewicz, 1990: Monotone finite-difference approximations to the advection-condensation problem. Mon. Wea. Rev., 118, 20822098, https://doi.org/10.1175/1520-0493(1990)118<2082:MFDATT>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Grabowski, W. W., and H. Morrison, 2008: Toward the mitigation of spurious cloud-edge supersaturation in cloud models. Mon. Wea. Rev., 136, 12241234, https://doi.org/10.1175/2007MWR2283.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • 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, 32493273, https://doi.org/10.1175/JAS-D-20-0099.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Hoffmann, F., 2016: The effect of spurious cloud edge supersaturations in Lagrangian cloud models: An analytical and numerical study. Mon. Wea. Rev., 144, 107118, https://doi.org/10.1175/MWR-D-15-0234.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Holdridge, D., J. Kyrouac, and E. Keeler, 2014: Balloon-Borne Sounding System (SONDEWNPN). Atmospheric Radiation Measurement (ARM), accessed 5 April 2019, https://doi.org/10.5439/1021460.

    • Crossref
    • Export Citation
  • Jiang, G.-S., and C.-W. Shu, 1996: Efficient implementation of weighted ENO schemes. J. Comput. Phys., 126, 202228, https://doi.org/10.1006/jcph.1996.0130.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Kent, J., P. A. Ullrich, and C. Jablonowski, 2014: Dynamical core model intercomparison project: Tracer transport test cases. Quart. J. Roy. Meteor. Soc., 140, 12791293, https://doi.org/10.1002/qj.2208.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Khairoutdinov, M., and Y. Kogan, 2000: A new cloud physics parameterization in a large-eddy simulation model of marine stratocumulus. Mon. Wea. Rev., 128, 229243, https://doi.org/10.1175/1520-0493(2000)128<0229:ANCPPI>2.0.CO;2.

    • Crossref
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