## 1. Introduction

Numerical model simulations of deep, moist convection are used for many purposes. One is to explore possible structures of convective clouds and mesoscale convective complexes. Given the difficulties and expense of gathering detailed observations within deep, precipitating convection, the numerical simulation is often considered an attractive method for testing hypotheses concerning the dynamics of clouds and storms. However, numerical models are ultimately based on approximations. Therefore, the model user must be aware of the limitations of numerical model simulations and must be able to identify features in model output that may not be physically realistic.

Recently, Takemi and Rotunno (2003, hereafter referred to as TR03) presented results from idealized numerical model simulations of squall lines in an environment without vertical wind shear. They documented vertical velocity fields that they note have an obviously unphysical organization. For example, they showed a regular pattern of convective cells at 3 km above ground. The updrafts and downdrafts were of approximately equal amplitude, were about three grid lengths wide, and were repeated for several wavelengths in both the along-line and cross-line directions.

Using a similar numerical model, this condition was found to be easily reproducible (Fig. 1). The numerical model, initial conditions, and configuration of this simulation will be detailed in section 2 of this paper. For the purpose of this introduction, it is sufficient to note that this simulation was conducted with a setup nearly identical to those of other idealized modeling studies, such as Rotunno et al. (1988) and Weisman (1993). That is, a squall line was triggered with a line thermal in a horizontally homogeneous environment using 1-km horizontal grid spacing.

At least two characteristics of this vertical velocity pattern make it unphysical. First, the cells are only two–three grid lengths wide. Such a small updraft size is unusual; rather, a minimum cell size of six–eight grid lengths is typically expected. The second artificial aspect is the regular, repeating pattern, which seems unrealistic. Such organization is not expected in a no-shear environment; rather, the pattern is expected to be more stochastic, with cell positions being more random in space across the line, and with cell structures being in various states of development along the line (TR03; Weisman and Rotunno 2004).

TR03 note that this spurious updraft pattern is paradoxical, given the high-order numerics inherent in their numerical model, a version of the Weather Research and Forecasting (WRF) Model. The formulation of the WRF Model allows it to be used without explicit artificial diffusive terms. Instead, WRF Model simulations typically use a small amount of flow-dependent implicit diffusion that acts primarily on small scales (e.g., Wicker and Skamarock 2002; TR03). Consequently, results from the WRF Model were expected to contain less numerical errors than results from older codes used in some previous studies.

After further analysis, TR03 conclude that most previous modeling studies had a greater amount of explicit artificial diffusion that probably prevented the appearance of such spurious vertical velocity patterns. This conclusion seems reasonable, given the need for artificial diffusion terms that prevent numerical errors from contaminating the solution in leapfrog-in-time models with (relatively) low-order numerical schemes, such as the Klemp–Wilhelmson (1978) model, the fifth-generation Pennsylvania State University–National Center for Atmospheric Research (Penn State–NCAR) Mesoscale Model (MM5; Dudhia 1993), and the Advanced Regional Prediction System (ARPS; Xue et al. 2000).

TR03 addressed the problem through the model’s subgrid turbulence scheme. By modifying certain tunable parameters within the turbulence scheme, they were able to obtain more reasonable looking results (e.g., Figs. 4 and 5 of TR03). In conclusion, they suggested optimum values for these parameters, which they argue are “appropriate for mesoscale cloud simulations” (i.e., simulations with horizontal grid spacing of order 1 km).

It is clear, given the simulations presented by TR03, that increasing the diffusiveness of a numerical model will damp the spurious updraft pattern, and that the results with more diffusion are better than the original results. It is also reasonable to argue that the proposed method of increasing diffusion—that is, via the subgrid turbulence scheme—is preferable to adding an artificial diffusion term, which acts to diffuse every small-scale perturbation throughout the domain, regardless of local environmental conditions (such as Richardson number). Thus, the most important conclusions of TR03 are not addressed by this new study.

However, it seems that the ultimate cause of the spurious updraft pattern was never established by TR03. It is possible that the cause of the spurious organization still remains in their simulations, but that the effects (i.e., spurious updraft patterns) are merely damped by the additional diffusivity of the model. A proper identification of the cause of the problem may lead to a better solution to the problem, which would then allow one to determine appropriate settings for other model components (such as values for turbulence parameters).

This paper is intended as a follow-up to the study of TR03. New insight is added based on simulations not conducted by TR03. Additionally, an explanation is offered concerning the cause of the spurious updraft pattern and concerning the methods one can use to prevent it from becoming a problem.

Ultimately, this new study suffers from the same limitation as that of TR03: the reliance on qualitative assessments. The absence of a “true” or “correct” solution prevents a conclusive decision on what the best result should be. However, it is hoped that the information provided herein will advance the discussion on model configurations suitable for the simulation of convection in the idealized framework used here.

## 2. Methodology

The nonhydrostatic numerical model described in Bryan and Fritsch (2002) is used in this study. This model is based on the third-order Runge–Kutta time-integration scheme for compressible equations (Wicker and Skamarock 2002). Consequently, the model used here is broadly similar to the height-based vertical coordinate version of the WRF Model used by TR03.

This study uses a “standard” equation set [i.e., equation set “A” of Bryan and Fritsch (2002)], mainly to facilitate comparison with other commonly used numerical models. In this formulation, the thermodynamic and pressure equations were developed assuming that the specific heat of liquid water is zero, and that the diabatic contribution to the pressure equation can be neglected. Thus, the model’s governing equations are essentially the same form as those in the Klemp–Wilhelmson model, MM5, and ARPS, among others. Unlike the WRF Model used by TR03, this formulation does not conserve total dry air mass during the simulation.

To be consistent with TR03, the horizontal advection is fifth order and the vertical advection is third order, based on the formulation provided by Wicker and Skamarock (2002). Also, the subgrid turbulence scheme is nearly identical to that used by TR03 and is based on the turbulence kinetic energy scheme of Deardorff (1980). Unless specified otherwise, the value used for the parameter *C _{k}* is 0.10. In all simulations, the value for

*C*is 0.93. (The turbulence parameters

_{e}*C*and

_{k}*C*are defined in TR03.) As in TR03, the stability dependence on these parameters is removed by specifying a constant length scale,

_{e}*l*= (Δ

*x*Δ

*y*Δ

*z*)

^{1/3}. These choices for turbulence parameters provided a minimal amount of diffusion from the subgrid turbulence closure, according to the results of TR03; they are chosen for the control simulations herein because they have emerged over the years as standard (i.e., commonly used) values.

The domain is 400 km long in the across-line (i.e., *x*) direction, 80 km long in the along-line (i.e., *y*) direction, and is 18 km deep. The domain is longer than that used by TR03 to accommodate a set of simulations in which the domain is not translated. Specifically, the increased length is necessary to keep the convective system inside the domain for all simulations.

In all other aspects, the simulations are configured as in TR03. Open lateral boundary conditions are used in the *x* direction, and periodic boundary conditions are used in the *y* direction. A Rayleigh damping layer is applied over the uppermost 6 km of the domain to minimize wave reflections from the upper boundary. The Kessler (1969) liquid-only microphysics scheme is used, the Coriolis parameter is zero, no atmospheric radiation scheme is used, and no surface fluxes or surface friction are applied. The horizontal grid spacing is 1 km, the vertical grid spacing is 500 m, the large time step is 6 s, and the small time step is 1.5 s. The simulations extend 4 h in time.

The initial thermodynamic conditions are horizontally homogeneous, based on the analytic sounding of Weisman and Klemp (1982) (Fig. 2a). A squall line is initiated by using a line thermal that has infinite size in the *y* direction, a 10-km radius in the *x* direction, a 1.5-km radius in the *z* direction, and a maximum potential temperature perturbation of 1.5 K at its center. Random potential temperature perturbations of maximum amplitude 0.1 K are inserted throughout the line thermal to allow three-dimensional structure to develop.

Four environmental wind profiles are studied, where only the low-level shear oriented perpendicular to the squall line is varied (Fig. 2b). The environmental conditions will be referred to by the magnitude of wind variation over the lowest 2.5 km (*U _{s}*), as indicated at the top of Fig. 2b. For the initial (i.e., “control”) simulations, a mean wind speed is subtracted from the initial wind profile to keep the squall line roughly in the middle of the domain during the entire simulation. In a second series of simulations, referred to as “shifted wind” simulations, the mean wind was chosen so that the individual convective cells remained nearly stationary relative to the grid. These effective domain translation speeds are listed in Table 1.

Sensitivity tests were conducted on the domain size and vertical resolution, to ensure that all conclusions drawn herein are insensitive to these choices. These tests were conducted for *U _{s}* = 0 and 20 m s

^{−1}and for both the control and shifted-wind cases. Simulations with a domain length of 1200 km confirm that the lateral boundaries are far enough away from the convection for all simulations, even with the 400-km length used herein. A second set of simulations with a domain depth of 36 km and a Rayleigh damper over the uppermost 24 km confirms that the Rayleigh damping layer is sufficient for the present work. Finally, a third set of simulations used vertical grid spacing of 62.5 m, and confirmed that the results herein are not sensitive to the vertical resolution. In summary, the qualitative and quantitative conclusions drawn about the spurious updraft pattern do not change when using these larger domain dimensions or higher vertical resolution.

As in TR03, vertical velocity spectra are computed from the model output to quantify details of the spurious updraft pattern. The spectra are computed over an 80 km × 80 km subdomain extending roughly 20 km ahead of the surface gust front to about 60 km behind the gust front. In contrast to TR03, herein a two-dimensional Fourier transform is performed. The results are presented one-dimensionally by averaging the two-dimensional spectrum in the *x* or *y* directions (i.e., in the across-line and along-line directions, respectively). This method allows for a relatively smooth spectral analysis using output from one output time.

Parcel trajectories were calculated during the model simulation to allow an assessment of conditions in a parcel-relative framework. To optimize accuracy of the calculations, the parcels were translated every time step during the model simulation. Environmental conditions were interpolated to the parcel position to evaluate the conditions experienced by the parcels as they pass through the convective region of the squall lines.

## 3. Dependence on environmental shear

One general conclusion of this study is that the spurious updraft pattern only occurs in certain environments (Fig. 3). As found by TR03, the simulation with no environmental shear contains obviously spurious organization (Fig. 3a). The dominant wavelength of the vertical velocity pattern is about 3Δ in both the *x* and *y* directions, where Δ refers to the horizontal grid spacing. For *U _{s}* = 10 m s

^{−1}, an unphysical pattern is still present (Fig. 3b); in this case, the updrafts and downdrafts in the across-line (i.e.,

*x*) direction have a wavelength of roughly 6Δ, but there is comparatively less variation in the along-line (i.e.,

*y*) direction.

For larger low-level shear, a spurious pattern is not apparent. In the *U _{s}* = 20 m s

^{−1}simulation, the low-level updraft is closer to the surface gust front and is nearly continuous along the line (Fig. 3c). A few, weak updrafts exist several kilometers behind the line, but there is no evidence of a regular pattern of poorly resolved cells. Similarly, the

*U*= 30 m s

_{s}^{−1}simulation does not show any obvious signs of an unphysical pattern, but instead could be characterized as a series of comparatively better resolved cells along the line (Fig. 3d). The model output was checked at other levels and at different times, and no obvious evidence for a spurious updraft pattern was discovered in these two larger-shear runs.

TR03 also found that the spurious pattern was not apparent in an environment with *U _{s}* = 20 m s

^{−1}. Their explanation for the lack of a problem in this environment was based on the implicit diffusion within the fifth-order advection scheme. For this scheme, the amount of diffusion is proportional to the grid-relative advective wind speed. Therefore, the

*U*= 0 m s

_{s}^{−1}simulation should have small implicit diffusion owing to small wind speeds, while the

*U*= 20 m s

_{s}^{−1}would have more diffusion owing to the higher wind speeds. Thus, TR03 conclude that the spurious pattern is suppressed by the greater implicit diffusion.

Although this explanation seems plausible, there is evidence that increased diffusion is not the reason why the spurious organization does not occur in larger-shear environments. Using the same model configuration, an additional set of simulations was conducted in which a different mean wind was subtracted from the initial wind profile (hereafter referred to as the shifted-wind simulations). The goal in these simulations was to have zero mean grid-relative horizontal flow in the region where the spurious updrafts are found, that is, at roughly the 3-km level and near the cold-pool head. Thus, the goal was to *decrease* the amount of implicit diffusion in all runs, and furthermore to have a comparable amount of implicit diffusion between runs with different shears. To keep the squall line inside the domain during the entire simulation, the initial line thermal was moved from the center of the domain to a location west of center. Thus, in these simulations the squall line moves several hundred kilometers eastward, as opposed to the control simulations in which the squall line remains near the center of the domain at all times. Because no surface friction is included in the simulations, the two results should be identical, owing to Galilean invariance. Of course, some differences are expected for numerical reasons, but these differences should be minor; however, it will be shown shortly that this is not the case.

*u*. This variable was calculated at 3 km above ground and in the region where updrafts occur; this location for the four shears is 10, 5, 2, and 0 km behind the surface gust front for the

*U*= 0, 10, 20, and 30 m s

_{s}^{−1}cases, respectively. The horizontal diffusion coefficient,

*α*, implied by the

*u*values is

*U*= 10 m s

_{s}^{−1}case. This simulation had an apparent spurious organization, despite having the largest implicit diffusion. The

*U*= 20 m s

_{s}^{−1}case had a comparable amount of implicit diffusion, yet has an apparently acceptable solution. Interestingly, the

*U*= 30 m s

_{s}^{−1}case had the least amount of implicit diffusion at 3 km, despite having a qualitatively acceptable solution. As designed, the shifted-wind simulations have less implicit diffusion compared to the control simulations, and roughly similar implicit diffusion among shears (Table 2).

Although the results should be similar, the patterns of *w* in the shifted-wind simulations are markedly different from the control results—and not in the manner that one might expect, considering the lower implicit diffusion. For the *U _{s}* = 0 and 10 m s

^{−1}runs, the spurious updraft patterns have been eliminated (Figs. 4a and 4b). The updrafts in the

*U*= 0 m s

_{s}^{−1}case remain poorly resolved, but there is no longer a regular, repeating pattern across or along the line. This spurious characteristic is also absent from the

*U*= 10 m s

_{s}^{−1}case (Fig. 4b).

In contrast, the results for *U _{s}* = 20 and 30 m s

^{−1}are qualitatively similar to the patterns in the control simulations. The

*U*= 20 m s

_{s}^{−1}has long segments of nearly continuous updraft (Fig. 4c), and the

*U*= 30 m s

_{s}^{−1}simulation has several comparatively better resolved cells along the line (Fig. 4d). Thus, the Galilean invariance is better captured in the larger-shear simulations.

To assist with the interpretation of results, vertical velocity spectra were computed from the fields shown in Figs. 3 and 4. Because only one output time was used for the computation, the spectra are for only the fields shown in Figs. 3 and 4 and allow for a quantification of patterns already discussed subjectively. The spectra are displayed in Fig. 5, with the along-line spectra multiplied by 100 to avoid overlap. On all panels, a vertical line highlights the scale corresponding to 6Δ. Recent studies have found that the effective resolution in this model configuration is approximately 6 times the grid spacing (Bryan et al. 2003; Skamarock 2004). Thus, anything to the left of this line in spectra should be interpreted as a physical solution that is unaffected, at least directly, by model diffusion. Anything to the right of this line in spectra is damped directly by the model’s diffusion, and should be interpreted as being poorly resolved.

The across-line spectrum for the *U _{s}* = 0 m s

^{−1}control simulation quantifies the spurious nature of the vertical velocity pattern; a pronounced increase in energy occurs at poorly resolved scales, with a peak at ∼3Δ (Fig. 5a). This peak does not occur in the shifted-wind simulation, even though less diffusion is implied by the lower horizontal wind speeds at this level. In contrast, the along-line spectra for the two simulations are practically identical at both well resolved and poorly resolved scales.

For the *U _{s}* = 10 m s

^{−1}environment, the “control” simulation also shows a distinct buildup of energy in the across-line direction, but in this case is maximized at 6Δ (Fig. 5b). The sharp increase of energy from 12Δ to 6Δ is suspicious, but is not as obviously problematic as the clearly underresolved peak in the

*U*= 0 m s

_{s}^{−1}case. On the other hand, the shifted-wind simulation does not contain such a peak near the model’s effective resolution. In fact, the difference in overall across-line spectral structures for the control and shifted-wind simulations is disturbing, given that the results should be identical, owing to Galilean invariance. For the along-line spectra, the results are broadly similar, especially at scales smaller than 10Δ.

The spectra from the two larger-shear cases do not reveal such distinct differences between simulations. For *U _{s}* = 20 m s

^{−1}, the well-resolved sections of the spectra are nearly identical (Fig. 5c). The only substantial difference occurs at the smallest scales in the across-line spectra, where the “control” run has notably lower energy. The

*U*= 30 m s

_{s}^{−1}spectra also have no obviously spurious features, and the spectra from the two runs are qualitatively similar (Fig. 5d).

Although the ultimate cause of the spurious updraft pattern has not been explained yet, it can be concluded that a lack of diffusion is not a candidate. When the implicit flow-dependent diffusion is reduced, the spurious pattern is avoided. Furthermore, the results suggest an environmental dependence that requires explanation; for some reason, the low-shear environments allow the spurious pattern, while the large-shear environments do not. Finally, there is a disturbing difference for results that should be very similar.

## 4. An explanation for the spurious pattern

### a. The cause of the spurious oscillations

The conditions in which the spurious updraft pattern exists are highlighted by a vertical cross section from the *U _{s}* = 0 m s

^{−1}control simulation (Fig. 6). From a system-relative-flow perspective, the spurious pattern first occurs where the air first becomes saturated—in this case ∼6 km to the east of the surface gust front. This is where subtle, small (2–3Δ) oscillations appear in the thermodynamic and kinematic fields. As this air ascends over the cold pool, the alternating updraft/downdraft pattern is amplified. Ultimately, the pattern ceases to exist when a more uniform vertical profile of equivalent potential temperature (

*θ*) is created through tilting of

_{e}*θ*by resolved cells and through diffusion from the turbulence code and from implicit numerical diffusion.

_{e}In these squall-line simulations, the spurious pattern forms, amplifies, and dissipates within deep (∼2–3 km) and wide (∼5–20 km) moist absolutely unstable layers (MAULs). A MAUL is a layer of air that is saturated and is statically unstable. Typically, these layers appear on a thermodynamic diagram (such as a skew *T*) as a saturated layer in which *θ _{e}* decreases with height (e.g., Fig. 7). In mesoscale convective systems, a MAUL often forms where an initially unsaturated and conditionally unstable environment is lifted to saturation over a deep layer by the ascent associated with cold pools (Kain and Fritsch 1998; Bryan and Fritsch 2000). This is the same region where the spurious updraft pattern occurs.

However, this analysis does not explain why the spurious pattern has a poorly resolved scale (3–6Δ), or whether such a pattern is actually physical. Two additional facts help address these points. First, the resolution used for these simulations (Δ = 1 km) is too coarse to adequately represent the processes that probably occur in actual MAULs. If one wanted to resolve explicitly the overturning in a 2-km-deep turbulent layer, one would need grid spacing on the order of 100 m (Bryan et al. 2003). With much higher resolution—and in the actual atmosphere—one should expect to see turbulent overturning at a spectrum of scales smaller than 2 km. It is probable that large updrafts would ultimately emerge from this turbulent layer, as in this coarse simulation; however, a regular, repeating series of 2–3-km-wide updrafts seems unlikely. As discussed in the introduction, a more stochastic distribution of cumulonimbus cells is expected.

Second, the coarse resolution introduces numerous numerical difficulties. Finite-difference techniques, such as those used in this numerical model, have difficulties translating poorly resolved features. It is probable that spurious perturbations are being introduced into the MAUL by dispersive errors in the advection scheme. Linear advection tests often reveal these perturbations as “overshoots” and “undershoots” (e.g., Fig. 1 of Wicker and Skamarock 2002). More relevant to this problem is the environmental conditions in which the perturbations are introduced. The perturbations will naturally damp in a statically stable environment; however, in statically unstable environments, such as MAULs, the perturbations will amplify.

To summarize, it is reasonable to conclude that the spurious updraft pattern is ultimately caused by a combination of imperfect numerics and the presence of a statically unstable environment. Thus, it is a combination of a numerical problem and a physical situation that allows the problem to amplify.

This hypothesis is difficult to prove because the use of imperfect numerics is unavoidable. On the other hand, it might be possible to see a trend toward decreasing spurious structures as numerical problems are reduced. For example, one could try using a monotonic advection scheme, which does not introduce spurious overshoots and undershoots. Such a scheme is examined in the following section, and shows some promising results. However, monotonic schemes are implicitly diffusive, especially for poorly resolved structures. This leaves open the question of whether one is simply adding greater diffusion to the solution by the use of a monotonic scheme.

Another alternative would be to conduct simulations with numerical models that are comparatively free of spurious oscillations. Examples might include spectral or psuedospectral models. These squall-line simulations could be repeated with such models in future studies, to see if they have an intrinsic advantage over finite-difference-based models.

Although it is difficult to prove that the errors inherent in model numerics are ultimately responsible for the spurious pattern, there is some circumstantial evidence to support the hypothesis. For example, in the *U _{s}* = 0 m s

^{−1}control run the updrafts are advected westward during the simulation by the system-generated circulation. A time–space (i.e., Hovmöller) diagram highlights how the cells move with the mean horizontal wind speed in the MAUL: about 5.5 m s

^{−1}(Fig. 8). In contrast, in the shifted-wind simulations, the initial wind is adjusted so that the horizontal flow on the grid is zero in this layer—and there is no spurious pattern in these simulations. It can be surmised that because the cells are not advected laterally, there is no introduction of spurious oscillations from the advection scheme. This is a likely explanation for the apparently superior results in the shifted-wind simulations, even though the implicit diffusion is lower. That is, if one does not introduce spurious perturbations, then one does not have to diffuse them.

### b. Shear dependence

The shear dependence documented in section 3 can now be explained. Because the spatial properties (e.g., depth and width) of MAULs vary with squall-line structure, the spurious organization problem should only be expected in certain types of convective systems. Analysis of the MAUL properties from the simulations reveals that MAULs are deepest and widest in strongly upshear-tilted convective systems (Fig. 9). For the simulations in this study, this system structure occurs when the low-level shear is weak. There are additional environments, not studied here, that can promote upshear-tilted convective systems, such as strong reverse shear (e.g., the “jet” profiles of Weisman et al. 1988).

The depth and width of the MAUL are not, necessarily, the relevant properties that can lead to spurious updraft patterns. Rather, these are properties that promote long parcel residence time within the MAUL. Sample parcel trajectories for the four control simulations are included in Fig. 9. For the *U _{s}* = 0 m s

^{−1}case (Fig. 9a), parcels ascend as they approach the system’s cold pool, but then move nearly horizontally through a MAUL. Not all parcels have a quasi-horizontal trajectory such as that shown in Fig. 9a. In fact, some parcels ascend more rapidly through vertically growing cells. However, quasi-horizontally moving parcels are more common in the upshear-tilted systems, including the

*U*= 10 m s

_{s}^{−1}case. In contrast, in the comparatively more upright system of the

*U*= 20 m s

_{s}^{−1}simulation (Fig. 9c), and in the downshear-tilted U

_{s}= 30 m s

^{−1}simulation (Fig. 9d), the parcels typically ascend continuously from low levels to the upper troposphere, with less horizontal advection through MAULs. Given that horizontal advection can be a source of small-scale, spurious perturbations, the upshear-tilted systems should be more susceptible to the spurious organizational problem.

Average parcel residence times and horizontal displacement within MAULs are listed in Table 3 for the control simulations and in Table 4 for the shifted-wind simulations. In general, these data show that parcels spend more time within MAULs in the upshear-tilted systems—that is, in the lower shear environments. However, there is a subtle difference between results from the two sets of simulations; the residence time trends downward monotonically for the shifted-wind simulations, but not for the “control” simulations. Because the shifted-wind simulations are not affected by the spurious pattern, these results are probably more realistic. Despite these details, it is clear that parcel residence time within MAULs trends downward as system structure becomes more upright.

The value for horizontal (*x*) displacement on the grid is dependent on the horizontal grid-relative wind speeds experienced by the parcels. Thus, the horizontal displacement within the MAUL is lower in the shifted-wind simulations (Table 4) than in the control simulations (Table 3). This result supports the hypothesis from the previous subsection that horizontal advection within MAULs is primarily responsible for the occurrence of the spurious updraft pattern.

These results contribute evidence, combined with theory from the previous subsection, that MAULs are a factor in the development of the spurious updraft pattern. The spurious pattern seems to be most likely for large parcel residence times in MAULs combined with horizontal grid-relative advection within the MAUL. This explains why the spurious updrafts are most likely in environments that encourage upshear-tilted convective systems, such as weak wind shear.

Earlier, it was noted that the 1-km grid spacing used herein is insufficient to resolve the turbulent overturning that occurs in MAULs. Consequently, the spatial properties of the MAUL in these simulations might not reflect the properties that would exist in higher-resolution simulations or in the actual atmosphere. However, ongoing studies with higher resolution confirm that MAULs occur, even with explicit representation of turbulence. For example, the simulations with 125-m grid spacing from Bryan et al. (2003) also produce MAULs with horizontal spatial scales of *O* (10 km), and with parcel residence times of *O* (10 min), even though some details of the MAUL structure are different. More importantly, they confirm that MAULs in low-shear squall lines are deeper, wider across the line, and more continuous along the line than in high-shear squall lines. In summary, all qualitative conclusions of this section have been confirmed with higher-resolution simulations.

### c. Structural differences

Although the conditions in the *U _{s}* = 0 and 10 m s

^{−1}simulations both favor the formation of spurious updraft patterns, there is clearly a structural difference in the pattern. The

*U*= 0 m s

_{s}^{−1}simulation is characterized by comparatively small cells on the order of 3Δ, while the

*U*= 10 m s

_{s}^{−1}simulation has a comparatively better resolved wavelength of approximately 6Δ (Figs. 3 and 5). This result is robust; it has been reproduced in several tests with the current model, in which numerical techniques such as advection schemes were varied.

Additionally, a model intercomparison has been conducted on similar squall-line simulations (Bryan et al. 2004); qualitatively similar results were found for simulations with the WRF Model and with ARPS. All three models used in this intercomparison vary in several ways, from the time-integration scheme, to the order of the spatial derivatives used in the advection scheme, to inevitable differences in how one applies physical processes. The fact that the spurious updraft pattern is similar in different models points to an environmental, rather than numerical, dependence to this scale.

The ultimate reason for the differences in spatial scales is possibly related to differences in cold-pool structure. For the *U _{s}* = 10 m s

^{−1}case, the ascent at the cold-pool head is deeper, stronger, and wider (across the line) than in the

*U*= 0 m s

_{s}^{−1}case. Therefore, a deeper and wider layer of high

*θ*air ascends at the cold-pool head in the

_{e}*U*= 10 m s

_{s}^{−1}case. Because this physically based scale varies with environmental conditions, the subsequent overturning within the MAUL is perturbed at different scales. Ultimately, however, the evidence points to inaccuracies in the advection scheme as the source for spurious perturbations within this otherwise physical flow.

## 5. Solutions to the problem

One solution to the spurious problem has already been presented. Apparently, one can avoid the problem by minimizing horizontal advection within the MAUL (e.g., the shifted-wind simulations; Fig. 4). However, this solution is not generally suitable to all applications. For example, there may be varying low-level shear along the line in some squall lines. This technique also requires a great deal of knowledge about how the system will evolve before the simulation is started. Solutions that are more general are necessary.

The following tests are not comprehensive. There may be solutions other than those presented here, and a variety of parameter settings (such as diffusion coefficients) are not investigated. Furthermore, different numerical modeling systems may be less susceptible to this problem in the first place. The main purpose of this section is to present possible solutions for finite-difference-based models in the context of the explanation offered earlier—that is, that numerically generated perturbations amplify in a statically unstable environment.

The following simulations were performed for all four shears. However, a general conclusion drawn from these studies is that the larger-shear environments are not sensitive to these changes; the same qualitative structure appears in all high-shear runs and is broadly similar to that in Figs. 3 and 4. In contrast, the low-shear simulations show a large variation in results. Thus, to focus the discussion, only the *U _{s}* = 0 and 10 m s

^{−1}results are shown in this section.

### a. Increased turbulence parameters

TR03 recommend increasing diffusion via the model’s subgrid turbulence parameterization. Based on a series of simulations, they recommend increasing *C _{k}* above its commonly used value of ∼0.10, while keeping

*C*at 0.93. Following their suggestion, results using

_{e}*C*= 0.20 in the Bryan–Fritsch model are shown in Fig. 10. As in TR03, the solution is more diffused, and the spurious updraft pattern is not apparent. However, the increased diffusion from the turbulence scheme acts to nearly eliminate all updrafts in the

_{k}*U*= 0 m s

_{s}^{−1}simulation (Fig. 10a). In contrast, the simulation with

*U*= 10 m s

_{s}^{−1}contains strong cells, without evidence of the spurious pattern (Fig. 10b). The spurious “spikes” in the across-line spectra have been removed by this technique (Figs. 10c and 10d). However, the large-scale energy in the

*U*= 0 m s

_{s}^{−1}simulation has been substantially reduced by this method.

To address whether *C _{k}* = 0.20 is too large for this numerical model, a series of simulations was conducted in which

*C*was increased in small increments from 0.10 to 0.20. Based on these results,

_{k}*C*of 0.14 was the minimum value necessary to eliminate the spurious features for

_{k}*U*= 0 m s

_{s}^{−1}, but

*C*of 0.18 was the minimum necessary for

_{k}*U*= 10 m s

_{s}^{−1}. This result, and the fact that the

*U*= 20 and 30 m s

_{s}^{−1}cases do not require an increase of

*C*, suggest that simply increasing

_{k}*C*is not a general solution that will be sufficient for all applications. In fact, there appears to be an environmental dependence to the “optimal” value of

_{k}*C*, which, again, means that a good deal of knowledge of the simulated results is needed before an appropriate value can be chosen. Furthermore, the shifted-wind simulations presented earlier do not show any obvious need for an increase in

_{k}*C*in the first place, further raising doubts about the need for increased turbulence parameters.

_{k}### b. Artificial diffusion

Historically, the inclusion of an artificial diffusion term has been a common method to control small-scale numerical noise. For example, artificial fourth-order diffusion is used in the Klemp–Wilhelmson, MM5, and ARPS models. Some modelers use the more scale-selective sixth-order diffusion (e.g., Carpenter et al. 1998; Lane et al. 2003). Application of artificial sixth-order diffusion would be very effective at removing the 3Δ structures of the *U _{s}* = 0 m s

^{−1}simulation. However, sixth-order diffusion acts primarily on scales less than 6Δ (e.g., Durran 1999, p. 84). Therefore, sixth-order diffusion would not damp the 6Δ structures in the

*U*= 10 m s

_{s}^{−1}simulation.

It has been argued that the higher accuracy of the advection schemes in this model configuration, combined with some implicit diffusion inherent in the odd-ordered advection schemes, obviates the need for artificial diffusion (e.g., TR03). However, given the results presented here, an argument could be made that some artificial diffusion would be prudent for simulations of squall lines.

To test the effect of artificial diffusion, the simulations were repeated with standard fourth- and sixth-order artificial diffusion (as in Durran 1999) applied only horizontally to all variables except pressure. The diffusion coefficient is 4.8% of one-dimensional stability for both schemes; thus, 2Δ structures are damped at the same rate in both schemes. This value for diffusion coefficient is the same as the default value for the MM5 and ARPS models and is slightly stronger than the “FILTER1” applied in tests by TR03.

As expected, the solutions with artificial diffusion (Figs. 11 and 12) are generally smoother than the results of the control simulation (Fig. 3). Furthermore, the fourth-order scheme clearly acts on scales larger than the sixth-order scheme. Curiously, for the *U _{s}* = 0 m s

^{−1}run, the fourth-order diffusion almost completely eliminates most updrafts (Fig. 11a), whereas the sixth-order diffusion allows many strong updrafts to persist in this environment (Fig. 12a).

Spectra from these simulations (Figs. 11–12) confirm that the application of standard sixth-order diffusion does not eliminate the problem; the across-line spectrum shows a questionable peak at ∼7–8Δ for both shears (Figs. 12c and 12d). It is tempting to surmise that the diffusion coefficient is not large enough to control the problem in this case. On the other hand, standard fourth- and sixth-order diffusion schemes can introduce spurious small-scale perturbations (e.g., Xue 2000). The hypothesis from the previous section argues that spurious oscillations introduced by numerical schemes are the ultimate cause of the spurious updraft pattern. In this context, the standard diffusion schemes may, in some instances, introduce more problems through their oscillatory nature than they solve through their diffusiveness.

Based on these results, the simulations were repeated using the “simple” monotonic diffusion scheme of Xue (2000). The results using monotonic fourth-order diffusion are similar to results with the standard scheme, and thus are not shown here. The results for the monotonic sixth-order scheme are shown in Fig. 13. The updraft patterns are generally smoother than in simulations without the monotonic limiter. Furthermore, the spectra for these simulations do not show any obviously unphysical patterns, and the peak at ∼7–8Δ is no longer present (Figs. 13c and 13d).

These results provide additional support to the argument that spurious perturbations introduced by numerical schemes play a role in the development of the spurious updraft problem. As a general rule, the “simple” monotonic flux limiter of Xue (2000) is recommended for simulations of mesoscale convective systems that require artificial diffusion.

### c. Monotonic advection

If horizontal advection is primarily responsible for introducing spurious perturbations into the MAUL, it is reasonable to conclude that the use of a monotonic advection scheme could largely prevent the problem. Furthermore, by limiting the introduction of spurious perturbations, a numerical model might not require enhanced diffusion from turbulence schemes or from artificial diffusive terms. To this end, some monotonic advection schemes were tested on this case. Here, results are presented using the weighted essentially nonoscillatory (WENO) advection scheme of Shu (2001). This scheme is applied only to the scalar fields: that is, potential temperature, all water mixing ratios, and turbulence kinetic energy. The advection of momentum uses the same fifth-order advection scheme as in all other simulations. Otherwise, the model is configured exactly as in the control simulations.

The results using WENO advection of scalars (Fig. 14) could be considered satisfactory, under the loose constraints of the qualitative assessment used in this study. Specifically, the spurious pattern is not evident, and no obviously spurious features are present in the spectra of these results. On the other hand, the updrafts are poorly resolved, as in the control and shifted-wind simulations.

Two benefits of using this monotonic advection are notable, based on this cursory study. First, the simulations with monotonic advection retain a higher effective resolution than simulations with diffusion increased via the subgrid model or via an artificial term. This factor offsets the additional cost of the scheme, which is the most expensive of all configurations studied in this paper. Second, this monotonic scheme does not require a user-determined turbulence parameter or diffusion coefficient that needs to be set carefully based on trial and error. This property makes the scheme attractive for use in environmental sensitivity studies.

## 6. Conclusions

This study analyzed a spurious updraft pattern that can occur in numerical simulations of squall lines. The problem was previously documented by TR03 for the case of no environmental wind shear. This new study investigated a broader range of environmental shear conditions and proposed an explanation for the ultimate cause of the problem. The main conclusions reached by this study are as follows:

The spurious pattern only occurs in certain environments. For the cases studied here, the pattern was found only in upshear-tilted convective systems.

A lack of diffusion is not ultimately responsible for the spurious pattern. In simulations in which the model’s implicit diffusion is reduced, the spurious pattern does not occur.

The ultimate cause of the spurious pattern is the presence of statically unstable layers (i.e., MAULs) and the introduction of spurious perturbations into these layers by numerical schemes.

Typically, the atmosphere is statically stable. Thus, for most model applications, the introduction of spurious perturbations by numerical schemes does not become a serious problem, because the perturbations naturally decay. The simulation of convective systems containing deep, broad MAULs appears to be a special circumstance that requires additional vigilance by numerical model designers and users.

This study also explored possible model formulations that can be used to diffuse and/or avoid the spurious pattern. This analysis was not exhaustive, but instead was included to illustrate how knowledge of the spurious pattern’s cause can be used to evaluate numerical techniques. The conclusions reached in this part of the study include the following:

The method advocated by TR03—that is, increased diffusion via the model’s subgrid turbulence scheme— has an environmental dependence. This property makes the technique difficult to use as a general solution to the problem.

The use of standard high-order diffusion schemes is problematic in some conditions because these schemes can introduce spurious small-scale perturbations. The addition of a monotonic limiter, such as that of Xue (2000), was found to be helpful.

The use of a monotonic advection scheme on scalars prevented the appearance of spurious updraft patterns, although the updrafts remained poorly resolved.

The simulations in the last section of this study generally support the hypothesis from the first part. That is, simulations with numerical schemes that introduce spurious perturbations are more susceptible to the spurious updraft pattern.

The lack of a definitive benchmark solution prevents an assessment of what is the best model configuration. However, as noted in TR03, several obviously bad model configurations have been noted. What can be concluded with certainty is that squall-line simulations with low-shear conditions are more sensitive to model formulation than simulations with high-shear conditions. Because of this sensitivity, more careful model setup is necessary for simulations of upshear-tilted convective systems.

Simulations with 125-m grid spacing are currently being conducted for the *U _{s}* = 0 m s

^{−1}environment. It is hoped that the higher-resolution runs will lead to a converged, benchmark solution, which could be used to revisit the issue of best model formulation with

*O*(1 km) grid spacing. However, preliminary results indicate that the high-resolution runs are also sensitive to model formulation. This result is not surprising. A deep, wide MAUL still exists with high resolution, and the overturning of this layer is still sensitive to how it is perturbed by necessarily imperfect numerical schemes. Furthermore, the highest resolution that is possible at this time does not show evidence of convergence (e.g., Bryan et al. 2003), leaving open the question of whether even higher, yet unobtainable, results would show different results. The most promising approach to obtain a converged solution might be direct numerical simulation (DNS), in which a constant eddy diffusivity is applied, and a subgrid model is not necessary. Unfortunately, a DNS of a squall line would require ∼1 m grid spacing, which is not possible with currently available computers, and the Reynolds number would be rather low compared to observed squall lines. Thus, a benchmark solution remains elusive at this time.

## Acknowledgments

I would like to acknowledge fruitful discussions on this topic with Richard Rotunno, Jason Knievel, Matthew Parker, Tetsuya Takemi, Richard James, Robert Fovell, and an anonymous reviewer. This work was supported by the Advanced Study Program of NCAR. All figures were created using the Grid Analysis and Display System (GrADS).

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Initial conditions for the squall-line simulations. (a) Initial thermodynamic conditions. (b) Initial wind profiles.

Citation: Monthly Weather Review 133, 7; 10.1175/MWR2952.1

Initial conditions for the squall-line simulations. (a) Initial thermodynamic conditions. (b) Initial wind profiles.

Citation: Monthly Weather Review 133, 7; 10.1175/MWR2952.1

Initial conditions for the squall-line simulations. (a) Initial thermodynamic conditions. (b) Initial wind profiles.

Citation: Monthly Weather Review 133, 7; 10.1175/MWR2952.1

Vertical velocity at 3 km and 4 h from simulations with (a) *U _{s}* = 0, (b)

*U*= 10, (c)

_{s}*U*= 20, and (d)

_{s}*U*= 30 m s

_{s}^{−1}. The contour interval is 1 m s

^{−1}for (a) and 2 m s

^{−1}for (b)–(d), with negative contours dashed and the zero contour excluded. The thick dashed contour is the position of the surface gust front.

Citation: Monthly Weather Review 133, 7; 10.1175/MWR2952.1

Vertical velocity at 3 km and 4 h from simulations with (a) *U _{s}* = 0, (b)

*U*= 10, (c)

_{s}*U*= 20, and (d)

_{s}*U*= 30 m s

_{s}^{−1}. The contour interval is 1 m s

^{−1}for (a) and 2 m s

^{−1}for (b)–(d), with negative contours dashed and the zero contour excluded. The thick dashed contour is the position of the surface gust front.

Citation: Monthly Weather Review 133, 7; 10.1175/MWR2952.1

Vertical velocity at 3 km and 4 h from simulations with (a) *U _{s}* = 0, (b)

*U*= 10, (c)

_{s}*U*= 20, and (d)

_{s}*U*= 30 m s

_{s}^{−1}. The contour interval is 1 m s

^{−1}for (a) and 2 m s

^{−1}for (b)–(d), with negative contours dashed and the zero contour excluded. The thick dashed contour is the position of the surface gust front.

Citation: Monthly Weather Review 133, 7; 10.1175/MWR2952.1

Same as in Fig. 3, except for shifted-wind simulations, in which the mean horizontal (*x*) flow at this level is approximately zero near the updrafts.

Citation: Monthly Weather Review 133, 7; 10.1175/MWR2952.1

Same as in Fig. 3, except for shifted-wind simulations, in which the mean horizontal (*x*) flow at this level is approximately zero near the updrafts.

Citation: Monthly Weather Review 133, 7; 10.1175/MWR2952.1

Same as in Fig. 3, except for shifted-wind simulations, in which the mean horizontal (*x*) flow at this level is approximately zero near the updrafts.

Citation: Monthly Weather Review 133, 7; 10.1175/MWR2952.1

Vertical velocity spectra at 3 km for control (thick solid) and shifted-wind (thin solid) simulations with (a) *U _{s}* = 0, (b)

*U*= 10, (c)

_{s}*U*= 20, and (d)

_{s}*U*= 30 m s

_{s}^{−1}. The along-line spectra have been multiplied by 100 to avoid overlap. Two thick gray lines illustrating

*κ*

^{−5/3}spectra have been included for reference. The vertical line denotes the 6Δ scale.

Citation: Monthly Weather Review 133, 7; 10.1175/MWR2952.1

Vertical velocity spectra at 3 km for control (thick solid) and shifted-wind (thin solid) simulations with (a) *U _{s}* = 0, (b)

*U*= 10, (c)

_{s}*U*= 20, and (d)

_{s}*U*= 30 m s

_{s}^{−1}. The along-line spectra have been multiplied by 100 to avoid overlap. Two thick gray lines illustrating

*κ*

^{−5/3}spectra have been included for reference. The vertical line denotes the 6Δ scale.

Citation: Monthly Weather Review 133, 7; 10.1175/MWR2952.1

Vertical velocity spectra at 3 km for control (thick solid) and shifted-wind (thin solid) simulations with (a) *U _{s}* = 0, (b)

*U*= 10, (c)

_{s}*U*= 20, and (d)

_{s}*U*= 30 m s

_{s}^{−1}. The along-line spectra have been multiplied by 100 to avoid overlap. Two thick gray lines illustrating

*κ*

^{−5/3}spectra have been included for reference. The vertical line denotes the 6Δ scale.

Citation: Monthly Weather Review 133, 7; 10.1175/MWR2952.1

A vertical cross section at *y* = 23 km and *t* = 4 h from the “control” simulation with *U _{s}* = 0 m s

^{−1}. Equivalent potential temperature (K) is shaded. Vertical velocity is contoured, with a contour interval of 1 m s

^{−1}, negative contours dashed, and the zero contour excluded.

Citation: Monthly Weather Review 133, 7; 10.1175/MWR2952.1

A vertical cross section at *y* = 23 km and *t* = 4 h from the “control” simulation with *U _{s}* = 0 m s

^{−1}. Equivalent potential temperature (K) is shaded. Vertical velocity is contoured, with a contour interval of 1 m s

^{−1}, negative contours dashed, and the zero contour excluded.

Citation: Monthly Weather Review 133, 7; 10.1175/MWR2952.1

A vertical cross section at *y* = 23 km and *t* = 4 h from the “control” simulation with *U _{s}* = 0 m s

^{−1}. Equivalent potential temperature (K) is shaded. Vertical velocity is contoured, with a contour interval of 1 m s

^{−1}, negative contours dashed, and the zero contour excluded.

Citation: Monthly Weather Review 133, 7; 10.1175/MWR2952.1

A sounding at *x* = 313 km, *y* = 23 km, and *t* = 4 h from the control simulation with *U _{s}* = 0 m s

^{−1}.

Citation: Monthly Weather Review 133, 7; 10.1175/MWR2952.1

A sounding at *x* = 313 km, *y* = 23 km, and *t* = 4 h from the control simulation with *U _{s}* = 0 m s

^{−1}.

Citation: Monthly Weather Review 133, 7; 10.1175/MWR2952.1

A sounding at *x* = 313 km, *y* = 23 km, and *t* = 4 h from the control simulation with *U _{s}* = 0 m s

^{−1}.

Citation: Monthly Weather Review 133, 7; 10.1175/MWR2952.1

A time–space (i.e., Hovmöller) diagram of vertical velocity (m s^{−1}) at *y* = 39 km and *z* = 3 km from the control simulation with *U _{s}* = 0 m s

^{−1}.

Citation: Monthly Weather Review 133, 7; 10.1175/MWR2952.1

A time–space (i.e., Hovmöller) diagram of vertical velocity (m s^{−1}) at *y* = 39 km and *z* = 3 km from the control simulation with *U _{s}* = 0 m s

^{−1}.

Citation: Monthly Weather Review 133, 7; 10.1175/MWR2952.1

A time–space (i.e., Hovmöller) diagram of vertical velocity (m s^{−1}) at *y* = 39 km and *z* = 3 km from the control simulation with *U _{s}* = 0 m s

^{−1}.

Citation: Monthly Weather Review 133, 7; 10.1175/MWR2952.1

Line-averaged plots of equivalent potential temperature (K) from control simulations with (a) *U _{s}* = 0, (b)

*U*= 10, (c)

_{s}*U*= 20, and (d)

_{s}*U*= 30 m s

_{s}^{−1}. The dashed contour encloses regions in which more than 50% of the volume is moist absolutely unstable. Sample parcel trajectories are shown by thick solid lines. The parcel trajectories begin at

*z*= 1.75 km and ∼10–20 km east of the squall lines.

Citation: Monthly Weather Review 133, 7; 10.1175/MWR2952.1

Line-averaged plots of equivalent potential temperature (K) from control simulations with (a) *U _{s}* = 0, (b)

*U*= 10, (c)

_{s}*U*= 20, and (d)

_{s}*U*= 30 m s

_{s}^{−1}. The dashed contour encloses regions in which more than 50% of the volume is moist absolutely unstable. Sample parcel trajectories are shown by thick solid lines. The parcel trajectories begin at

*z*= 1.75 km and ∼10–20 km east of the squall lines.

Citation: Monthly Weather Review 133, 7; 10.1175/MWR2952.1

Line-averaged plots of equivalent potential temperature (K) from control simulations with (a) *U _{s}* = 0, (b)

*U*= 10, (c)

_{s}*U*= 20, and (d)

_{s}*U*= 30 m s

_{s}^{−1}. The dashed contour encloses regions in which more than 50% of the volume is moist absolutely unstable. Sample parcel trajectories are shown by thick solid lines. The parcel trajectories begin at

*z*= 1.75 km and ∼10–20 km east of the squall lines.

Citation: Monthly Weather Review 133, 7; 10.1175/MWR2952.1

(a), (b) Vertical velocity at 3 km and 4 h and (c), (d) the corresponding spectra (thin lines) from simulations with *C _{k}* = 0.20 using

*U*= 0 [in (a) and c)] and

_{s}*U*= 10 m s

_{s}^{−1}[in (b) and (d)]. The spectra from the control simulation are included as thick lines in (c) and (d). The contouring for vertical velocity is the same as in Fig. 3. The convention for plotting spectra is the same as in Fig. 5.

Citation: Monthly Weather Review 133, 7; 10.1175/MWR2952.1

(a), (b) Vertical velocity at 3 km and 4 h and (c), (d) the corresponding spectra (thin lines) from simulations with *C _{k}* = 0.20 using

*U*= 0 [in (a) and c)] and

_{s}*U*= 10 m s

_{s}^{−1}[in (b) and (d)]. The spectra from the control simulation are included as thick lines in (c) and (d). The contouring for vertical velocity is the same as in Fig. 3. The convention for plotting spectra is the same as in Fig. 5.

Citation: Monthly Weather Review 133, 7; 10.1175/MWR2952.1

(a), (b) Vertical velocity at 3 km and 4 h and (c), (d) the corresponding spectra (thin lines) from simulations with *C _{k}* = 0.20 using

*U*= 0 [in (a) and c)] and

_{s}*U*= 10 m s

_{s}^{−1}[in (b) and (d)]. The spectra from the control simulation are included as thick lines in (c) and (d). The contouring for vertical velocity is the same as in Fig. 3. The convention for plotting spectra is the same as in Fig. 5.

Citation: Monthly Weather Review 133, 7; 10.1175/MWR2952.1

Same as in Fig. 10, except for simulations with regular fourth-order diffusion.

Citation: Monthly Weather Review 133, 7; 10.1175/MWR2952.1

Same as in Fig. 10, except for simulations with regular fourth-order diffusion.

Citation: Monthly Weather Review 133, 7; 10.1175/MWR2952.1

Same as in Fig. 10, except for simulations with regular fourth-order diffusion.

Citation: Monthly Weather Review 133, 7; 10.1175/MWR2952.1

Same as in Fig. 10, except for simulations with regular sixth-order diffusion.

Citation: Monthly Weather Review 133, 7; 10.1175/MWR2952.1

Same as in Fig. 10, except for simulations with regular sixth-order diffusion.

Citation: Monthly Weather Review 133, 7; 10.1175/MWR2952.1

Same as in Fig. 10, except for simulations with regular sixth-order diffusion.

Citation: Monthly Weather Review 133, 7; 10.1175/MWR2952.1

Same as in Fig. 10, except for simulations with monotonic sixth-order diffusion.

Citation: Monthly Weather Review 133, 7; 10.1175/MWR2952.1

Same as in Fig. 10, except for simulations with monotonic sixth-order diffusion.

Citation: Monthly Weather Review 133, 7; 10.1175/MWR2952.1

Same as in Fig. 10, except for simulations with monotonic sixth-order diffusion.

Citation: Monthly Weather Review 133, 7; 10.1175/MWR2952.1

Same as in Fig. 10, except for simulations with WENO advection on scalars.

Citation: Monthly Weather Review 133, 7; 10.1175/MWR2952.1

Same as in Fig. 10, except for simulations with WENO advection on scalars.

Citation: Monthly Weather Review 133, 7; 10.1175/MWR2952.1

Same as in Fig. 10, except for simulations with WENO advection on scalars.

Citation: Monthly Weather Review 133, 7; 10.1175/MWR2952.1

Domain translation speeds (m s^{−1}) for simulations in which the squall line remains near the center of the domain (“control”), and for simulations in which the mean wind is nearly zero in the region of interest (shifted wind).

Approximate coefficients of diffusion (× 10^{−5} s^{−1}) for the control and shifted-wind simulations.

Average statistics (3–4 h) for 80 parcels (released every 1 km in *y* at 1.25 km above ground) in the control simulations.