## 1. Introduction

Ensemble forecasting provides a way of accounting for the inherent uncertainty in numerical weather forecasts. One major source of such uncertainty arises from the inaccurate specification of initial conditions, and one of the most important ways of forming an ensemble is to generate a set of numerical simulations by starting from a set of plausible, but slightly different, initial conditions. As the forecasts from such initial-condition ensembles proceed, the ensemble typically becomes underdispersive in the sense that the spread among the ensemble members becomes too small to ensure that the verifying weather pattern is contained within the ensemble (e.g., Buizza et al. 2005; Raftery et al. 2005; Schwartz et al. 2014).

Several empirical methods have been proposed to increase ensemble spread and thereby reduce the problem of underdispersion, including stochastically perturbed parameterization tendencies (SPPT; Buizza et al. 1999) and stochastic kinetic energy backscatter (SKEB; Shutts 2005). At the European Centre for Medium-Range Weather Forecasts (ECMWF), SKEB is implemented in a configuration designed to account for unresolved deep convection in a manner conceptually similar to another alternative for increasing ensemble spread, the stochastic convective backscatter (SCB) algorithm (Shutts 2015). It has been supposed that if the spatial resolution is increased to the point where interactions between the convection and larger scales can be explicitly resolved, the need for parameterizations such as the ECMWF-type SKEB and SCB should be reduced (Leutbecher et al. 2017).

The assumption that SCB and ECMWF-type SKEB become less important at higher resolution is supported by previous studies following the evolution of pairs of simulations with nearly identical initial conditions and showing that the difference between the twins (the error) increases faster in fine-resolution convection-permitting simulations than in coarser-resolution simulations using convective parameterizations. Zhang et al. (2003) found more rapid error growth in convection-permitting 3.3-km-resolution simulations than in 30-km simulations with parameterized convection. Hohenegger and Schär (2007) found an order of magnitude difference in the doubling time of global errors between synoptic-scale (80-km resolution) and cloud-resolving (2.2-km resolution) models. Moving beyond pairs of near-twin experiments, similar results were obtained for larger ensembles by Clark et al. (2009, 2010), who showed that ensemble spread grows faster at convection-allowing 4-km resolution than at 20-km resolution with convective parameterization.

Deep convection is a highly nonlinear phenomenon with rapid error growth rates and short eddy turnover times. Consistent with these previous results, models with sufficient resolution to at least approximately include convective dynamics might naturally be expected to generate more rapid growth of ensemble spread than coarser-resolution models that rely on convective parameterizations. But how does the ensemble spread compare when all model simulations use a horizontal grid spacing sufficient to capture the basic dynamics of deep convection? Generalizing from the seminal idealized-model analysis of Lorenz (1969), differences between ensemble members are expected to grow most rapidly at the smallest scales. In particular, in homogeneous isotropic turbulence, dimensional analysis suggests that the kinetic energy of the perturbations about the ensemble mean (KE′) will grow at eddy turnover time scales that decrease as the wavelength of the perturbations decreases, provided the slope of the background kinetic energy (KE) spectrum is shallower than *k* is the wavenumber (Lorenz 1969; Lilly 1972; Palmer et al. 2014). Although deep convection is not homogeneous isotropic turbulence, the atmospheric kinetic energy spectrum has a

Nevertheless, the overall effect of numerical resolution on upscale KE′ growth is not immediately obvious. Suppose convection is being forecast with models having horizontal grid spacings of either 1 or 2 km and that, owing to the limitations of the data-assimilation procedure, the initial KE′ spectrum for both ensembles is saturated at all scales shorter than 8 km but is very small at all larger scales. Will the different spatial resolutions make a difference in the time required for KE′ at 16 km to become saturated in each ensemble? To the extent that KE′ grows to larger scales through a cascade involving just a few slightly shorter wavelengths, the smallest-scale contributions to KE′ in the higher-resolution model may not be of practical importance, and there may be little difference between the ensembles in the upscale growth of KE′.

To better illustrate the issue in the preceding question, Fig. 1 shows the evolution of the KE′ spectra in a close relative of Lorenz’s original 1969 turbulence model, the smooth-saturation Lorenz–Rotunno–Snyder (ssLRS) model (Durran and Gingrich 2014), configured for surface quasigeostrophic dynamics (Rotunno and Snyder 2008).^{1} The initial error is localized at the smallest retained wavelength of approximately 1 km (dashed blue curve), and the classic upscale cascade of initial errors is clearly apparent, with the most rapidly growing errors at any time having maximum amplitude at scales just slightly larger than those that are saturated. Now suppose that the initial KE′ spectrum is given instead by the dashed orange curve in Fig. 1 (which is the level to which errors would grow in 6 h if the initial KE′ distribution followed the dashed blue curve). That orange curve shows KE′ saturated at all scales for which *k* exceeds 10^{−3} m^{−1}. Suppose the cutoff wavelength in one ensemble is at 1 km (the end of the heavy black line at wavenumber

In this paper, we examine the influence of numerical resolution on the rate at which differences among ensemble members due to initial-condition perturbations increase in simulations of idealized mesoscale convective systems (MCSs) and in the ssLRS model of homogeneous surface quasigeostrophic turbulence. The rest of this paper is organized as follows. Section 2 provides details about the model configuration and the initial perturbations imposed to create ensembles of simulated MCSs. Section 3 describes an analysis of the perturbation growth and ensemble spread in the MCS simulations. Section 4 compares the upscale growth of initial perturbations in the MCS simulations with that in the ssLRS model. Last, section 5 contains the conclusions.

## 2. Model configuration and simulation strategy

The model configuration closely follows that in WD17. The same nonhydrostatic cloud-resolving model is used to generate ensembles of 1 control and 20 perturbed members, each at horizontal grid spacings of 1, 1.4, and 2 km. Additionally, an ensemble with 1-km grid spacing and quadrupled fourth-order numerical diffusion, designated “1 km-D,” is also included. The smallest resolvable wave (having a wavelength of ^{−1} above the layer of linear wind shear between the surface and 5 km. Individual simulations are made on a 512 km × 512 km square doubly periodic horizontal domain. The number of grid points in the *x* and *y* directions varies based on the resolution, with 512, 366, and 256 grid points at 1, 1.4, and 2 km, respectively. The integration time step is 2, 3, and 4 s at grid spacings of 1, 1.4, and 2 km, respectively. In all ensemble members, three identical bubbles 2 K warmer than the environment produce the initial updrafts, which subsequently evolve into an organized MCS owing to the wind shear. The bubbles are spheroidal with a 20-km horizontal radius and 1.4-km vertical radius, centered 1.4 km above the surface at (*x*, *y*) locations of (100, 250), (125, 175), and (150, 300) km. All other details of the model are as in WD17.

The synthetic composite reflectivity for the unperturbed control members in the 1-km, 1-km-D, and 1.4- and 2-km simulations are shown at 5 h in Fig. 2. In all cases, a strong north–south-oriented line of thunderstorms has developed. The reflectivity fields are very similar, with modest increases in the north–south extent of the line and losses in fine structure appearing as the grid spacing or the numerical smoothing increases; there are no major changes to the MCS structure or convective dynamics because of the differences in resolution.^{2}

*a*= 0.1 g kg

^{−1};

*L*= 128 km, giving a two-dimensional horizontal wavelength of 90.5 km; and the vertical

*e*-folding decay scale is

*H*=1 km. The phases

The initial perturbations are imposed on large scales for three reasons. First, it eliminates the sensitivity to the otherwise arbitrary factor by which the scale of the initial-condition perturbations exceeds the horizontal grid spacing in the different ensembles by making that factor very large. Second, it eliminates the sensitivity of the initial-condition perturbations to numerical dissipation; for example, if the initial perturbations were imposed at 8 km, this would be a 4Δ*x* wavelength in the 2-km ensemble and immediately subject to much more numerical dissipation than in the 1-km ensemble, where it would have a wavelength of 8Δ*x*. Finally, the third reason for imposing large-scale initial perturbations is that these may be a more important source of uncertainty than perturbations on the smallest resolved scales in very-high-resolution mesoscale models. In particular, recent work (Durran and Gingrich 2014; Durran and Weyn 2016; WD17) has highlighted a little-known result in Lorenz (1969) suggesting that initial large-scale errors can be as detrimental to forecasts as initial small-scale errors of the same absolute amplitude. Morss et al. (2009) used a similar strategy of imposing initial perturbations at large scales in an investigation of the influence of spatial resolution on the growth of perturbation KE in a dry quasigeostrophic model.

In addition to the ensembles with moisture perturbations present in the initial conditions, another set of ensembles is constructed by adding the same moisture perturbations when the MCSs and background circulations are well established, at 4 h into the simulation, thereby allowing the analysis of perturbation growth in a complex background state.

## 3. The MCS ensembles

To gain physical intuition about the spread of perturbations in the 1-, 1.4-, and 2-km ensemble simulations, the agreement among the members on the location of strong convective elements is shown in Fig. 3, which contours the number of ensemble members, including the control, having synthetic reflectivity matching or exceeding 45 dB*Z* at each spatial point. At *t* = 3 h into the simulations, there are a relatively large number of members in agreement on the locations of most convective cells. Nevertheless, the spread clearly decreases with increasing grid spacing, with the 2-km ensemble exhibiting nearly perfect alignment of every significant high-reflectivity cell. By *t* = 5 h, however, there is much less certainty in the location of the strongest convection in the 1-km ensemble, where there is a wide swath of points at which only 9–13 members agree on the location of the high-reflectivity cells. At 2-km resolution, on the other hand, the locations of these cells are much more similar across the ensemble members. At both 3 and 5 h, the spread in the 1.4-km ensemble lies in between that in the 1- and 2-km cases.

### a. Growth of ensemble variance

*ϕ*iswhere

*m*th ensemble member, the overbar denotes the ensemble mean (excluding the control), and the summation is over all

*x*and

*y*coordinates. The evolution of domain-averaged ensemble variance of hourly accumulated precipitation, water vapor mixing ratio

*q*

_{υ}at a height of

*z*= 2 km, and potential temperature

*θ*at

*z*= 2 km is shown in Fig. 4 for the ensembles with 1-, 1.4-, and 2-km horizontal grid spacing. Variance increases roughly exponentially at times up to about

*t*= 3 h (

*t*= 4 h for precipitation) and approximately linearly afterward. In general, the variance and its growth rate decreases as the horizontal grid spacing increases, and the differences over the first 4 h are statistically significant. An exception appears in the variance of

*q*

_{υ}for the 1.4-km ensemble, which exceeds that for the 1-km ensemble after 4 h. The variance in hourly precipitation increases particularly smoothly because every point (plotted at 10-min intervals) contains the accumulated precipitation over the previous 60 min.

To more quantitatively assess the dependence of the growth rate of domain-averaged ensemble variance on numerical resolution, Fig. 5 shows the evolution of Var(*q*_{υ}) at *z* = 2 km (blue curves) from each ensemble on a logarithmic scale. After about 1 h, log[Var(*q*_{υ})] increases almost linearly for a period of 2–3 h. For each ensemble, a least squares linear fit is calculated to log[Var(*q*_{υ})] beginning at 1 h into the simulations and ending at the first subsequent time for which the slope starts decreasing.^{3} The doubling time for Var(*q*_{υ}) is calculated for each ensemble from the slope of this linear fit and displayed, along with the standard deviation computed from the slopes of the individual ensemble members, in each panel of Fig. 5. These doubling times increase consistently with increases in horizontal grid spacing, and the differences between the doubling times at each step up in the grid spacing exceeds twice the standard deviation. While changes to the horizontal grid spacing of the model clearly result in faster ensemble spread at higher resolution, the 1-km-D simulations with increased diffusion have a doubling time very close to the 1.4-km simulations, suggesting that the factor-of-4 increase in numerical smoothing in the 1-km-D runs hinders the growth of Var(*q*_{υ}) in a similar manner to coarsening the resolution by a factor of

*q*

_{υ}) to MSD(

*q*

_{υ}) dominates that of Bias

^{2}(

*q*

_{υ}) until about

*t*= 3 h, while after

*t*= 4 h, Var(

*q*

_{υ}) is approximately equal to Bias

^{2}(

*q*

_{υ}). Hence, the growth in ensemble spread is largely responsible for the initial growth of errors relative to the control, while at later times, the control member is statistically indistinguishable from any individual ensemble member. Spatial maps of MSD(

*q*

_{υ}) at

*t*= 5 h are shown for all ensembles in Fig. 6. Noting the logarithmically spaced contour intervals, these maps illustrate the dominant contribution from the region of active convection (cf. Fig. 2) to the total MSD in all cases. Figure 6 also shows that, while the total MSD is similar in magnitude across the ensembles, the locally highest values occur in the ensemble with 1.4-km grid spacing.

*q*

_{υ}) grows faster as the resolution is made finer is remarkably similar to that which can be obtained using dimensional analysis to compare the growth of KE′ at different scales in homogeneous turbulence. As discussed in Lilly (1972) and Palmer et al. (2014), if

*E*(

*k*) denotes the background KE spectral density per unit wavenumber (m

^{3}s

^{−2}) in homogeneous isotropic turbulence, dimensional analysis yields a time scale for circulations at wavenumber

*k*ofAssuming

*E*follows a power law such that

*k*, the ratio of doubling times for circulations of scale

*λ*and

In Table 1, the ratios *α* equal to ^{4}

Ratio of variance doubling times at different pairs of scales from the dimensional analysis of homogeneous turbulence

### b. Perturbation kinetic energy spectra and their approach to saturation

*m*th ensemble member as

*ϕ*as

*x*and

*y*directions, respectively;

^{5}For comparison with the generalized Lorenz model,

Spectra of perturbation and total KE from the 1- and 2-km simulations with initial-condition perturbations in the humidity at 91-km wavelength are shown in Fig. 7. These spectra are vertically averaged over the layer

As an aggregate measure of the scale-dependent loss of predictability useful for interensemble comparisons, let

## 4. Interpretation via the ssLRS model

To investigate the influence of spatial resolution on the growth of the perturbation KE in a simpler framework, we consider the ssLRS model of Durran and Gingrich (2014); this is an extension of the homogeneous turbulence models developed in Lorenz (1969) and Rotunno and Snyder (2008) in which

Consistent with observations in the mesoscales, in the ssLRS model, we impose a background KE spectrum proportional to

Figure 9a shows evolving ^{6} except at wavenumber 16, which corresponds to approximately 100 km. The initial ^{7} Also of note is the decrease in

When the initial ^{8} *unsaturated* perturbations in the smallest scales. Specifically, the inclusion of wavelengths smaller than any given

The resolution dependence of the spectral evolution of the unsaturated

A plot of

## 5. Conclusions

We have shown that ensemble spread grows more rapidly in simulations of idealized mesoscale convective systems as the horizontal grid spacing decreases from 2, to 1.4, and finally, to 1 km. In contrast to previous studies, in each of these cases, the numerical resolution was sufficient to reasonably represent the dynamics of deep convection. The evolution of the spread in an ensemble of 1-km-resolution simulations with increased fourth-order numerical diffusion was similar to that in the 1.4-km ensemble, demonstrating that the changes in the growth rate respond to changes in the amplitude of the shortest waves and not simply the numerical resolution per se. These results support the idea that the need for stochastic parameterization methodologies such as ECMWF-type SKEB and SCB to artificially increase ensemble dispersion (Shutts 2005; Leutbecher et al. 2017) should be reduced as the resolution of cloud-resolving ensemble prediction systems increases.

The initial perturbations were introduced at a wavelength of 91 km, rather than in the shortest scales, to ensure that they could be represented with essentially identical accuracy independent of the numerical resolution and because initial errors on such scales are a potentially important source of forecast error (Durran and Gingrich 2014). The initial perturbations spread rapidly across all scales and grew significantly on all scales. Nevertheless, over the first 3 h, the most rapid growth occurred at wavelengths smaller than roughly 30 km. The perturbation kinetic energy

Encouraged by this similarity in the resolution dependence of the variance doubling times in our complex numerical simulations and a simple dimensional analysis of homogeneous turbulence models, we examined ^{9} Our results are also similar to those obtained by Morss et al. (2009) using a dry quasigeostrophic model at multiple horizontal resolutions, whose underlying dynamics are much closer to those in the ssLRS model than are the dynamics governing our cloud-resolving model for deep convection. Despite their varying dynamical complexity, all three systems show more rapid

Let us now return to the question posed in the introduction about the influence of numerical resolution on upscale perturbation growth when the initial perturbation

## Acknowledgments

This research is funded by Grants N000141410287 and N000141712660 from the Office of Naval Research (ONR). J. A. Weyn was partly supported by the Department of Defense (DoD) through the National Defense Science and Engineering Graduate (NDSEG) Fellowship Program. The simulations were conducted on Gordon, a supercomputer at the Navy DoD Supercomputing Resource Center (DSRC). The authors gratefully acknowledge the comments of Chris Snyder and two anonymous reviewers, which have greatly improved this manuscript.

## APPENDIX A

### A Flaw in the Lorenz Model

The experiment in Fig. 9a reveals a flaw in the Lorenz model. The perturbation kinetic energy at wavenumber bin 16 ^{−6} times the background saturation KE spectrum. Negative values of *k*) are not correct and must arise from inaccuracies in some combination of the closure approximations used in Lorenz’s model and the numerical calculations of the coefficient values for that model.

Evolution of perturbation KE at wavenumber 16

*k*th wavenumber bin,

*n*is the number of wavenumber bins in the model (Lorenz 1969; Rotunno and Snyder 2008). The values of

*k*, the tendency of the rate of error growth is negative. In the absence of sufficiently large errors at adjacent wavenumbers, this produces an initial decrease in

*k*will temporarily generate negative values of

As evident in Lorenz (1969) and in Tables 1–4 of Rotunno and Snyder (2008), the entries on the main diagonal of the ^{A1} In contrast to the total KE,

*only*at one isolated 2D wavenumber

While the single-wavenumber analysis suggests that

## APPENDIX B

### Comparison of 2.8-km-Resolution MCS Simulations to the Rest

An ensemble of simulations of MCSs with horizontal grid spacing of 2.8 km was also constructed; the results from this ensemble are shown in Fig. B1. The ensemble agreement at *t* = 5 h into the simulations for this ensemble (Fig. B1d) is qualitatively similar to that of the 2-km ensemble (Fig. 3f) except for one prominent cell that only has 16-member agreement at best. In contrast to the dependence on grid spacing in the finer-resolution cases, the variance growth rates relative to those in the 2-km ensemble did not significantly slow down. As shown in Fig. B1c, the variance doubling time at 2.8-km grid spacing was

We hypothesize that the 2.8-km grid is coarse enough to change the scaling of the perturbation growth rates by modifying the structure and growth of the convective elements. (One would certainly expect a major change in the fidelity of convective simulations, and therefore perturbation growth rates, at very coarse resolutions like 10 km.) Some brief quantitative support for this hypothesis is provided by the absolute maximum in both time and space of the updraft velocity in the control members of each ensemble. The maximum updraft velocity gradually decreases from 41.8 to 34.3 m s^{−1} as the resolution is coarsened from 1 to 2 km, but the maximum updraft velocity of the 2.8-km control simulation is only 24.4 m s^{−1}, a steep decline. The simulation with 2.8-km resolution is substantially reducing the updraft velocities and therefore not realistically representing the strong convection.

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^{2}

Additional simulations with a horizontal grid spacing of 2.8 km were also performed. These showed some important differences from the rest of the simulations and are discussed further in appendix B.

^{3}

Sensitivity tests where the slopes where calculated over hours 1–3 for all ensembles gave the same statistically significant relative rankings of the ensembles.

^{4}

This result may only hold for a relatively narrow range of horizontal grid spacings where the dynamics of the system are not significantly affected by the numerical resolution; see appendix B.

^{5}

Essentially identical results are obtained if the perturbations are computed as differences from the unperturbed control member, rather than the ensemble mean, except that as saturation occurs, the perturbation KE values approach twice those of the background KE.

^{6}

After rescaling to match the dimensional plots, the initial KE′ is *O*(10^{−7}) at all wavelengths.

^{7}

The KE′ growth in a 1.4-km ssLRS simulation (truncated at wavenumber 29) lies in between the coarse- and fine-resolution behaviors (not shown).

^{8}

To fix the negative value at wavenumber 16, discussed in appendix A, we set KE′ at wavenumber 16 to be the average of that at wavenumbers 15 and 17.

^{9}

These simulations also revealed a flaw in the original Lorenz model that is discussed in appendix A.

^{A1}

A few positive entries appear in these tables where Lorenz assumed the background KE started dropping toward zero at planetary-scale wavelengths.