## 1. Introduction

Lorenz (1963) introduced the notion of chaotic processes on the basis of an abstracted set of equations for the atmosphere. Far detached from abstraction, turbulence—ubiquitous to the physical atmosphere—has itself been long deemed chaotic (Ruelle and Takens 1971). The characteristic property of chaotic phenomena (i.e., an exponential growth of initial uncertainties) casts these systems into an extreme sensitivity to initial conditions. In a later paper, Lorenz (1969) generalized his treatment of chaotic behavior to a flow with multiple scales of motion, which arguably was a more accurate description of Earth’s atmosphere than his 1963 model. He postulated that the global circulation of air is bound by an inherent fundamental (and thus inescapable) prediction horizon of roughly 2 weeks, regardless of the quality of initial values and numerical integration. This ominous conclusion emerged from a combination of an infinitesimally small prediction horizon on the smallest scales and a constant time of upscale error propagation toward the largest scales. Together, these implications place immanent limitations on the system’s predictability. The term prediction horizon—the window of time within which a chaotic system can be appreciably predicted—is hence as relevant as it is appealing. Numerical weather prediction (NWP), however, is set on a trajectory of high-resolution forecasting, with its characteristic grid resolution having increased an order of magnitude in 30 yr. State-of-the-art limited-area models now feature kilometer-scale resolutions on a regular basis (Lean et al. 2008; Seity et al. 2011), and experiments have been performed with global weather forecasting below the 1-km scale (Miura et al. 2007). It would not be overly ambitious to conclude that this marks the onset of NWP models resolving part of the boundary layer turbulence spectrum, which is usually associated with a

Whether it is useful at all to study small-scale deterministic predictability closely relates to the availability of high-resolution observations at regular intervals. Unsurprisingly, with the advent of satellites, novel remote sensing techniques, cloud cameras, etc., observation of atmospheric variables appears to have become possible at a high spatiotemporal resolution, providing data with unprecedented detail that might be assimilated into high-resolution weather forecasting models. The question of predictability at these fine scales then becomes immediately relevant, as once initiated with these finescale observations, the deterministic predictions will be prone to diverge rapidly from the corresponding physical fields, even if the two were to remain statistically identical. This serves as motivation for our study, where we investigate the deterministic predictability of the finest scales of the atmospheric boundary layer (ABL), which will eventually bind the extent of high-resolution forecasts.

We address the apparent loss of deterministic predictability in a much more simplified setting of dry convective atmospheric boundary layers with homogeneous (i.e., uniform) surface boundary conditions. To this end, we use the framework of large-eddy simulations and study the evolution of slightly perturbed identical twin simulations.

It is important to stress that this study focuses on deterministic predictability and does not contest the skill of LESs in predicting statistical quantities like moments and fluxes of atmospheric variables. Rather, we treat LES as a kind of ground truth, assuming that the technique is sufficiently capable of representing the turbulent processes associated with boundary layer convection. This premise is corroborated by a number of (intercomparison) papers, in which LES models are tested against field observations and are demonstrated to predict means and even higher-order moments to a sufficient degree of accuracy (Moeng 1984; Siebesma et al. 2003; Sullivan and Patton 2011; Chung and Matheou 2014; Schalkwijk et al. 2015a).

So, at any point in time, the identical twin simulations studied here will have the same statistical properties, but, after a tiny perturbation, the actual turbulent realizations will begin to increasingly differ over time until eventually they become statistically uncorrelated. The rate at which these twin realizations diverge (i.e., the decorrelation time scale) is the topic that we wish to investigate. A comparable approach was recently followed by Lo and Ngan (2015), who reported on predictability in sheared urban boundary layer flow by studying LES of a simplified street canyon.

We begin with a description of our numerical setup in section 2, followed by the results from the LES studies in section 3. Discussing the limitations encountered in our preliminary LES studies, we move onto employing a simplified LES subgrid: in essence, a DNS at low Reynolds numbers in section 4. We conclude with a discussion of our results in section 5.

## 2. Numerical setup

### a. Large-eddy simulations

*K*is the eddy viscosity parameter given by

*e*is the subfilter-scale turbulent kinetic energy, which is calculated by solving an additional equation following Deardorff (1980).

*θ*profile taken as

### b. Perturbation

*x*and

*y*directions:

*z*, where

## 3. LES results

*t*), the volumetric error, after employing the perturbation given in (4) for

*k*

_{p}= 4 to 36 (corresponding to wavelengths between 100 m and 1 km). After the initial phase, the error evolution displays a well-behaved exponential growth rate, which can be approximated as

(top) Volumetric error growth for different perturbation wavenumbers

Citation: Journal of the Atmospheric Sciences 73, 7; 10.1175/JAS-D-15-0206.1

(top) Volumetric error growth for different perturbation wavenumbers

Citation: Journal of the Atmospheric Sciences 73, 7; 10.1175/JAS-D-15-0206.1

(top) Volumetric error growth for different perturbation wavenumbers

Citation: Journal of the Atmospheric Sciences 73, 7; 10.1175/JAS-D-15-0206.1

The exponential growth phase in Fig. 2 (top) shows the Lyapunov exponent to be independent of the perturbation wavenumber. This indicates a rapid redistribution of the error spectrum after seeding, after which a self-similar spectral shape governs the growth of error. The initial decrease in the error (see inset) is steeper for higher wavenumbers, indicating that small-scale errors decrease more significantly than large-scale errors. This is related to dissipation by the subgrid model, which acts more strongly on sharper gradients encountered for higher wavenumbers.

*θ*multiplied by 2. This marks the return-to-turbulence feature of the error energy, which attains the

### Resolution dependence

*x*= 6.25 to 100 m (maintaining a fixed aspect ratio of the grid boxes at

Volume-averaged error growth of buoyancy (solid lines) and velocity fields (dashed lines) for the LES upon successive resolution doubling Δ*x* = 100–6.25 m.

Citation: Journal of the Atmospheric Sciences 73, 7; 10.1175/JAS-D-15-0206.1

Volume-averaged error growth of buoyancy (solid lines) and velocity fields (dashed lines) for the LES upon successive resolution doubling Δ*x* = 100–6.25 m.

Citation: Journal of the Atmospheric Sciences 73, 7; 10.1175/JAS-D-15-0206.1

Volume-averaged error growth of buoyancy (solid lines) and velocity fields (dashed lines) for the LES upon successive resolution doubling Δ*x* = 100–6.25 m.

Citation: Journal of the Atmospheric Sciences 73, 7; 10.1175/JAS-D-15-0206.1

One might suggest that the deterministic error growth originates more strongly in regions near the surface and inversion layer, the characteristics of which are most sensitive to resolution changes. Figure 4 appears to confirm this reasoning; it shows the time evolution (taking the time of seeding as *t* = 0) of the horizontally averaged error profiles, which are normalized per time step to visualize the error distribution in the vertical direction. Indeed, after the initial phase, the error is more pronounced near the surface and inversion layers. The diagonal “filaments” arising from the surface illustrate the transport of error by the largest eddies in the bulk of the boundary layer. The slope of these filaments is found to be roughly 1.3 m s^{−1}, which corresponds to the convective velocity scale

(left) Slab-integrated error profiles for different-resolution LES cases (normalized and scaled for visualization). The vertical white lines mark the time of error saturation. Increasing the resolution adds finer features to the error profiles, which represent smaller scales, in turn intensifying the dominant error production regions (i.e., near the surface and near the inversion). (right) The dashed and solid lines show the vertical distribution of the normalized error, further averaged in time, for the period before and after saturation, respectively.

Citation: Journal of the Atmospheric Sciences 73, 7; 10.1175/JAS-D-15-0206.1

(left) Slab-integrated error profiles for different-resolution LES cases (normalized and scaled for visualization). The vertical white lines mark the time of error saturation. Increasing the resolution adds finer features to the error profiles, which represent smaller scales, in turn intensifying the dominant error production regions (i.e., near the surface and near the inversion). (right) The dashed and solid lines show the vertical distribution of the normalized error, further averaged in time, for the period before and after saturation, respectively.

Citation: Journal of the Atmospheric Sciences 73, 7; 10.1175/JAS-D-15-0206.1

(left) Slab-integrated error profiles for different-resolution LES cases (normalized and scaled for visualization). The vertical white lines mark the time of error saturation. Increasing the resolution adds finer features to the error profiles, which represent smaller scales, in turn intensifying the dominant error production regions (i.e., near the surface and near the inversion). (right) The dashed and solid lines show the vertical distribution of the normalized error, further averaged in time, for the period before and after saturation, respectively.

Citation: Journal of the Atmospheric Sciences 73, 7; 10.1175/JAS-D-15-0206.1

The results so far clearly reveal the following aspects of error evolution: the exponential growth in the early stages, the perturbation wavenumber independence of its growth rate, and the pronounced zones of error production in the boundary layer. However, the resolution dependence of the error growth rate reveals a poignant problem in our current setup: that is, changing the resolution modifies too many aspects of the flow at the same time; apart from better resolving smaller features of the flow, by increasing the resolution, one also changes the eddy viscosity given by (2). To decouple the effect of resolution and variations in eddy viscosity, we consider a revised setup in which we simplify the LES subgrid model to a fixed viscosity arrangement. This allows one to control the viscosity (Reynolds number) independent of the resolution.

## 4. LES with a simplified subgrid model

A direct numerical simulation (DNS) equivalent can be designed by studying a dimensionless version of the previous case and fixing the Reynolds number (Jonker et al. 2013, 2012; Garcia and Mellado 2014). But since we will continue to refer to the case as atmospheric boundary layer, we will address this approach as “LES with a simplified subgrid model” rather than DNS, as the latter term might misguide the reader into assuming that we resolve the atmospheric flow until the millimeter scale (which we clearly do not).

The simplified LES subgrid model is achieved here by replacing the eddy viscosity in (2) of DALES by a constant kinematic viscosity *ν*. The smallest scale of motion (Kolmogorov scale) now depends on the fixed viscosity; because of computing limits, only low to moderate Reynolds numbers are feasible. In contrast to the original LES setting, the fixed viscosity simulations will not be capable of sustaining the thermal inversion, which will diffuse away, owing to the large diffusive flux in the inversion zone. To overcome this issue, we remove the free-tropospheric region from the simulation and locate the domain top at the inversion; we mimic the penetrative boundary layer situation by prescribing the entrainment flux at the top. Specifically, the domain is limited to

Vertical profile of the turbulent flux of temperature in the simplified subgrid version of DALES. Total (solid black), resolved (dashed blue), and diffusive (red) fluxes depict flux profile; the dashed green line shows the total (subfilter plus resolved) original LES flux.

Citation: Journal of the Atmospheric Sciences 73, 7; 10.1175/JAS-D-15-0206.1

Vertical profile of the turbulent flux of temperature in the simplified subgrid version of DALES. Total (solid black), resolved (dashed blue), and diffusive (red) fluxes depict flux profile; the dashed green line shows the total (subfilter plus resolved) original LES flux.

Citation: Journal of the Atmospheric Sciences 73, 7; 10.1175/JAS-D-15-0206.1

Vertical profile of the turbulent flux of temperature in the simplified subgrid version of DALES. Total (solid black), resolved (dashed blue), and diffusive (red) fluxes depict flux profile; the dashed green line shows the total (subfilter plus resolved) original LES flux.

Citation: Journal of the Atmospheric Sciences 73, 7; 10.1175/JAS-D-15-0206.1

The hourly evolution of the vertical potential temperature profiles for the two LES versions is shown in Fig. 6, for simulations with Re = 30, starting with identical initial conditions. Although not a perfect copy, for our present purposes the simplified subgrid LES profiles compare well enough to their original counterparts of penetrative convection.

Evolution of the vertical potential temperature profile in the simplified LES (solid curves) closely follows the warming of the complete profiles obtained from the original LES (dashed curves) up to the inversion-layer height.

Citation: Journal of the Atmospheric Sciences 73, 7; 10.1175/JAS-D-15-0206.1

Evolution of the vertical potential temperature profile in the simplified LES (solid curves) closely follows the warming of the complete profiles obtained from the original LES (dashed curves) up to the inversion-layer height.

Citation: Journal of the Atmospheric Sciences 73, 7; 10.1175/JAS-D-15-0206.1

Evolution of the vertical potential temperature profile in the simplified LES (solid curves) closely follows the warming of the complete profiles obtained from the original LES (dashed curves) up to the inversion-layer height.

Citation: Journal of the Atmospheric Sciences 73, 7; 10.1175/JAS-D-15-0206.1

### a. Simulation results

*ν*is the fixed viscosity and

Comparison of error growth between simulations of identical Reynolds number (Re ≈ 200) while varying the resolution

Citation: Journal of the Atmospheric Sciences 73, 7; 10.1175/JAS-D-15-0206.1

Comparison of error growth between simulations of identical Reynolds number (Re ≈ 200) while varying the resolution

Citation: Journal of the Atmospheric Sciences 73, 7; 10.1175/JAS-D-15-0206.1

Comparison of error growth between simulations of identical Reynolds number (Re ≈ 200) while varying the resolution

Citation: Journal of the Atmospheric Sciences 73, 7; 10.1175/JAS-D-15-0206.1

The effect of the Reynolds number on error growth is now investigated by varying the viscosity while retaining a constant resolution. This is shown in Fig. 8 for a perturbation wavenumber

Error growth rates for various Reynolds numbers. Solid lines indicate the simulation results, and dashed lines represent the best-fit exponential growth curves.

Citation: Journal of the Atmospheric Sciences 73, 7; 10.1175/JAS-D-15-0206.1

Error growth rates for various Reynolds numbers. Solid lines indicate the simulation results, and dashed lines represent the best-fit exponential growth curves.

Citation: Journal of the Atmospheric Sciences 73, 7; 10.1175/JAS-D-15-0206.1

Error growth rates for various Reynolds numbers. Solid lines indicate the simulation results, and dashed lines represent the best-fit exponential growth curves.

Citation: Journal of the Atmospheric Sciences 73, 7; 10.1175/JAS-D-15-0206.1

Varying Reynolds number simulation details, including the grid spacing Δ*x*, Δ*y*, and Δ*z* (m), and the ratio of the characteristic grid length *z*_{i} = 960 m).

The error growth rate can be nondimensionalized as

The nondimensional growth rate

Citation: Journal of the Atmospheric Sciences 73, 7; 10.1175/JAS-D-15-0206.1

The nondimensional growth rate

Citation: Journal of the Atmospheric Sciences 73, 7; 10.1175/JAS-D-15-0206.1

The nondimensional growth rate

Citation: Journal of the Atmospheric Sciences 73, 7; 10.1175/JAS-D-15-0206.1

The effect of changing the convective time scale

The lack of convergence of the error growth rates observed in the original LES runs recurs in the Reynolds number dependence of Fig. 9. It is unclear how one must extrapolate these results to typical values for a convective atmospheric boundary layer (

The foregoing discussion implies that a detailed scale analysis of the error field is required to aid the interpretation of the results.

### b. Quantifying the prediction horizon per scale

*θ*fields of the twin simulations in time. As the simulations begin to diverge, the correlation between the twin simulations at a certain scale decreases at different rates. We calculate the spectral correlation coefficient between the

*θ*fields as follows:

*k*and time

*t*;

*α*as

As before, the error curve (Fig. 10) is divided into three intervals. Interval a marks the onset of the constant growth phase following the seeding. In interval b, the error is beginning to saturate, finally leading to interval c, representing the saturated phase of the error. These phases are divided into equally spaced temporal intervals (increasing in time from blue to red). The growth of the different scales is made more lucid in Fig. 11, showing the error energy spectrum corresponding to the intervals marked on the error curve.

Volumetric error using the spectral perturbation divided into three sections: seeding and attaining the constant exponent of growth (interval a), transition to saturation of the error (interval b), and saturated phase of error (interval c). The error energy spectra for the labeled intervals correspond to the panels in Fig. 11.

Citation: Journal of the Atmospheric Sciences 73, 7; 10.1175/JAS-D-15-0206.1

Volumetric error using the spectral perturbation divided into three sections: seeding and attaining the constant exponent of growth (interval a), transition to saturation of the error (interval b), and saturated phase of error (interval c). The error energy spectra for the labeled intervals correspond to the panels in Fig. 11.

Citation: Journal of the Atmospheric Sciences 73, 7; 10.1175/JAS-D-15-0206.1

Volumetric error using the spectral perturbation divided into three sections: seeding and attaining the constant exponent of growth (interval a), transition to saturation of the error (interval b), and saturated phase of error (interval c). The error energy spectra for the labeled intervals correspond to the panels in Fig. 11.

Citation: Journal of the Atmospheric Sciences 73, 7; 10.1175/JAS-D-15-0206.1

Evolution of the error energy spectrum for the three intervals marked in Fig. 10. The gray bands represent twice the *θ* spectrum values for each time instance from both the simulations. (a) The seeding (with the perturbation spectrum resembling the *θ* spectrum), (b) the successive saturation of slower-growing larger scales, and (c) the saturation of all scales except those larger than the inversion

Citation: Journal of the Atmospheric Sciences 73, 7; 10.1175/JAS-D-15-0206.1

Evolution of the error energy spectrum for the three intervals marked in Fig. 10. The gray bands represent twice the *θ* spectrum values for each time instance from both the simulations. (a) The seeding (with the perturbation spectrum resembling the *θ* spectrum), (b) the successive saturation of slower-growing larger scales, and (c) the saturation of all scales except those larger than the inversion

Citation: Journal of the Atmospheric Sciences 73, 7; 10.1175/JAS-D-15-0206.1

Evolution of the error energy spectrum for the three intervals marked in Fig. 10. The gray bands represent twice the *θ* spectrum values for each time instance from both the simulations. (a) The seeding (with the perturbation spectrum resembling the *θ* spectrum), (b) the successive saturation of slower-growing larger scales, and (c) the saturation of all scales except those larger than the inversion

Citation: Journal of the Atmospheric Sciences 73, 7; 10.1175/JAS-D-15-0206.1

Figure 11a shows the growth of the error spectrum immediately after the seeding. The initial blue curves following the seeding are seen to closely resemble the shape of the *θ* spectrum (gray band), shifted in amplitude. As the error begins to grow, the spectrum attains its self-similar growth profile (where the higher wavenumbers grow faster and dominate the growth rate). Figure 11b marks the onset of saturation of the error. Here, it is seen that the faster growing higher wavenumbers begin to saturate, attaining twice their *θ* spectrum values. This is reflected in the drop in the error growth rate seen in interval b in Fig. 10. The slower-growing larger scales are seen to successively attain their saturation values in this phase. Last, Fig. 11c marks the saturation of all scales, barring the very largest scales corresponding to

The spectral correlation calculated from (12) is shown over time for all wavenumbers in Fig. 12. The black line shows the correlation between the twin simulations just after seeding, *k*. The decorrelation time

The correlation coefficient *ρ* between the two simulations calculated using (12) is shown in time for all wavenumbers *k*. The initially imposed small decorrelation

Citation: Journal of the Atmospheric Sciences 73, 7; 10.1175/JAS-D-15-0206.1

The correlation coefficient *ρ* between the two simulations calculated using (12) is shown in time for all wavenumbers *k*. The initially imposed small decorrelation

Citation: Journal of the Atmospheric Sciences 73, 7; 10.1175/JAS-D-15-0206.1

The correlation coefficient *ρ* between the two simulations calculated using (12) is shown in time for all wavenumbers *k*. The initially imposed small decorrelation

Citation: Journal of the Atmospheric Sciences 73, 7; 10.1175/JAS-D-15-0206.1

The decorrelation time for all scales *ρ* dropping to a critical value

Citation: Journal of the Atmospheric Sciences 73, 7; 10.1175/JAS-D-15-0206.1

The decorrelation time for all scales *ρ* dropping to a critical value

Citation: Journal of the Atmospheric Sciences 73, 7; 10.1175/JAS-D-15-0206.1

The decorrelation time for all scales *ρ* dropping to a critical value

Citation: Journal of the Atmospheric Sciences 73, 7; 10.1175/JAS-D-15-0206.1

## 5. Discussion

The predictability of convective atmospheric boundary layers was investigated using turbulence-resolving identical twin simulations. The evolution of controlled perturbations in the atmospheric boundary layer exhibited hallmark characteristics of chaotic systems with limited predictability, distinguishable in two phases. The first phase comprises an exponential growth of error, approximated as

*l*, a different Lyapunov exponent can be associated,

*l*will first saturate before the error dynamics of the next larger scale comes into play, at least conceptually. Using

*l*to influence scale

*k*to influence

*T*for wavenumber

*ε*the dissipation rate of turbulent kinetic energy, and taking

*η*, with

Conceptual image describing the dynamics of error growth in

Citation: Journal of the Atmospheric Sciences 73, 7; 10.1175/JAS-D-15-0206.1

Conceptual image describing the dynamics of error growth in

Citation: Journal of the Atmospheric Sciences 73, 7; 10.1175/JAS-D-15-0206.1

Conceptual image describing the dynamics of error growth in

Citation: Journal of the Atmospheric Sciences 73, 7; 10.1175/JAS-D-15-0206.1

For very small

Another (deliberate) limitation of this study, already alluded to in the introduction, is related to the homogeneous surface conditions that were employed; a flow influenced by surface heterogeneity, such as that shown in Fig. 1, for example, might display more “resistance” to initial errors.

## Acknowledgments

Simulations were carried out using computing resources of SURFsara with funding by the Netherlands Organization for Scientific Research (NWO). One of us (HJ) gratefully acknowledges discussions with Richard Rotunno and Chris Snyder.

## REFERENCES

Chung, D., and G. Matheou, 2014: Large-eddy simulation of stratified turbulence. Part I: A vortex-based subgrid-scale model.

,*J. Atmos. Sci.***71**, 1863–1879, doi:10.1175/JAS-D-13-0126.1.Deardorff, J. W., 1980: Stratocumulus-capped mixed layers derived from a three-dimensional model.

,*Bound.-Layer Meteor.***18**, 495–527, doi:10.1007/BF00119502.Deardorff, J. W., G. Willis, and B. Stockton, 1980: Laboratory studies of the entrainment zone of a convectively mixed layer.

,*J. Fluid Mech.***100**, 41–64, doi:10.1017/S0022112080001000.De Roode, S. R., P. G. Duynkerke, and H. J. J. Jonker, 2004: Large-eddy simulation: How large is large enough?

,*J. Atmos. Sci.***61**, 403–421, doi:10.1175/1520-0469(2004)061<0403:LSHLIL>2.0.CO;2.Durran, D. R., and M. Gingrich, 2014: Atmospheric predictability: Why butterflies are not of practical importance.

,*J. Atmos. Sci.***71**, 2476–2488, doi:10.1175/JAS-D-14-0007.1.Fang, X., and Y.-H. Kuo, 2015: A new generic method for quantifying the scale predictability of the fractal atmosphere: Applications to model verification.

,*J. Atmos. Sci.***72**, 1667–1688, doi:10.1175/JAS-D-14-0112.1.Garcia, J. R., and J. P. Mellado, 2014: The two-layer structure of the entrainment zone in the convective boundary layer.

,*J. Atmos. Sci.***71**, 1935–1955, doi:10.1175/JAS-D-13-0148.1.Heus, T., and Coauthors, 2010: Formulation of the Dutch Atmospheric Large-Eddy Simulation (DALES) and overview of its applications.

*Geosci. Model Dev.*,**3**, 415–444, doi:10.5194/gmd-3-415-2010.Jonker, H. J. J., P. Duynkerke, and J. W. M. Cuijpers, 1999: Mesoscale fluctuations in scalars generated by boundary layer convection.

,*J. Atmos. Sci.***56**, 801–808, doi:10.1175/1520-0469(1999)056<0801:MFISGB>2.0.CO;2.Jonker, H. J. J., M. van Reeuwijk, P. Sullivan, and E. Patton, 2012: Interfacial layers in clear and cloudy atmospheric boundary layers.

,*Turbul. Heat Mass Transfer***7**, 3–14, doi:10.1615/ICHMT.2012.ProcSevIntSympTurbHeatTransfPal.10.Jonker, H. J. J., M. van Reeuwijk, P. P. Sullivan, and E. G. Patton, 2013: On the scaling of shear-driven entrainment: A DNS study.

,*J. Fluid Mech.***732**, 150–165, doi:10.1017/jfm.2013.394.Lean, H. W., P. A. Clark, M. Dixon, N. M. Roberts, A. Fitch, R. Forbes, and C. Halliwell, 2008: Characteristics of high-resolution versions of the Met Office Unified Model for forecasting convection over the United Kingdom.

,*Mon. Wea. Rev.***136**, 3408–3424, doi:10.1175/2008MWR2332.1.Lo, K., and K. Ngan, 2015: Predictability of turbulent flow in street canyons.

,*Bound.-Layer Meteor.***156**, 191–210, doi:10.1007/s10546-015-0014-z.Lorenz, E. N., 1963: Deterministic nonperiodic flow.

,*J. Atmos. Sci.***20**, 130–141, doi:10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2.Lorenz, E. N., 1969: The predictability of a flow which possesses many scales of motion.

,*Tellus***21A**, 289–307, doi:10.1111/j.2153-3490.1969.tb00444.x.Miura, H., M. Satoh, T. Nasuno, A. T. Noda, and K. Oouchi, 2007: A Madden–Julian Oscillation event realistically simulated by a global cloud-resolving model.

,*Science***318**, 1763–1765, doi:10.1126/science.1148443.Moeng, C.-H., 1984: A large-eddy-simulation model for the study of planetary boundary-layer turbulence.

,*J. Atmos. Sci.***41**, 2052–2062, doi:10.1175/1520-0469(1984)041<2052:ALESMF>2.0.CO;2.Morss, R. E., C. Snyder, and R. Rotunno, 2009: Spectra, spatial scales, and predictability in a quasigeostrophic model.

,*J. Atmos. Sci.***66**, 3115–3130, doi:10.1175/2009JAS3057.1.Ngan, K., P. Bartello, and D. Straub, 2009: Predictability of rotating stratified turbulence.

,*J. Atmos. Sci.***66**, 1384–1400, doi:10.1175/2008JAS2799.1.Rotunno, R., and C. Snyder, 2008: A generalization of Lorenz’s model for the predictability of flows with many scales of motion.

,*J. Atmos. Sci.***65**, 1063–1076, doi:10.1175/2007JAS2449.1.Ruelle, D., and F. Takens, 1971: On the nature of turbulence.

,*Commun. Math. Phys.***20**, 167–192, doi:10.1007/BF01646553.Schalkwijk, J., H. J. J. Jonker, A. P. Siebesma, and F. C. Bosveld, 2015a: A year-long large-eddy simulation of the weather over Cabauw: An overview.

,*Mon. Wea. Rev.***143**, 828–844, doi:10.1175/MWR-D-14-00293.1.Schalkwijk, J., H. J. J. Jonker, A. P. Siebesma, and E. van Meijgaard, 2015b: Weather forecasting using GPU-based large-eddy simulations.

,*Bull. Amer. Meteor. Soc.***96**, 715–723, doi:10.1175/BAMS-D-14-00114.1.Seity, Y., P. Brousseau, S. Malardel, G. Hello, P. Bénard, F. Bouttier, C. Lac, and V. Masson, 2011: The AROME-France convective-scale operational model.

,*Mon. Wea. Rev.***139**, 976–991, doi:10.1175/2010MWR3425.1.Siebesma, A. P., and Coauthors, 2003: A large eddy simulation intercomparison study of shallow cumulus convection.

,*J. Atmos. Sci.***60**, 1201–1219, doi:10.1175/1520-0469(2003)60<1201:ALESIS>2.0.CO;2.Simmons, A., and A. Hollingsworth, 2002: Some aspects of the improvement in skill of numerical weather prediction.

,*Quart. J. Roy. Meteor. Soc.***128**, 647–677, doi:10.1256/003590002321042135.Sullivan, P. P., and E. G. Patton, 2011: The effect of mesh resolution on convective boundary layer statistics and structures generated by large-eddy simulation.

,*J. Atmos. Sci.***68**, 2395–2415, doi:10.1175/JAS-D-10-05010.1.Sullivan, P. P., C.-H. Moeng, B. Stevens, D. H. Lenschow, and S. D. Mayor, 1998: Structure of the entrainment zone capping the convective atmospheric boundary layer.

,*J. Atmos. Sci.***55**, 3042–3064, doi:10.1175/1520-0469(1998)055<3042:SOTEZC>2.0.CO;2.Tribbia, J., and D. Baumhefner, 2004: Scale interactions and atmospheric predictability: An updated perspective.

,*Mon. Wea. Rev.***132**, 703–713, doi:10.1175/1520-0493(2004)132<0703:SIAAPA>2.0.CO;2.Vallis, G. K., 2006:

*Atmospheric and Oceanic Fluid Dynamics*. Cambridge University Press, 745 pp.