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    Fig. 1.

    Figure from Schalkwijk et al. (2015b) shows a high-resolution forecast based on a real weather case over the Netherlands. Copyright American Meteorological Society. Used with permission.

  • View in gallery
    Fig. 2.

    (top) Volumetric error growth for different perturbation wavenumbers . The exponential error growth rate is independent of and attains a constant value after a brief initial adjustment phase, shown in the inset, where the initial decrease in error is steeper for higher-wavenumber perturbations. (middle) Evolution of the error field (normalized for visualization) shown as a vertical cross section (in the middle of the LES domain) for the four instances marked on the error curve in the top panel; . (bottom) Corresponding evolution of the error power spectra during the following phases: (a) seeding, (b) exponential growth with a fixed growth rate, (c) transition to saturation, and (d) saturation. Time advances as the colors transition from blue to red.

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    Fig. 3.

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

  • View in gallery
    Fig. 4.

    (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.

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    Fig. 5.

    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.

  • View in gallery
    Fig. 6.

    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.

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    Fig. 7.

    Comparison of error growth between simulations of identical Reynolds number (Re ≈ 200) while varying the resolution . The resolution independence of the growth rates is clearly seen, as the measured Lyapunov exponent varies less than 6%.

  • View in gallery
    Fig. 8.

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

  • View in gallery
    Fig. 9.

    The nondimensional growth rate plotted against the Reynolds number. The blue points represent simulation results with , and red points represent . The dashed line shows a power law for reference (see the discussion section).

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    Fig. 10.

    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.

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    Fig. 11.

    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 .

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    Fig. 12.

    The correlation coefficient ρ between the two simulations calculated using (12) is shown in time for all wavenumbers k. The initially imposed small decorrelation is shown as the solid black line and a critical value as the solid gray line. The correlations are seen to drop in time and are faster for the higher wavenumbers.

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    Fig. 13.

    The decorrelation time for all scales , calculated as the time from seeding to the correlation ρ dropping to a critical value , is shown for different Reynolds number simulations. Scales smaller than show a Reynolds number dependence and decorrelate faster with increasing Re.

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    Fig. 14.

    Conceptual image describing the dynamics of error growth in turbulence.

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Predictability of Dry Convective Boundary Layers: An LES Study

Siddhartha MukherjeeDepartment of Geoscience and Remote Sensing, Delft University of Technology, Delft, Netherlands

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Jerôme SchalkwijkDepartment of Geoscience and Remote Sensing, Delft University of Technology, Delft, Netherlands

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Harmen J. J. JonkerDepartment of Geoscience and Remote Sensing, Delft University of Technology, Delft, Netherlands

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Abstract

The predictability horizon of convective boundary layers is investigated in this study. Large-eddy simulation (LES) and direct numerical simulation (DNS) techniques are employed to probe the evolution of perturbations in identical twin simulations of a growing dry convective boundary layer. Error growth typical of chaotic systems is observed, marked by two phases. The first comprises an exponential error growth as , with δ0 as the initial error, δ(t) as the error at time t, and Λ as the Lyapunov exponent. This phase is independent of the perturbation wavenumber, and the perturbation energy grows following a self-similar spectral shape dominated by higher wavenumbers. The nondimensional error growth rate in this phase shows a strong dependence on the Reynolds number (Re). The second phase involves saturation of the error. Here, the error growth follows Lorenz dynamics with a slower saturation of successively larger scales. An analysis of the spectral decorrelation times reveals two regimes: an Re-independent regime for scales larger than the boundary layer height and an Re-dependent regime for scales smaller than , which are found to decorrelate substantially faster for increasing Reynolds numbers.

Corresponding author address: Harmen J. J. Jonker, Department of Geoscience and Remote Sensing, Delft University of Technology, Stevinweg 1, 2628 CN Delft, Netherlands. E-mail: h.j.j.jonker@tudelft.nl

Abstract

The predictability horizon of convective boundary layers is investigated in this study. Large-eddy simulation (LES) and direct numerical simulation (DNS) techniques are employed to probe the evolution of perturbations in identical twin simulations of a growing dry convective boundary layer. Error growth typical of chaotic systems is observed, marked by two phases. The first comprises an exponential error growth as , with δ0 as the initial error, δ(t) as the error at time t, and Λ as the Lyapunov exponent. This phase is independent of the perturbation wavenumber, and the perturbation energy grows following a self-similar spectral shape dominated by higher wavenumbers. The nondimensional error growth rate in this phase shows a strong dependence on the Reynolds number (Re). The second phase involves saturation of the error. Here, the error growth follows Lorenz dynamics with a slower saturation of successively larger scales. An analysis of the spectral decorrelation times reveals two regimes: an Re-independent regime for scales larger than the boundary layer height and an Re-dependent regime for scales smaller than , which are found to decorrelate substantially faster for increasing Reynolds numbers.

Corresponding author address: Harmen J. J. Jonker, Department of Geoscience and Remote Sensing, Delft University of Technology, Stevinweg 1, 2628 CN Delft, Netherlands. E-mail: h.j.j.jonker@tudelft.nl

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 slope dependence. Indeed, recently tests have been conducted with turbulence-resolving forecasts at 100-m resolution (Schalkwijk et al. 2015a) using large-eddy simulations (LESs). In a follow-up study, Schalkwijk et al. (2015b) showed that it is also feasible to conduct such simulations on the scale of a small country. An example, based on a real weather case from July 2004 in the Netherlands, is given in Fig. 1. This picture contains several features that reinforce our motivation for this study. Comparison with the satellite image taken on that day shows that many features were simulated realistically (Schalkwijk et al. 2015b), the most prominent one being the effect of the land–sea transition on cloudiness. Thus, at this location, the cloud field is strongly tied to the surface characteristics. But farther inland, where the surface conditions become more homogeneous (at least in comparison to the strong inhomogeneity at the coast), the location of the scattered clouds should be interpreted in a stochastic sense. In other words, the skill of the simulation resides in predicting the cloud fraction but not the exact cloud locations.

Fig. 1.
Fig. 1.

Figure from Schalkwijk et al. (2015b) shows a high-resolution forecast based on a real weather case over the Netherlands. Copyright American Meteorological Society. Used with permission.

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

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

The LES model used here is the Dutch Atmospheric Large-Eddy Simulation (DALES), as detailed in Heus et al. (2010). The LES resolves three-dimensional turbulent motions of the atmospheric boundary layer and models the effect of the smallest scales by using a subfilter-scale model. DALES uses an eddy viscosity approach, where the subfilter-scale stress is modeled in terms of resolved velocities as
e1
where K is the eddy viscosity parameter given by
e2
with a proportionality constant and the filter length scale based on the grid resolution , , and in the horizontal and vertical directions. Further, e is the subfilter-scale turbulent kinetic energy, which is calculated by solving an additional equation following Deardorff (1980).
A case of a growing dry convective boundary layer driven solely by a constant surface heat flux is studied, based upon the study of Sullivan et al. (1998). We confine ourselves to their W06 case, a CBL capped by a weak inversion jump. The simulations proceed from an initial potential temperature θ profile taken as
e3
having a constant potential temperature in the mixed layer initially up to an inversion height , above which the atmosphere is stably stratified at a gradient of . The inversion jump is not specified and is allowed to develop naturally. The surface flux is fixed at , and all simulations are performed on a fixed domain of .

b. Perturbation

Error propagation is studied by performing identical twin simulations, a standard approach in predictability studies (Tribbia and Baumhefner 2004; Ngan et al. 2009). The simulations are allowed to evolve for a 1-h spinup period, after which one of them is perturbed. The perturbations employed in this study are two-dimensional sinusoidal fields added to the potential temperature field and have a fixed amplitude and (dimensionless) wavenumber in the x and y directions:
e4
for all z, where and are the potential temperature fields of the two separate (twin) simulations just after perturbation. In addition, these perturbations are independent of height; hence, the entire volumetric domain is perturbed during the seeding. As the simulations diverge after the perturbation is added, the adaptive time stepping of the twin simulations is synchronized by intercommunicating the smallest time step of the pair; such a synchronization greatly facilitated comparison.
The evolution of the perturbation is reported as a slab-averaged root-mean-square difference between the potential temperature fields in two simulations, given as
e5
where the angle brackets denote a volume average. The perturbation energy spectrum is calculated from the Fourier decomposition of the potential temperature difference in the two horizontal directions. The spectra are further averaged over the vertical direction.

3. LES results

The top panel of Fig. 2 shows the evolution of δ(t), the volumetric error, after employing the perturbation given in (4) for and , with the perturbation wavenumber varying from kp = 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
e6
where could be regarded as a global Lyapunov exponent, as it encompasses the integrated growth of error over all locations/scales present in the flow.
Fig. 2.
Fig. 2.

(top) Volumetric error growth for different perturbation wavenumbers . The exponential error growth rate is independent of and attains a constant value after a brief initial adjustment phase, shown in the inset, where the initial decrease in error is steeper for higher-wavenumber perturbations. (middle) Evolution of the error field (normalized for visualization) shown as a vertical cross section (in the middle of the LES domain) for the four instances marked on the error curve in the top panel; . (bottom) Corresponding evolution of the error power spectra during the following phases: (a) seeding, (b) exponential growth with a fixed growth rate, (c) transition to saturation, and (d) saturation. Time advances as the colors transition from blue to red.

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.

The evolution of the error field is shown as a vertical slice in the middle of the domain in Fig. 2 (middle panel) for the case of for the four intervals marked on the error growth curve in the top panel. The corresponding vertically averaged 2D power spectra of the error field are shown below the respective error field snapshots. The interval in Fig. 2a shows the error seed, which peaks at a single wavenumber, and is seen to immediately interact with all other scales that begin to grow, as seen from the upward-shifting error spectrum. The interval in Fig. 2b shows the error during the exponential growth phase following . Note that the shape of the power spectrum in this phase is self-similar and is dominated by the higher wavenumbers. It continually shifts upward with an equal distance (on a log scale), reflecting the constant growth rate. This feature is observed for all the simulations performed in this study; Fig. 2c shows the transition to saturation of the error energy in which the larger scales begin to saturate at a slower rate; Fig. 2d shows the fully saturated phase of the error along with the energy spectrum of θ multiplied by 2. This marks the return-to-turbulence feature of the error energy, which attains the slope after the two simulations have essentially become uncorrelated:
e7

Resolution dependence

In the LES, the smallest resolved scales are directly dependent on the grid resolution through the action of the subgrid model in (2). Simulations with increasing mesh refinement consequently resolve more turbulence, with an abundance of finer structures in the resolved flow. The eddy viscosity approach dissipates errors at the subfilter scale, confining their growth to the resolved flow. This presages a grid dependency of the error propagation. Indeed, the effect can be clearly seen in Fig. 3, where the LES resolution is varied from Δx = 6.25 to 100 m (maintaining a fixed aspect ratio of the grid boxes at and using ). Not all simulations are run for an equal time span because of the steep increase in computational cost (resolution doubling implies a factor of 16 in computational cost). For completeness, we have also shown in Fig. 3 the evolution of the error in the velocity fields:
e8
which, as expected from their tight dynamic coupling, is found to grow at the same pace as buoyancy.
Fig. 3.
Fig. 3.

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

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

It is apparent from these figures that the effect of increasing the resolution is very pronounced, as the rate of error propagation increases without any sign of convergence. This effect is surprising to the extent that most energetic eddies in the bulk of the boundary layer are resolved for and smaller; in other words, at this resolution, the first- and second-order statistics of an LES are reasonably well converged (Sullivan et al. 1998; Sullivan and Patton 2011). This highlights the distinction between the deterministic error in (5) between two realizations and the stochastic error between the statistics of two simulations. A metric for the latter would be
e9
where simulations 1 and 2 could have different resolutions.

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 . With a higher resolution, finer features are resolved, which intensify and concentrate the surface and inversion zones, as successively higher wavenumbers are available for a faster rate of error growth. However, a localized error analysis in horizontal planes at different heights shows that the error grows at the same rate, which could be understood as an effect of the strong vertical mixing in the boundary layer.

Fig. 4.
Fig. 4.

(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 (i.e., half the original LES domain in the vertical direction), and a fixed entrainment flux ratio, , is imposed at the top of the domain, which is close to shear-free convective boundary layer growth values from laboratory observations (Deardorff et al. 1980) and DNS simulations (Jonker et al. 2012; Garcia and Mellado 2014) and consistent with the original LES results (cf. Fig. 5).

Fig. 5.
Fig. 5.

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.

Fig. 6.
Fig. 6.

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

Hereinafter, the quantities presented are nondimensionalized by
e10a
e10b
e10c
e10d
The Reynolds number is given by
e11
where ν is the fixed viscosity and the fixed inversion height. By fixing the eddy viscosity of the LES subgrid, the Reynolds number [(11)] of the flow is no longer dependent on the resolution. This is first confirmed in Fig. 7, where the Reynolds number is fixed at Re ≈ 200, and the resolution is varied: . The exponential growth phase is seen to be independent of the resolution, yielding an almost identical value for the Lyapunov exponent, varying less than 6% for all the cases. In addition, these simulations show a similar independence of error growth rates to perturbation wavenumber (not shown) as was observed for the original LES runs (Fig. 2, top).
Fig. 7.
Fig. 7.

Comparison of error growth between simulations of identical Reynolds number (Re ≈ 200) while varying the resolution . The resolution independence of the growth rates is clearly seen, as the measured Lyapunov exponent varies less than 6%.

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 in all the simulations, the details of which are given in Table 1. The growth of error again shows a sensitive Reynolds number dependence, and its rate sharply increases for the higher-Re simulations, qualitatively comparable to the results in the previous section.

Fig. 8.
Fig. 8.

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

Table 1.

Varying Reynolds number simulation details, including the grid spacing Δx, Δy, and Δz (m), and the ratio of the characteristic grid length to the Kolmogorov scale (zi = 960 m).

Table 1.

The error growth rate can be nondimensionalized as . Figure 9 shows that strongly depends on Re without a clear sign of convergence. This behavior is qualitatively comparable to what was observed in the original LES runs.

Fig. 9.
Fig. 9.

The nondimensional growth rate plotted against the Reynolds number. The blue points represent simulation results with , and red points represent . The dashed line shows a power law for reference (see the discussion section).

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

The effect of changing the convective time scale is investigated by performing a second set of simulations with identical Re but with a lower surface heat flux , halving the value of and, consequently, doubling , which is also shown in Fig. 9. The growth rate of error is twice as slow in the second set of experiments: that is, is smaller to such an extent that the relationship remains similar to the first set of experiments, justifying, a posteriori, the choice of quantities in (10) to nondimensionalize the results.

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 (, , and ; hence and ); naive extrapolations yield very small error-doubling times, which are hardly reasonable because any larger-than-infinitesimal initial error would impact the entire system almost instantly. However, the volume-averaged error does inform on the specific scales being affected; after all, the volumetric error could be dominated by a huge number of tiny-amplitude errors on tiny scales. Such an error distribution is hardly relevant for the predictability one is typically interested in, that is, errors on length scales on the order of .

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

To quantify the prediction horizon for the different scales of the system, we find it useful to study the spectral decorrelation of the 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:
e12
where is the (vertically averaged) correlation at wavenumber k and time t; is the cospectrum of and ; and and are the power spectra of and , respectively. The power spectrum decomposes the variance , where can be expressed in terms of the two-dimensional Fourier transform by
e13
The cospectrum is obtained by replacing by the real part of in (13).
The perturbation field employed in the simulations below will no longer be monochromatic (i.e., characterized by a single wavenumber ), but rather characterized by a full spectrum chosen such that every wavenumber experiences the same initial decorrelation . This is accomplished by formulating the perturbation as a spectrally modulated version of a white noise field. Denoting the white noise spectrum as , we take
e14a
e14b
where denotes the spectral modulation yet to be determined. Because there is no correlation between and the white noise field, one can work out (12) to see that choosing the modulation α as
e15
causes the desired, spectrally uniform, initial decorrelation . Hereinafter, we take very close to unity, , so the perturbation decorrelates each scale only minutely.

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.

Fig. 10.
Fig. 10.

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

Fig. 11.
Fig. 11.

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 , which are seen to still grow at a very slow rate. These scales correspond to the largest structures in the simulated atmospheric boundary layer, with scales several times . The overall behavior of the error spectrum follows the Lorenz (1969) view on limited predictability systems, also seen in the study of Rotunno and Snyder (2008) for surface quasigeostrophic flow with a shallower than −3 spectral slope near the surface. We now investigate the spectral correlations in time, which brings us closer to estimating a time scale for saturation per scale.

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, . With time, the correlation is seen to drop for all wavenumbers at different rates, the higher wavenumbers rapidly dropping to within several . These are followed by the smaller wavenumbers, which remain correlated even after a long time. The solid line at marks a (admittedly arbitrary) cutoff value for the correlation used to estimate a decorrelation time for each scale. We are now in a position to reevaluate the effect of the Reynolds number on the error growth by studying individual scales. To this end, we analyze the decorrelation time per wavenumber k. The decorrelation time is defined as the time it takes for the spectral correlation to decrease from its seeding value to the critical value (see gray line in Fig. 12) {i.e., }. The obtained spectral decorrelation times are shown in Fig. 13 for various Reynolds numbers. The figure hints at two regimes. First, there appears to be a Reynolds-dependent regime, confined to scales smaller than the boundary layer depth, . The decorrelation time of these scales decreases with increasing Reynolds numbers. The larger scales , however, seem to represent a Reynolds-independent regime, as their decorrelation times are very similar for all Re values. A difficulty of our “pristine” dry convective boundary layer setup, however, is that there is hardly any “dynamics” at scales (e.g., Jonker et al. 1999). The conclusion for the large scales might therefore change for boundary layers that do have interesting dynamics on scales larger than , such as cloud-topped boundary layers (e.g., De Roode et al. 2004). In such situations, the decorrelation times of the large-scale structures might very well be faster than observed here for a dry convective boundary layer.

Fig. 12.
Fig. 12.

The correlation coefficient ρ between the two simulations calculated using (12) is shown in time for all wavenumbers k. The initially imposed small decorrelation is shown as the solid black line and a critical value as the solid gray line. The correlations are seen to drop in time and are faster for the higher wavenumbers.

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

Fig. 13.
Fig. 13.

The decorrelation time for all scales , calculated as the time from seeding to the correlation ρ dropping to a critical value , is shown for different Reynolds number simulations. Scales smaller than show a Reynolds number dependence and decorrelate faster with increasing Re.

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 , with the global Lyapunov exponent describing the constant rate of error growth. This growth rate was found to be independent of the perturbation wavelength, consistent with the findings of Lorenz (1969) in the range. The independence of the perturbation length scale was recently revisited by Durran and Gingrich (2014) and can be understood by the rapid downscale propagation of error to the smallest scales, which subsequently govern the error growth during the early stages. This notion led Durran and Gingrich (2014) to place the (ir)relevance of the proverbial butterfly in perspective, addressing predictability in comparison to the substantial uncertainties that are present in the larger scales.

During the early stages, the error growth in the turbulence-resolving twin simulations studied in this paper was found to display a strong dependence on the Reynolds number. An explanation will follow after looking more closely to the second phase of error evolution; here, the growth rate successively decreases as the faster-growing smaller scales begin to saturate, a result consistent with the Lorenz (1969) framework for flows with limited predictability. Similar behavior was observed for surface quasigeostrophic flow studies of Rotunno and Snyder (2008) and Morss et al. (2009), as well as resolution-dependent studies of Simmons and Hollingsworth (2002) on the ECMWF model. A similar notion was also recently propounded by Fang and Kuo (2015) using the general systems theory. The dynamics of error growth in the second regime has been conceptually presented in Fig. 14. One can attribute to an eddy with scale and velocity a time scale
e16
The important point is that to each scale of motion l, a different Lyapunov exponent can be associated, . As the error growth of large scales is slower, the error in scale l will first saturate before the error dynamics of the next larger scale comes into play, at least conceptually. Using as the time for scale l to influence scale (or wavenumber k to influence ), the total time T for wavenumber to affect via consecutive scale interactions [i.e., doublings], can then be written as an integral in wavenumber space (e.g., Vallis 2006):
e17
Upon inserting the 3D form of the energy spectrum, , with ε the dissipation rate of turbulent kinetic energy, and taking , one arrives at . For the situation studied in this paper, and ; hence, , or similarly, is constant (i.e., there is no Reynolds number dependence). Equation (17) provides a decent representation of the second stage, but it is not appropriate for the early stage of error evolution. During the initial stage the error growth is dominated by the Lyapunov exponent of the smallest scale of the flow, the Kolmogorov scale η, with . The second stage will only commence when the error in the smallest scale is approaching saturation. In our simulations, the twins were perturbed with a very small amplitude, so the early stage is relatively long and represented by a single Lyapunov exponent : that is, the error grows as . Using inertial subrange theory, one can show the Reynolds number dependence in the early stages of error evolution, since
eq1
This power-law dependence has been indicated for reference in Fig. 9.
Fig. 14.
Fig. 14.

Conceptual image describing the dynamics of error growth in turbulence.

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

For very small , the time of the early stage is . So in a realistic convective boundary layer with , the first stage of error growth is rapidly surpassed—after all, the time scale . However, even in state-of-the-art turbulence-resolving simulations, the Reynolds number is modest, and hence the first stage of error growth will be relatively inflated. Figure 8 is most instructive in this regard, where the second stage of error evolution is almost entirely dwarfed by the first stage. One could compensate for this effect by adapting the perturbation amplitude , using much higher values than we did, such that the error in the smallest resolved scales is already near to saturation. One then directly enters, arguably more relevant for realistic atmospheric turbulence, the Lorenz (1969) error dynamics.

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.

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