## 1. Introduction

In the last decade, interest has grown in the development and subsequent application of cloud-resolving models (see, e.g., Kain et al. 2006; Colle et al. 2005; Benoit et al. 2002) and of their associated data assimilation techniques (e.g., Ducrocq et al. 2000; Ferretti and Faccani 2005) and ensemble prediction systems (Kong et al. 2006). The increase in model resolution to a few kilometers allows an explicit (or partly explicit) treatment of moist convective processes in place of uncertain parameterizations (Molinari and Dudek 1992) and a better representation of topography and surface fields. It is hoped that cloud-resolving simulations will improve the forecast skill associated with smaller-scale phenomena, such as moist convection, and more generally of quantitative precipitation forecasting (QPF; see, e.g., Fritsch and Carbone 2004; Ebert et al. 2003; Mass et al. 2002).

The development of short-range storm-scale applications poses a number of computational and theoretical challenges, one of these concerning predictability. Smaller-scale phenomena may exhibit predictability horizons as short as 75 min for a 20-km wavelength (see Lorenz 1969; Lilly 1990). While confirmation of these theoretical estimates is not yet available, many real-case or idealized studies have documented the high sensitivity of simulated moist convection with respect to the environmental conditions, modeling choices, and computational procedures (see, e.g., Martin and Xue 2006; Farby 2006; Fuhrer and Schär 2005; Park and Droegemeier 2000; Crook 1996; Stensrud and Fritsch 1994).

The encountered sensitivities and the shorter predictability horizons call for application of the ensemble approach in short-range cloud-resolving quantitative precipitation forecasting (Brooks et al. 1992; Elmore et al. 2002; Kong et al. 2006). While ensemble prediction systems (EPS) are well established operationally for synoptic-scale medium-range weather forecasts (see, e.g., Molteni et al. 1996; Toth and Kalnay 1997; Houtekamer et al. 1996 and the reviews by Palmer 2000 and Ehrendorfer 1997), the design of short-range cloud-resolving EPS is problematic due to the limited CPU resources, the poor knowledge of the mechanisms promoting rapid error growth and propagation, and the various sources of uncertainty (Fritsch and Carbone 2004; Kong et al. 2006). In particular, several studies (e.g., Zhang et al. 2002, 2006a, b; Walser et al. 2004; Hohenegger et al. 2006) have underlined the importance of convective instability in disrupting mesoscale predictability. The existence of such alternate amplification mechanisms, in place of the baroclinic instability active for synoptic-scale medium-range forecasting (see, e.g., Molteni and Palmer 1993), may require the development of new EPS strategies.

It is the goal of this study to identify the mechanisms responsible for rapid perturbation growth in a mesoscale model case study involving moist convective instability. In particular, we seek to pinpoint propagation and amplification processes that may rapidly disrupt the predictability of the simulated precipitation field. We will also characterize the degree of linearity prevailing in our modeling system.

Our approach employs high-resolution cloud-resolving NWP integrations starting from different initial conditions, which are generated using three distinct perturbation methodologies. The first shifts the initialization time of the simulations, the second introduces Gaussian temperature disturbances at subjectively selected locations, and the third applies random numbers to the temperature field of a control integration. The main motivations for the chosen perturbation strategies reside in their easy implementation, their relatively large initial discrepancy, and the fact that they do not make any assumptions based on the simulated flow, especially concerning linearity. The Gaussian disturbances also represent a very convenient tool to trace the mechanisms promoting perturbation growth and propagation.

To simplify and clarify the analysis, we assume a perfect model and a perfectly predictable synoptic-scale situation, driving our model by identical lateral boundary conditions. Our study thus only investigates issues related to domain-internal error growth, up to the meso-*β* scale, and cannot address interactions between cloud and synoptic scales [see, e.g., Warner et al. (1997) for a general discussion of the effects of the lateral boundary conditions on a limited-area model solution]. Likewise, the generated ensemble and the employed sampling strategies are not meant for operational purposes. From a classical EPS perspective, the use of isolated Gaussian perturbations appears as particularly inappropriate due to the strong localization. An operational EPS would further need to account for both synoptic-scale uncertainties and model errors. These two terms might be sampled using a larger-scale ensemble prediction system, stochastic physics, and/or a multimodel approach (e.g., Molteni et al. 2001; Marsigli et al. 2001, 2005; Grimit and Mass 2002).

Nevertheless, it is known that the growth of domain-internal mesoscale perturbations alone is sufficient to disrupt the predictability. As shown in previous studies (see Hohenegger et al. 2006; Richard et al. 2007; Walser et al. 2004; Walser and Schär 2004), resulting predictability limitations are highly relevant for quantitative precipitation and flood forecasting. Depending on the simulated synoptic situation, meso-*γ*-scale initial uncertainties can grow upscale, contaminate the meso-*β* scale of the flow, and disrupt predictability even at scales encompassing the whole Alpine region (see in particular the July case studied by Walser et al. 2004). Laprise et al. (2000) even indicated that the smaller scales, which are not forced by the lateral boundary conditions and which are meant to be improved by a limited-area model, are inherently unpredictable. The importance of cloud-scale initial uncertainties has also been well recognized for data assimilation purposes (see, e.g., Ducrocq et al. 2000; Sun 2005).

The structure of this paper is as follows. Section 2 describes in more detail the experimental setup and the chosen perturbation methodologies. The results are presented in section 3. Section 3a introduces the simulated case, while sections 3b and 3c highlight the comparison of the sampling strategies and the related question of linearity. The results are interpreted in dynamical terms in section 4. Conclusions and some further remarks are given in section 5.

## 2. Methods

### a. Experimental setup

We use the nonhydrostatic Lokal-Modell (LM) developed by the German Weather Service (see Steppeler et al. 2003) in a similar setup as in Hohenegger et al. (2006) to simulate one case at high resolution taken from the Mesoscale Alpine Programme (MAP). The MAP field campaign (Bougeault et al. 2001) took place in the Alpine region from 7 September 1999 to 15 November 1999 and was designed to improve the understanding of mountain-related weather phenomena (see, e.g., Richard et al. 2007; Rotunno and Houze 2007). The chosen case is the MAP intensive observing period 3 (IOP3; 24–26 September 1999).

The employed LM domain covers most of the Alps (see, e.g., Fig. 1a) and includes 401 × 301 grid points in the horizontal as well as 45 sigma-pressure levels in the vertical. The horizontal grid spacing amounts to 0.02° (about 2.2 km), allowing the explicit treatment of moist convection. Note that a horizontal resolution of 2.2 km cannot resolve all the clouds, as the effective resolution of our model amounts to about 15 km (7Δ*x*; see Skamarock 2004; Hohenegger et al. 2006). Our simulations might thus be viewed as “convection permitting” rather than “convection resolving.”

The boundary conditions of the high-resolution integrations are derived from a coarser simulation. The latter is integrated over Europe from 1200 UTC 24 September to 1200 UTC 26 September with the LM at a resolution of 0.0625° (7 km). Its initial and lateral boundary conditions are obtained from the operational European Centre for Medium-Range Weather Forecasts (ECMWF) analysis. Because of its 7-km grid spacing, convective processes are parameterized after Tiedtke (1989).

### b. Perturbation methodologies

To isolate the key mechanisms related to error growth and propagation in a cloud-resolving model, three distinctive methodologies (see Table 1) have been applied to perturb the initial conditions of the 2.2-km LM integrations. All of these assume a perfectly predictable synoptic-scale situation (identical lateral boundary conditions) and a perfect model (see section 1).

*ϕ*(generally temperature).

*T*

_{CTRL}. The perturbed temperature is given as

*x*

_{o},

*y*

_{o}) indicates the location of the perturbation center (measured from the lower left corner of our computational domain),

*A*

_{o}its amplitude, and

*σ*its width. Two vertical profiles

*η*(

*z*) are considered:

*A*

_{o}yields GAUSS1_NEG. GAUSS1_x8 and GAUSS1_NRW are obtained by increasing the amplitude

*A*

_{o}and by narrowing the width

*σ,*respectively. The restriction to pure surface disturbances [Eq. (3)] gives GAUSS1_SFC, while GAUSS2 follows by shifting the location of the perturbation center. Following Eq. (1), the phase space divergence between perturbed

*ϕ*and control

_{p}*ϕ*simulations is measured by their absolute difference

_{c}The modification of the temperature is achieved before the standard initialization procedure of the LM, a digital filter after Lynch (1997), takes place. This automatically removes a part of the provoked unbalance and correspondingly adjusts temperature, pressure, and velocity fields. The initialization procedure alters the specified Gaussian functions, especially in GAUSS_SFC due to the missing coherence in the vertical structure. Concerning the remaining integrations, the digital filter tends to reduce the maximum amplitude *A*_{o} and to redistribute the imposed differences. A further set of simulations has been conducted with the initialization procedure switched off. Since they lead to equivalent results, the latter integrations are not considered here.

The third and last method makes use of a random number generator (simulation RAND; see Table 1). The random numbers are generated between −0.1 and 0.1 K with a mean of 0 K. They are added to the 3D temperature field of the control integration SHIFT6 before calling the initialization procedure.

## 3. Results

### a. The MAP IOP3 case

Figure 1 gives an overview of weather and predictability prevailing in MAP IOP3, as deduced from the shifting initialization technique. It pictures ensemble mean accumulated precipitation and precipitation rates averaged over the two subdomains Basel and Lago Maggiore.

MAP IOP3 is characterized by two elongated precipitation bands: one over the Massif Central–Jura region in France and one over the Ticino in Switzerland (see Fig. 1a). Observations (see, e.g., Fig. 3 of Hohenegger et al. 2006) support the presence of the two bands.

The first precipitation band is the result of frontal activity associated with a low pressure system located over the Atlantic Ocean. The front propagates in a southward direction and hits the Jura chain and the Massif Central during the late afternoon of 25 September. This produces locally heavy stratiform rainfalls, which succeed to a prefrontal convective episode (in the morning). The second rainband (over the Ticino) is related to the advection of warm, moist, and potentially unstable air ahead of the front. This southerly flow stems from the Mediterranean Sea, destabilizes the atmosphere, and triggers convective cells over the Alps during the evening of 25 September (see Pujol et al. 2005). Later, the precipitation amounts increase due to the arrival of the front.

The ability of the perturbed LM simulations in reproducing the convective and the stratiform episode can be derived from the corresponding time series in Figs. 1b and 1c. The convective activity, both in the morning of 25 September over the Basel area and during the evening over the Lago Maggiore subdomain, induces a larger spread between the members. For instance, the accumulated precipitation between 0300 and 1200 UTC over the Basel area (Fig. 1b) ranges between 9 and 18 mm. Walser et al. (2004) even noted that for this particular event, predictability limitations extend up to the scale of the whole mesoscale convective region (almost 200 km). The purely stratiform precipitation, in opposition, exhibits a smaller spread in Fig. 1b, with accumulated precipitation values ranging between 7.5 and 9 mm over the Basel subdomain from 1800 to 2200 UTC. The timing of the front almost coincides in the different simulations, while the precipitation peak slightly varies from member to member.

### b. Comparison of the perturbation methodologies

After having illustrated the typical behavior of the LM in simulating MAP IOP3, we devote our attention to a more detailed comparison of the perturbation methodologies defined in section 2b to assess the rapidity and linearity of error growth in our modeling system for the investigated case. As the LM is able to reproduce the turbulent kinetic energy spectrum down to a horizontal scale of 15 km (not shown), it is expected that the LM reasonably represents the type of error growth we are interested in. For investigation, we focus in the following on the temperature field. The consideration of the precipitation field alone is not conclusive since the computation of its spread is restricted to the raining regions.

Figure 2 shows the temperature spread *D* between GAUSS1 and SHIFT6 (middle column) and between RAND and SHIFT6 (right column). Despite the distinctive structures of the initial perturbations, comparison of the three columns reveals a surprisingly high level of agreement between the methodologies at later time. Already after 11 h, all of them serve to pinpoint the same regions of the flow as being of limited predictability, for example, southern Germany, central Switzerland, and parts of the Alps, thus revealing an extremely rapid error growth. The meso-*β*-scale patterns of predictability quickly converge with increasing lead time, while the finescale structure varies somewhat from methodology to methodology. In particular, the shifting initialization technique leads to a smoother field with smaller maxima and broader minima. This mainly follows from the comparison of a spread built upon six members against an absolute difference calculated between two simulations.

_{S}and, for the remaining simulations of Table 1, of the rms temperature difference

_{D}computed as

*η*and

_{i}*η*denote the number of grid points along the

_{j}*i*and

*j*directions.

Figure 3 pictures a rapid convergence of the rms values, confirming the visual impression of Fig. 2. Together, they demonstrate that after a transient phase, there is some equivalence between the three perturbation methodologies, in the sense that they all pinpoint the same meso-*β*-scale regions of the flow as suffering from predictability limitations. This behavior reflects the known property of the atmospheric dynamics to sustain error growth into a small subspace (see, e.g., Kalnay 2003), but the obtained convergence is extremely fast despite the large discrepancies existing at initial time between the perturbation methodologies. Consideration of the wind field instead of the mass field leads to similar conclusions, whereas the wind field tends to saturate 1–2 h before the temperature field (not shown). Note that _{D} converges toward _{S} in Fig. 3, which can be understood in terms of the statistics of random numbers. As shown in the appendix, the individual members essentially forget about their initial perturbations after some time and may be equivalently interpreted as random realizations.

Figure 3 further reveals that the obtained domain-mean error growth rates depend upon the initial perturbation amplitudes, as previously found with other models (see, e.g., Zhang et al. 2003). In particular, the shifting initialization technique yields almost no growth in an area-mean sense (as exhibited by Fig. 3) since the imposed initial differences are larger than the error saturation value. The error saturation value itself is determined by the synoptic-scale environment as communicated by the lateral boundary conditions. Locally, however, errors may grow even late in the integration period (see, e.g., Figs. 1c and 2). In opposition, GAUSS1_SFC exhibits both the smallest initial perturbation amplitude and the largest growth rate.

### c. Linearity

The rapid agreement between predictability measures in Figs. 2 and 3 implies large nonlinearities. In particular, Fig. 2 shows that a one-to-one mapping between initial uncertainty and future state appears lost by 11 h. To further address the degree of linearity in our modeling system, we consider the correlation between twin perturbations with inverted signs. Figure 4 shows the resulting time series of the correlation coefficient computed between the temperature differences of GAUSS1 and of GAUSS1_NEG with respect to SHIFT6. Both Gaussian perturbations are initially perfectly anticorrelated, consistent with their definition in Table 1. With increasing time, the initially negative correlation coefficient turns to positive values and ends up stagnating between 0.4 and 0.6 after 11 h of integration. This saturation of the correlation coefficient around 0.5 is as to be expected from random realizations (see appendix).

The initial phase of rapid anticorrelation decrease, from 0 to 11 h, suggests the presence of strong nonlinearities in our modeling system. This behavior is not due to the specifics of our perturbation approach that uses ad hoc perturbations, which probably do not project upon the most unstable subspace and thus might be washed out by numerical diffusion. To verify this, a further set of simulations has been conducted using temperature disturbances that had some time to grow (“bred” perturbations). The integrations start at 0000 UTC (instead of 2100 UTC) and are perturbed with the positive and negative differences obtained between GAUSS1 and SHIFT6 at 0000 UTC. Results are very similar to the ones shown in Fig. 4 and thus confirm that the exhibited nonlinearity is intrinsic and not due to spinup and diffusion effects.

## 4. Dynamical interpretation

Both the documented rapid error growth and strong nonlinearities are embedded into the dynamics of perturbation growth and propagation. The propagation and amplification of small-scale perturbations in our modeling system is the topic of this section. For investigation, we focus on the evolution of the temperature differences computed between GAUSS1 and SHIFT6. Similar experiments have been conducted with other Gaussian perturbations and led to equivalent conclusions.

The mechanisms acting to disperse initial localized perturbations in our modeling system are illustrated by Figs. 5 and 6. Figure 5 shows the spatial distribution of nonzero temperature differences captured after 0 and 6 min of integration time, respectively. Figure 6 contains two maps of the temperature differences obtained near the tropopause at 6 and 7 h with the corresponding vertical profile shown on the right-hand side.

Comparison of the evolution of the difference patterns in Figs. 5 and 6 reveals distinctive characteristics of the propagating signal. From Fig. 5, an isotropic expansion of the imposed temperature disturbance can be deduced. The signal propagates over approximately 110 km in 6 min, giving an estimated velocity of 306 m s^{−1}. The perturbations at the leading edge of this signal have amplitudes in the order of 0.001 K, compared to 1 K for the core of the initial perturbation. In opposition, Fig. 6 pictures a wavelike sequence of positive and negative differences in the middle to upper troposphere. In the lower layers (below 4 km), this signature is overpowered by some other perturbations. The amplitude of the wavelike signal is significantly larger than that shown in Fig. 5, but the associated phase velocities are slower (in the order of 60 m s^{−1}).

The small-amplitude, rapid, and isotropic spreading depicted by Fig. 5 corresponds to the typical properties of propagating sound waves, while the wavelike signature in Fig. 6 is reminiscent of gravity waves. Sound waves are part of the solution due to the use of a compressible model. It is clear that additional wave types are consistent with the characteristics derived from Figs. 5 and 6, for example, Lamb waves instead of sound waves. The exact nature of the signal is not decisive; the main result is the presence of both fast small-amplitude modes (sound like) as well as slow intermediate-amplitude modes (gravity wave like).

Numerical effects may also contribute to some propagation. In our model, numerical propagation is in principle feasible with velocities of 733 and 183 m s^{−1} given small and large time steps of 3 and 12 s, respectively. Detailed analysis of our simulations shows that some propagation occurs at speeds larger than sound, but the signal’s amplitude is very small and thus overpowered in Figs. 5 and 6 by slower modes.

Growing disturbances excite both sound and gravity waves, which propagate the information through our computational domain. Due to the use of perfect lateral boundary conditions and owing to the limited lifetime of the perturbations, error growth is best sustained if propagating disturbances are able to generate new perturbations. The triggering potential of gravity and especially of sound waves is examined in more detail in Fig. 7. The two panels show maps of the maximum (in the vertical) of the absolute temperature difference, valid after 1 and 2 h of integration, respectively. We are using maxima and absolute values since we are aiming at tracking the release of new disturbances by the propagating signal. Figures 7a,b point to the presence of numerous isolated perturbations. Some of them seem to pop up from the background noise level without evident precursors, while others grow in the vicinity of existing perturbations. Owing to these characteristics, the first type of perturbation growth results from sound wave activity, while the second one follows from gravity waves and possibly advective effects.

The hot spots of remote error growth (see Fig. 7b) are located over the Alpine ridge and over the Jura region. The latter region was characterized by convective instability in the morning of 25 September (see section 3a). Propagating small-amplitude perturbations may thus experience dramatic amplifications due to forced ascent over the topography and/or the triggering of moist convective instability. Note that the presence of convective instability is a necessary but not sufficient condition to sustain substantial error growth in our simulations (see Hohenegger et al. 2006).

As indicated by Fig. 7, the conjugate effect of sound and gravity wave propagation together with the triggering potential of the topography and/or of moist convective instability ensures a quick propagation of the perturbation signal throughout the computational domain followed by localized exponential growth. In this way, even locally imposed small-amplitude perturbations only need a few hours to contaminate the whole computational domain and to significantly disrupt the predictability of the simulated flow. This explains the obtained rapid agreement between predictability measures in section 3b, and the rapid dominance of nonlinear effects as documented in section 3c.

## 5. Conclusions and further remarks

High-resolution NWP integrations have been employed to investigate aspects of perturbation growth and propagation in a cloud-resolving model. We assumed a perfect model and a perfectly predictable synoptic-scale situation by using identical lateral boundary conditions. The initial perturbations of our experiments have been generated by shifting the initialization time of the simulations, by imposing Gaussian temperature disturbances, or by applying stochastic initial perturbations.

Analysis reveals an extremely fast error growth and propagation. Initial perturbations may propagate throughout the domain within a few hours and amplify at far remote locations. For the considered event, which involves convection of intermediate intensity, the generated meso-*β*-scale error patterns converge within 11 h, irrespective of vastly different initial perturbations. This rapid convergence further expresses the high degree of nonlinearity prevailing in our modeling system, as the individual ensemble members essentially forget about their initial perturbations within 11 h. It also confirms the general expectations that the degree of linearity decreases with increasing model mesh size (e.g., Gilmour et al. 2001; Ancell and Mass 2006).

Detailed analysis of the nature of perturbation growth and propagation isolates three elements: First, the rapid propagation of sound and gravity wave modes quickly communicates initial uncertainties throughout the computational domain. Second, the triggering of moist convective instabilities may lead to sizeable exponential amplifications of initially small-amplitude signals. Third, the meso- and synoptic-scale environment (as communicated by the lateral boundary conditions) determines the saturation amplitude of so-generated local perturbations and the preferred locations of error growth.

Sound waves, which are typically not allowed in anelastic or in lower-resolution hydrostatic models, accelerate the error growth in cloud-resolving simulations and thus shorten predictability. They have been traced back using Gaussian perturbations, but they are naturally excited by unbalances present in the initial conditions of an NWP model. It has also been shown in a linear framework that a convective cell, due to its unbalanced injection of mass and temperature into the atmosphere, generates a range of acoustic and gravity waves (Chagnon and Bannon 2005). As the atmosphere supports sound wave propagation, the question arises whether such features have to be carried along or should rather be filtered in the governing set of equations, depending on the expected effects of acoustic modes on the real weather and on its predictability. This question might nevertheless be a hypothetical one. In particular, anelastic models do not support sound propagation. Yet the anelastic equations involve a diagnostic elliptic equation, which in principle implies instantaneous communication of a local signal throughout the computational domain. Filtered models may thus permit even faster propagation than a compressible nonhydrostatic formulation as employed in our experiment. In practice, the specific nature of rapid propagation might not be so relevant (be it sound waves or some artificial propagation associated with filtered equations or numerical schemes), but the existence of rapid propagation beyond gravity wave propagation in an environment susceptible to the triggering of rapid error growth is.

From a practical point of view, it is suggested that the rapid propagation may complicate the design of ensemble prediction systems, data assimilation, and targeting techniques, since far remote initial uncertainties may disrupt the skill of a forecast within a few hours. These difficulties are emphasized by the underlying strong nonlinearities, which are likely to have a detrimental impact on the use of the tangent-linear approximation for cloud-resolving NWP. Also, the ensuing extremely short predictability horizons tend to confirm the pessimistic view of Lorenz (1969) concerning the ability of increased horizontal resolution to improve the forecast skill. This conclusion nevertheless has to be put into perspective, in the sense that not all the weather situations are subjected to moist convective instability.

## Acknowledgments

We are indebted to the German Weather Service (DWD) and MeteoSwiss for providing access to and support for the use of the LM. The numerical simulations have been performed on the NEC SX-5 at the Swiss National Supercomputing Centre (CSCS). The authors would also like to thank Oliver Fuhrer, Bodo Ahrens, and Reinhard Schiemann for constructive discussions as well as Daniel Lüthi for technical support. Comments provided by two anonymous reviewers and the editor substantially improved the manuscript and are gratefully acknowledged.

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## APPENDIX

### Statistical Derivation

The equivalence between the simulations in Fig. 2 and the saturation of the rms values and of the correlation coefficient in Figs. 3 and 4, respectively, may be interpreted with the following statistical model.

*X*

^{1}

_{i}, . . . ,

*X*

^{p}

_{i}be the temperature values obtained at a grid point

*i*(or over a larger grid box) in

*p*integrations (e.g., SHIFT6, GAUSS1, etc). These are decomposed as

*μ*denotes some constant value. Next, the deviations

_{i}*X*

^{1′}

_{i}, . . . ,

*X*

^{p′}

_{i}are assumed to be random realizations of an identical distribution

*, that is,*

_{i}*E*, Var, and Cov denoting expected value, variance, and covariance. The joint distribution

*N*grid points is

To test this concept, we estimate the spatially varying distribution parameters (i.e., the moments of the * _{i}*’s) from our simulations and create random realizations (pseudosimulations) therefrom. The

*’s are specified as normal distributions. The result of the procedure is illustrated in Fig. A1 with two maps of the differences obtained between the pseudointegrations*

_{i}*X*

^{2}−

*X*

^{1}and

*X*

^{3}−

*X*

^{1}valid by 22 h. Figure A1 is to compare and agrees qualitatively with Fig. 2, bottom row. In particular, Fig. A1 demonstrates that, even if at smaller scales the integrations are random realizations, there is equivalence at larger scales. The latter follows from the spatially varying variances Var〈

*X*

^{p′}

_{i}〉, which force divergence (large variance) over localized and congruent regions.

_{S},

_{D}, and

*X*

^{2′}−

*X*

^{1′}〉 = Var〈

*X*

^{2}′〉 + Var〈

*X*

^{1}′〉 = 2Var〈

*X*

^{1}′〉 [see Eq. (A5)]. Equations (A6) and (A7) show that, given random i.i.d. deviations,

_{D}should scale with

_{S}. This scaling was successfully applied in Fig. 3.

*U*−

*V*〉 = Var〈

*U*〉 + Var〈

*V*〉 − 2Cov〈

*U*,

*V*〉 and Cov〈

*U*−

*V*,

*W*−

*V*〉 = (½) (Var〈

*U*−

*V*〉 + Var〈

*W*−

*V*〉 − Var〈

*U*−

*W*〉), Eq. (A8) may be written as

Overview of the specified perturbation methodologies (see section 2b) with the corresponding simulation names used to perturb the initial conditions of the 2.2-km LM integrations. (Shift ini. = shifting initialization, Gauss. Pert. = Gaussian perturbations, and Rand. nbres = random numbers)