Impact of spectral nudging and domain size in studies of RCM response to parameter modification
Abstract
The paper aims at finding an RCM configuration that facilitates studies devoted to quantifying RCM response to parameter modification. When using short integration times, the response of the timeaveraged variables to RCM modification tend to be blurred by the noise originating in the lack of predictability of the instantaneous atmospheric states. Two ways of enhancing the signaltonoise ratio are studied in this work: spectral nudging and reduction of the computational domain size. The approach followed consists in the analysis of the sensitivity of RCMsimulated seasonal averages to perturbations of two parameters controlling deep convection and stratiform condensation, perturbed one at a time. Sensitivity is analyzed within different simulation configurations obtained by varying domain size and using the spectral nudging option. For each combination of these factors multiple members of identical simulations that differ exclusively in initial conditions are also generated to provide robust estimates of the sensitivities (the signal) and sample the noise. Results show that the noise magnitude is decreased both by reduction of domain size and the spectral nudging. However, the reduction of domain size alters some sensitivity signals. When spectral nudging is used significant alterations of the signal are not found.
Keywords
Regional climate models Parameter perturbations Internal variability Spectral nudging Domain size1 Introduction
Nested limitedarea Regional Climate Models (RCMs) are models that dynamically downscale global General Circulation Model (GCM) simulations or objective analyses to highresolution computational grids, using a highresolution representation of the surface forcing and model dynamics. RCMs require the information on some prognostic variables as their lateral boundary conditions (LBC). The choices of integration domains and nesting techniques are free parameters of RCMs. The optimal integration domain depends on the particular situation, although there are some general recommendations that can facilitate user’s judgment (e.g., Laprise et al. 2008). For example, Leduc and Laprise (2008) showed that the use of a too small domain could result in the simulations being deficient in finescale variance. It has been also noted that in large continentalscale domains RCM largescale variables can considerably drift from the driving fields, which can then result in appearance of large spurious gradients in the vicinity of the outflow boundaries. Spectral nudging (SN; Von Storch et al. 2000; Biner et al. 2000) has been employed to ensure that the model solution remains close to the largescale components of the driving fields over the entire domain. However, the use of spectral nudging remains an open issue. Alexandru et al. (2009) raised concern that the application of the SN could suppress the proper generation of finescale features. However, Colin et al. (2010) did not find SN to be detrimental on the modelling of extreme precipitation.
The choice of the integration domain and the use of spectral nudging can have a large impact on the RCM internal variability. Internal variability arises due to the nonlinear, chaotic nature of atmospheric models: any perturbation; however, small it is in magnitude, provokes the trajectories of the model solution in the phase space to diverge in time. In autonomous Global Circulation Models (GCMs) the difference between two simulations conducted with the same model but departing from initially slightly different states is on average as large as the difference between two randomly chosen GCM states, given a specific season. Internal variability also emerges in RCMs but, typically, it is smaller than in GCMs; the advection of information prescribed as the LBC keeps the evolution of the RCM internal variability somewhat bounded (e.g., Giorgi and Bi 2000; Caya and Biner 2004). However, intermittently in specific areas of the integration domain it can achieve values as large as in GCMs (Alexandru et al. 2007). Its time evolution appears to depend on the synoptic situation enforced by the driving fields (e.g., LucasPicher et al. 2008b; Nikiema and Laprise 2010) and is scale selective (Separovic et al. 2008). Reduction of domain size or the application of spectral nudging can both considerably reduce internal variability in RCMs (Alexandru et al. 2009). Thus, the average amplitude of internal chaotic variations appears to be in RCMs, to a certain extent, a controllable parameter. This fact may be of particular interest in studies oriented to RCM testing and modification.
The sensitivity of a RCM to any change in its structure and configuration, such as a modified parameterization or a perturbation of its tuneable parameters, generally consists of the response of the simulated variables to the modification (signal), as well as of internal variability noise. Since the work of Weisse et al. (2000) it has been widely acknowledged that estimation of the signal in the temporal evolution of the RCM variables requires ensemble simulations that can be generated, for example, by imposing perturbations to the initial conditions of both the control and the modified model versions. Internal variability deviations are partly filtered in the ensemble mean depending on the ensemble size, as the variance of the sample mean of a collection of independent and identically distributed random variables is inversely proportional to the sample size (e.g., Von Storch and Zwiers 1999). When the signal is small or the internal variability is large, ensembles of large size are needed in order to obtain statistically significant estimates of the simulation differences resulting from the model modifications. For sufficiently long integration times, internal variability deviations are substantially reduced in the time average. However, estimation of the time averages computed over shorter periods from years to a decade also necessitates sampling of the internal variability deviations, since it can be still nonnegligible in the time average of the single model run, especially for finescale variables such as precipitation (de Elia et al. 2008; LucasPicher et al. 2008a, b). When considering the difference between the time averages in the control and a modified model version, the variance introduced by the internal variability is twice as large as that in the time average in each model version, due to the aggregation of error through the difference terms.
Providing statistically significant estimates by means of ensemble simulations or longer integration periods for the control and modified model versions is hence computationally time consuming. While this issue might be of little relevance when the RCM is to be tested for a single modification, it can represent a hindrance in studies that require multiple testing of RCM response to modifications of a large number of parameters. This would typically be the situation in deliberate model tuning or in studies that address uncertainty originating in the RCM’s adjustable parameters wherein it is essential to identify in a highdimensional parameter space the plausible parameter perturbations that produce the largest response of the model (e.g., Sexton and Murphy 2003). The underlying methodological issue in such RCM studies is thus to optimize the use of computational resources by finding an appropriate test bed configuration (prototype simulation) that would be as inexpensive as possible in terms of the number of computational points and integration time and that can provide robust estimates of the model response to the modifications.
Our working hypothesis is that suppressing the internal RCM variability by means of domain size reduction or application of SN would allow for quantifying the signal with a smaller ensemble size and help to reduce the computational cost (Alexandru et al. 2007, Weisse and Feser 2003). The application of these methods to reduce internal variability noise requires better understanding of the ways they might alter the signal of RCM sensitivity to modification, e.g., by suppressing its magnitude. Too small domains are generally nonrecommended for climate simulations and sensitivity studies because of the spurious effects of the proximity of the lateral boundaries, finescale variance deficiency and lack of continentalscale interactions and feedback among the RCM variables (e.g., Jones et al. 1995; Seth and Giorgi 1998; Laprise et al. 2008). Results obtained in such domains are likely to be less realistic and difficult to extrapolate to the operational RCM simulations. However, when studying uncertainties originated in adjustable RCM parameters, a very large number of tests are required and the user may wish to conduct preliminary tests in a computationally inexpensive small domain. Outside this context the reduction of domain size and SN should not be considered as competing techniques to improve the signaltonoise ratio since the SN has not been shown to involve similar difficulties.
The manuscript is organized as follows. The model and the modifications performed on the model parameters in order to produce modified model versions and the experiments are described in Sect. 2. The analysis of model sensitivity to modification of parameters within different simulation configurations is carried out in Sect. 3. Summary and conclusions are provided in Sect. 4.
2 Experimental design
2.1 Model description
The model used in this study is the fifthgeneration Canadian Regional Climate Model (CRCM5; Zadra et al. 2008). It is a limitedarea version of the Canadian weather forecast model GEM (Côté et al. 1998); the model has a nonhydrostatic option, although this feature is not exploited here. GEM is a gridpoint model based on a twotimelevel semiLagrangian, semiimplicit time discretization scheme. The model includes a terrainfollowing vertical coordinate based on hydrostatic pressure (Laprise 1992) with 58 levels in the vertical, and the horizontal discretization on an Arakawa C grid (Arakawa and Lamb 1977) on a rotated latitudelongitude grid with a horizontal resolution of approximately 55 km and time step of 30 min. The nesting technique employed in CRCM5 is derived from Davies (1976); it includes a gradual relaxation of all prognostic atmospheric variables toward the driving data in a 10point sponge zone along the lateral boundaries. The lateral boundary conditions (as well as the initial conditions) are derived from ERA40 reanalysis (Uppala et al. 2005). Ocean surface conditions are prescribed from Atmospheric Model Intercomparison Project (AMIP) data (Fiorino 2004).
2.2 Experiments

P01—Threshold vertical velocity in the trigger function of the deep convection parameterization (Kain and Fritsch 1990).

P10—Cloud water to precipitation conversion time scale in the largescale condensation parameterization for stratiform precipitation (Sundqvist et al. 1989; Pudykiewicz et al. 1992).
Parameters’ settings used in different model versions
Model version  P10 (h)  P01 (m/s)  No. of ensemble simulations 

M00  2.8  3.4E−2  10 
M01  2.8  6.0E−2  5 
M10  10.0  3.4E−2  5 
Three sets of experiments are carried out in this study, all based on simulations conducted over a single year. For every model version multiple perturbed initialcondition ensemble simulations were performed. The initial conditions were perturbed initializing the model from November 01 1992 at 00UTC onward, 24 h apart. All the simulations, regardless of model version and initialization time, end on December 01 1993 at 00UTC. November 1992 is not considered in order to allow the spinup of the initial differences, thus leaving a 1year period for the analysis. The number of ensemble members is the same in all three sets; there are 10 members for the standard model version M00 and 5 members per each of the two perturbedparameter versions M01 and M10; the last column in Table 1 shows the ensemble size per each model version.
The second set of experiments, denoted as SYSN, is identical to SYNA in terms of its domain (NA; Fig. 1), model versions and number of ensemble members per every model version (Table 1); the only difference is that the spectral nudging (SN) was used. The nudging was only applied to the horizontal wind components, with the truncation at nondimensional wavenumber 4 (~1,500 km). The SN strength is set to zero below the level of 500 hPa and increases linearly with height, reaching 10% of the amplitude of the driving fields per time step at the top level. The choices of the truncation wavelength and the vertical profile of the nudging strength reflect the intention not to interfere with the model own interior dynamics at fine and intermediate spatial scales and in the lower half of the model’s atmosphere.
The third set of experiments, denoted as SYDS, consists in reducing the domain size. For every model version, the singleyear ensemble simulations are generated again, but over a domain of reduced size centred over the province of Quebec (without SN). The domain for the SYDS experiment consists of 70^{2} grid points and is shown in Fig. 1, including the 10point sponge zone.
3 Results
The variables selected for the analysis of results are seasonalaverage precipitation and 2 mtemperature. The analysis is focused on the influence of SN and domain size reduction on the model sensitivity to perturbations, internal variability noise and signaltonoise ratio. This section is organized as follows. Section 3.1 briefly reviews the sensitivities of CRCM5 seasonal averages to perturbations of the initial conditions and parameters, as a function of season and experimental configurations SYNA, SYSN and SYDS. Section 3.2 presents the spatial distribution of the internal variability noise in the three configurations. Sections 3.3 to 3.5 examine the spatial patterns of the sensitivity of CRCM5 seasonal averages to the parameter perturbations (signals), estimated with the difference of ensemble means of the control and modified model versions; these sections also provide the statistical significance of the sensitivity estimates and compare the signal patterns in the three simulation configurations. Section 3.6 examines the computational cost associated with different simulation configurations in terms of the minimum ensemble size necessary to achieve significant estimates.
3.1 Spread of differences excited by perturbations
Figure 2 shows that all rmsd exhibit an annual cycle with the maximum in summer and minimum in winter. The magnitude of the rmsd illustrates the physical significance of the model response to perturbations. The range of responses for precipitation and 2 mtemperature is 0–0.3 mm/day and 0–0.7°C in winter and 0.3–0.8 mm/day and 0.6–1.5°C in summer, respectively. Also the rmsd are in general the largest in the SYNA set and the smallest in the reduced domain size SYDS set. This holds for the three kinds of perturbations. The SYSN reduces internal variability noise (black marks) but it is less efficient in that than the reduction of domain size (SYDS); this being true for this case and different configurations of both spectral nudging and domain size could yield different results. The plots in Fig. 2 also provide a rule of thumb for the statistical significance of the response of the seasonal averages to the parameter perturbations: if differences between the control and perturbedparameter model version (red or blue marks) tend to lie above the maximum rmsd due to internal variability noise (black marks), given a season and simulation setup, this suggests the statistical significance of the corresponding model response to the parameter perturbation. As of precipitation (Fig. 2a), all signal rmsd in the SYNA setup are barely above noise level, except for condensationrelated parameter P10 in winter. The SN and reduction of domain size reduce the noise rmsd considerably but also the rmsd due to the parameter perturbations generally decreases. Thus, for precipitation in the SYSN and SYDS sets, the situation with statistical significance is not considerably changed. The exception is in summer when the convectionrelated parameter P01 produces significant rmsd, especially in the SYDS set. For 2 mtemperature (Fig. 2b) the responses to parameter perturbations are generally more statistically significant. Despite that, when the signal is weak, as P01 in winter, or noise very high, as in spring and summer, the parameterinduced rmsd appear not to be statistically significant. This also implies that the signaltonoise ratio varies for different CRCM5 variables.
It is difficult to infer from Fig. 2 whether the model response to parameter perturbations is on average smaller in the SYSN and SYDS sets or whether the lower rmsd in this set are a sole effect of reducing internal variability. We investigate this issue more thoroughly in the next subsections. Further, it can be seen that in winter (DJF), the perturbation P10 produces considerable and significant signals for both precipitation and temperature, while P01 produces a smaller response that is difficult to distinguish from internal variability. Perturbation P01 is related to the deep convection parameterization that is rarely active in winter over land. This perturbation produces a considerable and significant response over land only in the warmer half of the year.
The spatially averaged square differences may hide important information on the local behaviour of the CRCM5 response to the perturbations. In the following we begin the analysis of spatial patterns by first examining the noise level and then the spatial patterns of the model response to parameter perturbations are compared in the three experimental sets as a function of the parameter perturbation and season.
3.2 Noise level in the differences
The above considerations emphasize the need for ensemble integrations when studying RCM response to modification using singleyear simulations. It is not likely that any reasonable modification performed on the stateofthe art RCMs would produce larger differences in summer precipitation than the values of the noiseinduced standard deviation of the differences displayed in Fig. 3g. This implies a relative error of 100% in the estimates of the CRCM5 sensitivity to the parameter perturbations obtained without ensemble integrations. Time averaging over a season is not sufficient to ensure filtering of internal variability noise, and averaging over an ensemble or a longer period is required to assess the signal.
3.3 Signal P10 in winter
When the spectral nudging is applied (Fig. 4b,e) the statistical significance of the winter precipitation signal P10 is noticeably enhanced over the entire domain; it remains low only in the areas where the signal changes sign. The signal in the SYSN simulation is almost identical to that in SYNA over the west portion of the domain (Fig. 4a, b); these regions are closer to the inflow boundary and the SN is not likely to have a considerable impact on the largescale dynamics. Some differences between Fig. 4a, b appear over the eastern portion of the domain. When the SYDS setup is considered (Fig. 4c, f) a further increase in significance occurs: an almost 100% significance level can be seen over the entire domain. However, there is no signal of a magnitude larger than 0.2 mm/day in the SYDS domain, unlike in the other two setups over these regions.
Now we examine whether the use of SN or reduced domain size can produce a significant change in the signal induced by the perturbation P10. Thus, we aim at finding physically and statistically significant differences between the signals in the SYSN (SYDS) displayed in Fig. 4b, c and the signal in the SYNA set shown in Fig. 4a. The fact that at a given location the signals in the SYNA (SYDS) and SYSN are statistically significant does not imply that their difference is also statistically significant. To quantify the statistical significance of the difference of the signals we again apply the test for differences of means, but this time on the difference between the signals in the SYSN (SYDS) and SYNA (see Appendix 2 for details). The resulting fields of statistical significance of the signal’s differences are shown in Fig. 4g (for SYSNSYNA) and Fig. 4h (SYDSSYNA). The differences between the signals are not shown since they can be inferred from subtracting values from Fig. 4b, c from Fig. 4a. In Fig. 4g it can be seen that the SN yield statistically significant differences alterations of the signal at 90% level or higher only in small patchy areas; the exception is the north eastern part of the continent where the regions of significance occupy somewhat larger regions. The difference of the signals between the SYDS and SYNA sets (Fig. 4h) is similar to that between the SYSN and SYNA. From Fig. 4a, b it can be seen that the magnitudes of these alterations are not of large physical importance. It is also worth to note that even if the null hypothesis of no difference between the signals is true, it can be accidentally rejected. For the significance level of 90% the nominal rejection rate is 10% but larger rates are not unlikely; because of spatial correlation of the atmospheric variables, the nearby grid points tend to yield similar test results and the points that appear statistically significant only by chance can cluster, resulting in larger areas of apparent significance (Von Storch 1982; Livezey and Chen 1983).
3.4 Signal P10 in summer
For summer (JJA) precipitation despite a physically relevant magnitude of the model response to the perturbation P10 over many regions, the response is generally statistically insignificant, which is the major difference with respect to the winter case. This happens because the noise is very large in summer precipitation (as shown in Fig. 3g–i) and strong signals are required for significance, given our ensemble size. Because of the lack of significance the analysis of the summer precipitation response to P10 will not be presented. It is worth reminding that the lack of statistical significance is always a function of sample size and hence a consequence of the small sample used here. The smaller the signaltonoise ratio, the larger the sample needed to achieve significance.
We proceed to examine the model’s response to the perturbation of the threshold parameter for the onset of deep convection (P01 in Table 1). In winter, deep convection activity is at its minimum and is likely absent in higher latitudes of the domain. For this reason, the perturbation P01 produces almost no significant signal in winter (Fig. 2). Hence, for this perturbation, we focus on the summer months.
3.5 Signal P01 in summer
The results in the SYSN configuration show a substantial gain in statistical significance when SN is applied. The SYSN experiment reveals that the perturbation P01 mainly leads to a decrease in summer precipitation that varies from −0.2 in the northwest to below −2.0 mm/day in the southeast portion of the domain (Fig. 7b). The perturbation P01 also exhibits a strong effect on summer precipitation in the small SYDS domain (Fig. 7c); the model response is negative with values as small as −1.8 mm/day south of the Great Lakes. Further, the signal in the SYDS set is quite similar to the SYSN case, with somewhat smaller magnitudes. In other parts of the small domain, such as over the province of Quebec and off the East Coast, the signal is spatially variable, despite being highly statistically significant (Fig. 7e) and with considerable magnitudes of up to 1 mm/day. Since there are no remarkable topographic features in the small domain it can be argued that they are rather fingerprints of instantaneous weather patterns (storm tracks) that are not filtered out in 3month averages because of insufficient sample of the instantaneous atmospheric states and small variability between the ensemble members. This points to the fact that in such a small temporal sample, the ensemble means of the control M00 and perturbedparameter model M01 are dependent on the particular year. Figure 7g, h show that internal variability in summer is too large to permit the detection of the effect of the SN and domain size reduction on summer precipitation (if there is any) given the actual ensemble sizes.
3.6 Rule of thumb for the minimum ensemble size
It can be seen in Fig. 9 that the reduction of domain size (SYDS) is more efficient in reducing the noise level than the SYSN. It is worth reminding here that the SN parameters in the SYSN experiment were adjusted so that the SN forcing be rather weak and applied only in the upper levels. Alexandru et al. (2009) showed that a stronger nudging of large scales, applied at all levels, could substantially reduce internal variability noise. Whether this would change the magnitude of the signal cannot be inferred from the experiments considered here. The signal in the small SYDS domain has in most of the cases smaller magnitude than those in the large domain experiments SYNA and SYSN. Exceptions such as for parameter P01 for summer precipitation (Fig. 9a), could happen due to the contamination of the SYNA signal with noise, since the noise can alter the estimate of the magnitude of the signal in both ways—decreasing and increasing it. Similarly, the smaller signal magnitudes in SYDS domain in Fig. 9 do not prove that the small domain suppresses the signal but rather indicate that this could sometimes be the case. On the other hand, the SN is fairly efficient in reducing noise, while there is not much evidence that the model response is smaller.
Note that due to the properties of the Student’s distribution, if a small number of degrees of freedom was assumed instead of infinite number, the required critical value t _{0} that corresponds to the 95% significance would be larger, resulting in a more conservative (higher) demand for M _{min}. Due to some vagueness of the concept we rather intend to use M _{min} in relative terms, to compare the required sizes among different perturbations, simulation setups and seasons, than to recommend it in absolute terms for achieving specified significance levels.
In summer (JJA) the ensemble sizes required for the significant estimates of seasonal precipitation signals (Fig. 10a) are much larger. In the large domain with no SN (SYNA) the minimum number of members is about 25 for the perturbation P10 and 20 for P01, despite the latter exciting locally high sensitivities (see Fig. 7a). The spectral nudging (SYSN) almost halves the number of ensemble members needed to achieve statistical significance, while the reduction of domain size (SYDS) reduces the minimum number of members almost 5 times. Both methods of noise reduction appear to be very efficient for precipitation in summer. When summer 2m temperatures are considered (Fig. 10b) the SYSN and SYDS configuration are still efficient in reducing the minimum ensemble sizes but appear less sensitive to reduction of noise. This is due to the fact that signals in the SYNA configuration in summer temperatures are relatively strong (see Figs. 6a, 8a); so in that case the need for ensemble calculations is low in all the three configurations, as compared to the case of precipitation. In fact, in the case of 2 mtemperature the season that is associated with the largest computational cost of significant estimates is spring when the minimum ensemble sizes are 20 (Eq. 15) for the response to P10 (P01), respectively. Also the noise level in the SYNA setup in spring is slightly higher than in autumn.
4 Summary and conclusions
Development of RCMs and study of uncertainty related to the choices that must be made in constructing and applying RCMs often requires multiple testing of model response to a large number of modifications, which imposes a high demand on computational resources. A highresolution RCM simulation configuration, less computationally demanding than the operational RCM runs (in terms of the integration period, computational domain and internal variability noise), if used as test bed for RCM modification, would allow the allocation of the computational resources to testing a larger number of modifications. The objective of this work was to study the model response to RCM parameter perturbations using computationally less demanding configurations than the operational runs and eventually select an optimal configuration as a result of the tradeoff between the representativeness of results it may provide and its computational cost. The approach followed consisted of analysing sets of RCM simulations conducted for the three parameters’ settings, here referred to as the model versions: the control (unperturbed) model version and two modified versions in which two parameters that control deep convection and stratiform precipitation, respectively, were perturbed one at a time. These three model versions were used to generate RCM simulations within three setups, all with the integration period of a single year.
In the first setup, denoted as SYNA, we performed ensemble simulations with perturbed initial conditions over a large continentalscale domain with spectral nudging turned off. The parameter perturbations produced fairly large differences of ensemble means in 2 mtemperature, especially in summer. These differences were statistically significant in a large part of the domain. On the other hand, for precipitation the results in all seasons in the largest part of the domain were statistically insignificant, with exception of the topographically rich regions along the West Coast of North America.
In order to reduce internal variability noise—a nuisance at the time of quantifying the signal—, we performed perturbed parameter RCM simulations using two additional setups: (1) SYSN in which we used the same domain and number of ensemble members as in the previous two configurations but applied a weak spectral nudging (SN) at upper levels, and (2) SYDS in which the domain size is reduced. The main concern with these two configurations was that they might alter or even suppress the model sensitivity to parameter perturbations along with reduction of internal variability. However, the results of these two experiments when compared to the SYNA configuration showed that this concern was only justifiable in the case of a reduced domain. Not surprisingly, in the case of the largescale condensation parameter perturbation, the SYDS signal exhibited deviations of considerable magnitude from its counterpart in the SYNA set that is taken as reference here. These changes were statistically significant over larger areas near the inflow lateral boundaries. The use of the very small domains, such as SYDS, is known to be associated to several flaws, which was discussed in the Introduction section. The alteration of the responses to perturbations by the proximity of the lateral boundaries, noted in the SYDS, is in accord with the previous evidence. The SYDS domain may, however, be attractive for conducting fast and computationally inexpensive RCM sensitivity tests at the development stage of the model. The reduction of the computational cost when using the small SYDS domain is twofold: the integration area is much smaller (and hence computational cost) and the internal variability is low (hence potentially contributing to increasing statistical significance or reducing the need of large ensembles).
The model response to parameter perturbations in the SYNA and SYSN configurations was rather similar in pattern as well as in magnitude, and statistically significant only in rather small, scattered areas (which could be also a result of internal variability in case the null hypothesis of equal responses is true). Results did not provide evidence that the spectral nudging altered the mean model response to parameter perturbations. However, this should not be understood as a proof of SN not affecting the signal but rather as a consequence of the fact that the number of ensemble members was insufficient to identify the differences. In addition, the SN configuration used here was designed to minimally force the largescale flow and this only at upper levels. It is not known to the authors whether a stronger SN (that would better constrain internal variability deviations) would still exhibit little or no effects on the signal, as it is the case with the SN configuration used here.
Notes
Acknowledgments
The authors want to thank Ms Katja Winger for maintaining the CRCM5 at UQAM and for her technical help in performing simulations and the subsequent analysis of results. We thank Mr Mourad Labassi for maintaining a local computing environment at the Ouranos Consortium and Mr Alejandro Di Luca for his valuable comments. We also want to thank the two anonymous reviewers for their constructive suggestions.
Open Access
This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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