Kinetic energy (KE) spectra are applied to evaluate two convection-permitting models: the AROME numerical weather prediction operational model and the Meso-NH research model, that share the same physics and differ only in the dynamics (semi-Lagrangian semi-implicit versus Eulerian explicit schemes).
A first analysis of AROME spectra for winter and summer seasons shows that the model-derived spectra match the observational spectra well, including the transition between k−3 and k−5/3 regimes. The vertical distribution of the spectra is coherent with previous observational and numerical studies and the diurnal cycle has a strong impact on the amount of KE in the mesoscale during summer.
A comparative analysis of KE spectra for both models is then performed on a real case of individual convective cells that developed over plains, during the afternoon of 11 April 2007, characterized by a strong cold air outflow in the low levels.
AROME spectra are characterized by a coarser effective resolution than Meso-NH, even without explicit diffusion, revealing the impact of the implicit diffusion of the semi-implicit semi-Lagrangian scheme used in AROME. For large time steps, the damping increases and can be attributed preferentially to the SI part of the SISL (semi-implicit semi-Lagrangian) formulation. Adiabatic runs show that the transition to a shallow mesoscale regime is still apparent even if the mesoscale KE variance strongly depends on the presence of the physical processes.
Effective resolution of Meso-NH remains around 4–6Δx for horizontal grid spacings between 2.5 km and 250 m. The effects of subgrid mixing schemes are also investigated with Meso-NH at 500 m horizontal grid spacing in the grey zone for turbulence, illustrating the difficulty in finding a good equilibrium between resolved and subgrid mixing at this scale.
Observational studies (Nastrom and Gage, 1985; Lindborg, 1999) have shown that kinetic energy (KE) spectra follow a k−3 dependence on a large scale (where k is the wavenumber) dominated by rotational modes and a transition to a shallower k−5/3 dependence on mesoscale and smaller scales, dominated by divergent modes. The k−3 dependence on the large scale is well explained by 2D turbulence (Kraichnan, 1967) but an understanding of the mesoscale portion with the k−5/3 dependence is still under discussion. Gage and Nastrom (1986) and more recently Tung and Orlando (2003) argue that the −5/3 slope is produced primarily from a forward energy cascade (downscale, from larger to smaller scales), whereas Lilly (1983, 1989) argues that a small amount of the energy injected at small scales (convection or other sources) cascades upscale, producing the shallow spectra.
Spectral analysis is also a powerful tool to assess how the KE is distributed in atmospheric models according to spatial scales and their capacity to reproduce observational spectra (Koshyk et al., 1999). In particular, Skamarock (2004) used spectral analysis to define the effective resolution of the WRF model, i.e. the scale from which the model departs from the theoretical slope, also given by a simulation with a finer grid spacing. Indeed, a part of small-scale energy is damped as it is affected by implicit and explicit diffusion, the tail of spectra showing a rapid decrease of KE. Diffusion is needed to prevent spurious accumulation of energy at 2Δx wavelength, whereas, in nature, the dissipation of energy occurs at scales smaller than 1 mm, which is largely beyond the reach of the resolution used for numerical weather prediction. Skamarock (2004) found an effective resolution for the WRF model around 7Δx for simulations with different horizontal resolutions (varying from 22 to 4 km). KE spectra are also efficient for evaluating whether a model is too diffusive (with too coarse an effective resolution) or not sufficiently diffusive (accumulation of energy at the spectra tails). Whereas model-derived spectra with slopes matching observational spectra do not prove the simulations are realistic, it is a necessary condition but not a sufficient one. Moreover, as underlined by Skamarock (2004), a critically important measure of a numerical weather prediction (NWP) model's accuracy is its ability to resolve features at the limits of its grid resolution.
KE spectra have been largely used to evaluate general circulation models (e.g. Laursen and Eliasen, 1989) or mesoscale NWP models (e.g. Skamarock and Baldwin, 2003) over long periods with a broad range of meteorological situations, or on specific cases like idealized frameworks of squall lines (e.g. Takemi and Rotunno, 2003), supercells (e.g. Fiori et al., 2010), breezes (e.g. Knievel et al., 2007) or baroclinic waves (Waite and Snyder, 2009). However, very few of them deal with real convective cells. Moist convection is very instructive for characterizing a convection-permitting model in terms of filtering and effective resolution, as predictions are strongly dependent on the behaviour of the dynamics of the model, on the physics and also on the interaction between the two.
In this study, KE spectra are first examined from a climatological perspective for the convection-permitting NWP model AROME. Then, special attention is focused on KE spectra for individual convective cells over plains, which can lead to unrealistic strong outflow winds at the surface if strong explicit diffusion is imposed. One of the original features of our study is to compare in this case KE spectra for two mesoscale models–AROME and Meso-NH–which differ only in their dynamics. This allows separation of the numerical filters and subgrid mixing. Sensitivity tests on the explicit diffusion and time step, as well as on the transport and temporal schemes, are conducted to illustrate some damping mechanisms. Indeed, various dampings exist: damping from explicit diffusion, damping from the SI scheme, damping due to the choice of reference temperature profile and reference surface pressure (Simmons and Temperton, 1997), damping from subgrid-mixing, etc. Damping in the model must be controlled in order to obtain a numerical solution as accurate as possible; however, it is worth remembering that some effects are beneficial as damping contributes to ensure numerical stability. The initial spin-up is illustrated to verify that physically realistic fine-scale structures are generated. Finally, analysis of KE spectra for simulations with increasing horizontal grid spacings is conducted with the Meso-NH model on the same convective case, and sensitivity tests on the physical dissipation are led at 500 m horizontal grid spacing to explore the effects of subgrid model mixing in the grey zone of turbulence.
The outline of this paper is as follows. In section 2, the methodology used in this study is explained. In section 3, a climatology of KE spectra is presented for the AROME model. In section 4, KE spectra computed for the convective case are analysed for AROME and Meso-NH, and sensitivity tests on horizontal resolution and PBL parametrization are conducted with Meso-NH. The conclusions are given in section 5.
2. Models and set-up
2.1. Model configurations
Two mesoscale models are used in this study: Meso-NH and AROME.
Meso-NH (Lafore et al., 1998) is a research model that can simulate atmospheric motion ranging from the synoptic scale to large eddy simulation (LES). It is based on the aneslatic approximation of the pseudo-incompressible system of Durran (1989). It is a grid-point Eulerian model using a fourth-order centred advection scheme for the momentum components and the piecewise parabolic method (PPM) (Collela and Woodward, 1984) advection scheme for other variables, associated with leapfrog time marching. It uses a C-grid in the Arakawa convention (Mesinger and Arakawa, 1976) for both horizontal and vertical discretizations with a conformal projection system of horizontal coordinates and a Gal-Chen and Sommerville (1975) system of vertical coordinates. A spatial filter is applied to smooth the orography at the shortest scales. The model uses a fourth-order diffusion scheme to suppress very short wavelength modes, with an equivalent damping scale of 30 min for the 2Δx waves in this study. The physical package includes a mixed one-moment microphysical scheme (Pinty and Jabouille, 1998). It is a three-class ice parametrization coupled to a Kessler scheme for warm processes. It manages five prognostic variables of water condensates: cloud droplets, rain, ice crystals, snow and graupel mixing ratios in addition to water vapour. More details can be found in Lascaux etal. (2006). The representation of the turbulence in the planetary boundary layer is based on a prognostic equation for the turbulent kinetic energy, used with conservative variables for moist non-precipitating processes, combined with a diagnostic mixing length representing the size of the most energetic eddies (Cuxart et al., 2000). For hectometric horizontal resolutions, the scheme is used in its complete three-dimensional formulation (called ‘3D turbulence’) and the most energetic parametrized eddies are just a bit smaller than the grid mesh: the Deardorff length scale is used, corresponding to grid size but limited by the stability in the entrainment zone. When the scheme is used at the mesoscale (horizontal grid sizes larger than 2 km), it can be assumed that the horizontal gradients and turbulent fluxes are much smaller than their vertical counterparts, and can be neglected (called ‘1D turbulence’). The mixing length is parametrized in a physical way at every level according to Bougeault and Lacarrère (1989) (noted BL89). It is now largely accepted that at the mesoscale local turbulent schemes are not sufficient to parametrize the contribution of larger plumes in convective dry and cloudy boundary layers. Indeed, for dry thermals, the turbulence scheme, even with BL89 values, remains local and produces thermal vertical profiles without a countergradient zone. Moreover, it misrepresents shallow convective clouds. A mass flux formulation of convective mixing according to Pergaud et al. (2009) (noted EDMF hereafter, for ‘Eddy Diffusivity Mass flux’) has been introduced, based on a single updraught that reproduces the top entrainment for dry thermals and improves the representation of shallow convective clouds.
AROME (Seity et al., 2010) is the high-resolution operational forecast model of Meteo-France over France since 2008, with a 2.5 km horizontal grid spacing. In our study, the vertical grid includes 41 levels; the height of the lowest level is about 17 m above the ground and the uppermost level is located at about 1 hPa. It is a semi-implicit semi-Lagrangian (SISL) spectral model derived from the IFS/ARPEGE (Courtier et al., 1991) model for the dynamical part, with a dynamical core provided by ALADIN-NH (Bubnova et al., 1995) and based on fully compressible system equations. As for ALADIN-NH, various three-time-level (3-TL) and two-time-level (2-TL) SL schemes are implemented (Bénard et al., 2010). The operational version of AROME uses the 2-TL SLSI scheme following that of the IFS/ARPEGE model (Temperton and Staniforth, 2001) and uses the second-order extrapolation in the time scheme, the so-called ‘Stable Extrapolation Two-Time-Level Scheme’ or ‘SETTLS’ (Hortal, 2002). This option is used for computation of the trajectories and for treatment of the nonlinear residual. For computation of trajectories , an iterative algorithm determines the position of the origin point using tri-linear interpolations to compute winds at the origin point. Then, the advected variables (including winds) are interpolated to the origin point of the trajectory using a simplified Lagrange cubic scheme by neglecting the corner points (Ritchie et al., 1995; see Appendix B for more details), including quasi-monotone limiters for most quantities. The other terms in the dynamical equations are interpolated to the origin point using linear interpolations. The efficiency of the dynamics allows a time step of 60 s at 2.5 km resolution, when the Meso-NH time step is limited to 8 s at the same resolution. The AROME model uses an A-grid with a horizontal conformal projection and a mass-based terrain following hybrid vertical discretization. Similarly to Meso-NH, an equivalent spatial orography filter is applied to AROME with an additional spectral quadratic truncation at 3Δx. A spectral diffusion, which is a fourth-order linear one, is applied to all spectral prognostic variables except for the surface pressure (i.e. temperature, vorticity, divergence, vertical divergence), with the same strength. The characteristic damping time for the 4Δx waves is 2 h at the lowest level and the strength increases with height proportionally to the inverse of the pressure, replacing a top absorbing layer. For example, diffusion is twice as strong, around 500 hPa compared to diffusion near the surface and five times stronger around 200 hPa. For water condensates, a grid-point diffusion is applied inside the SL scheme and enforces a grid-point smoothing as a function of the local deformation tensor (the so-called SLHD scheme, i.e. semi-Lagrangian horizontal diffusion; Váňa et al., 2008). The specific content of water vapour and kinetic energy turbulence (TKE) are not diffused at all. The lateral boundary coupling (LBC) of AROME is performed using the Davies method. Its physical package comes from Meso-NH and is identical with the same microphysics, EDMF shallow convection and 1D turbulence schemes.
2.2. Numerical simulations
The climatological study presented in section 3 is conducted only with the operational 30 h forecasts of AROME beginning at 0000 UTC each day during winter 2009–2010 (from December to February) and summer 2009 (from June to August) periods. The operational domain, covering the whole of France, is displayed in Figure 1.
The case study of 11 April 2007, presented in section 4, corresponds to individual convective cells triggered in the afternoon on the plains of southwest France. This case has been chosen because a sensitivity test with a stronger numerical diffusion in the AROME forecasts (corresponding to the same values as those used in the ALADIN operational model, i.e. enhanced by a factor of 4 for the temperature, vorticity and specific humidity, and a factor of 20 for the horizontal divergence and vertical divergence compared to the operational values) induces what has been called ‘fireworks’ corresponding to strong unrealistic outflow in the low-level cooling of the convective cells. Indeed, the morphology and horizontal extent of the convective cells forecasted by AROME have been revealed as very sensitive to the intensity of explicit diffusion, making a KE spectra study instructive. First, simulations are performed with AROME and Meso-NH over the same domain covering southwest France with a 2.5 km horizontal grid spacing (southwest domain in Figure 1) starting from 0000 UTC 11 April 2007 with the same initial and coupling fields.
Different sensitivity tests on the SISL scheme are led by AROME. Then, complementary simulations are conducted with Meso-NH over a smaller domain centred over the convective cells (Girond domain) with increasing horizontal resolution from 2.5 km up to 250 m (Table 1) and then testing different turbulence options at 500 m horizontal grid spacing (Table 2). The same vertical discretization with 40 levels is used for all the Meso-NH simulations: 23 levels with 500 m vertical grid spacing between 3000 and 14 000 m and 17 levels below 3000 m with decreasing grid spacing as approaching the ground (grid spacing reaches 30 m for the lowest level above the ground), ensuring a very fine representation of the low levels. In the free troposphere, the vertical grid spacing is similar for AROME and Meso-NH simulations with 13 levels between 9 and 3 km.
Table 1. Configurations of the Meso-NH experiments.
Table 2. Configurations of the Meso-NH experiments with different turbulence options at 500 m horizontal grid spacing.
2.3. Spectra computation
An algorithm of spectral computation has been coded, based on Denis et al.'s (2002) study, which used a discrete cosine transform (DCT) (Ahmed and Rao, 1974) to convert grid-point fields into spectral ones. Indeed, spectral computation from a DCT is particularly well adapted for limited-area models to overcome the problem of non-periodic domain. Unlike spectral computation based on FFT, the meteorological fields do not need any trend removal for ensuring periodicity (Errico, 1985). Moreover, Denis et al. (2002) have shown that there is no aliasing on the large scale. It is worth noting that it is the same tool that is used to compute spectra for both models.
We have also introduced an additional option to the original algorithm to assign each element of the variance array to two wavenumbers instead of one. Indeed, each element is located in a band delimited by two ellipses corresponding to two wavenumbers (k and k + 1); the contribution to these two wavenumbers is distributed proportionally as a function of the distance to the two ellipses (see further details in Appendix A). Moreover, an additional smoothing using a Bezier approximation is also applied when plotting the spectra (except for Figure 12(a)).
KE spectra are computed every hour and for each model level using two-dimensional decomposition of the velocities (u,v,w) and are then averaged over selected periods and layers. As pointed out by Skamarock (2004), computing KE spectra on constant pressure or height surfaces only leads to small significant differences in the results.
3. Climatology for AROME
3.1. General characteristics
Figure 2(a) shows the KE spectra averaged over the free troposphere (between about 3 and 9 km) and over a 24 h period (hourly forecasts between 7 and 30 h in order to avoid model spin-up issues) during the summer (JJA) and winter (DJF) periods. Both curves represent the mean of more than 56 000 individual spectra.
For these periods AROME has its own analysis at 2.5 km resolution; therefore the spin-up duration is short (less than 3 h). Both spectra exhibit a k−3 dependence on the larger scales and a k−5/3 dependence on the mesoscale between about 100 and 25 km. At the highest wavenumbers, model dissipation removes energy and depresses the spectra. The effective resolution of AROME is about 9Δx (i.e. 22 km), which is coarser than the WRF effective resolution of 7Δx as shown in Skamarock (2004). The smallest wavelength represented in the model is 5 km, i.e. twice the grid resolution 2Δx. A decrease of energy can be seen at the wavelength corresponding to 3Δx due to the quadratic truncation applied to the orography. Thus at scales smaller than 3Δx there is an absence of variability generated by the orography. The comparison between winter and summer shows that there is more energy in the larger scales in winter, due to a larger number of synoptic lows and frontal systems. Conversely, there is more energy in the mesoscale part in summer, induced by convection and turbulence, until the shorter wavelengths. This impacts also the scales below 3Δx. This effect is much more obvious for spectra computed during the afternoon (Figure 2(b) at 1500 UTC). During winter, we can also note a small upturn at the end of the tail, meaning that there is a small but significant amount of energy being aliased to the longer modes (Skamarock, 2004), to which we shall return below.
3.2. Vertical distribution
The vertical distribution of KE spectra is displayed in Figure 3 for the summer and winter periods. Spectra have been averaged over three different layers of the atmosphere: the low troposphere (between about 0 and 2.5 km), the free troposphere (between about 3 and 9 km) and the upper troposphere (between about 11 and 15 km). In the low troposphere, more of the energy variance is in the mesoscale range, due to small-scale phenomena forced by terrain (e.g. mountain valley circulations) or surface heterogeneities (e.g. sea breezes) with a k−5/3 dependence and without k−3 transition. The free troposphere levels exhibit a more pronounced slope closer to −3 on the large scales (above 400 km), due to the existence of synoptic waves. The build-up of energy at the highest wavenumber, underlined in the previous section, mainly concerns the upper part of the troposphere and is more pronounced in winter than in summer. A detailed examination reveals that only a few days (12 days) in both periods are concerned with the 2Δx waves. These days are characterized by a very strong upper-level jet where the 2Δx waves are located, leading to the hypothesis that it could be a problem of stability. The problem is currently being examined, and this underlines the power of the KE spectra as a tool to detect the weaknesses of a model.
3.3. Diurnal cycle
Figure 4 shows the effect of the diurnal cycle on the spectra during summer in the free troposphere. The energy in the mesoscale, ranged up to about 200 km (but zoomed up to 50 km in Figure 4 for visibility), clearly increases during the day (from about 0900 UTC to 1800 UTC, with corresponding spectra represented by solid lines) as the convection and turbulence motions develop, then gradually decreases during the night (from about 1800 UTC to 0600 UTC, with spectra represented as dashed lines), reaching a minimum around 0600 UTC. In winter, the effect of the diurnal cycle on KE spectra is negligible (not shown). This impact is also discernible from wavelengths less than 7.5 km for which the variability from the orography is absent due to 3Δx truncation. This suggests that in this part of the spectra for AROME the KE is closely related to motions from the diurnal cycle.
4. A case study: convective cells over plain
4.1. Presentation of the case
A case with an observed moderate convective activity on 11 April 2007 over the southwest of France was chosen for this study as it has been identified as a case of ‘fireworks’, as defined in Section 2.2, during the pre-operational phase of the AROME model. Moreover, comparisons with Meso-NH forecasts have shown large differences.
Between a high-pressure system over the Netherlands and a low-pressure one over Portugal, a small convective front moves from the Mediterranean Sea to the Atlantic coasts in a southeasterly flow (not shown). The first convective cells develop along the low-level convergence line in a conditionally unstable atmosphere (with moderate values of CAPE between 1200 and 1500 J kg−1); subsequent cold pool dynamics induce lifting and triggering of new cells. The cumulative radar precipitation between 1300 and 1700 UTC (Figure 5(d)) indicates weak amounts of precipitation from the southeast to the northwest of the domain, up to 5 mm, with higher quantities around 25 mm only over the Gironde estuary. Three experiments are compared at 2.5 km resolution: AROME in its operational version (called ‘AROME oper’), AROME with a stronger diffusion (factor of 4 for temperature and vorticity and a factor of 20 for the divergence and vertical divergence), and Meso-NH (Figure 5(a–c)). All of them reproduce the convective area moving to the northwest but overestimate slightly the convective precipitation on the eastern part and shift the convective cells 80 km to the north.
Despite this, differences between the forecasts from AROME oper, AROME with a stronger explicit diffusion and Meso-NH are not negligible, especially on surface wind predictions (Figure 6): AROME develops larger area cells, associated with surface divergence and cooling induced by precipitation evaporation (not shown), especially with stronger diffusion, whereas convection in Meso-NH is split into small-scale cells. The intense outflow induced by AROME with strong diffusion is unrealistic in terms of cooling and wind intensity, and the convective cells progressively organize themselves into larger-scale systems with a tendency to structure into circular patterns. One of our objectives is therefore to analyse these differences in terms of KE spectra.
In contrast to the previous climatological study of section 3, for which forecasts start from AROME analysis, the simulations of 11 April 2007 start from an initial state provided by variational analysis from the ALADIN model (Fisher et al., 2005), with a coarser resolution of 10 km, as AROME did not possess its own data assimilation at this date. Therefore, there is a longer spin-up measurable on the spectra due to the scales that are not initialized. Figure 7 reveals that the mesoscale portion of the KE spectrum develops rapidly and reaches a fully developed state around 6 h, showing that the model is capable of generating a dynamically correct mesoscale energy spectrum even without its own initialization. More precisely, in the initial state, there is very little energy in the KE spectra at the mesoscale and below, beyond 100 km. The forecast model must therefore develop the mesoscale KE spectrum on its own by dynamical adaptation to a finer orography and to more detailed surface conditions. The mesoscale portion of the spectrum between 20 and 100 km develops rapidly as most of the energy is obtained after 3 h. Below 20 km, the increase of energy is high during the first 3 h. This increase is particularly strong during the first hour (solid line in Figure 7) at the smallest scales, due to temporary imbalances. KE at the spectrum tail then decreases up to 0600 UTC. It is likely that the energy removal between 0300 and 0600 UTC in the mesoscale part is partly due to dissipation mechanisms, to reverse cascade as the energy increases slightly at the larger scales, and to nocturnal effects as seen before on the diurnal cycle. After 0600 UTC, KE increases due to the diurnal cycle. The same behaviour is observed with the Meso-NH model (not shown).
4.2.1. Effective resolution and diffusion
Spectra are now averaged between 1300 and 1700 UTC over free troposphere, every 30 min, in order to catch the more active period of convection. A comparison between KE spectra for AROME (in its operational version) and Meso-NH models shows that the effective resolution is respectively about 9–10Δx (i.e. 22–25 km) and 5–6Δx (i.e. 12–15 km) (Figure 8). The variance loss is more pronounced and the spectral tail drops off more for AROME. If all explicit diffusion is suppressed into AROME (spectral diffusion and grid-point diffusion on hydrometeors), the Meso-NH spectrum remains the minimally dissipative member of the set with the most scale-selective dissipation. Therefore, as both models share the same physics, the energy loss with AROME can be directly linked to the implicit dissipation inherent to the SISL scheme. Conversely, in Meso-NH, the PPM scheme, as shown in Carpenter et al. (1990), and the fourth-order centred scheme exhibit a minimal numerical dissipation of resolvable modes even if a monotonicity constraint is applied to the PPM advection (Lin and Rood, 1997).
If now the explicit diffusion is increased with AROME, the spectral tail plunges even more, with an additional drop in effective resolution (about 30 km, i.e. 12Δx), confirming the qualitative impression of absence of fine-scale structure and overestimation of the large-scale organization of the convective cells (Figures 5(b) and 6(b)).
A sensitivity test on the time-step duration with AROME (Figure 9) shows that the dissipation is more pronounced with larger time steps. With a time step of 8 s, the dissipation at the smallest scales is weaker but remains higher than with Meso-NH (Figure 8) and the effective resolution is coarser. As a similar sensitivity test on the time step leads to the same effects for adiabatic runs (Figure 9, dashed lines), it is confirmed that the dissipation is related to the dynamics and not to the physics. The objective is therefore to try to understand the coarser effective resolution with AROME and the sensitivity to the time step. It is well known that the SL interpolations induce some damping or smoothing and conservation is not ensured (Staniforth and Côté, 1991). This effect is dramatically enhanced for experiments using only linear (instead of simplified cubic) interpolations, in agreement with McDonald (1984); indeed, the KE spectra are more damped up to 300 km, with a large loss of accuracy when using lower-order interpolations of advected variables to the origin point (Figure 10).
However, the damping inherent to SL transport cannot explain the sensitivity to the time step as the time-integrated damping in SL advection schemes generally increases with decreasing time step due to more frequent interpolations. A stability study in the simplified framework of the forced one-dimensional SL equation, as done for example in Hortal (2002) and in Durran and Reinecke (2004), suggests that the 2-TL SI ‘SETTLS’ scheme can induce some damping. It is worth noting that the same sensitivity to the time-step duration is found by using two other temporal schemes. Indeed, KE of higher wavenumbers is also higher when decreasing the time step with a 3-TL SI scheme (Figure 11(a)) or with a 2-TL ICI (iterative centred implicit) scheme with two iterations (not shown). The ICI scheme (Bénard, 2003) iterates the SI scheme in order to converge towards a fully centred implicit scheme. This allows second-order accuracy in time to be achieved without the SETTLS extrapolation. KE spectra for 2-TL SI and 2-TL ICI schemes are very similar (not shown as the spectra are almost identical), so sensitivity to the time-step duration is not an exclusive specificity of the SETTLS extrapolation.
Increasing the decentring of the 3-TL SI leapfrog scheme (Figure 11(b)) accentuates the damping slightly, especially for large time steps, while the effect of decentring is not discernible for small time steps. Damping is present even if there is no decentring (i.e. with decentring factor eps = 0) and no time filtering (not shown). The KE spectra for the 2-TL scheme is equivalent to those for the 3-TL scheme but with a time step twice as long, which illustrates the fact that the same level of accuracy is obtained but with an improvement in efficiency for a 2-TL scheme. Indeed, the temporal truncation errors are similar for a 3TL scheme and for the 2-TL scheme with a doubled time step as shown, for example, in Temperton and Staniforth (1987). Thus the sensitivity of KE to the time step is linked to the temporal truncation errors but is certainly not entirely explained by that. As suggested by Skamarock (2011), using too large time steps with the SLSI scheme can remove some small-scale variability. This is also in agreement with Cullen (2001) and Héreil and Laprise (1996), suggesting that long time steps can damp some physical modes.
In other respects, an important loss of energy variance can be noted in mesocale and small scales when suppressing the physics package (adiabatic runs), indicating that the physics contributes significantly to the amount of energy variance. This supports the conclusion of Hamilton et al. (2008) and Waite and Snyder (2009) that physical processes other than a downscale cascade are responsible for maintaining observed mesoscale KE levels. Moreover, the sensitivity of KE to diurnal cycle during summer, discussed in section 3.3, showing a diurnal increase and nocturnal decrease of KE levels, suggests an energy injection at small scales. Besides, the transition to the −5/3 slope still exists, at least up to 50 km, in adiabatic conditions (as in Hamilton et al., 2008, but not in Waite and Snyder, 2009). This suggests that to some extent the mesoscale KE derives from a nonlinear forward cascade.
We might suspect that the energy damping at high wavenumbers in AROME is partly due to the spatial orography filter. Indeed, the spectra on the orography (Figure 12(a)) with and without this filter show that a loss of variance of the orography is clearly discernible from about 30 km due to filtering. We note that the topography possesses a power spectrum that follows a k−2 behaviour as described, for example, by Balmino (1993). However, the KE spectra (Figure 12(b)) demonstrate that the effective resolution is the same in both cases, even if the amount of KE is very slightly higher in the mesoscale part without filter. Thus the orography contributes to the energy distribution but the representation of the orography in the model does not explain the energy damping at high wavenumbers. This is consistent with the Meso-NH/AROME comparison shown in Figure 8 where mean KE spectra of both models present a different damping while both have similar orography spectra. This is also in agreement with Hamilton etal. (2008), who show that over the mesoscale an aquaplanet model has less energy than a full model but the ratio of KE in the aquaplanet model to that in the full model is roughly constant, i.e. the slopes are roughly the same.
4. 3. Sensitivity to horizontal resolution with Meso-NH
In order to illustrate the degree of convergence, numerous studies have been led on the impact of horizontal resolution on idealized deep convection systems (Adlerman and Droegmeier, 2002; Bryan et al., 2003). Here a downscaling approach is conducted with Meso-NH, presenting the original study of a real case, and not idealized, of isolated convective cells. The domain width of 250 km × 250 km with horizontal grid spacings ranging from 2.5 km to 250 m is presented in Figure 1(c) (domain G). The configuration of each simulation is displayed in Table 1. KE spectra applied on the horizontal and vertical velocities are presented respectively in Figure 13(a, b). For the 500 m horizontal resolution, the impact of a 1D turbulence on KE spectra is weak in the free troposphere (not shown). Mechanically, the higher the resolution, the more small scales are represented. The effective resolution is a constant multiple of grid size, about 4–6Δx, which is very competitive compared to other models. The slope is a bit more than −5/3 for resolutions higher than 1 km but there is no accumulation of energy at 2Δx, showing that the explicit numerical diffusion value is sufficient for this case, and does not need to be increased with the higher resolution. This would mean that the resolved part of the flow is slightly overestimated at these smaller scales. Indeed, turbulence is well represented at very fine horizontal resolution from 10 m to 100 m, for which turbulent motions are mainly resolved, and by mesoscale models at resolution greater than 2 km, for which those motions are entirely parametrized, but the good equilibrium between resolved and unresolved contributions is still in the process of being defined at intermediate scales, the so-called ‘Terra Incognita’ (Wyngaard, 2004).
As already shown in other studies (Larsen, 1982), the spectral behaviour of the vertical velocity exhibits a much flatter wavenumber dependence compared to the horizontal velocity components (Figure 13(b)) and the energy contribution becomes significant at small scales, revealing several features. First, the peak of KE shifts towards the small scales with increasing horizontal resolution:
12 km for a 2.5 km horizontal grid spacing;
8–9 km for a 1 km horizontal grid spacing;
6 km for a 500 m horizontal grid spacing;
5.5 km for a 250 m horizontal grid spacing.
This peak characterizes the size of convective motions for this case. A convergence effect can be noticed around 5.5 km wavelength. For the 250 m horizontal resolution, an additional simulation has been performed with 350 m vertical grid spacing above 2000 m instead of 500 m (below 2000 m the same very fine resolution is kept). The characteristics of the KE spectra remain very similar (not shown), supporting that the convergence is not an effect of inadequate vertical resolution.
For idealized simulations of squall lines with increasing resolution (with grid spacing up to 125 m), Bryan et al. (2003) show that the convergence is not reached between the 250 m and 125 m simulations. However, in our real case of isolated convective cells, the 250 m simulation resolves an inertial subrange by a greater extent as shown by the energy spectra of Figure 13(b), suggesting that the LES regime could likely be reached with a coarser resolution than in Bryan et al.'s (2003) study. Indeed, Figure 14 shows that the convective systems are very similar for 500 m and 250 m grid spacing simulations. The benefit of an increase in horizontal resolution is particularly striking between the 2.5 km and 1 km grid spacing simulations. For the two highest horizontal resolutions, thunderstorm cells have the same spatial extent and the current density that goes against the easterly wind has the same extension. The cloud depth and phase speed of the convective cells are similar whereas the cloud top exceeds more than 10 000 m and the cells are located more eastward for 2.5 km and 1 km grid spacing simulations. This is also supported by Parodi and Tanelli (2010), who found that 333 m horizontal grid spacing simulations with the WRF model using an LES-type turbulence parametrization reproduces properly the physical processes and statistical properties of deep convection that developed in quasi-equilibrium conditions over the East Pacific Intertropical Convergence Zone (ITCZ).
Nevertheless, a simulation with horizontal grid spacing of the order of 100 m would be required to properly address this point, but it unfortunately exceeds our computer capabilities for the moment.
4.4. Sensitivity to parametrization
Additional sensitivity runs to different physical parametrizations have been performed at a 500 m horizontal resolution with Meso-NH (Table 2) in order to represent the resolved part of the flow and to evaluate the strengths and weaknesses of the parametrizations in the grey zone for turbulence.
Figure 15 represents KE spectra on horizontal and vertical velocities with Meso-NH at 500 m horizontal resolution in the boundary layer (BL) with 1D turbulence without EDMF, or 3D turbulence with or without EDMF, which are three alternatives of the model for the turbulence parametrization at this scale range. We focus on the BL because the differences in the free troposphere are weaker. Without horizontal turbulent mixing (with 1D turbulence) and without EDMF, there is a small excess of energy below 5 km wavelength on horizontal wind spectra as underlined before, meaning that the resolved part is too large and that the turbulence scheme does not mix the boundary layer well enough. When using a 3D turbulent scheme instead of a 1D scheme, again without EDMF, dissipation at the spectra tail is slightly reinforced: the KE spectrum on the horizontal wind approaches the −5/3 slope but remains above it. The same effect is discernible on the vertical velocity spectrum. However, the energy damping is too high when adding the mass-flux scheme, as it occurs from 20 km and removes too much small-scale variance. This is in agreement with Honnert etal. (2011), who find that the resolved part of the mixing in the boundary layer, compared to the adequate partial similarity function obtained from the LES cases, is too high at grid sizes smaller than 2 km when the mass-flux scheme is not activated. In contrast, the resolved part is too weak when the mass-flux scheme is activated, meaning that the latter can be improved at grid sizes smaller than 2 km. The authors suggest taking the mesh size into account, for example in the entrainment/detrainment rates. Further study will be needed to assess this new parametrization and adapt it for the free atmosphere. Nevertheless, this part clearly illustrates that physical parametrizations can induce significant damping. It is worth noting that the assessment of subgrid mixing schemes should take into account the other damping processes of the model and that a change in the dynamical core of a model (numerical filters in the model) could imply an adaptation of these parametrizations.
Kinetic energy spectra from mesoscale NWP forecasts with AROME (2.5 km horizontal grid spacing) have been examined over 6 months of winter and summer over France and then over a real convection case study. A comparison of the spectra with the Meso-NH model has been added in the convective case as both models present the same physics at 2.5 km resolution and are different only with regard to dynamics. This study allows us to underline the strengths and weaknesses of the mesoscale models and to determine their resolving capabilities. Both models reproduce the observed k−5/3 wavenumber dependence characteristic of the mesoscale and its transition from a steeper k−3 behaviour at larger scales. It is worth noting that a convection-permitting SLSI model is able to reproduce the transition to k−5/3 mesocale behaviour, in contrast to Shutts (2005), who documents the absence of this transition for the KE spectrum of the ECMWF model at coarser resolution. The AROME forecast spectra over long periods are consistent with the altitude and diurnal cycle effects. Diurnal variations of KE during summer suggest an injection of energy at small scales, supporting the existence of a reverse cascade. However, adiabatic runs show that the transition to a shallow mesocale regime is still apparent, suggesting that to some extent the mesocale regime also derives from a forward cascade even if the mesoscale KE amplitude strongly depends on the presence of the physical processes.
Spectral analysis in the real case of isolated convective cells confirms that the scale of the finest fully resolved modes is primarily dependent on implicit and explicit diffusion and shows that Meso-NH (effective resolution around 4–6Δx) is less dissipative than AROME (effective resolution around 9–10Δx) even if explicit diffusion is removed from AROME. Moreover, damping increases with the time step, regardless of the time schemes tested (3-TL SI, 2-TL SI, 2-TL ICI). This effect is also noted for adiabatic runs; therefore, the dynamical part of the model seems to be mainly responsible for the loss of accuracy at short wavelengths. For large time steps, the damping can be attributed preferentially to the SI part of the SISL formulation. Inversely, the relative contribution of the SL compared to SI increases at small time steps.
This illustrates the difficult challenge of NWP models to find a good compromise between efficiency and effective resolution, which concerns mainly the numerical treatment of propagating gravity waves and advection. The advection is challenging, not only for sharp gradients and discontinuities in scalar quantities such as cloud condensation nuclei concentrations and aerosols (Wang et al., 2009) but also for explicit moist convection which induces, besides strong gradients of hydrometeors, a strong convergence/divergence area.
On one hand, the SISL numerical schemes are the most efficient schemes, but the implicit diffusion of the SISL as it is presently used in AROME limits its resolving capability. Improvements are necessary to reduce the spatial filter, and one way is to take into account deformation associated with divergent/convergent motions present in convective cells (Malardel, 2009). Another way is likely to use higher-order spatial interpolations (Leslie and Dietachmayer, 1997) or to avoid the use of interpolation in the scheme, as suggested by Ritchie (1986). Besides, the choice of time step needs special care in order to maintain enough small-scale variability, as the use of too large time steps removes some KE variance and can damp some parts of the physical solution. As pointed out by Staniforth and Côté (1991), it would be important to increase the order of time discretization to keep the time-step advantage in comparison with an analogous Eulerian model for hectometric simulations.
On the other hand, even-order centred Eulerian schemes are scale selective because they contain very little implicit diffusion but are not sufficiently efficient when associated with leapfrog time marching as in Meso-NH. The necessary evolution of the Runge–Kutta time integration scheme (Wicker and Skamarock, 2002) improves efficiency (though less than SISL schemes) but the associated odd-order upwind-biased advection schemes are inherently diffusive and limit effective resolution somewhat.
Sensitivity to horizontal resolution with Meso-NH shows that the effective resolution remains around 4–6Δx. With increasing resolution, the smaller scales are better resolved and the peak of KE associated with vertical velocity shifts towards smaller scales and stabilizes around 5.5 km. Indeed, for the 250 m horizontal grid spacing simulation, this is well beyond the effective resolution as the wavelength of the peak represents more than 20Δx. The slope is slightly more than −5/3 at small scales, suggesting a slight overestimation of resolved motions. The effects of subgrid mixing have been investigated at 500 m horizontal resolution with Meso-NH, which is an intermediate scale between LES and current operational mesoscale predictions (around 2 km). With a 1D turbulence scheme, the resolved motions are too strong as the dynamics of the model tend to produce eddies to improve mixing in the BL. 3D turbulence instead of 1D improves the resolved/unresolved partition by removing a small part of the resolved eddies, while the mass-flux scheme produces excessive damping of the physical small-scale eddies and degrades the effective resolution. To obtain a better representation at these intermediate scales, the subgrid thermal of the mass-flux scheme should be weaker in order to simulate only a part of these thermals, as proposed by Honnert etal. (2011).
This study has focused on KE spectra; however, it would be interesting to evaluate how realistically AROME reproduces the variability of precipitation observed in nature. Further study will therefore compare rainfall spectra from the model and from the radar estimates of the Meteo-France mosaic. Further studies will also document the KE spectra for severe convection cases.
Acknowledgements The authors want to thank Antonio Parodi and another anonymous reviewer for their helpful suggestions. We are grateful to Karim Yessad, Pierre Bénard and Yann Seity for useful discussions on the SL scheme of the AROME model. We would like to thank Gaëlle Tanguy for her contribution in making the spectral tool available to Meso-NH users.
Appendix A. Contribution of each element of the variance array to two wavenumbers
Using the same formulation as Denis et al. (2002), the total variance can be expressed from spectral coefficients as
with (m,n)≠(0,0), Ni and Nj the number of points for the two dimensions of the domain.
Analysing data on rectangular domains requires the use of an elliptical truncation for constructing the power spectra. Normalization of the m and n wavenumber axes by Ni and Nj leads to an elliptic shape. Thus the normalized 2D wavenumber α(m,n) is defined as
An element in the variance array σ2(m,n) is located in a wavelength band between the limiting values of α′(k) and α′(k + 1) if α′(k) ≤ α(m,n) < α′(k + 1), with , k varying from 1 to min(Ni − 1,Nj − 1).
Instead of adding the contribution σ2(m,n) only to wavenumber k as in Denis etal. (2002), an alternative method is to distribute this contribution between the two wavenumbers k and k + 1 as follows:
the contribution to σ2(α′(k)) is: am,n × σ2(m,n);
the contribution to σ2(α′(k + 1)) is: bm,n × σ2(m,n);
am,n and bm,n representing weighting factors as a function of the distance to the limits α′(k) and α′(k + 1).
Appendix B. Simplified Lagrange cubic interpolations
For a 2D horizontal interpolation, four lines along the x-axis and four lines along the y-axis of the model grid surrounding the origin point are considered, forming a 16-point grid (4 × 4 points) with the origin point between the four inner points. For the second and third lines, cubic four-point interpolations are computed. For the first and fourth lines, only linear interpolations are computed using the second and third points of these lines. Then, a meridian cubic interpolation is computed from the four preceding interpolated values. Thus only the 12 closest points to the origin point are used to compute the interpolated data (instead of 16 points if cubic interpolations were used for each line).
For a 3D interpolation, the four levels surrounding the origin point are considered, i.e. four 16-point grids. For the second and third levels, the above-mentioned 2D horizontal interpolation is used. For the first and fourth levels, bi-linear interpolations are computed using the four inner points of the 16-point grid of these levels. Then, a vertical cubic interpolation is done using the interpolated values of the four levels. Thus only 32 points are used for each 3D interpolation (instead of 64 points), which combines four linear, two bi-linear and seven (quasi-monotone or standard) cubic Lagrange interpolations.