We consider two complementary approaches to stochastic parameterisation, both developed at ECMWF for IFS. The two schemes considered here are used operationally at weather and seasonal forecasting centres worldwide , and so have been extensively tested in a range of models on medium range and seasonal timescales. Following the “seamless prediction” paradigm, we choose to test these schemes here on climate timescales.

The first approach is the SPPT scheme , which uses multiplicative noise to represent model uncertainty due to the parameterisation process. The use of multiplicative noise has been motivated through several coarse-graining studies (; ). SPPT can be expressed as ∂X∂t=D+K+(1+μe)∑iPi, where ∂X∂t is the modelled total tendency in X. This is the sum of D, the dynamical tendency, K, the horizontal diffusion, and each Pi term, with Pi being the tendency from the ith physics scheme. The zero mean random perturbation, e, is constant in the vertical, but μ tapers the perturbation to zero close to the surface and in the stratosphere. The scheme acts on the tendencies of the physical fields (i.e. temperature, winds and specific humidity) resulting from the five main parameterisation schemes: radiation, turbulence and gravity wave drag, non-orographic gravity wave drag, convection, and large-scale water processes. All variables are perturbed with the same random number.

The perturbation, e, is generated using a spectral pattern generator , ensuring that it smoothly varies in space, while the patterns evolve in time following an AR(1) process. The perturbation at each time step is the sum of three independent random fields, which represent uncertainties on different temporal and spatial scales. The fields have horizontal correlations of 500, 1000 and 2000 km and temporal decorrelations of 6 h, 3 days and 30 days respectively, with associated standard deviations of 0.52, 0.18 and 0.06. The magnitude of the perturbation has been motivated through coarse-graining high-resolution model simulations , and a recent coarse-graining study has also provided justification for the noise temporal and spatial correlation scales . While the smallest scale (500 km and 6 h) dominates on weather forecasting timescales, it is expected that the larger scales will also be important on climate timescales. The SPPT scheme requires no retuning with changing horizontal resolution: the same noise characteristics are used operationally at ECMWF across model resolutions. This is because the multiplicative nature of the scheme applied to the total of all parameterised physical tendencies results in perturbations to the model that scale automatically as the parameterised tendencies scale with resolution. The resolution dependence of the individual contributions from the parameterisation schemes is implicitly dealt with at the (deterministic) parameterisation level.

In contrast to SPPT, the SKEB scheme aims to represent a physical process that is otherwise absent from the model . Kinetic energy loss is common in models due to numerical integration schemes and the parameterisation process . To counteract this, the SKEB scheme represents upscale transfer of kinetic energy by randomly perturbing the stream function.

Similar to SPPT, the SKEB scheme uses a spectral pattern generator to generate a spatially and temporally correlated perturbation field, which is added at each time step to the deterministic stream function tendency, ψt˙(ϕ,λ,t)=ψdet˙(ϕ,λ,t)+f(ϕ,λ,t), where ψt˙ is the total stream function tendency, ψdet˙ is the deterministic tendency and f is the additive perturbation field. The perturbation field is expressed in spherical harmonics, and each coefficient is evolved separately in time following an AR(1) process. A tuning parameter for the SKEB scheme, the backscatter ratio, is set to increase following resolution increase (see Table ; following practice at ECMWF) in order to improve the slope of the kinetic energy spectrum.

The standard SPPT and SKEB schemes were designed for use at NWP timescales. When implemented in the EC-Earth climate model, it was found that the SPPT scheme resulted in a large imbalance involving the water cycle, with a negative precipitation minus evaporation (P-E) that was 10 times larger than in the deterministic model associated with an anomalous latent heat flux and a negative net surface flux of about 2 Wm-2. This arises because the SPPT scheme was not designed specifically to conserve water vapour, and resulted in a water vapour sink in the atmosphere. A fix has been implemented, requiring that the global average of the tendencies (i.e. winds, temperature and more importantly specific humidity) before and after the SPPT perturbation is conserved. The new scheme removes the imbalance in P-E, which is now equal to that in the run where the SPPT scheme is disabled. This fix has subsequently been implemented at ECMWF .

Hereafter, Climate SPHINX simulations where stochastic parameterisation is operational will be defined as “stochastic” runs, while simulations where the scheme is deactivated will be mentioned as “deterministic” runs.

With respect to the previous version (v3.0.1), EC-Earth 3.1 shows a reduced radiative imbalance and an improved hydrological cycle . However, it still exhibits a cold bias in both its atmosphere-only and coupled configuration and a small imbalance in P-E. In a “present day” AMIP configuration, the 3.1 version is too cold, extracting heat from the underlying sea surface temperatures (SSTs) by about 1.5 Wm-2 and showing unrealistically high values for net SW (shortwave) and LW (longwave) fluxes at TOA (top of the atmosphere) (around 243–244 Wm-2). Thus, the first goal of the tuning was to provide reasonable radiative fluxes at TOA and at the surface for the standard deterministic version of the model (T255; see next paragraph for further description on the configurations adopted).

To improve the radiation budget, some of the convection and microphysical parameters from a more recent version of IFS (cy40r1) were retrieved. In addition to this, two standard tuning parameters have been modified see; the entrainment rate for organised convection (ENTRORG) was reduced from 1.75×10-4 to 1.5×10-4, and the rate of conversion of liquid water to rain (RPRCON) was reduced from 1.4×10-3 to 1.2×10-3.

The optimal choice of tuning parameters provides reasonable fluxes at the TOA (around 240 Wm-2) and a positive flux at the surface of about 0.6 Wm-2, in accordance with the best estimates from observations . It is important to note that the tuning of the radiative fluxes has been carried out only for the T255 deterministic model version: the radiative balance has not been tuned in the higher resolution or stochastic models. This ensures a clean comparison between simulations at different resolutions and with and without stochastic physics. If the model were re-tuned for each run, it is not possible to determine whether it is changing the tuning parameters, changing the resolution or including stochastic physics that are responsible . Including the SPPT scheme led to a negative bias in the surface heat fluxes of about 0.8 Wm-2, likely caused by a different distribution of the clouds.

The main radiative fluxes resulting after the complete tuning procedure are reported in Table . As shown in this table, the radiative balance of the model at higher resolution (and with stochastic physics) shows larger TOA SW and LW with increasing resolution. Net surface fluxes are highly variable, with higher values for coarser resolutions.

Finally, a supplementary modification – derived from a more recent IFS cycle – has been performed in order to produce a realistic quasi-biennial oscillation (QBO) at all resolutions. The EC-Earth 3.1 non-orographic gravity waves scheme is characterised by a momentum flux that is continuously launched in the mid-troposphere to simulate the effect of gravity waves. The latitudinal profile of this momentum flux governs the correct parameterisation of gravity waves: a too high amplitude of the momentum flux will disturb the QBO in equatorial zones, particularly at high resolutions, while a too low value will lead to unrealistic eddy-driven jets, especially in the Southern Hemisphere, where orographically induced wave drag is low. With the current latitudinal profile, the QBO was simulated only at standard resolution (T255 with 91 vertical levels). Following advice from ECMWF staff, a resolution-dependent parameterisation of non-orographic gravity wave drag replaced the version-dependent parameterisation present in EC-Earth 3.1 (an ad hoc parameterisation developed for the ECMWF System 4 seasonal forecast system). Namely, instead of using a low-momentum flux average value (GFLUXLAUN = 0.02) with a positive Gaussian peak at 50∘ S, we use a higher value (GFLUXLAUN = 0.0375), which is reduced with a Gaussian shape at the Equator. This negative peak is slightly deeper for stochastic runs than for deterministic simulations to compensate for the effect of the stochastic noise. The average value of the momentum flux was further reduced with increasing resolution (starting from T799) according to the ECMWF specification for IFS cy40r1 (see Table ).

The new non-orographic gravity wave scheme – a standard in the current operational forecast ECMWF model – allows for the simulation of the QBO at all the resolutions explored in Climate SPHINX, without deteriorating the jet streams. Given these positive results, the new parameterisation will be implemented in the upcoming EC-Earth 3.2 version.

Climate SPHINX simulations are grouped into three main blocks: present-day AMIP (PDA), future scenario AMIP (FSA) and past-to-future coupled (PFC). PDA and FSA are atmosphere-only simulations: 20 ensemble members are run at T159 (∼ 125 km), 20 at T255 (∼ 80 km), 12 at T511 (∼ 40 km), 6 at T799 (∼ 25 km) and 2 at T1279 (∼ 16 km) for both PDA and FSA experiments. For each resolution, half of the ensemble members have the stochastic physics parameterisations activated. All simulations have the same vertical grid with 91 levels (L91): these are hybrid levels with the last full level at 0.01 hPa. The number of ensemble members run and their resolution are also reported in Table .

The atmosphere-only experiments extend for 30 consecutive years, from 1979 to 2008 for PDA, while FSA experiments are run from 2039 to 2068.

PFC simulations are run with IFS at the T255L91 configuration, coupled with NEMO using the ORCA1 grid (a tripolar grid with resolution of 1∘ longitudinally and refinement to 1/3∘ at the Equator) with 46 vertical levels. The upper model level is at ca. 3 m and 10 levels are in the upper 100 m. Six ensemble members are run, three with the stochastic parameterisation active and three control members without stochastic parameterisation, from 1850 to 2100.

The initial conditions (ICs) in both the PDA and FSA experiments are taken from the ECMWF ERA-Interim Reanalysis for 1 January 1979. A first experiment is run at each resolution for a few days, and it is used to create the ICs for the other experiments. For instance, for T255 experiments, the ICs for the 10 ensemble members are extracted using the midnight values (00:00) from each of the first 10 days respectively, and then reassigned to 1 January.

The same ICs are used also for FSA: in order to account for the land-surface adjustment to the new forcing, a 1-year spin-up has been carried out for FSA (which is therefore starting from 2038).

For PFC simulations, given the different expected climatologies of integrations with/without stochastic physics, two 320-year spin-ups are carried out in coupled mode to equilibrate the ocean to the atmospheric forcing. Having spun-up, three oceanic states – from spin-up year 300, 310 and 320 – are coupled with three different atmospheric ICs; these are run in coupled mode for a further 10 years with fixed greenhouse gas (GHG) forcing for the year 1850. In this way the phase space distance between the simulations is 20 years and the atmosphere and land surface have had enough time to adjust to the new oceanic state.

Well-mixed GHGs, stratospheric ozone and volcanic aerosol concentrations have been set according to the CMIP5 protocol . Historical forcing is used for PDA experiments, whereas for the FSA experiments the high emission scenario (Representative Concentration Pathway 8.5, RCP8.5) is adopted. PFC simulations use the historical CMIP5 specification from year 1850 to year 2005 included; after that, the forcing is taken from the RCP8.5 scenario. Albedo, land use and vegetation patterns are set using the standard configuration of EC-Earth 3.1, which uses a MODIS-derived fixed climatological seasonal cycle for snow-free albedo and the leaf area index. The average yearly solar irradiance was set at 1368.2 Wm-2 with intrannual variations, following the standard EC-Earth 3.1 set-up. All the simulations of PDA and FSA experiments use this set-up. For the PFC simulations interannual variations following CMIP5 prescriptions (i.e. the 11-year solar cycle) have been added.

Given that both FSA and PDA simulations are atmosphere-only runs, a special effort has been taken to provide reliable SSTs in order to fully exploit the high resolution.

For PDA, SSTs have been obtained from the daily SST and SIC HadISST2.1.1, a pentad-based dataset with a resolution of 0.25∘×0.25∘ for SSTs and 1∘×1∘ for SIC . These are bilinearly interpolated onto the required reduced Gaussian grid for each resolution: climatologies for SST and SIC for the 1979–2008 period can be seen in the upper panels of Fig. .

A number of inconsistencies are found between the land–sea mask of IFS and of HadISST2.1.1; these are due to slightly different coastlines and a different representation of the lakes. For the different coastlines, linear extrapolation from HadISST2.1.1 has been performed. For the interior (i.e. lakes), a methodology similar to the one used in ERA20CM dataset has been adopted: 1-month lagged 2 m temperatures from the ERA-Interim monthly climatology of 1979–2008 are used as SST. Where the temperature is below zero, SIC is set to one, otherwise it is left at zero. This is interpolated in time on a daily basis and in space on the needed grid to create a smoothed seasonal cycle for lakes.

The creation of SST and SIC for the FSA experiment is more complex. We would like to consider the mean change and trend for the future climate (2038–2068) predicted by the state-of-the-art global coupled models (i.e. the CMIP5 models). However, the oceanic component of these models has generally a low horizontal (of the order of 1∘) and temporal (usually monthly) resolution, in which the oceanic circulation is not perfectly resolved. In order to improve our boundary conditions, we decided to take advantage of the high temporal and spatial resolution provided by the HadISST2.1.1. As a consequence, the SST for FSA experiments have been obtained as a combination of HadISST2.1.1 variability and the CMIP5 EC-Earth simulations ensemble mean trend.

First, the 1979–2008 HadISST2.1.1 SST has been detrended point by point to provide a set of anomalies with realistic variability. Second, the monthly seasonal cycle of the difference between the CMIP5 EC-Earth RCP8.5 ensemble mean over 2038–2068 (10 members) and the CMIP5 EC-Earth historical ensemble mean over 1979–2008 (10 members) has been computed. This provides for each grid point the average expected SST increase from the present-day to the future period according to a GCM (Global Circulation Model), as a function of calendar month. To account for changes in SST during the FSA period, for each grid point the average trend in SST from the CMIP5 EC-Earth RCP8.5 integrations for 2038–2068 was also extracted. All CMIP5 EC-Earth data were bilinearly interpolated in space on the HadISST2.1.1 grid and linearly in time to daily frequency.

Finally, a new Step1HadISST dataset has been created combining the detrended HadISST2.1.1 (expressing the high-resolution daily variability), the average daily change of CMIP5 EC-Earth (from RCP8.5 and historical, expressing the expected average temperature increase) and the linear trend of the CMIP5 EC-Earth RCP 8.5 (expressing the expected future trend in SST). The methodology used here, which shares the main characteristics with the method developed by , is sketched in Fig. .

However, the Step1HadISST reconstruction misses an important element; there is no information on the sea ice cover in the future. To account for this, we took data from the CMIP5 EC-Earth simulations as a reference for SIC. CMIP5 EC-Earth simulations show a considerable cold bias in SST with respect to HadISST2.1.1, but they show good sea ice coverage, especially for the Northern Hemisphere (see average Northern Hemisphere and Southern Hemisphere SIC in Fig. ).

Considering that an ensemble mean would be unrealistic, especially for a field with a large spatial variance as sea ice, we select a single ensemble member representative of the ensemble. Member “r8i1p1” has been chosen to characterise the ensemble, since its climatology shows the smallest SIC root mean square error (RMSE) when compared to the ensemble mean climatology in the time window 2038–2068. Clearly, using RMSE is only one of the possible metric to perform such selection; our main goal is to pick an ensemble member that is not an outlier when compared to the other EC-Earth CMIP5 ensemble members.

As a last step, we must evaluate SST for points where SIC coverage has disappeared in the future scenario. The lack of information about historical SST under sea ice results in undefined SST at these points using the methodology outlined above. We define these as “bare point”. For bare points, we want to make use of the model variability, but we do not want to have inconsistent SST at the boundaries (i.e. where bare points border the Step1Hadisst dataset).

Initially, we perform a linear extrapolation for bare points for Step1HadISST SSTs – which gives us a measure of the average SSTs at the bare points. However, these extrapolated values are missing a realistic spatial variability. We then mask the bare points also in the SST field of the CMIP5 EC-Earth ensemble member “r8i1p1”, and we subsequently linearly extrapolate new values. We then subtract from the original field of CMIP5 EC-Earth ensemble member “r8i1p1” these new extrapolated values, in order to obtain an anomaly field, which includes the spatial and temporal variability of the SST field over the bare points given by CMIP5 EC-Earth “r8i1p1”. This final field is then added to the linearly extrapolated Step1HadISST SST.

Hence, for each day, the SSTs for bare points are given by the EC-Earth CMIP5 RCP8.5 ensemble member “r8i1p1” SST minus extrapolated EC-Earth CMIP5 RCP8.5 SSTs plus extrapolated Step1HadISST. The methodology to obtain this specific SST reconstruction is illustrated in Fig. .

This provides a pattern of SSTs physically consistent with SICs; indeed, it avoids unrealistic values of SSTs in the proximity of the polar cap during winter and – using the CMIP5 EC-Earth data – it provides a reasonable distribution of SSTs in summer, where in the future scenario the sea ice coverage in the Northern Hemisphere often disappears. Moreover, there is no discontinuity at the border with Step1HadISST. The new dataset is defined as FutureHadISST2.1.1.

The same methodology used for the PDA simulations has been adopted also for FSA runs in order to solve the issues of the lakes and the different land sea mask; however, in this case we must account for the estimated temperature change over land. We consider the difference between the 1-month lagged 2 m surface temperature from CMIP5 EC-Earth RCP8.5 and the 1-month lagged 2 m surface temperature from CMIP5 EC-Earth historical ensemble (averaged over eight members). We then add this to the 1-month lagged 2 m surface temperature ERA-Interim monthly climatology of 1979–2008. This, analogous to what was done for SSTs, accounts for climate change.

The SST and SIC changes between FutureHadISST2.1.1 and HadISST2.1.1 are reported in Fig. ; as expected larger warming and sea ice retreat is seen in the Northern Hemisphere high latitudes. Figure  reports the time series and trends for SST (between 45∘ S and 45∘ N) and SIC (for both Northern and Southern hemispheres) for both FutureHadISST2.1.1 and HadISST2.1.1.

Simulations have been run on the 6.8 PFLOP SuperMUC IBM Petascale System at LRZ. The initial set-up and configurations have been performed on the Supermuc-I platform, based on Sandy Bridge-EP Xeon E5-2680 8C processors. For processor decomposition the Message Passage Interface (MPI) parallelism paradigm has been used. EC-Earth allows also for OpenMP/Shared memory parallelisation, which has been tested without showing any significant computational benefit.

An accurate scaling of the performance was performed during the first months of the simulations. However, a conservative choice has been undertaken, after considering that the wall time needed to run the simulations was not the main concern for the project success. The number of cores assigned to each experiments have been selected following the resolution of the model considered. Although stochastic physics experiments showed about 5–10 % decrease in performance (according to different resolutions), the same number of cores has been retained.

In summer 2015, a new Supermuc-II platform based on Haswell Xeon E5-2697 v3 processors was made available by the LRZ. The new HPC granted a reduction of about 5 % of the total core hours used, without affecting the wall time. About 75 % of the simulations have been run using the Haswell nodes. Details on the processor decomposition, computational costs and data outputs are reported in Table .

In Climate SPHINX, IFS has been set up to provide output in GRIB format every 3 h; however, the four T1279 simulations alone sum up to about 200 TB of raw data output. Summing together the restarts files and the output of all experiments the total amount of space occupied at the peak of the project (February 2016) reached about 1 PB. In order to reduce the size of the output and to increase the data accessibility to a larger audience, automatic post-processing routines have been implemented. At the end of each simulation leg, a script aimed at post-processing is launched; the script handles both the spectral and reduced Gaussian data from IFS and extracts and converts the requested variables from the default ECMWF format to a user-friendly, CMOR-like format on regular Gaussian grid. With this automatic procedure, more than 140 TB of post-processed data has been produced. A significant reduction of the data volume was obtained making use of the NetCDF-4 Zip format.

Monthly (MON), daily (DAY) and 6 h (6HRS) data for different subsets of variables have been produced. More than 50 fields have been stored at monthly frequency. In order to further reduce the space requirements, daily and 6 h three-dimensional (3-D) fields have been degraded to the spectral resolution of T255. Additional data at 3 h frequency have been stored for the Euro-Cordex domain (3HRS-CDX) and for a sub-domain including India, Tibet and Pakistan (3HRS-ITP). Total precipitation has also been saved over the global grid at full resolution with 3 h frequency (3HRS). Finally, synoptic monthly means have been stored for the main radiative variables (SMON). A few fields that are non-linear functions of the output (e.g. specific humidity) have been computed from the original 3 h output and then averaged at the required frequency in order to record them accurately. In addition to the atmospheric data, about 10 TB of oceanic output has been stored for PFC simulations. Data at daily and pentad frequency have been retained.

All the data, including raw output, post-processed data and restart files, have been archived on the tape archives of the Tivoli Storage Management Infrastructure of the LRZ.

Although a detailed analysis of the mean climate in all the simulations performed would be excessively long to be included in the present work, we introduce a couple of figures showing the sensitivity to resolution and stochastic physics parameterisation of the climatology of precipitation (Fig. ) and 200 hPa zonal wind (Fig. ). We compare the ensemble mean average fields of a low-resolution version (T255) with a high-resolution one (T799), in both its deterministic and stochastic configurations. Data have been interpolated on a common 2.5∘×2.5∘ grid.

The precipitation model bias – shown in Fig. , with respect to Global Precipitation Climatology Project (GPCP) dataset – is especially strong in Indian Monsoon region, with an excess of precipitation from the Indian Ocean to the western Pacific. More generally, EC-Earth tends to underestimate the precipitation over the continents and overestimate it over the oceans. When the comparison is carried out between stochastic and deterministic configurations, it is possible to see that SPPT and SKEB neither improve nor deteriorate the climatology at both T255 and T799 resolutions. Conversely, a slightly more evident change is seen comparing the high and low resolutions; here T799 shows a widespread increase of the extratropical precipitation. But again, when it is evaluated against the model bias such changes are minor.

Impacts on the upper-tropospheric zonal wind field are clearer and they are shown in Fig. . The T255 version – compared against ECMWF ERA-Interim reanalysis – shows too strong jets in both the hemispheres. The subtropical jet over Asia and the Pacific is also poleward displaced, while equatorial easterly jets are too weak. Again, stochastic physics bring minor changes, with a slightly stronger Atlantic jet, more penetrating over Europe. Conversely, the higher resolution leads to an overall weakening of the upper-tropospheric winds; this is especially true over North America and the Tibetan Plateau, suggesting that this change may be induced by the stronger surface drag caused by the more resolved (and thus higher) mean orography.

More generally, in these and other climatological fields (not shown) the impact of the two stochastic parameterisations and resolution appears to be small if compared to the model bias. Indeed, larger benefits from increasing resolution and stochastic physics are expected more in terms of variability rather than in terms of mean state e.g..

Therefore, in the following sections we will focus on a few selected features of climate variability. We will investigate the improvements and/or deteriorations following resolution increase and including the SPPT and SKEB stochastic parameterisations of three different phenomena: the distribution of the intensity of tropical rainfall, the tropical variability related to the Madden–Julian Oscillation and the mid-latitude variability associated with atmospheric blocking.

Climate models generally have too little tropical variability on timescales of several days e.g.. One aspect of the variability of particular interest is the occurrence of heavy-precipitation events, which can result in flooding, affect disease incidence and reduce crop yields . Changes in the frequency of these events can also affect trends in total precipitation due to non-linearity in land-surface processes .

Figure a shows the frequency distribution of daily-mean precipitation rates averaged over 2.5∘×2.5∘ grid boxes between 10∘ S and 10∘ N over the period 1998–2008 in data from GPCP, data from the Tropical Rainfall Measuring Mission (TRMM) 3B42 version 7 product and one ensemble member for each PDA run. Figure b shows the ratio of the frequency in each rain rate interval as a fraction of that in GPCP for each resolution.

Vertical bars in Fig.  show the 95 % confidence intervals of the frequencies associated with sampling uncertainty. These were calculated using a bootstrap method. For each dataset, a surrogate dataset was created by randomly sampling individual years of data with replacement. The frequency distribution of the surrogates and their frequency ratios with respect to the GPCP surrogate were calculated. This was repeated 1000 times to produce the distribution of the calculations associated with sampling uncertainty, from which the confidence intervals were derived.

At all resolutions, rain rates below 15 mmday-1 occur too often in the model data, by about 50 %, and rain rates between 20 and 60 mmday-1 occur too infrequently compared to both observational datasets. At rain rates near 30 mmday-1, the simulated frequencies are between about 35 and 50 % of the frequency in GPCP. At higher rain rates, the frequency differences between TRMM and GPCP become comparable in size to or larger than the differences between the modelled frequencies and the observational datasets. We do not know of a reason to strongly prefer one dataset over the other; therefore, we consider the model bias to be uncertain at these rain rates. The frequency of rain rates above 30 mmday-1 in the T159 and T255 models are below about 40 % of that in GPCP. At T511, T799 and T1279 the relative frequency difference compared to GPCP and TRMM decreases as the rain rate increases, and becomes comparable to that in GPCP in the 60–65 mmday-1 interval, though still much smaller than that in TRMM. Therefore, increasing the model resolution from T159 to T511 improves the simulated frequency of heavy-rainfall events compared to observational datasets, with the further improvements caused by increasing the resolution to T799 or T1279 being considerably smaller.

Figure c, d show the same data for the stochastic PDA runs. Stochastic physics has a similar effect at all resolutions. Frequencies of rain rates between 5 and 15 mm day-1 are reduced by about 10 % compared to those in the deterministic models, reducing the model bias. Frequencies above about ∼ 20 mmday-1 are substantially increased, by a larger factor at larger rain rates, up to a factor of ∼2.5 at rain rates around 60 mmday-1. This reduces the difference from GPCP and TRMM up to rain rates of 45 mmday-1 at all resolutions.

The higher-resolution stochastic models have rain rate frequencies between those of GPCP and TRMM at rates above 45 mmday-1, so they seem consistent with the observations given the observational uncertainty. The T255 stochastic model has rain rate frequencies closer to those in GPCP than any of the deterministic models in all but two of the 5 mmday-1 rain rate intervals shown. One hypothesis to explain this effect is that the stochastic perturbations sometimes increase the moistening tendency of the air, so that it occurs more often that there is a high amount of water vapour in the air and heavier rain events can occur, and there is a compensating decrease in the frequency of moderate rain events.

Therefore, stochastic physics brings this aspect of the simulations into better agreement with observations, suggesting that including a representation of unresolved variability and model error is important for simulating the statistics of extreme tropical precipitation events.

The MJO is the dominant mode of variability in the tropical region on sub-seasonal timescales . It is characterised by a strong interaction between tropical convection and the large-scale environment, manifest as a coherent eastward propagating pattern of precipitation followed by subsequent rainfall suppression . It is a challenge for the current generation of global climate and weather models to represent the dynamics and thermodynamics of the MJO realistically .

Here we use the technique to identify the dominant modes of variability in zonal winds and outgoing longwave radiation (OLR) in these model runs. Combined Empirical Orthogonal Functions (CEOFs) of intraseasonal OLR, U850 (zonal winds at 850 hPa) and U200 (zonal winds at 200 hPa) are computed for each of the runs. The first two leading modes (Realtime Multivariate MJOs: RMM1 and RMM2) correspond to MJO signatures in the tropical wind field and OLR. The amplitude A of the MJO is defined as A=(RMM1)2+(RMM2)2 and the phase Φ of the MJO is defined as Φ=tan⁡-1RMM2RMM1.

MJO occurrence is defined when the MJO A is > 1. Conventionally, eight phases of the MJO are defined . We reduce the eight phases to four phases respectively corresponding to the MJO being active in the Indian Ocean, Maritime Continent, Western Pacific and Western Hemisphere. We note that the Wheeler Hendon RMM index has been shown to be deficient in detecting MJO events when large-scale circulation signals of the MJO are missing . Figure  shows the frequency of occurrence vs. the mean amplitude of the MJO in the four different regions around the tropics for all the different runs (colours) and for ECMWF ERA-Interim Reanalysis grey; over the 1980–2001 period.

Overall the frequency of occurrence of the MJO in the different regions in the tropics for the different model resolutions is underestimated with respect to that of ERA-Interim. The MJO amplitude in the model simulations is lower than reanalysis over the Indian Ocean and the Western Hemisphere.

More importantly, increasing horizontal resolution does not seem to improve the representation of the phenomenon significantly. This may be explained considering that the simulation of the MJO in GCMs is influenced primarily by the representation of mesoscale dynamics and of convection. The simulation of mesoscale dynamics can be helped by increasing the resolution, while improvements in convection are driven by changes in physical parameterisations of the model. Yet, the coupling between the mesoscale dynamics and the convection is key for convectively coupled waves in the tropics . Therefore, increasing resolution alone may not be sufficient to improve the simulation of the MJO.

Conversely, the stochastic physics parameterisation improves the MJO frequency in all regions at all resolutions but T1279. It must be noted that this latter run was done with only one ensemble member compared to the other runs with three or more ensemble members over the same period; therefore, a sampling error due to natural variability should be considered. Above all, the best results are obtained for the T255 with stochastic physics, suggesting that the tuning of the mean state of the model might play a relevant role for a better MJO simulation.

Additionally, the stochastic physics climate runs show an improvement in the representation of the MJO propagation over the Maritime Continent (not shown). The lack of propagation of the MJO over the Maritime Continent into the western Pacific region is a known problem in GCMs . An improvement in the MJO propagation past the Maritime Continent due to SPPT has also been seen in the ECMWF seasonal forecasting system 4 . also showed an improved MJO propagation and improved probabilistic prediction skill for the MJO in the ECMWF system 4 when SPPT is active as compared to runs without stochastic physics. Such improved propagation in stochastic runs indicates either that there is an impact of the stochastic physics on the mean state in the region or that the variability in the region helps maintain the intraseasonal signal. The reasons for the change in MJO representation due to stochastic physics will be explored further in a more detailed future study by the authors.

One of the most important challenges for the current generation of climate models is the simulation of atmospheric blocking . Blocking is a recurrent weather pattern typically occurring in the Northern Hemisphere at the exit of the Atlantic and Pacific jet stream, more frequently during the winter season but observed throughout the year . It is characterised by a high-pressure, long-lasting low-vorticity anomaly that “blocks” the mid-latitude westerly flow, diverting synoptic disturbances poleward or equatorward . A blocking event can last several days or even weeks, and it may be associated with cold spells in winter and heat waves in summer .

Blocking here is diagnosed using the simple index introduced by , an extension of the better known index. This 1-D blocking index detects the reversal of the zonal flow measuring the geopotential height gradient at 500 hPa at 60∘ N, providing a binary blocking time series for each longitude. Although there is some evidence e.g. that stochastic physics may improve the blocking simulation, with the current diagnostic no statistically significant difference emerges – even at low resolution – when comparing deterministic and stochastic simulations. Therefore, the two simulations are combined together to provide an unique ensemble.

The upper panel of Fig.  shows the blocking frequency for the ERA-Interim Reanalysis (black) and the ensemble mean of the different horizontal resolutions (colours) of PDA experiments over the December–January–February (DJF) period. The common negative bias over the Atlantic and Pacific basins is clearly evident. Increasing the horizontal resolution leads to benefits over both the basins, with marked improvements especially for the Atlantic; here, T799 and T1279 runs show values comparable to the reanalysis. The largest improvement is, however, seen upgrading from T255 to T511, where the bias – measured as the relative difference between the blocking frequency averaged between 10∘ W and 30∘ E – is reduced from the 18 to 3 %.

Those clear improvements in blocking frequency are interestingly reflected by a change in the mean state. A simple way to represent the flow variability is to highlight a few isopleths of geopotential height, as done in the lower panel of Fig. . Indeed, the higher-resolution models show a strengthened pattern of the dominant Northern Hemisphere planetary waves, with marked ridges over the Rockies and Europe. Especially, the former over the Rockies suggests an important role of orography resolution in the representation of the eddy-driven jet stream and, indirectly, of Euro-Atlantic blocking frequencies.

Indeed, the reduction of the bias following resolution increase for winter Atlantic blocking (and not for Pacific blocking) seems to be a common feature of several GCMs . Such improvements have been associated with both better resolved transient eddy activity – which should sustain the blocking persistence – and with higher orography variance – which affects the mean state through planetary waves shaping . Conversely, Pacific blocking has been shown to be phenomenologically different and to be strongly affected by tropical dynamics e.g.; therefore, it is not surprising that the latter would be less affected by horizontal resolution changes.

A more detailed analysis of blocking and mid-latitude variability will be carried out by the authors in future studies.