• Arakawa, A., , J.-H. Jung, , and C.-M. Wu, 2011: Toward unification of the multiscale modeling of the atmosphere. Atmos. Chem. Phys., 11, 37313742, doi:10.5194/acp-11-3731-2011.

    • Search Google Scholar
    • Export Citation
  • Bechtold, P., , J.-P. Chaboureau, , A. Beljaars, , A. K. Betts, , M. Köhler, , M. Miller, , and J.-L. Redelsperger, 2004: The simulation of the diurnal cycle of convective precipitation over land in a global model. Quart. J. Roy. Meteor. Soc.,130, 3119–3137, doi:10.1256/qj.03.103.

  • Blackburn, M., and et al. , 2013: The Aqua-Planet Experiment (APE): Control SST simulation. J. Meteor. Soc. Japan, 91A, 17–56, doi:10.2151/jmsj.2013-A02.

    • Search Google Scholar
    • Export Citation
  • Buizza, R., , J.-R. Bidlot, , N. Wedi, , M. Fuentes, , M. Hamrud, , G. Holt, , and F. Vitart, 2007: The new ECMWF VAREPS (Variable Resolution Ensemble Prediction System). Quart. J. Roy. Meteor. Soc.,133, 681–695, doi:10.1002/qj.75.

  • Cohen, B. G., 2001: Fluctuations in an ensemble of cumulus clouds. Ph.D. thesis, University of Reading, 165 pp.

  • Cohen, B. G., , and G. C. Craig, 2006: Fluctuations in an equilibrium convective ensemble. Part II: Numerical experiments. J. Atmos. Sci., 63, 20052015, doi:10.1175/JAS3710.1.

    • Search Google Scholar
    • Export Citation
  • Côté, J., , S. Gravel, , A. Méthot, , A. Patoine, , M. Roch, , and A. Staniforth, 1998: The operational CMC–MRB global environmental multiscale (GEM) model. Part I: Design considerations and formulation. Mon. Wea. Rev., 126, 13731395, doi:10.1175/1520-0493(1998)126<1373:TOCMGE>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Craig, G. C., , and B. G. Cohen, 2006: Fluctuations in an equilibrium convective ensemble. Part I: Theoretical formulation. J. Atmos. Sci., 63, 19962004, doi:10.1175/JAS3709.1.

    • Search Google Scholar
    • Export Citation
  • Doms, G., and et al. , 2011: A description of the nonhydrostatic regional COSMO model. Part II: Physical parameterization. Consortium for Small-Scale Modelling Tech. Rep., 154 pp. [Available online at http://www.cosmo-model.org/content/model/documentation/core/cosmoPhysParamtr.pdf.]

  • Fox-Rabinovitz, M. S., 2000: Simulation of anomalous regional climate events with a variable-resolution stretched-grid GCM. J. Geophys. Res.,105, 29 635–29 645, doi:10.1029/2000JD900650.

  • Gregory, D., , J.-J. Morcrette, , C. Jakob, , A. C. M. Beljaars, , and T. Stockdale, 2000: Revision of convection, radiation and cloud schemes in the ECMWF integrated forecasting system. Quart. J. Roy. Meteor. Soc.,126, 1685–1710, doi:10.1002/qj.49712656607.

  • Groenemeijer, P., , and G. C. Craig, 2012: Ensemble forecasting with a stochastic convective parametrization based on equilibrium statistics. Atmos. Chem. Phys., 12, 45554565, doi:10.5194/acp-12-4555-2012.

    • Search Google Scholar
    • Export Citation
  • Hagos, S., , R. Leung, , S. A. Rauscher, , and T. Ringler, 2013: Error characteristics of two grid refinement approaches in aquaplanet simulations: MPAS-A and WRF. Mon. Wea. Rev., 141, 3022–3036, doi:10.1175/MWR-D-12-00338.1.

    • Search Google Scholar
    • Export Citation
  • Holloway, C. E., , S. J. Woolnough, , and G. M. S. Lister, 2012: Precipitation distributions for explicit versus parametrized convection in a large-domain high-resolution tropical case study. Quart. J. Roy. Meteor. Soc.,138, 1692–1708, doi:10.1002/qj.1903.

  • Hong, S.-Y., , and J. Dudhia, 2012: Next-generation numerical weather prediction: Bridging parameterization, explicit clouds, and large eddies. Bull. Amer. Meteor. Soc.,93, ES6–ES9, doi:10.1175/2011BAMS3224.1.

  • Honnert, R., , V. Masson, , and F. Couvreux, 2011: A diagnostic for evaluating the representation of turbulence in atmospheric models at the kilometric scale. J. Atmos. Sci., 68, 31123131, doi:10.1175/JAS-D-11-061.1.

    • Search Google Scholar
    • Export Citation
  • Horinouchi, T., 2002: Mesoscale variability of tropical precipitation: Validation of satellite estimates of wave forcing using TOGA COARE radar data. J. Atmos. Sci., 59, 24282437, doi:10.1175/1520-0469(2002)059<2428:MVOTPV>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Huffman, G. J., and et al. , 2007: The TRMM Multisatellite Precipitation Analysis (TMPA): Quasi-global, multiyear, combined-sensor precipitation estimates at fine scales. J. Hydrometeor., 8, 38–55, doi:10.1175/JHM560.1.

    • Search Google Scholar
    • Export Citation
  • Huffman, G. J., , R. F. Adler, , D. T. Bolvin, , and E. J. Nelkin, 2010: The TRMM Multi-Satellite Precipitation Analysis (TMPA). Satellite Rainfall Applications for Surface Hydrology, M. Gebremichael and F. Hossain, Eds., Springer Netherlands, 3–22, doi:10.1007/978-90-481-2915-7_1.

  • Jakob, C., , and A. P. Siebesma, 2003: A new subcloud model for mass-flux convection schemes: Influence on triggering, updraft properties, and model climate. Mon. Wea. Rev., 131, 27652778, doi:10.1175/1520-0493(2003)131<2765:ANSMFM>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Jung, J.-H., , and A. Arakawa, 2004: The resolution dependence of model physics: Illustrations from nonhydrostatic model experiments. J. Atmos. Sci., 61, 88102, doi:10.1175/1520-0469(2004)061<0088:TRDOMP>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Kain, J. S., 2004: The Kain–Fritsch convective parameterization: An update. J. Appl. Meteor., 43, 170181, doi:10.1175/1520-0450(2004)043<0170:TKCPAU>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Kain, J. S., , and J. M. Fritsch, 1990: A one-dimensional entraining/detraining plume model and its application in convective parameterization. J. Atmos. Sci., 47, 27842802, doi:10.1175/1520-0469(1990)047<2784:AODEPM>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Keane, R. J., , and R. S. Plant, 2012: Large-scale length and time-scales for use with stochastic convective parametrization. Quart. J. Roy. Meteor. Soc.,138, 1150–1164, doi:10.1002/qj.992.

  • Mishra, S. K., , and J. Srinivasan, 2010: Sensitivity of the simulated precipitation to changes in convective relaxation time scale. Ann. Geophys., 28, 18271846, doi:10.5194/angeo-28-1827-2010.

    • Search Google Scholar
    • Export Citation
  • Mlawer, E. J., , S. J. Taubman, , P. D. Brown, , M. J. Iacono, , and S. A. Clough, 1997: Radiative transfer for inhomogeneous atmospheres: RRTM, a validated correlated-k model for the longwave. J. Geophys. Res.,102, 16 663–16 682, doi:10.1029/97JD00237.

  • Neale, R. B., , and B. J. Hoskins, 2000a: A standard test for AGCMs including their physical parametrizations. I: The proposal. Atmos. Sci. Lett.,1, 101–107, doi:10.1006/asle.2000.0022.

  • Neale, R. B., , and B. J. Hoskins, 2000b: A standard test for AGCMs including their physical parametrizations. II: Results for the Met Office Model. Atmos. Sci. Lett.,1, 108–114, doi:10.1006/asle.2000.0024.

  • O’Brien, T. A., , F. Li, , W. D. Collins, , S. A. Rauscher, , T. D. Ringler, , M. Taylor, , S. M. Hagos, , and L. R. Leung, 2013: Observed scaling in clouds and precipitation and scale incognizance in regional to global atmospheric models. J. Climate, 26, 93139333, doi:10.1175/JCLI-D-13-00005.1.

    • Search Google Scholar
    • Export Citation
  • Orr, A., , P. Bechtold, , J. Scinocca, , M. Ern, , and M. Janiskova, 2010: Improved middle atmosphere climate and forecasts in the ECMWF model through a nonorographic gravity wave drag parameterization. J. Climate,23, 5905–5926, doi:10.1175/2010JCLI3490.1.

  • Plant, R. S., , and G. C. Craig, 2008: A stochastic parameterization for deep convection based on equilibrium statistics. J. Atmos. Sci., 65, 87105, doi:10.1175/2007JAS2263.1.

    • Search Google Scholar
    • Export Citation
  • Raschendorfer, M., 2001: The new turbulence parameterization of LM. COSMO Newsletter, No. 1, Deutscher Wetterdienst, Offenbach, Germany, 89–97. [Available online at http://www.cosmo-model.org/content/model/documentation/newsLetters/newsLetter01/newsLetter_01.pdf.]

  • Rauscher, S. A., , T. D. Ringler, , W. C. Skamarock, , and A. A. Mirin, 2013: Exploring a global multiresolution modeling approach using aquaplanet simulations. J. Climate, 26, 2432–2452, doi:10.1175/JCLI-D-12-00154.1.

    • Search Google Scholar
    • Export Citation
  • Sakradzija, M., , A. Seifert, , and T. Heus, 2014a: Shallow cumuli ensemble statistics for development of a stochastic parameterization. Geophysical Research Abstracts, Vol. 16, Abstract 7139-1. [Available online at http://meetingorganizer.copernicus.org/EGU2014/EGU2014-7139-1.pdf.]

  • Schumacher, C., , and R. A. Houze, 2003: Stratiform rain in the tropics as seen by the TRMM Precipitation Radar. J. Climate, 16, 17391756, doi:10.1175/1520-0442(2003)016<1739:SRITTA>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Seifert, A., 2008: A revised cloud microphysical parameterization for COSMO-LME. COSMO Newsletter, No. 8, Deutscher Wetterdienst, Offenbach, Germany, 25–28.

  • Shin, H. H., , and S.-Y. Hong, 2013: Analysis of resolved and parameterized vertical transports in convective boundary layers at gray-zone resolutions. J. Atmos. Sci., 70, 3248–3261, doi:10.1175/JAS-D-12-0290.1.

    • Search Google Scholar
    • Export Citation
  • Stephens, G. L., and et al. , 2010: Dreary state of precipitation in global models. J. Geophys. Res.,115, D24211, doi:10.1029/2010JD014532.

  • Tang, Y., , H. W. Lean, , and J. Bornemann, 2013: The benefits of the Met Office variable resolution NWP model for forecasting convection. Meteor. Appl.,20, 417–426, doi:10.1002/met.1300.

  • Tiedtke, M., 1989: A comprehensive mass flux scheme for cumulus parameterization in large-scale models. Mon. Wea. Rev., 117, 17791800, doi:10.1175/1520-0493(1989)117<1779:ACMFSF>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Walsh, K., , S. Lavender, , E. Scoccimarro, , and H. Murakami, 2013: Resolution dependence of tropical cyclone formation in CMIP3 and finer resolution models. Climate Dyn., 40, 585599, doi:10.1007/s00382-012-1298-z.

    • Search Google Scholar
    • Export Citation
  • Zängl, G., 2013: ICON: The icosahedral nonhydrostatic modelling framework of DWD and MPI-M. Proc. ECMWF Seminar on Numerical Methods for Atmosphere and Ocean Modelling, Reading, United Kingdom, European Centre for Medium-Range Weather Forecasts.

  • Zängl, G., , D. Reinert, , P. Rípodas, , and M. Baldauf, 2014: The ICON (ICOsahedral Non-hydrostatic) modelling framework of DWD and MPI-M: Description of the nonhydrostatic dynamical core. Quart. J. Roy. Meteor. Soc., doi:10.1002/qj.2378, in press.

  • View in gallery

    Snapshots of 6-h precipitation accumulation (mm) for (top) the Tiedtke–Bechtold scheme, (middle) the deterministic Plant–Craig scheme, and (bottom) the stochastic Plant–Craig scheme.

  • View in gallery

    Total precipitation (shown as an average precipitation; mm h−1) for (a) Tiedtke–Bechtold, (b) deterministic version of Plant–Craig, and (c) Plant–Craig, for different model resolutions.

  • View in gallery

    PDFs of total precipitation at three different resolutions, (left) at the original resolutions and (right) upscaled grid for three different schemes: (a) Tiedtke–Bechtold, (b) deterministic Plant–Craig, and (c) stochastic Plant–Craig. Also shown are the PDFs for only the first and last 15 days (two thin lines) and the middle 15 days (medium line).

  • View in gallery

    Precipitation estimates from TMPA 3B42RT version 7 for the year 2010 for 30°N–30°S, 180°–105°W. The data are aggregated onto three different resolutions.

  • View in gallery

    (a) PDFs for the three schemes, each at 80 km and (b) PDFs at three different resolutions, upscaled to 160 km, for the stochastic Plant–Craig scheme, with no input averaging.

  • View in gallery

    (a) PDFs of mass flux output from the closure scheme, at 40-km grid spacing, for five different amounts of averaging of the input to the closure scheme. The values taken are instantaneous, recorded every 6 h. The legend refers to the number of averaging iterations (see text). (b) PDFs of 6-h rainfall accumulation, at 40-km grid spacing, for five different amounts of averaging of the input to the closure scheme.

All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 66 66 6
PDF Downloads 44 44 6

The Plant–Craig Stochastic Convection Scheme in ICON and Its Scale Adaptivity

View More View Less
  • 1 Meteorologisches Institut, Ludwig-Maximilians-Universität München, Munich, and Deutscher Wetterdienst, Offenbach, Germany
  • | 2 Meteorologisches Institut, Ludwig-Maximilians-Universität München, Munich, Germany
  • | 3 Deutscher Wetterdienst, Offenbach, Germany
© Get Permissions
Full access

Abstract

The emergence of numerical weather prediction and climate models with multiple or variable resolutions requires that their parameterizations adapt correctly, with consistent increases in variability as resolution increases. In this study, the stochastic convection scheme of Plant and Craig is tested in the Icosahedral Nonhydrostatic GCM (ICON), which is planned to be used with multiple resolutions. The model is run in an aquaplanet configuration with horizontal resolutions of 160, 80, and 40 km, and frequency histograms of 6-h accumulated precipitation amount are compared. Precipitation variability is found to increase substantially at high resolution, in contrast to results using two reference deterministic schemes in which the distribution is approximately independent of resolution. The consistent scaling of the stochastic scheme with changing resolution is demonstrated by averaging the precipitation fields from the 40- and 80-km runs to the 160-km grid, showing that the variability is then the same as that obtained from the 160-km model run. It is shown that upscale averaging of the input variables for the convective closure is important for producing consistent variability at high resolution.

Corresponding author address: R. J. Keane, Deutscher Wetterdienst, Frankfurter Strasse 135, 63067 Offenbach, Germany. E-mail: richard.keane@dwd.de

Abstract

The emergence of numerical weather prediction and climate models with multiple or variable resolutions requires that their parameterizations adapt correctly, with consistent increases in variability as resolution increases. In this study, the stochastic convection scheme of Plant and Craig is tested in the Icosahedral Nonhydrostatic GCM (ICON), which is planned to be used with multiple resolutions. The model is run in an aquaplanet configuration with horizontal resolutions of 160, 80, and 40 km, and frequency histograms of 6-h accumulated precipitation amount are compared. Precipitation variability is found to increase substantially at high resolution, in contrast to results using two reference deterministic schemes in which the distribution is approximately independent of resolution. The consistent scaling of the stochastic scheme with changing resolution is demonstrated by averaging the precipitation fields from the 40- and 80-km runs to the 160-km grid, showing that the variability is then the same as that obtained from the 160-km model run. It is shown that upscale averaging of the input variables for the convective closure is important for producing consistent variability at high resolution.

Corresponding author address: R. J. Keane, Deutscher Wetterdienst, Frankfurter Strasse 135, 63067 Offenbach, Germany. E-mail: richard.keane@dwd.de

1. Introduction

Multiple-resolution atmospheric models are a relatively recent development in climate science, enabling, for example, a large area to be modeled but with the computational resources focused on a region of particular interest (Côté et al. 1998; Fox-Rabinovitz 2000; Rauscher et al. 2013; Hagos et al. 2013), and these models are an emerging tool in numerical weather prediction (Buizza et al. 2007; Tang et al. 2013). It is vital, in such multiple-resolution models, that the physical parameterizations adapt correctly to the different resolutions being used. On the one hand, an appropriate level of extra variability should be introduced at the finer resolution, but on the other hand the fundamental physics must be the same at all resolutions.

An important physical process, which must be parameterized at grid lengths on the order of tens of kilometers, is moist convection. It is well known that convection parameterizations do not adapt correctly to changes in resolution, yielding different behaviors at different resolutions (Keane and Plant 2012; Jung and Arakawa 2004; Walsh et al. 2013) and insufficient variability at finer resolutions (Holloway et al. 2012; Stephens et al. 2010). It may be expected that a convection scheme, specifically designed to adapt correctly to the model grid length, might perform better in a model with more than one resolution. Recent work by Keane and Plant (2012) has shown that such a scheme, the Plant and Craig (2008) stochastic convection scheme, does indeed improve on conventional convection parameterizations in its capability to adapt correctly to different scales. The present study takes this a step further by introducing the Plant–Craig scheme in a global circulation model (GCM) that is being developed for operational use with multiple resolutions.

The GCM used in the present study is the Icosahedral Nonhydrostatic GCM (ICON; Zängl et al. 2014; Zängl 2013) developed jointly by the Max Planck Institute for Meteorology and Deutscher Wetterdienst. The model operates on a triangular grid with a spherical surface, and grid lengths in the low tens of kilometers are planned. The present study uses ICON in the aquaplanet setup (Neale and Hoskins 2000a,b) in order to investigate the relationship between the rainfall variability and the model grid length for the Plant–Craig scheme and two deterministic convection schemes. The resolution is constant for each run, so this study represents an important step toward tests using more than one resolution in the same run.

The purpose of this study is to investigate how well the Plant–Craig scheme adapts to changes in grid spacing. Ideally, the scheme should yield significant extra variability with finer spacings, as compared to that with coarser spacings. However, the precipitation distribution should be independent of the grid spacing, when data from the different spacings are upscaled onto the same grid. The study investigates how well the Plant–Craig scheme adheres to these two criteria. In contrast to studies such as that of Hagos et al. (2013), which investigate spectra of various different quantities, we focus exclusively on frequency distributions of precipitation. The manuscript proceeds as follows. The numerical experiments performed in the study are detailed in section 2, which also gives an overview of the Plant–Craig scheme. The results are presented in section 3, which gives an overview of the total precipitation produced by the different setups, and in section 4, which provides a more detailed statistical investigation of the precipitation variability yielded by the different setups. The results are summarized in section 5, which also suggests directions for future research.

2. Methods

a. The Plant–Craig stochastic convection parameterization

The Plant–Craig scheme is used here, based on its implementation by Groenemeijer and Craig (2012), adapted to interface with ICON [the previous study used the limited-area Consortium for Small-Scale Modeling (COSMO) model, version 4.8], and with a few scientific modifications. A brief overview of its operation, along with the essential modifications from the study by Groenemeijer and Craig (2012), is given here; a fully comprehensive description is given by Plant and Craig (2008).

The scheme operates on each model grid point and uses as input six quantities, defined on the grid point in question, from the dynamical core: the two horizontal wind speeds, temperature, and the concentration of each of the three water phases. These quantities can be spatially averaged before they are input to the scheme: an investigation into the effects and desirability of this input averaging is given by Keane and Plant (2012), and some investigation into its advantages will also be presented here. The purpose of the input averaging is that the stochastic perturbations introduced by the scheme are around a true representation of the mean atmospheric state, and not about an atmospheric state that has already been perturbed during previous time steps. This makes explicit the “weakly interacting clouds” assumption of Cohen and Craig (2006), namely that clouds (and perturbations) affect the mean atmospheric state but do not directly affect each other, other than through their effect on the mean state.

The input averaging also has technical implications; in particular, it can require communication across processors when information from distant grid boxes is required. In this study, communication across processors was minimized by requiring that averaging could only be performed using neighboring grid boxes, and their nearest neighbors. When larger averaging areas were required, then an iterative procedure was used, whereby the averages obtained from nearest- and next-nearest neighbors were themselves averaged repeatedly until the required effective averaging area was obtained.

As well as the aforementioned input variables from the dynamical core, the scheme also reads in cloud properties from the previous time step. In this way, clouds can last for longer than a single time step and so the first task for the scheme is to remove clouds that have been present for longer than the prescribed cloud lifetime (this study follows the simple methodology of previous work in setting this cloud lifetime to a constant 45 min). A trigger procedure, described by Groenemeijer and Craig (2012), is then used to determine whether or not to add new clouds to the environment. If this trigger condition is fulfilled, then the plume model of the Kain–Fritsch scheme (Kain and Fritsch 1990; Kain 2004) is used to calculate the mass flux 〈M〉 that would remove 90% of the CAPE in a predefined closure time scale τc. This mass flux is used to scale a probability distribution of clouds, from which the scheme draws randomly. The distribution is consistent with a statistical-mechanics-based theory of weakly interacting clouds that is verified by cloud-resolving model simulations (Cohen and Craig 2006; Craig and Cohen 2006), and depends also on the mean mass flux per cloud 〈m〉 and the mean of the square of the cloud radius 〈r2〉; full details are given by Keane and Plant (2012). This scaling of the distribution with total mass flux 〈M〉 is central to the scale adaptivity of the Plant–Craig scheme: because the total mass flux (being an extensive variable) is proportional to the gridbox size, the scheme adapts automatically to the resolution of the underlying model grid.

Once plumes have been randomly selected for initiation, they are launched into the environment defined by the (possibly spatially averaged) inputs from the dynamical core, again based on the model in the Kain–Fritsch scheme. Their effects on tendencies of model grid values are summed, and added to those from preexisting clouds, to produce the feedback from the scheme to the dynamical core.

Groenemeijer and Craig (2012) reported that the Plant–Craig scheme produced far too low a proportion of convective precipitation when implemented into COSMO, and had to adjust three of the parameters from those suggested by Plant and Craig (2008) in order to yield a reasonable proportion of convective precipitation. The implementation in ICON, reported here, did not require such a large adjustment in these parameters in order to produce a reasonable proportion of convective precipitation, and so it was decided to use values closer to those originally put forward by Plant and Craig (2008). The three values used in the present study were

  • a value of 800 m for the root-mean-square average cloud radius ,
  • a value of 60 min for the closure time scale τc, and
  • a value of 5 × 107 kg s−1 for the average mass flux per cloud 〈m〉.

The reason for the slightly larger mass flux per cloud compared with Plant and Craig (2008) was computational: with a higher mass flux per cloud, fewer clouds are required for a given total mass flux, and so the memory requirements for recording clouds from time step to time step are lower.

b. Numerical model setup

The ICON dynamical core (Zängl et al. 2014; Zängl 2013) contains fully compressible equations of motion, conserving both air mass and tracer mass. An icosahedral grid is used in order to avoid polar singularities and to obtain approximately uniform horizontal resolution. There is a facility for static grid refinement by one- or two-way nesting, which was not implemented in this study, although the calculations for the radiation scheme were here always carried out on a grid with half the resolution (i.e., twice as coarse) of the rest of the model. As well as convection, other processes were parameterized, namely cloud microphysics (Doms et al. 2011; Seifert 2008), radiation (Mlawer et al. 1997), turbulence (Raschendorfer 2001), and nonorographic gravity wave drag (Orr et al. 2010).

To assess the scale adaptivity of the Plant–Craig scheme, it was used in ICON, in the aquaplanet setup (Neale and Hoskins 2000a), to parameterize convective precipitation over a 6-month period, at three different (constant) resolutions. At each resolution, the 6-h precipitation amount was recorded every 6 h, to produce datasets from which histograms can be produced to display the variability of the precipitation. The three resolutions are shown in Table 1.

Table 1.

ICON grid resolutions used in this study. The approximate grid spacings are used in the rest of the study to refer to the resolution in question.

Table 1.

The same experiments were also carried out for two deterministic schemes. The first of these is the Tiedtke–Bechtold scheme (Tiedtke 1989; Gregory et al. 2000; Jakob and Siebesma 2003; Bechtold et al. 2004), which is the standard ICON convection parameterization. This provides a control against which the Plant–Craig scheme can be compared. The second deterministic scheme is a deterministic version of the Plant–Craig scheme. To implement this, instead of sampling randomly from a distribution of clouds, the entire distribution is sampled and the mass fluxes are rescaled accordingly so that they sum to the input mass flux 〈M〉 obtained from the CAPE closure. In this way, the effects of the stochasticity of the Plant–Craig scheme can be isolated. For technical reasons, this “deterministic Plant–Craig parameterization” used a reduced cloud lifetime of 25 min, to reduce the memory requirements for storing clouds from previous time steps. The input averaging was not used for the deterministic schemes, since there are no artificial perturbations and the gridbox dynamics should therefore already obtain an estimate of the mean atmospheric state. Although it would be interesting to investigate the effect of input averaging on deterministic schemes, its use in this study, to prevent artificial perturbations feeding back on each other, is not appropriate to deterministic schemes.

Each experiment was spun up, either from rest or from output from a different previously run experiment, for at least 6 months, before the data were collected. Data were then collected for 6 months, in order to obtain a statistically representative sample of the model setup. The exceptions to this were as follows. First, for all the 40-km runs, data were collected for only 3 months—in this case it was verified that there were no significant differences between the beginning, middle, and end of the runs. Additionally, for the 40-km Plant–Craig runs (stochastic and deterministic), data collection was started from a model state that had been “spun across” for 1.5 months from a different Plant–Craig run (stochastic, with no input averaging) that had, in turn, been spun up for 4.5 months from rest.

3. General precipitation climatology

Although the focus of this study is on precipitation variability, this section briefly outlines the overall precipitation amounts and how they vary for the three schemes. If a stochastic parameterization is correctly implemented, then it should add variability without adversely affecting mean values. The stochastic Plant–Craig scheme is therefore compared to its deterministic counterpart, as well as to the Tiedtke–Bechtold scheme, and reference is made to a recent study by Blackburn et al. (2013), where an intercomparison between several GCMs run under aquaplanet conditions was carried out.

Figure 1 shows 6-h precipitation accumulation snapshots, as a function of latitude and longitude, for each of the Tiedtke–Bechtold scheme, the deterministic Plant–Craig scheme, and the stochastic Plant–Craig scheme. The grid spacing used to produce all three plots was 40 km. Figure 1 illustrates how the stochastic scheme adds variability at small scales that is not present in the deterministic schemes, although the difference between the two deterministic schemes is at least as large as the difference between the two versions of the Plant–Craig scheme.

Fig. 1.
Fig. 1.

Snapshots of 6-h precipitation accumulation (mm) for (top) the Tiedtke–Bechtold scheme, (middle) the deterministic Plant–Craig scheme, and (bottom) the stochastic Plant–Craig scheme.

Citation: Journal of the Atmospheric Sciences 71, 9; 10.1175/JAS-D-13-0331.1

The longitudinal-mean total precipitation as a function of latitude is plotted in Fig. 2, again for each of the three schemes, this time for three different grid spacings. These are within the range of variability of the models in Blackburn et al. (2013); the only major difference is the strength of the double peak for the Plant–Craig scheme (although note that the minimum between the peaks is very similar to that with Tiedtke–Bechtold). It should be noted, in particular, that the climate does not change significantly between the deterministic and stochastic versions of Plant–Craig, so the strong double peak is not caused by the stochastic perturbations.

Fig. 2.
Fig. 2.

Total precipitation (shown as an average precipitation; mm h−1) for (a) Tiedtke–Bechtold, (b) deterministic version of Plant–Craig, and (c) Plant–Craig, for different model resolutions.

Citation: Journal of the Atmospheric Sciences 71, 9; 10.1175/JAS-D-13-0331.1

Also plotted in Fig. 2 is the precipitation from the model dynamics (the total precipitation minus the precipitation from the convection scheme). As well as the variation of precipitation variability with resolution, the variation of total precipitation and fraction of convective precipitation with resolution is an important issue for the scale adaptivity of the model. The stochasticity of the Plant–Craig scheme cannot be expected to improve a GCM in this respect, but it must also not degrade it. In particular, the global total precipitation should not vary with resolution, since it is constrained by the global-mean tropospheric radiative divergence, and the convective fraction should not vary with resolution (for resolutions greater than roughly 10 km) because the convective processes take place on scales smaller than 10 km and therefore do not become resolved (O’Brien et al. 2013).

In terms of the global total precipitation, all three schemes conserve this equally well; the Tiedtke–Bechtold scheme produces roughly three-quarters as much precipitation as the other two schemes (0.14 vs 0.19 mm h−1), and this is consistent across the three resolutions. The value from Tiedtke–Bechtold is within the range of values found by Blackburn et al. (2013), while those for the two Plant–Craig schemes are rather larger. The convective fraction of precipitation decreases with resolution [as seen also by O’Brien et al. (2013)], although the Tiedtke–Bechtold scheme is rather better in this respect than the other two schemes. Importantly, however, the variation between the stochastic and deterministic versions of Plant–Craig is negligible compared with the variation between the two different deterministic schemes and so any scale incognizance in this respect is due to the underlying formulation of the convection scheme, and is not degraded by the stochasticity.

The rest of this study focuses on the latitude band between 20°N and 20°S, for which the precipitation from the model dynamics is negligible compared with that from the convection schemes. Although this is not in agreement with the observed average fraction of convective precipitation in the tropics (Schumacher and Houze 2003), this discrepancy between modeled and observed convective fractions is well known (Mishra and Srinivasan 2010), and the convective fraction here is within the range of values found in Blackburn et al. (2013) and, in any case, is not affected by the stochasticity of the Plant–Craig scheme.

4. Scale adaptivity of three different convection parameterizations

a. Precipitation variability

The effect of changing the grid resolution on the three schemes is indicated in Fig. 3. Each plot shows probability density functions (PDFs) of total precipitation between 20°N and 20°S, for grid spacings of each of 40, 80, and 160 km. For each of the three schemes, the raw data are plotted together on the left, then the data are all upscaled onto the 160-km grid and these are plotted together on the right. The precipitation data were simply divided by 6 to obtain values corresponding to millimeters per hour for ease of comparison with other data. To obtain the PDFs, these values were binned (into bins with width 0.2 mm h−1) and the fraction of values falling within each bin, divided by the bin width, was the PDF for that bin value [recall that, in the limit of infinitesimal bin width dR, the probability that the precipitation R falls between values R0 and R0 + dR is given by p(R0)dR where p(R) is the PDF, which here has units of hours per millimeter].

Fig. 3.
Fig. 3.

PDFs of total precipitation at three different resolutions, (left) at the original resolutions and (right) upscaled grid for three different schemes: (a) Tiedtke–Bechtold, (b) deterministic Plant–Craig, and (c) stochastic Plant–Craig. Also shown are the PDFs for only the first and last 15 days (two thin lines) and the middle 15 days (medium line).

Citation: Journal of the Atmospheric Sciences 71, 9; 10.1175/JAS-D-13-0331.1

The input averaging for the stochastic Plant–Craig scheme was implemented only for the 40-km grid spacing. This was done for each grid point by calculating an area-weighted average of each of the input fields to the scheme, across the grid point itself and the three neighboring grid points. The input fields for the 40- and 80-km grid spacings were thus essentially on the same spatial scale.

It is clear from the left panels of Figs. 3a and 3b that reducing the grid spacing yields very little extra variability for both the deterministic schemes (there are not many more instances of very high precipitation at finer scales than at coarser scales). This can also be seen on the right panels: By upscaling the data for the finer resolutions, very little variability is removed because the fields vary so smoothly, particularly for Tiedtke–Bechtold (see also Fig. 1). This lack of variability going from 160 to 40 km is undesirable, since significant extra variability is observed across this scale range in cloud-resolving model experiments (Cohen 2001).

The stochastic Plant–Craig scheme, in contrast, does produce significant extra variability at 40 km compared with 160 km. It is very similar to its deterministic counterpart at 160 km, as would be expected, but at 40 km the stochasticity of the scheme has a significant effect. When the data are upscaled (giving a direct comparison between the three different resolutions), the curves are close together, indicating that the scheme adapts well to the change in resolution. Although the two versions of the Plant–Craig scheme used a different cloud lifetime, this should not affect the conclusions since it was applied consistently across the resolutions (the different lifetime may affect the variability of the scheme, but should not significantly affect the increase in variability when the resolution is changed, if the same lifetime is used for the different resolutions).

Because the 6-month spinup of the experiments was somewhat short, the PDFs for the data at the beginning, middle, and end of the run were also isolated and plotted on the same graph, to investigate whether or not there were any systematic differences between them. It is clear that the general conclusions are not affected by looking at different parts of the run, suggesting that the spin-up time was sufficient, although the agreement between the resolutions on upscaling for the Tiedtke–Bechtold scheme is somewhat adversely affected.

b. Comparison with observations

There is some evidence from Horinouchi (2002) that extra variability is also observed across this scale range in the real atmosphere (in Fig. 2 of that paper, the PDF of observed rainfall on a 0.5°-resolution grid lies farther to the right than that of observed rainfall on a 2.0°-resolution grid). To investigate this further, we downloaded Tropical Rainfall Measuring Mission (TRMM) Multisatellite Precipitation Analysis (TMPA) data (Huffman et al. 2007) for the year 2010 (available at ftp://trmmopen.gsfc.nasa.gov/pub/merged/3B42RT/2010/) over 30°N–30°S, 180°–105°W. This corresponds to an area over the tropical Pacific Ocean, which may be considered to correspond somewhat to the tropical aquaplanet atmosphere. The data are based on merging passive microwave observations from various low-Earth-orbit satellites, and infrared observations from various geosynchronous satellites. They are also calibrated against rain gauge data on a monthly time scale. The TMPA dataset used was 3B42RT version 7 (Huffman et al. 2010).

The 3-hourly accumulations were aggregated to obtain 6-h accumulations, and were obtained on a 0.25° grid (roughly 28-km resolution at the equator, so on the same order as the highest resolution investigated in this study). A PDF of precipitation was obtained for the entire year, and is plotted in Fig. 4. Also plotted is the corresponding PDF obtained for the data aggregated onto a 0.5° grid and onto a 1° grid. It is clear from the figure that there is extra variability when the resolution is increased from around 100 km to the low tens of kilometers.

Fig. 4.
Fig. 4.

Precipitation estimates from TMPA 3B42RT version 7 for the year 2010 for 30°N–30°S, 180°–105°W. The data are aggregated onto three different resolutions.

Citation: Journal of the Atmospheric Sciences 71, 9; 10.1175/JAS-D-13-0331.1

Although these observations do not correspond directly to the aquaplanet simulations, it is possible to estimate whether or not the correct proportion of variability is being added as the resolution doubles. To do this, we looked at the values of the precipitation for which the PDF took values of 10−4 and 10−3 h mm−1. For both the observed dataset, and for the dataset obtained using the stochastic Plant–Craig scheme, these values both increased by a factor on the order of 1.1–1.4 as the resolution was doubled, although it is true that the factor between the two finer resolutions was somewhat greater than that between the two coarser resolutions using the Plant–Craig scheme, while these factors were more comparable using the observational dataset.

It is also interesting to note that the difference between the two deterministic schemes is greater than the difference between the two versions of Plant–Craig: This is illustrated in Fig. 5a, where some of the data from Fig. 3 are displayed slightly differently, so that all three schemes are together on the same plot with only one resolution (here an 80-km grid spacing, not upscaled). This indicates that, while introducing stochasticity appears to be essential for a scheme to adapt correctly to variation in resolution, the precise variability produced by a given scheme depends as much on the assumptions used to design the scheme as on whether it is stochastic or deterministic.

Fig. 5.
Fig. 5.

(a) PDFs for the three schemes, each at 80 km and (b) PDFs at three different resolutions, upscaled to 160 km, for the stochastic Plant–Craig scheme, with no input averaging.

Citation: Journal of the Atmospheric Sciences 71, 9; 10.1175/JAS-D-13-0331.1

c. Effects of input averaging on scale adaptivity

The input averaging makes an important contribution to the scale adaptivity of the Plant–Craig scheme, and this is demonstrated here. Figure 5b shows the upscaled PDFs for the stochastic Plant–Craig scheme with no input averaging at any of the resolutions (cf. Fig. 3c, with input averaging). Although the agreement is still reasonably good, it is significantly worse than that with the input averaging implemented, demonstrating that the input averaging improves the scale adaptivity of the scheme.

It might be expected that input averaging would decrease convective variability since the input fields to the parameterization are smoother. In fact, the input averaging does reduce the variability of the mass flux output from the closure calculation, but has the opposite effect on the variability of the 6-h accumulated precipitation. This is illustrated in Fig. 6a, where PDFs of instantaneous mass flux output from the closure are plotted for the 40-km grid spacing and five different amounts of input averaging: no averaging (0), the averaging amount used in the rest of the study [1 (nn); i.e., only the three neighboring grid points and a single iteration], and three larger amounts of averaging. For the larger amounts of averaging, the input fields were averaged over the nearest neighbors and also their nearest neighbors (leading to 10 grid points in total, including the central grid point). This averaged field was then itself averaged in the same way (over the 10 nearest- and next-nearest-neighbors) and this process was iteratively repeated. The number in the legend shows how many iterations were used for each curve. The curves fall farther to the left (indicating less variability) the more input averaging is implemented, as would be intuitively expected.

Fig. 6.
Fig. 6.

(a) PDFs of mass flux output from the closure scheme, at 40-km grid spacing, for five different amounts of averaging of the input to the closure scheme. The values taken are instantaneous, recorded every 6 h. The legend refers to the number of averaging iterations (see text). (b) PDFs of 6-h rainfall accumulation, at 40-km grid spacing, for five different amounts of averaging of the input to the closure scheme.

Citation: Journal of the Atmospheric Sciences 71, 9; 10.1175/JAS-D-13-0331.1

Careful examination shows that the extreme values of 6-h rainfall accumulation are a result of large values of rainfall lasting for more than one time step. In the absence of input averaging, a large amount of convection at one time step (because of the instability of the environment, but also randomly due to the stochastic nature of the scheme) acts to stabilize the environment, reducing the CAPE and therefore reducing the amount of convective rain produced for the following time step. The input averaging strongly mitigates this effect, since the input to the CAPE closure calculation at the following time step is based not only on the stabilized environment of the grid point itself, but also on surrounding grid points that may not have been stabilized by the convection scheme (and are likely to be relatively unstable given the slowly varying nature of the forcing of the convection). This is illustrated in Fig. 6b, where PDFs for the same five averaging amounts as in Fig. 6a, with 40-km grid spacing, are shown, this time for 6-h rainfall accumulation. The curves now fall farther to the right (indicating more variability) the more input averaging is implemented. Note that the data for Fig. 6 are for much shorter runs, sometimes started directly from states that had been spun up with a different amount of input averaging, and run for a few tens of days. It was found, however, that this was always sufficient to obtain an idea of how the precipitation variability would behave over a full 3-month (or 6-month) run, and is certainly sufficient to illustrate why an increase of input averaging leads to more variability.

It is also apparent from Fig. 6b that, when too much input averaging is implemented, there is an irregularity in the tail of the distribution. This was also found at coarser resolutions, and found to be due to very large values of the grid-scale precipitation, and not convective precipitation (not shown; it should be noted that the grid-scale precipitation in all other cases did not contribute to the tail of the distribution of total precipitation). The inference is that, when the input environment is too strongly smoothed, then local environmental instabilities are not input to the convection scheme and so the scheme does not act to stabilize the environment; this must then be done by the grid-scale precipitation, leading to extreme values of grid-scale precipitation at individual grid points.

5. Conclusions

The ability of a stochastic convection parameterization, based on equilibrium statistics of convective mass flux, to adapt correctly to changes in model grid spacing on the order of tens of kilometers has been demonstrated. The stochastic nature of the scheme has been shown to be essential to add variability at finer resolutions, while the scheme converges correctly to its deterministic counterpart at coarser resolutions. The scale adaptivity has been demonstrated by averaging the precipitation data, for three different resolutions, onto the same grid and comparing its statistical distribution at the three resolutions. The three statistical distributions were found to be essentially similar, showing that the extra variability added at finer resolutions was due solely to the increased precision of the model and not due to fundamentally different physics at the different resolutions.

It is clear that the averaging of the input variables improves the scale adaptivity of the scheme. It would be expected, intuitively, that this would be the case since, in the absence of input averaging the stochastic perturbations simply feed back on each other positively, leading to erroneously high variability at finer scales. However, it is interesting that, somewhat counterintuitively, the absence of input averaging actually leads to an erroneously low variability at finer scales, since the stochastic perturbations feed back negatively as explained in the previous section of this study.

It is possible, then, to use the input averaging as a tuning parameter in order to improve the scale adaptivity of the scheme. A second factor that could be used in this way is the average mass flux per cloud 〈m〉: if clouds are, for example, set to be smaller on average then, for a given mass flux, there will be more clouds and therefore less variability. These two tuning parameters could be used to constrain the scheme, since there are two aspects to the scale adaptivity (adding the correct amount of extra variability at higher resolution, and agreement of the results on coarse graining). However, this would require a closer comparison with observations than was possible in this study.

The next step toward further investigating the applicability of the Plant–Craig scheme in ICON (and of such schemes in variable-resolution setups in general) is to extend the experiments in this study to setups with finer resolution over a limited portion of the domain, and coarser resolution over the rest of the domain (with possibly an intermediate resolution in between). The same exercise (upscaling the precipitation statistics onto the same grid spacing) could be performed to investigate the ability of the Plant–Craig scheme to yield consistent statistical behavior in the different parts of the domain, in comparison with conventional schemes.

Global models are now starting to reach resolutions where it is not clear which aspects of convection to parameterize (Hong and Dudhia 2012; Honnert et al. 2011; Shin and Hong 2013), and this is another aspect of scale adaptivity that has not been addressed in this study. In particular, there should be a smooth transition from GCM-type vertical moisture profiles to cloud-resolving-model-type profiles as the resolution is increased from 10 to 1 km (Arakawa et al. 2011). It is also desirable to gradually reduce the convective fraction until clouds are fully resolved. Mishra and Srinivasan (2010) showed that the convective fraction could be varied by adjusting the convective closure time scale, a tunable parameter in the Plant–Craig scheme, and so a pragmatic way of achieving scale adaptivity at these resolutions may be to relate this time scale to an extensive variable (one that is directly related to the gridbox area) such that it increases as the gridbox size decreases, reducing the convective fraction appropriately.

Another promising strategy for achieving scale adaptivity at kilometer-scale resolutions is to apply underlying theories on which the Plant–Craig scheme is based, to develop a scheme for shallow convection, and to allow the model dynamics to handle the deep convection. It has been shown by Sakradzija et al. (2014a,b, manuscript submitted to Nonlinear Processes Geophys.) that the theory carries over, with some adjustment, to shallow convection.

Finally, it should be noted that this study investigated only the effect of the convection scheme and not other schemes such as radiation or cloud microphysics, and it therefore addresses only one part of the problem of scale adaptivity of atmospheric models. However, it is hoped that the general principle of using a stochastic parameterization with some intrinsic relationship to the grid size may carry over to be applicable to other phenomena.

Acknowledgments

The authors thank Dmitrii Mirinov, Pieter Groenemeijer, and Bob Plant for useful discussions about the planning and execution of this project. This work was funded by Deutscher Wetterdienst in Offenbach, Germany, within the framework of an extramural research program. We would also like to thank the three anonymous reviewers for a thorough assessment of the manuscript and for providing many insightful suggestions for improving the paper.

REFERENCES

  • Arakawa, A., , J.-H. Jung, , and C.-M. Wu, 2011: Toward unification of the multiscale modeling of the atmosphere. Atmos. Chem. Phys., 11, 37313742, doi:10.5194/acp-11-3731-2011.

    • Search Google Scholar
    • Export Citation
  • Bechtold, P., , J.-P. Chaboureau, , A. Beljaars, , A. K. Betts, , M. Köhler, , M. Miller, , and J.-L. Redelsperger, 2004: The simulation of the diurnal cycle of convective precipitation over land in a global model. Quart. J. Roy. Meteor. Soc.,130, 3119–3137, doi:10.1256/qj.03.103.

  • Blackburn, M., and et al. , 2013: The Aqua-Planet Experiment (APE): Control SST simulation. J. Meteor. Soc. Japan, 91A, 17–56, doi:10.2151/jmsj.2013-A02.

    • Search Google Scholar
    • Export Citation
  • Buizza, R., , J.-R. Bidlot, , N. Wedi, , M. Fuentes, , M. Hamrud, , G. Holt, , and F. Vitart, 2007: The new ECMWF VAREPS (Variable Resolution Ensemble Prediction System). Quart. J. Roy. Meteor. Soc.,133, 681–695, doi:10.1002/qj.75.

  • Cohen, B. G., 2001: Fluctuations in an ensemble of cumulus clouds. Ph.D. thesis, University of Reading, 165 pp.

  • Cohen, B. G., , and G. C. Craig, 2006: Fluctuations in an equilibrium convective ensemble. Part II: Numerical experiments. J. Atmos. Sci., 63, 20052015, doi:10.1175/JAS3710.1.

    • Search Google Scholar
    • Export Citation
  • Côté, J., , S. Gravel, , A. Méthot, , A. Patoine, , M. Roch, , and A. Staniforth, 1998: The operational CMC–MRB global environmental multiscale (GEM) model. Part I: Design considerations and formulation. Mon. Wea. Rev., 126, 13731395, doi:10.1175/1520-0493(1998)126<1373:TOCMGE>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Craig, G. C., , and B. G. Cohen, 2006: Fluctuations in an equilibrium convective ensemble. Part I: Theoretical formulation. J. Atmos. Sci., 63, 19962004, doi:10.1175/JAS3709.1.

    • Search Google Scholar
    • Export Citation
  • Doms, G., and et al. , 2011: A description of the nonhydrostatic regional COSMO model. Part II: Physical parameterization. Consortium for Small-Scale Modelling Tech. Rep., 154 pp. [Available online at http://www.cosmo-model.org/content/model/documentation/core/cosmoPhysParamtr.pdf.]

  • Fox-Rabinovitz, M. S., 2000: Simulation of anomalous regional climate events with a variable-resolution stretched-grid GCM. J. Geophys. Res.,105, 29 635–29 645, doi:10.1029/2000JD900650.

  • Gregory, D., , J.-J. Morcrette, , C. Jakob, , A. C. M. Beljaars, , and T. Stockdale, 2000: Revision of convection, radiation and cloud schemes in the ECMWF integrated forecasting system. Quart. J. Roy. Meteor. Soc.,126, 1685–1710, doi:10.1002/qj.49712656607.

  • Groenemeijer, P., , and G. C. Craig, 2012: Ensemble forecasting with a stochastic convective parametrization based on equilibrium statistics. Atmos. Chem. Phys., 12, 45554565, doi:10.5194/acp-12-4555-2012.

    • Search Google Scholar
    • Export Citation
  • Hagos, S., , R. Leung, , S. A. Rauscher, , and T. Ringler, 2013: Error characteristics of two grid refinement approaches in aquaplanet simulations: MPAS-A and WRF. Mon. Wea. Rev., 141, 3022–3036, doi:10.1175/MWR-D-12-00338.1.

    • Search Google Scholar
    • Export Citation
  • Holloway, C. E., , S. J. Woolnough, , and G. M. S. Lister, 2012: Precipitation distributions for explicit versus parametrized convection in a large-domain high-resolution tropical case study. Quart. J. Roy. Meteor. Soc.,138, 1692–1708, doi:10.1002/qj.1903.

  • Hong, S.-Y., , and J. Dudhia, 2012: Next-generation numerical weather prediction: Bridging parameterization, explicit clouds, and large eddies. Bull. Amer. Meteor. Soc.,93, ES6–ES9, doi:10.1175/2011BAMS3224.1.

  • Honnert, R., , V. Masson, , and F. Couvreux, 2011: A diagnostic for evaluating the representation of turbulence in atmospheric models at the kilometric scale. J. Atmos. Sci., 68, 31123131, doi:10.1175/JAS-D-11-061.1.

    • Search Google Scholar
    • Export Citation
  • Horinouchi, T., 2002: Mesoscale variability of tropical precipitation: Validation of satellite estimates of wave forcing using TOGA COARE radar data. J. Atmos. Sci., 59, 24282437, doi:10.1175/1520-0469(2002)059<2428:MVOTPV>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Huffman, G. J., and et al. , 2007: The TRMM Multisatellite Precipitation Analysis (TMPA): Quasi-global, multiyear, combined-sensor precipitation estimates at fine scales. J. Hydrometeor., 8, 38–55, doi:10.1175/JHM560.1.

    • Search Google Scholar
    • Export Citation
  • Huffman, G. J., , R. F. Adler, , D. T. Bolvin, , and E. J. Nelkin, 2010: The TRMM Multi-Satellite Precipitation Analysis (TMPA). Satellite Rainfall Applications for Surface Hydrology, M. Gebremichael and F. Hossain, Eds., Springer Netherlands, 3–22, doi:10.1007/978-90-481-2915-7_1.

  • Jakob, C., , and A. P. Siebesma, 2003: A new subcloud model for mass-flux convection schemes: Influence on triggering, updraft properties, and model climate. Mon. Wea. Rev., 131, 27652778, doi:10.1175/1520-0493(2003)131<2765:ANSMFM>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Jung, J.-H., , and A. Arakawa, 2004: The resolution dependence of model physics: Illustrations from nonhydrostatic model experiments. J. Atmos. Sci., 61, 88102, doi:10.1175/1520-0469(2004)061<0088:TRDOMP>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Kain, J. S., 2004: The Kain–Fritsch convective parameterization: An update. J. Appl. Meteor., 43, 170181, doi:10.1175/1520-0450(2004)043<0170:TKCPAU>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Kain, J. S., , and J. M. Fritsch, 1990: A one-dimensional entraining/detraining plume model and its application in convective parameterization. J. Atmos. Sci., 47, 27842802, doi:10.1175/1520-0469(1990)047<2784:AODEPM>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Keane, R. J., , and R. S. Plant, 2012: Large-scale length and time-scales for use with stochastic convective parametrization. Quart. J. Roy. Meteor. Soc.,138, 1150–1164, doi:10.1002/qj.992.

  • Mishra, S. K., , and J. Srinivasan, 2010: Sensitivity of the simulated precipitation to changes in convective relaxation time scale. Ann. Geophys., 28, 18271846, doi:10.5194/angeo-28-1827-2010.

    • Search Google Scholar
    • Export Citation
  • Mlawer, E. J., , S. J. Taubman, , P. D. Brown, , M. J. Iacono, , and S. A. Clough, 1997: Radiative transfer for inhomogeneous atmospheres: RRTM, a validated correlated-k model for the longwave. J. Geophys. Res.,102, 16 663–16 682, doi:10.1029/97JD00237.

  • Neale, R. B., , and B. J. Hoskins, 2000a: A standard test for AGCMs including their physical parametrizations. I: The proposal. Atmos. Sci. Lett.,1, 101–107, doi:10.1006/asle.2000.0022.

  • Neale, R. B., , and B. J. Hoskins, 2000b: A standard test for AGCMs including their physical parametrizations. II: Results for the Met Office Model. Atmos. Sci. Lett.,1, 108–114, doi:10.1006/asle.2000.0024.

  • O’Brien, T. A., , F. Li, , W. D. Collins, , S. A. Rauscher, , T. D. Ringler, , M. Taylor, , S. M. Hagos, , and L. R. Leung, 2013: Observed scaling in clouds and precipitation and scale incognizance in regional to global atmospheric models. J. Climate, 26, 93139333, doi:10.1175/JCLI-D-13-00005.1.

    • Search Google Scholar
    • Export Citation
  • Orr, A., , P. Bechtold, , J. Scinocca, , M. Ern, , and M. Janiskova, 2010: Improved middle atmosphere climate and forecasts in the ECMWF model through a nonorographic gravity wave drag parameterization. J. Climate,23, 5905–5926, doi:10.1175/2010JCLI3490.1.

  • Plant, R. S., , and G. C. Craig, 2008: A stochastic parameterization for deep convection based on equilibrium statistics. J. Atmos. Sci., 65, 87105, doi:10.1175/2007JAS2263.1.

    • Search Google Scholar
    • Export Citation
  • Raschendorfer, M., 2001: The new turbulence parameterization of LM. COSMO Newsletter, No. 1, Deutscher Wetterdienst, Offenbach, Germany, 89–97. [Available online at http://www.cosmo-model.org/content/model/documentation/newsLetters/newsLetter01/newsLetter_01.pdf.]

  • Rauscher, S. A., , T. D. Ringler, , W. C. Skamarock, , and A. A. Mirin, 2013: Exploring a global multiresolution modeling approach using aquaplanet simulations. J. Climate, 26, 2432–2452, doi:10.1175/JCLI-D-12-00154.1.

    • Search Google Scholar
    • Export Citation
  • Sakradzija, M., , A. Seifert, , and T. Heus, 2014a: Shallow cumuli ensemble statistics for development of a stochastic parameterization. Geophysical Research Abstracts, Vol. 16, Abstract 7139-1. [Available online at http://meetingorganizer.copernicus.org/EGU2014/EGU2014-7139-1.pdf.]

  • Schumacher, C., , and R. A. Houze, 2003: Stratiform rain in the tropics as seen by the TRMM Precipitation Radar. J. Climate, 16, 17391756, doi:10.1175/1520-0442(2003)016<1739:SRITTA>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Seifert, A., 2008: A revised cloud microphysical parameterization for COSMO-LME. COSMO Newsletter, No. 8, Deutscher Wetterdienst, Offenbach, Germany, 25–28.

  • Shin, H. H., , and S.-Y. Hong, 2013: Analysis of resolved and parameterized vertical transports in convective boundary layers at gray-zone resolutions. J. Atmos. Sci., 70, 3248–3261, doi:10.1175/JAS-D-12-0290.1.

    • Search Google Scholar
    • Export Citation
  • Stephens, G. L., and et al. , 2010: Dreary state of precipitation in global models. J. Geophys. Res.,115, D24211, doi:10.1029/2010JD014532.

  • Tang, Y., , H. W. Lean, , and J. Bornemann, 2013: The benefits of the Met Office variable resolution NWP model for forecasting convection. Meteor. Appl.,20, 417–426, doi:10.1002/met.1300.

  • Tiedtke, M., 1989: A comprehensive mass flux scheme for cumulus parameterization in large-scale models. Mon. Wea. Rev., 117, 17791800, doi:10.1175/1520-0493(1989)117<1779:ACMFSF>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Walsh, K., , S. Lavender, , E. Scoccimarro, , and H. Murakami, 2013: Resolution dependence of tropical cyclone formation in CMIP3 and finer resolution models. Climate Dyn., 40, 585599, doi:10.1007/s00382-012-1298-z.

    • Search Google Scholar
    • Export Citation
  • Zängl, G., 2013: ICON: The icosahedral nonhydrostatic modelling framework of DWD and MPI-M. Proc. ECMWF Seminar on Numerical Methods for Atmosphere and Ocean Modelling, Reading, United Kingdom, European Centre for Medium-Range Weather Forecasts.

  • Zängl, G., , D. Reinert, , P. Rípodas, , and M. Baldauf, 2014: The ICON (ICOsahedral Non-hydrostatic) modelling framework of DWD and MPI-M: Description of the nonhydrostatic dynamical core. Quart. J. Roy. Meteor. Soc., doi:10.1002/qj.2378, in press.

Save