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  • View in gallery
    Fig 1.

    System characterized by double-potential (a),(c) or single-potential well (e),(g) and their associated PDFs (b),(d),(f), and (h). (a) If the noise is sufficiently small and under appropriate initial conditions, the system will stay in the deeper potential well and the associated probability density function will have a single maximum (b). (c) As the amplitude of the noise increases, the system can undergo a noise-induced transition and reach the secondary minimum in the potential, leading to a shifted mean and increased variance in the associated probability density function (d). A linear system characterized by a single potential well and forced by additive white noise (e) will have a unimodal PDF (f). However, when forced by multiplicative (state-dependent) noise (g), the noise changes the effective potential of the unforced system, so that the associated PDF becomes bimodal (h).

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    Fig. 2.

    Top-of-the-atmosphere net longwave radiation (outgoing longwave radiation; W m−2) in Dec–Feb for the period 1981–2010. (left) StochphysOFF minus ECMWF interim reanalysis (ERA-Interim), (center) System 4 (S4) minus reanalysis, and (right) S4 minus stochphysOFF. Significant differences at the 95% confidence level based on a two-sided Student’s t test are hatched. Adapted from Weisheimer et al. (2014).

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    Fig. 3.

    Relative frequencies of MJO events in each of the eight MJO phases. From Weisheimer et al. (2014).

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    Fig. 4.

    Power spectra of average sea surface temperature in the Niño-3.4 region in 135-yr-long simulations with the Community Atmosphere Model (CAM) coupled to an ocean model. (left) Compared to Hadley Centre Sea Ice and Sea Surface Temperature dataset (HadISST) observations (blue), the simulation has 3 times more power for oscillations with periods between 2 and 4 years. (right) When the simulation is repeated with the stochastic parameterization SPPT, the temperature variability in this range is reduced, leading to a better agreement between the simulated and observed spectra. Adapted from Christensen et al. (2017).

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    Fig. 5.

    Niño-3.4 SST root-mean-square error (lines) and ensemble spread (dots) according to forecast time in EC-EARTH 3 seasonal reforecast experiments initialized in May 1993–2009 with SR or HR atmosphere and ocean components, with and without activating a three-scale SPPT perturbation method in the atmosphere.

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    Fig. 6.

    Forecast diagnostics as a function of time for the operational (black), fixed perturbed parameter (blue), and stochastically varying perturbed parameter (red) ensemble forecasts. (top) Forecast bias for (a) temperature at 850 hPa (T850) and (b) zonal wind at 850 hPa (U850) shown as a fraction of the bias for the operational system (BIAS/BIASoper). (bottom) Root-mean-square ensemble spread (dashed lines) and root-mean-square error (solid lines) for (c) T850 and (d) U850. Diagnostics are averaged over the region 10°S–20°N, 60°E–180°. Adapted from Christensen et al. (2015b).

  • View in gallery
    Fig. 7.

    The right tail of the PDF of summer-season hourly precipitation from a 50-member ensemble of 1-yr single-column model simulations with stochastic (blue) and conventional parameterizations (black) and 15 years of observations (green) over a model grid box encompassing the U.S. Department of Energy’s Atmospheric Radiation Measurement program’s site in Lamont, OK. The large-scale forcing for the single-column model simulations are generated from a present-day CESM simulation at a spatial resolution of about 2.8° × 2.8°. Adapted from Langan et al. (2014).

  • View in gallery
    Fig. 8.

    (a) Map of the century-mean net upward water flux (mm day–1) at the sea surface in a control integration of a coupled climate model. (b) Difference from the control for an experiment in which the net freshwater flux across the air–sea interface is stochastically perturbed before being passed to the ocean. (c) Difference from the control for an experiment in which the net heat flux across the air–sea interface is stochastically perturbed before being passed to the ocean. From Williams (2012).

  • View in gallery
    Fig. 9.

    (a) Amplitude of fluctuations of the eddy forcing as measured by the standard deviation of divergence of eddy flux in a 1/10° ocean GCM. (b) Mean eddy forcing measured by the magnitude of the mean divergence of eddy heat flux. The amplitude of the fluctuations is about one order of magnitude larger than the mean eddy forcing. Adapted from Li and von Storch (2013).

  • View in gallery
    Fig. 10.

    Difference in mean standard deviation of sea ice thickness forecasts (m) between ensembles generated by stochastic ice strength as well as atmospheric initial perturbations and ensembles generated solely by atmospheric initial perturbations, averaged for days (left) 1–10, (center) 11–30, and (right) 31–90 after initialization at 0000 UTC 1 Jan. Stippled areas indicate differences statistically significant at the 5% level, using a two-tailed F test. Note the different contour intervals. Adapted from Juricke et al. (2014).

  • View in gallery
    Fig. 11.

    Histograms of the subgrid cloud-base mass flux, resulting from the stochastic shallow cumulus cloud scheme (STOCH) and coarse-grained LES, are compared for three horizontal grid resolutions of (left) 1.6, (center) 3.2, and (right) 12.8 km. Adapted from Sakradzija et al. (2015).

  • View in gallery
    Fig. 12.

    Snapshot of the spatial field of convective states obtained from LES data. The distinction between the various convective states was based on cloud-top height and rainwater content. Adapted from Dorrestijn et al. (2013a).

  • View in gallery
    Fig. 13.

    Climate responses of global-mean temperature to a CO2 doubling (2 × CO2 minus 1 × CO2) obtained from the ECHAM5/Max Planck Institute Ocean Model experiments with different representations of small-scale fluctuations: “diffus” refers to experiments in which the strength of horizontal diffusion is varied, and “noise” refers to experiments in which white noise is added to small scales of the atmospheric model ECHAM5. From Seiffert and von Storch (2008).

  • View in gallery
    Fig. 14.

    (left) The response in mean streamfunction variance of a barotropic vorticity equation to an anomalous vorticity forcing at 45°N, 150°W projected onto 90 empirical orthogonal functions (EOFs) and the simulation of this response by (center) a 90-EOF climate model with unmodified subgrid-scale parameterization (relative error = 0.527) and (right) a climate model with subgrid-scale parameterization corrected by FDT (relative error = 0.342). Adapted from Achatz et al. (2013).

  • View in gallery
    Fig. 15.

    Comparison of the upper-level kinetic energy spectra of a two-level benchmark simulation (dashed line) with associated LES (solid line) at various resolutions for atmospheric isotropic stochastic (isoS) LES, atmospheric isotropic deterministic (isoD) LES, atmospheric deterministic scaling law (lawD) LES, oceanic stochastic scaling law (lawS) LES, and oceanic deterministic scaling law LES. The top spectrum has the correct kinetic energy, with the others shifted down for clarity.

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Stochastic Parameterization: Toward a New View of Weather and Climate Models

Judith BernerNational Center for Atmospheric Research,* Boulder, Colorado

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Ulrich AchatzInstitut für Atmosphäre und Umwelt, Goethe-Universität, Frankfurt am Main, Germany

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Lauriane BattéCNRM-GAME, Météo-France/CNRS, Toulouse, France

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Lisa BengtssonSwedish Meteorological and Hydrological Institute, Norrköping, Sweden

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Alvaro de la CámaraNational Center for Atmospheric Research,* Boulder, Colorado

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Hannah M. ChristensenAtmospheric, Oceanic and Planetary Physics, University of Oxford, Oxford, United Kingdom

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Matteo ColangeliGran Sasso Science Institute, L’Aquila, Italy

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Danielle R. B. ColemanNational Center for Atmospheric Research,* Boulder, Colorado

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Daan CrommelinCentrum Wiskunde en Informatica, and Korteweg-de Vries Institute for Mathematics, University of Amsterdam, Amsterdam, Netherlands

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Stamen I. DolaptchievInstitut für Atmosphäre und Umwelt, Goethe-Universität, Frankfurt am Main, Germany

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Christian L. E. FranzkeMeteorological Institute, and Centre for Earth System Research and Sustainability, University of Hamburg, Hamburg, Germany

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Petra FriederichsMeteorological Institute, University of Bonn, Bonn, Germany

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Peter ImkellerInstitut für Mathematik, Humboldt-Universität zu Berlin, Berlin, Germany

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Heikki JärvinenDepartment of Physics, University of Helsinki, Helsinki, Finland

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Stephan JurickeAtmospheric, Oceanic and Planetary Physics, University of Oxford, Oxford, United Kingdom

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Vassili KitsiosOceans and Atmosphere Flagship, CSIRO, Aspendale, Victoria, Australia

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François LottLaboratoire de Météorologie Dynamique (CNRS/IPSL), Ecole Normale Supérieure, Paris, France

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Valerio LucariniMeteorological Institute, and Centre for Earth System Research and Sustainability, University of Hamburg, Hamburg, Germany, and Department of Mathematics and Statistics, University of Reading, Reading, United Kingdom

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Salil MahajanOak Ridge National Laboratory, Oak Ridge, Tennessee

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Timothy N. PalmerAtmospheric, Oceanic and Planetary Physics, University of Oxford, Oxford, United Kingdom

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Cécile PenlandPhysical Sciences Division, NOAA/Earth System Research Laboratory, Boulder, Colorado

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Mirjana SakradzijaMax Planck Institute for Meteorology, and Hans-Ertel-Centre for Weather Research, Deutscher Wetterdienst, Hamburg, Germany

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Jin-Song von StorchMax Planck Institute for Meteorology, Hamburg, Germany

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Antje WeisheimerNational Centre for Atmospheric Science, and Atmospheric, Oceanic and Planetary Physics, University of Oxford, Oxford, and ECMWF, Reading, United Kingdom

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Michael WenigerMeteorological Institute, University of Bonn, Bonn, Germany

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Paul D. WilliamsDepartment of Meteorology, University of Reading, Reading, United Kingdom

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Jun-Ichi YanoGAME-CNRM, CNRS, Météo-France, Toulouse, France

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Full access

Abstract

The last decade has seen the success of stochastic parameterizations in short-term, medium-range, and seasonal forecasts: operational weather centers now routinely use stochastic parameterization schemes to represent model inadequacy better and to improve the quantification of forecast uncertainty. Developed initially for numerical weather prediction, the inclusion of stochastic parameterizations not only provides better estimates of uncertainty, but it is also extremely promising for reducing long-standing climate biases and is relevant for determining the climate response to external forcing. This article highlights recent developments from different research groups that show that the stochastic representation of unresolved processes in the atmosphere, oceans, land surface, and cryosphere of comprehensive weather and climate models 1) gives rise to more reliable probabilistic forecasts of weather and climate and 2) reduces systematic model bias. We make a case that the use of mathematically stringent methods for the derivation of stochastic dynamic equations will lead to substantial improvements in our ability to accurately simulate weather and climate at all scales. Recent work in mathematics, statistical mechanics, and turbulence is reviewed; its relevance for the climate problem is demonstrated; and future research directions are outlined.

The National Center for Atmospheric Research is sponsored by the National Science Foundation.

© 2017 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

CORRESPONDING AUTHOR E-MAIL: Judith Berner,berner@ucar.edu

Abstract

The last decade has seen the success of stochastic parameterizations in short-term, medium-range, and seasonal forecasts: operational weather centers now routinely use stochastic parameterization schemes to represent model inadequacy better and to improve the quantification of forecast uncertainty. Developed initially for numerical weather prediction, the inclusion of stochastic parameterizations not only provides better estimates of uncertainty, but it is also extremely promising for reducing long-standing climate biases and is relevant for determining the climate response to external forcing. This article highlights recent developments from different research groups that show that the stochastic representation of unresolved processes in the atmosphere, oceans, land surface, and cryosphere of comprehensive weather and climate models 1) gives rise to more reliable probabilistic forecasts of weather and climate and 2) reduces systematic model bias. We make a case that the use of mathematically stringent methods for the derivation of stochastic dynamic equations will lead to substantial improvements in our ability to accurately simulate weather and climate at all scales. Recent work in mathematics, statistical mechanics, and turbulence is reviewed; its relevance for the climate problem is demonstrated; and future research directions are outlined.

The National Center for Atmospheric Research is sponsored by the National Science Foundation.

© 2017 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

CORRESPONDING AUTHOR E-MAIL: Judith Berner,berner@ucar.edu

Stochastic parameterizations—empirically derived or based on rigorous mathematical and statistical concepts—have great potential to increase the predictive capability of next-generation weather and climate models.

THE NEED FOR STOCHASTIC PARAMETERIZATIONS.

Numerical weather and climate modeling is based on the discretization of the continuous equations of motion. Such models can be characterized in terms of their dynamical core, which describes the resolved scales of motion, and the physical parameterizations, which provide estimates of the grid-scale effect of processes, that cannot be resolved. This general approach has been hugely successful in that skillful predictions of weather and climate are now routinely made (e.g., Bauer et al. 2015). However, it has become apparent through the verification of these predictions that current state-of-the-art models still exhibit persistent and systematic shortcomings due to an inadequate representation of unresolved processes.

Despite the continuing increase of computing power, which allows numerical weather and climate prediction models to be run with ever-higher resolution, the multiscale nature of geophysical fluid dynamics implies that many important physical processes (e.g., tropical convection, gravity wave drag, and microphysical processes) are still not resolved. Parameterizations of subgrid-scale processes contain closure assumptions and related parameters with inherent uncertainties. Although increasing model resolution gradually pushes these assumptions further down the spectrum of motions, it is realistic to assume that some form of closure or physical parameterization will be present in simulation models into the foreseeable future.

Moreover, for climate simulations, a decision must be made as to whether computational resources should be used to increase the representation of subgrid physical processes or to build a comprehensive Earth system model by including additional climate components such as the cryosphere, chemistry, and biosphere. In addition, the decision must be made about whether computational resources should go toward increased horizontal, vertical, and temporal resolution or additional ensemble members.

Additional challenges are posed by intrinsically coupled phenomena like the Madden–Julian oscillation (MJO) and tropical cyclones. Correctly simulating these tropical multiscale features requires resolving or accurately representing small-scale processes such as convection in addition to capturing the large-scale response and feedback. Many of the Coupled Model Intercomparison Project, phase 5 (CMIP5), climate models still do not properly simulate the MJO and convectively coupled waves (Hung et al. 2013).

A great challenge is posed by the representation of partially resolved processes (either in the time or space domain). The range of scales on which a physical process is only partially resolved is often referred to as the “gray zone” (e.g., Gerard 2007). In this gray zone, the number of eddies in each grid box is no longer large enough to fulfill the “law of large numbers” underlying deterministic bulk parameterizations, and a stochastic approach becomes essential. An example for a partially resolved process is convection, which is often split into a resolved (large scale) and parameterized component (e.g., Arakawa 2004). The equilibrium assumption no longer holds when the model resolution is increased such that a clear scale separation between convection and larger scales no longer is valid (e.g., Yano and Plant 2012a,b). In this case, the subgrid-scale parameterization takes a prognostic form rather than being diagnostic, as explicitly shown for the mass-flux formulation by Yano (2014).

As the next generation of numerical models attempts to seamlessly predict weather as well as climate, there is an increasing need to develop parameterizations that adapt automatically to different spatial scales (scale-aware parameterizations). A big advantage of the mathematically rigorous approach is that the subgrid model is valid for increasing spatial resolutions within a range of scales that is obtained as part of the derivation.

Mathematical approaches to stochastic modeling rely on the assumption that a physical system can be expressed in terms of variables of interest and variables that one does not want to explicitly resolve. In the mathematical literature this is usually referred to as the operation of coarse graining and is performed through the method of homogenization (Papanicolaou and Kohler 1974; Gardiner 1985; Pavliotis and Stuart 2008). The goal is then to derive an effective equation for the slow predictable processes and to represent the effect of the now unresolved variables as random noise terms.

Many stochastic parameterizations are based on the assumption of a scale separation between the temporal decorrelation rates between the rapidly fluctuating processes represented by a white noise and the slow processes of interest (e.g., Gardiner 1985; Penland 2003a). In geophysical applications, there is often—but not always—a relationship between spatial and temporal scales of variability, with fast processes associated with small scales and slow processes associated with large scales. If this is the case, separating physical processes by time scales can result in decomposing small-scale features from large-scale phenomena, and spatial and temporal scale separation become equivalent.

Such thinking underlies the pioneering study of Hasselmann (1976), who split the coupled ocean–atmosphere system into a slow ocean and fast weather fluctuation components and subsequently derived an effective equation for the ocean circulation only. One finds that the impact of the fast variables on the dynamics of the slow variables boils down to a deterministic correction—a mean field effect sometimes referred to as noise-induced drift or rectification—plus a stochastic component, which is a white random noise in the limit of infinite time-scale separation.

A simple example demonstrating noise-induced transitions and drifts is presented in Fig. 1. Assume that the unforced nonlinear climate system can be described by a double-well potential (Fig. 1a). If the noise is sufficiently small (short red arrows) and under appropriate initial conditions, the system will stay for a finite time in the deeper potential well and the associated probability density function of states will have a single maximum (Fig. 1b). As the amplitude of the noise increases (long arrows in Fig. 1c), the system can undergo a noise-induced transition and reach the secondary potential well. The resulting probability density function (PDF) will exhibit two local maxima (Fig. 1d), signifying two different climate regimes, rather than a single maximum, as in the small-noise scenario. Note that the stochastic forcing not only changes the variance, but also the mean.

Fig 1.
Fig 1.

System characterized by double-potential (a),(c) or single-potential well (e),(g) and their associated PDFs (b),(d),(f), and (h). (a) If the noise is sufficiently small and under appropriate initial conditions, the system will stay in the deeper potential well and the associated probability density function will have a single maximum (b). (c) As the amplitude of the noise increases, the system can undergo a noise-induced transition and reach the secondary minimum in the potential, leading to a shifted mean and increased variance in the associated probability density function (d). A linear system characterized by a single potential well and forced by additive white noise (e) will have a unimodal PDF (f). However, when forced by multiplicative (state-dependent) noise (g), the noise changes the effective potential of the unforced system, so that the associated PDF becomes bimodal (h).

Citation: Bulletin of the American Meteorological Society 98, 3; 10.1175/BAMS-D-15-00268.1

But even a linear system characterized by a single potential when unforced can change the mean, if forced by multiplicative or state-dependent white noise (Figs. 1e–h). The noise is called “multiplicative” if its amplitude is a function of the state, which is denoted by the red arrows of different length in Fig. 1g. The noise-induced drift changes the single-well potential of the unforced system (Fig. 1e), so that the effective potential including the effects of the multiplicative noise has multiple wells (not shown) and the associated PDF becomes bimodal (Fig. 1h). Note that in this example the shift in the mean compared to the unforced PDF (Fig. 1f) is caused by the noise, which is referred to as “noise-induced drift” (e.g., Sardeshmukh et al. 2001; Berner 2005; Sura et al. 2005).

Operational weather and climate centers now use stochastic parameterization schemes routinely to make ensemble predictions from short-range to seasonal time scales (e.g., Berner et al. 2009; Weisheimer et al. 2014). Most ensembles suffer from underdispersion, which means that, on average, the observed state is more often outside the cone of forecasts than can be statistically justified. Stochastic perturbations introduce more diversity among the forecasts, which helps to ameliorate this problem and result in more skillful ensemble forecasts.

A fundamental argument that has been often overlooked is that the merit of stochastic parameterization goes far beyond providing uncertainty estimations for weather and climate predictions but may be also needed for better representing the mean state (e.g., Sardeshmukh et al. 2001; Palmer 2001; Berner et al. 2008) and regime transitions (e.g., Williams et al. 2003, 2004; Birner and Williams 2008; Christensen et al. 2015a) via inherent nonlinear processes. This is especially relevant for climate projections, which have long-standing mean-state errors, such as a double intertropical convergence zone (e.g., Lin 2007) and erroneous stratocumulus cloud cover, that play a crucial role in the climate response to external forcing.

Mechanisms for how Gaussian zero-mean fluctuations can change the mean state (see Fig. 1) have been discussed in Tompkins and Berner (2008) and Beena and von Storch (2009). Tompkins and Berner (2008) introduce perturbations to the humidity field and find that positive perturbations are more likely to trigger a convective event than negative perturbations can suppress convection. Beena and von Storch (2009) study the ocean response air–sea flux perturbations and similarly find that negative buoyancy anomalies result in an altered stratification, while positive anomalies tend to sustain the existing stratification. Insofar as stochastic parameterizations can change the mean state, they have the potential to affect the response to changes in the external forcing (e.g., Seiffert and von Storch 2008).

In mathematical terms, this is the question how a stochastic forcing affects the invariant measure of a deterministic dynamical system (Lucarini 2012) and how the climate response to such a forcing can be framed as a problem of nonequilibrium statistical mechanics (Colangeli et al. 2012, 2014; Lucarini and Sarno 2011; Lucarini et al. 2014a,b).

Here we argue that stochastic parameterizations are essential for

  • estimating uncertainty in weather and climate predictions,

  • reducing systematic model errors arising from unrepresented subgrid-scale fluctuations,

  • triggering noise-induced regime transitions, and

  • capturing the response to changes in the external forcing

and should be applied in a systematic and consistent fashion, not only to weather, but also to climate simulations.

Several studies have identified the assessment of the benefits of stochastic closure schemes as a key outstanding challenge in the area of mathematics applied to the climate system (Palmer 2001, 2012; Palmer and Williams 2008; Williams et al. 2013). For accessible reviews of rigorous mathematical approaches applied to weather and climate, we refer to Penland (2003a,b), Majda et al. (2008), and Franzke et al. (2015). The current study focuses on recent developments and successful applications of empirical and rigorous approaches to the subgrid-parameterization problem in weather and climate models.

REPRESENTING UNCERTAINTY IN COMPREHENSIVE CLIMATE AND WEATHER MODELS.

Adding uncertainty a posteriori: The stochastically perturbed parameterization tendency scheme and the stochastic kinetic energy backscatter scheme.

Stochastic parameterizations are based on the notion that, as spatial resolution increases, the method of averaging (Arnold 2001; Monahan and Culina 2011) is no longer valid and the subgrid-scale variability should be sampled rather than represented by the equilibrium mean. In addition, unrepresented interactions between unresolved subgrid-scale processes with the large-scale flow might affect the resolved dynamics.

The former is addressed by the stochastically perturbed parameterization tendency (SPPT) scheme, which perturbs the net tendencies of the physical process parameterizations (convection, radiation, cloud physics, turbulence, and gravity wave drag). One essential feature for its success is that the noise is correlated in space and time. SPPT has a beneficial impact on medium-range, seasonal, and climate forecasts (Buizza et al. 1999; Teixeira and Reynolds 2008; Palmer et al. 2009; Weisheimer et al. 2014; Christensen et al. 2015b; Dawson and Palmer 2015; Batté and Doblas-Reyes 2015). SPPT tends to be most active in the tropics and near the surface, where the parameterized tendencies are large.

The stochastic kinetic energy backscatter scheme (SKEBS) aims to represent model uncertainty arising from unresolved subgrid-scale processes and their interactions with larger scales by introducing random perturbations to the streamfunction and potential temperature tendencies. For this purpose, the scheme reinjects a small fraction of the dissipated energy into the resolved flow. Originally developed in the context of large-eddy simulations (LESs; Mason and Thomson 1992), it was adapted by Shutts (2005) for numerical weather prediction (NWP).

Depending on the details of the implementations, SKEBS tends to have the most impact in the storm-track regions and in the free atmosphere above the boundary layer and permits the physical parameterization schemes to adjust to a slightly perturbed large-scale background flow. Its beneficial impact on weather and climate forecasts are reported, for example, in Berner et al. (2011, 2015), Tennant et al. (2011), Weisheimer et al. (2014), Sanchez et al. (2016), while Shutts (2013) criticizes the arbitrary nature of some of the design features based on coarse graining, high-resolution simulations to compute the backscatter term. His stochastic convective backscatter scheme (Shutts 2015) includes a phase relationship between flow and perturbations and adds additional perturbations to the divergent flow to remedy some of the identified shortcomings.

While these schemes are motivated by physical reasoning and scheme parameters are informed in some manner, for example, by coarse graining, high-resolution output (Shutts and Palmer 2007; Shutts and Callado Pallarès 2014) or comparison with observations (Watson et al. 2015), the perturbations are essentially empirical constructs. For example, the amplitude of the perturbations is typically determined as the value that satisfactorily reduces the ensemble underdispersion. Obviously, such an approach is only possible for forecast ranges where verification is possible, such as for short-term, medium-range, and seasonal forecasts. A common criticism of this approach is that the improved skill is solely the result of the increase in spread. However, Berner et al. (2015) found that the merits of stochastic parameterization go beyond increasing spread and can account for structural model uncertainty.

In the following examples, we show recent results that demonstrate the potential of stochastic parameterizations to improve the mean state representation and variability as well as the skill of seasonal forecasts.

First, we present recent results from the seasonal forecasting system, System 4 (S4), at the European Centre for Medium-Range Weather Forecasts (ECMWF). In the simulations with both SPPT and SKEBS, excessively strong convective activity over the Maritime Continent and the tropical western Pacific is reduced, leading to smaller biases in outgoing longwave radiation (Fig. 2, adapted from Weisheimer et al. 2014), cloud cover, precipitation, and near-surface winds when compared to a simulation without stochastic parameterization, stochphysOFF. The stochastic schemes also lead to an increase in the frequency (Fig. 3, from Weisheimer et al. 2014) and amplitude of MJO events, which is an improvement. A reduction of excessive amplitudes in westward-propagating convectively coupled waves in an earlier model version is reported in Berner et al. (2012).

Fig. 2.
Fig. 2.

Top-of-the-atmosphere net longwave radiation (outgoing longwave radiation; W m−2) in Dec–Feb for the period 1981–2010. (left) StochphysOFF minus ECMWF interim reanalysis (ERA-Interim), (center) System 4 (S4) minus reanalysis, and (right) S4 minus stochphysOFF. Significant differences at the 95% confidence level based on a two-sided Student’s t test are hatched. Adapted from Weisheimer et al. (2014).

Citation: Bulletin of the American Meteorological Society 98, 3; 10.1175/BAMS-D-15-00268.1

Fig. 3.
Fig. 3.

Relative frequencies of MJO events in each of the eight MJO phases. From Weisheimer et al. (2014).

Citation: Bulletin of the American Meteorological Society 98, 3; 10.1175/BAMS-D-15-00268.1

Another example of the positive impact of stochastic schemes is evident in climate simulations with the Community Earth System Model (CESM). Compared to observations, the modeled spectrum of average sea surface temperature in the Niño-3.4 region has 3 times more power for periods between 2 and 4 years (Fig. 4, adapted from Christensen et al. 2017). SPPT markedly reduces the temperature variability in this frequency range, leading to a much better agreement with nature (Christensen et al. 2017). Interestingly, in these examples, adding stochasticity results in reduced variability, which is a nontrivial response.

Fig. 4.
Fig. 4.

Power spectra of average sea surface temperature in the Niño-3.4 region in 135-yr-long simulations with the Community Atmosphere Model (CAM) coupled to an ocean model. (left) Compared to Hadley Centre Sea Ice and Sea Surface Temperature dataset (HadISST) observations (blue), the simulation has 3 times more power for oscillations with periods between 2 and 4 years. (right) When the simulation is repeated with the stochastic parameterization SPPT, the temperature variability in this range is reduced, leading to a better agreement between the simulated and observed spectra. Adapted from Christensen et al. (2017).

Citation: Bulletin of the American Meteorological Society 98, 3; 10.1175/BAMS-D-15-00268.1

Along with the improved model climate, stochastic perturbations benefit probabilistic forecast performance on seasonal time scales. This has been reported in a number of studies using earlier versions of ECMWF’s seasonal system (Berner at al. 2008; Doblas-Reyes et al. 2009; Palmer at al. 2009) and has recently been confirmed in the newest version (Weisheimer et al. 2014) and in the EC-Earth Consortium (EC-EARTH) system model (Batté and Doblas-Reyes 2015). Figure 5 shows ensemble mean and spread in forecasts for Niño-3.4-area sea surface temperatures with the EC-EARTH model, run at a standard horizontal resolution (SR; ∼60 km for the atmospheric and ∼100 km for the ocean component) and at high resolution (HR; ∼40 km for the atmospheric component and 25 km for the ocean.) For both resolutions, the introduction of SPPT perturbations increases the ensemble spread. Furthermore, SPPT reduces the mean error in the standard resolution, but not as much as increasing horizontal resolution.

Fig. 5.
Fig. 5.

Niño-3.4 SST root-mean-square error (lines) and ensemble spread (dots) according to forecast time in EC-EARTH 3 seasonal reforecast experiments initialized in May 1993–2009 with SR or HR atmosphere and ocean components, with and without activating a three-scale SPPT perturbation method in the atmosphere.

Citation: Bulletin of the American Meteorological Society 98, 3; 10.1175/BAMS-D-15-00268.1

A number of studies have found evidence for stochasticity leading to noise-induced transitions in midlatitude circulation regimes, especially over the Pacific–North America region (Jung et al. 2005, Berner et al. 2012, Dawson and Palmer 2015, Weisheimer et al. 2014). These results suggest that stochastic parameterizations are also relevant for the prediction of the dominant modes of atmospheric variability, such as the North Atlantic Oscillation and the Pacific–North America pattern (J. Berner et al. 2016, unpublished manuscript).

Adding uncertainty a priori: Perturbed parameter approaches for the atmospheric component.

While the performance of the stochastic schemes discussed in the last section is undisputed, the schemes have been criticized in that they are added a posteriori to models that have been independently developed and tuned. Ideally, stochastic perturbations should represent model uncertainty where it occurs. One obvious way to represent uncertainty at its source rather than a posteriori is the perturbed parameter approach, which perturbs the closure parameters in the physical process parameterizations. There are two variants: the parameter can be fixed throughout the integration but vary for each ensemble member (e.g., Murphy et al. 2004; Hacker et al. 2011) or vary randomly with time (e.g., Bowler et al. 2008, 2009; Ollinaho et al. 2013; Jankov et al. 2016, manuscript submitted to Mon. Wea. Rev.). Strictly, the first variant is not a stochastic parameterization, but an example for a multimodel, since each ensemble member has a different climatology. However, since stochastic parameter perturbations are routinely compared to fixed-parameter schemes, this section discusses both.

While perturbed parameter ensembles typically outperform unperturbed ensembles on weather time scales, they typically cannot sufficiently account for all deficiencies in the spread (Hacker et al. 2011; Reynolds et al. 2011; Christensen et al. 2015b) and do not lead to the same reliability as the a posteriori schemes discussed above (Berner et al. 2015). Presumably, this is because a posteriori schemes are designed to encapsulate all model uncertainty, of which parameter uncertainty is only one contributor.

An ensemble system is considered statistically reliable when a predicted probability for a particular event (e.g., temperature exceeding 17°C) compares well with the observed frequencies. Another limitation of this approach is that the parameter uncertainty estimates are subjective, and information about parameter interdependencies is not included.

The following studies are examples for applications of the perturbed parameter approach to physical process parameterizations and perturbing the interface between different model components. We start with results pertaining to perturbations in the atmospheric component and move to those of other model components, such as land and ocean models, which are more relevant for climate applications.

A number of studies report on improved skill due to parameter perturbations to boundary layer and convection schemes (Hacker et al. 2011; Reynolds et al. 2011; Jankov et al. 2016, manuscript submitted to Mon. Wea. Rev.). Recently, a stochastic “eddy diffusivity–mass flux” parameterization has been developed (Suselj et al. 2013, 2014) that combines an eddy diffusivity component with a stochastic mass-flux scheme. The resulting scheme unifies boundary layer and shallow convection and was operationally implemented in the operational Navy Global Environmental Model.

Christensen et al. (2015b) used an objective covariance estimate of parameter uncertainty (Järvinen et al. 2012; Ollinaho et al. 2013) for four convection closure parameters and developed both a fixed-parameter and a stochastically varying perturbation scheme. Both schemes improved the forecast skill of the ECMWF ensemble prediction system, with a larger impact observed for the fixed perturbed parameter scheme (Fig. 6, adapted from Christensen et al. 2015b). In addition, for some variables such as wind at 850 hPa, the scheme leads to a reduction in bias (Fig. 6, adapted from Christensen et al. 2015b).

Fig. 6.
Fig. 6.

Forecast diagnostics as a function of time for the operational (black), fixed perturbed parameter (blue), and stochastically varying perturbed parameter (red) ensemble forecasts. (top) Forecast bias for (a) temperature at 850 hPa (T850) and (b) zonal wind at 850 hPa (U850) shown as a fraction of the bias for the operational system (BIAS/BIASoper). (bottom) Root-mean-square ensemble spread (dashed lines) and root-mean-square error (solid lines) for (c) T850 and (d) U850. Diagnostics are averaged over the region 10°S–20°N, 60°E–180°. Adapted from Christensen et al. (2015b).

Citation: Bulletin of the American Meteorological Society 98, 3; 10.1175/BAMS-D-15-00268.1

Recently, a body of work proposes stochastic approaches for another atmospheric parameterization, namely, nonorographic gravity waves (Lott et al. 2012; Lott and Guez 2013; de la Cámara and Lott 2015). Observational studies indicate that the gravity wave field is very intermittent and only predictable in a statistical sense. Recently, de la Cámara et al. (2014) informed the free parameters of the stochastic gravity wave scheme using momentum flux measurements.

Uncertainty in land surface, ocean, and coupled component models.

Physical parameters of land surface models are often not well constrained by observations. A recent study by MacLeod et al. (2015) introduced parameter perturbations to key soil parameters and compared their impact with stochastic perturbations of the soil moisture tendencies in seasonal forecasts with the ECMWF coupled model. Both the perturbed parameter approach and the stochastic tendency perturbations improved the forecasts of extreme air temperature for the European heat wave of 2003.

A shortcoming in land models stems from the omission of subgrid land heterogeneity, which impacts the surface heat flux. Langan et al. (2014) retained the subgrid variability by drawing the area for each plant functional type at each time step from a Dirichlet distribution, rather than using constant area weights. First, results with a single-column model version of CESM reveal an increase in the variability as well as larger extreme values in convective precipitation (Fig. 7, adapted from Langan et al. 2014).

Fig. 7.
Fig. 7.

The right tail of the PDF of summer-season hourly precipitation from a 50-member ensemble of 1-yr single-column model simulations with stochastic (blue) and conventional parameterizations (black) and 15 years of observations (green) over a model grid box encompassing the U.S. Department of Energy’s Atmospheric Radiation Measurement program’s site in Lamont, OK. The large-scale forcing for the single-column model simulations are generated from a present-day CESM simulation at a spatial resolution of about 2.8° × 2.8°. Adapted from Langan et al. (2014).

Citation: Bulletin of the American Meteorological Society 98, 3; 10.1175/BAMS-D-15-00268.1