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.
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
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).
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
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.
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.
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