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Chapter 5

Circulation modelling

CHAPTER
COORDINATORS
Stefania Ciliberti
CHAPTER
AUTHORS
Nadia Ayoub, Jérôme Chanut, Mauro Cirano13, Anne Delamarche, Pierre De Mey-Frémaux, Marie Drevillon, Yann Drillet, Helene Hewitt, Simona Masina, Clemente Tanajura, Vassilios Vervatis, and Liying Wan

5.6 Ensemble modelling

Numerical models, applied to nonlinear dynamical systems such as the ocean, inevitably approximate the solution of the so-called Navier-Stokes shallow-water equations, because of limitations in computer power to resolve the whole spectrum of geophysical processes. In addition, numerical modelling is subject to numerous inherent uncertainties related to modelling parameters, to forcing functions, to initial and boundary conditions. This is why a single forecast is, to some extent, uncertain, and we use ensemble modelling to answer how uncertain a forecast is. Ensemble prediction systems (EPS) are well-known in atmospheric science communities for more than 25 years (Palmer, 2018) but are more recent in operational oceanography, with marked advances in the last decade (e.g., TOPAZ system, Sakov et al., 2012). EPS uses ensemble modelling and adds other components, such as probabilistic outputs and soon machine learning under varying flavours, with prediction as objective. In most cases, EPS also incorporates ensemble-based data assimilation (DA) to decrease forecast errors.

 

Figure 5.10. (a) Schematic of an ensemble simulation with equiprobable forecasts (blue trajectories); the forecast pdf gives an indication of the likelihood of occurrence of the different states; (b) Schematic of the flow-dependent ensemble spread in relation to the ensemble mean, an individual member, the unperturbed deterministic run, and the climatology - credits: https://confluence.ecmwf.int/

Due to the chaotic nature of the ocean, the probabilistic approach is an interesting alternative beyond the classic deterministic approach, and it can help users to interpret model predictions supplemented by their uncertainties. Ensemble modelling consists of possible ocean states using Monte Carlo techniques to sample the probability density function (pdf) of the model forecast. Each model simulation is called an ensemble member. This approach is illustrated in Figure 5.10a. The ensemble is initialised by a sample of different initial conditions (e.g. perturbed analyses in DA). The model operator (which can be also perturbed during integration) is then used to bring forward in time each member and produce an ensemble of model simulations. The ensemble members may diverge radically or remain broadly similar, resulting in a forecast PDF. A quantitative assessment of the ensemble is depicted in Figure 5.10b. The ensemble mean and spread (estimating model uncertainty) are calculated as first and second order statistical moments from the members, and can be compared with the unperturbed deterministic simulation and the climatology (and to observations, if available). The ensemble spread is flow-dependent and varies for different state variables. Ensemble forecasting aims at quantifying this flow-dependent uncertainty. EPS are highly demanding systems in terms of computational resources and can be run efficiently in HPC facilities. A major challenge for the next generation of OOFSs is to improve their services by integrating ensemble capabilities in their systems.

5.6.1 Basic concepts

There are three main categories of ocean model ensembles: (a) multi-model ensembles, e.g. Copernicus Marine Service multi-model products and CMIP6 coupled models for climate studies  (🔗5 ); (b) stochastic model ensembles, used in research e.g. the OCCIPUT project (Penduff et al., 2014), and less frequently in operational oceanography due to their computational cost; and (c) ocean model response to an atmospheric EPS, e.g. using the ECMWF-EPS atmospheric forcing(🔗6 ). 

The focus here is on the practical aspects for the implementation of a stochastic ocean model, mainly for short- to medium-range forecasting applications. The notion “stochastic model” for a system exhibiting chaotic behaviour can be defined by the partial differential Fokker-Planck equation, describing the temporal evolution of the state pdf, controlled by stochastic diffusion and advection processes, and local model tendencies. Stochastic modelling is used to represent model errors and as an ulterior step can be integrated in ensemble-based DA. Several methods and tools to produce stochastic model ensembles have been discussed in the literature following the SANGOMA project (🔗7 ). 

The main objectives of (ensemble) stochastic modelling are: (a) the estimation of model uncertainties providing realistic error bars and confidence intervals at useful ranges for ocean predictions; and (b) using model uncertainties in a DA framework to enrich background error covariances with flow-dependent errors and improve model prediction at the range of the outer loop of the DA scheme. The most useful statistical properties are the ensemble mean, the covariances and spread given by the diagonal of the covariance matrix, and sometimes the higher order moments (Quattrocchi et al., 2014). Stochastic ensembles are not used solely for DA but can be applied also as a machine learning base for artificial intelligence applications, guiding observational strategies based on array design (Charria et al., 2016; Lamouroux et al., 2016), and enabling probabilistic forecasting (Cheng et al., 2020), e.g. occurrence of ocean upwelling or bloom events, occurrence of sea level and storm surge exceeding a particular threshold, sea ice concentration, etc. 

The main elements to be decided and identified when generating an ocean model ensemble are: (a) the relevant quantities to perturb; (b) the stochastic parameterizations; and (c) the dynamical balances that must be preserved, if any (which in turn influence choices in (a)). These notions are often combined under the term “stochastic physics”. 

The ensemble verification is an important integral part of the ensemble modelling and EPS-developing process. An ensemble empirical consistency aims at verifying a posteriori the model pdf approximated by the ensemble of forecasts, with respect to existing observations and their pdfs. The underlying notion is the model and data joint probability on the right-hand-side of the equal sign in the Bayes theorem. Empirical consistency can be explored by specific criteria and analysis tools, e.g. from rank histograms being the simplest measuring “reliability” (Candille and Talagrand, 2005) to Desroziers et al. (2005) consistency diagnostics on innovations and ensemble pattern-selective consistency analysis (Vervatis et al., 2021a). The “reliability” measures to which degree the forecast probabilities agree with outcome frequencies and is an important attribute for the development of probabilistic scores. Such scores are for example the Continuous Rank Probability Score (CRPS) (Hersbach, 2000; Candille and Talagrand, 2005) and the Brier Score measuring, in addition to “reliability”, the attribute of “resolution”. For a reliable EPS, “resolution” is the ability to separate cases when an event occurs or not, i.e. probabilities being close to 0 or 1. The ensemble consistency evaluation framework provides important information to test the relevance of an EPS when the system is set-up (e.g. the ensemble size).

5.6.2 Ocean model uncertainties

Ocean model uncertainties emerge from sources of errors relevant to the ocean state, including physics, biogeochemistry, and sea ice, as well as errors in the initial state and boundary conditions (i.e. atmospheric forcing and lateral open boundary conditions). Model uncertainties in ocean physics have a significant impact in all other system components as, for example, in biogeochemistry and sea ice. The choice of the perturbed model quantities depends: (a) on the ocean application, e.g. global vs regional and coastal configurations, and short- to medium- or seasonal-range forecasts; (b) on the processes resolved by the model (or not, such as subgrid scale processes); (c) on choices in the DA framework, e.g. variational and Kalman filter approaches, variables and parameters included in the control vector, assimilated observations etc.; and (d) on the dynamical balances the user wants to preserve in the perturbation space, e.g. leaving velocities unperturbed tends to preserve the degree of geostrophy of the ocean state. 

Recent advances in NEMO incorporated an easy-to-use modelling framework for the production of ocean model ensembles (Brankart et al., 2015), including the following schemes applied also in NWP systems: SPPT (Buizza et al., 1999), SPUF (Brankart, 2013), SPP (Ollinaho, et al., 2017) and SKEB (Berner et al., 2009). The stochastic parameterizations in all schemes are implemented via first-order autoregressive Markov processes, i.e. a statistical model based on the assumption that the past value of uncertainty determines the present within some error. Several studies expand the NEMO ensemble framework (Bessières et al., 2017; Vervatis et al., 2021b), incorporating a stochastic ocean physics package (Storto and Andriopoulos, 2021). 

The SPPT perturbs the net parameterized model tendencies, assumed to contain upscaled ocean model errors due to subgrid parameterizations. The SPUF scheme is based on random walks sampling gradients (which represent the sub-grid unresolved scales) from the state vector and adding them to the models’ solution; the random walks consist of independent consecutive steps in all directions. The SPP introduces perturbations at each time step to parameters within the model parameterization schemes. The SKEB adds perturbations to the barotropic stream function, upscaling a fraction of the dissipated energy back to the resolved flow, which is often useful assuming that the inverse cascade of energy is underestimated in ocean models due to unresolved sub-grid processes. 

Selecting the appropriate perturbation scheme and properly tuning the stochastic parameterizations for the autoregressive processes (for each of the perturbed model quantities) are essential steps to produce meaningful stochastic ensembles. All stochastic perturbation schemes have their advantages and disadvantages (e.g. energy and mass conservation laws, production of over/under-dispersive ensembles, etc.), though the SPPT scheme appears to be the most effective (in terms of generating sufficient model spread) and easiest to use (in terms of stochastic parameterizations) for many model quantities. 

A common approach to generate stochastic ocean model ensembles is by using a pseudorandom combination of multivariate empirical orthogonal functions (EOFs) to perturb the wind forcing (Vervatis et al., 2016). The wind has a large impact on upper-ocean model uncertainties because it controls the Ekman and geostrophic components of the Sverdrup dynamics; it also largely drives the shelf-seas dynamics in addition to tides. In general, all surface atmospheric forcing variables constitute major sources of ocean model uncertainties. Momentum, heat, and freshwater fluxes are key quantities coupling the air-sea processes and are parametrized in terms of bulk coefficients. These model parameters can also be stochastically perturbed (in addition to atmospheric forcing) with spatiotemporal varying patterns (or by applying simple Gaussian noise, if there is no information available regarding their scales). 

Complementary to stochastic approaches, ocean model uncertainties can be introduced by making use of an atmospheric ensemble. Using an atmospheric EPS does not necessarily outperform the stochastic modelling approach in terms of ocean model spread. In general, it takes longer for the ensemble to spin-up and increase its spread, and the method also requires a large amount of data to process. On the other hand, the main advantage of using an atmospheric EPS is the realism of the fields in terms of conserved quantities. A common approach of a marine EPS generated by an atmospheric EPS, used successfully at operational centres, is the ocean wind-wave ensemble forecasting (Janssen, 2004). 

In the ocean interior below the seasonal thermocline, model uncertainties can be introduced effectively by perturbing the ocean boundary conditions and the water column properties. Such perturbations are usually difficult to implement because of the need to ensure physical consistency in the water column, and because errors in the prescribed boundary fields are usually unknown. A favourable solution for the open boundaries is when a coarse parent ensemble is available providing uncertainty estimates to the nested child model (Ghantous et al., 2020). 

Methods incorporating polynomial chaos expansions along with EOF-based perturbations of temperature and salinity profiles in isopycnal coordinate space, can be applied efficiently in estimating model error propagation in the open boundaries (Thacker et al., 2012). Model uncertainties affecting also the water column properties can be applied in the equation of state by perturbing the temperature and salinity state, using the SPUF method aimed at representing sub-grid unresolved scales. Other quantities that can be perturbed in the ocean interior and its boundaries are the model bathymetry influencing the barotropic and baroclinic states (Lima et al., 2019), the bottom drag coefficient affecting the bottom Ekman flow transport and tidal mixing in shelf-seas (Vervatis et al., 2021b), and the SSH together with depth integrated velocities in tidal open boundaries (Barth et al., 2009). 

Inflation methods and bred vectors for short-range ocean prediction systems can be used to initialise an ensemble of forecasts. The choice of perturbing initial conditions is also relevant to DA, for example using ensemble-based hybrid variational methods such as the 4D-EnsVar controlling (possibly among other quantities) the initial conditions. 

Ensemble-based DA methods are used to improve the predictive skill of biogeochemical and sea-ice models. In these Earth systemcomponents,model errors stemfromunresolved diversity, unresolved scales, and multiple model parameterizations. The unresolved diversity refers for example to the biodiversity restriction, including only a few species in the biogeochemical model, and to restrictions in the categorization of sea-ice in an effort to reduce complexity and state variables. These diversity restrictions often lead to missing model processes that are instead approximated by parameterizations. On the other hand, the unresolved scales depend on the model resolution (in a way similar to the unresolved scales for physics). 

In this context, the most common quantities to perturb in biogeochemical models are the sources and sinks (e.g. photosynthesis, respiration, death, and grazing), and the biogeochemical parameters controlling some of these processes (e.g. growth and mortality rates, nutrient limitations, grazing, etc.) (Santana-Falcón et al., 2020). Other biogeochemical model state uncertainties depend on the water column optical properties and the penetrative solar radiation, affecting photosynthesis and primary production (Ciavatta et al., 2014). An anamorphosis transformation in lognormal space is required for any use of the stochastic biogeochemical outputs that involve Gaussian distributions, such as variance analysis or DA (Simon and Bertino, 2009). This latter attribute of selecting a positive distribution function to introduce model uncertainties is also followed for sea-ice perturbations, e.g. using a gamma distribution for the sea-ice strength variable to improve DA and model performance for sea-ice concentration and sea-ice thickness (Juricke et al., 2013).

5.6.3 Towards ocean EPS

A summary of the practical aspects and challenges of a roadmap towards ocean probabilistic forecasting for the next generation of OOFS is as follows. Initially, ensemble forecasting should be developed and tested without the use of DA. This will allow operational centres to coordinate their activities, such as: (a) preparing OGCM platforms for the production of ensembles, e.g. several choices among regional centres tuning the stochastic parameterizations; (b) integrating ensembles in their operational chain assessing the computational cost (doubled for DA) and which variables are essential to archive; and (c) providing tools for the interpretation of uncertainty estimates as well as guiding users to extract information from ensembles, e.g. ocean indices forthe probabilistic detection of events. An open issue in this first step, without DA, is how ensembles are going to be initialised in an operational context. In a second step, within a DA framework, the initialization of the ensemble is part of the DA process.

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