Abstracts for WS1
Title: Climate sensitivity; timescales and late tipping points
Abstract: Climate response metrics are used to quantify the Earth's climate response to anthropogenic changes of atmospheric CO2. Equilibrium Climate Sensitivity (ECS) is one such metric that measures the equilibrium response to CO2 doubling. However, both in their estimation and their usage, such metrics make assumptions on the linearity of climate response, although it is known that, especially for larger forcing levels, response can be nonlinear. Such nonlinear responses may become visible immediately in response to a larger perturbation, or may only become apparent after a long transient. In this paper, we illustrate some potential problems and caveats when estimating ECS from transient simulations. We highlight ways that very slow timescales may lead to poor estimation of ECS even if there is seemingly good fit to linear response over moderate timescales. Moreover, such slow timescale might lead to late abrupt responses ("late tipping points") associated with a system's nonlinearities. We illustrate these ideas using simulations on a global energy balance model with dynamic albedo. We also discuss the implications for estimating ECS for global climate models, highlighting that it is likely to remain difficult to make definitive statements about the simulation times needed to reach an equilibrium. [joint work with Robbin Bastiaansen and Anna von der Heydt]
Reyk Börner
Title: Why does the Atlantic ocean circulation avoid tipping over the saddle point?
Abstract: The Atlantic Meridional Overturning Circulation (AMOC), a major global ocean current, could abruptly shift from its current state to a different flow regime. Predicting the risk of such a critical transition, or climate tipping point, requires a better understanding of the Earth system's global stability properties. Here we study noise-induced transitions in the bistable dynamical landscape of a simple box model of the AMOC. Specifically, we analyse simulated transition paths and their mean exit time distribution under different stochastic forcing, relating these results to the Freidlin-Wentzell theory of large deviations. Our results surprisingly show that, for weak noise, transition paths do not cross the basin boundary near the saddle point (Melancholia state), which may be imagined as the "mountain pass" of the system's dynamical landscape, or quasi-potential. We discuss potential reasons for the observed saddle-point avoidance. [Joint work with Valerio Lucarini and Larissa Serdukova]
Piermarco Cannarsa
Title: Parameter determination for Energy Balance Climate Models with memory
Abstract: This talk willaddress two EB models with memory arising in climate dynamics, which consist of a 1D degenerate parabolic equation involving a memory term and a reaction term (of Sellers/Budyko type, in the usual terminology). Beside discussing existence and regularity of solutions, we will provide uniqueness and stability estimates that can be used to reconstruct the insolation function in Sellers' model with memory. We will also briefly discuss ongoing related research concerning a two-layer EB model.
Alberto Carrassi
Title: Using machine learning in geophysical data assimilation
(some of the issues and some ideas)
Abstract: In recent years, data assimilation, and more generally the climate science modelling enterprise have been influenced by the rapid advent of artificial intelligence, in particular machine learning (ML), opening the path to various form of ML-based methodology.
In this talk we will schematically show how ML can be included in the prediction and DA workflow in three different ways. First, in a so-called “non-intrusive” ML, we will show the use of supervised learning to estimate the local Lyapunov exponents (LLEs) based exclusively on the system’s state [1]. In this approach, ML is used as a supplementary tool, added to the given physical model. Our results prove ML is successful in retrieving the correct LLEs, although the skill is itself dependent on the degree of local homogeneity of the LLEs on the system’s attractor.
In the second and third approach, ML is used to substitute fully [4] or partly [5]a physical model with a surrogate one reconstructed from data. Nevertheless, for high-dimensional chaotic dynamics such as geophysical flows this reconstruction is hampered by (i) the partial and noisy observations that can realistically be gathered, (ii) the need to learn from long time series of data, and (iii) the unstable nature of the dynamics. To achieve such inference successfully we have suggested to combine DA and ML in several ways. We will show how to unify these approaches from a Bayesian perspective, together with a description of the numerous similarities between them [2,3]. We will show that the use of DA in the combined approach is pivotal to extract much information from the sparse, noisy, data. The full surrogate model achieves prediction skill up to 4 to 5 Lyapunov time, and its power spectra density is almost identical to that of the original data, except for the high-frequency modes which are not well captured [4]. The ML-based parametrization of the unresolved scales in the third approach [5] is also extremely skilful. This has been studied using a coupled atmosphere-ocean model and again the use of coupled DA [6] in the combined DA-ML method makes possible to exploit the data information from one model compartment (e.g., the ocean) to the other (e.g., the atmosphere).
[1] Ayers D, J Amezcua, A Carrassi and V Ohija, 2022. Supervised machine learning to estimate instabilities in chaotic systems: estimation of local Lyapunov exponents, Under Review, Available at https://arxiv.org/abs/2202.04944
[2] Bocquet M, Brajard J, A Carrassi, and L Bertino, 2019. Data assimilation as a learning tool to infer ordinary differential equation representations of dynamical models, Nonlin. Proc. Geophys.,26, 175–193
[3] Bocquet M, Brajard J, A Carrassi, and L Bertino, 2020. Bayesian inference of chaotic dynamics by merging data assimilation, machine learning and expectation-maximization, Found. Data Sci.,2, 55–80
[4] Brajard J, A Carrassi, M Bocquet and L Bertino, 2020. Combining data assimilation and machine learning to emulate a dynamical model from sparse and noisy observations: a case study with the Lorenz 96 model., J. Comp. Sci.,44, 101171
[5] Brajard J, A Carrassi, M Bocquet and L Bertino. 2021. Combining data assimilation and machine learning to infer unresolved scale parametrisation., Phil. Trans A of the Roy Soc., 379 (2194). 20200086
[6] Tondeur M, A. Carrassi, S. Vannitsem and M. Bocquet: On temporal scale separation in coupled data assimilation with the ensemble Kalman filter. J. Stat. Phys., 44. 101171, 2020
Ruth Chapman
Title: Stochastic data adapted AMOC box models
Abstract: he Atlantic Meridional overturning Circulation is responsible for the comparatively temperate climate found in western Europe, and it’s previous collapse thought to have triggered glacial periods seen in the paleo data. This is a system that has multiple stable states- referred to as ‘on’ when the circulation is strong as in the current climate, and ‘off’ when it is much weaker. The AMOC has tipping points between these states. Tipping points occur when a rapid shift in dynamics happens in response to a relatively small change in a parameter. Making future projections of AMOC response to the climate is essential for avoiding any anthropogenic caused tipping, but it is computationally expensive to calculate the full hysteresis for different scenarios. This work looks at a conceptual box model of the AMOC which is easy to understand and cheap to implement. Previous work has considered bifurcation and rate-dependent tipping. This current work looks to estimate a realistic amount of noise from various GCM data sets and apply this to the model. This allows noise tipping to be studied, alongside a wide variety of different hosing functions to force the model. Ongoing works is looking to calibrate the model to further GCMs, and estimate escape times and transition rates between the stable states.
Mickaël D. Chekroun
Title: Separating the role of noise from that of memory effects in data-driven turbulent closures
Abstract: Slow-fast systems arise in many scientific applications, in particular in atmospheric and oceanic flows with fast inertiaâgravity waves and slow geostrophic motions. When the slow and fast variables are strongly coupled---symptomatic of breakdown of slaving relationships---it remains a challenge to derive reduced systems able to capture the dynamics.
In this talk, I will discuss the generic ingredients that can guide to an efficient closure in such challenging situations. The approach relies on a filtering operated through a nonlinear parameterization that separates the full dynamics into its slow motion and fast residual dynamics.
Dynamical and geometric properties of this residual that dictate its stochastic modeling will be identified and consequences to neural network parameterizations, discussed. The paradigmatic atmospheric Lorenz 80 model will serve as an illustration of the concepts. This talk is based on a joint work with James Mc Williams (UCLA) and Honghu Liu (Virgina Tech).
Bibliography
M. D. Chekroun, H. Liu, and J. C. McWilliams. Stochastic rectification of fast oscillations on slow manifold closures. Proc. Natl. Acad. Sci. USA, 118:e2113650118, 2021.
M. D. Chekroun, H. Liu, and J. C. McWilliams. Variational approach to closure of nonlinear dynamical systems: Autonomous case. J. Stat. Phys., 179:1073--1160, 2020.
M. D. Chekroun, A. Tantet, H. A. Dijkstra, and J. D. Neelin. Ruelle--Pollicott resonances of stochastic systems in reduced state space. Part I: Theory. J. Stat. Phys., 179:1366--1402, 2020.
A. Tantet, M. D. Chekroun, H. A. Dijkstra, and J. D. Neelin. Ruelle-Pollicott resonances of stochastic systems in reduced state space. Part II: Stochastic Hopf Bifurcation. J. Stat. Phys. 179:1403--1448, 2020.
D. Kondrashov, M. D. Chekroun, and P. Berloff. Multiscale Stuart-Landau emulators: Application to wind-driven ocean gyres. Fluids, 3:21, 2018.
M. D. Chekroun, H. Liu, and J. C. McWilliams. The emergence of fast oscillations in a reduced primitive equation model and its implications for closure theories. Computers & Fluids, 151:3-22, 2017.
M. D. Chekroun and D. Kondrashov. Data-adaptive harmonic spectra and multilayer Stuart-Landau models. Chaos, 27:093110, 2017.
M. D. Chekroun, J. D. Neelin, D.~Kondrashov, J. C. McWilliams, and M. Ghil. Rough parameter dependence in climate models: The role of Ruelle-Pollicott resonances. Proc. Natl. Acad. Sci. USA, 111:1684--1690, 2014.
Yumeng Chen
Title: Inferring the instability of a dynamical system from the skill of data assimilation exercises
Abstract: Data assimilation (DA) aims at optimally merging observational data and model outputs to create a coherent statistical and dynamical picture of the system under investigation. Indeed, DA aims at minimizing the effect of observational and model error and at distilling the correct ingredients of its dynamics. DA is of critical importance for the analysis of systems featuring sensitive dependence on the initial conditions, as chaos wins over any finitely accurate knowledge of the state of the system, even in absence of model error. Clearly, the skill of DA is guided by the properties of dynamical system under investigation, as merging optimally observational data and model outputs is harder when strong instabilities are present. In this paper we reverse the usual angle on the problem and show that it is indeed possible to use the skill of DA to infer some basic properties of the tangent space of the system, which may be hard to compute in very high-dimensional systems. Here, we focus our attention on the first Lyapunov exponent and the Kolmogorov–Sinai entropy and perform numerical experiments on the Vissio–Lucarini 2020 model, a recently proposed generalization of the Lorenz 1996 model that is able to describe in a simple yet meaningful way the interplay between dynamical and thermodynamical variables. Based on these numerical experiments, we illustrate a possible way to infer instability properties from the skill of data assimilation.
Bin Cheng
Title: Near Resonance Approximation of Rotating Navier-Stokes Equations
Abstract: We formalise the concept of near resonance for the rotating Navier-Stokes equations, based on which we propose a novel way to approximate the original PDE. The spatial domain is a three-dimensional flat torus of arbitrary aspect ratios. We prove that the family of proposed PDEs are globally well-posed for any rotation rate and initial datum of any size. Such approximations retain much more 3-mode interactions, thus more accurate, than the conventional exact resonance approach. Our approach is free from any limiting argument that requires physical parameters to tend to zero or infinity, and is free from any small divisor argument (so estimates depend smoothly on the torus' aspect ratios). The key estimate hinges on counting of integer solutions of Diophantine inequalities rather than Diophantine equations. The main results and ingredients of the proofs can form part of the mathematical foundation of a non-asymptotic approach to nonlinear oscillatory dynamics in real-world applications.
Daan Crommelin
Title: Machine Learning and Resampling for Stochastic Parameterization with Memory
Abstract: For parameterization of unresolved processes, data-based methods relying on machine learning (ML) techniques are rapidly gaining ground. Ususally the ML-based parameterization is deterministic and ignores uncertainty in the feedback from the small-scale (unresolved) to the large-scale processes. By considering stochastic rather than deterministic parameterization, this uncertainty can be taken into account. In this talk I will discuss recent work constructing data-based stochastic parametrizations with memory, using resampling. A straightforward approach to implement resampling is by binning. In case of long memory, resampling by binning is hampered by curse of dimension. To overcome this, a neural network for probabilistic classification can be used in combination with resampling. I will discuss both approaches and show their performance on test problems. .
Gianmarco Del Sarto
Title: Stochastic modelling with application to tipping point and extreme events - PhD project presentation
Abstract: The term tipping point is used when a climate system undergoes a large, non-linear change, sometimes catastrophic. We describe two main research questions identified about this topic. First, considering a one-dimensional EBM perturbed by noise, we would like to investigate how the covariance operator's spectrum of the linearized equation is affected by the bifurcation parameter change. Second, we consider a general SDE on a daily time scale and we want to understand under which assumptions its invariant measure, depending on a scaling parameter, converges (to the long-time dynamics).
Henk A. Dijkstra
Title: Transition Probabilities in the Wind-driven Ocean Circulation
Abstract: The Kuroshio Current in the North Pacific displays path changes on an interannual-to-decadal time scale. In an idealised barotropic quasi-geostrophic model of the double-gyre wind-driven circulation under stochastic wind-stress forcing, such current path variability can occur due to transitions between different equilibrium states. Due to the high-dimensionality of the problem, it is challenging to determine the probability of these transitions under the influence of stochastic noise. Here we present a new method to determine estimates of these transition probabilities, using a Dynamical Orthogonal (DO) field approach. The stochastic partial differential equations system is decomposed using a Karhunen-Loeve expansion resulting in coupled equations for the ensemble mean state and the so-called DO modes. The original DO method is first reformulated in a matrix approach which is much broader applicable to various (geophysical) problems. Using this matrix-DO approach, we are able to determine transition probabilities in the double-gyre problem and to identify transition paths between different equilibrium states. The analysis leads to the understanding which conditions are most favourable for transition.
Simon Driscoll
Title: Sensitivity Analysis and Machine Learning of a Sea Ice Melt Pond Parametrisation
Abstract: Sea ice plays an essential role in global ocean circulation and in regulating Earth's climate and weather. Melt ponds that form on the ice have a profound impact on the Arctic's climate, and their evolution is one of the main factors affecting sea-ice albedo and hence the polar climate system. Parametrisations of these physical processes are based on a number of assumptions and can include many uncertain parameters that have a substantial effect on the simulated evolution of the melt ponds. Using the state-of-the-art sea ice column physics model, Icepack, we conduct a global sensitivity sensitivity analysis of all its melt pond parameters. Results from the sensitivity analysis indicate that parameters controlling the amount of melt water allowed to run off to the ocean plays a substantial effect on the total sea ice volume, and its albedo. The results reveal that meltwater added to the melt ponds in early melting season has a greater role in influencing the sea ice properties.
We perform simulations of the Icepack model forced by hourly data from the Climate Forecasting System Version 2 reanalysis dataset, over a range of Arctic locations, and then assess if neural networks can learn and emulate the level-ice melt pond parametrisation of the Icepack model. Neural networks demonstrate the ability to learn and predict output given by the level-ice melt pond parametrisation, and furthermore do not suffer from drift or instability when used in the Icepack model replacing the melt pond parametrisation itself. With uncertainty around the precise values of many sea ice parameters, our work opens the possibility of, for example, applying hybrid data assimilation and machine learning techniques that have been used to incorporate direct (often spare and noisy) data to infer unresolved scale parametrisations, in sea ice models, whilst future work is aimed at using neural networks to model the entire Icepack model (i.e. a neural network representing all major column/1D thermodynamic, biogeochemical and ridging sea ice processes).
Charlie Egan
Title: Optimal transport and the Eady problem
Abstract: First introduced by Eady in 1949, the Eady slice equations model the formation and evolution of weather fronts. The strong discontinuities in the temperature and velocity profiles associated with weather fronts make these equations challenging to solve numerically. This poster describes Cullen and Purser’s ‘geometric method’ that was developed to overcome this issue, highlighting its relation to optimal transport. It includes results from our recent implementation of this numerical method, which uses state-of-the-art techniques from the area of semi-discrete optimal transport. This is joint work with David Bourne (Heriot-Watt), Colin Cotter (Imperial), Mike Cullen (Met Office), Beatrice Pelloni (Heriot-Watt), Steve Roper (University of Glasgow) and Mark Wilkinson (Nottingham Trent).
Benedetta Ferrario
Title: Stationary solutions for the equations of 2D fluids
Abstract: Some results about stationary solutions for 2D fluids are presented, both for the viscous and inviscid fluids, obtained by means of stochastic analysis.
Michael Ghil
Title: Atmospheric low-frequency variability and topological tipping points (G. D. Charó1, Y. Baouche2, D. Sciamarella1 and M. Ghil2,3)
Abstract: Increasing attention has been given recently to the effect of time-dependent forcing on the climate system on interannual and multidecadal time scales. The recent increase in the number, severity, and duration of extreme events — like heat waves, droughts, and floods — suggests that more attention to the effect of slow climate change on extended weather prediction might be in order.
An interesting development in applying nonautonomous and random dynamical systems (NDS and RDS) theory to the climate system is the study of topological tipping points (TTPs). This study arises from bringing the tools of algebraic topology to bear on RDSs like the stochastically perturbed Lorenz (1963) convection model and the Rössler model. More precisely, we discuss herein the homology groups of the random attractors — called LORA and RORA, respectively — that are obtained by adding multiplicative noise to the corresponding deterministic models.
Charó et al. (Chaos, 2021) have shown that Branched Manifold Analysis of Homologies (BraMAH) applied to LORA documents rapid changes in the homological properties of the associated complex, most strikingly so in the number of “holes.” Work in progress avers that these changes do, in fact, correspond to TTPs for both LORA and RORA, and we discuss their predictability. Lifting these types of results to the level of intermediate models of atmospheric low-frequency variability (LFV) seems possible via empirical model reduction (EMR) that provides a numerically efficient implementation of the Mori-Zwanzig approach (Vissio et al., GRL, 2020). It thus appears that a new road opens to connect extreme events in physical space to abrupt occurrences in phase space.
1 Buenos Aires, Argentina
2 Paris, France
3 Los Angeles, CA, USA
Tobias Grafke
Title: Metastable Atmospheric Jets: Large Deviations for Stochastic Averaging
Abstract: In stochastic systems with time-scale separation the average behavior of the effective slow variables can be computed rigorously by stochastic averaging and homogenization. The result is a law-of-large-number type result describing the limiting evolution equation in the limit of infinite time-scale separation. Similarly, typical fluctuations around this most likely behavior can be quantified through a central limit theorem, so that effective stochastic dynamics for the slow degrees of freedom can be derived. Here we show that these considerations do not contain extremely long-time behavior, such as metastability on the slow time scale, which necessitates macroscopic fluctuations of the slow variable away from its typical behavior. Classical results prove that the corresponding large deviation principle (LDP) is different in general from that of any stochastic differential equation (SDE) one would write for the slow variables alone.
This situation is not only of mere theoretical interests, but occurs in real-world systems such as atmospheric flow: Here, the slow evolution of atmospheric jets is driven by fast fluctuations of the turbulent flow, and can be approximated by quasi-linear theory. In certain parameter regimes, different arrangements of planetary jets are simultaneously locally stable, and rare fluctuations can drive the system between these states on very long time-scales.
Francesco Grotto
Title: Burst and Collapses of Surface Quasi-Geostrophic Vortices
Abstract: By means of the arguments developed in joint works with Umberto Pappalettera, it is possible to exhibit configurations of Surface Quasi-Geostrophic Point Vortices that lead to collapse in finite time, as well as vortex dynamics in which a single vortex (in a system of many) bursts into three new ones. These kinds of dynamics can be used to produce (wildly non-unique, measure-valued) weak solutions for fluid-dynamical PDEs. We will discuss properties of such solutions, in an attempt to shed light on whether they might be employed to model features of complex phenomena such as turbulent flows.
Peter Koltai
Title: The birth and death of coherence: Semi-material coherent sets
Abstract: The decomposition of the state space of a dynamical system into almost invariant sets is important for understanding its essential macroscopic behavior. The concept is reasonably well understood for autonomous dynamical systems, and recently a generalization appeared for non-autonomous systems: coherent sets. Methods for identifying coherent sets, and Lagrangian coherent structures more generally, rely on coherence being present throughout a specified time interval. In reality, coherent structures are ephemeral, continually appearing and disappearing. We present a new construction, based on the dynamic Laplacian, that relaxes this materiality requirement in a natural way, and provides the means to resolve the births, lifetimes, and deaths of coherent structures..
Salah Kouhen
Title: Predictability and the kinetic energy power spectrum in simple models
Abstract: Since the work of Lorenz 1969 a link between the KE spectral slope of the atmosphere and long-term predictability has been established. The relationship between short-term predictability and variations in the KE inertial range slope has not received attention. I have been analysing alpha-turbulence models (Pierrehumbert et al 1994) to see if a shallower than average spectral slope corresponds with decreased predictability. The work of Selz et al 2019 has shown that there is significant variation in the power spectrum of the atmosphere, so this research could have implications for ensemble forecasts.
Boualem Khouider
Title: Toward a Stochastic Relaxation for the Quasi-Equilibrium Theory of Cumulus Parameterization
Abstract: The representation of clouds and organized convection in the tropics remains one of the biggest sources of uncertainties in climate and long-term weather prediction models. Some of the most common cumulus parameterization schemes, namely, mass-flux schemes, rely on the quasi-equilibrium (QE) closure, which assumes that convection consumes large-scale instability and restores large-scale equilibrium instantaneously. However, the QE hypothesis has been challenged both conceptually and in practice. Subsequently, the QE assumption was relaxed, and instead, prognostic equations for the cloud work function (CWF) and the cumulus kinetic energy (CKE) were derived and used. It was shown that even if the CWF kernel serves to damp the CWF, the prognostic system exhibits damped oscillations on a timescale of a few hours, giving parameterized-cumulus-clouds enough memory to interact with each other, with the environment, and with stratiform anvils in particular. Herein, we show that when cloud-cloud interactions are reintroduced into the CWF-CKE equations, the coupled system becomes unstable. Moreover, we couple the CWF-CKE prognostic equations to dynamical equations for the cloud area fractions, based on the mean field limit of a stochastic multicloud model (SMCM) that represents the main cloud types that characterize organized tropical convections and their two-way interactions with the large scale dynamics across multiple temporal and spatial scales. Qualitative analysis and numerical simulations show that the CKE-CWF-cloud area fraction equations exhibit interesting dynamics including multiple equilibria, limit cycles, and chaotic behavior both when the large-scale forcing is held fixed and when it oscillates with various frequencies. This is representative of cumulus convection variability, and its capability to transition between various regimes of organization at multiple scales and regimes of scattered convection, in an intermittent and chaotic fashion. It is work mentioning that the SMCM has been used elsewhere as a stand alone cumulus parameterization in a comprehensive global climate model and this implementation led to dramatic improvements in terms of the simulations of the key tropical modes of variability at synoptic and intra-seasonal scales including the Madden-Julian Oscillation and monsoonal variability.
Frank Kwasniok
Title: The structure of predictability in an intermediate-complexity atmospheric model: covariant Lyapunov vectors and finite-time Lyapunov exponents
Abstract: A comprehensive investigation of the predictability properties in a three-level quasi-geostrophic atmospheric model with realistic mean state and variability is performed. The full spectrum of covariant Lyapunov vectors and associated finite-time Lyapunov exponents (FTLEs) is calculated. The statistical properties of the fluctuations of the FTLEs as well as the spatial localisation and entanglement properties of the covariant Lyapunov vectors are studied. We look at correlations between the FTLEs by means of a principal component analysis, identifying modes of collective excitation across the Lyapunov spectrum. We also investigate FTLEs conditional on underlying weather regimes. An advanced clustering algorithm is employed to decompose the state space into weather regimes associated with specific predictability properties as given by the FTLEs. Finally, the extreme value properties of the FTLEs are studied using generalised Pareto models for exceedances above a high and below a low threshold. Return levels as well as upper and lower bounds on the FTLEs are determined and extremely unstable or stable atmospheric states are identified.
Oana Lang
Title: A pathwise parameterisation for a stochastic 2D Euler vorticity equation
Abstract: In this talk I will introduce a probabilistic pathwise approach to effectively calibrate the stochastic parameters in a 2D vorticity equation. More precisely, I will show that the calibration can be performed in an optimal way to match a set of given data, and the model is robust with respect to the stochastic parameters. In the second part of the talk I will briefly present the analytical properties of the underlying 2D Euler model. This is joint work with Wei Pan and Dan Crisan and it is based on:
[1] A pathwise parameterisation for stochastic transport, to appear in the STUOD Springer Proceedings (with W. Pan).
[2] Well-posedness for a stochastic 2D Euler equation with transport noise, Stochastics and Partial Differential Equations: Analysis and Computations, 1-48 (2022) (with D. Crisan).
Francesca Leonelli
Title: Detecting oceanic Extreme Events in a changing climate: the Mediterranean Marine Heatwaves study-case
Abstract: Marine heatwaves (MHWs) are persistent and intense seawater temperature anomalies exceeding typical conditions, generally defined upon a reference climatology. The presence of a trend in temperature can potentially bias climatologies towards more intense and frequent MHWs events. We performed MHWs detection on original and detrended Mediterranean Sea Surface Temperature (SST) data covering the satellite period (1982-2021), the latter obtained by removing the slow component of SST variability (i.e., the trend) from the whole time series, allowing to better disentangle long-term temperature changes from transient ones (obtaining a new concept of MHW, the effective MHWs). Results show that the sharp increase of Mediterranean MHWs intensity, evidenced in the original SSTs, is largely reduced when detrending is applied. Overall, the shift of anomalies' distribution, revealed on the original SST, is removed by this approach. The next steps of improving MHW detection will be introduced, including the investigation of prominent climate modes' impact on detection (such as El Niño), and introduction of other oceanic variables to the analysis. This work is part of the CAREHeat project, funded by the European Space Agency (ESA).
Chiara Cecilia Maiocchi
Title: Decomposing the Dynamics of the Lorenz 1963 model using Unstable Periodic Orbits: Averages, Transitions, and Quasi-Invariant Sets
Abstract: Unstable periodic orbits (UPOs) are a valuable tool for studying chaotic dynamical systems, as they allow one to distill their dynamical structure. We consider here the Lorenz 1963 model with the classic parameters' value. We investigate how a chaotic orbit can be approximated using a complete set of UPOs up to symbolic dynamics' period 14. At each instant, we rank the UPOs according to their proximity to the position of the orbit in the phase space. We study this process from two different perspectives. First, we find that longer period UPOs overwhelmingly provide the best local approximation to the trajectory. Second, we construct a finite-state Markov chain by studying the scattering of the orbit between the neighbourhood of the various UPOs. Each UPO and its neighbourhood are taken as a possible state of the system. Through the analysis of the subdominant eigenvectors of the corresponding stochastic matrix we provide a different interpretation of the mixing processes occurring in the system by taking advantage of the concept of quasi-invariant sets.
Vincent Martinez
Title: Relaxation-based parameter recovery for hydrodynamic systems
Abstract: We will present an algorithm designed to recover unknown parameters of a system from large scale observations. The algorithm is based on a feedback-control system that attempts to reconstruct unobserved scales by enforcing asymptotic synchronization with the observed scales. In an ideal setting of error-free models and observations, the algorithm, properly tuned, can be guaranteed to converge to the true values of the parameters provided that sufficiently many, but ultimately finite number of length scales are observed.
Ivo Pasmans
Title: Tailoring data assimilation for discontinuous Galerkin models
Abstract: Satellites provide observational coverage of sea ice motion, coverage and thickness, often at resolutions finer than the grid resolution of the current generation of sea ice models. Data assimilation (DA) aims to combine these observations with model output to arrive at a better estimate of the true sea-ice state. As part of the Scale Aware Sea-Ice Project (SASIP), a new sea ice model is being developed using a discontinuous Galerkin (DG) approach to solve the model equations. Contrary to the current generation of finite-difference and finite-volume models, DG models are able to resolve the model solution on a subgrid scale. In this work, we explore the possibility to exploit this ability of DG models to improve the DA using an idealized 1D elastoviscous sea ice model and a Finite-Size Ensemble Transform Kalman Filter (ETKF-N). In particular, we look at 1) the benefits of using higher-order polynomial interpolation over linear interpolation in the observation operator, 2) the possibility to assimilate multiple observations per grid cell instead of averaging all observations in a grid cell into one superobservation as is the current practice, 3) the idea to remove spurious long-distance correlations stemming from the use of a limited-size ensemble by constructing a scale-sensitive DG localisation operator. This localisation operator splits the model solution by polynomial order and applies localisation with different length scales to the different polynomial orders. A more detailed description of these ideas and some preliminary results will be presented.
Sahani Pathiraja
Title: New approaches to Monte Carlo based nonlinear filtering
Abstract: Ensemble Kalman type methods have arguably been the method of choice for Monte Carlo based sequential data assimilation in many large scale geophysical applications. Despite their desirable computational robustness properties, they are not consistent with Bayes theorem for non-linear, non-Gaussian systems which characterise many geophysical systems. Recently, a range of controlled particle filters have been proposed which aim to emulate the structure of Ensemble Kalman type methods whilst simultaneously providing consistent samples in the asymptotic limit. More specifically, such filters involve constructing a control law to steer particles such that the corresponding probability distribution satisfies a variational Bayes formula.
We first provide an overview of this new class of filters and how they can be used for nonlinear ensemble data assimilation. A framework which allows to derive these filters will be explored, which will also highlight the main differences among them. We then discuss recent analytic and numerical work on a diffusion map based approximation of one such filter, namely the Feedback Particle Filter.
Greg Pavliotis
Title: Eigenfunction martingale estimating functions and filtered data for drift estimation of discretely observed multiscale diffusions
Abstract: We propose a novel method for drift estimation of multiscale diffusion processes when a sequence of discrete observations is given. For the Langevin dynamics in a two-scale potential, our approach relies on the eigenvalues and the eigenfunctions of the homogenized dynamics. Our first estimator is derived from a martingale estimating function of the generator of the homogenized diffusion process. However, the unbiasedness of the estimator depends on the rate with which the observations are sampled. We therefore introduce a second estimator which relies also on filtering the data, and we prove that it is asymptotically unbiased independently of the sampling rate. A series of numerical experiments illustrate the reliability and efficiency of our different estimators. This is joint work with the late Assyr Abdulle and Andrea Zanoni.
Beatrice Pelloni
Title: Semi-discrete optimal transport techniques for the solution of the semi-geostrophic system
Abstract: I will present the main ideas for approaching the existence (and possibly uniqueness) of solutions for the semi-geostrophic system using a constructive (and numerically implemented) methodology based on discretising space variable. This can be implemented for the incompressible was well as the compressible system, and yields an alternative proof of the celebrated existence theorem of Benamou and Brenier. This is work in collaboration wiht my PhD students Charlie Egan and Theo Lavier, and my colleagues David Bourne, MArk Wilkinson, Steve Roper and Colin Cotter.
Joran Rolland
Title: Fast sampling and resampling of rare multistability events in turbulent geophysical flows.
Abstract: Many geophysical fluid mechanics models can reproduce atmospheric and oceanic multistability, which are possible scenarii for sudden climate changes. However, even in these models, sampling these multistability events numerically can prove to be computationally expensive, owing to the very long necessary time series. In this talk, I will present rare events simulation methods that have been used to bypass this problem and efficiently sample rare transitions in multistable turbulent flows like zonal jets and stable wall flows. I will also present methods that can be used to construct low dimensional stochastic models of these GFD systems from short datasets. These models can capture the main features of the dynamics and can then be used to resample long realistic time series.
These works has been performed with Eric Simonnet, Freddy Bouchet, Dario Lucente, Corentin Herbert, Antoine Barlet, Indra Kanshana, Christophe Cuvier and Pierre Brangaça.
Gabriele Sbaiz
Title: Interaction of multi-scales for fluids in fast rotation
Abstract: In this talk, we are interested in the study of geophysical fluids (like the atmosphere and oceans) on large scales. In this context, we observe the duality between the fast rotation (due to the Coriolis force), that drives the motion to horizontal planes, and the gravitation, that acts against this process by driving the fluid to a vertical stratification. Moreover, different scales, where some effects are predominant with respect to the others, interact and contribute to the average dynamics of the flow. We will review these features for a multi-scale and heat-conducting model (called Navier-Stokes-Fourier system) where we take into account the competition of several external forces as the Coriolis, the gravitational and the centrifugal forces. For this multi-scale problem, we have two main goals: the first aim is to understand the limit dynamics when the Coriolis term tends to infinity; the second one is to find a robust method that allows to deal with all the possible scenarios appearing due to the different scaling. In addition, our approach allows us also to handle a critical scaling in which some low stratification effects due to the gravity appear in the limit.
This is a joint work with Daniele Del Santo, Francesco Fanelli and Aneta Wr'oblewska-Kami'nska.
Miguel Teixeira
Title: Mathematically singular behaviour of trapped orographic gravity waves
Abstract: Gravity waves need to be parametrized in numerical weather prediction models, as they decelerate the mean atmospheric circulation. Among these, waves that are trapped near the surface due to vertical wind and temperature variations are typically not represented, due to a number of technical difficulties.
These waves, known as trapped lee waves, are mathematically interesting, even in their simplest representation using linear theory, corresponding to discrete eigenmodes of the wave equation.
As for all gravity waves, the linear theory of trapped lee waves only allows analytical solutions in conjugation with Fourier analysis. In this approach, the stationary trapped lee wave modes, may be viewed as singularities in the continuous wave spectrum induced by a multi-scale orography.
Scorer (1949) first calculated the velocity field associated with trapped lee waves under inviscid and stationary conditions for a simple two-layer atmosphere. These monochromatic waves extend indefinitely downstream of the source orography, but Scorer's asymptotic solution was only formally valid relatively far away from the source orography.
More recently, Teixeira et al. (2013) calculated the total drag produced by these waves (which is one of the quantities that needs to be parametrized in weather prediction models). This drag comes from a singularity in the integral defining the total pressure force exerted on the terrain.
However, one apparent paradox, which hampers their parametrization, is that trapped waves are unable to exert a reaction force on the atmosphere in the inviscid approximation, as the existence of that force (expressed by the divergence of the wave momentum flux) requires the existence of a dissipation mechanism for the waves. This paradox has recently been resolved in a paper currently in press in QJRMS, where infinitesimally small dissipation has been adopted. This approach allows the derivation of wave momentum flux profiles that unambiguously define the reaction force exerted by the orography that generates the waves on the atmosphere. It turns out that the behaviour of the trapped lee waves, revealed by this approach, is singular, in the sense that solutions assuming from the outset that friction is zero are different from the physically consistent solutions obtained when friction is assumed to be infinitesimal small, but non-zero.
Milo Viviani
Title: On the infinite dimension limit of invariant measures and solutions of Zeitlin's 2D Euler equations
Abstract: In this talk we consider a finite dimensional approximation for the 2D Euler equations on the sphere, proposed by V. Zeitlin, and show their convergence towards a solution of the Euler equations with marginals distributed as the enstrophy measure. The method relies on nontrivial computations on the structure constants of the Poisson algebra of functions on $\mathbb{S}^2$, that appear to be new. Finally, we discuss the problem of extending our results to Gibbsian measures associated with higher Casimirs, via Zeitlin’s model.