Numerical schemes for various Hydrodynamic models
We will introduce Euler time numerical schemes for some models in Hydrodynamic with a special focus on stochastic Navier-Stokes equations. We will also discuss various rates of convergences in probability and in mean square.
Diagnostic use of the semi-geostrophic model
The talk illustrates the importance of simple models in understanding
production models. I first summarise the principles that have to be
followed. I then show how these can be used to replicate the large-scale
behaviour of a shallow water model, but then point out why 3 dimensional
simple models are required to explain the observed large-scale behaviour of
the atmosphere. I then show how the semi-geostrophic model can be used
diagnostically to understand the source of errors in the subtropical
circulation of the Met Office model.
Mean-field approach to Bayesian estimation of Markovian signals
Estimating Markovian signals X from noisy observations is an impor- tant problem in the natural and engineering sciences. Within the Bayesian approach the underlying mathematical problem essentially consists in the (stochastic) analysis of the conditional law of X with a view towards its efficient numerical approximation.
In this talk I will discuss mean-field type descriptions of the conditional law of X, when X is the solution of a stochastic differential equation, and present recent results on corresponding ensemble-based numerical approxi- mations in the case with correlated observation noise.
The talk is based on joint work with S. Ertel, S. Pathiraja and S. Reich.
1. S. Ertel, W. Stannat: Analysis of the Ensemble Kalman–Bucy Filter for correlated observation noise arXiv:2205.14253
2. S. Pathiraja, S. Reich, W. Stannat: McKean-Vlasov SDEs in nonlinear filtering, SIAM J. Control Optim. 59 (2021), no. 6, 4188–4215.
3. S. Pathiraja, W. Stannat: Analysis of the feedback particle filter with diffusion map based approximation of the gain, Foundations of Data Science 3 (2021): 615-645.
A local ergodic result to link weather and climate - Research question description
I am going to describe one of the problems my supervisor and I have identified in the last months, and that I would like to address during my PhD. Considering a general SDE with additive noise on a weather time scale depending on a scaling parameter, it consists in proving a local ergodic result for the corresponding process on the climate time scale. To formulate the problem, I will review the framework and the main ideas developed in [1], where a result about the convergence of time averages for RDS is proved.
Reference
[1] Flandoli, Franco & Pappalettera, Umberto & Tonello, Elisa. (2021). Nonautonomous attractors and Young measures. Stochastics and Dynamics. 22. 10.1142/S0219493722400032.
Forecasting localised enhancements to atmospheric turbulence
It is well known that the atmosphere exhibits turbulence with a -5/3 energy spectrum on small scales (on the order of 100 km and smaller). It is also well known (especially to airline passengers) that transient localised enhancements to this background turbulence are hazardous to aircraft. Understanding this aviation-affecting subset of turbulence is an important and challenging application of geophysical fluid dynamics. This talk will describe a relatively new theory and mechanism for the generation of aviation turbulence. It will also test the skill of the mechanism by comparing observations of turbulence with predictions made from the theory. These tests are successful, to the extent that the algorithm is now being used operationally to predict turbulence for the aviation sector every day. Finally, I will explain why climate change is strengthening aviation turbulence, potentially leading to bumpier flights in future.
An involution framework for Metropolis-Hastings algorithms on general state spaces
We consider a general framework for Metropolis-Hastings algorithms used to sample from a given target distribution on general state spaces. Our framework is based on a fundamental involution structure, and shown to encompass several popular algorithms as special cases, both in the finite- and infinite-dimensional settings. In particular, random walk, preconditioned Crank-Nicolson (pCN), schemes based on a suitable Langevin dynamics such as the Metropolis Adjusted Langevin algorithm (MALA), and also ones based on Hamiltonian dynamics including several variants of the Hamiltonian Monte Carlo (HMC) algorithm. In addition, we provide an abstract framework for algorithms that generate multiple proposals at each iteration, which yield efficient sampling schemes through the use of modern parallel computing resources. Here we derive several generalizations of the aforementioned algorithms following as special cases of this multiproposal framework. To illustrate effectiveness of these sampling procedures, we present applications in the context of some Bayesian inverse problems in fluid dynamics.
Wetropolis flood investigator: analysis of flooding under simplified weather and climate
The goal of the presentation is to discuss how Wetropolis can be used to study the impact of weather, including climate change, on extreme flooding and drought events.
In 2015, people from environmental and hydraulic consulting bodies requested a portable demonstrator to publicly visualise what a return period or Annual Event Probability (AEP) is for an extreme flooding event. They faced difficulties conveying the concept of AEP, given that for a certain return period extreme events are generally not separated by regular time intervals and given that the public tends to find associated statistical concepts difficult to comprehend.
In 2016, the Wetropolis flood demonstrator was realised based on a mathematical-modelling inspired design [1]. It has been showcased many times since, including the Mathematics of Planet Earth 2020 and 2022 exhibitions in London. The weather, with climate and drought options, is generated every 10s-Wetropolis’ day via skewed Galton boards acting as pinball machine. Each daily weather outcome triggers rainfall varying both in location (rain in a moor, a reservoir, or both moor and reservoir, or no rain) and amount (10%, 20%, 40%, or 90% of a set rain volume per 10s-day), by triggering suitable pump actions. The miniature landscape in Wetropolis consists of a porous groundwater moor, reservoirs upstream and near the city further downstream, a constant upstream influx of river water, and a winding downslope river with a flood plain. Wetropolis’ weather and flooding are by design conceptual. The viewing audience should preferably not turn to dust by having to wait, on average, 100 years for extreme flooding in the city to occur. Via scaling, Wetropolis’ return period is therefore reduced to 6:06min, i.e. ~36 Wetropolis’ days. After a year of drought in 2018, people asked for a drought demonstration within Wetropolis, as well as a climate-change visualisation. Climate change has been added with a switch leading to increased averaged daily rainfall and more severe flooding events. Droughts are modelled by using the Galton-board statistics to randomly prolong some of the “no-rain” days.
In addition to outreach, Wetropolis has inspired a novel flood-mitigation cost-effectiveness tool, which was used in the EU [2]. While the weather in Wetropolis is straightforward, the resultant probability distribution of flooding events in the city is more complex. It is a spatial-temporal modification of that weather given the groundwater and river dynamics between the distributed rainfall and the city’s river levels. Several people have suggested to use Wetropolis scientifically, as flood investigator. After introducing Wetropolis [3], the goal of the presentation is to discuss how the Wetropolis laboratory set-up and a Numerical Wetropolis Prediction model can be used to understand: extreme flooding and drought statistics, rare-event simulations (for events of “intermediate rarity”), flood control, data assimilation, and flood-mitigation measures.
References
Bokhove, Hicks, Kent, Zweers 2020: Wetropolis extreme rainfall and flood demonstrator: from mathematical design to outreach and research. Hydrology and Earth System Sciences 24, 2483-2503. Full design & more (videos): https://github.com/obokhove/wetropolis20162020
Bokhove, Kelmanson, Hicks, Kent: 2022: Flood mitigation: from outreach demonstrator to a graphical cost-effectiveness diagnostic for policy makers. UK Research Excellence Framework Impact Case Study. (Click on flood mitigation case:) https://results2021.ref.ac.uk/impact/submissions/1eedb5bd-8f92-4737-a6f0-1e61c997e4f0/impact
Bokhove 2022: Wetropolis YouTube video made for all as part of the EPSRC project “Data Assimilation for the Resilient city” (DARE) of Prof. Sarah Dance. Link imminent.
A data-driven approach to transitions in multiscale systems via covariant Lyapunov vectors
We study in detail the role of covariant Lyapunov vectors and their respective angles for detecting transitions between metastable states in dynamical systems, as recently discussed in several atmospheric science applications. The underlying models are built from data by the dynamical clustering method, called FEM-BV-VAR, and the Lyapunov vectors are approximated based on these models. We test this data-based numerical approach at the hand of three well-understood example systems with increasing dynamical complexity, identifying crucial properties that allow for a successful application of the method.
Multi-scale stochastic systems, fractional averaging, and dynamics
Multi scales in time are prevalent, so is autocorrelated noise, yet we focus on stochastic equations models based Brownian motion, whose independent increments property renders the Markov property of the solutions, when dealing with interacting evolutions of random variables in multi-scales. Recent developments in mathematical techniques now allow to address this problem.
Semi-discrete optimal transport methods for the semi-geostrophic equations
The semi-geostrophic (SG) equations model large-scale atmospheric flows. They are used by meteorologists at the UK Met Office to diagnose errors in General Circulation Models, and have been established as one of the foremost models for the formation and evolution of atmospheric fronts. Atmospheric fronts include the boundaries between warm and cool air masses. They are small scale features (100 kilometres or less) which exist in and influence large-scale flows (on the order of 1000 kilometres). This interaction between multiple scales poses significant modelling and computational challenges. Efforts to overcome these difficulties have prompted a rich exchange of knowledge between meteorological and mathematical communities, with the theory of optimal transport at its heart. In this talk, I aim to illuminate both directions of this knowledge exchange through the use of semi-discreteoptimal transport. I will describe the geometric method - an algorithm for solving the SG equations, designed by meteorologists Mike Cullen and Jim Purser (1984). I will highlight its relation to optimal transport, and I will present results from our recent implementation of this numerical method. This is joint work with David Bourne (Heriot-Watt), Colin Cotter (Imperial), Mike Cullen (Met Office - retired), Beatrice Pelloni (Heriot-Watt), Steve Roper (University of Glasgow) and Mark Wilkinson (Nottingham Trent).
Dynamics and Predictability of the coupled ocean-atmosphere system: A reduced-order model perspective
During the last decade, a physically-based reduced-order coupled ocean-atmosphere model has been developed allowing for exploring the qualitative properties of the coupled dynamics of such a system. Different versions have been designed that are now incorporated on a flexible Python platform named qgs (Demaeyer et al, 2020), which allows in addition to couple a land to an atmosphere. Moreover, the number of modes can be selected at will. The coupled ocean-atmosphere model has been used (i) to explore the emergence of low-frequency variability in the atmosphere and the ocean due to their coupling, (ii) to investigate the way to develop optimal ensemble prediction systems and data assimilation schemes in the context of multiscale coupled models; (iii) to clarify the influence of the tropical regions on the extratropics; and (iv) to evaluate the limits of predictability of its components. All these analyses performed using tools from dynamical systems theory, are reviewed, the lessons learned summarized, and the potential for new lines of research discussed.
A particular attention will be devoted to a recent work done on the impact of the El-Nino-Southern-Oscillation (ENSO) on the extratropical regions (Vannitsem et al, 2021). In this context, unidirectional ENSO forcing is used to mimic the atmospheric bridge between the tropics and the extratropics. The variability of the coupled ocean-atmosphere extratropical module is then investigated through the analysis of its pullback attractors (PBAs). This analysis focuses on two types of ENSO forcing generated by the tropical module, one periodic and the other aperiodic. For a substantial range of the ENSO forcing, two chaotic PBAs are found to coexist for the same set of parameter values. Different types of extratropical low-frequency variability (LFV) are associated with either PBA over the parameter ranges explored. For periodic ENSO forcing, the coexisting PBAs exhibit only weak nonlinear instability. For chaotic forcing, though, they are quite unstable and certain extratropical perturbations induce transitions between the two PBAs. These distinct stability properties may have profound consequences for extratropical climate predictions: in particular, ensemble averaging may no longer help isolate the LFV signal.
References
Demaeyer, J., L. De Cruz and S. Vannitsem (2020). qgs: A flexible Python framework of reduced-order multiscale climate models. Journal of Open Source Software, 5(56), 2597, https://doi.org/10.21105/joss.02597
Vannitsem, S., J. Demaeyer, and M. Ghil (2021). Extratropical low-frequency variability with ENSO forcing: A reduced-order coupled model study. Journal of Advances in Modeling Earth Systems, 13, e2021MS002530. https://doi.org/10.1029/2021MS002530
The Inviscid Limit for 2D Incompressible Fluid with Unbounded Vorticity
In this talk we review some recent results concerning the inviscid limit for the 2D Euler equations with unbounded vorticity. In particular, by using techniques from the theory of transport equation with no smooth vector fields, we show that the solutions obtained in the vanishing viscosity limit satisfiy a representation formula in terms of the flow of the velocity and that the strong convergence of the vorticity has a log rate in term on the viscosity. The talk is based on results obtained in collaboration with Gianluca Crippa (Univ. Basel) and Gennaro Ciampa (Università Milano Statale).
Ergodic properties for a stochastic two-layer model of geophysical fluid dynamics
A two-layer quasi-geostrophic model for geophysical flows is studied, with the upper layer being perturbed by additive noise. This model is popular in the geosciences, for instance to study the effects of a stochastic wind forcing on the ocean. A rigorous mathematical analysis however meets with the challenge that in the model under study, the noise configuration is spatially degenerate as the stochastic forcing acts only on the top layer. Exponential convergence of solutions laws to the invariant measure is established, implying a spectral gap of the associated Markov semigroup on a space of Hölder continuous functions.
Limited computer resources lead to a simplified representation of unresolved small-scale processes in weather and climate models, through parameterisation schemes. Among the parameterised processes, turbulent fluxes exert a critical impact on the exchange of heat, water and carbon between the land and the atmosphere. Turbulence theory was, however, developed for homogeneous and flat terrain, with stationary conditions. The theory fails in unsteady flow contexts or with heterogeneous landscapes, but no alternative, viable theory is available. This is not only a source of error in forecasts or climate scenarios, but also a source of model uncertainty which should be characterised and considered when using weather and climate models.
I will present a data-driven approach to develop a stochastic parameterisation of turbulent fluxes, thereby representing the model uncertainty arising from the incomplete representation of our unsteady atmosphere. This is particularly relevant in cold environments or at nighttime, where the atmospheric boundary layer is stably stratified. Turbulence then coexists with non-turbulent motions from the grey zone between the largest turbulent eddies and smallest mesoscale motions, traditionally specified to be 2km horizontal scale. These non-turbulent motions can include density currents, wave-like motions or two-dimensional modes and represent a non-stationary forcing of turbulence. The stochastic parameterisation extends conventional models and enables the representation of the resulting unsteady, intermittent fluxes. It could help overcome some of the limitations of weather and climate models to represent mixing in the stable boundary layer.
A generic approach to stochastic climate modelling is developed for the example of an idealized Atmosphere-Ocean model that rests upon Hasselmann’s paradigm for stochastic climate models. Namely, stochasticity is incorporated into the fast moving atmospheric component of an idealised coupled model by means of stochastic Lie transport, while the slow moving ocean model remains deterministic. A remarkable property of the model is that the dynamics of its higher moments are governed by deterministic equations obtained by replacing the drift velocity of the stochastic Lie transport vector field by its expected value.
A preprint of this work is available at:
D. Crisan, D. D. Holm, P. Korn, Hasselmann's Paradigm for Stochastic Climate Modelling based on Stochastic Lie Transport. arXiv:2205.04560
Based on a novel treatment of near resonances, we introduce a new approximation for the rotating stratified Boussinesq system on three-dimensional tori with arbitrary aspect ratios. The rotation and stratification parameters are arbitrary and not equal. We obtain global existence for the proposed nonlinear system for arbitrarily large initial data. This system is sufficiently accurate, with an important feature of coupling effects between slow and fast modes. The key to global existence is a sharp counting of the relevant number of nonlinear interactions. An additional regularity advantage arises from a careful examination of some mixed type interaction coefficients. In a wider context, the significance of our near resonant approach is a delicate balance between the inclusion of more interacting modes and the improvement of regularity properties, compared to the well-studied singular limit approach based on exact resonance.
John Methven, Paul Berrisford, Thomas Frame, Dominic Jones, Lina Boljka, Carlo Cafaro
In the two decades, western Europe has seen a number of extreme seasons. For example, anomalously high precipitation totals for the summers of 2007 and 2012, and winter 2013/14, as well as the exceptionally high temperature of summer 2003 and the coldness of winter 2009/10. One common feature in all these examples is the existence of persistent, near stationary Rossby wave patterns on the tropopause. Here a rigorous framework is used to extract these "slow modes of variability" from data and examine their interaction with the background state jet.
The theory of wave-mean flow interaction requires a partition of the atmospheric flow into a notional background state and perturbations to it. The evolution of both components and their diagnosed ``interaction'' depends upon the partition. Here, the background state is defined in terms of two fundamental integral properties of the full flow: mass and circulation enclosed by potential vorticity (PV) contours within isentropic layers. The background state is defined as a zonally symmetric re-arrangement of the full state, such that PV contours enclose the same mass and circulation as the full state. For adiabatic and frictionless flow, the integrals are all invariant and therefore the MLM state is a steady solution of the primitive equations. Modes of variability are extracted using the empirical normal mode (ENM) technique. This combines amounts to EOF analysis using pseudomomentum as the norm. Although the system is nonlinear with large-amplitude distubances interacting across scales, only a few ENMs dominate the variability and the ratio of pseudoenergy to pseudomomentum yields a unique theoretical phase speed for each. The dominant baroclinic wave structures propagate at the predicted rate, illustrating the relevance of modes despite the nonlinear interactions. It also clarifies how the dominant modes depend on the background state jet.