MPE CDT Student Cohort 2017

Principal Supervisor: Prof. Pavel Berloff (Department of Mathematics, Imperial College London)
Co-advisor: Peter Dueben, (ECMWF)

Summary: The oceanic turbulent circulation exhibits multiscale motions on very different space and time scales interacting with each other; e.g., jets, vortices, waves, and large-scale variability. In particular, mesoscale oceanic eddies populate nearly all parts of the ocean and need to be resolved in order to represent their effects on the general ocean and atmosphere circulations. However, capturing effects of these small-scale flows is highly challenging and requires non-trivial approaches and skills, especially when it comes to representing their effects in non-eddy resolving ocean circulation models. Therefore, the main goal of my project is to develop data-driven eddy parameterizations for use in both eddy-permitting and non-eddy-resolving ocean models. Dynamical models of reduced complexity will be developed to emulate the spatio-temporal variability of mesoscale eddies as well as their feedbacks across a large range of scales. These can serve as a low-cost oceanic component for climate models; and therefore the final aim of this project is to use the existing observational data to feed eddy parameterizations in comprehensive ocean circulation and climate models such as the ones used in global weather forecasts or in Climate Model Intercomparison Project(CMIP) models like CMIP7.

We will employ a variety of both common and novel techniques and methods of statistical data analysis and numerical linear algebra to extract the key properties and characteristics of the space-time correlated eddy field. The key steps involved in this framework are, a) first, find the relevant data-adaptive basis functions, i.e. the decomposition of time evolving datasets into their leading spatio-temporal modes using, for example, variance-based methods such as Principal Component Analysis (PCA) and, b) once the subspace spanned by above basis functions are obtained, we derive the evolution equations that emulate the spatio-temporal correlations of the system using methods such as nonlinear autoregression, artificial neural network, Linear Inverse Modelling (LIM), etc.

The proposed new science will help develop a state-of-the-art data-adaptive modelling framework for evaluation and application of Machine Learning and rigorous mathematical theory for dynamical and empirical reduction within the hierarchy of existing oceanic models.

Supervisors: Matthew Piggott (Lead supervisor, Department of Earth Science & Engineering, Imperial College London) and Colin Cotter (Department of Mathematics, Imperial College London). Industry supervisor: Dr Catherine Villaret (East Point Geo Consulting).

Summary: An estimated 250 million people live in regions that are less than 5 metres above sea level. Hence with sea level rise and an increase in both the frequency and severity of storms as a result of climate change, the coastal zone is becoming an ever more critical location for the application of advanced mathematical techniques. Models are currently used to assist in the design of coastal zone engineering projects including flood defences and marine renewable energy arrays. There are many challenges surrounding the development and application of appropriate coupled numerical models because they include both hydrodynamic and sedimentary processes and need to resolve spatial scales ranging from sub-metre to 100s of kilometres.

My project aims to develop and use advanced numerical modelling and statistical tools to improve the understanding of hazards and the quantification and minimisation of erosion and flood risk. Throughout this project, I will consider the hazards in the context of idealised as well as real world scenarios.
The main model I will use in my project is XBeach, which uses simple numerical techniques to compute dune erosion, scour around buildings and overwash. XBeach is also currently used, to a limited degree, with Monte Carlo techniques to generate a large number of storm events with different wave climate parameters. Uncertain atmospheric forcing is very important in erosion/scour processes and flood risk, which are intimately linked in many situations and cannot be considered in isolation. In my project I will explore how the new technique of Multi-level Monte Carlo simulations can be combined with XBeach to quantify erosion/flood risk. I am not only interested in the effects of extreme events, but also the cumulative effect of minor storm events for which Monte Carlo techniques are particularly appropriate. I will also explore how an adaptive mesh approach can be coupled with the statistical approach to assess the risk to coastal areas.

My project aims to develop and use advanced numerical modelling and statistical tools to improve the understanding of hazards and the quantification and minimisation of erosion and flood risk. Throughout this project, I will consider the hazards in the context of idealised as well as real world scenarios. The main model I will use in my project is XBeach, which uses simple numerical techniques to compute dune erosion, scour around buildings and overwash. XBeach is also currently used, to a limited degree, with Monte Carlo techniques to generate a large number of storm events with different wave climate parameters. Uncertain atmospheric forcing is very important in erosion/scour processes and flood risk, which are intimately linked in many situations and cannot be considered in isolation. In my project I will explore how the new technique of Multi-level Monte Carlo simulations can be combined with XBeach to quantify erosion/flood risk. I am not only interested in the effects of extreme events, but also the cumulative effect of minor storm events for which Monte Carlo techniques are particularly appropriate. I will also explore how an adaptive mesh approach can be coupled with the statistical approach to assess the risk to coastal areas.

Supervisors: Martin Rasmussen (Imperial College London, Department of Mathematics), Jochen Broeker (University of Reading, Department of Mathematics and Statistics), Pavel Berloff (Imperial College London, Department of Mathematics)

Summary: The concept of a tipping point (or critical transition) describes a phenomena where the behaviour of a physical system changes drastically, and often irreversibly, compared to a small change in its external environment. Relevant examples in climate science are the possible collapse of the Atlantic Meridional Overturning Circulation (AMOC) due to increasing freshwater input, or the sudden release of carbon in peatlands due to an external temperature increase. The aim of this project is to develop the mathematical framework for tipping points and therefore contribute to a deeper understanding of them.

A number of generic mechanisms have been identified which can cause a system to tip. One such mechanism is rate-induced tipping, where the transition is caused by a parameter changing too quickly - rather than it moving past some critical value. The traditional mathematical bifurcation theory fails to address this phenomena. The goal of this project is to use and develop the theory of non-autonomous and random dynamical systems to understand rate-induced tipping in the presence of noise. A question of particular practical importance is whether it is possible to develop meaningful early-warning indicators for rate-induced tipping using observation data. We will investigate this question from a theoretical viewpoint and apply it to more realistic models.

Supervisors: Xue-Mei Li (Department of Mathematics, Imperial College London, Lead supervisor), Darryl Holm ( Department of Mathematics, Imperial College London), Dan Crisan (Department of Mathematics, Imperial College London)

Summary: The Gulf Stream can be thought of as a giant meandering ribbon-like river in the ocean which originates in the Caribbean basin and carries warm water across the Atlantic to the west coast of Europe, keeping the European climate relatively mild. In spite of its significance to weather and climate, the Gulf Stream has remained poorly understood by oceanographers and fluid dynamicists for the past seventy years. This is largely due to the fact that the large-scale flow is significantly affected by multi-scale fluctuations known as mesoscale eddies. It is hypothesised that the mesoscale eddies produce a backscatter effect which is largely responsible for maintaining the eastward jet extensions of the Gulf Stream and other western boundary currents.

The difficulty in modelling such currents lies in the high computational cost associated with running oceanic simulations with sufficient resolution to include the eddy effects. Therefore approaches to this problem have been proposed which involve introducing some form of parameterisation into the numerical model, such that the small scale eddy effects are taken into account in coarse grid simulations.

There are three main approaches we may consider in including this parameterisation: the first is stochastic advection, the second is deterministic roughening and the third is data-driven emulation.

These approaches have all be explored for relatively simple quasi-geostrophic ocean models, but we shall attempt to apply them to more comprehensive primitive equation models which have greater practical applications in oceanography. In particular we shall be using the MITgcm and FESOM2 models, to which we shall apply our parameterisations and run on a low-resolution grid and compare the results with high-resolution simulations.

Supervisors: Nikolas Kantas (Department of Mathematics, Imperial College London, Lead supervisor), Professor Alistair Forbes (NPL)

Summary: This project aims to develop new methodology for performing statistical inference in environmental modelling applications. These applications require the use of a large number of sensors that collect data frequently and are distributed over a large region in space. This motivates the use of a space time varying stochastic dynamical model, defined in continuous time via a (linear or non-linear) stochastic partial differential equation, to model quantities such as air quality, pollution level, and temperature. We are naturally interested in fitting this model to real data and, in addition, on improving on the statistical inference using a carefully chosen frequency for collecting observations, an optimal sensor placement, and an automatic calibration of sensor biases. From a statistical perspective, these problems can be formulated using a Bayesian framework that combines posterior inference with optimal design.

Performing Bayesian inference or optimal design for the chosen statistical model may be intractable, in which case the use of simulation based numerical methods will be necessary. We aim to consider computational methods that are principled but intensive, and given the additional challenges relating to the high dimensionality of the data and the model, must pay close attention to the statistical model at hand when designing algorithms to be used in practice. In particular, popular methods such as (Recursive) Maximum Likelihood, Markov Chain Monte Carlo, and Sequential Monte Carlo, will need to be carefully adapted and extended for this purpose.

Supervisors: Prof Axel Gandy (Statistics Section, Department of Mathematics, Imperial College London), Dr David Brayshaw (Department of Meteorology, University of Reading)

In the face of climate change, considerable efforts are being undertaken to reduce carbon emissions. One of the most promising pathways to sustainability is decarbonising electricity generation and electrifying other sources of emissions such as transport and heating. This requires a near-total decarbonisation of power systems in the next few decades.

Making strategical decisions regarding future power system design (e.g. what power plant to build) is challenging for a number of reasons. The first is their complexity: electricity grids can be immensely complicated, making the effect of e.g. an additional power plant difficult to estimate. The second is the considerable uncertainty about future technologies, fuel prices and grid improvements. Finally, especially as more weather-dependent renewables are added, there is climate-based uncertainty: we simply don’t know what the future weather will be, or how well times of high demand will line up with times of high renewable output.

This project aims to both understand the effect of climate-based uncertainty on power system planning problems and develop methodologies for robust decision-making under these unknowns. This will be done in the language of statistics, using techniques such as uncertainty quantification, data reduction and decision-making under uncertainty. Furthermore, this investigation will employ power system models, computer programs simulating the operation of an electricity grid.

When modelling complicated physical systems such as the ocean/atmosphere system with relatively simple mathematical models based on (ordinary/partial, deterministic/stochastic) dierential equations, we expect some discrepancy between the mathematical model and the actual physical system. It is by now well understood that model error, plays an important role in the delity of the mathematical model and on its predictive capabilities. Model uncertainty, together with additional sources of randomness due, e.g. to incomplete knowledge of the current state of the system, sensitive dependence on initial conditions, parameterization of the small scales etc, should be taken into account when making predictions about the system under investigation.

In addition, many climatological models exhibit 'tipping points' - critical transitions where the output of the model changes disproportionately compared to the change in a parameter. [LHK+08] documents several, the most pertinent to British weather being the Stommel-Cessi box model for Atlantic thermohaline circulation, which suggests the collapse of the Atlantic Meridional Overturning Circulation, upon small changes in freshwater input.

Weather forecasting bodies overcome these inherent difficulties ensemble techniques (or probabilistic forecasting), running multiple simulations accounting for the range of possible scenarios. A forecast should then skilfully indicate the confidence the forecaster can have in their prediction, by accurately representing uncertainty [AMP13]. Clearly, model uncertainty can have a dramatic effect on the predictive capabilities of our mathematical model when we are close to a noise induced transition, a tipping point or a phase transition. This poses an important mathematical question: how can we systematically quantify the propogation of uncertainty through the model, from model parameters and initial conditions, to model-output, even in cases of 'tipping'?

[LHK+08] Timothy M. Lenton, Hermann Held, Elmar Kriegler, Jim W. Hall, Wolfgang Lucht, Stefan Rahmstorf, and Hans Joachim Schellnhuber. Tipping elements in the earth's climate system. Proceedings of the National Academy of Sciences, 105(6):1786{1793, 2008.

[AMP13] H. M. Arnold, I. M. Moroz, and T. N. Palmer. Stochastic parametrizations and model uncertainty in the Lorenz '96 system. Philosophical Transactions of the Royal Society of London Series A, 371:20110479{20110479, April 2013.

Supervisor: G.A. Pavliotis (Imperial College London); V. Lucarini (U. Reading)