**Thomas
Bendall**

#### Project Title:

On coupling resolved and unresolved physical processes in finite element discretisation of geophysical fluids

#### Project Details:

In numerical weather forecasts and climate models, the resolved fluid processes are called the ‘dynamics’, and are separated from the unresolved processes (known as the ‘physics’). To reflect the behaviour of the Earth System, these processes communicate to one another through a process called ‘coupling’. Currently the UK Met Office is developing a new numerical discretisation for its dynamical core, known as Gung Ho!. This will use a finite element method, which presents a host of issues for the coupling of the new dynamical core with the physics processes. For my project, I will be exploring some of these issues and attempting to find solutions for them.

#### Supervisor:

Prof Colin Cotter

**Rose
Gibbins**

#### Project Title:

Earth’s Entropy Budget

#### Project Details:

In his landmark book “The Theory of Heat Radiation” Planck (1913) showed that associated with any flux of electromagnetic radiation there is a flux of entropy and, through using an inversion of the black body function for temperature as a function of intensity, the spectrum of radiant entropy can be deduced from the spectrum of radiant energy. In a climatic steady state the global annual average solar energy absorbed by the Earth is exactly balanced by an equal emission to space of heat energy. It is well established, however, that the entropy flux associated with the emitted radiation is far larger than that of the absorbed solar radiation, with the balance being created by non-equilibrium processes within the climate system.

Different states of the climate are thus reflected in different fluxes of entropy and in this project we investigate the impact on the entropy budget of factors that produce radiative perturbations to the climate, for example changes to the concentration of atmospheric carbon dioxide or to solar irradiance. To date there has been very little work published on spectral entropy fluxes, and some that has is misleading or of low accuracy. In this project we will use state-of-the-art computer models of atmospheric radiative transfer to calculate the radiation and entropy spectra for various climate change scenarios and thus deduce the entropy production required of the Earth .We will complement the radiative point of view on entropy production with that based on the analysis of irreversible processes inside the geophysical fluids. We shall look at the impact of performing spatial and temporal coarse graining in the climatic fields resulting from general circulation models as well as in the spectral resolution of the radiative models. This will inform us on the degree of convergence of the simulated properties with resolution.We expect to use the output of GCMs available through the PCMDI as well as high resolution runs available in Reading. We also expect to use data resulting from radiative transfer models. An additional approach will be to calculate the entropy fluxes associated with radiation spectra measured from satellites, or calculated within global climate models.

#### Supervisor:

Prof Joanna Dorothy Haigh

**Thomas
Gibson**

#### Project Title:

Higher order finite element methods for numerical weather prediction

#### Project Details:

This project is developing computational algorithms for numerical weather prediction and climate models. Recently, the supervisors have developed new numerical approximations of the equations of motion for the atmosphere, based on the mathematical framework of compatible finite element methods. These approximations allow use to be made of massively parallel supercomputers, and thus are being used in the “Gung Ho” project to make a new dynamical core (the part of the model that predicts winds, temperatures and pressures) for the Met Office and the UK climate science and meteorology community. Such a model has many different aspects that must be designed, analysed, and implemented, and this project will concentrate on a few of those.

This project is about numerical model development for weather prediction models. Over the last five years, the “Gung Ho” dynamical core project has settled upon using compatible finite element methods in a new dynamical core for the Met Office. This choice has been motivated by the need to move away from the latitude-longitude grid in current use, because it inhibits parallel scalability. This calls for discretisation methods that can be used on pseudo-uniform grids. Compatible finite element methods have been shown to satisfy the geophysical balance requirements of the current Endgame dynamical core in the Met Office Unified Model, whilst remaining consistent and accurate on the more general grids required for massively parallel simulation, and hence currently form the basis of development on the Gung Ho project.

This project will contribute new discretization methods and solution algorithms, their analysis, and implementation in software through the Firedrake project.

#### Supervisor:

Dr David Ham

**Zoe
Goss**

#### Project Title:

Second-generation design tools for the optimal exploitation of tidal renewable energy

#### Project Details:

The regularity of the tides makes them very attractive for renewable power production. At multiple locations around the UK the local acceleration of tidal currents provides suitable sites for the installation of large arrays of tidal turbines. However, these arrays will only be developed if they can be shown to be economically viable, feasible from engineering perspectives, and with acceptable environmental impacts. Optimisation of the array size, location and its precise configuration is essential to achieve this, and sophisticated numerical models are required to assist developers in these tasks.

To date our work in this area has focussed on the use of automatic code-generation techniques for the numerical solution of the depth-averaged shallow water equations and, uniquely, the development of associated adjoint-based optimisation algorithms. A current PhD project is working towards the inclusion of uncertainty quantification techniques within both power yield estimates as well as ‘robust’ array optimisation algorithms, and another is considering the trade-off between maximising power or profit within an array design and potential environmental impacts.

The focus of this project will be to take the next steps towards the development of a so-called ‘second-generation’ design tool of value to industrial users. In particular, we will address the fact that little effort to date has gone into the parameterisation of subgrid-scale processes, including turbulence. This is of vital importance for reliable estimated of how turbines interact with one another within an array and respond to and alter the ambient environment. This effort will build upon a current MRes project considering ‘horizontal large eddy simulations (HLES)’ methods, and extend to three-dimensional RANS and LES methods using the numerical capabilities developed in a parallel ongoing EPSRC project “A new simulation and optimisation platform for marine technology”.

#### Supervisor:

Prof Matthew Piggott

**Josephine
Park**

#### Project Title:

Material Transport and Stirring in the Ocean

#### Project Details:

Ocean currents and eddies constantly transport and stir huge amounts of water masses and their properties. An efficient way of detecting these processes is by releasing neutral floats and tracking their trajectories. An alternative diagnostics is provided by tracking passive tracer concentrations. On the one hand, the observations show enormous complexity of the eddy-induced transport, which turns out to be not only spatially inhomogeneous and anisotropic but also significantly non-diffusive. On the other hand, general circulation models routinely approximate this transport as homogeneous and isotropic diffusion. This dire situation ensures great potential not only for upgrading the diffusion approach, but also for developing new, physically consistent and much more accurate, simple models of material transport. The goal of this Project is to investigate material transport and stirring properties in idealized but dynamically consistent and structurally rich eddying-flow simulations, and to use these analyses for developing new generation of simple transport models based on other principles. The Project will involve simulations of several types of idealized geostrophic turbulence.

Subsequent kinematic analyses of their transport and stirring properties will be used for developing simple stochastic and deterministic models of the transport and stirring for practical applications.

#### Supervisor:

Prof Matthew Piggott

**Josephine
Park**

#### Project Title:

Material Transport and Stirring in the Ocean

#### Project Details:

Ocean currents and eddies constantly transport and stir huge amounts of water masses and their properties. An efficient way of detecting these processes is by releasing neutral floats and trackingtheir trajectories. An alternative diagnostics is provided by tracking passive tracer concentrations. On the one hand, the observations show enormous complexity of the eddy-induced transport, which turns out to be not only spatially inhomogeneous and anisotropic but also significantly non-diffusive. On the other hand, general circulation models routinely approximate this transport as homogeneous and isotropic diffusion. This dire situation ensures great potential not only for upgrading the diffusion approach, but also for developing new, physically consistent and much more accurate, simple models of material transport. The goal of this Project is to investigate material transport and stirring properties in idealized but dynamically consistent and structurally rich eddying-flow simulations, and to use these analyses for developing new generation of simple transport models based on other principles.

The Project will involve simulations of several types of idealized geostrophic turbulence. Subsequent kinematic analyses of their transport and stirring properties will be used for developing simple stochastic and deterministic models of the transport and stirring for practical applications.

#### Supervisor:

Prof Matthew Piggott

**Michael
Haigh**

#### Project Title:

Gulf Stream Dynamics in the Shallow Water model

#### Project Details:

This Project is about understanding fluid dynamics of the Gulfstream, which is powerful and important ocean current. The novelty is in terms of extending existing theories from multi-layer quasigeostrophic to multi-layer shallow-water approximation. The challenge is (a) in computing eddy-resolving multi-layer shallow-water solutions of the midlatitude ocean gyres with the western boundary currents and their eastward jet extensions, and (b) in interpreting these solutions with a theory of the nonlinear eddy backscatter. The latter requires mathematical analyses of the ocean circulation responses to transient forcing functions that represent mesoscale eddy stresses.

The mighty Gulf Stream current originates in the Caribbean basin, then follows the eastern coast of the United States up to the Cape Hatteras, where it separates from the coast. After separation, the Gulf Stream continues as the north-eastward jet extension that carries relatively warm water across the North Atlantic towards Europe. The Gulf Stream can be viewed as a giant, ribbon-like “river in the ocean” that meanders, sheds enormous vortices called “rings”, radiates complicated planetary waves and stirs the North Atlantic water masses. Properties of the Gulf Stream flow system are heavily controlled by the Earth rotation and sphericity, by the water density distribution across the ocean, and by the shapes of the coasts and bottom topography. Because of all these factors, dynamical understanding of the Gulf Stream structure, spatio-temporal variability, transport and other properties remains vague, despite more than 50 years of vigorous research on this challenging topic.

This Project will focus on isopycnal (i.e., multi-layer shallow-water) primitive-equation model with just a few layers of constant density — this is perfect intermediate set-up between “theoretical” QG and “comprehensive” primitive-equation models. Proposed analyses will be organized into 2 interconnected work programs.

#### Supervisor:

Dr Pavel Berloff

**Matthew
Garrod**

#### Project Title:

Climate change and climate sentiment

#### Project Details:

Public sentiment regarding climate change is an important factor in enhancing and slowing progress towards climate goals. Recent work suggests that in the US, climate change has actually had a beneficial effect with regard to the weather characteristics Americans perceive as important. Climate sentiment itself is a stochastic process running on a social network: large social media studies have observed climate sentiment but the characteristics of the participants (GPS, socio-economic characteristics) are only partly observed and likewise the social network on which the sentiment dynamics unfolds. The student will investigate the control and forecasting of climate sentiment on spatially embedded networks with partly observed node-locations and unobserved links. Models where individuals are not only influenced by their neighbours but also by the climate field associated with their (partially observed) node-location will be considered. The work will help create understanding of how climate sentiment unfolds and provide limits on how it can be influenced. While the thrust of this project will be to investigate the theory of dynamics and control on partially conditioned soft geometric graphs, with a view to understanding limits on prediction and influence of climate sentiment, the candidate will be encouraged to investigate other climate related datasets including call record data in systems with carefully monitored climatic conditions (SAFE project) to other large social datasets on climate sentiment (IEA Reading).

#### Supervisor:

Dr Nick Jones

**Aythami
Bethencourt de Leon**

#### Project Title:

Stochastic GFD: Uncertainty modelling for weather & climate models

#### Project Details:

In a series of preparatory discussions with Piotr Smolarkievicz, Nils Wedi and other scientific leaders at ECMWF, and then after participating in a week-long invited workshop on model error computations and analysis, Cotter, Crisan and Holm believe we have made enough contact with our partners at ECMWF to embark on a collaborative study of a priori introduction of stochasticity for GFD models at various levels of approximation, by using the systematic approach of Holm [2015] and with the intention that this approach will also be tested on ECMWF codes for numerical simulation of aspects of weather and climate. Holm’s method matches Smolarkievicz’s MPDATA algorithm closely, so a priori stochastic transport may be included in that research test bed at ECMWF to assess its effectiveness at an early stage. The parallel development and study at Imperial College will follow the sequence of work packages of a recent EPSRC Standard Grant to Cotter, Crisan and Holm, lasting until 2019, whose numerical implementation centres on Finite Element Methods and whose data assimilation step (for identifying most likely solution paths) is based on particle filtering methods. Representations of uncertainty due to the model dynamics and physics in weather and climate models is a high priority, and is also one of the greatest challenges in modern computational science and mathematics. Currently, the design of such models distinguishes between their dynamical cores (that compute processes represented on the model grid) and their multiscale, multiphysics subgrid scale (unresolved) physical parameterisations. In this distinction, the physical parameterisations are traditionally deterministic formulas that provide estimates of the grid-scale effect of processes (e.g., deep convection) that cannot be resolved by the dynamical core. We propose to pursue stochastic transport algorithms which model uncertainty dynamically and, thus, introduce stochasticity into the advected variables which enter the physical parameterisations. This approach will introduce stochasticity via the inputs to the physical parameterisations which will be self-consistent with the stochastically perturbed transport terms. Consequently, the introduction of any further stochasticity into the physical parameterisations can be performed for physically motivated reasons, instead of being arbitrarily assigned. The controlled introduction of stochastic transport based on velocity correlation EoFs obtained from data assimilation instead of solely random displacements of initial conditions will also be a new feature, in addition to the effects of stochastic tansport on advected variables appearing in the physical parameterisations.

#### Supervisor:

Prof Darryl D Holm

**Oana
Lang**

#### Project Title:

Data Assimilation for Stochastic Partial Differential Equations

#### Project Details:

Climate change is one of the most challenging problems that we are currently facing, fundamental research in this area being, therefore, of crucial importance. Large-scale circulatory motions in oceans and atmosphere involve complex geophysical phenomena that have a determinant influence on climate dynamics. Although it seems impossible to model reality in its entire complexity, the introduction of stochasticity into ideal fluid dynamics appears as a realistic tool for modelling the sub-grid scale processes that cannot otherwise be resolved. We investigate a data assimilation problem for an infinite dimensional model reflecting the motion of an incompressible fluid below a free surface when the vertical length scale is much smaller than the horizontal one. From a mathematical point of view this approach involves a stochastic filtering problem with an infinite dimensional signal modelled by a stochastic partial differential equation, and a large, but finite, observation process. The signal describes the evolution of a rotating two-dimensional shallow water system with the primitive equations. Even if the model is highly simplified in the sense that it does not have the full stratification of the real atmosphere and it involves only a single layer of incompressible fluid, the motions that it supports have close analogues in the real atmosphere and ocean: it allows for processes like gravity and Rossby waves, eddy formation and geophysical turbulence. We will study the influence of missing physics via model noise, while at the same time restricting the evolution by conditioning it on observations of an underlying true system. The observations will be either Eulerian or Lagrangian. The objective of the research is to produce a quantitative and qualitative analysis of the posterior distribution of the state, given the data. The relevance of the results will be investigated by exploring first steps of their realization in an operational framework.

#### Supervisor:

Prof Dan Crisan

**Paulina
Rowinska**

#### Project Title:

Stochastic modelling and estimation of renewable energy production data with applications in operational decision making

#### Project Details:

Weather variables have a huge impact on energy generation based on renewables such as wind and solar which are getting increasingly important in the pursuit of sustainable economic growth. This project aims at constructing stochastic models for renewable energy production data based on appropriate weather variables which are tailored to operational planning and risk management purposes. In addition, efficient methods for statistical inference will be developed which will ensure the applicability of the new models. The scientific contributions of this project will encompass new stochastic models for wind and solar, methodological developments in statistical inference for such models as well as detailed empirical studies.

Climate change threatens the economic prosperity of future generations which makes it urgent to strive for sustainable economic growth, which is in fact one of the key priorities within the UK Sustainable Development Strategy which has been drawn up as a response to Agenda 21 of the United Nations. Mathematics and Statistics play a key role in tackling this challenge and can deliver the reliable tools for risk assessment, which are urgently needed.

The ultimate objective of this PhD project is to develop new technologies in stochastic modelling and statistical inference to reliably quantify risk and uncertainty related to renewable sources of energy production such as wind and solar. While there is a plethora of weather modelling and forecasting methodologies around, see e.g. for a recent survey, such models are typically not tailored to applications in operational decision making which limits their practical appeal in this context.

This project aims to tackle this challenge through a collaborative effort with EDF who will provide expert advice from the perspective of the world-leading power company.

#### Supervisor:

Dr Almut Veraart

**Tasmin
Symons**

#### Project Title:

Some Functional Methods for Meterological Time Series: Analysis & Prediction

#### Project Details:

Recent years have seen the (contested) emergence of a new flavour of El Nino, in the central Pacific. This has led to the suggestion that the atmosphere-ocean coupling in the tropical Pacific has changed: how can we tell? This project will utilise the sophisticated mathematical techniques of Functional Data Analysis to answer questions about the El Nino Southern Oscillation (ENSO). In this set-up, data points are seen as observations of a curve. Although this may appear to be a more complicated setting in which to work, we now have access to powerful mathematical tools, improving our analysis.

The aim of this project is to develop this theory further, and apply it to answer questions about ENSO and some related climate processes.

This project aims to develop new techniques in functional data analysis in order to address challenging questions about the El Niño Southern Oscillation (ENSO), one of the most important drivers of the climate system..

#### Supervisor:

Professor N. Bingham

**Riccardo
Passeggeri**

#### Project Title:

Multivariate ambit fields: Theory, stochastic simulation, statistical inference and meteorological applications

#### Project Details:

Multivariate spatio-temporal stochastic processes are of key importance in various applications, including in particular atmospheric and environmental sciences. The aim of this project is to develop the new class of so-called multivariate ambit fields (MAFs) which constitute a flexible, but yet analytically tractable class of multivariate random fields, and derive suitable stochastic simulation and statistical inference tools tailored to applications in the environmental sciences.

The first part of the project consists of deriving the theoretical properties of MAFs, which are defined via kernel smoothing of a multivariate volatility modulated Levy basis; see [BNBV2015, CK2015]. As such MAFs are generally non-Gaussian and allow for stochastic variability both in time and potentially in space. We will in particular study different approaches to parameterising the cross-sectional dependence structure in a parsimonious way.

The second part of the project will focus on developing efficient simulation schemes for MAFs. We will extend methods based on Fast Fourier techniques which are powerful in the univariate set-up, see [NV2016], and quantify the corresponding numerical approximation error.

The third part of the project consists of developing statistical inference tools for MAFs. We will compare composite-, quasi- and simulated likelihood methods in detailed simulation studies.

Finally, we will carry out various data examples: E.g. we will focus on modelling temperature and pressure over a wider region in a bivariate model based on MAFs, see e.g. [GKS2010] for a study of the corresponding forecast errors. Also, we will tailor MAFs to obtain a joint model for solar and wind power production in the European energy market.

#### Supervisor:

Dr Almut Veraart

**Jemima
Tabeart**

#### Project Title:

On the treatment of correlated observation errors in data assimilation

#### Project Details:

Numerical weather forecasts are obtained by evolving forward the current atmospheric state using computational techniques that solve equations describing atmospheric motions and other physical processes. The current atmospheric state is estimated by a mathematical technique known as data assimilation. Data assimilation blends previous forecasts with new atmospheric observations, weighted by their respected uncertainties. The uncertainty in the observations is not well understood, and currently up to 80% of observations are not used in the assimilation because these uncertainties cannot be properly quantified and accounted for. This project will investigate mathematical methods to approximate observation uncertainty that preserve observation information content while being sufficiently efficient for practical use in operational weather prediction.

In Numerical Weather Prediction (NWP), large computational models simulate the complex nonlinear equations of motion of the atmosphere. Forecast accuracy is still constrained by uncertainties in the initial conditions, known as the analysis. Variational data assimilation techniques are often used to compute the analysis by minimizing a nonlinear cost function. This is essentially a weighted measure of the distance from forecast states (the background) and the available observations over a fixed time window, weighted by the uncertainties in the data. Thus for good results, accurate specification of the forecast and observation error distributions is vital.

It is becoming increasingly important to use observation data from remote sensing instruments (e.g., satellites and ground based-radar) providing detailed information about the current state of the atmosphere on fine scales. Although it is well known that these data have spatially correlated errors, data assimilation algorithms have typically treated the errors as white noise. This approximation is made since the details of the correlation structure are often unknown. It also allows a simplification of the calculations and a reduction in computational cost. Unfortunately, these measures do not fully exploit the observations and significant information may be lost in the assimilation.

More recently, we have shown that it is possible to estimate observation error correlations. A proper treatment of observation error correlations results in more accurate analyses and improvements in forecast skill. However, estimates of observation error correlations are often noisy, and it is unclear the best way to regularize these to ensure speedy convergence of the iterative scheme used to minimize the cost function and preserve the maximum amount of observation information content.

This PhD project will investigate methods of covariance regularization that preserve observation information content. The first stage of the project will be to consider existing covariance regularization schemes applied to some typical noisy estimated observation error covariance matrices and compare their effects. Metrics to consider in the comparison include observation information content, analysis accuracy and minimization convergence speed. This initial work is expected to provide a basis for the development of a new regularization method or variational assimilation pre-conditioning technique. The project lies within numerical linear algebra and optimization and will consist of theoretical work supported by numerical computations in an idealized framework. There will also be the opportunity to work with real observation and model data from the Met Office and gain an understanding of practical operational issues.

#### Supervisor:

Dr Sarah Louise Dance

**Lea
Oljaca**

#### Project Title:

Probabilistic and uncertainty information in suboptimal filters applied to dissipative PDE’s

#### Project Details:

The term Data assimilation refers to methods whereby noisy observations of a physical system are incorporated into a dynamical model of that system, for instance a PDE, in order to reconstruct the current state of the system or even entire past orbits. Filters are a subclass of data assimilation algorithms which provide an estimate of the current state based on present and past (but not future) observations. Further, filters are often designed to work recursively, using new observations to update on previous estimates.

Although optimal filters exist in theory, they are essentially infinite dimensional in nature which renders application of these hypothetical algorithms infeasible in practice. A very large number of suboptimal but practically feasible filtering approaches have been proposed, most of them being in one way or another reminiscent of the Kalman Filter, which is the optimal filter in the case of linear systems with Gaussian perturbations. A better understanding of these filters is of utmost importance in all fields of applications such as weather forecasting, but it also leads to interesting mathematical questions related to stochastic dynamical systems and probability theory.

The analysis of filtering algorithms (optimal or suboptimal) essentially evolves around the following three core questions:

1. Is the filter stable in the sense that the internal parameters of the filter stay within the relevant range, or do they diverge to machine infinity in finite time with large probability (see[4])?

2. Is the filter stable in the sense that the initial values of certain parameters such as the initial estimate of the state become irrelevant as time progresses (see[2,3])?

3. What is the accuracy of the filter, either in absolute terms or relative to other algorithms, for instance the optimal filter (see[2,3])?

In the context of geophysically relevant models such as the Navier Stokes equations and its various relatives, all three questions have been studied in the literature.

The accuracy of certain filtering algorithms (Question 3) has also been the subject of the MPECDT MRes project “Data assimilation in the 2–dim incompressible Navier–Stokes Equation”, which is a predecessor to this PhD project. That project (and various publications), exploit a remarkable fact shared by many dissipative PDE’s relevant in geophysical fluid dynamics, namely that the projection of the solution onto a suitable finite dimensional space will eventually determine the whole (infinite dimensional) solution.

As far as we can see however, performance analysis has mainly focussed on the filters’ ability to estimate the current state. The potential to provide useful aposteriori error information has often been mentioned and was in fact a major driver behind the development of various filter variants, but there is not much in terms of rigorous analysis of whether this information is in fact reliable. Moreover, it seems that there is no generally accepted methodology whereby such a question could be addressed (see however [1]). The aim of this project is to further develop such a methodology and contributed to filling this gap.

#### Supervisor:

Dr Jochen Broecker