### Maha Hussein Kaouri

**Based at:**University of Reading

Research Project: My research will focus on variational data assimilation schemes where we aim to approximately minimize a function of the residuals of a nonlinear least-squares problem by using newly developed, advanced numerical optimization methods. As the function usually depends on millions of variables, solving such problems can be time consuming and computationally expensive. A possible application of the method would be to estimate the initial conditions for a weather forecast. Weather forecasting has a short time window (the forecast will no longer be useful after the weather event occurs) and so it is important to choose a method which gives the most optimal solution in the given time. This is why the analysis of the efficiency of new techniques is of interest. In summary, the aim of my PhD research is to apply the latest mathematical advances in optimization in order to improve the forecast made by environmental models whilst keeping computational cost and calculation time to a minimum.

MPE CDT Aligned student

Research Project: In Neuroscience, mean-field models are nonlinear dynamical systems that are used to describe the evolution of mean neural population activity, within a given brain region such as the cortex. Mean-field models typically contain 10-100 unknown parameters, and receive high-dimensional noisy input from other brain regions. The goal of my PhD is to develop statistical methodology for inferring mechanistic parameters in this type of differential equation model.

Research Project: In Neuroscience, mean-field models are nonlinear dynamical systems that are used to describe the evolution of mean neural population activity, within a given brain region such as the cortex. Mean-field models typically contain 10-100 unknown parameters, and receive high-dimensional noisy input from other brain regions. The goal of my PhD is to develop statistical methodology for inferring mechanistic parameters in this type of differential equation model.

MPE CDT Aligned student

Research Project: Fast, approximate methods for electromagnetic wave scattering by complex ice crystals and snowflakes. The goal of my PhD is to develop a method to approximate the scattering properties of ice particles in clouds. This could be used to improve scattering models that are available, and therefore allow more precise retrievals of ice cloud properties. These retrievals could be used to evaluate model-simulated clouds and identify problems that exist in the model, thus enabling improvements to be made to the parameterization of ice processes.

Research Project: Fast, approximate methods for electromagnetic wave scattering by complex ice crystals and snowflakes. The goal of my PhD is to develop a method to approximate the scattering properties of ice particles in clouds. This could be used to improve scattering models that are available, and therefore allow more precise retrievals of ice cloud properties. These retrievals could be used to evaluate model-simulated clouds and identify problems that exist in the model, thus enabling improvements to be made to the parameterization of ice processes.

MPE CDT Aligned student

Research project: Next-generation atmospheric models are designed to be more flexible than previous models, so that the choice of mesh and choices of numerical schemes can be deferred or changed during operation (Ford et al., 2013; Theurich et al., 2015). My PhD project seeks to make numerical weather and climate predictions more accurate by developing new meshes and numerical schemes that are suitable for next-generation models. In particular, the project addresses the modelling of orographic flows on arbitrary meshes, focusing on three aspects: first, how orography is best represented by a mesh; second, how to accurately advect quantities over orography and, third, how to avoid unphysical solutions in the vertical balance between pressure and temperature.

Research project: Next-generation atmospheric models are designed to be more flexible than previous models, so that the choice of mesh and choices of numerical schemes can be deferred or changed during operation (Ford et al., 2013; Theurich et al., 2015). My PhD project seeks to make numerical weather and climate predictions more accurate by developing new meshes and numerical schemes that are suitable for next-generation models. In particular, the project addresses the modelling of orographic flows on arbitrary meshes, focusing on three aspects: first, how orography is best represented by a mesh; second, how to accurately advect quantities over orography and, third, how to avoid unphysical solutions in the vertical balance between pressure and temperature.

MPE CDT Aligned student

Research project: Atmospheric convection occurs on length scales far smaller than the grid scales of numerical weather prediction and climate models. However, as the resolution of modern models continues to increase, the local convective effects become evermore significant and, thus, there is a demand for new convection schemes which can produce accurate results at these new scales. One such candidate is through conditional averaging, an approach in which grid boxes are split into convectively stable and unstable regions where separate differential equations are solved for each. The scheme incorporates mass transport by convection and memory which are features often ignored in current models. There is thus a possibility to better represent convection using this new approach.

Research project: Atmospheric convection occurs on length scales far smaller than the grid scales of numerical weather prediction and climate models. However, as the resolution of modern models continues to increase, the local convective effects become evermore significant and, thus, there is a demand for new convection schemes which can produce accurate results at these new scales. One such candidate is through conditional averaging, an approach in which grid boxes are split into convectively stable and unstable regions where separate differential equations are solved for each. The scheme incorporates mass transport by convection and memory which are features often ignored in current models. There is thus a possibility to better represent convection using this new approach.

MPE CDT Aligned student

Research project: my research is divided into two closely related parts. In the first part, I consider two-dimensional models of stochastically driven limit cycles, which are used to describe oceanic weather fluctuations, and study the effect of the interaction between the noise excitation and a phase amplitude coupling, also called shear. I can show that for a certain class of such models there is a transition from noise-induced synchronisation to noise-induced chaos depending on the level of shear.

The synchronisation means convergence to one trajectory for all initial conditions, given a certain noise realisation. The chaos is measured by a positive stability exponent, the Lyapunov exponent, positive entropy and certain properties of the invariant random measure. In models without shear, we can show that noise destroys the bifurcation from equilibria to limit cycles, i.e. Hopf bifurcations, with respect to the attracting objects but that the bifurcation still manifests itself in terms of loss of hyperbolicity.

The second part consists in comparing such bifurcation phenomena for unbounded noise with scenarios for bounded noise. The first approach is to look at killed processes, i.e. to study only trajectories that survive on a bounded domain for potentially unbounded noise. The second approach is to model the bounded noise as a function of a fast chaotic variable perturbing the slow variable where the noise converges to Brownian motion in the scaling limit. I can use recent important results concerning stochastic limits of fast-slow systems.

Research project: my research is divided into two closely related parts. In the first part, I consider two-dimensional models of stochastically driven limit cycles, which are used to describe oceanic weather fluctuations, and study the effect of the interaction between the noise excitation and a phase amplitude coupling, also called shear. I can show that for a certain class of such models there is a transition from noise-induced synchronisation to noise-induced chaos depending on the level of shear.

The synchronisation means convergence to one trajectory for all initial conditions, given a certain noise realisation. The chaos is measured by a positive stability exponent, the Lyapunov exponent, positive entropy and certain properties of the invariant random measure. In models without shear, we can show that noise destroys the bifurcation from equilibria to limit cycles, i.e. Hopf bifurcations, with respect to the attracting objects but that the bifurcation still manifests itself in terms of loss of hyperbolicity.

The second part consists in comparing such bifurcation phenomena for unbounded noise with scenarios for bounded noise. The first approach is to look at killed processes, i.e. to study only trajectories that survive on a bounded domain for potentially unbounded noise. The second approach is to model the bounded noise as a function of a fast chaotic variable perturbing the slow variable where the noise converges to Brownian motion in the scaling limit. I can use recent important results concerning stochastic limits of fast-slow systems.

MPE CDT Aligned student

Supervisor: Pavel Berloff (Imperial College London, Department of Mathematics)

Research project: Alternating jets have been well observed in Earth's oceans and planetary atmospheres like Jupiter. These are elongated structures in the zonal velocity contours, which can be few hundreds to thousands of kilometres long and can stay for a time period of few weeks to months. There are many mechanisms proposed to explain the phenomenon however not all characteristics of these jets are explained and the reality is rather more complex. The aim of my project is to understand the phenomenon using more complex mathematical models, which incorporate real world parameters. In particular, I am studying what effects does bottom topography have on the stability of the jets

Supervisor: Pavel Berloff (Imperial College London, Department of Mathematics)

Research project: Alternating jets have been well observed in Earth's oceans and planetary atmospheres like Jupiter. These are elongated structures in the zonal velocity contours, which can be few hundreds to thousands of kilometres long and can stay for a time period of few weeks to months. There are many mechanisms proposed to explain the phenomenon however not all characteristics of these jets are explained and the reality is rather more complex. The aim of my project is to understand the phenomenon using more complex mathematical models, which incorporate real world parameters. In particular, I am studying what effects does bottom topography have on the stability of the jets

Supervisors: Horatio Boedihardjo (Lead Supervisor, Department of Mathematics and Statistics, Reading University), Jochen Broecker (Department of Mathematics and Statistics, Reading), Martin Rasmussen (Department of Mathematics, Imperial College London)

Summary of MRes project: It has long been conjectured (and in some cases proved) that percolation models share a number of universal characteristics. However, several important questions remain open, and percolation is still not completely understood from a mathematical point of view. Due to the complexity and incompleteness of analytic approaches to percolation, explicit simulations of percolation are used in physical models. This means that the success of percolation as a model for physical phenomena, although intuitively plausible, is to some extent coincidental. Originally incepted as a simple model for liquid flow through porous media, it also plays an important role in modelling floods and land slides as percolation contributes to the depletion of surface water. Further, percolation of sea water into ice floes and the layer of snow often found on such floes is a major mechanism for sea ice development. This project will contribute to a deeper understanding of percolation, and of phenomena in which percolation is relevant. Recently, a connection was established between percolation and Schramm–Loewner evolutions (SLE’s). SLE describes the percolation interface (in the limit of very small cell size) as the set of all initial conditions which make the solutions to a simple stochastic differential equation diverge. Thereby, understanding SLE is the key to a number of qualitative and quantitative aspects of percolation phenomena. Further, it offers the possibility to simulate the percolation interface numerically without the need to simulate percolation itself. A well known long standing problem in the field of SLE is to establish the existence of SLE curves by analytic (rather than stochastic) means. This has been partly - but by no means completely - solved, and current approaches make strong assumptions about the local configuration of percolation. Progress in this problem feeds back to our understanding of phase transitions and percolation. The MRes project will focus on the simulation of SLE curves (in particular the limiting behaviour of percolation).

Summary of MRes project: It has long been conjectured (and in some cases proved) that percolation models share a number of universal characteristics. However, several important questions remain open, and percolation is still not completely understood from a mathematical point of view. Due to the complexity and incompleteness of analytic approaches to percolation, explicit simulations of percolation are used in physical models. This means that the success of percolation as a model for physical phenomena, although intuitively plausible, is to some extent coincidental. Originally incepted as a simple model for liquid flow through porous media, it also plays an important role in modelling floods and land slides as percolation contributes to the depletion of surface water. Further, percolation of sea water into ice floes and the layer of snow often found on such floes is a major mechanism for sea ice development. This project will contribute to a deeper understanding of percolation, and of phenomena in which percolation is relevant. Recently, a connection was established between percolation and Schramm–Loewner evolutions (SLE’s). SLE describes the percolation interface (in the limit of very small cell size) as the set of all initial conditions which make the solutions to a simple stochastic differential equation diverge. Thereby, understanding SLE is the key to a number of qualitative and quantitative aspects of percolation phenomena. Further, it offers the possibility to simulate the percolation interface numerically without the need to simulate percolation itself. A well known long standing problem in the field of SLE is to establish the existence of SLE curves by analytic (rather than stochastic) means. This has been partly - but by no means completely - solved, and current approaches make strong assumptions about the local configuration of percolation. Progress in this problem feeds back to our understanding of phase transitions and percolation. The MRes project will focus on the simulation of SLE curves (in particular the limiting behaviour of percolation).

Supervisors: Colin Cotter (Lead Supervisor, Department of Mathematics, Imperial College London), Tommaso Benacchio (Met Office), Werner Bauer (Department of Mathematics, Imperial College London)

Summary of MRes project: The compatible finite element approach has recently been proposed as a discretisation method for numerical weather prediction and is currently being developed for the GungHo dynamical core (the fluid dynamics component of an atmosphere model) at the Met Office. In this project, we will apply the approach of asymptotic limit analysis for investigating numerical discretisations, using an adaptation of compatible finite elements to a hydrostatic model.

In the context of asymptotic limit analysis, this model can be seen as a simplification achieved by a limiting process of parameters from a more general parent model. The resulting simplified model exhibits numerical discretisations that are accurate over longer times than an equivalent numerical discretisation of the parent model, and thus can be used as a numerical reference solution.

The aim of the project is to then take a sequence of numerical solutions from the parent model following the limit, and compare them with the hydrostatic reference solution. This will enable us to investigate the quality of the numerical discretisation of the parent model.

Summary of MRes project: The compatible finite element approach has recently been proposed as a discretisation method for numerical weather prediction and is currently being developed for the GungHo dynamical core (the fluid dynamics component of an atmosphere model) at the Met Office. In this project, we will apply the approach of asymptotic limit analysis for investigating numerical discretisations, using an adaptation of compatible finite elements to a hydrostatic model.

In the context of asymptotic limit analysis, this model can be seen as a simplification achieved by a limiting process of parameters from a more general parent model. The resulting simplified model exhibits numerical discretisations that are accurate over longer times than an equivalent numerical discretisation of the parent model, and thus can be used as a numerical reference solution.

The aim of the project is to then take a sequence of numerical solutions from the parent model following the limit, and compare them with the hydrostatic reference solution. This will enable us to investigate the quality of the numerical discretisation of the parent model.

Supervisors: David Ham (Lead Supervisor, Department of Mathematics, Imperial College London), Erik van Sebille, (Department of Physics and Grantham Institute, Imperial College London), Michael Lange (Department of Earth Science and Engineering, Imperial College London)

Summary of MRes project: Lagrangian particle tracking is a key analytic tool in oceanography. This project will apply the techniques of symbolic numerical mathematics to express Lagrangian particle problems in a high level mathematical syntax. The project will employ the Firedrake automatic numerical PDE system to represent the Eulerian background fields in parallel and will couple this to the recently developed Parcels Lagrangian ocean particle tracking code to create the fully capable system. Mathematically, this requires understanding the graph theory of parallel meshes, the functional analysis of Eulerian PDE methods and the linear analysis of numerical ODE solvers. This understanding must be used to create and compose symbolic code objects with the correct mathematical properties and operations.

Summary of MRes project: Lagrangian particle tracking is a key analytic tool in oceanography. This project will apply the techniques of symbolic numerical mathematics to express Lagrangian particle problems in a high level mathematical syntax. The project will employ the Firedrake automatic numerical PDE system to represent the Eulerian background fields in parallel and will couple this to the recently developed Parcels Lagrangian ocean particle tracking code to create the fully capable system. Mathematically, this requires understanding the graph theory of parallel meshes, the functional analysis of Eulerian PDE methods and the linear analysis of numerical ODE solvers. This understanding must be used to create and compose symbolic code objects with the correct mathematical properties and operations.

Supervisors:Davoud Cheraghi (Lead Supervisor, Department of Mathematics, Imperial College London) and Gabriel Rooney (Met Office)

Summary of the MRes project: Convection is the dominant form of mass motion in the atmosphere. Attempts for a mathematical model for thermal convection eventually led to the equations of Lorenz, (Lorenz, 1963). This work raised questions that have motivated the developments of techniques in dynamical systems, and has fundamentally affected the approach of weather and climate prediction. It is well-known that the flow of a Lorenz model may have a non-periodic chaotic attractor (Tucker, 1998). This results in sensitive dependence on initial conditions along most orbits of the flow, now well-known as "butterfly effect". Having an attractor means that there are contractions along parts of the flow, while being chaotic means that there are also large expansions along parts of the same flow. This interaction has led to the realization that the use of "probabilistic methods" and "stochastic approaches" is required for the study of these systems and the techniques for numerical weather predictions (Slingo and Palmer, 2011). However the development of probabilistic ensemble forecasting systems depends on our understanding of the Lorenz model and the associated "Lorenz maps". The main objective of this project is to study the connections between "predictability limit of low-dimension convective models" and the "fine-scale dynamical behavior of the associated Lorenz maps".

Summary of the MRes project: Convection is the dominant form of mass motion in the atmosphere. Attempts for a mathematical model for thermal convection eventually led to the equations of Lorenz, (Lorenz, 1963). This work raised questions that have motivated the developments of techniques in dynamical systems, and has fundamentally affected the approach of weather and climate prediction. It is well-known that the flow of a Lorenz model may have a non-periodic chaotic attractor (Tucker, 1998). This results in sensitive dependence on initial conditions along most orbits of the flow, now well-known as "butterfly effect". Having an attractor means that there are contractions along parts of the flow, while being chaotic means that there are also large expansions along parts of the same flow. This interaction has led to the realization that the use of "probabilistic methods" and "stochastic approaches" is required for the study of these systems and the techniques for numerical weather predictions (Slingo and Palmer, 2011). However the development of probabilistic ensemble forecasting systems depends on our understanding of the Lorenz model and the associated "Lorenz maps". The main objective of this project is to study the connections between "predictability limit of low-dimension convective models" and the "fine-scale dynamical behavior of the associated Lorenz maps".

Supervisors: Danica Vukadinovic Greetham (Lead Supervisor, Department of Mathematics and Statistics, University of Reading), Claudia Neves (Department of Mathematics and Statistics, University of Reading)

Industrial Partner: Dr. Maciej Fila (Scottish and Southern Electricity Networks, Reading)

Summary of MRes project: Low carbon technologies such as electric vehicles, heat pumps, solar and wind generation represent an exciting opportunity for a reduction in emissions of carbon dioxide into our atmosphere, but on the other hand represent a huge challenge to the electricity distribution organisations, in particular on the low voltage network level. The new loads created by these technologies in combination with their uptake result in the need for a detailed prediction of individual daily and seasonal loads several years in advance. With this in mind, the MRes will focus on two tasks: i) optimising forecasting techniques to model energy demand and, ii) deducing the properties of the tail behaviour of energy demands for both the present and the future, using Extreme Value Theory. The second part of this aims to answer find the upper limits of energy demands from household.

Industrial Partner: Dr. Maciej Fila (Scottish and Southern Electricity Networks, Reading)

Summary of MRes project: Low carbon technologies such as electric vehicles, heat pumps, solar and wind generation represent an exciting opportunity for a reduction in emissions of carbon dioxide into our atmosphere, but on the other hand represent a huge challenge to the electricity distribution organisations, in particular on the low voltage network level. The new loads created by these technologies in combination with their uptake result in the need for a detailed prediction of individual daily and seasonal loads several years in advance. With this in mind, the MRes will focus on two tasks: i) optimising forecasting techniques to model energy demand and, ii) deducing the properties of the tail behaviour of energy demands for both the present and the future, using Extreme Value Theory. The second part of this aims to answer find the upper limits of energy demands from household.

Supervisors: Ted Shepherd (Lead Supervisor, Department of Meteorology, University of Reading), Sebastian Reich (Potsdam University and Department of Mathematics and Statistics at the University of Reading) and Martin Leutbecher (ECMWF)

Summary of the MRes project: In the MRes project we shall take an idealised model called 'α-turbulence' as a toy problem. This is a generalisation of 2D incompressible Navier-Stokes (2DNS) equations (which approximate the atmosphere reasonably well) that recovers the surface quasi-geostrophic (SQG) model by taking another choice of the parameter α. Moreover, the α-turbulence problem is mathematically well-posed and could be easily solved numerically. The goal of the project is to explore the role of initial and model errors in the uncertainty of weather forecasts in the context of α-turbulence, particularly the contrast between 2DNS and SQG. The mathematics involved includes uncertainty quantification, error cascades, probabilistic prediction, geophysical fluid dynamics and dynamical systems.

We expect this project to provide guidance on the importance of representing these errors at scales close to truncation scales of current operational numerical weather prediction (NWP) ensembles, and on how important it is to correctly model the background energy spectrum in the atmosphere in order to realistically simulate error growth in NWP ensembles.

Summary of the MRes project: In the MRes project we shall take an idealised model called 'α-turbulence' as a toy problem. This is a generalisation of 2D incompressible Navier-Stokes (2DNS) equations (which approximate the atmosphere reasonably well) that recovers the surface quasi-geostrophic (SQG) model by taking another choice of the parameter α. Moreover, the α-turbulence problem is mathematically well-posed and could be easily solved numerically. The goal of the project is to explore the role of initial and model errors in the uncertainty of weather forecasts in the context of α-turbulence, particularly the contrast between 2DNS and SQG. The mathematics involved includes uncertainty quantification, error cascades, probabilistic prediction, geophysical fluid dynamics and dynamical systems.

We expect this project to provide guidance on the importance of representing these errors at scales close to truncation scales of current operational numerical weather prediction (NWP) ensembles, and on how important it is to correctly model the background energy spectrum in the atmosphere in order to realistically simulate error growth in NWP ensembles.

Supervisors: David Ferreira (Lead Supervisor, Department of Meteorology, University of Reading), Tobias Kuna, (Department of Mathematics and Statistics, University of Reading).

Summary of MRes project: For the last 3 million years, Earth climate has been oscillating between interglacial states (like today’s climate) and glacial states (when ice sheets covered North America and Scan- dinavia). Various observations establish a statistical link between the Glacial-Interglacial Cycles (GIC) and the Milankovitch cycles, the millennial oscillations of Earth’s orbital pa- rameters - eccentricity (100 kyr), obliquity (41 kyr), and precession (23 kyr) -that perturb the incoming solar radiation on Earth. However, we do not have a well-established theory for this link, revealing a critical gap in our understanding of the climate system and rais- ing questions on our ability to predict its future evolution. In the 80s, Nicolis, Benzi and collaborators proposed a novel hypothesis: Earth’s climate is in stochastic resonance with he Milankovitch cycles (e.g. Benzi, 2010). This project will examine the transition between the multiple states of the Earth via the theory of stochastic resonance.

Summary of MRes project: For the last 3 million years, Earth climate has been oscillating between interglacial states (like today’s climate) and glacial states (when ice sheets covered North America and Scan- dinavia). Various observations establish a statistical link between the Glacial-Interglacial Cycles (GIC) and the Milankovitch cycles, the millennial oscillations of Earth’s orbital pa- rameters - eccentricity (100 kyr), obliquity (41 kyr), and precession (23 kyr) -that perturb the incoming solar radiation on Earth. However, we do not have a well-established theory for this link, revealing a critical gap in our understanding of the climate system and rais- ing questions on our ability to predict its future evolution. In the 80s, Nicolis, Benzi and collaborators proposed a novel hypothesis: Earth’s climate is in stochastic resonance with he Milankovitch cycles (e.g. Benzi, 2010). This project will examine the transition between the multiple states of the Earth via the theory of stochastic resonance.

Supervisors: John Methven (Lead Supervisor, Department of Meteorology, University of Reading), Hilary Weller (Department of Meteorology, University of Reading) and Tristan Pryer (Department of Mathematics and Statistics, University of Reading)

It can be useful to split atmospheric flow into a background state and perturbations to it. This is important for climate dynamics: there is a slowly varying background flow (the climate) which “interacts” with quickly varying phenomena (the weather). Usually climate is identified through statistical averages but we are looking for a new definition in which the climate obeys physical laws and evolution equations. Here, a background state is defined in terms of two fundamental integral properties of the full flow: mass and circulation enclosed by potential vorticity (PV) contours.

In this MRes project, I will be looking at a vortex in a single layer fluid, described by the shallow water equations. The background state will be calculated using an optimal transport method. This method allows the mass and circulation integrals to be conserved, as required. The method also minimises the total energy so that a unique and stable background state is obtained.

It can be useful to split atmospheric flow into a background state and perturbations to it. This is important for climate dynamics: there is a slowly varying background flow (the climate) which “interacts” with quickly varying phenomena (the weather). Usually climate is identified through statistical averages but we are looking for a new definition in which the climate obeys physical laws and evolution equations. Here, a background state is defined in terms of two fundamental integral properties of the full flow: mass and circulation enclosed by potential vorticity (PV) contours.

In this MRes project, I will be looking at a vortex in a single layer fluid, described by the shallow water equations. The background state will be calculated using an optimal transport method. This method allows the mass and circulation integrals to be conserved, as required. The method also minimises the total energy so that a unique and stable background state is obtained.

Research interests: stochastic processes, dynamical systems, complex systems mechanics, populations dynamics, climate change and human activities impact.

Supervisors: Richard Everitt (Lead Supervisor, Department of Mathematics and Statistics, University of Reading), Heather Graven (Department of Physics, Imperial College London)

Summary of the MRes project: In the study of weather and climate data, accounting accurately for its spatial (and often spatio-temporal) variation is important to guarantee the accuracy of inferences made from the data. An area well-suited for further development and application of new statistical mapping approaches is the estimation of the global distribution of dissolved carbon dioxide (CO2) in the surface ocean using available sparse observations. The ocean presently absorbs approx. 25% of the emissions of CO2 from human activities, which are the primary cause of climate change. One of the main techniques for estimating ocean CO2 uptake makes use of ship-based observations of dissolved CO2 in the surface ocean from research cruises and commercial “ships-of-opportunity”. My MRes project proposes to model spatial dependence through the use of Gaussian Markov random fields, applied to ocean CO2 data. This is a mathematically rich subject area that involves ideas from statistical inference, stochastic PDEs and numerical linear algebra. Continuously indexed Gaussian fields (GFs) are a cornerstone of spatial statistics, but suffer from the “big n problem”, referring to the high computational cost (O(n3)) when factorising covariance matrices of dimension n*n, where n is large. Some GFs may be seen to be solutions to linear stochastic partial differential equations, and it has recently been shown (Lindgren, Rue, and Lindström 2011) that approximate stochastic weak solutions to these SPDEs are given by discretely indexed Gaussian Markov random fields (GMRFs). GMRFs are special cases of multivariate Gaussian distributions that have the property that many of the entries in their precision matrices are zero. This makes doing computations using GMRFs more feasible than GFs, using ideas from numerical linear algebra to significantly simplify the matrix manipulations.

Summary of the MRes project: In the study of weather and climate data, accounting accurately for its spatial (and often spatio-temporal) variation is important to guarantee the accuracy of inferences made from the data. An area well-suited for further development and application of new statistical mapping approaches is the estimation of the global distribution of dissolved carbon dioxide (CO2) in the surface ocean using available sparse observations. The ocean presently absorbs approx. 25% of the emissions of CO2 from human activities, which are the primary cause of climate change. One of the main techniques for estimating ocean CO2 uptake makes use of ship-based observations of dissolved CO2 in the surface ocean from research cruises and commercial “ships-of-opportunity”. My MRes project proposes to model spatial dependence through the use of Gaussian Markov random fields, applied to ocean CO2 data. This is a mathematically rich subject area that involves ideas from statistical inference, stochastic PDEs and numerical linear algebra. Continuously indexed Gaussian fields (GFs) are a cornerstone of spatial statistics, but suffer from the “big n problem”, referring to the high computational cost (O(n3)) when factorising covariance matrices of dimension n*n, where n is large. Some GFs may be seen to be solutions to linear stochastic partial differential equations, and it has recently been shown (Lindgren, Rue, and Lindström 2011) that approximate stochastic weak solutions to these SPDEs are given by discretely indexed Gaussian Markov random fields (GMRFs). GMRFs are special cases of multivariate Gaussian distributions that have the property that many of the entries in their precision matrices are zero. This makes doing computations using GMRFs more feasible than GFs, using ideas from numerical linear algebra to significantly simplify the matrix manipulations.

Supervisors: Horatio Boedihardjo (Lead Supervisor, Department of Mathematics and Statistics, University of Reading), Andrea Moiola (Department of Mathematics and Statistics, University of Reading), Chris Westbrook (Department of Meteorology, University of Reading), Dan Crisan (Department of Mathematics, Imperial College London), Jochen Broecker (Department of Mathematics and Statistics, University of Reading)

Summary of the MRes project: The mathematical description of meteorological fields such as snow cover and clouds remains a serious challenge due to the highly irregular structure of these fields. In Physics, the Kardar-Parisi-Zhang (KPZ) equation is a stochastic partial differential equation (SPDE) which describes the universal large scale behaviour of surface growth. It is therefore natural to conjecture that KPZ may be applied to model clouds and snow as well as other ``growing surfaces’’ in atmospheric physics, such as the height of the convective boundary layer.

In order to investigate the suitability of the KPZ equation as a model for meteorological phenomena, a robust way to compute numerical solutions is needed. The key objective of the MRes project will be to develop an efficient numerical scheme for approximating the solution to the KPZ equation. It should be possible to convert Martin Hairer's theory of regularity structures, which opens the door for a mathematical study of the KPZ equation which was not possible before, into a practical numerical scheme.

Summary of the MRes project: The mathematical description of meteorological fields such as snow cover and clouds remains a serious challenge due to the highly irregular structure of these fields. In Physics, the Kardar-Parisi-Zhang (KPZ) equation is a stochastic partial differential equation (SPDE) which describes the universal large scale behaviour of surface growth. It is therefore natural to conjecture that KPZ may be applied to model clouds and snow as well as other ``growing surfaces’’ in atmospheric physics, such as the height of the convective boundary layer.

In order to investigate the suitability of the KPZ equation as a model for meteorological phenomena, a robust way to compute numerical solutions is needed. The key objective of the MRes project will be to develop an efficient numerical scheme for approximating the solution to the KPZ equation. It should be possible to convert Martin Hairer's theory of regularity structures, which opens the door for a mathematical study of the KPZ equation which was not possible before, into a practical numerical scheme.

Supervisors: Clare Watt (Lead Supervisor, Department of Meteorology, University of Reading), Mathew Owen (Department of Meteorology, University of Reading) and Tristan Pryer, (Department of Mathematics and Statistics, University of Reading)

Summary of the MRes Project: In the Earth’s radiation belts, the evolution of the phase-averaged phase-space density for high-energy electrons (with energy >500keV) can be described by a diﬀusion equation [2]. Rather than being a differential equation with respect to the underlying spatial variables, it depends on the adiabatic invariants of charged particle motion [1]. Such variables are awkward to make use of and, in particular, to visualise, thus a change of variable is utilised. This change of variable results in the transformation of the problem into a high dimensional advection-diffusion problem.

The goal of this project is to derive, implement and test a numerical scheme for this problem to allow for various tests to be conducted against real in-situ data obtained from the NASA Van Allen Probes mission. This numerical scheme will then be used to make predictions of the form of phase-space density in the Earth’s radiation belts which are cutting edge.

Summary of the MRes Project: In the Earth’s radiation belts, the evolution of the phase-averaged phase-space density for high-energy electrons (with energy >500keV) can be described by a diﬀusion equation [2]. Rather than being a differential equation with respect to the underlying spatial variables, it depends on the adiabatic invariants of charged particle motion [1]. Such variables are awkward to make use of and, in particular, to visualise, thus a change of variable is utilised. This change of variable results in the transformation of the problem into a high dimensional advection-diffusion problem.

The goal of this project is to derive, implement and test a numerical scheme for this problem to allow for various tests to be conducted against real in-situ data obtained from the NASA Van Allen Probes mission. This numerical scheme will then be used to make predictions of the form of phase-space density in the Earth’s radiation belts which are cutting edge.

Supervisors: Pavel Berloff (Lead Supervisor, Department of Mathematics, Imperial College London) and Erik van Sebille (Grantham Institute and Department of Physics, Imperial College London)

Summary of MRes project: An estimated 250,000 metric tonnes of plastic debris are floating on the surface of the ocean at this moment, and this pollution greatly harms the environment and marine animals. In order to increase our understanding of how marine animals interact with the plastic, it is important to know how it is moved around by the oceanic circulation, involving flow features ranging from microscale to the basin scale. On the simplest level of approximation, the plastic can be modelled as purely passive tracer, represented either by continuous concentration fields or by ensembles of infinitesimal Lagrangian particles. In this project, the plastic particles are viewed as finite-size and finite-mass objects, which interact with ambient fluid and, therefore, experience flow drag. These “inertial particles” remain much less explored, especially in the oceanographic context.

The goal of this project is to investigate material transport and stirring properties of “inertial particles” in idealised but dynamically consistent and structurally rich eddying flows. Questions that will be addressed in particular are: How robust is the “inertial” effect, and how does it depend on mesoscale and sub-mesoscale oceanic turbulence? What are sensitivities to the involved physical characteristics of the inertial particles? What are the fundamental mechanisms affecting the evolution of the inertial particles?

The results of the Project will have a direct impact on our understanding of the fate of plastic pollution in the ocean, and help to improve models for tracking other inertial objects in the ocean (e.g., lost persons, wreckage, icebergs).

Summary of MRes project: An estimated 250,000 metric tonnes of plastic debris are floating on the surface of the ocean at this moment, and this pollution greatly harms the environment and marine animals. In order to increase our understanding of how marine animals interact with the plastic, it is important to know how it is moved around by the oceanic circulation, involving flow features ranging from microscale to the basin scale. On the simplest level of approximation, the plastic can be modelled as purely passive tracer, represented either by continuous concentration fields or by ensembles of infinitesimal Lagrangian particles. In this project, the plastic particles are viewed as finite-size and finite-mass objects, which interact with ambient fluid and, therefore, experience flow drag. These “inertial particles” remain much less explored, especially in the oceanographic context.

The goal of this project is to investigate material transport and stirring properties of “inertial particles” in idealised but dynamically consistent and structurally rich eddying flows. Questions that will be addressed in particular are: How robust is the “inertial” effect, and how does it depend on mesoscale and sub-mesoscale oceanic turbulence? What are sensitivities to the involved physical characteristics of the inertial particles? What are the fundamental mechanisms affecting the evolution of the inertial particles?

The results of the Project will have a direct impact on our understanding of the fate of plastic pollution in the ocean, and help to improve models for tracking other inertial objects in the ocean (e.g., lost persons, wreckage, icebergs).

Supervisors: Xuesong Wu (Lead Supervisor, Department of Mathematics, Imperial College London), John Methven (Department of Meteorology, University of Reading) , Andrew Charlton-Perez (Department of Meteorology, Reading University)

Summary of the MRes project: The aim of the present project is (a) to investigate ﬁrst the key fundamental aspects in this coupling, namely, generation, reﬂection and breakdown of Rossby waves as well as their back action on the troposphere, (b) to integrate these fundamental processes in a uniﬁed framework, thereby constructing a self-consistent physics-based model for troposphere-stratosphere coupling, and (c) to diagnose how processes such as Rossby wave reﬂection and its interference with upwards radiation inﬂuence the predictability of both the ﬂow in stratosphere and troposphere

Summary of the MRes project: The aim of the present project is (a) to investigate ﬁrst the key fundamental aspects in this coupling, namely, generation, reﬂection and breakdown of Rossby waves as well as their back action on the troposphere, (b) to integrate these fundamental processes in a uniﬁed framework, thereby constructing a self-consistent physics-based model for troposphere-stratosphere coupling, and (c) to diagnose how processes such as Rossby wave reﬂection and its interference with upwards radiation inﬂuence the predictability of both the ﬂow in stratosphere and troposphere

Supervisors: Matthew Piggott (Lead Supervisor, Department of Earth Science & Engineering, Imperial College London), David Ham (department of Mathematics, Imperial College London)

Summary of the MRes project: The tsunami which hit Fukushima, Japan, in March 2011, following from an earthquake off the Pacific coast of Tōhoku, was in many ways a disastrous event. It caused 15,894 deaths, many more injuries and an enormous amount of damage to public and private property. The direct damage caused by the tsunami also led to a level 7 meltdown at the Fukushima Daiichi nuclear power plant, causing further destruction. In order to mitigate similar disasters in the future around the world, it is crucial that efforts are made to improve early warning systems, such as those which already exist in the Pacific ocean, in order to make rapid assessments of the potential inundation caused by a particular tsunami and plan for immediate evacuation of populated areas, if necessary. Efficient and accurate modelling tools are a key component of these systems. Numerically modelling a tsunami event is mathematically interesting because it involves the consideration of physical processes at a wide range of scales, including propagation across ocean basins as well as (relatively) small-scale sources and key target locations such as population centres or key infrastructure. For the scope of an MRes project it is sufficient to consider the socalled two-dimensional shallow water equations, since (in the deep ocean) the height of a tsunami wave is of the order of one metre with wavelengths of 100s of kms, whilst the depth of the ocean is of the order of only thousands of metres. The need for a rapidly calculated solution means that, in constructing a numerical simulation of a tsunami, we would like to have an effective algorithm which is very efficient. One means of obtaining this is to use adaptive mesh techniques combined with the finite element discretization method. By doing this, we may resolve the features of the wave more accurately, and save computer power by using a coarser resolution where the ocean is calmer. By making use of adaptivity, we may also consider goal-based mesh refinement, wherein we would like to analyse the inundation at a particular location of interest. In response to the situation of the 2011 tsunami, we could for example focus upon the inundation at the Fukushima Daiichi nuclear power plant and study how well the disaster can be re-emulated and thereby studied.

Objectives and methodology: The initial aim of this MRes project will be to consider the propagation of a wave in an idealised domain on a fixed mesh, and to examine the resulting approximation using numerical analysis, e.g. convergence rates as the mesh resolution and time step are refined. Following this, a longer term aim will be to investigate the use of adaptive mesh methods such as adaptive mesh refinement (AMR) [Jansson et al, 2012] or mesh optimization techniques [Barrall et al, 2016] to the example test problem provided by the 2011 Japanese tsunami. Success of the developed approach will be benchmarked against fixed mesh simulations as well as real world observations. A key difficulty to be overcome in this project will be to consider the interplay between the adaptivity procedure and the error analysis. In the longer term there is a possibility of being able to input observational data from measurements at the coast upon the arrival of the tsunami wave and to study the inverse problem, providing improves estimates of the nature of the source of the wave - namely its location, the size of the fault and the time period over which it formed. Upon obtaining a numerical solution to the example problem by these means, a final objective is to compare the results with those obtained from more sophisticated methods and to establish the best error measure to use.

Summary of the MRes project: The tsunami which hit Fukushima, Japan, in March 2011, following from an earthquake off the Pacific coast of Tōhoku, was in many ways a disastrous event. It caused 15,894 deaths, many more injuries and an enormous amount of damage to public and private property. The direct damage caused by the tsunami also led to a level 7 meltdown at the Fukushima Daiichi nuclear power plant, causing further destruction. In order to mitigate similar disasters in the future around the world, it is crucial that efforts are made to improve early warning systems, such as those which already exist in the Pacific ocean, in order to make rapid assessments of the potential inundation caused by a particular tsunami and plan for immediate evacuation of populated areas, if necessary. Efficient and accurate modelling tools are a key component of these systems. Numerically modelling a tsunami event is mathematically interesting because it involves the consideration of physical processes at a wide range of scales, including propagation across ocean basins as well as (relatively) small-scale sources and key target locations such as population centres or key infrastructure. For the scope of an MRes project it is sufficient to consider the socalled two-dimensional shallow water equations, since (in the deep ocean) the height of a tsunami wave is of the order of one metre with wavelengths of 100s of kms, whilst the depth of the ocean is of the order of only thousands of metres. The need for a rapidly calculated solution means that, in constructing a numerical simulation of a tsunami, we would like to have an effective algorithm which is very efficient. One means of obtaining this is to use adaptive mesh techniques combined with the finite element discretization method. By doing this, we may resolve the features of the wave more accurately, and save computer power by using a coarser resolution where the ocean is calmer. By making use of adaptivity, we may also consider goal-based mesh refinement, wherein we would like to analyse the inundation at a particular location of interest. In response to the situation of the 2011 tsunami, we could for example focus upon the inundation at the Fukushima Daiichi nuclear power plant and study how well the disaster can be re-emulated and thereby studied.

Objectives and methodology: The initial aim of this MRes project will be to consider the propagation of a wave in an idealised domain on a fixed mesh, and to examine the resulting approximation using numerical analysis, e.g. convergence rates as the mesh resolution and time step are refined. Following this, a longer term aim will be to investigate the use of adaptive mesh methods such as adaptive mesh refinement (AMR) [Jansson et al, 2012] or mesh optimization techniques [Barrall et al, 2016] to the example test problem provided by the 2011 Japanese tsunami. Success of the developed approach will be benchmarked against fixed mesh simulations as well as real world observations. A key difficulty to be overcome in this project will be to consider the interplay between the adaptivity procedure and the error analysis. In the longer term there is a possibility of being able to input observational data from measurements at the coast upon the arrival of the tsunami wave and to study the inverse problem, providing improves estimates of the nature of the source of the wave - namely its location, the size of the fault and the time period over which it formed. Upon obtaining a numerical solution to the example problem by these means, a final objective is to compare the results with those obtained from more sophisticated methods and to establish the best error measure to use.

Supervisors: Arnaud Czaja (Lead Supervisor, Department of Physics, Imperial College London) and David Ferreira (Department of Meteorology, University of Reading)

Summary of MRes project: Fluid motions in the atmosphere and the ocean are inherently unstable. Due to instabilities, small perturbations can grow exponentially. A type of fluid instability, known as ‘baroclinic instability’, gives rise to storms and weather systems in the atmosphere, as well as equivalent ‘storms’ in the ocean. In the ocean, these storms are called eddies.

The ocean circulation is a crucial component in the Earth’s climate system, continuously transporting heat away from the equator and towards higher latitudes. The UK would be far colder than it is without this heat transport from the ocean. The oceanic eddy field is a major component of the ocean circulation. This field is important to understand as eddies transport anomalously warm or cool water along with them, significantly modulating the ocean’s sea surface temperature distribution. However, oceanic eddies are currently not well-represented in climate models.

The goal of my MRes project is to revisit the oceanic eddy field using a very different theoretical framework originally developed for the atmosphere. By analysing high-resolution simulations from MIT’s global circulation model, we will look at the convective motion of ocean fluid parcels, as well as looking for evidence of oscillations in the heat flux and the associated timescales. We hope this will shed new light on the dynamics of the oceanic eddy field and help inform its future representation in climate models.

Summary of MRes project: Fluid motions in the atmosphere and the ocean are inherently unstable. Due to instabilities, small perturbations can grow exponentially. A type of fluid instability, known as ‘baroclinic instability’, gives rise to storms and weather systems in the atmosphere, as well as equivalent ‘storms’ in the ocean. In the ocean, these storms are called eddies.

The ocean circulation is a crucial component in the Earth’s climate system, continuously transporting heat away from the equator and towards higher latitudes. The UK would be far colder than it is without this heat transport from the ocean. The oceanic eddy field is a major component of the ocean circulation. This field is important to understand as eddies transport anomalously warm or cool water along with them, significantly modulating the ocean’s sea surface temperature distribution. However, oceanic eddies are currently not well-represented in climate models.

The goal of my MRes project is to revisit the oceanic eddy field using a very different theoretical framework originally developed for the atmosphere. By analysing high-resolution simulations from MIT’s global circulation model, we will look at the convective motion of ocean fluid parcels, as well as looking for evidence of oscillations in the heat flux and the associated timescales. We hope this will shed new light on the dynamics of the oceanic eddy field and help inform its future representation in climate models.