Transition to Turbulence in Flow over Rough Topography in the Southern Ocean
It is an emerging picture that deep ocean turbulence exerts a control over the climate system through regulating the oceanic uptake and redistribution of heat, carbon, nutrients and other tracers. Observations of such turbulence, and our ability to model it numerically, however, have been limited if non-existent until very recently. The challenge ahead is to understand physics of such turbulence to help represent them properly in climate models that are coarse resolution, hence incapable of resolving such processes.
The aim of this project is to better quantify the energy pathways from the mean geostrophic flow to dissipation and mixing through interactions with bottom topography, first through the generation of overturning lee waves above ridge-like topography and later through the generation of lee vortices at seamounts. Recent realistic and high resolution simulations of the Drake Passage provide an invaluable case study of flow topography interactions in an energetic and dynamically relevant area of the Southern Ocean. We will isolate these key processes from the simulations, analyse them for their contribution to energy pathways in the Southern Ocean, and create theoretical frameworks in order to enable better parametrisation of the processes in coarse resolution models.
Dr Ali Mashayek
What determines the ability of the jet stream to shift in response to external forcing?
A major challenge for climate science is to predict how atmospheric circulation will change as the planet warms. Current climate models generally predict a poleward shift of the midlatitude jet streams and storm tracks with warming. However, the climate science community faces two major difficulties in predicting future jet stream changes. First, the projected jet responses to greenhouse gas forcing are not robust in climate models; second, the dynamical mechanisms responsible for future jet shifts remain uncertain. Taken together, these two obstacles lead to particularly low confidence in climate model projections of jet stream changes.The approach of my PhD project is to use a hierarchy of general circulation models to better understand the relationship between the initial state of the jet and its response to forcing. At the lowest order of this hierarchy we will use a barotropic model which represents the interaction of Rossby waves with the mean flow. The barotropic model findings will be used to interpret results from more realistic climate models. The model results will be interpreted physically using linear Rossby wave theory.
Dr Paulo Ceppi
Next generation numerics for global ocean modelling
Building an ocean model using compatible finite element numerics. This will begin with the following steps:
1. extend the linear incompressible nonhydrostatic Boussinesq implicit solver to include free surface (using the combined pressure-free surface approach.
2. Plug this solver into the Gusto modelling system and use existing Gusto advection solvers to construct a semi-implicit nonlinear equation solver loop using the linear Boussinesq solver as the implicit solver step.
At this stage we will have a full numerical scheme for the nonlinear nonhydrostatic incompressible Boussinesq equations and will be able to investigate performance and accuracy through standard test problems. Going beyond that we could then work on a number of different topics establishing this system as a useable ocean modelling research tool: salinity and equation of state, turbulence closures e.g. Gent-McWilliams, work towardswind driven baroclinic gyres, representation of bathymetry, parallel scalability, stability over large timesteps, etc. The supervisory team includes a researcher who leads Gusto code development (Shipton) and an external collaborator who has been using Firedrake to develop a coastal ocean model.
Professor Colin Cotter
Atlantic-Pacific Decadal Teleconnections in Coupled Ocean-Atmosphere Model
The impact of ocean-atmosphere coupling at decadal timescales still remain poorly understood despite recent advances in climate modelling. This is largely due to the fact that General Circulation Models (GCMs) are still unable to fully resolve the dynamics of ocean mesoscale eddies (or the “oceanic weather”). Furthermore, the complexity and the sheer number of process that are modelled in GCMs make it extremely challenging to disentangle mechanisms and causalities. The approach taken in my PhD is to use an idealized coupled ocean-atmosphere model that fully resolves the dynamics of these mesoscale eddies, collect high-frequency data over long time-scales, and model only a subset of physical processes that are deemed to be the most important in our climate system.
Dr Pavel Berloff
SPDEs in fluid dynamics and their application to ocean debris
The issue of ocean plastics has recently been much discussed by academics, policy makers, and environmental campaigners. The mathematical models which are used to describe the advection of plastics have largely ignored key factors such as sub-grid-scale dynamics and the mass of the debris. This raises the interesting question of how inertial particles move in a fluid governed by a SPDE. Using recent developments in stochastic fluid equations [Holm 2015] as a springboard, we will explore how the introduction of transport noise affects the properties (such as well posedness and smoothness) of a fluid model. In particular, can this type of noise restore uniqueness to a model? Furthermore, we will input the velocity field of the fluid into an equation which will return the velocity of the debris [Maxey & Riley, 1983], exploring the validity of doing this and whether this accurately models reality. Such a model would have applications in predicting the motion of ocean debris (such as icebergs, plastics, or aircraft wreckage) and, considering the model as an inverse problem, calibrating ocean models from drifter buoy data by understanding how the movement of the buoys differs from that of the fluid.
Professor Dan Crisan
Eco-systemic response to climate change
The aim of the project is to develop forecasting techniques in order to predict the onset of ecological crises. It is well known that these critical transitions (also known as tipping points) happen in many different areas, from ecological and natural systems to social and political ones. As a parameter is changed, for example the emission of CO2 or the global average temperature, the system becomes more and more unstable, meaning that a small fluctuation can lead it to a different, and usually more disastrous, state.
The project will proceed on two parallel paths. On one hand, since we still lack of a coherent mathematical framework describing these phenomena, we are going to borrow ideas from well-developed theories, such as statistical mechanics, the theory of phase transitions and the theory for fast/slow systems, to construct a solid toolbox to interpret critical transitions.
On the other hand, we are going to investigate methodologies that provide us a way to detect in time series data the onset of a crisis. Some work has already been done in this direction; however, robust data-driven indicators for these critical transitions are still missing. We are going to use the ideas developed in the more theoretical part of the project firstly on simulated data from models and then to real world data sets.
Professor Henrik Jensen
Hybrid numerical-asympotic boundary element methods for multiple scattering problems
Linear wave scattering problems are ubiquitous in science and engineering applications. In the atmosphere, visible and ultraviolet radiation from the sun is scattered by ice crystals in cirrus clouds in the cold upper troposphere. These same clouds also scatter and absorb infrared radiation emitted from the earth’s surface and lower troposphere. Together, these effects exert an important influence on the earth’s radiation balance, and must be represented correctly in numerical climate models.
There are a number of unresolved problems with computing light scattering from an ice crystal in this regime. Firstly, the size of the ice particle is typically large compared to the wavelength of light illuminating it. This means that conventional numerical methods for such problems are prohibitively expensive. As a result the state-of-the-art in ice cloud radiative transfer is the use of ray-tracing (geometric optics). However, this approach cannot capture the effects of diffraction at the corners and edges of the ice crystal. This is typically overcome via a crude correction after computing the ray-tracing solution, but it is not clear whether this is accurate.
A second unresolved problem lies in the nature of the crystal surfaces themselves. Geometric optics is valid for surfaces which are flat and smooth. Real ice particles often have imperfections (roughness): such as steps, or pits, or in some cases may be rounded lumps. This roughness is in fact one of the leading order controls on the far field scattering pattern, and representing it properly is therefore a high priority. We would like to understand the influence of this roughness from a fundamental level, and therefore better constrain the way it is represented in radiative transfer.
The main tool for investigating these questions is the development of numerical methods for simulating light scattering that are able to capture the effects of diffraction whilst remaining computationally tractable across the frequency spectrum. One promising approach is the development of the class of methods known as “hybrid numerical-asymptotic boundary element methods” (HNABEM), which have been proven to be exceptionally efficient at solving a range of scattering problems, with various boundary conditions and geometries. To date though, the range of geometries for which these methods have been shown to be applicable is insufficient to address realistic questions in atmospheric science. This project aims to develop algorithms closely related to HNABEM yet containing key new ideas that will allow them to be applied to more general scattering problems, and to use these algorithms to investigate questions such as those posed above.
Firstly we will continue previous work on problems of scattering by multiple screens, with a goal being to consider scattering by a rough needle like ice crystals in two dimensions, and to compare qualitative results obtained with similar results for a standard three- dimensional solver so as to understand the potential usefulness of studying related two-dimensional problems. Secondly we will compare results obtained by state of the art approaches for high frequency three dimensional electromagnetic transmission problems in atmospheric physics (e.g., Physical Geometric-Optics Hybrid (PGOH) methods) with those obtained by standard numerical solvers, such as BEM++, so as to understand the significance of the diffraction missed by PGOH, and to develop ideas for approximating this difference via HNABEM.
The EPSRC research area that is of most relevance to the project is Numerical Analysis, with the project focused on the development, analysis and implementation of numerical methods for the solution of problems arising in atmospheric physics.
Professor Simon Neil Chandler-Wilde
Reformulating aircraft routing algorithms to reduce fuel burn
With the accuracy of transatlantic navigation set to improve dramatically, aircraft flying across the Northern Atlantic will no longer have to rely on procedural flying, which involves following widely spaced tracks at fixed speed and altitude. The satellite system currently being tested will allow much better aircraft-to-aircraft and Air Traffic Control-to-aircraft tracking, so that this vast air space can be used far more effectively. As the aviation industry is responsible for 5% of all anthropogenic climate change, the need to reduce emissions has never been so pressing. At the heart of my PhD project is the aim to make long-haul flying more environmentally friendly, in a simple to realise, timely and economically efficient manner.
This PhD follows my MRes project in which it was shown that there is potential for airlines to reduce fuel burn and thus CO2 emissions, by changing the routes currently flown between London Heathrow Airport and John F. Kennedy Airport, in New York. In this preliminary study a time minimisation approach was used alongside some new fuel burn rate calculations. Only the heading angle of the aircraft was varied in creating optimal paths, with air speed and altitude remaining fixed. Comparing the new routes with current flight records, it was found that fuel and emissions could be reduced by between 5 and 17% depending on direction being flown and model of aircraft flying.
In the new study these ideas will be taken further, so that routes flown can be fitted into the airline schedules more easily. Start and end times and positions for a route will be given as boundary conditions, whilst the air speed and altitude will be varied along with the heading angle, to allow the aircraft to complete the given route in the correct time window, burning the minimum amount of fuel possible. This will involve using Optimal Control Theory to set up simulations of single routes across daily wind fields.
The cruise phase alone will be examined, as this makes up the vast majority of the fuel requirement for long-haul flights. The key innovation will be in using the new fuel burn calculations to make up part of the cost functional, so that optimisation revolves around fuel burn and thus emissions.
Another part of this investigation will be to look at turbulence avoidance, which is set to become increasingly important due to the effects of climate change. In order to allow pilots to know which route to fly, once they have had to change course, particularly in avoiding difficult weather conditions, it will be useful to have a map of routes rather than just a single path. In order to provide this, dynamic policy programming techniques will be used, so that the most fuel-efficient path from every point on a given map is calculated. This involves deriving the Hamilton-Jacobi-Bellman equation to describe the conditions encountered at each point on the map and the motion of the plane. This equation is then applied recursively until the destination airport is reached. Then by tracing the path back again, the optimal feedback control will give heading angles, air speeds and altitudes and thus all points travelled through during the journey. Once simulations have been run using the new models, fuel usage and emissions generation can be checked against current flight data to quantify what savings in both fuel and emissions can be attained. Results will be shared with both the academic world and industry via published articles and conference presentations.
Dr Paul Williams
Advection and convection for weather and climate models
Everything in the atmosphere is transported by the wind – temperature, pollutants, moisture, clouds and even the wind itself (non-linear advection). The aim of this research is to investigate implicit time stepping methods for advection schemes, with implication to climate models. The current time stepping method used for transport by the Met office and ECMWF is the Semi-Lagrangian method. It is remarkably stable and accurate for long time steps however it isn’t conservative and relies on the uniformity of the mesh.
An alternative method for long time stepping is implicit time stepping, this method is less accurate for larger time steps, but remains stable on non-uniform meshes, and can be made conservative. Additionally, the accuracy of implicit time stepping can be improved, and fixed-point methods can be implemented to more accurately capture the nonlinear effects. Newton and quasi-newton Methods will be developed, for Rayleigh Bernard convection. The applicability of these methods will be addressed with regard to climate modelling. In the atmosphere the Courant number is only big in a few regions, so the accuracy gained by the Semi-Lagrangian method may be outweighed by the loss of conservativity, we investigate implicit time stepping as a potential conservative alternative
Dr Hilary Weller
Morphological instabilities in geophysical flows
We will begin with deposition problems and morphological instabilities, termed “crenulations”, over calcite surfaces such as stalagmites, and also consider related dissolution problems. The formation mechanism is driven by a liquid film flowing down the structure and carrying slightly supersaturated calcium carbonate that deposits onto the wall. The change in wall-shape affects the flow and the flow affects the shape, albeit at different time scales. The Stokes/Navier-Stokes equations need to be addressed along with the underlying chemistry and surface growth models. We will use mathematical modelling, asymptotic analysis of partial differential equations, and numerical computations to advance the field in several crucial directions not considered previously.
In what follows we provide an overview of some of the mathematics along with an itemised work plan of the proposed thesis:
Realistic geometries: The state-of-the-art consists of modelling the stalagmite as a flat plate with an under-lying liquid film that naturally lead to Rayleigh-Taylor instabilities and absolute-convective transitions. The correct conical geometry is quite different from the flat plate picture.
New mathematical approaches: We will use full 3D axisymmetric geometries and analyse flows over slowly varying conical geometries using a combination of asymptotics and numerical simulations using finite-element and/or finite-volume methods. 3D axisymmetric geometries support waves that are not found in 2D and we propose to fully quantify the impact of such axial non-uniformities on the crenulation patterns. Our theoretical objectives are to analyse the flows in the appropriate geometries encountered in the field.
Novel instability mechanisms: The non-uniform 3D axisymmetric flows (that will be calculated during the project) over tapering conical geometries that are undergoing crenulation modulations are expected to support instabilities that have not been addressed previously. The spatiotemporal fluid structure interaction is expected to support new instabilities that will be analysed and compared with geomorphological features. Both linear and nonlinear analyses will be carried out.
Dissolution flows: A related parallel study of dissolution flows is also proposed. Here the flow sculpts the boundary at a rate that depends on the local fluid shear at the wall, leading to a dynamic fluid-structure interaction that is important in determining the global flow behaviour, for example laminar to turbulent transitions. Several mathematical issues remain to be studied, such as corner formation in dissolution flows past solid objects. In the vicinity of such features we propose to use matched asymptotic expansions in the spirit of triple-deck theory in order to provide the correct regularisation of the singularity and evaluate its effect on the global dynamics.
Dr Alex Lukyanov
Air-sea heat exchange, sea ice leads and submesoscale eddies
Because of surface winds and ocean currents, the sea ice cover in the Arctic and Antarctic is continuously pushed and stretched, generating ridges and leads. The latter denotes the gaps and cracks found in the sea ice cover, which expose the “warm” ocean (~-2oC) to the cold atmosphere (-10s of oC). Because of the large air-sea temperature contrast, sensible and latent heat exchanges between the two fluids above leads can be a 100 times larger than heat fluxes between ocean and atmosphere in the presence of sea ice.
These sharp contrasts in heat exchanges can create contrasts in the properties of water masses in the near-surface ocean. The latter can give rise to sharp density gradients which are prone to instabilities, and can become the source of a vigourous turbulence in the upper ocean, the so-called submesoscale eddies.
Submesoscale eddies drive large horizontal exchanges between ice-free and ice-covered ocean. They have relatively fast time scales (hours to days) and length scales of about 1km. While these eddies have been observed and modelled before, they have received little attention comparatively to mesoscale eddies (length scales of hundreds of kms), notably in the context of sea ice covered regions.
The scale of leads ranges from a few meters to kilometres. Climate models we rely on for climate change and decadal predictions (relevant to policy makers) employ in their ocean component, grid scales between 25 and 100 km. Their grid sizes are much larger than the scale of sea ice leads and submesoscale eddies. As a result, climate models ignore submesoscale processes due to the presence of leads, hence also ignoring their impact on the ocean properties and air-sea exchanges.
The overarching aim of the proposed work is to understand and evaluate the impact of submesoscale eddy dynamics on air-sea exchanges in leads. Specifically, we aim to address the following questions: What are the properties (time and space scales, magnitude, etc) of the density contrasts generated in the vicinity of leads? Under which conditions can these become unstable and generate a submesoscale eddy field? How are air-sea exchanges in open ocean modulated by the development of submesoscale eddies in the near-surface layer? Should, and if so how, these processes be included in climate models?
We will first build a mathematical model (heat and freshwater budget) of this situation to develop an understanding of the timescales and magnitudes of the density front development, with guidance from available observations. We will then use this basis to explore the linear instability of the submesoscale eddies (instability conditions, growth rate) and compare with published numerical simulations of developing instabilities, as well as with developed and tested parameterizations of submesoscale eddies in climate models in a mid-latitude context. We will extend the mathematical model by including the submesoscale eddy parameterization and interactive air-sea exchanges. After this the PhD will move on to developing and analyzing non-linear 3-d numerical simulations of the above problem with a state-of-the art General Circulation Model.
Dr David Ferreira
Response Theory in Multiscale GFD Systems
The investigation of the response of GFD systems to perturbations is a key area of research within the mathematics of climate. Problems as different as the prediction of climate change, the understanding of the coupling between different climatic subsystems, the generation of low-frequency atmospheric variability from high-frequency weather noise, and the construction of parametrizations can be addressed using such a point of view . Multiscale systems feature specific mathematical challenges in the construction of response operators, as a result of the presence of slow decay of correlations for some variables, like the ocean in the case of climate.Using techniques from ergodic theory and chaos , we will be looking at the response of a multiscale system of geophysical relevance to perturbations in the forcing. Moreover we intend to investigate from both from the point of view of deterministic  and stochastic dynamics . For example we aim to use Ruelle Response theory  to gain insight in the deterministic case. Additionally, we will look into the properties of multiscale systems. These are topics of great relevance in dynamical systems theory, GFD, and statistical mechanics.
We aim at using the modular MAOOAM climate quasi-geostrophic model , which allows for a great flexibility of configurations and is able to describe both atmospheric and oceanic dynamics in a simplified yet meaningful way. Understanding the geometry of the tangent space is key to the Ruelle formalism and we hope to understand this through the use of covariant Lyapunov vectors . This will allow us to project the response along the stable and unstable directions in the tangent space and so elucidate the applicability of the fluctuation-dissipation theorem. An example of a climatic problem we aim to investigate within this quasi-geostrophic framework would be the coupling between high-frequency, synoptic-scale weather noise and low-frequency, planetary-scale circulation variability . Note this is underpinned by a multiscale asymptotic theory . This project has strong links with the ongoing debate on state dependent response in the context of climate research. Further areas for exploration would be the construction of unstable periodic orbits  for the system and analysis of how they can be used to reconstruct the invariant measure of the system and its response. A third line of investigation deals with the study of the response of the correlations of the fields, rather than observables  and the investigation of the nearing to the tipping points.
Dr Valerio Lucarini
Finding unstable periodic orbits in a chaotic dynamical system using rare events algorithm
Unstable periodic orbits (UPOs), have been proved to be a relevant mathematical tool in the study of Climate Science. In a recent paper Lucarini and Gritsun provided an alternative approach for understanding the properties of the atmosphere. Climate can be interpreted as a non-equilibrium steady state system and, as such, statistical mechanics can provide us with tools for its study. UPOs decomposition plays a relevant role in the study of chaotic dynamical system. There is an intrinsic difficulty in sampling UPOs. Newton-like approaches have been proposed in the literature. The issue with these methods is that they are computationally expensive and do not guarantee convergence. During the PhD we would like to develop a new methodology to sample UPOs.
The idea is to populate with trajectories particular areas of interest of the attractor, reducing computational costs at a minimum. Ragone et al. developed a large deviation algorithm to sample rare events in a climate model. An ensemble simulation is performed, where the trajectories start from independent initial conditions that sample the invariant measure of the model. After a fixed time period the simulation is stopped and a score function is associated to each trajectory, depending on the dynamic up to that point. The trajectories that are going towards the region of interest, according to the score function, are copied, and the ones that perform badly are killed. At this stage, the surviving trajectories are slightly perturbed, so that they can evolve differently, and resampling is iterated another time. We believe we could propose a similar approach for sampling UPOs in a climate model. Positive results of UPOs search algorithms can be found in the literature. In particular, Gritsun and Lucarini first implemented this approach in a geophysical setting. Once computed the UPOs, we would like to use them to reconstruct the invariant measure of the system and study the response of the system to perturbations. A well known result shows that it is possible to evaluate the expectation value of a reference function as an average over the various UPOs, where each one is weighted depending on its instability. The least unstable orbit will have the largest contribution, since it affects more heavily the dynamic of the system. Pollicott and Vytnova provided an alternative decomposition of the average of a reference function in power series, where the coefficient can be computed in terms of periodic points. This decomposition provides a computational more efficient means of approximation. We would like to investigate further this approach on the chosen climate model.
In a first phase of the project the student will familiarise with the mathematical background, working with a simpler model such as Lorenz ’96. The main aim will be understanding and applying known methodologies for finding UPOs. Those techniques will then be extended to the more complex model MAOOAM, which has been found to exhibit chaotic behaviour for some values of the parameters. The challenge will be implementing a multiscale approach on this mode, as we have that ocean dynamics and atmosphere dynamics run on completely different time scale. In the last part of the project, the computed UPOs will be used to reconstruct the invariant measure of the system and study its response to external forcing. The innovation in the project is brought by the idea of finding a connection between rare events algorithms literature and UPOs. As far as we know this has never been done.
Professor Valerio Lucarini
Storyline descriptions of climate variability and change
Predictions of future climate change are usually represented as best estimates with an uncertainty range. However, at the regional scale, changes in atmospheric circulation play a large role and several outcomes may be possible. Under such conditions, an aggregate approach does not provide informative statements about risk. Storylines provide a way to represent the uncertainty in climate change itself, but need to be embedded within a probabilistic framework to account for the uncertainty in the particular realization of climate that will occur.
In this project we build on the candidate’s MRes project which applied a data-driven analysis method to identify wintertime circulation regimes in the Euro-Atlantic sector, as these are known to be a primary determinant of weather and climate extremes over Europe. A novelty of that work, which involved only atmospheric reanalysis data, was the use of spatially unfiltered data rather than EOFs, and the use of a temporal persistence criterion rather than temporally smoothed data. The first step of the PhD project is to build a simple Bayesian model that can be used to predict regime transitions. The model will be developed using atmospheric reanalysis and tested using output from the ECMWF seasonal prediction system, which provides a large data set (roughly 1000 winters) and is considered to be quite realistic.
The aim is to develop a general mathematical framework for a probabilistic treatment of atmospheric circulation regimes as a mediator between slow drivers of climate variability and change, and regional weather and climate extremes. It will thus significantly generalize the deterministic (and linear) storyline approach of Zappa & Shepherd (2017). One direction of research is to couple the circulation regime to a statistical model of particular regional weather and climate extremes, in a manner where the dynamic (circulation) and thermodynamic (temperature) aspects can be separately controlled. Another direction is to use the regime-based framework as a way of factorizing the probabilities of predictions on the subseasonal to seasonal (S2S) timescale, e.g. P(C,T)=P(T|C)P(C) where C is the circulation regime and T the temperature extreme, thereby assessing model skill in a suitably probabilistic manner. Finally, different storylines of climate change could be developed in terms of changes in the loadings of different regimes, rather than simple shifts in the mean, as a way of developing suitable stress tests for climate risk.
In developing the framework based on data analysis, dynamical relationships derived from the equations of motion can be used as expert knowledge. To derive the equations for these dynamical relationships, different methods can be used. Methods that will be considered are asymptotic methods and the Mori-Zwanzig. The Mori-Zwanzig formalism gives a formal rewriting of the system in terms of a Markovian, noise and memory term. When the memory is not short, the memory term can under certain assumptions be interpreted as a delay term, where care needs to be taken when the model is nonlinear. Delay models have been used previously in climate systems, but mainly focussed on the El Nino Southern Oscillation and Energy Balance Models. Because delay models are infinite-dimensional systems, they can potentially convey more information compared to an ODE, while still being formulated in terms of a small number of variables. The presence of a delay in the system can also be derived from data and is a key ingredient when studying causal relationships in climate.
Professor Theodore Gordon Shepherd