Nonconservative action principles from mean-flow interactions

Hamilton's principle plays a fundamental role in classical mechanics. Dating back to W.R. Hamilton, the dynamics of a physical system are captured in a single functional, called the action functional S, which by a variational principle yields the equations of motion of the system.
However, there are physical laws which can not be captured by the traditional Hamilton's principle: If the system involves non-conservative components the necessary time-symmetry is broken and Hamilton's principle is not valid. One example of such dynamics is a viscous fluid with heat conduction, which can be described by Navier-Stokes equations.
A recently developed formalism by Galley allows the formulation of a variational principle for non-conservative systems, using only initial boundary conditions by doubling the variables.
We are going to use this novel framework and develop it further, specifically for fluid dynamics.

A central modeling question is how to choose the nonconservative potential function, which describes the nonconservative effects in the system.
Interactions in a fluid systems happen on different scales, some of which are too small to numerically (or practically) capture.
Interactions below or above the captured scale still contribute to the dynamics, but cannot directly be integrated in the model.
Our central idea is to use an averaging method to integrate out small degrees of freedom and obtain an accurate model of the interaction with the mean (larger scale) flow. In particular, we want to use the Generalised Lagrangian Mean (GLM), as it preserves geometrical properties and conserved quantities of the system, opposed to Eulerian averages which are mostly used in practice.

Our main goals for future steps in this project are to (i) look deeper into geometrical aspects arising in the combination of the GLM with the non-conservative action principle, and (ii) apply the GLM to a fluid dynamical system in order to integrate out small-scale variables and obtain an expression for the non-conservative potential.
Our applications are focused on astrophysics, with a focus on turbulent flows which occur in a range of astrophysical phenomena on every scale, such as stars, planetary and black hole discs, and interstellar medium.
Possible applications could be the PT-symmetric system to which the non-conservative formalism for field theories has been succesfully applied by Kevredikis, or non-linear Schrödinger equations to which the formalism has been applied by Rossi et al. Once these steps are made and we gained a deeper understanding of the properties of the combined formalism, we want to move on to more complicated examples of systems which arise in
astrophysical phenomena, in particular turbulent flows.

Our research has direct application to astrophysics and will help understand phenomena arising from turbulent flows.
Moreover, our research has the potential to be beneficial in a broad range of scientific problems due to its theoretical and fundamental nature.

Combining specialised modelling techniques with complex data analysis in order to deliver prediction with quantified uncertainties lies at the heart of many of the major challenges facing UK industry and society over the next decades. Indeed, the recent Government Office for Science report "Computational Modelling, Technological Futures, 2018" specifies putting the UK at the forefront of the data revolution as one of their Grand Challenges.

The beneficiaries of our research portfolio will include a wide range of UK industrial sectors such as the pharmaceutical industry, risk consultancy, telecommunications and advanced materials, as well as government bodies, including the NHS, the Met Office and the Environment Agency.

Examples of current impactful projects pursued by students and in collaboration with stake-holders include:

- Using machine learning techniques to develop automated assessment of psoriatic arthritis from hand X-Rays, freeing up consultants' time (with the NHS).

- Uncertainty quantification for the Neutron Transport Equation improving nuclear reactor safety (co-funded by Wood).

- Optimising the resilience and self-configuration of communication networks with the help of random graph colouring problems (co-funded by BT).

- Risk quantification of failure cascades on oil platforms by using Bayesian networks to improve safety assessment for certification (co-funded by DNV-GL).

- Krylov regularisation in a Bayesian framework for low-resolution Nuclear Magnetic Resonance to assess properties of porous media for real-time exploration (co-funded by Schlumberger).

- Machine learning methods to untangle oceanographic sound data for a variety of goals in including the protection of wildlife in shipping lanes (with the Department of Physics).

Future committed partners for SAMBa 2.0 are: BT, Syngenta, Schlumberger, DNV GL, Wood, ONS, AstraZeneca, Roche, Diamond Light Source, GKN, NHS, NPL, Environment Agency, Novartis, Cytel, Mango, Moogsoft, Willis Towers Watson.

SAMBa's core mission is to train the next generation of academic and industrial researchers with the breadth and depth of skills necessary to address these challenges. SAMBa's most sustained impact will be through the contributions these researchers make over the longer term of their careers. To set the students up with the skills needed to maximise this impact, SAMBa has developed a bespoke training experience in collaboration with industry, at the heart of its activities. Integrative Think Tanks (ITTs) are week-long workshops in which industrial partners present high-level research challenges to students and academics. All participants work collaboratively to formulate mathematical
models and questions that address the challenges. These outputs are meaningful both to the non-academic partner, and as a mechanism for identifying mathematical topics which are suitable for PhD research. Through the co-ownership of collaboratively developed projects, SAMBa has the capacity to lead industry in capitalising on recent advances in mathematics. ITTs occur twice a year and excel in the process of problem distillation and formulation, resulting in an exemplary environment for developing impactful projects.

SAMBa's impact on the student experience will be profound, with training in a broad range of mathematical areas, in team working, in academic-industrial collaborations, and in developing skills in communicating with specialist and generalist audiences about their research. Experience with current SAMBa students has proven that these skills are highly prized: "The SAMBa approach was a great template for setting up a productive, creative and collaborative atmosphere. The commitment of the students in getting involved with unfamiliar areas of research and applying their experience towards producing solutions was very impressive." - Dr Mike Marsh, Space weather researcher, Met Office.