Transport Properties of Incompressible Field-Guided MHD Turbulence

As a flow moves, it carries fluid from one location to another. In this way, a tiny parcel of fluid gradually wanders away from its initial position. In the fluid dynamics laboratory, this can be visualised by injecting dye with a syringe at a specified point in the flow. If dye is continuously released then one can trace the dye's trajectory. Instead, if at a specific point in time a small patch of fluid is dyed, then one can then study the rate of spreading of the patch. The transport of a passive contaminant in a turbulent (disordered) flow is a topic that is widely studied in fluid dynamics research and the topic can have great relevance to our everyday lives, especially when the contaminant represents an industrial pollutant dispersing in the Earth's atmosphere, viruses that can pose threats to our immune systems, or volcanic ash that can disrupt flight schedules for weeks. In magnetohydrodynamic (MHD) turbulence, transport studies are concerned with electrically conducting flows that interact with magnetic fields. Gaining knowledge of the fundamental properties of such flows is necessary in order to understand how geomagnetic storms behave (these can disrupt communication and navigation satellites and cause black outs in power grids) and how instabilities develop in laboratory plasma experiments studying magnetic confinement fusion for clean energy production purposes.

Progress with understanding such complicated physical systems relies, first and foremost, on establishing a solid theoretical foundation for more simplified mathematical models. While there are a number of physical situations in which it is important to be able to understand turbulent transport in magnetised plasmas, it is also the case that applied mathematicians can learn a great deal about the fundamental dynamics and structure of an electrically conducting flow by studying its transport properties. It is this particular aspect of turbulent transport that motivates our proposed work.

Over recent years, significant progress has been made with the fundamental theory of MHD turbulence. The success is largely a result of a massive increase in computational power that has enabled a series of high-resolution numerical simulations to be performed. The numerical results have been used to test competing theoretical predictions and the findings have spawned many new avenues of research. Of particular interest is the discovery of the intricate highly-aligned structure that field-guided MHD turbulence takes. Herein we propose to further our investigations into this intriguing structure and its effects. Through a series of high-resolution numerical simulations and theoretical studies of MHD turbulence, we will study the efficiency of transport by monitoring the trajectories of tracer particles. We anticipate that the our results will provide important information for developing a comprehensive phenomenological model of strong field-guided MHD turbulence, for designing future numerical simulations of plasma turbulence, and ultimately for interpreting observations and experiments.

The proposed work aims to contribute to the development of a strong theoretical foundation for incompressible, field-guided MHD turbulence, this being the simplest possible mathematical framework for describing magnetised plasma turbulence.
The proposed topic is of fundamental importance in fluid dynamics, and the approach that we take is the typical applied mathematics approach. A complicated physical system is broken down into a series of simplified components. We seek to develop a strong theoretical foundation for the simplest component first, before we progressively add more complicated physics. In this way, while the impact of the individual components may be difficult to quantify when they are considered in isolation, understanding each individually is crucial for the development of the whole, and collectively the ultimate impact will be great.

In the above-mentioned manner, the work has the potential to impact a wide variety of application areas. This includes a diverse range of phenomena that occur in the presence of astrophysical turbulence, in addition to turbulent processes operating in some laboratory plasma physics experiments. For example, turbulence in the solar convection zone is believed to play a crucial role in the generation of the large-scale, ordered magnetic fields that eventually appear at the solar surface in the form of sunspots. Understanding how the large-scale solar dynamo operates remains one of the most important unsolved problems in astrophysical fluid dynamics. Such solar magnetic activity spawns violent events such as coronal mass ejections and solar flares, which propagate through interplanetary space and can significantly disrupt the magnetic environment of the Earth, leading to black outs in power grids and causing disruptions to communication and navigation satellites. Understanding the coronal heating problem (why the solar corona is much hotter than the visible surface of the Sun) also requires an understanding of magnetised turbulence, while turbulent mixing also affects the chemistry of the interstellar medium and plays a role in star formulation in molecular clouds. Furthermore, understanding field-guided MHD is crucially important to laboratory plasma physicists studying magnetic confinement fusion.

It is clear that a complete understanding of these very complicated physical systems requires the generation and amalgamation of a great deal of knowledge across a number of different research disciplines (applied mathematics, astrophysics, plasma physics, theory, observations and high-performance computing). While this will take place over decadal time frames, being able to establish a strong theoretical foundation for incompressible field-guided MHD turbulence forms one of first steps in this process. With fusion energy likely to become the energy resource of the future, and with the significant efforts currently underway to understand solar variability and space weather, it is clear that the knowledge gained from this study has the potential to have considerable impact on society and the economy over the long term.

During the two year time frame of the proposed study, the most direct impact of the work on society and the economy would be through communication and engagement activities that would be aimed at widening participation in maths and science education, and raising the public's awareness of the importance of maths and science research. Through the use of the UK's high-performance computing facilities, the work would also contribute to economic prosperity by promoting the development of the UK's technological base. The project will generate a large amount of numerical data that will be used in the undergraduate mathematics curriculum at the University of Exeter. The results will therefore contribute to the training of skilled persons for the non-academic careers in which logical argument, physical reasoning and technological proficiency are required.