Analysis of waves, instabilities and mean flows in MHD systems
Lead Research Organisation:
University of Exeter
Department Name: Mathematics
Abstract
Magnetic fields are ubiquitous in the universe: for our own Sun, fields are generated in the deep interior and as they emerge from the visible surface they create spectacular phenomena such as sunspots and solar flares. The Earth's magnetic field has been a source of mystery for centuries, in particular the knowledge that the North and South poles move in time, and sometimes reverse completely. The field is now known to be generated from deep within the Earth by the motion of liquid metal, and the changes in magnetic field direction are not due to any solid body moving within the Earth, but arise from a reorganisation of the electrical currents associated with the magnetic field. Related to this are attempts to confine a fusion plasma using magnetic fields in the laboratory, for example in the ITER tokomak.
Behind all these processes are the interactions of magnetic field and flow of a conducting fluid, liquid iron in the core of the Earth and ionised hydrogen plasma in the Sun. On these geophysical and astrophysical scales (much bigger than available in the laboratory), magnetic fields taken on a life of their own. Magnetic field lines have elastic properties and waves - Alfven waves - can travel, transferring energy and momentum, create instabilities, and may generate or suppress large-scale fluid, jet-like motions.
The research proposed is to understand the fundamental mathematics behind these processes: rather than numerically modelling the Earth or Sun by including all possible relevant phenomena, the goal in theoretical fluid mechanics is to take problems apart into their simplest components, so as to understand basic processes, parameter regimes and how to model them simply when studying more complex phenomena. We propose a range of projects that involve studying first, magnetic field instabilities - can magnetic fields give rise to growing waves in a fluid system? Secondly, how do instabilities and waves generate mean flows, for example jet-like structures (these are famously seen as the coloured bands on Jupiter, though magnetic fields are not thought to be a key component of the physics in this instance). Third, we plan to understand how one can use mathematics to describe a whole sea of magnetic waves and their effects, using methods from differential geometry (which lies at the base of the theory of general relativity). Finally, all these theoretical questions cannot exist in a vacuum and we propose numerical simulations of carefully designed systems - numerical experiments - to motivate and validate theory, and to extend our understanding to regimes that are not easily accessible to theory.
In this way, we propose to advance the theory, methods of modelling and numerical simulation, for a wide range of geophysical and astrophysical systems.
Behind all these processes are the interactions of magnetic field and flow of a conducting fluid, liquid iron in the core of the Earth and ionised hydrogen plasma in the Sun. On these geophysical and astrophysical scales (much bigger than available in the laboratory), magnetic fields taken on a life of their own. Magnetic field lines have elastic properties and waves - Alfven waves - can travel, transferring energy and momentum, create instabilities, and may generate or suppress large-scale fluid, jet-like motions.
The research proposed is to understand the fundamental mathematics behind these processes: rather than numerically modelling the Earth or Sun by including all possible relevant phenomena, the goal in theoretical fluid mechanics is to take problems apart into their simplest components, so as to understand basic processes, parameter regimes and how to model them simply when studying more complex phenomena. We propose a range of projects that involve studying first, magnetic field instabilities - can magnetic fields give rise to growing waves in a fluid system? Secondly, how do instabilities and waves generate mean flows, for example jet-like structures (these are famously seen as the coloured bands on Jupiter, though magnetic fields are not thought to be a key component of the physics in this instance). Third, we plan to understand how one can use mathematics to describe a whole sea of magnetic waves and their effects, using methods from differential geometry (which lies at the base of the theory of general relativity). Finally, all these theoretical questions cannot exist in a vacuum and we propose numerical simulations of carefully designed systems - numerical experiments - to motivate and validate theory, and to extend our understanding to regimes that are not easily accessible to theory.
In this way, we propose to advance the theory, methods of modelling and numerical simulation, for a wide range of geophysical and astrophysical systems.
Planned Impact
This research project concerns fundamental problems in theoretical fluid mechanics: how can we understand and quantify the way in which magnetic fields interact with waves and mean flows? This is an relevant topic to a wide range of application areas as we indicate below. Because the proposed research is geared towards basic understanding of fundamental phenomena, its impact will be broad and long-term, appropriate to the EPSRC Mathematical Sciences Programme. Impact is in the following areas:
* developing new mathematical tools for the analysis of waves and instabilities, and their interactions with mean flows and magnetic fields,
* modelling space weather, in particular parameterising the effects of a sea of Alfven waves on flows and transport,
* interpreting observations and simulations for the magnetic fields of Earth, Sun and other geophysical and astrophysical systems,
* modelling waves and transport phenomena in fusion plasmas,
* on our understanding of fundamental phenomena in fluid dynamics, relevant to a very broad academic and modelling community.
Impact will be provided via:
* publications in highly respected journals such as the Journal of Fluid Mechanics, Physics of Fluids, Physics of Plasmas. Publications will also be available via Open Access on the University of Exeter repository,
* dissemination by means of seminars and conference / workshop presentations,
* organising a workshop on the mathematical analysis of waves and mean flows, bringing together scientists working in related areas of fluid mechanics with and without magnetic fields,
* Training a PDRA to a high level in analytical and numerical methods relevant to a wide range of applications, for example to modelling weather and climate.
* developing new mathematical tools for the analysis of waves and instabilities, and their interactions with mean flows and magnetic fields,
* modelling space weather, in particular parameterising the effects of a sea of Alfven waves on flows and transport,
* interpreting observations and simulations for the magnetic fields of Earth, Sun and other geophysical and astrophysical systems,
* modelling waves and transport phenomena in fusion plasmas,
* on our understanding of fundamental phenomena in fluid dynamics, relevant to a very broad academic and modelling community.
Impact will be provided via:
* publications in highly respected journals such as the Journal of Fluid Mechanics, Physics of Fluids, Physics of Plasmas. Publications will also be available via Open Access on the University of Exeter repository,
* dissemination by means of seminars and conference / workshop presentations,
* organising a workshop on the mathematical analysis of waves and mean flows, bringing together scientists working in related areas of fluid mechanics with and without magnetic fields,
* Training a PDRA to a high level in analytical and numerical methods relevant to a wide range of applications, for example to modelling weather and climate.
Publications
Algatheem A
(2023)
Zonostrophic instabilities in magnetohydrodynamic Kolmogorov flow
in Geophysical & Astrophysical Fluid Dynamics
Childress S
(2021)
Area waves on a slender vortex revisited
in Fluid Dynamics Research
Gilbert A
(2023)
A Geometric Look at Momentum Flux and Stress in Fluid Mechanics
in Journal of Nonlinear Science
Pramy F
(2022)
A computer-assisted proof of dynamo growth in the stretch-fold-shear map
in Dynamical Systems
Wang C
(2022)
Critical-layer instability of shallow-water magnetohydrodynamic shear flows
in Journal of Fluid Mechanics
Wang C
(2022)
An analytical study of the MHD clamshell instability on a sphere
in Journal of Fluid Mechanics
| Description | The award concerns magnetic and fluid flows - our universe is threaded by active magnetic fields: we are familiar with the Earth's field and some of its effects (for example the aurora borealis) and the Sun's field, reponsible for sunspots and solar flares, yet many aspects of these systems remain poorly understood and challenging. We have analysed key problems concerning types of instabilities that can occur when magnetic fields and fluid flows interact, and shown how so-called critical layers play a key role in driving instability. These are localised regions where waves are very strongly amplified in the fluid flow. We have further provided the first detailed analysis and understanding of the so-called clamshell instability, relevant to solar and stellar magnetic fields. We have also studied some of the mathematical aspects of the equations for magnetic fields and fluid flows using so-called differential geoemetry. This provides not only a deep level of understanding of fundamental properties of these systems, but also provide practical tools for numerical simulations, which is a topic we will plan to pursue in future work. We have also studied non-magnetic fluid systems using new powerful geometrical methods to understand key aspects of how vortices evolve and interact. Finally the problem of the generation of magnetic field by fluid flows - the so-called dynamo problem - remains an important topic for mathematical analysis, and with colleagues (Dr. Mestel. Ms. Pramy) at the Open University, we have studied idealised models for this process, using computer assisted proof techniques, to give a rigorous analysis of growth of magnetic field. |
| Exploitation Route | Our research will be used by others in astrophysics and in applied mathematics - our studies of instabilities are relevant to astronomers and astrophysicists aiming to understand our universe. Our work on the fundamental mathematics of magnetic fields and fluid flows is relevant to the development of modern numerical methods, in areas such as simulations for space weather. |
| Sectors | Education |
| Description | Our findings will have mostly academic impact: in applied mathematics we often study idealised problems which generate understanding that then feeds into many other areas. In this present grant, the applied mathematics links to primarily to astrophysics and astronomy: in particular understanding magnetic fields in the universe including Sun and Earth, and space weather, where the Sun and Earth's magnetic field interact via solar flares and coronal mass ejections. We have provided the first study of the so-called clamshell instability and developed geometric methods for magnetohydrodynamics relevant to numerical simulations. Although the impact in mathematics is quite slow to be taken up, we are confident that the several substantial papers emerging from our work on this grant will have impact in these areas. |
| First Year Of Impact | 2021 |
| Sector | Education |
| Impact Types | Societal |
| Description | Collaboration with Prof. Steve Tobias, University of Leeds. |
| Organisation | University of Leeds |
| Department | School of Mathematics Leeds |
| Country | United Kingdom |
| Sector | Academic/University |
| PI Contribution | We are working together on problems involving MHD waves and instabilities, using analytical and numerical techniques. |
| Collaborator Contribution | Prof. Steve Tobias in Leeds has been involved in project discussions and problem formulation; he has read draft papers and advised on numerical methods. Now that covid is receding he is planning to host a short visit to Leeds from the project PDRF, Dr. Chen Wang. This collaboration continues since our work on a joint paper: Gilbert, A.D., Mason, J. & Tobias, S.M. 2016 Flux expulsion with dynamics. J. Fluid Mech. 791, 568-588. |
| Impact | No outputs yet, at this point in the grant. |
| Start Year | 2014 |
| Description | Critical-layer instabilities in MHD shear flows, invited seminar, Chen Wang, University of Leeds, May 2022 |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | Regional |
| Primary Audience | Professional Practitioners |
| Results and Impact | This was a seminar entitled ``Critical-layer instabilities in MHD shear flows'' given by Dr. Chen Wang at the University of Leeds, given to Professor Steve Tobias and his students. The talk relates to Dr. Chen Wang's research on magnetic critical layers of astrophysical relevance. |
| Year(s) Of Engagement Activity | 2022 |
| Description | Geometric fluid mechanics and generalised Lagrangian mean (GLM) theory, invited seminar, Andrew Gilbert, Faculty of Mathematics, University of Warsaw |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | Seminar entitled ``Geometric fluid mechanics and generalised Lagrangian mean (GLM) theory'' given by Andrew Gilbert to the Faculty of Mathematics, University of Warsaw. This relates to the geometric methods used to study fluid dynamical and magnetohydrodynamical systems developed with Jacques Vanneste, and being further developed under the present grant. |
| Year(s) Of Engagement Activity | 2021 |
| Description | Instability in shallow-water magnetohydrodynamics with magnetic shear, contributed talk, Chen Wang, British Applied Mathematics Colloquium, Edinburgh, 2021 |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | National |
| Primary Audience | Professional Practitioners |
| Results and Impact | This was a talk entitled ``Instability in shallow-water magnetohydrodynamics with magnetic shear '' given by Dr. Chen Wang at the British Applied Mathematics Colloquium hosted in Edinburgh, given to academics, postdoctoral research associates/fellows, and postgraduate students. The talk relates to Dr. Chen Wang's research on MHD instability of geophysical / astrophysical relevance. |
| Year(s) Of Engagement Activity | 2021 |
| URL | https://www.icms.org.uk/workshops/2021/bmc-bamc-2021 |
| Description | MHD flow on a sphere: semi-circle rules and clamshell instability, contributed talk, Chen Wang, UKMHD 2022, Sheffield |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | National |
| Primary Audience | Professional Practitioners |
| Results and Impact | This was a seminar entitled ``MHD flow on a sphere: semi-circle rules and clamshell instability '' given by Dr. Chen Wang at UKMHD annual meeting in Sheffield, given to academics, postdoctoral research associates/fellows, and postgraduate students. The talk relates to Dr. Chen Wang's research on MHD instability of geophysical / astrophysical relevance. |
| Year(s) Of Engagement Activity | 2022 |
| URL | https://ukmhd.sites.sheffield.ac.uk/ |
| Description | Nonlinear dynamics of forced baroclinic critical layers, contributed talk, Chen Wang, IX International Symposium on Stratified Flows. September 2022, Cambridge, UK |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | This was a seminar entitled ``Nonlinear dynamics of forced baroclinic critical layers'' given by Dr. Chen Wang at IX International Symposium on Stratified Flows hosted at Cambridge University, given to academics, postdoctoral research associates/fellows, and postgraduate students. The talk relates to Dr. Chen Wang's research on critical layers in non-magnetic systems of geophysical / astrophysical relevance. |
| Year(s) Of Engagement Activity | 2022 |
| Description | Nonlinear dynamics of forced baroclinic critical layers, invited seminar, Chen Wang, Imperial College London |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | Regional |
| Primary Audience | Professional Practitioners |
| Results and Impact | This was a seminar entitled ``Nonlinear dynamics of forced baroclinic critical layers'' given by Dr. Chen Wang at Imperial College London, given to academics, postdoctoral research associates/fellows, and postgraduate students. The talk relates to Dr. Chen Wang's research on critical layers in non-magnetic systems of geophysical / astrophysical relevance. |
| Year(s) Of Engagement Activity | 2021 |
| URL | https://www.youtube.com/watch?v=x7S0iP_ERW8 |