Analysis of waves, instabilities and mean flows in MHD systems

磁场在宇宙中无处不在：对于我们自己的太阳来说，磁场是在内部深处产生的，当它们从可见的表面出现时，就会产生太阳黑子和太阳耀斑等壮观的现象。几个世纪以来，地球的磁场一直是一个神秘的来源，特别是关于南北两极随时间移动，有时完全相反的知识。现在我们知道，磁场是由地球深处液态金属的运动产生的，磁场方向的变化不是由于地球内部的固体运动，而是由于与磁场有关的电流的重组而产生的。与此相关的是试图在实验室中使用磁场限制聚变等离子体，例如在ITER托科马克中。在所有这些过程的背后是磁场和导电流体流动的相互作用，地球核心的液态铁和太阳中的电离氢等离子体。在这些地球物理和天体物理的尺度上（比实验室中可用的要大得多），磁场有了自己的生命。磁力线具有弹性特性，而波——阿尔芬波——可以传播，传递能量和动量，产生不稳定性，并可能产生或抑制大规模的流体射流运动。提出的研究是了解这些过程背后的基本数学：而不是通过包括所有可能的相关现象对地球或太阳进行数值模拟，理论流体力学的目标是将问题分解成最简单的组成部分，以便了解基本过程，参数制度以及如何在研究更复杂的现象时简单地建模。我们提出了一系列涉及研究的项目，首先是磁场的不稳定性——磁场能否在流体系统中产生不断增长的波？其次，不稳定性和波动是如何产生平均流的，例如射流状结构（这些结构被认为是木星上的彩色带，尽管磁场不被认为是这种情况下物理的关键组成部分）。第三，我们计划了解如何使用微分几何（这是广义相对论的基础）的方法，用数学来描述整个电磁波的海洋及其效应。最后，所有这些理论问题都不可能存在于真空中，我们提出了精心设计的系统的数值模拟-数值实验-来激励和验证理论，并将我们的理解扩展到理论不容易接近的制度。在此基础上，我们提出了对广泛的地球物理和天体物理系统进行理论、建模和数值模拟的方法。

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.