New methods to study the phenomenon of turbulence

Many processes in nature can be described by differential equations, but only a few of them can be solved explicitly with solutions in formulas. This is the motivation for developing numerical equations to find approximate solutions. The numerical equations developed in Michael’s thesis have a particular focus on geometric properties. Though the thesis is mathematical, the problems it addresses originate in physics and mainly have to do with magnetohydrodynamic (MHD) turbulence.

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– It is difficult to define turbulence rigorously. Intuitively you can think of the turbulent behaviour when a fluid moves, but it is very hard to predict how it will behave in the future. It looks chaotic though there is no randomness in the models of motion.

Space is full of plasma

MHD turbulence occurs in fluids whose particles are electrically charged. Such fluids are called plasmas and are rarely encountered on Earth, but in space – actually, 99% of the observable universe consists of plasma. From a mathematical perspective, equations that are used to model plasmas have a lot of conservation laws and a generic method, for example forward or backward Euler method, is not consistent with the conservation properties. 

In the thesis, the methods that are developed are consistent with conservation properties, meaning that they preserve everything that is preserved by exact equations. This is done in two steps: first the equations are discretized in space, using an approach called matrix hydrodynamics. The key idea is to replace unknown functions with matrices and Poisson brackets with a matrix commutator. Then the matrix equations are discretized in time, using the discrete Lie–Poisson reduction. This results in a numerical scheme that is fully compatible with the geometric structures of the original equations. In the later articles of the thesis, these new methods are applied to study the dynamics and the long-time behaviour of solutions.

Geometry brings beauty

Michael has been interested in mathematics since early schooldays. His master’s degree was in Physics, but the project was about geometric PDE theory, so rather mathematical. He realised that he wanted to do a PhD and started to look for positions. A former advisor had been a professor in Norway and had spoken well about the Nordic countries, so Michael was glad to accept the offer from Chalmers. He has found it exciting to work on the project and what he has liked most is that the methods that have been used all comes with a geometric flavour – it is the geometry that brings beauty into the methods and the research area, Michael thinks.

– When I came here, everything was new and unusual, it was like being on another planet. It is a huge blessing to be able to build a career in a free and open society. I also enjoy the Swedish nature very much. The Swedes seem to be very close to nature.

Michael’s position will end in August, and he will then continue his career as a postdoc. He will not travel far, however: just across the courtyard, to the Plasma Theory research group at the Department of Physics!