University of Edinburgh June 8, 2015

**Fresh theoretical understanding of the behaviour of turbulent plasmas could inform potential applications, from tokamak fusion reactors to new understanding of magnetic fields in cosmology. Researchers at the School of Physics & Astronomy have developed a new mathematical description of the energy flow of a turbulent plasma, and how the loss of energy from a plasma can be controlled.**

The study, led by Prof. Arjun Berera and PhD student Moritz Linkmann using the ARCHER supercomputer, has led to the first simplified formula to quantify these effects in plasmas affected by magnetic fields. The work also offers new insights into energy flows between fluids and magnetic systems, aiding understanding of how magnetic energy can grow at large scales in a plasma.

##### Novel insights into turbulent plasma

Recent work at the School of Physics and Astronomy has added novel insights into how the growth, flow and decay of energy in a turbulent plasma can be controlled by the plasma viscosity, the state of magnetic helicity (internal angular momentum and degree of tangledness of the magnetic field) and the state of cross helicity (correlation between the magnetic field fluctuations and the fluctuations of kinetic energy inherent in a turbulent plasma).

A new formula is obtained for understanding the flow of energy out of, and therefore the energy maintained in, a turbulent plasma, which depends on the state of magnetic and cross helicities contained in the magnetic field-fluid system. These results, obtained by a combination of theoretical work and numerical simulations using the ARCHER supercomputer, show how this energy flow can be controlled, leading to the first simplified formula to quantify these effects in magnetofluids. Understanding has also been obtained in how energy flows between the fluid and the magnetic field, adding new insights on how magnetic energy can grow at large length scales in a turbulent magnetofluid. These theoretical results are fundamental steps towards potential practical applications in areas as varied as controlling the plasma in a tokamak fusion reactor and understanding the presence and growth of magnetic fields in galaxies, galaxy clusters and even at the scale of the entire Universe.

This work has come out in a Physical Review Letter and an earlier Physical Review E Rapid Communication. Both figures below contain results from the two publications, all obtained from medium to high resolution simulations carried out on ARCHER.

Results shown in Fig.1 extend the accuracy and extent of detail from previous results in the literature, while results shown in Fig. 2 had been anticipated in terms of qualitative expectations but were never studied systematically before. Their papers have in turn proposed a new way of looking at the problem and by doing so obtained a simple expression derived from the underlying equations that can predict and explain the behaviour seen in these figures, which is the main significant new advance from this work. Their systematic studies have been made possible to a large extent through access to ARCHER, which enabled them to probe a significant section of parameter space.

It has been known for some time that certain correlations between the velocity and magnetic vector fields alter the dynamics of turbulent magnetofluids. What is new from our work is it predicts with a simple expression how the flow of energy out of a turbulent plasma can be controlled based on the viscosity and state of angular momemtum in the magnetic-fluid system. This is a fundamental step toward potential practical applications in areas as varied as controlling the plasma in a tokamak fusion reactor, understanding the presence and growth of magnetic fields in galaxies, galaxy clusters and even at the scale of the entire Universe.” Arjun Berera

“Much work still needs to be done before quantitative theoretical predictions can be made. Our results are fundamental in the sense that they apply to turbulent magnetofluids far from the boundaries of a containing vessel. Therefore this does not give the full details for specific geometries, such as of a fusion reactor, but our results describe the general behaviour of evolution of the plasma far away from any boundaries, thus are of general applicability to a range of plasmas systems.” Moritz Linkmann

This work has been published in a Physical Review Letter and an earlier Physical Review E Rapid Communication:

“Magnetic helicity and the evolution of decaying magnetohydrodynamic turbulence”

Arjun Berera and Moritz Linkmann

Phys. Rev. E 90, Rapid Communication, 041003(R) – Published 22 October 2014

https://journals.aps.org/pre/abstract/10.1103/PhysRevE.90.041003

“Nonuniversality and Finite Dissipation in Decaying Magnetohydrodynamic Turbulence”

M. F. Linkmann, A. Berera, W. D. McComb, and M. E. McKay

Physical Review Letters 114, 235001 (2015) – Published 11 June 2015

https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.114.235001

**Figure 1**

Decay exponent of magnetic energy as function of degree of turbulence (Reynolds number) of the plasma for maximally helical initial magnetic fields. The inset shows the decay of the ratio Γ(t) of magnetic and kinetic energies, showing that magnetic energy decays slower than kinetic energy.

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**Figure 2**

Decay rate of total energy in the plasma as function of the degree of turbulence, magnetic field strength and electrical conductivity (described by a generalised Reynolds number) within the plasma for different states of cross and magnetic helicity.