Abstract

Nuclear fusion promises an idealistic future of clean and essentially limitless energy production. By heating Hydrogen gas to enormous temperatures such that the kinetic energy of the nuclei is enough to overcome Coulomb repulsion, nuclei are fused together releasing energy. At such a large temperature, the gas is fully ionised becoming a plasma, which is manipulated with magnetic fields to maintain the extreme conditions required for fusion.

Keeping the plasma at such extreme temperatures for long enough to produce energy remains the largest challenge for fusion energy. The most promising candidate is the toroidal confinement device, the tokamak. While showing many desired features, the plasma inside a tokamak is still subject to instabilities.

A common plasma instability is the Shear Alfv´en wave (SAW), analogous to a wave on a string, SAWs travel along the magnetic field with a velocity similar to the velocity of the energetic particles released in the fusion reaction. This can lead to resonance between the particles and the wave, resulting in exponential growth of the wave amplitude. An uncontrolled, exponentially growing wave is clearly a problem for the delicate stability of the plasma, and hence determining the spectrum of SAWs is of crucial importance.

The spectrum is typically split into two types of SAWs. The first and most common are continuum solutions, which are highly localised, with a frequency that depends on the local plasma parameters creating a continuous spectrum. The second, and much more dangerous, are discrete or global solutions, which have a much broader structure. 

When continuum solutions are excited by energetic particles, the energy is dispersed between neighbouring solutions, which typically overcomes the growth. However, discrete solutions can exist inside gaps in the frequency spectrum, so do not experience this damping, and therefore pose a serious threat to future experiments.

An example of a typical spectrum is shown in figure 1a, most solutions are part of the continuum which varies smoothly with r due to the smooth variation in plasma parameters. We can also see an example of a discrete solution, the toroidal Alfv´en eigenmode (TAE), which exists outside the continuum. Figure 1b shows an example of the typical continuum mode structure, highlighting the sharp peak and localised structure. In contrast, figure 1c shows the TAE, with a much broader structure.

In idealised, theoretical scenarios, such as the one used to construct figure 1,

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Figure 1: Typical example of the shear Alfv´en wave spectrum in tokamak plasmas.

we construct coordinates that align with the magnetic field structure. Because SAWs propagate along the magentic field and can be highlighly localised, these coordinates drastically simplify their dynamics, allowing reasonably straightforward computation of the SAW spectrum.

These coordinates rely on the symmetry present in an ideal toroidal device. For more realistic fusion devices, the practical constraints of construction, perturbations and sometimes deliberate efforts can break this symmetry. In some cases, there is enough symmetry present that magnetic coordinates can still be created, however, they often have limitations and cannot be used in general scenarios. In other cases, such as when the magnetic field becomes chaotic, magnetic coordinates are not constructable and computing the SAW spectrum becomes numerically infeasible.

For such cases, instead of using magnetic coordinates, we can take advantage of the residual structure in chaotic systems to construct pseudo magnetic coordinates. An example is shown in figure 2, the left figure shows an example of a system with a chaotic magnetic field. Overlaid on this chaotic field are the surfaces that cling to the residual structures that we use to create a coordinate system. In the right figure, the same magnetic field is shown in the new pseudo magnetic coordinates showing the recovery of some of the structure seen in simpler theoretical cases.

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Figure 2: Example of a chaotic magnetic field and the use of pseudo magnetic coordinates to simplify the structure..

With these coordinates, we numerically solve the reduced Magneto-Hydrodynamic (MHD) equations in 3 dimensions using finite elements. We show that many of the properties seen in the spectrum of a symmetric case still persist when symmetry is broken and also show examples where this is not true. This makes steady progress towards understanding how symmetry breaking perturbations effect the shear Alfv´en spectrum in fusion plasmas.

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