Neo-classical Tearing Modes

Tokamak plasmas are toroidally symmetric: magnetic field lines wind around toroidal magnetic flux surfaces to create a system with good confinement properties. In this state, all field lines on a given magnetic flux surface carry the same current. However, under certain situations the current flows in filaments: some field lines carry more current than others. This causes corrugations in the flux surfaces resulting from magnetic islands. Magnetic field lines follow these corrugations, rapidly transporting heat and particles across that region. The result is that the pressure gradient is removed from inside the island, leading to a loss in the central pressure which, for a fusion power plant, would mean a reduction in fusion energy. Our work in this field concentrates on two main areas of research: experimental measurements of the plasma pressure distribution in the vicinity of a magnetic island, and theory and computational modelling of neoclassical tearing modes.

Temperature and density measurements near an island

In a £2M collaborative project with CCFE, we are significantly upgrading the Thomson Scattering diagnostic on the MAST tokamak (sited at Culham Science Centre in Oxfordshire). This equipment measures the electron temperature and density with high accuracy and high resolution inside the plasma. The project is important because theory predicts that whether a magnetic island grows, further degrading confinement, or shrinks depends on whether the density and temperature gradients are removed from inside the island. This is a key question for ITER.

The physics is the following. For very small magnetic islands, the diffusion across the magnetic flux surfaces dominates over that along the magnetic field lines. In this situation, a pressure gradient is maintained across a magnetic island, in which case the island usually decays away again and there is no harm done. However, if the island is somewhat larger than a threshold width, which is typically around 1cm, then the transport along magnetic field lines is dominant and the pressure gradient is removed from inside the island. Now, there is a current that flows along magnetic field lines, called the bootstrap current. This bootstrap current is proportional to the pressure gradient, and is therefore removed from within these larger magnetic islands. The resulting loss of current magnifies the island, removing more pressure gradient, further enhancing the loss of bootstrap current and increasing the island size still further. The resulting magnetic island, which is called a neoclassical tearing mode, can be huge: several 10’s of centimetres, leading to a major loss of confinement or even a loss of control and termination of the discharge in a disruption.

The aim of this project is actually two-fold: (1) we wish to see if the observed threshold for island growth is indeed due to this reduction of pressure gradient inside the island, and (2) with these measurements we are able to probe the transport properties inside a tokamak and, in particular, measure the ratio of transport perpendicular and parallel to magnetic field lines both for particles and heat, and compare the two. Theory predicts they should be different, but the difference will depend on the turbulent processes which are thought to dominate the cross-field transport.

Variation with temperature across the plasma.
The variation of temperature across the plasma, measured with the ruby Thomson Scattering system on the MAST tokamak at Culham. The vertical lines denote where the safety factor q=2 and mark the position of the neoclassical tearing mode. Note that the temperature gradient is zero here, which is particularly evident on the inner edge.

Theoretical modelling of neo-classical tearing modes

In a long-term project, supported by EPSRC, we are developing a theory to understand the threshold mechanism for neoclassical tearing modes. There are two primary models that we are exploring. The first is to look in more detail at the threshold predicted by a balance between transport parallel and perpendicular to magnetic field lines (see above). In particular, we are calculating the self-consistent flows and modifications to the pressure gradient around a magnetic island, and assessing the implications for stability to the ion temperature gradient (ITG) mode, which is an instability widely believed to be the cause of turbulent cross-field diffusion in a tokamak. We find, not surprisingly, that the island has a stabilising influence on the mode. In addition, we find that it significantly modifies its mode structure, tending to localise it somewhere in the vicinity of the X-point (but shifted slightly). These results show that the island itself can influence the characteristics of the turbulence and small magnetic islands may even be beneficial for suppressing turbulent transport losses. We have developed both a WKB theory, and a full 2-D solution for the ITG stability and mode structure, and are in the process of comparing the two.

In the second approach, we are exploring a different mechanism for the threshold: ion polarisation current. This arises when the magnetic island moves through the plasma. The ions and electrons oscillate up and down as the island flows through them but, because of the greater inertia of the heavy ions, they do not oscillate together. The result is a current called the polarisation current. This polarisation current is only important for small magnetic islands with a width around 1cm. However, the theory is complicated and there remains uncertainty regarding whether the polarisation current is stabilising or destabilising. Our theoretical studies aim to address this important question. One area of study, in collaboration with CCFE and the University of Bristol, is to explore the impact of cross-(magnetic) field diffusion on the distribution of electrons and ions. This provides a prediction of the electric field near to the island. The modifications to the electric field caused by the cross-field diffusion are expected to have a significant impact on the polarisation current: results are expected soon. In parallel with this, we are also exploring the effect of toroidal geometry on the polarisation current. In particular, we are calculating how the polarisation current depends on the plasma collisionality. The results of this will feed into a project to build a new computer code which will include as many of the physics mechanisms described in this section as possible. This state of the art code is a major undertaking, supported by EPSRC, and the resulting simulations are likely to require high performance parallel computers.