# Non-diffusive transport

It has long been known that the standard model for transport in magnetically confined plasmas (Neoclassical transport) often fails to provide an accurate description of experimental results: it tends to underestimate transport by one order of magnitude or more, typically; the non-Neoclassical component of transport is called "anomalous". This is a very disappointing situation with a view to constructing a fusion reactor, since worse confinement means that an eventual reactor will need to be bigger and more expensive. Therefore, the search for the cause of this failure (and for methods to restore transport to its Neoclassical value) is one of the main issues of fusion research.

The standard Neoclassical model is a collisional (diffusive) model, which means that transport is characterised by typical scale lengths, both for space and time, so that the effective diffusion coefficient is essentially the mixing length value: $D = \Delta r^2 / \Delta t$, where $\Delta r$ is the typical step size and $\Delta t$ the typical waiting time.

In recent years, it has been suggested that the plasma may contain phenomena that invalidate this picture. There may turbulent eddies in which particles become trapped for some time, and there certainly are transport barriers, associated with rational magnetic surfaces, and stochastic regions of the magnetic field. [1] This could cause the waiting time distribution to become non-exponential; and thus the motion would be non-Markovian. Likewise, the phenomenon of streamers, appearing in many models of plasma turbulence, could carry particles across long distances in the radial direction, and the distribution of particle steps could then also be deformed and develop long tails. Consequently, the transport would then be non-local. Nobody knows exactly how important these phenomena are in the global transport picture.

Whatever the case, a well-established methodology exists to describe this deviation from standard diffusive transport (with characteristic scales): the Continuous Time Random Walk (CTRW) model. [2] The CTRW model provides a mathematical framework for handling non-diffusive transport (arising as a generalisation of the diffusive transport when eliminating the stated characteristic scales), but it does not explain why such non-diffusive transport should arise: answering the latter requires detailed computer simulations of turbulence and experimental observations.

However, even without fully understanding the origin of the non-diffusive behaviour, it is possible to construct models based on these ideas, and see whether these models fare better in predicting the global transport properties of plasmas than the standard diffusive models. [3] [4] Note that there is another ingredient that may be essential to explain deviations from the standard transport model: self-organisation; we will not discuss this here.

Another approach is to test whether non-diffusive transport phenomena actually occur in simulations and experiment. To do so, tracer particles are injected into the plasma fluid, and their evolution is followed in time. The tracer trajectories can be analyzed by means of several analysis techniques, e.g. by calculating the particle distribution probability function, or by detecting velocity correlations along trajectories. The application of this method has yielded clear indications that plasma turbulence induces non-diffusive transport in simulations. [5] [6] [7] [8] [9] No significant data are as yet available from actual experiments, due to the considerable experimental difficulty of injecting and following tracer particles, although some experiments in this sense are planned.

## References

1. J.H. Misguich at el., Plasma Phys. Controlled Fusion, 44, L29 (2002)
2. R. Balescu, Aspects of Anomalous Transport in Plasmas, Institute of Physics Pub., Bristol and Philadelphia, 2005, ISBN 9780750310307
3. B.Ph. van Milligen, R. Sánchez, and B.A. Carreras, Phys. Plasmas 11, 2272 (2004)
4. B.Ph. van Milligen, B.A. Carreras, and R. Sánchez, Phys. Plasmas 11, 3787 (2004)
5. B. Carreras, V. Lynch, and G. Zaslavsky, Phys. Plasmas 8, 5096 (2001)
6. L. García and B. Carreras, Phys. Plasmas 13, 022310 (2006)
7. D. del Castillo-Negrete, B. Carreras, and V. Lynch, Phys. Plasmas 11, 3854 (2004)
8. J. Mier, R. Sánchez, L. García, D. Newman, and B. Carreras, Phys. Plasmas 15, 112301 (2008)
9. G. Sánchez Burillo, B.Ph. van Milligen, A. Thyagaraja, Phys. Plasmas 16, 042319 (2009)