# Transport Modelling in Nuclear Fusion

# 1 - Background

To make predictions of confinement in devices such as, for example, ITER one normally uses one of the physics-based transport models available to the fusion community. Such models, usually based on a simple quasi-linear approach to plasma turbulence, provide an estimation of turbulence driven fluxes. These models include among others the Multi-mode Model developed in Lehigh University (US) and Gyro-Landau-Fluid (GLF) model developed at General Atomics (also US); they are based on electrostatic turbulence such as Ion Temperature Gradient (ITG) driven turbulence, Trapped Electron Mode (TEM) and Electron Temperature Gradient (ETG) driven turbulence. However, it is important to benchmark them using well-diagnosed experimental profile data. So the validation of the theoretical transport models for thermal energy and particles on the database of the different tokamaks must be investigated. In particular, they must be extensively tested on high confinement discharges (H-mode) and hybrid scenarios in various tokamaks. The hybrid scenario is a new and attractive operational regime where the MHD activity, typical in many H-mode discharges, is avoided by proper shaping of the current density profile. This regime is being explored on JET and considered as a reference operational regime for ITER.

# 2 - International Profile Database

The international, multi-tokamak, confinement profile database, hereafter referred to as PDB, has been developed by the world-wide fusion community to provide a convenient facility for testing transport models across a wide variety of tokamaks and operational regimes. An earlier version was released to the public in 2000. The web-based database contains the well-diagnosed plasma profiles and heating sources needed for meaningful model testing. It falls under the auspices of the Confinement and Database Topical Group of the International Tokamak Physics Activity (ITPA) and has been hosted and managed, scientifically and technically, at Culham since July 2001. In parallel the Transport Topical Group collected a profile database so as to compare internal transport barriers (ITBs) on different machines, and it was natural in the main to adopt the structures of the original PDB. Some global variables were added so as to characterise some of the specific properties of discharges with transport barriers. The addition of new variables and independent management led to some divergence from the PDB. It was agreed that the ITB profile database should join the PDB at Culham, where efforts to merge the frameworks together could be undertaken.

The strategy to merge the database structures followed the following basic steps:

- Store ITB data in the same structures as PDB;
- Specify a common set of variables for both ITB and PDB databases;
- Ensure full compatibility between submitted discharges and the common set of variables.

Steps 1 and 2 have been achieved, driven by Culham but with substantial contributions from key users and data-providers , but step 3 proved much more difficult. It was essential to ask data providers for some rather basic information to improve the data which had already been submitted to PDB and Culham effort was directed to make this task as simple as possible, but some cases required resubmissions by the data-providers delaying the process. At present, of 306 discharges in the original PDB and ITB databases, 216 comply with the new combined database framework. On satisfactory completion of the above steps the data could, at least in principle, allow public release of a combined database, providing a further valuable facility for testing models for predicting the performance of devices such as ITER, that would be accessible to modellers outside the ITPA groups.

# 3 - Collisional Transport

Magnetic confinement can be understood from either the particle or fluid pictures of plasmas. In either case, non-ideal effects will tend to spoil confinement. Collisions cause particles to diffuse across the magnetic field by a random walk process. In the MHD model, the equivalent process is described by the diffusion of a conductive fluid through a magnetic field. Note that in ideal MHD, where resistivity is assumed to go to zero, the diffusion rate also goes to zero; the magnetic field is said to be frozen into the fluid. Taking Ohm’s law and crossing it with B gives

Combining this with the force balance equation and solving for perpendicular velocity gives

The first term describes plasma diffusion, while the second term is the fluid equivalent to the E × B particle drift. A little algebra will show, not surprisingly, the diffusion rate for the fluid is the same as that derived above for particles. For a typical fusion plasma, this classical diffusion coefficient is on the order 0.001 m2/sec, far too small to present any problem for magnetic confinement.

The collisional diffusion coefficient, just calculated, is correct only for a plasma confined by a magnetic field that is straight and uniform. In a torus, diffusion is significantly enhanced by the particle drifts. The rotational transform forces the drifts to cancel when averaged over complete orbits, however, collisions can disrupt the orbit and cause the cancellation to be incomplete. The theory for collisional diffusion in axisymmetric toroidal geometry is called neoclassical transport and has been extensively developed for tokamaks. Consider a particle with its parallel motion in the same direction as the plasma current, which follows the field lines in a right hand spiral. In the poloidal cross section, such a particle will travel in a clockwise circle. The drift will move such a particle off its flux surface, toward the plasma center while it is near the bottom of its orbit and away from the center while it is near the top. The maximum displacements occur when the particle is on the horizontal midplane, resulting in an orbit that is displaced outward, away from the torus center. This result holds whether the toroidal field and plasma current are in the same or in opposite directions. Ions circulating opposite to the plasma current will always be shifted inwards.

For non-axisymmetric plasmas, collisional diffusion can be larger. In an axisymmetric system, particles are governed by conservation of canonical angular momentum. As long as the particle’s kinetic momentum is not too large, they are constrained to stay close to flux surfaces. (This is, in fact an alternative picture for describing particle orbits in a torus.) This constraint is absent if the system lacks axisymmetry where certain classes of particles can make large radial excursions or leave the plasma entirely without undergoing any collisions. The loss rate for the plasma as a whole is then governed by the rate at which the hole in the distribution function is filled in. These effects are most important in intermediate collisionality regimes, where particles on lost orbits can travel significant distances without colliding, but where collisions are still able to fill in the losses from the background plasma. For stellarators, whose helical field breaks the axisymmetry, these losses can dominate transport; modern stellarators are carefully designed to minimize their magnitude. The fields of nominally axisymmetric devices, like a tokamak, have a small asymmetry due to the finite number of toroidal coils. This periodic ripple can cause significant loss of energetic particles, particularly those with mostly perpendicular energy, since parallel motion tends to average out the asymmetries.