With the continuing progress of plasma modelling in recent years the desire arose to develop simplified approaches to the kinetic description of the electron component in weakly ionized plasmas. Methods which are based on the direct solution of the Boltzmann equation in some limiting situations, namely the 'nonlocal' approximation in a weakly collisional plasma and the 'local' approximation in a highly collisional plasma, may be much more efficient than conventional simulation techniques. In this paper the foundation of these approaches is reviewed. Their quantitative accuracy and their applicable range is examined on the basis of a comparison to the numerical solution of the complete, space-dependent Boltzmann equation for a positive column plasma.

For fusion reactors, based on the principle of magnetic confinement, it is important to avoid so-called magnetic islands or tearing modes. They reduce confinement and can be the cause of major disruptions. One class of magnetic islands is that of the perturbation field driven modes. This perturbation field can, for example, be the intrinsic error field. Theoretical work predicts a strong relationship between plasma rotation and the excitation of perturbation field modes. Experimentally, the theory on mode excitation and plasma rotation has been confirmed on several tokamaks. In those experiments, however, the control over the plasma rotation velocity and direction, and over the externally applied perturbation field was limited. In this paper experiments are presented that were carried out at the TEXTOR tokamak. Two tangential neutral beam injectors and a set of helical perturbation coils, called the dynamic ergodic divertor (DED), provide control over both the plasma rotation and the external perturbation field in TEXTOR. This made it possible to set up a series of experiments to test the theory on mode excitation and plasma rotation in detail. The perturbation field induced by the DED not only excites magnetic islands, it also sets up a layer near the plasma boundary where the magnetic field is stochastic. It will be shown that this stochastic field alters both the rotational response of the plasma on the perturbation field and the threshold for mode excitation. It therefore has to be included in an extended theory on mode excitation.

The first results of the Dynamic Ergodic Divertor in TEXTOR, when operating in the m=n 3=1 mode configuration, are presented. The deeply penetrating external magnetic field perturbation of this configuration increases the toroidal plasma rotation. Staying below the excitation threshold for the m=n 2=1 tearing mode, this toroidal rotation is always in the direction of the plasma current, even if the toroidal projection of the rotating magnetic field perturbation is in the opposite direction. The observed toroidal rotation direction is consistent with a radial electric field, generated by an enhanced electron transport in the ergodic layers near the resonances of the perturbation. This is an effect different from theoretical predictions, which assume a direct coupling between rotating perturbation and plasma to be the dominant effect of momentum transfer. Helical magnetic field perturbations are introduced in tokamak plasmas to study, on the one hand, the ergodic divertor concept [1,2] and, on the other hand, the interaction of such perturbations with the magnetohydrodynamics (MHD) stability of the plasma [3,4]. Recent experiments, for instance, suggest a control method to mitigate edge localized modes while maintaining the pedestal pressure and thus plasma confinement [5][6][7]. However, open questions remain, in particular, with regard to the influence on the momentum transport of the plasma. Indeed, one motivation to equip the tokamak TEXTOR with the Dynamic Ergodic Divertor (DED) [8] was to be able to study the interaction between helical magnetic field perturbations and plasma transport and stability.The DED consists of 16 magnetic perturbation coils (four quadruples), plus two additional coils for the compensation of the magnetic field imperfections at the feeder regions of the coils. The coils wind helically around the inner side of the torus (major radius: R 1:75 m; minor radius of the circular plasma cross section typically a 0:47 m) with a pitch corresponding to the magnetic field lines of the magnetic flux surface with a safety factor of q 3. Depending on the choice of coil connections to the power supplies, base modes with different poloidal and toroidal mode numbers can be produced. For the DED these are m=n 12=4, 6=2, and 3=1. The penetration depth into the plasma strongly depends on the mode numbers: While the m=n 12=4 affects the edge plasma only, the m=n 3=1 mode reaches into the plasma center (the maximum radial magnetic field component achievable by the DED at the q 2 surface is 10 ÿ3 of the total magnetic field).In this Letter we present results obtained by the m=n 3=1 mode operation. Covering about one-third of the poloidal cross section of the torus, the mode spectrum of the DED does not contain many sidebands. For the m=n 3=1 configuration the three dominant resonant components inside the plasma are m 1, 2, and 3. In Fig. 1 their strengths at the respective resonances are PRL 94, 015003 (2005) P H Y S I C A L

The ergodization of the magnetic field lines imposed by the dynamic ergodic diverter (DED) in TEXTOR can lead both to confinement improvement and to confinement deterioration. The cases of substantial improvement are in resonant ways related to particular conditions in which magnetic flux tubes starting at the X points of induced islands are connected with the wall. This opening process is connected with a characteristic modification of the heat deposition pattern at the divertor target plate and leads to a substantial increase and steepening of the core plasma density and pressure. The improvement is tentatively attributed to a modification of the electric potential in the plasma carried by the open field lines. The confinement improvement bases on a spontaneous density built up due to the application of the DED and is primarily a particle confinement improvement.

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