Flow and transport in the edge of fusion devices

The main topic of the Thesis is the study of the electrostatic, plasma response (Er and flows) to magnetic islands embedded in the chaotic edge of a tokamak, when resonant magnetic perturbations (RMPs) are applied. Results are compared with the known phenomenology and theory in the chaotic edge of a reversed-field pinch (RFP). Proxies of the two configurations are the tokamak TEXTOR, with the application of the dynamic ergodic divertor (DED); and the RFX-mod RFP. The main tool used for simulations of islands, two-fluid transport (electrons and ions), and ambipolar Er field, is the Hamiltonian guiding-center code Orbit. As an initial step, to validate the reconstruction of the edge topology of TEXTOR and RFX, the Orbit code has been validated against the volumepreserving code Nemato [24]. In the limit of low energy, Orbit can be used to trace the magnetic field topology, in a way in all respects similar to field line-tracing codes. Nemato is a field-line tracing code, implemented to integrate solenoidal flows for incompressible fluid dynamics, with automatic volume preservation [47]. The question is, how accurate is the description of the magnetic field with Orbit, given that it is a Hamiltonian code (therefore, with a simplectic matrix), but it uses a Runge-Kutta (RK) integrator instead of a fully implicit solver (which is the case of Nemato). Besides this, Orbit describes perturbations in terms of a scalar field , such that ~B = r × ~B0, with B0 the equilibrium field. The two codes are validated on the structure of the q = 0 island chain, which characterizes the multi helicity (MH) configuration in RFP. As input for both codes we use the snapshot of a cylindrical 3D nonlinear, magnetohydrodynamic (MHD) visco-resistive simulation (SpeCyl code [20]). The first benchmarking test employs a Hamiltonian (single-mode) magnetic field configuration. Both codes successfully yield field lines which follow flux surfaces in both the m = 1 and m = 0 cases. The comparison between the codes is then extended to a chaotic magnetic field configuration, by including many modes. The result is that the scalar field representation of Orbit and the RK integrator do not include measurable differences in the Poincar ́e maps and in the calculation of the correlation length of the chaotic field. As a second step in this Thesis, in order to develop a common picture of particle transport with edge magnetic islands in Tokamaks and RFPs, test particle transport simulations are carried out in TEXTOR following the steps of the study performed in RFX-mod [123]. The RMP configurations studied are the m/n = 12/4 and 3/1. The Orbit code has been adapted to the equilibrium of TEXTOR (circular equilibrium with pressure [144]), and a proper form for the eigenfunctions has been developed, on the basis of the analytical formula used in TEXTOR using as input the current flowing in the DED. The resulting Poincar ́e plots show the well-known, basic features of TEXTOR stochastic edge, such as the inner island chain, the remnant islands, and the laminar flux tubes embedded in the ergodic fingers, which is consistent with previous works on this subject. Maps in the (r, ) plane of the electron and ion parallel connection length to the wall, Lk(r, ), highlight the properties of the magnetic structures observed in the Poincar ́e plot: while ions, having a large Larmor radius, are weakly affected by the magnetic topology, electron trajectories are linked to the magnetic field lines. The behavior of Lk entails a characteristic modulation of the radial electric field Er with large positive values in the zone with electron Lk 0 (the so-called laminar flux tubes, which occupy a region in between the main island chain and the remnant islands). As a further step, the evaluation of the local radial transport of particles, i.e. ion and electron diffusion coefficients (Di and De, respectively), has been performed along a helical path from the O-point (OP) through the X-point (XP) of an m/n = 4/1 remnant island. The result shows that Di is rather constant along the path, and it is almost neoclassical, while De is larger (4 ÷ 40 m2/s), and is strongly modulated (larger at the XP, lower at the OP), consistently with the Lk maps. Finally, an analytic 3D formulation of the ambipolar potential for the 3/1 DED configuration is developed on the basis of the geometry of the m/n = 4/1 magnetic island, that balances electron and ion fluxes inside the island. The result is compared to measurements of plasma potential inside an m/n = 4/1 island in the edge of the TEXTOR device and with the analysis on RFX-mod edge. In RFX-mod Orbit predicts the potential well to stay in proximity of the OP of the main island (m/n = 0/1), while measurements show the potential well near the XP. In the TEXTOR experiment fast Mirnov probe measurements show that the potential well corresponds to the XP of the m/n = 4/1 island, i.e. the region with larger De, consistently with Orbit results. The difference between RFX and TEXTOR could be ascribed to a collisional dependence (the case of RFX is highly collisional, contrary to TEXTOR); to a different level of chaos, in RFX compared to TEXTOR; or to a more pronounced plasma-wall interaction. The final, main message of the Thesis is that RMPs in tokamaks, even if induced as static perturbation, are capable of producing a pattern of large, radial electric field Er, which is the footprint of the underlying topology. The pattern of Er can be successfully reproduced by Orbit. The comparison with RFX shows that collisional effects can be important in determining amplitude and phase of this electrostatic potential.

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