Stability of Tokamak and RFP Plasmas with an Extended Region of Low Magnetic Shear

It has been observed experimentally that magnetically confined plasmas, characterised by the safety factor q with a small or slightly inverted magnetic shear, have good confinement properties. Such plasmas typically have no internal transport barrier, operate with q95 around 4 and are good candidates for long pulse operation at high fusion yield in the reactor ITER. These hybrid scenarios are an intermediate step between the reference standard H-mode (high confinement) scenario with monotonic q and inductive current, and advanced scenarios with strongly reversed magnetic shear in which the entire plasma current is ideally generated non-inductively. This thesis focuses on the study of the dynamics of hybrid plasmas, with weak or almost zero magnetic shear, in tokamak and Reversed Field Pinch (RFP) configurations, when q in the central region assumes values close to one (tokamaks) or to a rational number (tokamaks, RFPs), though the exact resonance is avoided. The first part of this thesis is focused on the study of tokamak and RFP equilibria with slightly reversed shear when an extremum in the safety factor is close to a low order rational. These equilibria are characterised by the possible presence of internal helical cores, although the plasma edge is symmetric in the toroidal direction. Such 3D structures can be understood as the result of the nonlinear saturation of ideal MHD modes. The amplitude of large scale m=1 helical displacements in tokamak and RFP plasmas is investigated using contrasting approaches, namely 3D equilibrium and non-linear stability codes. The non-linear amplitude of such saturated modes obtained with the stability code is compared both with the helical core structure resulting from equilibrium numerical calculations, and with analytic predictions which extend the nonlinear treatment of reversed q plasmas to arbitrary toroidal mode numbers. A preliminary study of the impact of a n=1 RMP on the equilibrium helical distorsion is also presented. The second part of the thesis is devoted to the analytical and numerical study of the stability of an initially axisymetric tokamak configuration when the safety factor is almost flat and very close to a rational value over a macroscopically extended region in the plasma centre. Such conditions typically occur either in hybrid scenarios or following reconnection of a global instability such as a sawtooth. This configuration is characterised by non-negligible coupling between a fundamental mode and its Fourier adjacent modes. A dispersion relation has been derived both for ideal and resistive modes, with additional non-MHD effects such as plasma diamagnetism, viscosity and equilibrium velocity flows. The analytical results show that the resistive sidebands coupled to a core kink-like mode exhibit extremely fast growth, though additional non-MHD effects tend to moderately reduce the extreme growth rate of the resistive modes. The existence of such modes has been confirmed numerically, where the sensitivity of the growth rate to changes in resistivity and two-fluid effects has been demonstrated, and thus in turn provides generally good agreement with the analytical theory developed. A family of modes are obtained, including modes with novel scaling against plasma resistivity, some of which rotate in the electron diamagnetic direction, and others in the ion diamagnetic direction, consistent with experimental observations in e.g. TCV during hybrid-like operation.