An analytical theory of the nonlocal magnetorotational instability (MRI) is developed for the simplest astrophysical plasma model. It is assumed that the rotation frequency profile has a steplike character, so that there are two regions in which it has constant different values, separated by a narrow transition layer. The surface wave approach is employed to investigate the MRI in this configuration. It is shown that the main regularities of the nonlocal MRI are similar to those of the local instability and that driving the nonaxisymmetric MRI is less effective than the axisymmetric one, also for the case of the nonlocal instability. The existence of nonlocal instabilities in nonmagnetized plasma is predicted.

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Measuring the value of the safety factor (q) in the core during sawtooth cycles is still an open issue. A new method to measure q in Tore Supra plasma core is presented here. It relies on the analysis of the time evolution of a set of MHD modes detected after the sawtooth crashes. These modes are in the frequency range of previously observed Beta-induced Alfvén Eigenmodes, but with a frequency declining in time. The mode frequency analysis shows that the q profile is reversed when we have ICRH, after the sawtooth crash. In high current discharges ( I p > 1.15 MA), the q-profile remains reversed for a longer time compared with lower plasma current discharges. Non-linear 3D MHD simulations of sawteeth performed with the XTOR-2F code (Lütjens and Luciani 2010 J. Comput. Phys. 229 8130-43) exhibit features that are similar to these observations.