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Turbulence and the nonlinear dynamics of sawteeth in tokamaks

Tokamaks typically show cyclic behaviour (i.e. sawtooth oscillations) and this suggests strongly that these systems may be describable in terms of two reduced degrees of freedom, or equivalently two 'effective' physical variables. This leads us to further explore an earlier model which is based on a thermal instability within a central core region which is assumed to be within the q=1 surface, as suggested by experimental and theoretical considerations. The proposed model should be regarded as a 'proof of principle' calculation which shows that sawtooth oscillations need not necessarily result from gross linear MHD instabilities (e.g. M=1) but may be a consequence of thermal instabilities associated with plasma turbulence. To identify the relevant dynamical variables and derive the equations of motion which they satisfy, we consider a full set of conservation equations along with certain constitutive relations applying to the turbulent transport of energy and particles. In particular, we formulate and apply a formal invariance principle relating to the structure of turbulent correlations. We assume the existence of the q=1 surface at r=r1 and that there are no particle sources within this core region. Adopting simple models for the spatial profiles of density, temperature and current and spatially averaging the equations within the core(r<or=r1), we develop two coupled equations for the time evolution of the averaged poloidal beta (equivalently the temperature) and the magnetic fluctuation level. Analytical and numerical studies show that the solutions of these equations exhibit limit-cycle behaviour. This is the result of the nonlinear interaction between the internal energy (i.e. pressure) of the plasma within the core and the magnetic turbulence. This interaction is the combined effect of the conservation laws, the turbulent constitutive relations governing the transport and crash physics, and the boundary conditions at r1.