Analytical study of growth estimates, control of fluctuations, and conservative structures in a two-field model of the scrape-off layer

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Anomalous transport, turbulence, and generation of large-scale, collective structures (so-called blobs) in the scrape-off layer (SOL) of tokamaks are some of the main issues that control the machine performance and the life expectancy of plasma-facing components, and here one tries to achieve some understanding of these questions through a theoretical, analytical study of a reduced two-dimensional two-field (density plus vorticity) model of the SOL. The model is built around a conservative system describing transport perpendicular to the magnetic field in a slab geometry, to which terms are added to account for diffusion and parallel losses (both for particles and current) and to mimic plasma flow from the core (in the form of a source). Nonlinear estimates for the growth rates are derived, which show the growth in the density gradient to be bounded above by the vorticity gradient, and vice-versa, therefore suggesting a nonlinear instability in the model. The possibility of controlling fluctuations by means of a biasing potential is confirmed (negative polarisations being shown to be more effective in doing so, thus providing an explanation for what is seen in experiments), as well as the advantage in reducing the inhomegeneity of the magnetic field in the SOL to decrease the plasma turbulence there. In addition, focusing on the conservative part of the equations, exact solutions in the form of travelling waves are obtained which might be the conservative ancestors of the blobs that are observed in experiments and in numerical simulations.