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Spatially non-local aspects of turbulent transport in tokamak plasmas are examined with global gyrokinetic simulations using the ORB5 code. Inspired by very accurate measurements in the TCV tokamak in L-mode, we initialise plasma profiles with constant logarithmic gradients in the core and constant linear gradients in the 'pedestal' ($\rho \in [0.8,\,1]$). The main finding is that transport in the core is strongly affected by the presence of pedestal gradients. This non-local pedestal-core coupling appears to be correlated with the appearance of repetitive avalanches that propagate across both pedestal and core regions. Below a certain threshold value in pedestal gradient, no well-defined frequency is found for avalanches. Above this threshold, a well-defined frequency shows up, which roughly matches that of the local geodesic acoustic mode (GAM) frequency near the plasma edge and is thus well below the local GAM frequency in the core: this behaviour is very similar to the global coherent mode structure observed in TCV. Above this threshold in pedestal gradient, the core transport increases sharply: there is therefore a non-locality in marginality. The probability density functions (PDFs) of density, temperature, temperature gradient and potential are found to have nearly Gaussian distributions, whereas the heat flux can have, in the presence of avalanches, a more or less strongly positively skewed PDF, which could be fitted by a log-normal distribution. The skewness of the heat flux is found to be radially and non-locally dependent: its value in the plasma core critically depends on the presence of gradients in the pedestal. The relation flux versus gradient is examined in detail. The local instantaneous flux versus gradient relation shows a hysteresis behaviour during an avalanche but no clear correlation, unlike the flux and zonal flow (ZF) shearing rate, which are anti-correlated: flux is higher when shearing rate is lower. This leads to corrugated time-averaged radial profiles of transport, heat and temperature gradient, with heat diffusivity having local maxima where the ZF shearing rate goes to zero and temperature gradient has local minima. Finally, we show how the flux versus gradient relation can be analysed locally for series of simulations with different averaged gradients.