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Abstract:

The absence of greenhouse gas emissions, the widely available fuel, the inherent safety and the limited nuclear waste make nuclear fusion an attractive option for our future electricity supply. However, a number of technological challenges still stand, one of which is the high heat and particle load on the divertor. Turbulent transport processes largely determine the outward heat and particle fluxes and thus the efficiency of the plasma confinement. Taking the turbulence into account is of crucial importance to predict the load on the divertor. Mean-field transport codes remain a key tool for designing the plasma facing components and the divertor. However, the description of turbulent transport in these codes remains lacking. Typically, the turbulent fluxes are described via an ad hoc diffusive approximation, in which the diffusion coefficients are determined for a particular experiment. This poses severe limitations on the predictive capabilities of these mean-field codes. As a solution to this, this thesis proposes an approach for modelling the turbulent fluxes that is inspired by the Reynolds-Averaged Navier-Stokes methods commonly used in hydrodynamic turbulence modelling. In this approach, still only the average of the turbulent flow field is resolved to maintain a tractable computational cost, but the turbulent fluxes are modelled by relating them to quantities characteristic of the turbulence. Firstly, this thesis establishes an analytical framework by deriving the mean-field equations through a rigorous averaging procedure. This provides an exact interpretation for the quantities in the mean-field equations and identifies the closure terms. The turbulent ExB fluxes of heat and particles are demonstrated to be the crucial closure terms to be modelled. It is proposed to retain the diffusive model structure commonly used in mean-field codes, but to relate the transport coefficients to averaged quantities characterising the turbulence. In particular, it is found that the average kinetic energy (k) and enstrophy (ζ) in the ExB drift fluctuations, which are exactly the fluctuations causing the ExB turbulent transport, can be used to robustly capture the turbulent transport coefficients. Hence, to complete the analytical framework, equations for k and ζ are derived. This allows to pinpoint and physically interpret their sources, sinks and transport terms. Most of these terms are in turn closure terms which require modelling. To develop a practical, self-consistent model, this analytical framework is subsequently applied to the basic case of 2D electrostatic plasma edge turbulence, for which reference data is provided by the TOKAM2D turbulence code. The focus lies on the interchange-dominated sheath-connected scrape-off layer case. The extension to a core region with drift wave-like dynamics is likewise considered. A Bayesian inference framework for model comparison and parameter estimation supports the development of models for important closure terms. In particular, these Bayesian inference methods are used to select the best performing model for the relation between k (and ζ) and the transport coefficients. With the above methodology, a self-consistent model for the turbulent ExB fluxes is established. The physics of this model is that mean-field density and temperature gradients lead to turbulent heat fluxes down the gradient. If this flux is in the direction opposite to the magnetic field strength gradient, this drives the interchange source of k (through a generally valid analytical relation), which in turn leads to increased transport, etc. The turbulence saturates when the mean-field gradients are such that a balance is established between this interchange source and the sheath losses, which were found to provide the dominant sink of k. Note that all elements of this model derive from specific terms in the analytical framework, giving a clear analytical basis and physical meaning to the model. Forward 1D mean-field simulations with this k model are capable of reproducing the profiles of the averaged TOKAM2D reference data very well. While it is shown that including the enstrophy has the potential to further improve the turbulent transport description, it is found that the additional complexity of closing the enstrophy equation may render the resulting mean-field model less accurate at present. Even though important physics ingredients such as drift waves, flow shear and neutrals are still missing and extensive testing is required to establish its predictive capabilities, the model already provides a large improvement over the current practise for modelling the turbulent transport in mean-field codes. Its implementation in SOLPS-ITER shows that transport patterns which are novel for mean-field codes can be achieved. The results of this thesis directly lead to ballooned transport combined with fast parallel spreading of it due to fast ``anomalous'' transport of k through parallel current fluctuations. In this way, this work provides a crucial stepping stone for the development of self-consistent turbulent transport models for mean-field simulations, both in terms of analytical mean-field equations as background and by suggesting a concrete model for a subset of the dynamics involved in future fusion reactors.