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Drift wave theory for transport in tokamaks

Review article, 2019

We revisit the theory of tokamak transport due to drift waves. We recall the whole development, starting with simple theories from the 1950s till today advanced theories using advanced fluid models taking full account of kinetic effects in the frequency regime of drift waves, also including the effects of zonal flows, and also fully nonlinear kinetic theory itself. Traditionally drift waves have been described by either kinetic theory or expanded fluid theories. The expansions have usually been made in the ratio of magnetic drift frequency and frequency. As was just shown toroidal drift waves are mainly driven by the magnetic drift resonance so such an expansion is usually not allowed. It is, in fact, natural that toroidal effects play an important role since they originate from bending the system to a torus, thus eliminating the third direction in which the magnetic field does not confine plasma. Due to an exact fluid closure, we are now able to use fluid theory completely without expansion, thus maintaining the fluid resonances due to magnetic drifts in the denominators. This gives us a new normalization of drift wave equations and this enables us to recover nonlinear (Dimits) upshifts, spinup of poloidal rotation in internal transport barriers and the L–H transition. The principle of our reactive closure is that we include all moments with sources in the experiment. Here, it is usually enough to use the diamagnetic heat flow as closure term but more general cases are, of course, possible and remain parts of our general closure. The fact that the model is self-consistent then also leads to the overall experimental power scaling τe∼P−2/3. Using a full transport matrix we are then also able to obtain adequate particle pinches and apparently nonlocal phenomena such as the heat pinch on DIII-D. Recently an extension of the derivation of the fluid closure also showed that quasilinear theory works well for the real part of the eigenfrequency in our fluid description of driftwaves. A fluid description using an exact closure with a fully valid quasilinear approach also lets us cover cases with nonlinear thresholds for gradients then including also sand pile thresholds. Finally, we use a correlation length at the maximum of the E × B drift in k-space. Then we can go beyond the gyro-Bohm scaling if we allow the parameter kρ to vary. Thus, we are, in practice, able to relax all limitations of traditional drift wave theories. This means that we can recover all aspects of low-frequency tokamak transport by a systematic derivation from first principles. Our model is not fitted numerically to any other model.

Transport matrix

Turbulent transport

Fluid closure

L–H transition

Magnetic drift resonance