Second stability region for gyrokinetics and the L-H transition

Despite enormous progress in modelling and simulating the L-H transition, no broad consensus has emerged on the underlying physical mechanism — which is perhaps surprising given how robustly this phenomenon is attained in tokamak experiments. We speculate that this is partly because many such simulations are performed with a fixed magnetic field. In an actual tokamak plasma, the equilibrium magnetic field responds to changes in the pressure gradient: as the latter grows, the Shafranov shift increases and the local magnetic shear on the outboard side of the torus tends to decrease — a circumstance instrumental to the establishment of the 'second stability region' for MHD-ballooning modes (Lortz Nührenberg 1978). Furthermore, an increased pressure gradient drives more bootstrap current, which also modifies the magnetic field and tends to reduce the global magnetic shear in the H-mode pedestal. Using the well-known s-α model for the magnetic geometry (Connor et al. 1978), we show that a second-stability region exists for linear gyrokinetics as well, accessed when the collisionality is sufficiently small and α⪆1. Electrostatic simulations of more realistic L- and H-mode geometries demonstrate that the combined action of the Shafranov shift and bootstrap current is sufficient to reduce transport levels by well over an order of magnitude. We also calculate the transport along a path connecting the L- and H-mode equilibria, finding non-monotonic behaviour as a function of the pressure gradient — mirroring results obtained on DIII-D for an internal transport barrier (McClenaghan et al. 2019). We suggest that a transport simulation in which the magnetic geometry is evolved self-consistently with the plasma profiles should exhibit a bifurcation as the heating power is increased.