Up-down asymmetric tokamaks
  Bulk toroidal rotation has proven capable of stabilising both dangerous MHD modes and turbulence. In this thesis, we explore a method to drive rotation in large tokamaks: up-down asymmetry in the magnetic equilibrium. We seek to maximise this rotation by finding optimal up-down asymmetric flux surface shapes.   First, we use the ideal MHD model to show that low order external shaping (e.g. elongation) is best for creating up-down asymmetric flux surfaces throughout the device. Then, we calculate realistic up-down asymmetric equilibria for input into nonlinear gyrokinetic turbulence analysis. Analytic gyrokinetics shows that, in the limit of fast shaping effects, a poloidal tilt of the flux surface shaping has little effect on turbulent transport. Since up-down symmetric surfaces do not transport momentum, this invariance to tilt implies that devices with mirror symmetry about any line in the poloidal plane will drive minimal rotation. Accordingly, further analytic investigation suggests that non-mirror symmetric flux surfaces with envelopes created by the beating of fast shaping effects may create significantly stronger momentum transport.   Guided by these analytic results, we carry out local nonlinear gyrokinetic simulations of non-mirror symmetric flux surfaces created with the lowest possible shaping effects. First, we consider tilted elliptical flux surfaces with a Shafranov shift and find little increase in the momentum transport when the effect of the pressure profile on the equilibrium is included. We then simulate flux surfaces with independently-tilted elongation and triangularity. These two-mode configurations show a $60\%$ increase over configurations with just elongation or triangularity. A rough analytic estimate indicates that the optimal two-mode configuration can drive rotation with an on-axis Alfven Mach number of $1.5 \%$ in an ITER-like machine. 
Justin Ball
physics.plasm-ph