  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.
