  Breaking the up-down symmetry of tokamaks removes a constraint limiting
intrinsic momentum transport, and hence toroidal rotation, to be small. Using
gyrokinetic theory, we study the effect of different up-down asymmetric flux
surface shapes on the turbulent transport of momentum. This is done by
perturbatively expanding the gyrokinetic equation in large flux surface shaping
mode number. It is found that the momentum flux generated by shaping that lacks
mirror symmetry (which is necessarily up-down asymmetric) has a power law
scaling with the shaping mode number. However, the momentum flux generated by
mirror symmetric flux surface shaping (even if it is up-down asymmetric) decays
exponentially with large shaping mode number. These scalings are consistent
with nonlinear local gyrokinetic simulations and indicate that low mode number
shaping effects (e.g. elongation, triangularity) are optimal for creating
rotation. Additionally it suggests that breaking the mirror symmetry of flux
surfaces may generate significantly more toroidal rotation.
