**Justin Ball, Felix I. Parra, and Michael Barnes**

Rudolf Peierls Centre for Theoretical Physics, University of Oxford, Oxford OX1 3NP, United Kingdom

Culham Centre for Fusion Energy, Culham Science Centre, Abingdon OX14 3DB, United Kingdom

E-mail: Justin.Ball@physics.ox.ac.uk

**Abstract.**

A particular translational symmetry of the local nonlinear *δf* gyrokinetic model is demonstrated analytically and verified numerically. This symmetry shows that poloidally translating all the flux surface shaping effects with large poloidal mode number by a single tilt angle has an exponentially small effect on the transport properties of a tokamak. This is shown using a generalization of the Miller local equilibrium model to specify an arbitrary flux surface geometry. With this geometry specification we find that, when performing an expansion in large flux surface shaping mode number, the governing equations of gyrokinetics are symmetric in the poloidal translation of the high order shaping effects. This allows us to take the fluxes from a single configuration and calculate the fluxes in any configuration that can be produced by translating the large mode number shaping effects. This creates a distinction between tokamaks with mirror symmetric flux surfaces and tokamaks without mirror symmetry, which is expected to have important consequences for generating toroidal rotation using up-down asymmetry.