The gradient flow coupling in the Schr\"{o}dinger Functional.

We study the perturbative behavior of the Yang-Mills gradient flow in the Schr\"odinger Functional, both in the continuum and on the lattice. The energy density of the flow field is used to define a running coupling at a scale given by the size of the finite volume box. From our perturbative computation we estimate the size of cutoff effects of this coupling to leading order in perturbation theory. On a set of Nf=2 gauge field ensembles in a physical volume of L ~ 0.4 fm we finally demonstrate the suitability of the coupling for a precise continuum limit due to modest cutoff effects and high statistical precision.Comment: LaTeX, 27 pages, 7 figures. Added section on gauge fixing. Typos corrected. Results unchange