  We consider the Navier-Stokes system in two and three space dimensions
perturbed by transport noise and subject to periodic boundary conditions. The
noise arises from perturbing the advecting velocity field by space-time
dependent noise that is smooth in space and rough in time. We study the system
within the framework of rough path theory and, in particular, the recently
developed theory of unbounded rough drivers. We introduce an intrinsic notion
of a weak solution of the Navier-Stokes system, establish suitable a priori
estimates and prove existence. In two dimensions, we prove that the solution is
unique and stable with respect to the driving noise.
