  We introduce a rough perturbation of the Navier-Stokes system and justify its
physical relevance from balance of momentum and conservation of circulation in
the inviscid limit. We present a framework for a well-posedness analysis of the
system. In particular, we define an intrinsic notion of solution based on ideas
from the rough path theory and study the system in an equivalent vorticity
formulation. In two space dimensions, we prove that well-posedness and
enstrophy balance holds. Moreover, we derive rough path continuity of the
equation, which yields a Wong-Zakai result for Brownian driving paths, and show
that for a large class of driving signals, the system generates a continuous
random dynamical system. In dimension three, the noise is not enstrophy
balanced, and we establish the existence of local in time solutions.
