  We develop a framework for event-by-event ideal hydrodynamics to study the
differential elliptic flow which is measured at different centralities in Au+Au
collisions at Relativistic Heavy Ion Collider (RHIC). Fluctuating initial
energy density profiles, which here are the event-by-event analogues of the eWN
profiles, are created using a Monte Carlo Glauber model. Using the same event
plane method for obtaining $v_2$ as in the data analysis, we can reproduce both
the measured centrality dependence and the $p_T$ shape of charged-particle
elliptic flow up to $p_T\sim2$~GeV. We also consider the relation of elliptic
flow to the initial state eccentricity using different reference planes, and
discuss the correlation between the physical event plane and the initial
participant plane. Our results demonstrate that event-by-event hydrodynamics
with initial state fluctuations must be accounted for before a meaningful lower
limit for viscosity can be obtained from elliptic flow data.
