The effects of event-by-event fluctuations in the initial geometry of the colliding nuclei are important in the analysis of final flow observables in relativistic heavy-ion collisions. We use hydrodynamic simulations to study the amplitude correlations between different orders of event-by-event fluctuating anisotropic flow harmonics... While the general trends seen in the experimental data are qualitatively reproduced by the model, deviations in detail, in particular for peripheral collisions, point to the need for more elaborate future calculations with a hybrid approach that describes the late hadronic stage of the evolution microscopically. It is demonstrated explicitly that the observed anti-correlation between $v_2$ and $v_3$ is the consequence of approximately linear hydrodynamic response to a similar anti-correlation of the corresponding initial eccentricities $\epsilon_2$ and $\epsilon_3$. For $n{\,>\,}3$, the hydrodynamic correlations between $v_{2,3}$ and $v_n$ deviate from the rescaled correlations among the corresponding initial eccentricities, demonstrating nonlinear mode coupling effect in higher order flows. (read more)