  We present the first very long-term simulations (extending up to ~140 ms
after merger) of binary neutron star mergers with piecewise polytropic
equations of state and in full general relativity. Our simulations reveal that
at a time of 30-50 ms after merger, parts of the star become convectively
unstable, which triggers the excitation of inertial modes. The excited inertial
modes are sustained up to several tens of milliseconds and are potentially
observable by the planned third-generation gravitational-wave detectors at
frequencies of a few kilohertz. Since inertial modes depend on the rotation
rate of the star and they are triggered by a convective instability in the
postmerger remnant, their detection in gravitational waves will provide a
unique opportunity to probe the rotational and thermal state of the merger
remnant. In addition, our findings have implications for the long-term
evolution and stability of binary neutron star remnants
