Inertial Oscillations in Geostrophic Flow: Is the Inertial Frequency Shifted by ζ/2 or by ζ?

The short answer to the question posed in the title is that it depends on the frame of reference chosen to describe the motions. In the inertial limit, the frequency in a rotating frame of reference corresponds to the rotation rate of the inertial current vectors relative to that frame. When described in a reference frame rotating with a geostrophic flow having a relative vertical vorticity zeta, inertial oscillations have a frequency f + zeta, equal to twice the fluid's rotation rate around the local vertical axis. From a nonrotating frame of reference, one would measure only half this frequency; the other half arises from describing inertial motions in a reference frame rotating with the background flow. However, when described in a reference frame rotating with Earth, hence rotating at -zeta/2 relative to the geostrophic frame, inertial oscillations have a frequency reduced to f + zeta/2.