According to Coriolis force an air parcel is deflected in the absence of other forces by the acceleration (Holton, 4th Edition):
These equations lead to circular motion.
This figure is from a Matlab script, provided by Holton:
But how is this consistent with the picture, where an object moves along a straight line in the inertial system and the deflection is only apparently because we are in a rotating frame? If the object moves along a straight line in the inertial system, the curve cannot be circular in the rotating frame.
It's due to the disappearance of vertical motion, I think. But in Holton, the third equation is simply left out without comment, and instead only the above two equations are finally written. It is clear that the above equations come out when the vertical velocity is zero, but this is not justified in the explanations. Is this just an empirical observation?