SBharris, I have just looked at both of your links. In both cases, contrary to what you say above, the derivation requires that the term  only applies in the limit. In the former paper (arXive) it occurs at equation (13). In the latter paper (J price) this is clearly stated in the last sentence of the wording beside fig. 5 on page 14 of the pdf file.

This means that the  term must be exclusively a tangential term. Hence the other v term in the Coriolis force must be a radial term. In other words, these transformation equations only apply to co-rotating radial motion such as would occur if an object were to be constrained to move in a radial groove on a rotating turntable.

As regards circular motion, it appears that both you and Brews are unfamiliar with planetary orbital theory. A circular motion occurs when the inward gravity force (centripetal force) always exactly cancels with the outward centrifugal force.

You both insist that a circular motion involves a net inward centripetal force. If that were so, the radius would shrink and we wouldn't have a circular motion.

You are basing your beliefs in this regard upon the restricted high school derivation of centripetal force in circular motion. The force needed to rotate the velocity vector inwards shows up to be the standard expression for centripetal force. However, if we examine the exact same vector diagram for a straight line motion measured relative to the same origin, we will obtain the exact same expression, only radially outwards. The centripetal force in circular motion is therefore merely cancelling with the centrifugal force. This fact is totally overlooked in high school physics courses.

You must never lose track of the physical meaning behind the maths symbols which you use. These transformation equations cannot be applied to a particle which is stationary in the inertial frame. The entire derivation of those equations is based on the actual components of its actual acceleration. One component leads to the Coriolis force and the other component leads to the centrifugal force. If there is no actual acceleration, then these two components will both be either zero, or equal and opposite. There will be no net inward centripetal force no matter what frame of reference we view it from.

As regards the merry-go-round example, it actually undermines your point and bolsters the very point that I have been trying to make. The quote The radial Coriolis force associated with azimuthal motion is much like an increase or slackening of the centrifugal force tends to confirm my view that the so-called radial Coriolis force is in fact a centrifugal force and that in meteorology it would invoke Archimedes' principle and invoke a torque on a cyclone, due to differential centrifugal force on the east-west wind at one side, and the west-east wind at the other side.