Coriolis acceleration

Coriolis acceleration is often referred to as an imaginary, or fictitious acceleration.  This is only half true, well actually, three-quarters true.

If we observe a force-free inertial trajectory from a rotating reference frame, the trajectory only appears to be curved.  In these cases (50%), the above characterization is appropriate.

If, however, the trajectory is constrained relative to the rotating reference frame, then any velocity component in the plane perpendicular to the axis of rotation will alter the inertial acceleration in this same plane. This velocity component can be further separated into two perpendicular components; one, along a radial line towards or away from the axis of rotation; the other, tangent to the circular arc of the reference frame's motion. The radial component of velocity creates a true Coriolis acceleration.  The tangential component of velocity (another 25%) only alters centripetal acceleration and its Coriolis term involves mathematical smoke and mirrors.

Coriolis acceleration due to radial velocity

A thought experiment will illustrate this effect.  Imagine a room which can be rotated by a vertical axle located beneath the center of the room.  The whole room is like a merry-go-round or carrousel, and can be rotated counter-clockwise.  You are standing close to one wall, facing the center of the room.  Ahead of you is a long narrow carpet running straight through the center of the room and ending at the opposite wall.

The room slowly starts to rotate, driven by a silent and smooth drive mechanism.  As the room gathers angular velocity, it feels as if the floor in front of you is tilting up - you lean forward to keep your balance.  This effect stabilizes at, for example, when the perceptual uphill incline in front of you is ten percent. (Centripetal acceleration is 3.22 ft/sec^2)

You start to walk toward the center of the room.  Now it feels as if the room is tilting to your right.  The faster you walk, the more the room tilts.  You find a pace at which you can keep your balance by leaning to your left.

Keeping that same pace, you find that the sideways tilt of the floor is constant, but your forward progress becomes easier as you approach the center of the room.  At the very center of the room the floor feels level in the forward direction, but is has the same tilt to the right.  Be careful!  If you go faster, the room will tilt further to the right, and even though the floor feels level ahead, you are about to start downhill.

As you continue past the center of the room, the floor starts tilting down in front of you, requiring you to lean back against the perceived hill to keep from going faster.  You stop just before reaching the wall.  The sideways tilt disappears, but the perception of a downhill slope remains.

The above thought experiment demonstrates the effects of our inertial attachment to the larger outside world when our local reference frame (the room) in a state of rotation. 

The uphill and downhill perception is simply a matter of centripetal acceleration and the center of the room (the axis of rotation) is always the top of the hill.  For a given angular rate, the outward centrifugal force we feel is proportional to the radial distance from the axis.
The sideways perceived tilt to the right is a little trickier and has two equal sources: 

 At the beginning of the thought experiment, the angular acceleration of the room was so gradual, we had no perception of the sideways velocity (and momentum) we were accumulating toward our right side.  As we moved toward the center this velocity decreased.  Just as we are thrown forward in a car when the brakes are applied, this sideways deceleration tries to throw us to the right. (The rate of deceleration is proportional to our velocity toward the center of rotation.)
Since we are walking in a straight line relative to the room, our velocity vector is changing direction at the same rate.  (The tip of our velocity vector is moving to the left.)  The counter clockwise rotation of our velocity vector is equivalent to being in a left turn.  This throws your body to the right just like a left turn in a car.
For both of the above sources, the magnitude is exactly the same for the leftward acceleration; it is the product of the velocity and the angular rate in radians per unit time.  This is why the equation for Coriolis acceleration has a factor of two.

Coriolis acceleration due to a radial velocity is always in a tangential direction. 

If Vr is negative, (towards the axis of rotation), then at is negative, meaning the tangential inertial velocity is decelerating.  If the tangential velocity of a mass is decelerating, it applies an accelerating torque to the disk.

Anyone who thinks that all Coriolis accelerations are imaginary or fictitious should carefully consider the mechanism by which ice skaters accelerate their spin by pulling in their arms.  (Or, how hurricanes and tornadoes accelerate due to the inward flow of air.)  

Coriolis due to tangential velocity

This is where Coriolis gets its dubious reputation.  The Coriolis acceleration for tangential velocities involves a bit of mathematical sleight of hand.

Basically, the net tangential velocity only produces centripetal acceleration.

In the above equation, omega R is the tangential velocity of the rotating reference frame, and vt' is the tangential velocity relative to the rotating reference frame.  Whether the relative tangential velocity is in the same direction as the reference frame, (+), or in the opposite direction, (-), the value of centripetal acceleration is always less than (or equal) to zero.  i.e. it can never be directed outward, which would cause an inward centrifugal force.

We can, however, expand the above equation into three components:  

The first term is the centripetal acceleration of a stationary point on the rotating reference frame, and the third term is the centripetal acceleration that the relative tangential velocity would have if the reference frame were not rotating.

Voilà!  The second term has exactly the same form as Coriolis acceleration for radial velocities.  If the relative tangential velocity is in the same direction as the reference frame tangential velocity, then the Coriolis term enhances the inward centripetal acceleration.  If the relative tangential velocity is in the opposite directing, the Coriolis term opposes the reference frame centripetal acceleration.

The inclusion of the Coriolis term for tangential velocity allows us to simply take the cross product of the rotation vector and reference frame velocity vector without breaking out the radial and tangential components.  This mathematical trick corrects the overall centripetal acceleration while the reference frame tangential velocity is treated as an inertial tangential velocity.

Applying Coriolis acceleration to total relative velocity is purely a mathematical convenience.  It is only valid if the other two terms are also included - the centripetal acceleration of the reference frame itself, and the centripetal acceleration that the tangential velocity would have in a non-rotational frame.

Consider the two cases in which the relative tangential velocity has exactly the same magnitude of the reference frame tangential velocity.

If the relative velocity is in the same direction as the reference frame, then the inertial tangential velocity is doubled and the centripetal acceleration increases by a factor of four.

If the relative velocity is in the opposite direction as the reference frame, then the inertial tangential velocity is zero and the centripetal acceleration is also zero.

The Coriolis term makes both of these corrections automatically.