Coriolis Synthesis: The inertial circle (Free Oscillations or Inertial Oscillations)
Real forces:PGF, Friction, Gravity
Apparent forces:coriolis, centrifugal
You will see in the literature (including Holton) a discussion of what is referred to as an “inertial circle” which is really a constant (i.e., conserved) angular momentum (L = mVr, dL/dt = 0) flow. The inertia circle is often invoked as a way to explain the Coriolis force. The reason for this is that it is a simple flow whereby all other “real” forces are presumed to vanish i.e., no pressure gradient, friction – but what about gravity (a real force)? Well, if you read Dale Durran’s 1993 BAMS article he states that:
When viewed in a non-rotating reference frame (e.g., from outer space), the inertial oscillation also appears to be oscillatory. This is a clue, because if an object appears to be accelerating from a non-rotating reference frame, then it must be acted upon by a real force (it’s a non-inertial frame!).
Because gravity plays a role in establishing the oscillation (as a restoring force much like a spring system, or wave clouds that undulate downstream of a mountain range), the oscillation best illustrated with a curved surface (parabolic dish) rather than a flat surface (merry-go-round) because the latter has no gravitational component in the direction opposite of the centrifugalforce. But why might it be important to use a model where there is a component of gravity opposite that of the centrifugal force? Well, I’ve included a figure from Durran’s article below– with some annotation on my part – it’s really the exact same figure that is included in Holton (4th edition, Fig. 1.6 page 13) that should help answer this question. Objects fixed on the surface of the earth do not “fall” poleward because a poleward component of truegravity (g) is exactly balanced by an equatorward component of the centrifugal force (a result of the solid body nature of the earth).It is because of this that the centrifugal force does not appear in the “equations of motion on a rotating sphere”. The poleward component of true gravity is a result of the rotation – which shears the earth’s surface creating a bulge at the equator (centrifuge effect – the centrifugal force is largest at the equator and zero at the poles – imagine if you spun some clay on a pottery wheel – it would deform by an amount that depends on the rotation rate). The “new” deformed surface is what we refer to as a “geopotential surface”. A component of the true gravity (that due to the mass of the earth only – sans rotation) is parallel to the deformed surface while the apparent (or effective) gravity is perpendicular (hence there is no component of apparent gravity along the geopotential – and thus no ball rolling toward the equator!).On a flat turntable or merry-go-round – a co-rotating object would be flung off, just like you would be if gravity were suddenly shut down here on ye ol’ earth, if the string of a ball were cut loose, or the friction on a rotating disc was removed.
This equilibrium whereby the poleward (true) gravitational component is counter-balanced by the centrifugal effect can be disrupted! For example – what if we turned off the rotation but maintained the deformed shape (just a hypothetical!)? There would be no centrifugal component in the direction of the equator and thus a ball located on the surface of the deformed earth would begin to roll poleward! Since this is not a likely scenario, let’s consider an alternative way to disrupt this balance, for example consider a ball placed at the north pole of a frictionless, rotating orb. If this ball is forcibly displaced a small amount (small enough that we can assume that f, the Coriolis parameter, is a constant – known as the f-plane approximation), in any direction, what happens?The ball has zero angular momentum (L = m Vr = mr2 = 0 [solid body] since r, the distance to the axis of rotation is zero) to begin with and, in the absence of any torques (e.g., pressure gradient, friction) – its angular momentum must remain zero (i.e., it is conserved). Another way to look at this is that the ball is NOT rotating around the earth’s polar axis[1] (like a ball on a string) and will never do so (unless there is some external torque applied). Hence, there can not be an equatorward centrifugal force acting on the ball. If there is no centrifugal force acting on the displaced ball, then there is no force to counterbalance the poleward component of true gravity (recall the pole is downhill). The displacement of the ball from the stationary (equilibrium) state disrupts the balance. Thus, a small “tweak” in any direction from the pole immediately results in a “restoring” force whereby the ball rolls back downhill toward the pole. As a result, the ball that has been shoved away from it’s equilibrium position at the pole begins to slow, stops at some distance away from the pole (that depends on how hard it was pushed), and then begins to slide back downhill (toward the pole, i.e., its initial position). However, the ball has momentum as it slides back toward the pole (just like a frictionless pendulum!!!) and thus doesn’t stop – it overshoots its original position (i.e., pole) and continues until it comes to a stop due to the poleward component of true gravitation! In the absence of friction – the ball goes back and forth – ad infintum! This is why it is better to think of this oscillation in terms of a rotating parabolic dish rather than a merry-go-round – the dish is able to reproduce a component of gravity opposite that of the centrifugal – just like the earth scenario. Wikipedia ( actually has a nice picture of a parabolic dish and an animation illustrating this very concept for one particular “shove” of a frictionless puck (the tweak shown on the Wikipedia home page produces an ellipsoidal orbit in the non-rotating [inertial] frame and a circular orbit in the rotating frame. Another example (a different shove) produces a linear motion in the non-rotating frame (again, much like a pendulum). See an online avi file at:
[1] Note that it is important to distinguish between rotation about the earth’s polar axis (ball on string analogy) with that of “local” rotation – which is a measure of the earth’s vorticity (spin) the latter of which is defined as f = 2sin (= 2 at the poles – definitely not zero unless you are at the equator!).