In this section, we give a mathematical, although elementary, derivation for the magnitude of the Coriolis force.
Consider a disk rotating with an angular velocity X, so that at a radius r from the axis of rotation the disk has a tangential velocity ud = Xr. Suppose that an observer is sitting on the disk and is thus stationary in the disk's frame of reference. As we saw in the previous section, in the rotating frame there is a centrifugal force of magnitude Q2r trying to push the observer out, and in this case the centrifugal force is balanced by the friction between the observer and the disk.
Now let us suppose that the observer is passed by another body (a cyclist, for example) who is moving along a radius of the disk at a velocity u relative to the disk. The cyclist's total velocity, in the inertial frame, is U = ud + u = Xr + u, and his total acceleration toward the axis of rotation, Ac, as measured in the inertial frame, is
This acceleration must be caused by a centripetal force directed toward the axis of rotation, equal to the mass of the cyclist (M, say) times his acceleration, and this acceleration is provided by the friction of the wheels against the surface and by the cyclist leaning in. That is, the inward force is given by Fc = MAc = M(X2r + u2/r + 2Xu), where M is the mass of the cyclist.
Let us now consider how the forces and accelerations appear to the observer sitting in the rotating frame. The inward force on the cyclist is still Fc because the frictional force and the force due to the cyclist leaning in are still present. There is also an outward centrifugal force on the cyclist equal to MX2r because we are measuring everything in the rotating frame. Thus, the net apparent inward force on the cyclist, as measured by the observer, is Fr - MX2r = M(u2/r + 2Xu).
Regarding the acceleration, the observer measures the cyclist to be moving in a circle with a speed u, and therefore with an apparent acceleration toward the center of the disk of just u2/r. Thus, in the rotating frame, even after properly taking into account the centrifugal force, there appears to be a mismatch between the inward forces per unit mass (u2/r + 2Xu) and the inward acceleration u2/r. We reconcile ourselves to this mismatch by saying that, in the rotating frame, there is an additional force on a moving body equal to 2Xu per unit mass. This force is the Coriolis force, and it acts at right angles to a body moving in a rotating frame of reference.
To summarize, in the inertial frame there is an inward centripetal force on the cyclist given by u 2
c r and a consequent inward acceleration given by u 2
In the rotating frame, there is an inward acceleration given by u2
The inward force is caused by the real frictional and leaning forces on the cyclist, Fc, as given by equation 3.18, but this force is greater than that required to produce a balance in the rotating frame. Such a balance is achieved by positing outward centrifugal and Coriolis forces:
Coriolis force for a body moving meridionally
We now consider the Coriolis force for a body moving in the meridional direction, that is, along a line of longitude. As in the previous section, we consider flow on a rotating disk for which meridional flow corresponds to radial flow.
Suppose a body is sitting on the rotating disk and so is stationary in the rotating frame of reference, held in place by friction against the centrifugal forces pushing it out. Let us now suppose that we push it toward the axis of rotation. One of the fundamental consequences of Newton's laws of motion is that, unless tangential forces are acting on the body, its angular momentum, m, around the axis of rotation is conserved. We express this mathematically as m / Ur (Xr + u) r Xr2 + ur constant, (3.22)
where X is the angular velocity (proportional to the rate of rotation) of the disk, r is the distance from the axis of rotation, and u is the velocity in the tangential direction. The quantity U = Xr + u is just the tangential velocity in the absolute (or nonrotating) frame of reference. The initial velocity is in the radial direction, so the tangential velocity is initially zero, but we may anticipate that the Coriolis force will deflect the body in the tangential direction. Let us differentiate equation 3.22 with respect to time:
dm dr2 du dr dr du dr dt dt
Now, because m is constant, dm/dt = 0, but both r and u may change. The derivative of r with respect to time is just v, the velocity in the radial direction, so that equation 3.23 becomes du
dt or, rearranging the terms, dU 2Xv UVV. (3.25)
The term on the left-hand side is the apparent acceleration of the body, as seen in the frame of reference of the rotating disk. The right-hand side must then be equal to the forces acting on the body, divided by the mass of the body. How do we interpret these forces? The first term is just the Coriolis force caused by the rotation of the disk itself; it is our old friend 2Xv, with the minus sign just giving us the direction of the force. If a flow is toward the axis of rotation, then v = dr/dt is negative, which produces a force pushing the body to the right, with a positive u. The second term on the right-hand side, uv/r is usually much smaller than the Coriolis term and is another apparent force that arises because of the movement of the body itself. (It is sometimes called a metric term, as it only appears in non-Cartesian coordinates.)
We have thus shown that when a body moves in the radial direction it experiences a Coriolis force, equal to twice the angular velocity multiplied by the velocity in the radial direction. This force deflects the body in the tangential direction, perpendicular to the direction in which the body is moving. We saw earlier in this chapter that when a body is moving in the tangential direction it experiences an apparent force in the radial direction, now equal to twice the rotation rate multiplied by the velocity in the tangential direction. Now, the velocity in an arbitrary direction can always be decomposed into a velocity in the radial direction plus a velocity in the tangential direction, and we may conclude that, in general, a body moving in a rotating frame experiences a force at right angles to the direction of its velocity, and of magnitude equal to twice the rotation rate times the speed of the body. This force is the Coriolis force, and it deflects bodies to the right in the Northern Hemisphere and to the left in the Southern Hemisphere.
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