Apparent Forces

The motion referred to by Newton’s First Law was for a coordinate system fixed in space (inertial motion). This isn’t valid for motions in our atmosphere since the earth is moving underneath.

We must take into account the accelerations of the coordinate reference frame in addition to those of the fluid itself. Apparent Forces are used to include the effects of an accelerating coordinate reference frame (e.g. rotation of the earth).

Two apparent forces: Centrifugal force and Coriolis force

Centrifugal Force

Consider a ball attached to a string or length r, whirling around at a constant angular velocity (ω).

From an inertial (fixed coordinate) frame of reference, the speed of the ball is constant but the velocity is changing (since velocity is a vector that includes direction). To calculate the acceleration that the ball experiences,

v VD −= ω 2 r Dt

This is the expression for the centripetal acceleration. The force causing the acceleration is the string pulling on the ball.

Now, if we have the same situation but with the coordinate system rotating with the ball, it would appear to us (stationed on the rotating coordinate system) that the ball would not be moving at all.

But there is still a force acting on the ball (pull of the string). So we need to balance the centripetal acceleration by an equal and opposite force. This is called the centrifugal force.

The centrifugal force is an apparent force. It is needed to so that we may still use Newton’s laws of motion even with a rotating frame of reference. It’s magnitude is given 2 by Ω RO, where RO is the distance from a point on the earth to the spin axis.

Effective Gravity (g)

The true gravitational acceleration (g*) pulls all mass towards the center of mass of the earth. However, the centrifugal force (Ω2R) pulls all objects outward from the axis of planetary rotation. The effective gravity (g) is the vector sum of these two forces. It does not point directly at the center of earth mass:

Recall that geopotential (Φ) at any point is defined as the work that must be done against gravity in order to raise a mass of 1 kg from sea level to that point. Geopotential by convention = 0 on the surface of the earth. Surfaces of constant geopotential (e.g. the surface of the earth) are normal (perpendicular) to the effective gravity. Since effective gravity does not point directly at the center of the earth’s mass, the constant geopotential surface takes the shape of an oblate ellipsoid instead of a perfect sphere, shown by the red line in the figure below:

So the earth “bulges” at the equator. When we use gravity in the equations of motion later on, the centrifugal force remains integrated with g and is not included explicitly.

Coriolis Force

If an object is moving at the same speed as the earth’s rotation rate (e.g. calm wind), then we only need to invoke the apparent centrifugal force to satisfy Newton’s laws of motion. However, if the object (or air molecules, for instance) are moving faster or slower than the rotation rate of the earth, a second apparent force must be introduced to explain it’s motion. This force is called the Coriolis Force (CF).

Note that the earth rotates in a counter-clockwise fashion from a northern hemisphere frame of reference (get a globe, look down from the North Pole, and spin the globe counter clockwise). If we have westerly wind flow (wind from the west), it means that the air is moving faster than the angular velocity of the earth. Therefore, the wind has a higher centrifugal force than that of the earth at that latitude:

Ω = Ω + u air earth 2 2 ()earth Ru O Ω>+Ω ROearth

If u > 0 and RO is constant

So there is a net force acting radially outward from the axis of rotation. In the horizontal plane, there is a southward component (in the northern hemisphere) that acts perpendicular and to the right of the zonal wind, “pulling” the wind to the right.

Note that at the Equator, the centrifugal force is perpendicular to the axis of rotation and therefore there is no southward component. Winds do experience a vertical CF at the equator but none in the horizontal plane. The vertical component is very small, however, and need not be considered any further.

The same arguments can be made for easterly flow. The wind is moving slower than the angular velocity of the earth, thus the air will have a weaker centrifugal force than that of the earth. This results in a radially inward directed CF, pulling the wind toward the north. Note that this is still perpendicular and to the right of the flow.

Through angular momentum arguments and several assumptions (see Holton for complete derivation), the amount of displacement in the zonal, meridional, and vertical directions is given by:

Du uv vsin2 +Ω= tanφφ Dt a Dv u 2 u sin2 −Ω−= tanφφ Dt a Dw u 2 u cos2 φ +Ω= Dt a

Where u = zonal wind, v = meridional wind, w = vertical wind, φ = latitude, and a = radius of earth. The second terms on the right side arise from curvature effects, and can be safely neglected through scale analysis since u << ΩR. Also, the quantity 2Ωsinφ is abbreviated with the letter ‘f’, and is called the Coriolis parameter. It ranges from 0 at the equator to +- 1.46 x 10-4 s-1 at the poles. Thus, for horizontal motion,

Du = fv Dt

Dv −= fu Dt

In vector notation this is written as:

CF = -fk x V

Where V is the 3-D velocity vector. As we will see later, vertical Coriolis Forces are much smaller than other terms that give rise to vertical motion and will be neglected.

In the southern hemisphere, deflection is to the left of the wind. Spin your globe again and turn it upside down to see why.

The CF is negligible for motions that have time scales that are much shorter than the period of the earth’s rotation. So you can’t blame it on your bad basketball shooting.

The CF does not alter the speed of motion, only the direction.

The Coriolis Force is named after French mathematician Gaspard Gustave de Coriolis (1792-1843).