Consider a circular region of radius r. Let A, C and B be the three water molecules at the nnorthern edge, center and southern edge of this circle with latitude angles of λ_A, λ_C, and λ_B, respectively.Their velocities if the region is nnot rotating wirh respect to the Earth are given by the formula Rρ where R is the distance to Earth's aaxis and ρ is Earth's rate of rotation of 2π radians per day.

The distance to the Earth's axis of a point at latitude λ is given by the formula R_Ecos(λ) where R_E is Earth's radius.

Since λ_A>λ_C>λ_B the velocities in the absence of any rotation of the region satisfy the condition

u_A<u_C<u_B

Let the ratio of the northern velocity to the southern velocity of the unrotating region be denoted as σ; i.e.,

σ = u_A/u_B

If the ratio of the actual north/south velocities exceeds σ the region is rotating clockwise; if it falls short the region is rotating counterclockwise.

Angular Momentum

A body of mass m situated at a distance R from the Earth's axis has an angular momentum of mRρ due to its rotation with the Earth. If it moves to a different point its angular momentum will remain the same but its velocity will change; i.e.;

uR = (u + Δu)(R + ΔR)
and hence
uΔR + ΔuR + ΔuΔR = 0
and thus
Δu = −uΔR/(R+ΔR)
or, equivalently
Δu/u = −Δ/(R+ΔR)

Rising or Falling Air Masses

Relatively warm, moist air at the Earth's surface will rise and cool until it reaches a level where it is in equilibrium with the ambient atmosphere. Suppose that is an altitude of h. Then the increase in in the distance to Earth's axis is

ΔR = h·cos(λ)

The quantity h·cos(λ) is a larger proportion change for the R of the northern point of the region than for the center point and southern point ofthe region, Therefore the the rising relatively warm, moist air of hurricanes and typhoons produces a counterclockwise rotaion.

On the other hand a mass of cold dense air at an altitude h falling to the Earth's surface produces an increase in air velocity which is greater at the northern point of the region that at the center and southern point. Therefore the atmospheric region rotates clockwise. It is an anticyclone.