﻿ An Explanation of the General Path of Hurricanes Due to Their Angular Momentum and the Rotation of the Earth

San José State University

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Thayer Watkins
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 Why do the paths of tropical cyclones such as hurricanes and typhoons recurve?

 The Global Torque on a Hurricane Resulting from the Forced Precession of its Angular Momentum with the Rotation of the Earth

Abstract: When an entity with angular momentum about its axis, such as a gyroscope, is subjected to a torque it precesses. If it is forced to precess it experiences a torque. Tropical cyclones (hurricanes, typhoons etc.) have angular momenta with respect to their axes. They turn with the Earth's rotation and are thus forced to precess. This results in their being subject to a global torque which accelerates them toward the Earth's pole in their hemisphere. Hurricanes are thus accelerated toward the North Pole.

There appears to be no other explanation of
why tropical storms move poleward.

These tropical cyclones also have angular momentum with respect to the Earth's axis. The conservation of this angular momentum means that as they move to higher latitudes they experience an acceleration to the east. Thus a hurricane that develops in the low latitude Atlantic and appears to move west begins to turn north. It thus recurves to the east as it moves north. A Southern Hemispheric tropical cyclone such as an Australian one moves west and then recurves to the southeast.

The purpose of this analysis is to establish a reason why hurricanes and other meteorological systems which possess angular momentum from spin about their axes tend to move toward the Earth's poles. They move toward the pole where their angular momentum vector will be aligned with the angular momentum vector of the Earth.

When an object with angular momentum, such as a gyroscope, is subjected to a torque it precesses; i.e., it executes a circular motion at a right angle to the torque. However if an object with angular momentum is forced to precess it experiences a torque at a right angle to its precession. The explanation for the movement of tropical cyclones (hurricanes, typhoons, etc.) is that they are forced to precess because of the rotation of the Earth while the verticality of their angular momentum is enforced by the rising of the warm, moist air at their centers.

Some historical data on hurricane paths are shown below. To simplify the presentation the term hurricane will be used instead of the more proper tropical cyclone. Let L be the magnitude of the angular momentum of a hurricane or other spinning structure. In the following vector quantities are shown in red. Vectorially angular momentum is Lk, where k is the local unit vertical vector. When the hurricane moves to a different location the vertical unit vector is different so there is a vectoral change in angular momentum even though the magnitude does not change.

Newton's Second Law in terms of angular momentum is:

#### dL/dt = T

where T is the torque vector which in this case is r×F, where r is the radial vector from the center of the Earth to the center of the base of the hurricane and can be presented as rk where r is the radius of the Earth and F is a force perpendicular to r.

From vector analysis it is known that the vectorial time rate of change of a unit vertical vector on a rotating sphere is given by

#### dk/dt = Ω×k

where Ω is the vector of the angular velocity of the sphere; i.e., a vector directed along the spin axis of the sphere with a magnitude equal to its angular velocity.

Since the magnitude of the angular momentum vector is constant

Thus

#### T = rk×F = L(Ω×k) which can be rearranged to -r(L(F×k) = L(Ω×k) or (L(Ω+rF)×k = 0

This last equation implies that

#### LΩ + rF = λkand hence rF = -LΩ + λk

This means that the horizontal component of F, Fh, is the same as the horizontal component of -(L/r)Ω; i.e.,

#### Fh = -(L/r)Ωcos(φ),

where φ is the latitude.

Thus if the hurricane is to be carried along at the same latitude there must be a force of -(LΩ/r)cos(φ) applied to it. In the absence of such a force the hurricane will move poleward with an acceleration equal to this force divided by the mass of the hurricane.

Normally a hurricane does not follow the Earth's rotation exactly and so the effective rotation rate is somewhat different from that of the Earth. If u is the west-to-east velocity of the hurricane and R=rcos(φ) is the distance to the Earth's axis then the effective rotation rate of the hurricane with respect to the Earth's axis is:

#### Ω' = Ω + u/R = Ω + u/(rcos(φ)) and thus the force on the hurricane is (L/r)(Ω + u/rcos(φ))cos(φ) or equivalently (L/r)Ωcos(φ) + Lu/r2

At this point it is necessary to take note of the fact that there are two angular momenta associated with a hurricane. The angular momentum considered above is that due to the spin of the hurricane. There is also the angular momentum of the mass of the hurricane moving around the Earth's axis. This rotational angular momentum is equal to the mass of the hurricane times the velocity of the hurricane with respect an inertial coordinate system times the distance to the axis of rotation. The absolute velocity of the hurricane is:

#### ΩR + u so if m is the mass then the angular momentum is m(ΩR+u)R = m(Ωrcos(φ) + u)rcos(φ)

When a hurricane moves north its distance from the Earth's axis is decreased so that in order to conserve angular momentum the velocity u must increase. If u0 is the velocity at φ0 then, since the mass is constant,

#### (Ωrcos(φ) + u)cos(φ) = (Ωrcos(φ0) + u0)cos(φ0) and hence u = (Ωrcos(φ0) + u0)(cos(φ0)/cos(φ)) - Ωrcos(φ)

Thus while u0 might be negative a movement north may increase u to a positive level. That is to say a hurricane initially moving west may stop and eventually start moving east. This is called recurvature.

For a more detailed analysis of the path of a hurricane or other cyclonic system see Recurvature.

The acceleration on the hurricane toward the pole is then:

#### dv/dt = (L/m)(1/r)Ωcos(φ) + (L/m)(u/r2)

The second term, (L/m)(u/r2, is of a smaller order of magnitude than the first term, (L/m)(1/r)Ωcos(φ). The ratio is u/(Ωrcos(φ), the ratio of the relative velocity of the hurricane to the velocity of the surface of the Earth due to its rotation. At φ=30° of latitude this ratio would be on the order of 0.03.

When the hurricane consists of a cylindrical windwall at radius a the moment of inertia I = ma2 so the angular momentum L = Iω = ma2ω = maq where q is the wind velocity at the windwall. Therefore (L/m) = aq. Thus the rate of change of the poleward velocity is:

#### dv/dt = q(a/r)Ωcos(φ) + uqa/r2

It is important to take note of how great are the velocities of objects on the Earth's surface due simply to the rotation of the Earth. The table below shows those velocities as a function of latitude. At the equator objects motionless with respect to the Earth's surface are traveling at about 463 meters per second. This is 1042 miles per hour. An object has to be traveling at this rate to cover the 25 thousand miles of Earth's circumference in 24 hours.

Absolute Velocity
Due to Rotation
of Earth
Latitude Easterly
Velocity
(degrees) (m/s)
0 463.0
5 461.2
10 455.9
15 447.2
20 435.0
25 419.6
30 400.9
35 379.2
40 354.6
45 327.4
50 297.6
55 265.6
60 231.5
65 195.6
70 158.3
75 119.8
80 80.4
85 40.4
90 0

Consider a hurricane at latitude 20° N moving west at about 15 miles per hour (24 km per hour). This speed is about 6.7 meters per second. However, due to the Earth's rotation that hurricane's absolute velocity is 435 m/s minus 6.7 m/s or 428.3 m/s. If that hurricane moves to a higher latitude its distance from Earth's axis of rotation is less so in order to preserve angular momentum its easterly velocity must increase. As it moves north it switches from having a westward movement relative to the Earth's surface to having an eastward movement.

At 20° latitude the distance to the Earth's axis is 5982 km or 5.98 million meters. The angular momentum per unit mass is this figure times the velocity of 428.3 m/s or 2.56×109 m2/s. At 25° latitude the distance to the Earth's axis is 5.77 million meters, about 3.7 percent less than at 30°. The absolute wind velocity must therefore increase about 3.7 percent to a value of 444.1 m/s. This is 24.5 m/s faster than the surface of the Earth is traveling at 25° latitude. The hurricane would therefore be traveling with an easterly component of velocity of 24.5 m/s. This means that a hurricane which is moving west at 24 km/hr (6.7 m/s) at 20° latitude would have to have a west-to-east velocity of 24.5 m/s at 25° to preserve angular momentum. This is a quite high velocity of about 90 km/hr or 56 miles/hr. If the prevailing winds at 25° latitude N are not traveling at that velocity the frictional effects would slow the hurricane down. The frictional effects would not necessarily slow the hurricane's velocity to the same level as the prevailing winds but the deviation would be restricted.

Likewise the northward velocity of the hurricane would be restricted by the frictional effects of its traveling through prevailing winds of a different velocity and direction. Thus the global torque on a hurricane due to its forced precession with the Earth creates a northward acceleration which leads to a northward movement. The movement of the hurricane to higher latitudes creates an acceleration to the east. So an initially westward moving hurricane, on average, begins to veer to the north and as it does so it westward movement slows and eventually turns into an eastward movement. The net result is the hurricane, on average, veers to the north and recurves to the northeast if it lasts long enought to do so.

The analysis would also apply to anti-cyclones. Although of a much smaller magnitude a high pressure system would experience a global torque which would accelerate it toward the equator. But as it moves toward the equator its eastward velocity would decrease and it could in principle recurve in the Northern Hemisphere to the southwest just as a cyclone recurves to the northeast. In the Southern Hemisphere the recurving would be to the northwest.

For a continuation of the analysis see the Poleward Acceleration of Meteorological Vortices.