|San José State University|
Silicon Valley &
An Explanation of
the General Paths of Hurricanes
Due to the Forced Precession of
Their 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
while the verticality of their angular momentum is enforced by the rising of the warm, moist air
at their centers.
Their angular momentum vectors 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 willy-willy moves west and then recurves to the southeast.
The purpose of this analysis is to establish the reason why hurricanes and other tropical cyclones which possess angular momentum from spin tend to move toward the Earth's pole in their hemisphere. Since they start out appearing to move west there is a recurving of their paths. Some historical data on hurricane paths are shown below.
Typhoons are the same phenomena as hurricanes, but located in the northwestern Pacific Ocean area. (The name comes from Mandarin da-feng (great wind)). Here is an illustration of their general paths.
The typhoons which strike south China and Vietnam generally dissipate before they can recurve. A typhoon which made it into north central China in August of 1975 brought heavy rains which broke a series of dams and brought devastation to the area. For more on this see August 1975.
The analysis also implies that high pressure systems are subject to an acceleration toward the equator. If they are initially moving east they will recurve. An eastward moving high pressure system in the Northern Hemisphere will move southward and recurve to the southwest. A westward moving high pressure system will not recurve but will move southwestward.
The analysis proceeds basically as does that which establishes the nature and magnitude of centrifugal force.
A body moving in a circular orbit experiences centripetal acceleration and therefore for it to move in a circular orbit there must be a force on the body equal to the mass of the body times its centripetal acceleration. The force on the body must be directed inward to the center of curvature of the circular path but the structure which supplies this force on the body experiences it as an outward force which is called the centrifugal force. Likewise a body with spin angular momentum will rotate around the Earth's axis only if it is subject to a torque. In the absence of an externally imposed torque it will move as though it is subject to a force which creates a global torque upon it.
Let L be the magnitude of the angular momentum of a hurricane or other spinning structure. Vectorially angular momentum is L k, 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:
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 and can be presented as rk 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
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
This last equation implies that
This means that the horizontal component of F is the same as the horizontal component of -(L/r)Ω; i.e., Fj = -(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:
At this point it is necessary to take note of the fact that there are two angular momenta associated with a hurricane. The angular momnentum 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 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:
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
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 eventually start moving east.
The acceleration on the hurricane toward the pole is then:
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.
Due to Rotation
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 gloval 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.
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