& Tornado Alley
and Other Tropical Cyclones
Abstracting from the variations in pressure gradients, the topography and the frictional forces acting on a hurricane its motion is governed by the conservation of its angular momentum with respect to the Earth's axis and the forced precession of its angular momentum with respect to its own axis.
Let u be the eastward velocity of the center of the hurricane relative to the Earth's surface and v its corresponding northward velocity. Let φ be the latitude angle and θ the longitude angle. If r is the radius of the Earth then the distance from the center of the hurricane to the Earth's axis rcos(φ). Let Ω the angular velocity of the Earth's rotation. Then the absolute velocity of the center of the hurricane is
The angular momentum per unit mass of the hurricane is then
Conservation of angular momentum requires this to be constant so
The relative velocity is given by u=rcos(φ)(dθ/dt). Entering this expression into the above equation gives:
This equation can be solved for the rate of change of longitude, dθ/dt.
The equation for dφ/dt can be derived from an equation for the poleward acceleration of the hurricane due to its forced precession with Earth's rotation. This equation, which is derived elsewhere is:
where v=r(dφ/dt), q is the wind velocity at the windwall and a is the radius of the windwall.
The above equation reduces to
The two equations determining φ and θ are
These may be reduced to
This last equation may be multiplied by (dφ/dt) and the result integrated to give
where C is a constant of integration. (The integration constant C can be expressed in terms of the initial latitude and the initial rates of change of latitude and longitude.) Thus, from the above, the equation for dφ/dt is
The equation for dθ/dt is
By dividing the equation for dθ/dt by the equation for dφ/dt one obtains the equation for the trajectory of a hurricane; i.e.,
The recurving latitude occurs where dθ/dφ = 0; i.e.,
Since Λ depends upon the initial latitude φ0 and the initial relative velocity rcos(φ0)(dθ/dt)0 the recurving latitude φ* also depends upon those variables. The relationship is Λ = [rcos(φ0)]²(Ω+(dθ/dt)0). The dependence is
If the initial rate of change of latitude of the hurricane is zero then the constant of integration appearing in the previous equations reduces to
and hence the equation for dφ/dt becomes
Using this expression plus the expression for Λ allows the equation for dθ/dφ to be expressed as
A rough approximation of the equation is
where A and B are constants.
Thus the general shape of the trajectory is of the form
As can be seen in the chart the latitude of recurvature φ* is about 30°N.
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