Thayer Watkins
Silicon Valley
& Tornado Alley

The Equations of Motion for Hurricanes
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.

The Equation for the Rate of Change of Longitude

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

Ωrcos(φ) + u

The angular momentum per unit mass of the hurricane is then

rcos(φ)[Ω(rcos(φ) + u]

Conservation of angular momentum requires this to be constant so

rcos(φ)[Ω(rcos(φ) + u] = Λ

The relative velocity is given by u=rcos(φ)(dθ/dt). Entering this expression into the above equation gives:

(rcos(φ))²(Ω + dθ/dt) = Λ

This equation can be solved for the rate of change of longitude, dθ/dt.

The Equation for the Rate of Change of Latitude

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:

dv/dt = q(a/r)Ωcos(φ) + uqa/r²

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

rd²φ/dt² = q(a/r)Ωcos(φ) + cos(φ)(dθ/dt)q(a/r)
= q(a/r)cos(φ)[Ω + dθ/dt]

The Mutual Determination of
the Rates of Change of Latitude and Longitude

The two equations determining φ and θ are

(rcos(φ))²(Ω + dθ/dt) = Λ
rd²φ/dt² = q(a/r)cos(φ)[Ω + dθ/dt]

These may be reduced to

rd²φ/dt² = q(a/r)cos(φ)Λ/(rcos(φ))²
rd²φ/dt² = q(a/r)Λ/r²cos(φ)

This last equation may be multiplied by (dφ/dt) and the result integrated to give

½[dφ/dt]2 = C + [qaΛ/r4][ln|sec(φ) + tan(φ)|]

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

dφ/dt = [2C + 2(qaΛ/r4)[ln|sec(φ) + tan(φ)|]]1/2

The equation for dθ/dt is

dθ/dt = (Λ/r²)sec²(φ) - Ω

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.,

dθ/dφ = [(Λ/r²)sec²(φ) - Ω]/[2C + 2(qaΛ/r4)[ln|sec(φ) + tan(φ)|]]1/2

The Point of Recurvature of the Hurricane Path

The recurving latitude occurs where dθ/dφ = 0; i.e.,

(Λ/r²)sec²(φ*) - Ω = 0
or, equivalently
cos²(φ*) = Λ/(Ωr²)

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

cos²(φ*) = cos²(φ0)[1 + (dθ/dt)0/Ω]
or, equivalently
cos(φ*) = cos(φ0)[1 + (dθ/dt)0/Ω]1/2

Simplification of the Equation
for the Trajectory of a Hurricane

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

C = - [qaΛ/r4][ln|sec(φ0) + tan(φ0)|]

and hence the equation for dφ/dt becomes

dφ/dt = √2[qaΛ/r4ln{|sec(φ) + tan(φ)|/|sec(φ0) + tan(φ0)|}]1/2

Using this expression plus the expression for Λ allows the equation for dθ/dφ to be expressed as

dθ/dφ =
                         (1/√2)[(Ω+(dφ/dt)0)cos²(φ0)/cos²(φ) − Ω]                         
[(qa/r²)(Ω+(dφ/dt)0)cos²(φ0))ln{|sec(φ) + tan(φ)|/|sec(φ0) + tan(φ0)|}]1/2

A rough approximation of the equation is

dθ/dφ = A*sec²(φ)+B

where A and B are constants.

Thus the general shape of the trajectory is of the form

θ = A*tan(φ)+B*φ + C

This can take the form of a skewed quadratic of θ in terms of φ. The nature of the actual trajectories of hurricanes is illustrated below:

As can be seen in the chart the latitude of recurvature φ* is about 30°N.

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