Rankine Vortices

One approach to modeling tropical cyclones is as rankine vortices. A Rankine vortex is one in which the inner part turns as a disk and in the outer part the velocity is inversely proportional to the distance from the center; i.e.,

v(r) = v_a(r/a) for r≤a
v(r) = v_a/(r/a) for r≥a
 

where v_a is the tangential velocity at r=a. In a cyclone a would be the radius of the eyewall.

In computing the angular momentum in a Rankine vortex each band of width Δr in the outer portion has the same angular momentum so a Rankine vortex that extends to infinity would have an infinite angular momentum. Or, stated another way, the angular momentum of a Rankine vortex is directly proportional to the outer limit of the vortex even though the velocity at the outer limit may be negligibly small. But tropical cyclones are on the sphere of the Earth and therefore necessarily of finite extent.

A Rankine Vortex on a Sphere

Let θ be the angle from the center of the vortex to a point a distance r from the center and let R be the radius of the sphere. The distance r is then Rθ and hence the formula for the velocities in a Rankine vortex on a sphere is:

v(θ) = v_a(Rθ/a) for Rθ≤a
v(θ) = v_a/(Rθ/a) for Rθ≥a
 

The angular momentum of an infinitesimal element with respect to the spin axis of the vortex is the areal density ρ times the velocity times the distance s of the element from the spin axis. The distance s of a point at angle θ from the spin axis is Rsin(θ). The area of a band of with dθ centered at an angle θ is (2πs)Rdθ=(2πRsin(θ)dθ.

The angular momentum L is then the integral of the angular momenta in bands of width Rdθ; i.e.,

L = ∫₀^π(ρv(θ)Rsin(θ))(2πRsin(θ))Rdθ
= 2πρR³∫₀^πv(θ)sin²(θ)dθ
 

In meteorology quantities are usually stated per unit mass. The mass of the vortex is ρ(4πR²) so the angular momentem of the vortex per unit mass is

L/M = (R/2)∫₀^πv(θ)sin²(θ)dθ
 

The integral must be evaluated for the inner and outer parts of the vortex separately. For the inner part of the vortex the angle θ ranges from 0 to a/R so the integral is

∫₀^a/Rv_a(Rθ/a)sin²(θ)dθ
 

The indefinite integral of

θsin²(θ)
is
θ²/4 - (θ/4)sin(2θ)-(1/8)cos(2θ)
 

so the angular momentum of the inner portion of the vortex evaluates to

(R/2)v_a(R/a)[θ_a²/4 - (θ_a/4)sin(2θ_a)-(1/8)cos(2θ_a) + (1/8)]
where θ_a=a/R.
 

The evaluation of the angular momentum for the outer part of the vortex involves the integral of sin²(θ)/θ. The indefinite integral of this quantity is

ln(θ)/2 - [cos(2θ) - 1]
 

Thus the angular momentum per unit mass for the outer portion of the vortex evaluates to

(R/2)v_a(a/R)[ln(π)/2 - ln(θ_a)/2 + (cos(2θ_a)-1)]
= v_a(a/2)[ln(π/θ_a)/2 + (cos(2θ_a)-1)]
 

where again θ_a is the angular distance of the eyewall of the vortex from the center of the sphere; i.e., a/R.

Representative Values
of the Parameters and
Values for Cyclones

In SI units the typical values for a hurricane are:

• a = 3×10⁵ m
• v_a = 50 m/s
• θ_a = 3×10⁵/6.37×10⁶ = 0.0471 radians = 2.7°
• inner L/M = 7.5×10⁶ m²/s
• outer L/M = 15.8×10⁶ m²/s
• total L/M = 23.3×10⁶ m²/s