The horizontal momentum equations for air flow in natural (or path-following) coordinates are:

dV/dt = -(1/ρ)(∂p/∂s)
 
V²/r + fV = -(1/ρ)(∂p/∂n)
 

where s denotes distance along the path, n distance perpendicular to the path, and r is the radius of curvature of the path. In the equations, V is wind velocity, p is pressure, ρ is air density, f is the Coriolis parameter and t time.

For an axially symmetric cyclone the first equation reduces to 0=0 and the second to:

V²/r + fV = (∂Φ/∂r)
 

where r is now the distance to the center of the cyclone and Φ is the geopotential. This equation is of interest because it establishes a connection between the the variation in velocity with distance from the center of the cyclone and a quantity which can, through the hydrostatic balance equation, be linked to the vertical profile of temperature in the cyclone. But first consider the dependence of Φ on distance from the center for a known velocity profile.

Consider a Rankine vortex in which

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

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

Thus for a cyclone in the region within the eyewall

∂Φ/∂r = (v_a/a)²r + fv_a(r/a)
and hence
Φ = Φ(0) + [(v_a/a)² + fv_a/a]r²
 

At the eyewall and outside of it

∂Φ/∂r = a²v_a²/r³ + fav_a/r
and hence
Φ(r) = C - a²v_a²/(3r²) + fav_aln(r)
 

where C is a constant of integration. This constant C may be evaluated in terms of the geopotential at the eyewall; i.e.,

C = Φ(a) + av_a²/3 - fav_aln(a)
 

In the general case, hydrostatic balance requires that

∂Φ/∂z* = RT/H
 

where T is the absolute temperature, z* is the geopotenial height, H is the scale height and R is the gas constant. The geopotential height z* is the vertical coordinate in the log-pressure coordinate system and is equal to -Hln(p/p₀). The geopotential height z* comes close to being the same as the physical height.

If the equation

V²/r + fV = (∂Φ/∂r)
 

is differentiated with respect to z* the result is

[2V/r + f]∂V/∂z* = (∂/∂z*)(∂Φ/∂r)
but
(∂/∂z*)(∂Φ/∂r) = (∂/∂r)(∂Φ/∂z*) = (R/H)(∂T/∂r)
 

so the above reduces to

[2V/r + f]∂V/∂z* = (R/H)(∂T/∂r)
 

This equation connects the vertical variation in horizontal wind velocity to the variation in temperature with distance from the center of the cyclone.

Since ∂V/∂z*<0 above the boundary layer, ∂T/∂r must also be negative above that boundary layer. Thus the temperature is at its maximum at the center of the cyclone, at least in its upper levels.

Sources:

James R. Holton, An introduction to dynamic meteorology, Elsevier Academic Press, Burlington, MA, 2004.

Charney and Eliassen, "On the growth of the hurricane depression," Journal of Atmospheric Science, vol. 21 (1964), pp. 68-75.