Dynamics and Thermodynamics of Hurricane

Typhoon is a typical storm in the region of the Indian or western Pacific oceans. Hurricane is a storm with a violent wind, in particular a tropical cyclone in the Caribbean. Both of them form over hot oceans at 10 - 30 degree of latitude. The Coriolis force (resultant force between the centrifugal force and gravitational force of earth) in this region drives the wind toward the eye. The result is the angular momentum associated with the earth's rotation is concentrated into angular momentum associated with hurricane winds. The cyclone grow with the energy gained from expansion and compression of vapor over the sea.

The hurricane's eye is formed when the centrifugal force of the air equal the inward-directed pressure gradient force. The place where the forces balanced each other is called the eye wall. The pressure inside the eye is the lowest and increase rapidly outside the wall. By the simplified model of pressure distribution, the maximum tangential winds are found at the outer side of the eye which is twice the radius of the eye.

Because the mechanical energy of the hurricane is come from the heat of the ocean, it can analogous to Carnot heat engines. A Carnot process is a closed cycle consists two expansion phases and two compression phases.

1. The air in the boundary layer spirals in from the place over the sea-level toward the eye wall isothermally (constant temperature ~26-28 C). Pressure decreases to ~P=90 kPa as the air approaches the low-pressure eye. Evaporation from the sea surface causes the increase of entropy. It is the major source of energy for the storm. (isothermal expansion)
2. Inside the wall, vertical air flow rises moist adiabatically to the top of the eye wall (~15km above sea surface). At the top of the wall, temperature drops to -18C and pressure is as low as 25 kPa. In this adiabatic process, entropy conserves because air expand. (adiabatic expansion)
3. Once the cloudy air reaches the top, it spirals outward at constant altitude (travel to the edge of the hurricane ~700 km). There is a pressure gradient between the eye (25 kPa) and the atmosphere (20 kPa). During this high-altitude outflow, air rapidly loses heat due to infrared radiation, causing its temperature to decrease from -18 C to -83 C (away from the eye wall). Entropy decrease because the cooling converts more water vapor into precipitation. (isothermal compression) For the temperature of the core, see the note below.
4. Finally, at the edge of the hurricane, air subsides dry adiabatically. Temperature increases adiabatically to 28 C, due to the compression as air descends into higher pressure (normal level). This dry adiabatic process preserves entropy. (adiabatic compression)

The gain of entropy at one temperature and loss at a different temperature allows the Carnot engine to produce mechanical energy.

Note:
Because of the latent heating, the centre of hurricane is warmer than the surrounding. Latent heat warms the whole depth of the hurricane core. While the eye might be only ~2 C warmer near the sea surface, it can be 10 C warmer at 12 km altitude. Because warm layers of air have greater pressure, it will spread out horizontally at the top of the hurricane.

Summary
The angular momentum of the winds is maintained by the Coriolis force of the earth. If it exist long enough then boundary-layer air is drawn toward the eye. The traveling air carries quite amount of water vapor due to evaporation as it blows over the increasingly rough sea. The cyclonically-spiraling air (with heavy moist) reaches the eye wall and rises to produce heavy rain. The warm water vapor is the main source of energy for the hurricane. At the top of the hurricane, the rising air spreads out, creating anticyclonic outflow that is observed by satellite. The outflow eventually subsides back toward the sea surface.