1. 5.6.1 Hurricane : introduction
Chap Hurricanes
5.6.1 Hurricane : introduction
5.6.2 Hurricane structure
5.6.3 Hurricane : theory
5.6.4 Forecasting of hurricane
sommaire chap.5
sommaire

2. Two important dynamic quantities : - angular momentum
Hurricanes : theory
Two important dynamic quantities :
- angular momentum
- asolute vorticity
Formation of tornadoes
Formation of eyewall and eye
Development of tropical cyclone
Sommaire hurricane
sommaire

3. 5.6.3 Hurricanes : theory conservation of angular momentum
‣ for a unit mass of air at distance r from the center of a
tropical storm, absolute angular velocity is :
: tangential wind
: radial distance of the air parcel
from the hurricane eye
‣ Its absolute angular momentum, m, about the axis of
cylindrical coordinate is :
Magnitude scale : ⇒
‣ In hurricane, the quantity rvθ , is constant for any given
air parcel (can differ from parcel to parcel) ⇒

4. Two important dynamic quantities : - angular momentum
Hurricanes : theory
Two important dynamic quantities :
- angular momentum
- asolute vorticity
Formation of tornadoes
Formation of eyewall and eye
Development of tropical cyclone
Sommaire hurricane
sommaire

5. 5.6.3 Hurricanes : theory absolute vorticity : inner eyewall
Absolute vorticity about the axis of cylindrical coordinate is :
constant
VΘmax
rmax ω
constant
∂ V⍬/ ∂ r
Inner Eyewall (R<40 km)
From the centre to the radius rmax of maximum wind V⍬max :
- the tangential flow can be represented as a solid rotation
with angular velocity , ω =V⍬max/ rmax , constant
- ∂ V⍬/ ∂ r is constant
⇒ ζa is constant inner eyewall
numerical
application
Tangential
Wind (m/s)
Source : Sheets, 80
Inner eyewall

6. 5.6.3 Hurricanes : theory absolute vorticity :inner eyewall
Numerical application of ζa at 20°N ( )
inner eyewall, at 40 km :
- V⍬max = 40 m/s at rmax=40 km ⇨ ⇨
ζa constant and maximum inner eyewall 40 O
Inner wall
40 km

7. 5.6.3 Hurricanes : theory absolute vorticity : outer eyewall
Absolute vorticity about the axis of cylindrical coordinate is :
Outer Eyewall (R>40 km)
Outside rmax, the radial variation of V⍬ can generally represented as: ⇨
⇨ Proceeding outwards, ζa decrease exponentially outer eyewall
numerical
application
VΘmax Vθ
Tangential
Wind (m/s)
Source : Sheets, 80
rmax
rmax
Outer eyewall
Outer eyewall

8. 5.6.3 Hurricanes : theory absolute vorticity : outer eyewall
Numerical application of ζa at 20°N ( )
outer the eyewall, at 80 km :
- V⍬max = 40 m/s at rmax=40 km ⇨ ⇨ ⇨
⇨ Proceeding outwards, ζa decrease exponentially outer
eyewall 40
3.5 O
40 km
80 km
Outer wall

9. Two important dynamic quantities : - angular momentum
Hurricanes : theory
Two important dynamic quantities :
- angular momentum
- asolute vorticity
Formation of tornadoes
Formation of eyewall and eye
Development of tropical cyclone
Sommaire hurricane
sommaire

10. 5.6.3 Hurricanes : theory Formation of tornadoes
Knowing that the angular momentum, rV⍬ is constant for a given air parcel :
- what happens as an air parcel spirals inward toward the center of the hurricane?
- rV⍬ constant simply means that r1V⍬1 = r2V⍬2
Numerical application :
let V⍬1 = 10 kts r1 = 500 km
If r2 = 30 km, then using the equation
r1V⍬1 = r2V⍬2 we find that V⍬2 = (V⍬1 .r1)/r2 = 167 kts!!!
Eye r1 v1
Eye r2 v2
Hurricane

11. 5.6.3 Hurricanes : theory Formation of tornadoes
The same mechanism is at work in tornadoes
Note spiral bands converging toward the center
Source : Image satellite de la NOAA

12. 5.6.3 Hurricanes : theory Formation of tornadoes
Hurricanes often produce tornadoes :
distribution
Location of all hurricane-spawned tornadoes relative to hurricane center and motion.
Source : McCaul,91

13. Two important dynamic quantities : - angular momentum
Hurricanes : theory
Two important dynamic quantities :
- angular momentum
- asolute vorticity
Formation of tornadoes
Formation of eyewall and eye
Development of tropical cyclone
Sommaire hurricane
sommaire

14. 5.6.3 Hurricanes : theory formation of the eyewall
Reminder : knowing that the angular momentum, rV⍬ is constant
for a given air parcel, V⍬ increase as the air flows towards the center
Radial equation of motion (disregarding friction)
Centrifugal
Force
Coriolis
Force
Pressure
force
Proceeding inwards, V⍬ and even more V⍬2/r increase and to
balance, the pressure gradient must increase too (MSPL fall
inwards).
Inward from a critical radius, rcr, any pressure force can’t
anymore balance the fast increasing centrifugal force V⍬2/r.
We also say that, the flow V⍬ , is becoming supergradient.
Inwards rcr :
⇨ ∂ Vr/ ∂ r becomes positive providing for outwards acceleration
Inwards rcr :

15. 5.6.3 Hurricanes : theory formation of the eyewall : angular momentum
Inward from critical radius : supergradient-wind z
Northern
Hemisphere
r <rcr
Pressure
force ⇨
Convergent flow can’t
go further inwards
and resulting in strong
upwards motions =
Birth of the Eyewall !!
Coriolis
Force
Centrifugal
Force

16. 5.6.3 Hurricanes : theory Vertical tilt of the eyewallz
Northern
Hemisphere
As the inward directed pressure gradient force decrease
with height, the outward directed radial acceleration, ∂ Vr/ ∂t,
increases, so that the rising parcel is thrust outward, which in
turn entails a widening of the eye with height (Hastenrath, p.216)

17. 5.6.3 Hurricanes : theory formation of the eyewall : pumping Ekman
⇨ In addition to the angular momentum implications for
the formation of eyewall, Anthes (82) point out that
for a circular vortex in solid rotation, Ekman pumping
(which is maximum when ζa is maximum) becomes inefficient
near the axis of rotation.
⇨ The max. upward motion occur at some distance outward
from the center
⇨ this boudary layer processes would be further conducive
to the development of an eyewall
inefficient 40
3.5
80 km O
40 km
= Ekman pumping

18. 5.6.3 Hurricanes : theory Formation of the eye
Northern
Hemisphere z
400 km
The strong divergence in upper troposphere is divided into 2
branches :
⒈one part of the airstream is strongly subsiding (+ 3m/s)
inward the eyewall originating the eye
⒉the other part of the airstream is spiraling outward the
eyewall with light subsidence outwards the hurricane (400 km
from center)

19. Two important dynamic quantities : - angular momentum
Hurricanes : theory
Two important dynamic quantities :
- angular momentum
- asolute vorticity
Formation of tornadoes
Formation of eyewall and eye
Development of tropical cyclone
Sommaire hurricane
sommaire

20. Hypothesis of Kerry Emmanuel (JAS, 86, p.586) :
Hurricanes : theory Development of tropical cyclone : Carnot cycle
Hypothesis of Kerry Emmanuel (JAS, 86, p.586) :
Tropical cyclones are developped, maintained and intensified by self-induced anomalous fluxes of moist enthalpy (sensible and latent heat transfer from ocean) with neutral environment, i.e. with no contribution from preexisting CAPE.
In this sense, storms are taken to result from an air-sea interaction instability, which requires a finite amplitude initial disturbance.
2. K. Emmanuel demonstrates that a weak but finite
amplitude vortex (wind variation at least 12m/s over a
radius of 82 km) can grow in a conditional neutral
environmnent.
3. These precedings points suggest that the steady tropical cyclone may be regarded as a simple Carnot heat engine in which :
- air flowing inward in the boudary layer acquires moist enthalpy from the sea surface,
- then ascends (eyewall),
- and ultimately gives off heat at the much lower
temperature of the upper troposphere

21. 5.6.3 Hurricanes : theory Development of a hurricane : Carnot cycle
In other words, the Carnot heat engine convert thermal
energy (enthalpy) into kinetic energy (wind)
the Carnot cycle is defined by : 2 isothermals
: 2 adiabatics
A schematic of the heat engine ‘Carnot’
Source : Emanuel, 91
Moist
Adiabatic
expansion
Isothermal as
compressional heating is
balanced by radiational
heat loss into space
Dry Adiabtic
Compression
Isothermal

22. 5.6.3 Hurricanes : theory Development of tropical storm : Carnot cycle
The Carnot cycle gives the best efficiency for a ‘heat engine’ :
W: work produced
Q : heat furnished
T2 : cold source = temperature at tropopause
T1 : hot source = Sea SurfaceTemperature
⇨ The efficiency of the Carnot cycle depends of the vertical gradient
of temperature between Ttropopause and SST.
⇨ Greater this difference is, greater the conversion of enthalpic
energy into kinetic energy is and fall pressure is
⇨ Under climatological SST and Ttropopoause , it can be calculated
the minimum sustainable central pressure of tropical cyclones (hPa) :
AUGUST
FEBRUARY
Source : Emanuel, 91

23. Two important dynamic quantities : - angular momentum
Hurricanes : theory
Two important dynamic quantities :
- angular momentum
- asolute vorticity
Formation of tornadoes
Formation of eyewall and eye
Development of tropical cyclone
Sommaire hurricane
sommaire

24. 5.6.3 Hurricanes : theory angular momentum
Source : Emanuel, 86
Absolute angular momentum (103m2s-1) in hurricane (JAS, kerry, 86, p.585)
⇒ In a hurricane, airstream follow iso-m = inertial stability

25. Equation of vertical velocity at top of Ekman layer,
Hurricanes : theory ‘pumping Ekman’
Reminder :
Both, convection and friction forces in the boundary layer
generates convergent low-level fields
- The equation of absolute vorticity explains why inflow produces
cyclonic spin-up in proportion to the existing environmental
vorticity field
Equation of vertical velocity at top of Ekman layer,
called ‘Ekman pumping’ :
wH: Vertical velocity at the top of Ekman layer : Ekman Pumping
K: coeff. of eddy viscosity
α0 : angle of inflow between observed wind and geostrophic wind at
the bottom of Ekman layer
ζg: geostrophic vorticity
⇨ Vertical velocity at the top of Ekman layer, wH, is proportionnal
to the geostrophic vorticity
⇨ We can also add that vertical velocity, w, increase with height
inside the boundary layer (not explained with this equation) and is
maximum (wH) at the top of the Ekman layer

26. References
Anthes, R. A., 1982 : ‘Tropical cyclones, their evolution, structure and effects’. Meteorological Monographs, Vol.19, n°41, Amer. Meteor. Soc., Boston, 208p.
Carlson, T. N.and J. D. Lee : Tropical meteorological. Pennsylvania State University, Independent Study by Correspondence, University Park, Pennsylvania, 387 p.
Eliassen, A., 1971 :’On the Ekman layer in a circular vortex’. J. Meteor. Soc. Japan, 49, special isuue, p
Emanuel, Kerry A., 1986 : An Air-sea Interaction theory for tropical cyclone; pt1; steady state maintenance. J. of Atm. Science, Boston, vol. 43, n°6, p
Emanuel, Kerry A., 1991, The theory of hurricane : Annual review of Fluid Mechnics, Palo Alto, CA. Vol.23, p
McCaul, E. W. Jr., 1991 : ‘Buoyancy and shear characteristics of hurricane-tornado environments’. Mon. Weather Rev., MA. Vol.119, n°8, p
Merrill, R. T., 1993 : ‘Tropical Cyclone Structure’ –Chapter 2, Global Guide to Tropical Cyclone Forecasting, WMO/Tropical Cyclone- N°560, Report N° TCP-31, World Meteorological Organization; Geneva, Switzerland
Palmen, E. and C. W. Newton, 1969 : Atmospheric circulation systems. Academic Press, New York and London, 603p.
Sheets, R. C., 1980 : ‘Some Aspects of tropical cyclone modification’. Australian Meteorological magazine, Canberra, vol. 27, n°4, pp