Chap. 5.6 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
5.6.3 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
5.6.3
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 :
v
v : tangential wind
va   sin   
r
r : radial distance of the air parcel
from the hurricane eye
‣ Its absolute angular momentum, m, about the axis of
cylindrical coordinate is :
fr 2
m  r (r sin   v )  (
 rv )
2
Magnitude scale :
104
⇒ m  rv
106
‣ In hurricane, the quantity rvθ , is constant for any given
air parcel (can differ from parcel to parcel)
⇒
Dm
Dt
0
5.6.3 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
5.6.3
Hurricanes : theory
absolute vorticity : inner eyewall
Absolute vorticity about the axis of cylindrical coordinate is :
a
• Inner Eyewall (R<40 km)
v
v


r
r
constant constant
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
numerical
application
⇒ ζa is constant inner eyewall
ω
Tangential
Wind (m/s)
VΘmax
Source : Sheets, 80
∂ V⍬/ ∂ r
rmax
rmax
Inner eyewall
5.6.3
Hurricanes : theory
absolute vorticity :inner eyewall
Numerical application of ζa at 20°N ( f  5.105 s 1 )
inner eyewall, at 40 km :
- V⍬max = 40 m/s at rmax=40 km
v max
40
3 1



1
.
10
s
3
rmax 40 .10
v
40
3 1


1
.
10
s
3
r 40.10
⇨ a
 2.103 s 1  40 f
⇨ ζa constant and maximum inner eyewall
f
40
O
Inner wall
40 km
5.6.3
Hurricanes : theory
absolute vorticity : outer eyewall
Absolute vorticity about the axis of cylindrical coordinate is :
a
v
v


r
r
• Outer Eyewall (R>40 km)
Outside rmax, the radial variation of V⍬ can generally represented as:
 
rmax 0.5
 max r
v  v
⇨
v
r 
 a   max  max 
2r  r 
⇨ Proceeding outwards, ζa
0.5
numerical
decrease exponentially outer eyewall
application
VΘmax
Tangential
Wind (m/s)
Source : Sheets, 80
Vθ
Vθ
rmax
Outer eyewall
rmax
Outer eyewall
5.6.3
Hurricanes : theory
absolute vorticity : outer eyewall
Numerical application of ζa at 20°N ( f  5.105 s 1 )
outer the eyewall, at 80 km :
- V⍬max = 40 m/s at rmax=40 km
0.5
⇨
v max  rmax 
a 


2r  r 
⇨
40  40.103 


a 
3 
3 
160.10  80.10 
0.5
⇨  a  0.176104 s  3,5 f
⇨ Proceeding outwards, ζa decrease exponentially outer
1
eyewall
f
40
3.5
O
40 km
80 km
Outer wall
5.6.3 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
5.6.3
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!!!
v1
Eye
r1
Eye
r2
v2
Hurricane
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
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
5.6.3 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
5.6.3
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
n
n
Radial equation of motion (disregarding friction)
v2
vr
1 p

 fv 
0
t
r
 0 r
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.
v2
1 p
Inwards rcr :
 fv 
r
0 r
⇨
∂ Vr/ ∂ r becomes positive providing for outwards acceleration
Inwards rcr :
vr
0
t
5.6.3
Hurricanes : theory
formation of the eyewall : angular momentum
Inward from critical radius : supergradient-wind
z
Northern
Hemisphere
 
V e

Fp
v2
1 p
 fv 
r
0 r
Centrifugal Coriolis
Force
Force

Fie

Fch
Pressure
force

Fp
vr
0
t

Fch

Fie
⇨
⇨

er
vr
0
t
Convergent flow can’t
go further inwards
and resulting in strong
upwards motions =
Birth of the Eyewall !!
5.6.3
Hurricanes : theory
Vertical tilt of the eyewall
z
Northern
Hemisphere
vr
0
t

Fp

Fch

Fp

Fie
vr
0
t

Fch

Fie

er
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)
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
f
40
3.5
O
40 km
= Ekman pumping
80 km
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)
5.6.3 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
5.6.3
Hurricanes : theory
Development of tropical cyclone : Carnot cycle
Hypothesis of Kerry Emmanuel (JAS, 86, p.586) :
1.
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
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’
Moist
Adiabatic
expansion
Source : Emanuel, 91
Isothermal as
compressional heating is
balanced by radiational
heat loss into space
Dry Adiabtic
Compression
Isothermal
5.6.3
Hurricanes : theory
Development of tropical storm : Carnot cycle
• The Carnot cycle gives the best efficiency for a ‘heat engine’ :
W
T2
 1
Q
T1
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
5.6.3 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
5.6.3
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
5.6.3
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’ :
K
wH 
. sin 2 0 . g
2f
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
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.784-789
-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. 585-604
- Emanuel, Kerry A., 1991, The theory of hurricane : Annual review of
Fluid Mechnics, Palo Alto, CA. Vol.23, p.179-196
- McCaul, E. W. Jr., 1991 : ‘Buoyancy and shear characteristics of
hurricane-tornado environments’. Mon. Weather Rev., MA. Vol.119, n°8, p.
1954-1978
- 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. 259-280