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 : theoryconservation of angular momentum
‣ for a unit mass of air at distance r from the center of atropical storm, absolute angular velocity is :
0DtDm
)2
()sin(2
rvfr
vrrm
Magnitude scale : 104 106
rvm
‣ In hurricane, the quantity rvθ , is constant for any given air parcel (can differ from parcel to parcel)
‣ Its absolute angular momentum, m, about the axis of cylindrical coordinate is :
r
vva
sin v : tangential wind
r : radial distance of the air parcel from the hurricane eye
⇒
⇒
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 :
• 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
Inner eyewall
r
v
r
va
TangentialWind (m/s)
constant
VΘmax
rmax rmax
ω
numericalapplication
constant
∂ V / ∂ r⍬
Source : Sheets, 80
5.6.3 Hurricanes : theory absolute vorticity :inner eyewall
fsa 4010.2 13 ⇨
ζa constant and maximum inner eyewall
40 km
f
Numerical application of ζa at 20°N ( ) inner eyewall, at 40 km :
- V⍬max = 40 m/s at rmax=40 km
1510.5 sf
40
⇨
OInner wall
133
max
max 10.110.40
40 sr
v
133
10.110.40
40
sr
v
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:
r
v
r
va
TangentialWind (m/s)
VΘmax
rmax rmax
5.0
maxmax
rrvv
Outer eyewallOuter eyewall
numericalapplication
5.0
maxmax
2
r
r
r
va
⇨ ⇨ Proceeding outwards, ζa decrease exponentially outer eyewall
Vθ Vθ
Source : Sheets, 80
5.6.3 Hurricanes : theory absolute vorticity : outer eyewall
fsa 5,310176.0 14 ⇨
Numerical application of ζa at 20°N ( ) outer the eyewall, at 80 km :
- V⍬max = 40 m/s at rmax=40 km
40
Outer wall
5.0
maxmax
2
r
r
r
va
⇨
5.0
3
3
3 10.80
10.40
10.160
40
a
⇨
1510.5 sf
40 km
f
O80 km
3.5
⇨ Proceeding outwards, ζa decrease exponentially outer
eyewall
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 : theoryFormation of tornadoes
Knowing that the angular momentum, rV ⍬ is constant for a given air parcel :
EyeEye
Hurricane
r1
v1
r2
v2
- rV ⍬ constant simply means that r1V⍬1 = r2V⍬2
- what happens as an air parcel spirals inward toward the center of the hurricane?
Numerical application : let V⍬1 = 10 kts r1 = 500 km
If r2 = 30 km, then using the equationr1V⍬1 = r2V⍬2 we find that V⍬2 = (V⍬1 .r1)/r2 = 167 kts!!!
5.6.3 Hurricanes : theoryFormation of tornadoes
The same mechanism is at work in tornadoes
Note spiral bands converging towardthe center
Source : Image satellite de la NOAA
5.6.3 Hurricanes : theoryFormation of tornadoes
Hurricanes often produce tornadoes :
distribution
Location of allhurricane-spawnedtornadoes relativeto hurricane centerand 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 : theoryformation 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
CoriolisForce
Pressureforce
• 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.
01
0
2
r
pfv
r
v
t
vr
0t
vr
r
pfv
r
v
0
2 1
Inwards rcr :
⇨ ∂ Vr/ ∂ r becomes positive providing for outwards acceleration
Inwards rcr :
5.6.3 Hurricanes : theoryformation of the eyewall : angular momentum
z
e
re
Inward from critical radius : supergradient-wind
r
pfv
r
v
0
2 1
ieF
Centrifugal Force
ieF
chF
CoriolisForce
chF
pF
Pressureforce
pF
NorthernHemisphere
⇨
0t
vr
0t
vr
⇨ Convergent flow can’tgo further inwards and resulting in strongupwards motions =Birth of the Eyewall !!
Vr <rcr
5.6.3 Hurricanes : theoryVertical tilt of the eyewall
z
re
chF
pF
NorthernHemisphere
0t
vr
ieF
pF 0
t
vr
chF
ieF
As the inward directed pressure gradient force decreasewith height, the outward directed radial acceleration, ∂ Vr/ ∂t,
increases, so that the rising parcel is thrust outward, which inturn entails a widening of the eye with height (Hastenrath, p.216)
5.6.3 Hurricanes : theoryformation of the eyewall : pumping Ekman
f
40 kmO 80 km3.5
40
⇨ 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
= Ekman pumping
z
The strong divergence in upper troposphere is divided into 2branches :
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)
NorthernHemisphere
5.6.3 Hurricanes : theoryFormation of the eye
400 km
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 : theoryDevelopment 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 finiteamplitude 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 : theoryDevelopment 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’
MoistAdiabaticexpansion
Dry AdiabticCompression
Isothermal
Isothermal ascompressional heating is
balanced by radiationalheat loss into space
Source : Emanuel, 91
5.6.3 Hurricanes : theoryDevelopment of tropical storm : Carnot cycle
1
21T
T
Q
W
• The Carnot cycle gives the best efficiency for a ‘heat engine’ :
W: work producedQ : heat furnishedT2 : cold source = temperature at tropopauseT1 : 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) :
AUGUSTFEBRUARY
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
Absolute angular momentum (103m2s-1) in hurricane (JAS, kerry, 86, p.585)
⇒ In a hurricane, airstream follow iso-m = inertial stability
Source : Emanuel, 86
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
gH f
Kw .2sin.
2 0
wH: Vertical velocity at the top of Ekman layer : Ekman PumpingK: coeff. of eddy viscosityα0 : angle of inflow between observed wind and geostrophic wind at the bottom of Ekman layerζg: 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
Equation of vertical velocity at top of Ekman layer,called ‘Ekman pumping’ :
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