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