5.6.1 Hurricane : introduction 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

Two important dynamic quantities : - angular momentum 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 : : 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 : 104 106 ⇒ ‣ In hurricane, the quantity rvθ , is constant for any given air parcel (can differ from parcel to parcel) ⇒

Two important dynamic quantities : - angular momentum 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 : 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

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

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

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

Two important dynamic quantities : - angular momentum 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!!! Eye r1 v1 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

Two important dynamic quantities : - angular momentum 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 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 :

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

5.6.3 Hurricanes : theory Vertical tilt of the eyewall z 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)

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

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)

Two important dynamic quantities : - angular momentum 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

Hypothesis of Kerry Emmanuel (JAS, 86, p.586) : 5.6.3 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

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

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

Two important dynamic quantities : - angular momentum 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

Equation of vertical velocity at top of Ekman layer, 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’ : 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