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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

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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

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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 : Magnitude scale : 10 4 10 6 ‣ In hurricane, the quantity rv θ, 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 : : tangential wind : radial distance of the air parcel from the hurricane eye ⇒ ⇒

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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

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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 r max of maximum wind V ⍬ max : - the tangential flow can be represented as a solid rotation with angular velocity, ω =V ⍬ max / r max, constant - ∂ V ⍬ / ∂ r is constant ⇒ ζ a is constant inner eyewall Inner eyewall Tangential Wind (m/s) constant V Θmax r max ω numerical application constant ∂ V ⍬ / ∂ r Source : Sheets, 80

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

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

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

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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

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

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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

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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

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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

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5.6.3 Hurricanes : theory formation of the eyewall n Reminder : knowing that the angular momentum, r V ⍬ is constant for a given air parcel, V ⍬ increase as the air flows towards the center n 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, r cr, 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 r cr : ⇨ ∂ V r / ∂ r becomes positive providing for outwards acceleration Inwards r cr :

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

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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, ∂ V r / ∂t, increases, so that the rising parcel is thrust outward, which in turn entails a widening of the eye with height (Hastenrath, p.216)

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5.6.3 Hurricanes : theory formation of the eyewall : pumping Ekman 40 km O 80 km 3.5 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 ζ a 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

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

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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

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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

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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 Dry Adiabtic Compression Isothermal Isothermal as compressional heating is balanced by radiational heat loss into space Source : Emanuel, 91

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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 T tropopause and SST. ⇨ Greater this difference is, greater the conversion of enthalpic energy into kinetic energy is and fall pressure is ⇨ Under climatological SST and T tropopoause, it can be calculated the minimum sustainable central pressure of tropical cyclones (hPa) : AUGUSTFEBRUARY Source : Emanuel, 91

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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

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5.6.3 Hurricanes : theory angular momentum Absolute angular momentum (10 3 m 2 s -1 ) in hurricane (JAS, kerry, 86, p.585) ⇒ In a hurricane, airstream follow iso-m = inertial stability Source : Emanuel, 86

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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 w H : Vertical velocity at the top of Ekman layer : Ekman Pumping K: coeff. of eddy viscosity α 0 : 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, w H, 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 (w H ) at the top of the Ekman layer Equation of vertical velocity at top of Ekman layer, called ‘Ekman pumping’ :

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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

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