The Tropical Atmosphere: Hurricane Incubator

Transcription

1 The Tropical Atmosphere: Hurricane Incubator Images from journals published by the American Meteorological Society are copyright AMS and used with permission.

2 A One-Dimensional Description of the Tropical Atmosphere

3 Elements of Thermal Balance: Solar Radiation Luminosity: 3.9 x J s -1 = 6.4 x 10 7 Wm -2 at top of photosphere Mean distance from earth: 1.5 x m Flux density at mean radius of earth S 0 L 0 4π d 2 2 = 1370 Wm

4 Stefan-Boltzmann Equation: F =σt4 σ = Wm 2 K 4 Sun: T 4 = σ Wm T 6,000 K

5 Disposition of Solar Radiation: Total absorbed solar radiation = S 0 1 a p π r p 2 a p planetary albedo ( 30%) Total surface area = 4π r 2 p Absorption per unit area = S 0 1 a 4 p Absorption by clouds, atmosphere, and surface

6 Terrestrial Radiation: Effective emission temperature: σt 4 S 0 1 a e 4 p Earth: T = 255K = e 18 C Observed average surface temperature = 288 K = 15 C

7 Highly Reduced Model Transparent to solar radiation Opaque to infrared radiation Blackbody emission from surface and each layer

8 Radiative Equilibrium: Top of Atmosphere: S 4 σt = a = σt A 4 p e T A = T e Surface: σt s 4 = σt A 4 + S 1 4 T = 2 4 T s e 0 ( 1 a p ) = 2σT e 4 = 303 K

9 Surface temperature too large because: Real atmosphere is not opaque Heat transported by convection as well as by radiation

10 Extended Layer Models TOA : σt 2 4 = σt e 4 T 2 = T e Middle Layer : 2σT 1 4 = σt 2 4 +σt s 4 = σt e 4 +σt s 4 Surface : σt s 4 = σt e 4 +σt T s = 3 4 T e T 1 = 2 4 T e

11 Effects of emissivity<1 Surface : 2ε σt 4 = ε σ T 4 + ε σ T 4 A A A 1 A s ) 1 4 A e s T = ( 5 T 321K < T 2 Stratosphere : 2εσT 4 = ε σ T 4 t t A 2 ) 1 4 t e e T = ( 1 T 214K < T 2

12 Full calculation of radiative equilibrium:

13 Problems with radiative equilibrium solution: Too hot at and near surface Too cold at a near tropopause Lapse rate of temperature too large in the troposphere (But stratosphere temperature close to observed)

14 Missing ingredient: Convection As important as radiation in transporting enthalpy in the vertical Also controls distribution of water vapor and clouds, the two most important constituents in radiative transfer

15 When is a fluid unstable to convection? Pressure and hydrostatic equilibrium Buoyancy Stability

16 Hydrostatic equilibrium: Weight : gρδxδ yδz Pressure : pδ δ x y ( p +δ p ) δ xδ y F = MA: ρδx δy δ z dw = gρδxδyδ z δ pδxδy dt dw p 1 dt = g α z, α = = specific volume ρ

17 Pressure distribution in atmosphere at rest: RT R * Ideal gas : α =, R p m Hydrostatic : 1 p ln( p) g = = p z z RT Isothermal case : p = p e z H, H RT = "scale height " 0 g Earth: H~ 8 Km

18 Buoyancy: Weight g ρ δ x δy δz : b Pressure : pδxδ y ( p + δ p) δxδ y dw F = MA : ρ bδxδyδ z = gρ b δ xδyδ z δ pδxδ y dt dw p p = g α but = g b dt z z α e dw = g α b α e B dt α e

19 Buoyancy and Entropy Specific Volume: α = 1 ρ Specific Entropy: s α T Maxwell : α =α( ps, s = p p s B = g (δα ) α T = p δ s = δ s s p p s (δα ) p g T = δ s = α α p s T z s δ s Γ δ s

20 The adiabatic lapse rate: First Law of Thermodynamics : Q ds rev dt dα = T = c v + p dt dt dt dt d (α p) dp = c v + α dt dt dt = (c + R) dt α dp v dt dt = c dt α dp p dt dt Adiabatic : c p dt αdp = 0 Hydrostatic : c dt + gdz = 0 p ( dt dz) s = g cp Γ d

21 Γ= g c p Earth s atmosphere: Γ= 1 K 100 m

22 Model Aircraft Measurements (Renno and Williams, 1995)

23 Radiative equilibrium is unstable in the troposphere Re-calculate equilibrium assuming that tropospheric stability is rendered neutral by convection: Radiative-Convective Equilibrium

24 Better, but still too hot at surface, too cold at tropopause

25 Above a thin boundary layer, most atmospheric convection involved phase change of water: Moist Convection

26 Moist Convection Significant heating owing to phase changes of water Redistribution of water vapor most important greenhouse gas Significant contributor to stratiform cloudiness albedo and longwave trapping

27 Stability Assessment using Tephigrams: Pressure (mb) Temperature (C)

28 Stability Assessment using Tephigrams: Convective Available Potential Energy (CAPE): p CAPE i (α α i p n p e )dp p = i R d (T ρ T )d ln ( ) p p ρ e p

29 Tropical Soundings: Virtually no CAPE

30 Annual Mean Kapingamoronga

31 Tropical Cyclones: Steady State Physics. Where does the Energy Come From?

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33 Angular momentum per unit mass M = rv + Ωsin (θ )r 2

34 Energy Production

35 Carnot Theorem: Maximum efficiency results from a particular energy cycle: Isothermal expansion Adiabatic expansion Isothermal compression Adiabatic compression Note: Last leg is not adiabatic in hurricane: Air cools radiatively. But since environmental temperature profile is moist adiabatic, the amount of radiative cooling is the same as if air were saturated and descending moist adiabatically. Maximum rate of working: Ts T W = o Q T s

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37 Total rate of heat input to hurricane: Q = 2π 0 r 0 ρ C V (k * k ) + C V 3 rdr k 0 D Surface enthalpy flux Dissipative heating In steady state, Work is used to balance frictional dissipation: W = 2π 0 r 0 ρ C D V 3 rdr

38 0 Plug into Carnot equation: r 0 T T ρ V 3 s rdr = r o 0 C ρ C ( * D k V k 0 k) rdr T o 0 If integrals dominated by values of integrands near radius of maximum winds, 2 C k T s T o ( * V k max 0 k ) C D T o

39 Graph of Solution for particular values of coefficients:

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43 00 GMT 21 March o N 20 o N 0 o 20 o S 40 o S 0 o 60 o E 120 o E 180 o W 120 o W 60 o W 0 o

44 Numerical simulations

45 Relationship between potential intensity (PI) and intensity of real tropical cyclones

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48 Wind speed (m/s) Potential wind speed (m/s) V (m/s) Time after maximum intensity (hours) 100

49 Evolution with respect to time of maximum intensity

50 Evolution with respect to time of maximum intensity, normalized by peak wind

51 Evolution curve of Atlantic storms whose lifetime maximum intensity is limited by declining potential intensity, but not by landfall

52 Evolution curve of WPAC storms whose lifetime maximum intensity is limited by declining potential intensity, but not by landfall

53 CDF of normalized lifetime maximum wind speeds of North Atlantic tropical cyclones of tropical storm strength (18 m s 1) or greater, for those storms whose lifetime maximum intensity was limited by landfall.

54 CDF of normalized lifetime maximum wind speeds of Northwest Pacific tropical cyclones of tropical storm strength (18 m s 1) or greater, for those storms whose lifetime maximum intensity was limited by landfall.

55 Evolution of Atlantic storms whose lifetime maximum intensity was limited by landfall

56 Evolution of Pacific storms whose lifetime maximum intensity was limited by landfall

57 Composite evolution of landfalling storms

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