Impact of Turbulence on the Intensity of Hurricanes in...
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Impact of Turbulence on the Intensity of Hurricanes in Numerical Models*
Richard Rotunno NCAR
*Based on:
Bryan, G. H., and R. Rotunno, 2009: The maximum intensity of tropical cyclones in axisymmetric numerical model simulations. Mon. Wea. Rev., in press.
Bryan, G. H., and R. Rotunno, 2009: Evaluation of an analytical model for the maximum intensity of tropical cyclones. J. Atmos. Sci., in press.
Rotunno, R., Y. Chen, W. Wang, C. Davis, J. Dudhia and G. J. Holland, 2009: Resolved turbulence in a three-dimensional model of an idealized tropicaI cyclone. Bull. Amer. Met. Soc., in press.
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mt = rv + 1
2fr 2
θe = const
Eliassen & Kleinschmidt (1957)
Steady, Axisymmetric Tropical Cyclone
Interior
Boundary Layer
Ftotal = −
dpρ∫
Δ
q2
2+ gz
+Δpρ+ F = 0
Bernoulli Eq:
p − pressureρ − densityq − speedv − azimuthal vel.f − Coriolis param.g − gravitymt − ang.mom.θe − equiv.pot.temp.(r, z) − (rad.,alt.)
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1
3
2
θe = const
Steady, Axisymmetric Tropical Cyclone
Interior
Boundary Layer
−dpρ∫ = Tds∫
s ≡ cp lnθe
Ftotal ≈ CD h−1q3
1
2
∫ dt= T ds∫ ≈ (Tsea − Tout )(s2 − s1) ≈ CE h−1q (ssea − sair )1
2
∫ dt
p − pressureρ − densityq − speedT − temperatureh − PBL heightθe − equiv.pot.temp..cp − spec.heat
s − moist entropy(r, z) − (rad.,alt.)CD ,CE − drag,transfer
vm
2 ~CE
CD
(Tsea − Tout ) (ssea − sair ) @ r = rm
Emanuel (1986)
rm
Assume integrals dominated by contributions at r=rm
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dx = dr er+dz ez (ω a × u = -
12∇q2 - ρ -1∇p - g∇z) • dx
In the r-z plane and above PBL :
qv− vapor mix.rat.L0 − latent heatcp − spec.heat ct. pres.
u = (u,v,w) − velocityωa = (ξ ,η,ζ ) − vorticityp − pressureρ − densityq − speedg − gravityM − ang.mom.ψ − str. fcn.s − entropy(r, z) − (rad.,alt.)
(F = 0)
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dx = dr er+dz ez (ω a × u = -
12∇q2 - ρ -1∇p - g∇z) • dx
In the r-z plane and above PBL :
η wdr - u dz( ) - v ζdr - ξdz( ) = -
12
dq2 - ρ -1dp - gdz
qv− vapor mix.rat.L0 − latent heatcp − spec.heat ct. pres.
u = (u,v,w) − velocityωa = (ξ ,η,ζ ) − vorticityp − pressureρ − densityq − speedg − gravityM − ang.mom.ψ − str. fcn.s − entropy(r, z) − (rad.,alt.)
(F = 0)
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dx = dr er+dz ez (ω a × u = -
12∇q2 - ρ -1∇p - g∇z) • dx
In the r-z plane and above PBL :
η wdr - u dz( ) - v ζdr - ξdz( ) = -
12
dq2 - ρ -1dp - gdz
Axisymmetry: ρu = r −1 ∂ψ
∂z; ρw = −r −1 ∂ψ
∂r; ξ = −r −1 ∂Μ
∂z; ζ = r −1 ∂Μ
∂r
qv− vapor mix.rat.L0 − latent heatcp − spec.heat ct. pres.
u = (u,v,w) − velocityωa = (ξ ,η,ζ ) − vorticityp − pressureρ − densityq − speedg − gravityM − ang.mom.ψ − str. fcn.s − entropy(r, z) − (rad.,alt.)
(F = 0)
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dx = dr er+dz ez (ω a × u = -
12∇q2 - ρ -1∇p - g∇z) • dx
In the r-z plane and above PBL :
η wdr - u dz( ) - v ζdr - ξdz( ) = -
12
dq2 - ρ -1dp - gdz
Axisymmetry:
−ρd
−1dp = Tds − cpdT − L0dqvPsuedo-Adiabatic Thermodynamics (Bryan 2008): ρu = r −1 ∂ψ
∂z; ρw = −r −1 ∂ψ
∂r; ξ = −r −1 ∂Μ
∂z; ζ = r −1 ∂Μ
∂r
qv− vapor mix.rat.L0 − latent heatcp − spec.heat ct. pres.
u = (u,v,w) − velocityωa = (ξ ,η,ζ ) − vorticityp − pressureρ − densityq − speedg − gravityM − ang.mom.ψ − str. fcn.s − entropy(r, z) − (rad.,alt.)
(F = 0)
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dx = dr er+dz ez (ω a × u = -
12∇q2 - ρ -1∇p - g∇z) • dx
In the r-z plane and above PBL :
η wdr - u dz( ) - v ζdr - ξdz( ) = -
12
dq2 - ρ -1dp - gdz
Axisymmetry:
−ρd
−1dp = Tds − cpdT − L0dqvPsuedo-Adiabatic Thermodynamics (Bryan 2008): ρu = r −1 ∂ψ
∂z; ρw = −r −1 ∂ψ
∂r; ξ = −r −1 ∂Μ
∂z; ζ = r −1 ∂Μ
∂r
−ηρdr
dψ =1
2r 2 dM 2 + Tds − dE −12
d fΜ( )Bister & Emanuel (1998)
E ≡
q2
2+ gz + cpdT + L0dqv
qv− vapor mix.rat.L0 − latent heatcp − spec.heat ct. pres.
u = (u,v,w) − velocityωa = (ξ ,η,ζ ) − vorticityp − pressureρ − densityq − speedg − gravityM − ang.mom.ψ − str. fcn.s − entropy(r, z) − (rad.,alt.)
(F = 0)
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−ηρdr
dψ =1
2r 2 dM 2 + Tds − dE −12
d fΜ( )
−
ηm
ρdmrm
dψ ≈1
2rm2 dM 2 + Tsea,m − Tout ,∞( )ds
1
2rm2 = − Tsea,m − Tout ,∞( ) ds
dM 2
r= rm
vm
2 =CE
CD
Tsea,m − Tout ,∞( )(ssea,m* − sair ,m ) ≡ (E − PI )2 Emanuel-Potential
Intensity
Emanuel (1986)
Gradient-Wind Balance
Integrate over control volume (shaded) rm
u = (u,v,w) − velocityωa = (ξ ,η,ζ ) − vorticityρ − densityT − temperatureM − ang.mom.E − total energyf −Coriolis param.ψ − str. fcn.s − moist entropy(r, z) − (rad.,alt.)
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Persing and Montgomery (2003)
Rotunno and Emanuel (1987) Model
Sensitive to Grid Resolution
lh = 3000mRE87 paper RE87 code lh = 0.2 × Δh
Bryan and Rotunno (2009 MWR):
Sensitive to lh
7.5 15
7.5 15
Vmax
Vmax
E − PI
Vmax −max vel.lh − hor mix.lengthΔh − hor.grid length
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Vmax , E-PI vs lh
Bryan and Rotunno (2009 JAS)
RE87 PM03
Vmax −max vel.E - PI - Emanuel Pot. Intensitylh − hor mix.length
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Sensitivity of Wind Speed to Turbulence Mixing Length lh
Bryan and Rotunno (2009 MWR)
Vmax −max vel.lh − hor mix.length(r, z) − rad.,vert.
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Structure of and Angular Momentum M θe
Bryan and Rotunno (2009 MWR)
(lh = 750m)
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Horizontal Diffusion Weakens Gradients of and θe M
M , θe at z = 1.1km
Bryan and Rotunno (2009 MWR)
lh − hor.mix.lengthM − ang.mom.θe − equiv.pot.temp.(r, z) − (rad.,alt.)
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Horizontal Diffusion Weakens Gradients of and θe M
M , θe at z = 1.1km
Bryan and Rotunno (2009 MWR)
lh − hor.mix.lengthM − ang.mom.θe − equiv.pot.temp.(r, z) − (rad.,alt.)
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Horizontal Diffusion Weakens Gradients of and θe M
M , θe at z = 1.1km
Bryan and Rotunno (2009 MWR)
v2
r
gθ 'θ0
rz
lh − hor.mix.lengthM − ang.mom.θe − equiv.pot.temp.(r, z) − (rad.,alt.)
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Components of E-PI
1. Moist slantwise neutrality
2. PBL Model
3. Gradient-wind and hydrostatic balance
Bryan and Rotunno (2009 JAS)
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Flow with small horizontal diffusion not in gradient-wind balance
Bryan and Rotunno (2009 JAS)
v − azimuthal velvg −gradient wind
(r, z) − (rad.,alt.)
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Rotating Flow Above a Stationary Disk*
* Bödewadt (1940) (Schlichting 1968 Boundary Layer Theory); see also Rott and Lewellen (1966 Prog Aero Sci)
zνω
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What is ?
There are no observations of radial turbulent fluxes in a hurricane
lh
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Marks et al. (2008, MWR)
Reflectivity (dBZ) at 1726 UTC
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Marks et al. (2008, MWR)
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37km
Numerical Models
Turbulent fluxes represented by mixing-length theory
Axisymmetric 3D
Turbulent fluxes computed, but high resolution required
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Davis et al. (2008 MWR)
Δ = 1.33kmΔ = 4.0km
Weather Research and Forecast (WRF) Forecasts
Radar Reflectivity at z=3km a) WRF b) WRF c) ELDORA
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1/Δmeso 1/ΔLES 1/l
Mesoscale limit
LES limit
the “terra incognita”
k
F(k)
Wyngaard (2004 JAS)
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1/Δmeso 1/ΔLES 1/l
Mesoscale limit
LES limit
the “terra incognita”
k
F(k)
Wyngaard (2004 JAS)
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Idealized TC: f-plane zero env wind fixed SST
Nested Grids
WRF Model Physics: WSM3 simple ice No radiation Relax to initial temp. CD (Donelan) CE (Carlson-Boland) CE/CD ~ 0.65 YSU PBL LES PBL
(Δ ≥1.67km)(Δ <1.67km)
Domain 6075km
1500km 1000km
333km 111km
37km
(Δ = 15km)
(Δ = 5km)(Δ = 1.67km)
(Δ = 556m) (Δ = 185m)
(Δ = 62m)
50 vertical levels Δz=60m~1km
Ztop=27km
Rotunno et al. (2009 BAMS)
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(Δ = 1.67km) (Δ = 556m)
(Δ = 185m)
2020
20
20
−20−20
−20
−20
00
0
0
x[km] x[km]
y[km
] y[
km]
(Δ = 62m)
10-m Wind Speed t=9.75d
max=61.5
max=121.7 max=86.2
max=86.7
Rotunno et al. (2009 BAMS)
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10-m Wind Speed
37km 37 km
Max=85.5 Max=82.3 Max=83.7
t = 9.75d , Δ = 62m
instantaneous 1-min average
max=121.7 max=78.8
Rotunno et al. (2009 BAMS)
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Vorticity Magnitude t = 9.75d , Δ = 62m
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Vorticity Magnitude t = 9.75d , Δ = 62m
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10-m Tangentially Averaged Wind Speed vs Grid Interval
Δ Rotunno et al. (2009 BAMS)
ms
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A very high-resolution simulation • Stretched structured grid • In center: Δx = Δy = Δz = 62.5 m • Initialized from 1-km simulation
49 km
49 km
Courtesy of G. Bryan
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w (m/s) at z = 1 km
Δ = 1000 m Δ = 62.5 m
Courtesy of G. Bryan
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Courtesy of G. Bryan
v(r, z)
s(r, z)
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Conclusions
• Treatment of turbulence in numerical models of hurricanes is as important for intensity prediction as other factors (e.g. sea-surface transfer, ocean-wave drag, cloud physics…)
• Quantitative information needed on turbulent transfer in hurricanes
• Observations are expensive, LES so far inconclusive