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.

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.)

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

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)

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)

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)

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)

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)

−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.)

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

Vmax , E-PI vs lh

Bryan and Rotunno (2009 JAS)

RE87 PM03

Vmax −max vel.E - PI - Emanuel Pot. Intensitylh − hor mix.length

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.

Structure of and Angular Momentum M θe

Bryan and Rotunno (2009 MWR)

(lh = 750m)

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.)

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.)

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.)

Components of E-PI

1. Moist slantwise neutrality

2. PBL Model

3. Gradient-wind and hydrostatic balance

Bryan and Rotunno (2009 JAS)

Flow with small horizontal diffusion not in gradient-wind balance

Bryan and Rotunno (2009 JAS)

v − azimuthal velvg −gradient wind

(r, z) − (rad.,alt.)

Rotating Flow Above a Stationary Disk*

* Bödewadt (1940) (Schlichting 1968 Boundary Layer Theory); see also Rott and Lewellen (1966 Prog Aero Sci)

zνω

What is ?

There are no observations of radial turbulent fluxes in a hurricane

lh

Marks et al. (2008, MWR)

Reflectivity (dBZ) at 1726 UTC

Marks et al. (2008, MWR)

37km

Numerical Models

Turbulent fluxes represented by mixing-length theory

Axisymmetric 3D

Turbulent fluxes computed, but high resolution required

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

1/Δmeso 1/ΔLES 1/l

Mesoscale limit

LES limit

the “terra incognita”

k

F(k)

Wyngaard (2004 JAS)

1/Δmeso 1/ΔLES 1/l

Mesoscale limit

LES limit

the “terra incognita”

k

F(k)

Wyngaard (2004 JAS)

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)

(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)

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)

Vorticity Magnitude t = 9.75d , Δ = 62m

Vorticity Magnitude t = 9.75d , Δ = 62m

10-m Tangentially Averaged Wind Speed vs Grid Interval

Δ Rotunno et al. (2009 BAMS)

ms

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

w (m/s) at z = 1 km

Δ = 1000 m Δ = 62.5 m

Courtesy of G. Bryan

Courtesy of G. Bryan

v(r, z)

s(r, z)

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