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.
Steady, Axisymmetric Tropical Cyclone θ e = const Interior Boundary Layer m t = rv + 1 2 fr 2 Eliassen & Kleinschmidt (1957) Bernoulli Eq: Δ 2 + gz + Δp ρ + F = 0 q2 F total = dp ρ p pressure ρ density q speed v azimuthal vel. f Coriolis param. g gravity m t ang.mom. θ e equiv.pot.temp. (r,z) (rad.,alt.)
Steady, Axisymmetric Tropical Cyclone 3 θ e = const Interior r m Boundary Layer dp = Tds ρ s c p lnθ e 1 2 2 F total C D h 1 q 3 dt = T ds Assume integrals dominated by contributions at r=r m 1 v m 2 (T sea T out )(s 2 s 1 ) C E h 1 q(s sea s air ) ~ C E C D (T sea T out ) (s sea s air ) 2 1 @ r = r m Emanuel (1986) dt p pressure ρ density q speed T temperature h PBL height θ e equiv.pot.temp.. c p spec.heat s moist entropy (r, z) (rad.,alt.) C D,C E drag,transfer
In the r-z plane and above PBL (F = 0) : (ω a u = - 1 2 q2 - ρ -1 p - g z) dx dx = dr e r +dz e z q v vapor mix.rat. L 0 latent heat c p spec.heat ct. pres. u = (u,v,w) velocity ω a = (ξ,η,ζ ) vorticity p pressure ρ density q speed g gravity M ang.mom. ψ str. fcn. s entropy (r,z) (rad.,alt.)
In the r-z plane and above PBL (F = 0) : (ω a u = - 1 2 q2 - ρ -1 p - g z) dx dx = dr e r +dz e z η( wdr - u dz) - v( ζdr - ξdz) = - 1 2 dq2 - ρ -1 dp - gdz q v vapor mix.rat. L 0 latent heat c p spec.heat ct. pres. u = (u,v,w) velocity ω a = (ξ,η,ζ ) vorticity p pressure ρ density q speed g gravity M ang.mom. ψ str. fcn. s entropy (r,z) (rad.,alt.)
In the r-z plane and above PBL (F = 0) : (ω a u = - 1 2 q2 - ρ -1 p - g z) dx dx = dr e r +dz e z η( wdr - u dz) - v( ζdr - ξdz) = - 1 2 dq2 - ρ -1 dp - gdz ψ Axisymmetry: ρu = r 1 z ; ρw = r ψ 1 r ; ξ = r Μ 1 z ; ζ = r Μ 1 r q v vapor mix.rat. L 0 latent heat c p spec.heat ct. pres. u = (u,v,w) velocity ω a = (ξ,η,ζ ) vorticity p pressure ρ density q speed g gravity M ang.mom. ψ str. fcn. s entropy (r,z) (rad.,alt.)
In the r-z plane and above PBL (F = 0) : (ω a u = - 1 2 q2 - ρ -1 p - g z) dx dx = dr e r +dz e z η( wdr - u dz) - v( ζdr - ξdz) = - 1 2 dq2 - ρ -1 dp - gdz Axisymmetry: ψ ρu = r 1 z ; ρw = r ψ 1 r ; ξ = r Μ 1 z ; ζ = r Μ 1 r Psuedo-Adiabatic Thermodynamics (Bryan 2008): ρ d 1 dp = Tds c p dt L 0 dq v q v vapor mix.rat. L 0 latent heat c p spec.heat ct. pres. u = (u,v,w) velocity ω a = (ξ,η,ζ ) vorticity p pressure ρ density q speed g gravity M ang.mom. ψ str. fcn. s entropy (r,z) (rad.,alt.)
In the r-z plane and above PBL (F = 0) : (ω a u = - 1 2 q2 - ρ -1 p - g z) dx dx = dr e r +dz e z η( wdr - u dz) - v( ζdr - ξdz) = - 1 2 dq2 - ρ -1 dp - gdz Axisymmetry: ψ ρu = r 1 z ; ρw = r ψ 1 r ; ξ = r Μ 1 z ; ζ = r Μ 1 r Psuedo-Adiabatic Thermodynamics (Bryan 2008): ρ d 1 dp = Tds c p dt L 0 dq v Bister & Emanuel (1998) η ρ d r dψ = 1 2r 2 dm 2 + Tds de 1 2 d E q2 2 + gz + c p dt + L 0 dq v ( f Μ ) q v vapor mix.rat. L 0 latent heat c p spec.heat ct. pres. u = (u,v,w) velocity ω a = (ξ,η,ζ ) vorticity p pressure ρ density q speed g gravity M ang.mom. ψ str. fcn. s entropy (r,z) (rad.,alt.)
η ρ d r dψ = 1 2r 2 dm 2 + Tds de 1 2 d ( f Μ ) Integrate over control volume (shaded) r m u = (u,v,w) velocity ω a = (ξ,η,ζ ) vorticity ρ density T temperature M ang.mom. E total energy f Coriolis param. ψ str. fcn. s moist entropy (r,z) (rad.,alt.) η m dψ 1 ρ dm r m 2r dm 2 + ( T 2 sea,m T ) out, ds m Gradient-Wind Balance 1 ( ) 2r m 2 = T sea,m T out, ds dm 2 r = r m v m 2 = C E C D ( * T sea,m T out, )(s sea,m s air,m ) (E PI) 2 Emanuel-Potential Intensity Emanuel (1986)
Persing and Montgomery (2003) Rotunno and Emanuel (1987) Model V max Sensitive to Grid Resolution 7.5 15 Bryan and Rotunno (2009 MWR): RE87 paper RE87 code V max Sensitive to l h = 3000m l h = 0.2 Δ h l h E PI V max max vel. l h hor mix.length Δ h hor.grid length 7.5 15
V max, E-PI vs l h PM03 RE87 V max max vel. E - PI - Emanuel Pot. Intensity l h hor mix.length Bryan and Rotunno (2009 JAS)
Sensitivity of Wind Speed to Turbulence Mixing Length l h V max max vel. l h hor mix.length (r, z) rad., vert. Bryan and Rotunno (2009 MWR)
Structure of θ e and Angular Momentum M (l h = 750m) Bryan and Rotunno (2009 MWR)
Horizontal Diffusion Weakens Gradients of and θ e M M, θ e at z = 1.1km l h hor.mix.length M ang.mom. θ e equiv.pot.temp. (r, z) (rad.,alt.) Bryan and Rotunno (2009 MWR)
Horizontal Diffusion Weakens Gradients of and θ e M M, θ e at z = 1.1km l h hor.mix.length M ang.mom. θ e equiv.pot.temp. (r, z) (rad.,alt.) Bryan and Rotunno (2009 MWR)
Horizontal Diffusion Weakens Gradients of and θ e M M, θ e at z = 1.1km gθ ' θ 0 z r v 2 r l h hor.mix.length M ang.mom. θ e equiv.pot.temp. (r, z) (rad.,alt.) Bryan and Rotunno (2009 MWR)
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 v azimuthal vel v g gradient wind (r, z) (rad.,alt.) Bryan and Rotunno (2009 JAS)
Rotating Flow Above a Stationary Disk* z ν ω * Bödewadt (1940) (Schlichting 1968 Boundary Layer Theory); see also Rott and Lewellen (1966 Prog Aero Sci)
l h What is? There are no observations of radial turbulent fluxes in a hurricane
Reflectivity (dbz) at 1726 UTC Marks et al. (2008, MWR)
Marks et al. (2008, MWR)
Numerical Models Axisymmetric 3D 37km Turbulent fluxes represented by mixing-length theory Turbulent fluxes computed, but high resolution required
Weather Research and Forecast (WRF) Forecasts Radar Reflectivity at z=3km a) WRF Δ = 1.33km b) WRF Δ = 4.0km c) ELDORA Davis et al. (2008 MWR)
F(k) Mesoscale limit the terra incognita LES limit 1/Δmeso 1/l 1/ΔLES k Wyngaard (2004 JAS)
F(k) Mesoscale limit the terra incognita LES limit 1/Δmeso 1/l 1/ΔLES k Wyngaard (2004 JAS)
Domain Idealized TC: f-plane zero env wind fixed SST Nested Grids WRF Model Physics: WSM3 simple ice No radiation Relax to initial temp. C D (Donelan) C E (Carlson-Boland) C E /C D ~ 0.65 YSU PBL LES PBL (Δ 1.67km) (Δ < 1.67km) 6075km (Δ = 15km) 1500km (Δ = 5km) 1000km (Δ = 1.67km) 333km (Δ = 556m) 50 vertical levels Δz=60m~1km Z top =27km 111km (Δ = 185m) 37km (Δ = 62m) Rotunno et al. (2009 BAMS)
10-m Wind Speed t=9.75d (Δ = 1.67km) (Δ = 556m) 20 max=61.5 max=86.7 y[km] y[km] 0 20 20 0 (Δ = 185m) max=86.2 (Δ = 62m) max=121.7 20 20 0 x[km] 20 20 0 x[km] 20 Rotunno et al. (2009 BAMS)
10-m Wind Speed instantaneous t = 9.75d, Δ = 62m 1-min average max=121.7 max=78.8 Max=85.5 Max=82.3 Max=83.7 37km 37 km 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 m s Δ Rotunno et al. (2009 BAMS)
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
v(r, z) s(r, z) Courtesy of G. Bryan
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