Thermodynamics of Tropical Cyclogenesis1

David J. Raymond

New Mexico TechSocorro, NM, USA

May 23, 2011

1This work supported by the US Office of Naval Research and NationalScience Foundation.

Collaborators

I Carlos LópezI Sharon SessionsI Saška Gjorgjievska

Flux Form of Vorticity Equation (Haynes and McIntyre 1987)

Vertical component of absolute vorticity:

∂ζz∂t

+ ∇h · Z + k̂ ·∇hθ ×∇hΠ = 0

Horizontal flux of vertical vorticity:

Z = Z 1 + Z 2 + Z f = vhζz − ζhvz + k̂ × F

Vertical component of baroclinic generation (ignore):

k̂ ·∇hθ ×∇hΠ ≈ 0

Circulation Tendency Equation

dΓ

dt= −˛

vnζzdl +

˛ζnvzdl +

˛Ftdl

Γ =

ˆζzdA

n

t

A

Top and bottom-heavy mass flux profiles

strong downdrafts weak downdrafts

updraft and downdraft mass flux

Typhoon Nuri overview

120 125 130 135 140 145 150 155 160longitude (deg E)

0

5

10

15

20

25

30

lati

tude (

deg N

)

TWTDTSTY

Nuri track, sea surface temperature (C)

26

27

28

29

30

31

Nuri 1 (tropical wave)

10 m/s

143 144 145 146 147 148 149 150longitude (deg E)

11

12

13

14

15

16

17

18

lati

tude (

deg N

)

z = 1.2 km

−0.3

−0.2

−0.1

0.0

0.1

0.2

0.3

143 144 145 146 147 148 149 150longitude (deg E)

11

12

13

14

15

16

17

18

lati

tude (

deg N

)

z = 5.0 km

−0.3

−0.2

−0.1

0.0

0.1

0.2

0.3

Nuri 1: relative winds, absolute vorticity (ks−1 )

Nuri 2 (tropical depression)

10 m/s

138 139 140 141 142 143longitude (deg E)

13

14

15

16

17

18

lati

tude (

deg N

)

z = 1.2 km

−0.3

−0.2

−0.1

0.0

0.1

0.2

0.3

0.4

138 139 140 141 142 143longitude (deg E)

13

14

15

16

17

18

lati

tude (

deg N

)

z = 5.0 km

−0.3

−0.2

−0.1

0.0

0.1

0.2

0.3

0.4

Nuri 2: relative winds, absolute vorticity (ks−1 )

Nuri 1 vorticity budget

0 5 10 15 20 25

circulation (km2 /s)

0

4

8

12

16

heig

ht

(km

)

−30 −15 0 15 30

circ tend (km2 /s/day)

0

4

8

12

16

convergence

tilting

friction

net

−5 0 5 10 15

mass flux (109 kg/s)

0

4

8

12

16

Nuri 1

Nuri 2 vorticity budget

0 5 10 15 20 25

circulation (km2 /s)

0

4

8

12

16

heig

ht

(km

)

−30 −15 0 15 30

circ tend (km2 /s/day)

0

4

8

12

16

convergence

tilting

friction

net

0 5 10 15

mass flux (109 kg/s)

0

4

8

12

16

Nuri 2

Tropical wave TCS030 overview

120 125 130 135 140 145 150 155 160longitude (deg E)

0

5

10

15

20

25

30

lati

tude (

deg N

)

TW

TCS030 track, sea surface temperature (C)

26

27

28

29

30

31

TCS030 (Weak tropical wave)

10 m/s

140 141 142 143 144 145 146 147longitude (deg E)

10

11

12

13

14

15

16

17

lati

tude (

deg N

)

z = 1.2 km

−0.3

−0.2

−0.1

0.0

0.1

0.2

0.3

140 141 142 143 144 145 146 147longitude (deg E)

10

11

12

13

14

15

16

17

lati

tude (

deg N

)

z = 5.0 km

−0.3

−0.2

−0.1

0.0

0.1

0.2

0.3

TCS030: relative winds, absolute vorticity (ks−1 )

TCS030 vorticity budget

0 5 10 15 20 25

circulation (km2 /s)

0

4

8

12

16

heig

ht

(km

)

−30 −15 0 15 30

circ tend (km2 /s/day)

0

4

8

12

16

convergence

tilting

friction

net

−5 0 5 10 15

mass flux (109 kg/s)

0

4

8

12

16

TCS030

Summary 1

I Circulations spin up when the positive circulation tendency dueto the convergence of vorticity exceeds the negative tendencydue to friction.

I Top-heavy convective mass flux profiles generate mid-levelspinup.

I Bottom-heavy mass flux profiles generate low-level spinup.

Question: What controls the magnitude and shape of mass fluxprofiles?

Over the Pacific at dawn

Nice clouds

Rough ocean

Thermodynamic study of 5 Pacific systems

I Nuri (1 and 2) developing typhoonI TCS025 (1 and 2) strong waveI TCS030 weak waveI TCS037 developing midget tropical cycloneI Hagupit developing typhoon (very early stage)

Soundings in moist entropy form – TCS030

200 220 240 260 280 300 320moist entropy (J/K/kg)

0

2

4

6

8

10heig

ht

(km

)Mean Soundings

TCS030

Soundings in moist entropy form – Nuri1

200 220 240 260 280 300 320moist entropy (J/K/kg)

0

2

4

6

8

10heig

ht

(km

)Mean Soundings

TCS030

Nuri1

Soundings in moist entropy form – Nuri2

200 220 240 260 280 300 320moist entropy (J/K/kg)

0

2

4

6

8

10heig

ht

(km

)Mean Soundings

TCS030

Nuri1

Nuri2

Thermodynamic Effect of Vortices

Developing disturbanceWest Pacific wave

PV anomaly

warmPV anomaly

warmReed and Recker (1971)

cool

Rain and Mass Flux Profiles (Raymond and Sessions 2007)

­2 ­1 0 1 2

θ perturbation (K)

0

5000

10000

15000

20000

he

igh

t (m

)

unperturbed

δθ = ± 0.5 K

δθ = ± 1.0 K

δθ = ± 2.0 K

A

0 0.5 1

moisture perturbation (g/kg)

unperturbed

δr = 0.25 g/kg

δr = 0.5 g/kg

δr = 1.0 g/kg

B

Rain and Mass Flux Profiles (cont...)

5 10 15 20

imposed windspeed (m/s)

0

10

20

30

40

50

60

rain

ra

te (

kg

/m2/d

ay)

unperturbed

δθ = ± 2.0 K

δr=1.0 g/kg

A

5 10 15 20

imposed windspeed (m/s)

0

0.2

0.4

0.6

0.8

1

NG

MS

unperturbed

δθ = ± 2.0 K

δr = 1.0 g/kg

B

Rain and Mass Flux Profiles (cont...)

0 0.01 0.02 0.03

mass flux (kg/m2s)

0

5000

10000

15000

20000

he

igh

t (m

)

unperturbed

δθ = ± 0.5 K

δθ = ± 1.0 K

δθ = ± 2.0 K

vy=7m/s

A

0 0.01 0.02 0.03

mass flux (kg/m2s)

unperturbed

δr = 0.25 g/kg

δr = 0.5 g/kg

δr = 1.0 g/kg

vy=7m/s

B

Rain and vorticity convergenceWater budget in a control volume:

d [r ]

dt= − [∇h · (vhr)]− g(R − F rs)

Mixing ratio: r ; Rainfall rate: R ; Surface evaporation rate: F rs ;Area average: X ; Area average and pressure integral: [X ];Use to estimate rainfall rate (steady state):

R ≈ F rs − [∇h · (vhr)]

Boundary layer vorticity convergence and rainfall rate:

−∇h · (vhζz)bl

ζz= −C

[∇h · (vhr)]

W

Precipitable water: W

Rain and vorticity convergence observed

−0.3 0.0 0.3 0.6 0.9scaled moist conv (day−1 )

−0.4

0.0

0.4

0.8

1.2

1.6

2.0

2.4

scale

d v

ort

conv (

day−1)

slope = 1.05

Mid-level vortex and low-level spinup

0.03 0.06 0.09 0.12 0.15mid-level vorticity (ks−1 )

−0.04

0.00

0.04

0.08

0.12

0.16

vte

nd

low

(ks−1day−1)

Summary 2

I A mid-level circulation provides a thermodynamic environmentwhich promotes bottom-heavy convective mass flux profilesand more intense rain.

I Intense rain is correlated with strong vorticity convergence.I These factors promote low-level spinup and consequent

formation of a warm-core cyclone.

Question: What limits mid-level vortices?

Clouds from G-V over Atlantic

Visible image of Hurricane Earl (2010)

View from the hotel

Rain and Saturation Fraction (Raymond, Sessions, andFuchs 2007)

0 0.2 0.4 0.6 0.8 1

saturation fraction

300

290

280

270

260

250

240

230

220

210

200IR

brightn

ess tem

pera

ture

(K

)EPIC soundings

EPIC dropsondes

ECAC soundings

5

10

15

20

25E

PIC

P­3

radar ra

inra

te (m

m/d

)

Saturation Fraction and Moist Entropy

Saturation fraction (S):

S =[r ]

[r∗]=

[s − sD ]

[s∗ − sD ]

Approximate moist entropy (s) and saturated moist entropy (s∗):

s = CP ln(T/TF )− R ln(p/pR) + Lr/TF = sD + Lr/TF

s∗ = sD + Lr∗(sD , p)/TF

Moist entropy budget in a control volume:

d [s]

dt= − [∇h · (vhs)] + [G ] + g

(F es − F et

)Area average: X ; Area average and pressure integral: [X ];Irreversible generation: [G ]; Upward fluxes: F es − F et

Environmental injection of dry airNormalized Okubo-Weiss parameter (measure of rotation vs.horizontal strain; Dunkerton et al. 2009):

N =ζ2r − σ2

1 − σ22

ζ2r + σ2

1 + σ22

where

ζr =∂v∂x− ∂u∂y

σ1 =∂v∂x

+∂u∂y

σ2 =∂u∂x− ∂v∂y

Averaged over middle levels (3-5 km).

vorticity strain

Entropy tendency and Okubo-Weiss

0.3 0.5 0.7 0.9 1.1

normalized Okubo-Weiss

−1.8

−1.2

−0.6

0.0

0.6

ent

tend (

J/K

/m2

/s)

Summary 3

I The entropy tendency should be a measure of futureintensification, since this tendency is related to the saturationfraction tendency, and hence the prospects for future rainfall.

I A small value of the normalized Okubo-Weiss parameter islikely to be correlated with the import of dry air via strainflow, and hence a negative entropy tendency.

I Entropy tendency and normalized Okubo-Weiss are wellcorrelated and the three systems undergoing intensificationhad positive entropy tendency and N > 0.8.

The NCAR Gulfstream-V aircraft in St. Croix