Richard Rotunno

NCAR

Bryan, G. H., and R. Rotunno, 2009: The maximum intensity of tropical cyclones in axisymmetric

numerical model simulations. Mon. Wea. Rev..

Bryan, G. H., and R. Rotunno, 2009: Evaluation of an analytical model for the maximum intensity of

tropical cyclones. J. Atmos. Sci.

Comparison of an analytical and a

numerical model for hurricane potential

intensity

EMC/MMM/DTC Joint Hurricane Science Workshop

Vmax , E-PI vs lh

Bryan and Rotunno (2009 JAS)

Vmax

max vel.

E - PI - Emanuel Pot. Intensity

lh hor mix.length

vm

2 C

E

CD

Tsea T

out ssea

* sair

r r

m

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.

Emanuel Potential Intensity TheorySteady, inviscid,

axisymmetric,

pseudo-adiabatic

equations of

motion projected onto

r-z plane

dr

r2

dM 2

d T T

out ds

d

v Ź= azi. velocity

ŹŹazi.vorticity

dŹdry airŹdensity

T temperature

M ang.mom.

E total energy

f Coriolis param.

str. fcn.

s moist entropy

(r, z) (rad.,alt.)

Assume

s s( )ŹŹŹM M ( )

r

Tout

ds

d

dE

d

f

2

dM

d

dr

r2

dM 2

d T

ds

d

dE

d

f

2

dM

d

Emanuel Potential Intensity Theory

(r

m,h)

dr

m

rm

2

dM 2

d T

sea T

out ds

d

v Ź= azi. velocity

ŹŹazi.vorticity

dŹdry airŹdensity

T temperature

M ang.mom.

E total energy

f Coriolis param.

str. fcn.

s moist entropy

(r, z) (rad.,alt.)

Assume

s s( )ŹŹŹM M ( )

r r

m, z h

Assume

Gradient Wind Balance

rm

2

dM 2

d T

sea T

out ds

d

Assume

Gradient Wind Balance

ŹŹŹŹŹr r

m, z h

Emanuel Potential Intensity Theory

v Ź= azi. velocity

ŹŹazi.vorticity

dŹdry airŹdensity

T temperature

M ang.mom.

E total energy

f Coriolis param.

str. fcn.

s moist entropy

(r, z) (rad.,alt.)

Assume

s s( )ŹŹŹM M ( )

Assume

Mixed Layer PBL

ds

dM

s

M z0

r rm

C

E

CD

ssea

* sair

rv

r rm

d

dM 2

rm

2 T

sea T

out ds

dM 2

Assume

s s( )ŹŹŹM M ( )

Assume

Gradient Wind Balance

ŹŹŹŹŹr r

m, z h

Assume

Mixed Layer

vm

2 C

E

CD

Tsea T

out ssea

* sair

r r

m

Emanuel (1986 JAS)

Components of E-PI

1. Moist slantwise neutrality

2. PBL Model

3. Gradient-wind and hydrostatic balance

Bryan and Rotunno (2009 JAS)

s s( M )

Structure of and Angular Momentum Me

Bryan and Rotunno (2009 MWR)

(lh 750m)

Components of E-PI

1. Moist slantwise neutrality

2. PBL Model

3. Gradient-wind and hydrostatic balance

Bryan and Rotunno (2009 JAS)

ds

dM

zhrr

m

s

M

z0rr

m

ds

dM

zhrr

m

s

M

z0rr

m

?

Bryan and Rotunno (2009 JAS)

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 vel

vggradient 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)

z

Vmax , E-PI, PI+ vs lh

Bryan and Rotunno (2009 JAS)

Vmax

max vel.

E - PI - Emanuel Pot. Intensity

lh hor mix.length

rm rad. V

max

m azi.vort.at r

m

wm vert.vel.at r

m

PI (E PI )2 rmmwm

Components of E-PI

1. Moist slantwise neutrality

2. PBL Model

3. Gradient-wind and hydrostatic balance

Bryan and Rotunno (2009 JAS)

Conclusions

• Simulated TC Vmax > E-PI for weak horizontal diffusion

• Weak horizontal diffusion Violation of gradient wind

• Horizontal diffusion & non-gradient wind in natural TCs

not generally known

e, M , const

Eliassen & Kleinschmidt (1957):

free atmosphere

PBL

“From these conditions it is probably possible to

explain the essential features of the hurricane…This

problem, however, has not been solved.”

Emanuel (1986) solved it.

Postscript: E-PI

Assume

s s( )ŹŹŹM M ( )

Assume

s s( )ŹŹŹM M ( )

Assume

Gradient Wind Balance

ŹŹŹŹŹr r

m, z h

Assume

Mixed Layer

Gradient Wind Balance

dx = dr e

rdz e

z (

a u = -

1

2q2 - -1p - gz) dx

In the r-z plane and above PBL :

qv vapor mix.rat.

L0 latent heat

cp spec.heat ct. pres.

u = (u,v,w) velocity

a = ( ,, ) vorticity

p pressure

density

q speed

g gravity

M ang.mom.

str. fcn.

s entropy

(r, z) (rad.,alt.)

(F 0)

dx = dr e

rdz e

z (

a u = -

1

2q2 - -1p - gz) dx

In the r-z plane and above PBL :

wdr - u dz - v dr - dz = -

1

2dq2 - -1dp - gdz

qv vapor mix.rat.

L0 latent heat

cp spec.heat ct. pres.

u = (u,v,w) velocity

a = ( ,, ) vorticity

p pressure

density

q speed

g gravity

M ang.mom.

str. fcn.

s entropy

(r, z) (rad.,alt.)

(F 0)

dx = dr e

rdz e

z (

a u = -

1

2q2 - -1p - gz) dx

In the r-z plane and above PBL :

wdr - u dz - v dr - dz = -

1

2dq2 - -1dp - gdz

Axisymmetry:

u r1

z; w r1

r; r1

z; r1

r

qv vapor mix.rat.

L0 latent heat

cp spec.heat ct. pres.

u = (u,v,w) velocity

a = ( ,, ) vorticity

p pressure

density

q speed

g gravity

M ang.mom.

str. fcn.

s entropy

(r, z) (rad.,alt.)

(F 0)

dx = dr e

rdz e

z (

a u = -

1

2q2 - -1p - gz) dx

In the r-z plane and above PBL :

wdr - u dz - v dr - dz = -

1

2dq2 - -1dp - gdz

Axisymmetry:

d

1dp Tds cpdT L

0dq

vPsuedo-Adiabatic Thermodynamics (Bryan 2008):

u r1

z; w r1

r; r1

z; r1

r

qv vapor mix.rat.

L0 latent heat

cp spec.heat ct. pres.

u = (u,v,w) velocity

a = ( ,, ) vorticity

p pressure

density

q speed

g gravity

M ang.mom.

str. fcn.

s entropy

(r, z) (rad.,alt.)

(F 0)

dx = dr e

rdz e

z (

a u = -

1

2q2 - -1p - gz)

In the r-z plane and above PBL :

wdr - u dz - v dr - dz = -

1

2dq2 - -1dp - gdz

Axisymmetry:

d

1dp Tds cpdT L

0dq

vPsuedo-Adiabatic Thermodynamics (Bryan 2008):

u r1

z; w r1

r; r1

z; r1

r

qv vapor mix.rat.

L0 latent heat

cp spec.heat ct. pres.

u = (u,v,w) velocity

a = ( ,, ) vorticity

p pressure

density

q speed

g gravity

M ang.mom.

str. fcn.

s entropy

(r, z) (rad.,alt.)

(F 0)

dr

r2

dM 2

d T

ds

d

dE

d

f

2

dM

d

E q2

2 gz c pT L0qv

dx

Persing and Montgomery (2003)

Rotunno and Emanuel (1987) Model

Sensitive to Grid Resolution

lh 3000mRE87 paper

RE87 code lh 0.2 h

Bryan and Rotunno (2009 MWR):

Sensitive to lh

7.5 15

7.5 15

Vmax

Vmax

E PI

Vmax

max vel.

lh hor mix.length

h hor.grid length

Vmax     E PI Emanuel Potential

Intensity