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