Tropical Cyclone Axisymmetric Physics Tropical Cyclone Axisymmetric Physics Kerry Emanuel Lorenz Center, MIT
Tropical Cyclone Axisymmetric Physics
Tropical Cyclone Axisymmetric Physics
Kerry EmanuelLorenz Center, MIT
Overview: What is a Tropical Cyclone?
A tropical cyclone is a nearly symmetric, warm-core cyclone powered by wind-induced enthalpy fluxes from the sea surface
Global Climatology
Tracks of all tropical cyclones in the historical record from 1851 to 2010. The tracks are colored according to the maximum wind at 10 m altitude, on the scale at lower right.
View of the eye of Hurricane Katrina on August 28th, 2005, as seen from a NOAA WP-3D hurricane
reconnaissance aircraft.
Physics of Mature Hurricanes
References:
Emanuel, J. Atmos. Sci., 1986
Rousseau-Rizzi & Emanuel, J. Atmos. Sci., 2019 (in early online release)
Cross-section through a Hurricane & Energy Production
Nearly isothermal expansion
Isothermal compression
eye edge
Low entropy
Hei
ght a
bove
oce
an
Ocean surfaceRadius (km)
Carnot Theorem: Maximum efficiency results from a particular energy cycle:
Isothermal expansionAdiabatic expansionIsothermal compressionAdiabatic compressionNote: Last leg is not adiabatic in hurricanes: Air cools radiatively. But since the environmental temperature profile is moist adiabatic, the amount of radiative cooling is the same as if air were saturated and descending moist adiabatically.
Maximum rate of energy production:
s o
s
T TP QT
Total rate of heat input to hurricane:
0 * 300
2 | | | |r
k DQ C k k C rdr V V
Surface enthalpy flux Dissipative heating
In steady state, energy production is used to balance frictional dissipation:
0 3
02 | |
r
DD C rdr V
3 *0| | | |s o
D ko
max maxT TC C k k
T V V
2 *| | 0maxC T Tk s oV k kC TD o
s o
s
T TD QT
Note that this is valid between ANY two streamlines in the region of ascent
Internally determined
3 * 30 max| | | | | |s o
D k Dmaxs
maxT TC C k k C
T V V V
Simulations with cloud-permitting axisymmetric model for different horizontal mixing lengths
Rousseau-Rizzi and Emanuel, J. Atmos. Sci., 2019
Derivation of gradient wind potential intensity from thermal wind balance
2 22
3
14
g gV MfV f r
r r r
Gradient balance
p
Hydrostatic balance
3
2 ** p
g gM M sr p r s r
212g gM rV rf
Thermal wind
3*
2 *
s
g g g
g
M M MT dsr p p dM r
3*
2 *
s
g g g
g
M M MT dsr p p dM r
*3
1 1 *2
gM sg g
r ds Tr p M dM p
Integrate in pressure:
2 2
*g gb o
b go
M M dsT Tr r dM
*gb
b g
gob o
o
V V dsT Tr r dM
(1)
*b
ggb b o
dsV r T TdM
Convective criticality: * bs s
g
bgb b b o
dsV r T TdM
(1)
Define outflow to be where 0oV
Put (1) in differential form:
Integrate entropy equation through depth of boundary layer:
(2)
(3)
Integrate angular momentum equation through depth of boundary layer:
(4)
03*1 | | | |k D
s
dsh C k k Cdt T
V V
2 0.g gb o
M dMdsT Tdt r dt
| |gD
dMdMh h C rVdt dt
V
Substitute (3) and (4) into (2) and equate
2 *| | 0C T Tk s o k kC TD o
V (5)
Same answer as from Carnot cycle. This is still not a closed expression, since we have not determined the boundary layer enthalpy, k or the outflow temperature, To
| |:V Vwith
Angular momentum surfaces plotted in the V-T plane. Red curve shows shape of balanced M surface originating at radius of maximum winds. Dashed red line is ambient
tropopause temperature.
2
2 *
.d
d ms dsrz dMRi M
zz
V
2 *
.t m
c
dsrM dMz Ri
* * ,s ds Mz dM z
Implications for Outflow Temperature
22 *
* .t m
c
dsrs dMz Ri
But the vertical gradient of saturation entropy is related to the vertical gradient of temperature:
* ,** *s
p
TpT Tss sp
2
2
* *1
p
p v
v p
TcT
s L qR c T
Use definition of s* and C.-C.:
(6)
(7)
(5)
2
2
.** *1
p m
v
v p
TcT
ss L qzR c T
Substitute (16) into (15) and use hydrostatic equation:
Substitute (5) into (8):2
2 .* *o c
t
T Ri dMs r ds
Gives dependence of Outflow T on s*
2 2 22
2 2
*1b o v bb p c
b v p t
T T L q rV c RiT R c T r
If
(8)
(9)
we can neglect first term on left of (8)
,* *
T T dMs M ds
2 .*
o c
t
T Ri dMM r ds
Using
We can re-write (9) as (10)
2 1 *2b b b o
dsM r f T TdM
We can also re-write (1) as
(11)
3
*0
| || |bk b d
b
dsh C s s Cdt T
VVBoundary layer
entropy:(12)
| |dMh r Vdt
VBoundary layer angular momentum: (13)
Combine (12) and (13):
* 20 | |bb k
D b
s sds CdM C rV T rV
V
Let *, | | ,b b bs s V V r rV
*0 ** k b
D b b b b
s sds C VdM C rV T r
Balance condition (1):
*bb o
b
V dsT Tr dM
(14)
(15)
Eliminate Vb between (14) and (15):
*20
2
** b k
o D b b o
s sds T CdM T C r T T
Eliminate rb2 between (11) and (16):
2* *2 0,b o
ds ds fdM dM T T
where 0 * *2
b k
o D
T C s sT C M
Remember that2 *
o c
t
T Ri dMM r ds
(16)
(17)
(10)
inward from some outer radius ro, defined such that
0 oV at r r
In general, integrating this system will not yield To=Tt at r=rmax. Iterate value of rt until this condition is met.
If V >> fr, we ignore dissipative heating, and we neglect pressure dependence of s0*, then we can derive an approximate closed-form solution.
2
2
2
2,
2
k
D
CC
m
m k k
D D m
rrM
M C C rC C r
Assuming that Ri is critical in the outflow leads to an equation for To that, coupled to the interior balance equation and the slab boundary layer lead (surprisingly!) to a closed form analytic solution for the gradient wind (as represented by angular momentum, M, at the top of the boundary layer:
(18)
2
22
2
2.
22
k
D
oCC
mo
m m k k o
D D m
rrfr
V r C C rC C r
1
22 11 12 2
k
D
Ck C
m o mD
Cr fr VC
22
2
2k
kD
D
CC CC m
m po
rV Vfr
The maximum wind speed, , found from maximizing the radial dependence of wind speed in the solution (18) on previous slide is given by
mV
(19)
(20)
(21)
Evaluate at ro:
For :o mr r
20 * *k
p b t eD
CV T T s sC
22 2 12
k
D
k
D
CC
Ck C
m pD
CV VC
Can be calculated directly from SST and soundings
Substituting (20) into (21) gives
(22)
Numerical integrations with RE87 model (no dissipative heating, no pressure dependence of k0*) : Left, regular variables; Right: Velocity scaled by (31) and time scaled by the inverse square-root of the enthalpy exchange coefficient.
0o 60oE 120oE 180oW 120oW 60oW
60oS
30oS
0o
30oN
60oN
0 10 20 30 40 50 60 70 80
Annual Maximum Potential Intensity (m/s)
Thermodynamic disequilibrium is necessary to maintain ocean heat balance:
Ocean mixed layer Energy Balance (neglecting lateral heat transport):
*0| |k entrainC k k F F F
sV
2
| |entrains o
po D s
F F FT TVT C
V
Greenhouse effect decreases this
Mean surface wind speedWeak explicit
dependence on Ts
Ocean mixed layer entrainment
Relationship between potential intensity (PI) and intensity of
real tropical cyclones
(Following slides from Emanuel, K.A., 2000: A statistical analysis of hurricane intensity. Mon. Wea. Rev., 128, 1139-1152.)