T
T dM

,
s * M ds *
Using
We can re-write (18) as
To
Ri  dM 
  2c 
.
M
rt  ds * 
ds * 
1
We can also re-write (6) M b  r  f   Tb  To 

2
dM


as
2
b
Boundary layer
entropy
Boundary layer
angular momentum
3
V
dsb
|
|
 Ck | V |  s0*  sb   Cd
h
dt
Tb
dM
h
 rr | V | V
dt
1
(19)
(20)
(21)
(22)
Combine (21) and (22):
*
s
dsb
Ck  0  sb  | V |2


dM
CD rV
V
Tb rV
V
Let
sb  s*, | V | V  Vb , r 

rb
*
s
ds *
Ck  0  s * Vb



dM
CD rV
Tb rb
b b
(23)
Balance condition (8):
Vb
ds *
   Tb  To 
rb
dM
2
(24)
Eliminate Vb between (23) and (24):
*
s
 0  s *
2
 ds *  Tb Ck

 
2
dM
T
C
r


o
D b  Tb  To 
( )
(25)
Eliminate rb2 between (20) and (25):
ds *
f
 ds * 

 0,,

  2
dM Tb  To
 dM 
2
where
Remember that
(26)
Tb Ck s0 * s *

To CD 2M
To
T
Ric  dM 
 2 

M
rt  ds * 
3
(19)
Integrate (26) and (19) inward from some outer radius ro, defined such that
V 0
at
r  ro
In general,, inte grating
g this system will not yield To=Tt at
r=rmax. Iterate value of rt until this condition is met.
If V >> fr,
fr we ignore dissipative heating
heating, and we neglect
neglect
pressure dependence of s0*, then we can derive an
approximate closed-form solution.
4
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
represented
by angular momentum, M, at the top of the boundary layer:
2
 M 


 Mm 
C
2 k
CD
 r
2 
rm 


,
2
Ck Ck  r 
2


 
CD CD  rm 
5
(27)
2
 fro2 


 2Vm rm 
C
2 k
CD
 ro 
2 
rm 


.
2
Ck Ck  ro 
2

 
CD CD  rm 
1 2 1  1 Ck 
rm  fro Vm 

2
2
C
D 

1
C
2 k
CD
(28)
(29)
The maximum wind speed,V m , found from maximizing the radial dependence of wind speed in the solution (27) is given by
C
2 k
CD
m
V
 2rm 
V  2 
 fro 
2
p
6
Ck
CD
(30)
Ck
V 
Tb  Tt  s0 * se *
CD
2
p
Substituting (29) into (30) gives
 1 Ck 
V V 

 2 CD 
2
m
2
p
7
Ck
CD
C
2 k
CD
(31)
Substituting (31) into (29) gives
1
rm   
2
3
2
2
o
fr
Tb  Tt  s0 * se *
Also,
CD
rt  r
Ric
Ck
2
2
m
8
9
70
Control
With dissipative heating
60
V (m/s)
50
40
30
20
10
0
0
100
200
300
400
radius
di (km)
(k )
10
500
600
700
11
Numerical integrations with RE87 model (no dissipative heating
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
coefficient.
12
Effects of Pressure
Pressure--Dependence of
of
Surface Saturation Enthalpy
Enthalpy
13
14
15
16
17
18
00 GMT 27 April 2007
40oN
20oN
o
0
20oS
40oS
o
60oE
0
10
20
120oE
30
180oW
40
50
19
120oW
60
60oW
70
80
o
0
90
Relationship between potential intensity (PI) and intensity of real tropical cyclones
cyclones
20
21
22
Wind speed (m/s)
Potential wind speed (m/s)
80
70
60
50
V (m/s)
40
30
20
10
-80
-60
-40
-20
0
20
40
60
Time after maximum intensity (hours)
23
80
100
Evolution with respect to time of maximum intensity
24
Evolution with respect to time of maximum intensity,
normalized by peak wind
25
Evolution curve of Atlantic storms whose lifetime maximum
intensity is limited by declining potential intensity, but not by
landfall
26
Evolution curve of WPAC storms whose lifetime maximum
intensityy is limited byy declining
g potential
p
intensityy, but not by
y
landfall
27
CDF of normalized lifetime maximum wind speeds of North
Atlantic tropical cyclones of tropical storm strength (18 m s−1) or
greater, for those storms whose lifetime maximum intensity was
greater
limited by landfall.
28
CDF of normalized lifetime maximum wind speeds of
Northwest Pacific tropical cyclones of tropical storm strength
(18 m s−1)
s 1) or greater,
greater for those storms whose lifetime
maximum intensity was limited by landfall.
29
Evolution of Atlantic storms whose lifetime maximum intensity
was limited byy landfall
30
Evolution of Pacific storms whose lifetime maximum intensity
was limited byy landfall
31
Composite evolution of landfalling storms
32
MIT OpenCourseWare
http://ocw.mit.edu
12.811 Tropical Meteorology
Spring 2011
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33