Tropical Cyclone Inner Core
Core
Dynamics
Dynamics
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Assumptions
• Axisymmetric flow
• Gradient and hydrostatic balance above
PBL
• Troposphere neutral to slantwise moist
convection
ti outside
t id eye
• Moist adiabatic lapse rate in eye above
inversion
ped anticy
yclone at storm top
p
• Well develop
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Local energy balance (from previous lecture):
M
ds *
  Tb  To 
2
rb
dM
Definitions:
f 2
f 2
R  M  rV  r
2
2
 *  Ts  To   s*  sa* 
  Ts  To  s  sa 
“potential radius”
*
s

s
 a a
*
 s  Ts  Tt   s0a
 sa   constant
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Scaling:
 *,    s   *,  
R, r 
s
f
Ts  To

Ts  Tt
 R,
r 
Scaled equations:
1
2  
 *
*
 
3
,
2
r
R R
R 2  2rV  r 2  2rV
R  *
 V 
2 R
2
4
(core)
(1)
Conservation of angular momentum (dimensional):
 
dM
 gr
dt
p
p
Integrate over depth of PBL:
dM
2
pb
  gr  s   gr  s CDV
dt
ps
2
 ggr
CDV
Rd Ts
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Scaling for time:
t  CD
1
Rd Ts
1
pb  s 2 t
p
gps
Nondimensional ang
gular momentum eq
quation:
dR
V2
 r
dt
R
R2
But r 
2V
dR
1

  RV
dt
2
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(2)
Nondimensional PBL entropy equation:
d  Ck

V   0*     V 3  Fb ,
dt CD
Ts  Tt

Ts
Time derivative in R space:
d





R
 
dt 
R
P
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(3)
Assume
use
 well mixed in boundary layer,
1

R   RV :
2
 1
 Ck
*
3
 RV

V   0     V
 2
R CD
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R  *
R 
But V  

2 R
2 R
2
in eyewall
  Ck
*
3

  V   0     1 /    V 
  CD

(Steady state solution:)
Ck *
V 
0   

1 /    CD
2
1
9
(4)
R 
:
Differentiate (4) with respect to R and use V  
2 R
R
2


V


 1 R V
  4 V R



Ck *
2
3 1 /    V  C   0   
D


1 2  Ck 1 R  
 V 2
 2
4  CD   R 
First two terms: Propagation;
Second term: damping; Third term: Possible amplification
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From previously derived dependence of outflow
t
temperature
t
on angullar momentum:
t

1 To
Ric 3 

 2 R
R
Ts  Tt R
rt
2V 2
Thus development equation becomes



 1 R V 
Ck *
2
  4 V R 3 1 /    V  C   0   

D


1 Ck 1 2 1 Ric 4

V 
R
2
2 CD 
8 rt
V


11
1 2
Note that first term steepens V gradient when V  Vmax
3

 0 necessary for amplification
R
2
V gradient cannot steepen indefinitely:
V V
V
  
,
r r
 R 
r  V V  
r


 1  
 1   

r r R R 
r r  R R
R
V r
V
    1   
r R
R
V r V

  r R R
r V
1
R R
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   when
V
R 2V
 
R
r
R
Eyewall undergoes frontal collapse!
This can onlyy be prevented by
y 3-D eddies
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1

R   RV
2
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Simplified amplification model:
  Ck
*
3
  V   0     1 /    V  ,
  CD

1 2

  1  AP  1  A
A    V

2 

R  *
2
V 
2 R

1 3 Ric 1
 R  2 2
*
0
R
2
rt V Enforce    *
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How to handle frontal collapse in model?
Three methods:
1: Zero diffusion model:
For r < rm:
V 0
   *  constant
Does not prevent frontal collapse
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2. Minimum diffusion model: Just enough radial diffusion to prevent failure of coordinate transformation
transformation.
From expression for vertical component of vorticity, we
enforce
V 2V

R
R
Inside the outermost radius, Rcrit, where this is violated, we
take
V 2V

R
R
 R 
 V  Vcrit 

 Rcrit 
2
18
By integration of
 *   crit
*  
 *
2V 2

R
R
4

 R  
1 2
 for R  Rcrit ,
 Vcrit 1 

  Rcrit  
2


for R  Rcrit
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20
3. Maximum diffusion model: 3-D turbulence perfectly efficient in
establishing constant angular velocity inside rm:
V  Vmax
r
R
 Vmax
rm
Rm
  R 2 
2
 *   m Vmax 1    for R  Rm ,

  Rm  
*  
for R  Rm
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http://ocw.mit.edu
12.811 Tropical Meteorology
Spring 2011
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