Time--dependent, axisymmetric
Time
mo
mod
del ph
hrase
rased
d in R space
• Hydrostatic and gradient balance above
PBL
pse rates on M surfaces
• Moist adiabatic lap
above PBL
quasi equilibrium
• Boundary layer quasi-equilibrium
• Deformation-based radial diffusion
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Potential Radius:
f 2
f 2
R  M  rV  r
2
2
(1)
Local energy conservation:
2 Ts  Tt  
s *
1 1 2
 2 
2
rb rt
f 2 R3
R
(2)
Differentiate in time:
1 rb 1 rt Ts  Tt   s *
 3

3
2 3
rb  rt 
f R R 
2
(3)
Mass continuity:

ru 
,
pp

r  
r
Transform to potential radius coordinates:
 r r dR r dP  
dr
ru  r
r 

,

dt
R dt
d
P dt
d  p
  R
r 
r   r 
r

 r



 p
P p r P P
3
r


 2

P
2
Define ψ0 as streamfunction at top of boundary layer
and
d use siimplle fifiniitte diff
difference in verti
ticall:
rb
 2  0   ,

2
rt
 2

2
4
(4)
PBL flow:
r
r dR 
r
r

R dt P

Angular momentum balance:
 
f dR 2
 gr
2 dt
P
f R2  r 2
V
2
r
r 2  

r 2 g

,
R f P
P
2
2 r 2
2 r 2
 0   g 2  s  g 2  s CD | V | V
f R
f R
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Saturation entropy:
sd Q rad 
s *  d 



 M u  M d  w 
  m 
z
T 
Downdraft:
M d  1   p  M u
Boundary layer entropy:
hs
s
s
 Ck | V |  s0*  s   CD | V |3   M u  w0  s  sm   CD r | V |V
|V

fRR
U d tto define
Used
d fi M ueq when
h  0;
0 otherwi
th ise, equati
tion integrat
t
tedd for s
6
Relaxation equation:
M u M ueq  M u


c
Precipitation efficiency:
sm  s
sm0
p 
s  sm0 Middle troposphere entropy:

sm
s
 M u  s 
sm   Q rad

7
Radiation:
Q rad    s * s *(t  0) 
Radial diffusion added to equations for rb , rt , s*,, and ss
m
2


1  2  R 
Db  
 rb b
 2 
R R 
rb  rb  
2


1  2  R 
Dt  
 rt t
 2 
R R 
rt  rt  
 
s * 
Ds*  2  rbb

rb 
rb 
sm 
 
Dsm  2  rm
b

rm 
rm 
8
2



R
2
i  l ri  2 
ri  ri 
Note that surface saturation entropy depends on pressure,
which is calculated from gradient wind balance using V
Complete equations summarized in Emanuel (1995), posted on course web page.
page.
9
Model behavior
behavior
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Saturate troposphere inside 100 km in initial state:
state:
11
Character of control simulation
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MIT OpenCourseWare
http://ocw.mit.edu
12.811 Tropical Meteorology
Spring 2011
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