Dynamics of the ZonalZonal-Mean,
Mean,
Time-Mean Tropical Circulation
TimeCirculation
1
First consider a hypothetical
yp
planet like Earth, but
with no continents and no seasons and for which
the only friction acting on the atmosphere is at the
surfface.
This planet has an exact nonlinear equilibrium
solution for the flow of the atmosphere,
characterized by
1. Every column is in radiative-convective
equilibrium,
2. Wind vanishes at planet’s surface
3. Horizontal pressure gradients balanced by
3
Coriolis accelerations
2
Hydrostatic balance:
p
  g
z
I pressure coordinat
In
di tes:

  ,
p
where
where
  1  specific volume,

  gz  geopotential
3
Geostrophic balance on a sphere:
u
  
tan 

  2 sin  urel 
a
 a  p
2
rel
Can be
C
b phrased
h
d iin tterms off absolute
b l t angular
l momentum per unit mass:
M  a cos   a cos   u 
 M   a cos  


  sin  

2
3

a cos 


2
4
2
4
4
Eliminate  using hydrostatic equation:
1 tan  M
    T   s * 

 
 

2
2
a cos  p
   p  p  s*    p
2
(Thermal wind equation)
Take s*  s *( )
(convective neutrality)
Integrate upward from surface (p=p0),
taking
g u=0 at surface:
5
s * 
 2 2
2
M  a cos   a cos   cot  Ts  T 





ss *
2
 u  2a cos  u  cot  Ts  T 
 0.

2
2
2
Ts  T  s *

u
2 sin  y
Implies strongest west-east winds where
entropy gradient is strongest, weighted toward
equator
6
Two potential problems with this solution:
1. Not enough angular momentum available for
required west-east wind,
2 Equilibrium solution may be unstable
2.
7
How much angular momentum is need
ded?
d?
At the tropopause,
s * 
 2 2
2
M  a cos   a cos   cot  Ts  Tt 





2
2
2
How much angular momentum is available?
M eq   a
2
s *
 Ts  Tt 


2
4
 2 a 2 tan  1 cos 4  
cos 2 
8
More generally, we require that angular
momentum decrease away from the equator:
1 M 2
0
sin  
1   2
s * 
 4 a cos  
cos  cot  Ts  Tt 
 0.


 
sin   
2
2
3
Let
y  sec 2 
  Ts  Tt s * 
 1 y
 2 2
0
yy   a y 
3
9
What happens when this condition i viol
is
i lated?
d?
Overturning circulation must develop
Acts to drive actual entropy distribution
back toward criticality
Postulate constant angular momentum at
tropopause:
M t  a
10
2
Integrate thermal wind equations down from tropopause:
t
s *
M   a  a cos  cott  T  Tt 

2
2
4
2
2
T  Tt s *
a  1

 u
 cos   

3
2  cos 
 2a sin  
Note:
N
t u att surfface allways < 0 ffor supercriti
iticall
entropy distribution
11
12
Critical entropy distribution:

Ts  Tt s * 1  1
  2 1
2 2
 a yy 2  y

Ts  Tt
Ts  Tt *  1 
1 

s*  2 2 seq 1
1  y   
2 2
a
a
y 
 2
Ts  Tt *  1  2
1
 2 2 seq 1  cos  
2
cos 
a
 2
13



Violation results in large-scale overturning
circulation, known as the Hadley Circulation, that
transports heat poleward and drives surface entropy
gradient back toward its critical value
From "The General Circulation of the Tropical Atmosphere and Interactions with Extratropical
Latitudes - Vol. 1" by MIT Press. Courtesy of MIT Press. Used with permission.
14
15
Simulations by Held and Hou (1980)
16
17
Zonal wind at tropopause
18
The cross-equatorial Hadley Circullatiion
Ci
19
Numerical simulation using a 2-D primitive equation
equat
o model
ode with
t pa
parameterized
a ete ed co
convection
ect o
(Pauluis, 2004)
Imposed sea surface temperature
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12.811 Tropical Meteorology
Spring 2011
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