Tropical Cyclones: Steady State Physics
Physics
1
Energy
Energy Production
Production
2
Carnot Theorem: Maximum efficiency results from a pa
particular
t cu a e
energy
e gy cyc
cycle:
e
•
•
•
•
Isothermal expansion
Adiabatic expansion
I th
Isothermal
l compressiion
Adiabatic compression
Note: Last leg is not adiabatic in hurricane: Air cools radiatively. But since
environmental temperature profile is moist adiabatic, the amount of
radiative cooling
g is the same as if air were saturated and descending
g moist
adiabatically.
Maximum rate of working:
Ts 
To  W
Q
Ts
3
Total rate of heat input to hurricane:
*
3


Q  2   Ck | V |  k0  k   CD | V |  rdr
0
r0
Surface enthalpy flux
Dissipative
heating
In steady state, Work is used to balance frictional dissipation:
W  2 
r0
0
3

 CD | V |  rdr
4
Plug into Carnot equation:

r0
0
Ts  To
 CD | V |  rdr 
To
3

r0
0
 Ck | V |  k0*  k   rdr
If integ
grals dominated byy values of integ
grands near
radius of maximum winds,

C
T

T
 | Vmax |2  k s o k0*  k
C D To
Not a closed expression since k at radius of
maximum winds is unknown
5

Problems w
with
ith Energy Bound:
Bound:
• Implicit assumption that all irreversible entropy
production is by dissipation of kinetic energy. But
outside of eyewall, cumuli moisten
environment....accounting for almost all entropy
production there
• Approximation of integrals dominated by high wind
region is crude
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Local energy balance in eyewall regi
region:
on:
T = To
Height
Ψ1
Ψ2
T = Tb
h
Radius
Image by MIT OpenCourseWare.
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Definition of streamfunction, ψ:

,
 rw 
rr

 ru  
z
Flow parallel to surfaces of constant ψ, satisfies mass
continuity:
1 
1 
  ru  
  rw   0
r r
r z
8
Variables conserved (or else constant along
streamlines) above PBL,
PBL where flow is considered
reversible, adiabatic and axisymmetric:
Energy:
Entropy:
Angular
Momentum:
1
2
E  c pT  Lv q  gz  | V |
2
Lv q *
s*  c p ln T  Rd ln p 
T
1 2
M  rV  fr
2
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First definition of s*:
Tds*
d *  c p ddT  Lv dq
d *  ddp
(1)
Steadyyflow:
p
p
 dp   dr   dz
rr
z
Substitute from momentum equations:
V 2

 dpp  dz  g  V w  dr   fV
f  V u 
 r

10
((2))
Identity:
1
1
2
2
 Vw  dz   Vu  dr  d  u  w    d ,
2
r
where
u w
 
z r
azimuthal vorticity
Substituting (3) into (2) and the result into (1) gives:
V 2

1
Tds*
Tds
*  dE VdV
VdV  
 fV  dr 
 d
r
 r

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(4)
(3)
One more identity:
V 2

M 1 
 fV  dr   2  f  dM
VdV  
2 
r
 r

Substitute into (4):
Tds*
M
1
1


dM   d  d  E  fM  ,
2
2
r
r


Note that third term on left is very small: Ignore
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(5)
Integrate (5) around closed circuit:
T = To
Height
Ψ1
Ψ2
T = Tb
h
Radius
Image by MIT OpenCourseWare.
Right side vanishes; contribution to left only from end points
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1 1
Tb  To  ds *   2  2  MdM  0
 rb ro 
1
1
ds *
 2  2  Tb  To 
rb ro
MdM
Mat re storm
Mature
storm:
( )
(6)
ro  rb :
M
ds *
 2   Tb  To 
rb
dM
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(7)
In inner core,
V  fr
 M  rV
ds *
Vb  rb Tb  To 
dM
Convective criticality:
(8)
s*  sb
dsb
 Vb  rb Tb  To 
dM
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(9)
dsb /dM determined by boundary layer processes:
T = To
Height
Ψ1
Ψ2
T = Tb
h
Radius
Image by MIT OpenCourseWare.
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Put (7) in differential form:
ds M dM
 0,
Tb  To   2
dt r dt
(10)
Integrate entropy equation through depth of boundary layer:
layer:
ds 1
*
3

h
 Ck | V |  k s  k   CD | V | F
 Fb  ,
dt Ts
(11)
where Fb is the enthalpy flux through PBL top. Integrate
angular momentum equation through depth of boundary layer:
dM
h
 CD r | V | V ,
dt
17
(12)
Substitute (11) and (12) into (10) and set Fb to 0:


Ck Ts  To
2
*
k0  k
 |V | 
C D To
(13)
Same answer as from Carnot cycle
cycle. This is still not a closed
expression, since we have not determined the boundary layer
enthalpy, k
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What Determines Outflow Tempera
emperatture?
ure?
19
Simulations with CloudCloud-Permitting,
Permitting, Axisymme
symmettric Model
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Saturation entropy (contoured) and V=0 line (yellow)
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Streamfunction (black contours), absolute temperature (shading) and V=0 contour(white)
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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. 23
Rich
Ri
hard
dson Numb
ber (capped
d att 3)
3). Box sh
hows area used
d
for scatter plot. 24
Vertical Diffusivity (m2s-1)
25
Ri=1
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Imp
plications
cat o s for
o Out
Outflow
o Temperature
e pe atu e
ds *
r m
dM .
Ri 
M
z
2
ds *
rt  m
M
M
dM .


z
Ric
2
s * ds * M

,
z dM z
27
2
 ds * 
rt 
m 

s *
dM

 .


z
Ric
2
(14)
But the vertical gradient of saturation entropy is related to the vertical gradient of temperature:
Use definition of
ss* and C
C.-C
-C.::
 T 


T  T   p  s*

  s * ,
s *  s *  p
p
(15)
T
cp
 T 
,

 
 s * 
p 
L2v q * 
1 
2
R
c
T
v p


28
(16)
Substitute (16) into (15) and use hydrostatic equation:
T
cp

m
T


.
 * 
s
L2v q *  s *
1 
2
z
R
c
T
v p


If
Tb  To 

2
Vb  c p
Tb
2
(17)

L2v q *  
rb2  we can neglect first term
Ric 2  on left of (17)
 1 

2 

rt 
 Rv c pT  
Substitute ((14)) into ((17):
)
To
Ric  dM 
 2 
 .
s *
rt  ds * 
2
(18)
Gives dependence of Outflow T on s*
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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
30
(19)
(20)
(21)
(22)
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12.811 Tropical Meteorology
Spring 2011
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