The Tropical Atmosphere: Hurricane Incubator
Incubator
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A One-Dimensional Description
Description
of the Tropical Atmosphere
Atmosphere
Elements of Thermal Balance:
Solar Radiation
• Luminosity: 3.9 x 1026 J s-1 = 6.4 x 107 Wm-2
at top of photosphere
• Mean distance from earth: 1.5 x 1011 m
• Flux density at mean radius of earth
L
−2
0
S ≡
= 1370 Wm
0
2
4π d
Stefan-Boltzmann Equation: F =σ T 4
−8
−2
−4
σ =5.67×10 Wm K
Sun:
→
7
4
−2
σ T =6.4×10 Wm
T 6,000 K
Disposition of Solar Radiation:
⎛
⎞
2
Total absorbed solar radiation = S ⎜1− a ⎟ π r
0⎝
p⎠ p
a ≡ planetary albedo ( 30% )
p
Total surface area = 4π r 2
p
S
⎛
⎞
0
Absorption per unit area =
⎜1− a p ⎟
4 ⎝
⎠
Absorption by clouds, atmosphere, and surface
Terrestrial Radiation:
Effective emission temperature:
S
⎛
⎞
4
0
σT ≡
1− a ⎟
⎜
e
p⎠
4 ⎝
Earth: T = 255K = − 18°C
e
Observed average surface temperature = 288 K = 15°C
Highly Reduced Model
• Transparent to solar radiation
• Opaque to infrared radiation
• Blackbody emission from
surface and each layer
Radiative Equilibrium:
Top of Atmosphere:
S
⎛
⎞
4
4
0
1− a ⎟ = σ T
σT =
⎜
A
p⎠
e
4 ⎝
→
TA = Te
Surface:
S0
4
σ Ts = σ TA + (1− a p ) = 2σ Te
4
4
4
1
→
Ts = 2 4 T
e = 303 K
Surface temperature too large
large
because:
because:
• Real atmosphere is not opaque
• Heat transported by convection as well as
by radiation
Extended Layer Models
Models
TOA : σ T2 4 = σ Te 4 → T2 = Te
Middle Layer : 2σ T = σ T2 + σ Ts = σ Te + σ Ts
4
1
4
Surface : σ Ts = σ Te + σ T
4
1
→ Ts = 3 4 Te
4
4
1
1
4
T1 = 2 Te
4
4
4
Effects of emissivity<1
Surface : 2ε Aσ TA 4 = ε Aσ T14 + ε Aσ Ts 4
( 2)
→ TA = 5
1
4
Te 321K
< Ts
Stratosphere : 2ε tσ Tt = ε Aσ T2
4
( 2)
→ Tt = 1
1
4
Te 214K
4
< Te
Full calculation of radiative equilibrium:
Problems with radiative
radiative
equilibrium solution:
solution:
• Too hot at and near surface
• Too cold at a near tropopause
• Lapse rate of temperature too large in the
troposphere
• (But stratosphere temperature close to
observed)
Missing ingredient: Convection
• As important as radiation in transporting
enthalpy in the vertical
• Also controls distribution of water vapor
and clouds, the two most important
constituents in radiative transfer
When is a fluid unstable to
to
convection?
convection?
• Pressure and hydrostatic equilibrium
• Buoyancy
• Stability
Hydrostatic equilibrium:
equilibrium:
Weight : − g ρδ xδ yδ z
Pressure :
pδ xδ y − ( p + δ p ) δ xδ y
dw
F = MA : ρδ xδ yδ z
= −g ρδ xδ yδ z − δ pδ xδ y
dt
dw
∂p
α = 1 ρ = specific volume
= −g − α ,
dt
∂z
Pressure distribution in atmosphere at rest:
RT
R*
,
R≡
Ideal gas : α =
p
m
1 ∂p ∂ ln( p )
g
=
=−
Hydrostatic :
∂z
p ∂z
RT
Isothermal case :
p = p0 e
−z
H
,
RT
= "scale height "
H≡
g
Earth: H~ 8 Km
Buoyancy:
Weight : − g ρbδ xδ yδ z
Pressure :
pδ xδ y − ( p + δ p ) δ xδ y
dw
F = MA : ρbδ xδ yδ z
= − g ρbδ xδ yδ z − δ pδ xδ y
dt
dw
∂p
∂p
but
= − g − αb
=−g
αe
dt
∂z
∂z
αb − αe
dw
→
=g
≡B
αe
dt
Buoyancy and Entropy
α = 1ρ
Specific Volume:
Specific Entropy:
s
α = α ( p, s
⎛ ∂α ⎞ ⎛ ∂T ⎞
Maxwell : ⎜
⎟
⎟ =⎜
⎝ ∂s ⎠ p ⎝ ∂p ⎠ s
⎛ ∂T ⎞
⎛ ∂α ⎞
=⎜
⎟⎟ δ s
⎟ δ s = ⎜⎜
⎝ ∂s ⎠ p
⎝ ∂p ⎠ s
B=g
(δα ) p
(δα ) p g ⎛ ∂T ⎞
α
=
⎜⎜
α ⎝ ∂p
⎛ ∂T ⎞
⎟⎟ δ s = − ⎜
⎟ δ s ≡ Γδ s
⎝ ∂z ⎠ s
⎠s
The adiabatic lapse rate:
First Law of Thermodynamics :
dsrev
dT
dα
= cv
+p
Q =T
dt
dt
dt
dT d (α p )
dp
= cv
+
−α
dt
dt
dt
dT
dp
= ( cv + R )
−α
dt
dt
dT
dp
−α
= cp
dt
dt
Adiabatic : c p dT − α dp = 0
Hydrostatic : c p dT + gdz = 0
→
(
dT
dz )
s
=− g
cp
≡ −Γ d
g
Γ= c
p
Earth’s atmosphere:
Γ = 1 K 100 m
Model Aircraft Measurements
(Renno and Williams, 1995)
Radiative equilibrium is unstable in the troposphere
Re-calculate equilibrium assuming that tropospheric stability is rendered neutral by convection: Radiative-Convective Equilibrium
Better, but still too hot at surface, too cold at tropopause
Above a thin boundary layer, most atmospheric convection involved phase change of water:
Moist Convection
Convection
Moist Convection
Convection
• Significant heating owing to phase
changes of water
• Redistribution of water vapor – most
important greenhouse gas
• Significant contributor to stratiform
cloudiness – albedo and longwave
trapping
Stability Assessment using Tephigrams:
100
Pressure (mb)
200
300
400
500
600
700
800
900
1000
-40
-30
-20
-10
0
10
Temperature (C)
20
30
40
50
Stability Assessment using Tephigrams:
Convective Available Potential Energy
(CAPE):
CAPEi ≡ ∫ (α p − α e )dp
pi
pn
(
)
= ∫ Rd Tρ − Tρ d ln ( p )
pi
p
p
e
Tropical Soundings: Virtually no CAPE
CAPE
Annual Mean Kapingamoronga
Kapingamoronga
Tropical Cyclones: Steady State
State
Physics. Where does the Energy
Energy
Come From?
From?
Angular
momentum
per unit mass
M = rV + Ωsin (θ ) r 2
Energy Production
Production
Carnot Theorem: Maximum efficiency results from
from
a particular energy cycle:
cycle:
•
•
•
•
Isothermal expansion
Adiabatic expansion
Isothermal compression
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 is the same as if air were saturated and descending moist
adiabatically.
Maximum rate of working:
Ts −
To W=
Q
T
s
Total rate of heat input to hurricane:
*
3
Q = 2π ∫ ρ ⎡⎣Ck | V | ( k0 − k ) + CD | V | ⎤⎦ rdr
0
r0
Dissipative
heating
Surface enthalpy flux
In steady state, Work is used to balance frictional dissipation:
r0
W = 2π ∫ ρ ⎡⎣CD | V | ⎤⎦ rdr
0
3
Plug into Carnot equation:
∫
r0
0
Ts − To
ρ ⎡⎣CD | V | ⎤⎦ rdr =
To
3
∫
r0
0
*
⎡
ρ ⎣Ck | V | ( k0 − k ) ⎤⎦ rdr
If integrals dominated by values of integrands near
radius of maximum winds,
(
Ck Ts − To
2
*
k0 − k
→ |Vmax | ≅
C D To
)
Graph of Solution for particular values of coefficients:
coefficients:
00 GMT 21 March 2008
40oN
20oN
o
0
20oS
40oS
o
60oE
0
0
10
20
120oE
30
40
180oW
50
120oW
60
70
60oW
80
90
o
0
100
Numerical simulations
Relationship between potential
potential
intensity (PI) and intensity of
of
real tropical cyclones
cyclones
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)
80
100
Evolution with respect to time of maximum intensity
Evolution with respect to time of maximum intensity,
normalized by peak wind
Evolution curve of Atlantic storms whose lifetime maximum
intensity is limited by declining potential intensity, but not by
landfall
Evolution curve of WPAC storms whose lifetime maximum
intensity is limited by declining potential intensity, but not by
landfall
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
limited by landfall.
CDF of normalized lifetime maximum wind speeds of
Northwest Pacific tropical cyclones of tropical storm strength
(18 m s−1) or greater, for those storms whose lifetime
maximum intensity was limited by landfall.
Evolution of Atlantic storms whose lifetime maximum intensity
was limited by landfall
Evolution of Pacific storms whose lifetime maximum intensity
was limited by landfall
Composite evolution of landfalling storms
MIT OpenCourseWare
http://ocw.mit.edu
12.103 Science and Policy of Natural Hazards
Spring 2010
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