Spring School on Fluid Mechanics of Environmental Hazards
Kerry Emanuel
Massachusetts Institute of Technology
Lecture 2:
Physics
SteadyState Energetics
Energy Production
Distribution of Entropy in Hurricane Inez, 1966
Source: Hawkins and Imbembo, 1976
Total rate of heat input to hurricane:
Q
2
0 r
0
C k
 V 
k
*
0
C
D
 V 
3
Surface enthalpy flux
Dissipative heating rdr
In steady state, Work is used to balance frictional dissipation:
W
2
0 r
0
C
D
 V 
3 rdr
Plug into Carnot equation:
0 r
0
C
D
 V 
3
rdr
T s
T o
T o
0 r
0
C k
 V 
k
*
0
k
rdr
If integrals dominated by values of integrands near radius of maximum winds,
 V max

2
C k
T s
T o
C
D
T o
k
0
* k
Theoretical Upper Bound on Hurricane
Maximum Wind Speed:
Surface temperature
 V pot

2
k s o
C
D
T o
Ratio of exchange Outflow coefficients of enthalpy and temperature momentum
k
* k
0
Airsea enthalpy disequilibrium
60 o
N
Annual Maximum Potential Intensity (m/s)
30 o
N
0 o
30 o
S
60 o
S
0 o
0 10
60 o
E
20
120 o
E
30
180 o
W
40 50
120 o
W
60
60 o
W
70 80
Observed Tropical Atlantic Potential Intensity
Emanuel, K., J. Climate, 2007
Data Sources: NCAR/NCEP reanalysis with pre1979 bias correction, UKMO/HADSST1
Thermodynamic disequilibrium necessary to maintain ocean heat balance:
Ocean mixed layer Energy Balance (neglecting lateral heat transport):
C k
 V s

k
*
0
k
F
F
F entrain
2
V pot
T s
T o
T o
Ocean mixed layer entrainment
Greenhouse effect
F
F
F entrain
C
D
 V s

Weak explicit dependence on T s
Mean surface wind speed
Dependence on Sea Surface Temperature
(SST):
Relationship between potential intensity (PI) and intensity of real tropical cyclones
Why do real storms seldom reach their thermodynamic potential?
One Reason: Ocean Interaction
Strong Mixing of Upper Ocean
NearInertial Oscillations of the Upper
Ocean
NavierStokes equations for incompressible fluid, omitting viscosity and linearized about a state of rest:
u
t
v
t
1
1
p
x
p
y
fv
F
fu
F y x
0
1
p
z
g
u x
v y
w z
0 f
Special class of solutions for which p=w=0:
u
t
v
t
fv
F x
fu
F y
Unforced solution:
2 t
2 u
2 f u
fF y
F x
t u
A sin
B cos
Mixing and Entrainment:
Mixed layer depth and currents
SST Change
Comparison with same atmospheric model coupled to 3D ocean model; idealized runs:
Full model (black), string model (red)
Computational Models of Hurricanes:
A simple model
• Hydrostatic and gradient balance above PBL
• Moist adiabatic lapse rates on M surfaces above PBL
• Parameterized convection
• Parameterized turbulence
Transformed radial coordinate:
Potential Radius: f
2
2 f
R M rV r
2
2
Example of Distribution of R surfaces
Model behavior
Comparing Fixed to Interactive SST:
A good simulation of Camille can only be obtained by assuming that it traveled right up the axis of the Loop Current:
2. Sea Spray
3. Wind Shear
Effects of Environmental Wind Shear
• Dynamical effects
• Thermodynamic effects
• Net effect on intensity
Streamlines (dashed) and θ surfaces (solid)
Mean Absolute Error of NOAA/NHC Tropical Cyclone Intensity Forecasts
Tropical Cyclone Motion
Tropical cyclones move approximately with a suitably defined vertical vector average of the flow in which they are embedded
35
30
25
20
15
145
50
40
30
150
20
155
Longitude
160 165
Lagrangian chaos:
“Beta Gyres”
Operational prediction of tropical cyclone tracks: