Spring School on Fluid Mechanics of Environmental Hazards
Kerry Emanuel
Massachusetts Institute of Technology

Lecture 2:
Physics

Steady-State 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

Air-sea 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 re-analysis with pre-1979 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

Near-Inertial Oscillations of the Upper
Ocean

Navier-Stokes 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 3-D 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: