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First Law of Thermodynamics
The internal energy dU changes when:
1. heat dQ is exchanged between a parcel and its environment
2. work is done by a parcel on its environment (or vice-versa)
dQ  dU  dW
For a cyclic process in which the internal energy does not change (you end up where
you started), then any heating dQ must be balanced by work done to or by the system
dW. In this case

dW  dQ
W total  Qtotal
This merely says that the total work done by the system is balanced by the total
heating done through the cycle.
This cycle has a maximum efficiency, the Carnot engine. This is the physics that sets
the speed limit on how intense a tropical storm can be.

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Carnot cycle
The temperature returns to its starting point, so there is no change in the internal
energy over the course of this cycle. All the work done must be balanced by all the
heating that occurs.
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Gas is heated (Q > 0), so piston rises to reduce the pressure. This pressure drop would
cause the temperature to go down, but heat added compensates exactly so that T is
constant at Thigh. This step is isothermal expansion.
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The heat source is turned off (Q = 0 now), but the piston keeps moving up. Now without
any heat source to balance, the continued decrease in pressure results in the gas cooling
(T goes down). This step is adiabatic expansion.
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In the reverse of step 1, the a cold source is applied (Q < 0). The piston falls, which
would lead to a warming but for the loss of heat by the system. The temperature is fixed
at Tlow. This step is isothermal compression.
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With the cooling source removed (Q = 0 again), the piston continues to drop back to its
starting point. With the increasing pressure and smaller volume comes a rise in
temperature. This is adiabatic compression.
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Carnot cycle
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Wtotal  Qtotal  Qhigh  Qlow 
An ideal hurricane
A natural coordinate system for working with hurricanes is cylindrical coordinates. All
equations are written in terms of radius, angle, and height, as opposed to Cartesian
coordinates (distance east, distance north, height).
Cartesian coordinates are (x, y, z). Cylindrical coordinates are (r, q, z).
An ideal hurricane does not vary azimuthally.
That means that it is axisymmetric.
That means that all the structure can be viewed in the (r, z) plane.
Real hurricanes are not ideal.
But the strongest ones follow most closely to these constraints. And we’re using this
analysis to find out how intense a storm could possibly be.
Deviations from this structure can only be detrimental to a storm’s health.
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An ideal hurricane
The total work done must balance the heat input at the surface:
T  T 
T

high
low
surface  Toutflow
W total  


 T
Qhigh  
 T
Qinput at surface
 high 

surface


The heat input at the surface comes from two sources:
1. Fluxes of heat--actually, enthalpy, k--from the ocean
2. Dissipative heating from recycled frictional dissipation
Qinput at surface 
2  ro

r *
 Ck  V kocean  krdrdq  D
0 0

Where D is the second source, the dissipative heating from frictional dissipation. This
is the same expression as the total work done by the environment (friction at the
surface):
2  ro
3
W total  D 

r
 CD  V rdrdq
0 0
An ideal hurricane
Putting all of this together and solving for the wind speed V, the highest wind speed
possible in a hurricane is:
Vmax
Ck Ts  To *

kocean  k

CD To
This says that the maximum wind speed possible is a function of the exchange
coefficients Ck and CD, the sea surface temperature Ts, the temperature at the top of
the convection To, and the thermodynamic disequilibrium between the saturated
ocean surface
 and the overlying marine boundary layer.
Notice that while the ocean surface temperature affects the outcome, the critical
thing is not the actual value of the water temperature, but rather the difference
between it and the temperature at which convection terminates.
This is a point lost on too many meteorologists. There is not anything magic about
water that is 79oF or 26.5oC. This particular isotherm happens to live in the same
neighborhood as the boundary where it becomes very difficult for convective motions
originating at the surface to puncture through the trade inversion and produce deep
convection that penetrates into the upper troposphere.
http://wxmaps.org/pix/hurpot.html
Potential intensity of tropical cyclones
Vmax
Ck SST  To *

(k0  k)
CD
To
• Emanuel (1986), Bister and Emanuel (1998), Holland (1997)
• A hurricane’s maximum intensity is controlled by:
  Ratio of surface exchange coefficients (~1)
 Surface thermodynamic disequilibrium (evaporative fluxes)
 Difference between SST (hot reservoir) and convective
outflow temperature, To.
The temperature and level of convective outflow play a very
strong role.
Latitude
Storm Season potential intensity and SST
Longitude
Potential intensity (m/s)
Joint distribution
SST (oC)
Level of neutral buoyancy (hPa)
Joint distribution
SST (oC)
Level of neutral buoyancy (hPa)
Joint distribution
Potential intensity (m/s)
Level of neutral buoyancy (hPa)
Joint distribution
Low potential
intensities coincide
with regions whose
soundings are
unsuitable for deep
convection.
High potential
intensities exist only
where deep
convection can.
Potential intensity (m/s)
Latitude
Storm Season potential intensity and LNB
Longitude
Latitude
Preindustrial era control
Longitude
Latitude
Last Glacial Maximum
Longitude
Latitude
LGM – 0ka potential intensity and DSST
Longitude
An ideal hurricane meets reality
Of course there are many things that inhibit actual storms from reaching the
maximum speed permitted by the laws of thermodynamics.
But interestingly, the strongest storms we observe on Earth are the same magnitude
as that predicted by these thermodynamic principles, providing an important
validation of this theory.
Storms cannot be as strong as they like: they cannot exceed the values set by the
relationship between the input temperature, the outflow temperature, and the
strength of the surface fluxes. So a hurricane with 600 mph winds is not possible on
Earth, given the relationship of these parameters.
Real hurricanes are limited by many factors; these include: incorporating drier air into
the circulation (this inhibits convection, raising the outflow temperature), land
(reduces the surface fluxes), wind shear (axisymmetry is violated and deep convection
is limited), and interactions with the ocean itself.
These actually produce fascinating coupled atmosphere-ocean dynamics, and the
result can be seen in cooler sea surface temperatures left in a storm’s wake.
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time = 1
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time = 2
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1
time = 3
2
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2
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t
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time = 3
2
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1
t
Mixed-layer depth in a hurricane’s wake
meters
SST change in a hurricane’s wake
oC