Hurricanes and the Carnot cycle

We are going to show that hurricanes are ( in good approximation ) a natural realization of the

Carnot cycle

.

rare South Atlantic tropical cyclone viewed from the International Space Station on March 26, 2004.

Some basics about tropical cyclones for details read • tropical cyclones: a storm system with a closed circulation (cyclonic) around a center of low pressure that originates over tropical oceans and is driven principally by heat transfer from the ocean counterclockwise circulation in the Northern Hemisphere •Categorization of tropical cyclones: maximum averaged wind speed 17 m/s or less 18 to 32 m/s 33 m/s or greater tropical depression tropical storm called

hurricanes

in the western North Atlantic and eastern North Pacific regions typhoons in the western North Pacific severe tropical cyclones elsewhere

Structural elements of a tropical cyclone •Basic flows Primary circulation origin of circulation is the Coriolis force

F C

 2

m v

velocity in the rotating frame   angular velocity of rotating frame (earth) Fictitious force in the rotating reference frame of the earth Low pressure region Brief reminder to the Coriolis force: 

xt

 

vt

2  1 2  2  a c 2 Exact:

d dt inertial

  

rot v inertial a inertial



v rot



a rot r

 2  

v rot



m a rot



m a inertial

Coriolis force  2

m

 

v rot



m

 

r

 

r



•Eye, Eyewall and Rainbands Secondary circulation click for animation

The hurricane as a Carnot heat engine see for details A  B: air undergoes isothermal expansion as it flows toward the lower pressure of the storm center while in contact with the surface of the ocean (heat bath @ T s  300K) B  C: Adiabatic (very fast) ascent of the air C  D: air flows out at the top of its trajectory and is incorporated from the extreme low pressure region into other weather systems via an isothermal compression (heat bath @ T 0  200K) D  A: air undergoes an adiabatic compression when loosing altitude fast

PV-Diagram of the hurricane Carnot engine

A D B

T s =300K

C

T 0 =200K

C B D A

Where does the work go which the hurricane produces from the heat of the ocean Work drives the wind with surface speed

v

Devastation by hurricane Katrina, City of Huntington Beach stationary state: Generated work per time dissipated (friction) • dissipation   3

dW dt

 

F drag v

  3 because

F drag



v

2 • rate of heat transfer from the ocean to the atmosphere   quantifies the thermodynamic disequilibrium

b

between the ocean and atmosphere

We know the textbook efficiency of a Carnot engine : 

Carnot

: 

W Q in



T S



T S T

0 important difference to textbook Carnot cycle textbook Carnot cycle tropical cyclones

W

=work done on environment work used for turbulent dissipation transformed back into heat @T s heat from the ocean

Q in



a v

3 

b v

Heat from turbulent dissipation  

a v a v

3 3 

b v

and back into the front end of the Carnot cycle

W



a v

3  

a v

2 

b



v



T s



T

0

E T

0 

a v

2

a v

2  1   

b v

 1   

E

where E:=b/a theoretical upper bound on hurricane wind speed note T 0