In this issue of Math in the News we look at the science and math of heat waves.

1. 8/1/11

2. Heat Waves
A Bermuda High is a
high-pressure air system
that starts, and stays,
along the East Coast.

3. Heat Waves
Think of the draft of air
created by a helicopter’s
propellers.

4. Heat Waves
A similar phenomenon
happens with the
swirling air of the high
pressure front.

5. Heat Waves
The downward force
comes, in part, from the
torque created by the
force of the wind (F) and
the radius (r) of the high-
pressure system.

6. Heat Waves
The direction of torque
is always
perpendicular to the
plane formed by the
radius and the force.

7. Heat Waves
You can think of the
high-pressure system
as a rotating cylinder
of humid air.

8. Heat Waves
Torque is a force and
this force is applied
over the column of air.
Since pressure is force
divided by area, then
the overall effect is to
increase the air
pressure.

9. Heat Waves
The downward pressure
pushes out cooler air,
and this persists during
the high-pressure
system. This causes a
temperature increase.

10. Heat Waves
The downward pressure
also keeps cooler air
fronts from breaking
through the high-
pressure system. This
keeps temperatures
warm.

11. Heat Waves
A continual flow of
warm, moist air from
the Caribbean keeps
the high-pressure
system intact. This
prolongs the heat.

12. Heat Waves
The warm, moist air
from Caribbean is
heavier than cool, dry
air. This causes an
increase in pressure
and temperature.

13. Heat Waves
But how much pressure
does the high-pressure
system add to the
regions affected by this
system?

14. Heat Waves
Pressure is the ratio of
the downward force
and the area of the
high-pressure system.
For a cylinder the
pressure formula
changes to this.

15. Heat Waves
As we saw earlier, the
downward force from
the pressure system is
the torque. Express
torque in force units.

16. Heat Waves
Torque is the product
of angular momentum
(I) and angular
acceleration (α). For a
rotating disk, the
torque formula is as
shown here.

17. Heat Waves
Use this formula for
torque in the pressure
equation.

18. Heat Waves
We cannot directly
measure the mass but
we can estimate mass
using the density of air
and volume of the
cylinder.

19. Heat Waves
Inputting the mass
equation into the
pressure equation and
simplifying yields a
fairly simple equation.

20. Heat Waves
This graph shows the
pressure as a function
of radius, with
parameters for h, α,
and ρ. The horizontal
graph shows standard
pressure.

21. Heat Waves
As the radius
increases, so does the
air pressure, and as a
result, so does the
temperature.