How does the spinning of a tornado aﬀect its shape and the path of its
movement? In severe

storms, a tornado causes the surrounding winds to blow ﬁercely.
As a result, the strong

winds push the tornado along, hopefully in a nondestructive
path. To understand weather

patterns during strong storms, meteorologists (both professional
and amateur) can use the

motion of tornados to predict the direction of the overall
storm. A similar approach is used

to model ﬂuid motion in turbulent regimes. By studying the
motion of small swirling eddies,

we can understand the general motion of the ﬂuid.

Without the cost of real-life experiments, mathematical modeling
of ﬂuid turbulence can

allow engineers and scientists to predict and reduce the aerodynamic
drag on an airplane or

an automobile, understand the ﬂow of blood in the human heart,
or decrease the amount of

pollutants produced by a combustion engine. Under suitable
conditions, some ﬂuid motion

is such that the velocity (the direction and speed of the ﬂuid)
at any given time and position

is not found to be the same when it is measured several times
under seemingly identical

situations. Fluctuating motions of this kind are said to be
turbulent. Turbulence is composed

of eddies or vortices (think of little tornados) that zigzag
and whirl around the overall

direction of motion. The mathematical equations that govern
ﬂuid turbulence, called the

Navier-Stokes equations, are known and reasonably well understood,
but they are extremely

diﬃcult to solve except in rare instances. Even accurate numerical
solutions cannot be

achieved for many realistic ﬂows important to industrial applications.

One approach taken to understanding turbulence is the development
of mathematical

models for the movement of vortex ﬁlaments. The vorticity of
a ﬂuid is a measure of how

much the ﬂuid is rotating or spinning. A vortex ﬁlament is
a thin tube created when the

ﬂuid rotates or spins very intensely about a single curve.
When a ﬁlament rotates it changes

the ﬂuid movement around it. The induced motion of the ﬂuid
then transports the ﬁlament.

Therefore the evolution equation for a vortex ﬁlament depends
on the ﬂuid motion it has

created. The best example of a vortex ﬁlament is a tornado.
The rotation of the tornado

about its axis causes the air around it to move quickly. The
air motion causes high winds

that change the shape of the tornado and puts the tornado into
motion along a path.