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.