The divergence was the `collective' property of flow fields that was simple, and that characterized their importance in transporting something from one place to another. A complementary property, called circulation, tells what is special about flow when it simply moves something around and around in circles, with no net transport accomplished on large scales. The circulation is just what the name suggests, and that we think about naturally in terms of whirling eddies in water. We need to define it carefully, however, to talk about how it can be related to measurements. In the process, we will find that this characteristic of the flow field has some very remarkable, and perhaps not obvious, features.
Giving a meaningful notion to the size and ``direction'' of a
circulating flow is straightforward. A weathervane or paddle-wheel
spins around in a tornado, even if the wind is blowing sideways as
well as helically, as
long as the wind is blowing faster on one side of the wheel
than on the other. Moreover, there is an amount of circulation
(which may be zero) no matter which direction we point the axis
of the paddle-wheel, because it either spins or it does not. The
speed at which it spins when the axis points in any direction is
just determined by how much the circulation ``points'' along
that direction, as drawn in fig. 7.6.
What the paddle-wheel is doing is allowing only one direction for the motion of each of its paddles, around the edge of the circle pinioned by the axis. A paddle catches the wind, and tells how fast the wind is moving along that direction at that place. Since the wheel itself is stiff, each of its paddles cannot move independently, so they move in response to the sum of the speeds on all the protruding surfaces. This is why a sideways-blowing wind does not cause the paddle-wheel to spin, unless the motion is faster on one side than on another. There is a net circulation around some axis when the sum of all the effects causes the wheel to spin as it is pinioned about that axis.
Thus, much as we used a sphere to define a divergence, by
summing the lengths of components of arrows outward from its
surface, we can use a circle to define the circulation
of
a flow field about some axis. We put the circle perpendicular
to the axis of interest, map out the flow field where the circle
is, and add the lengths of all the components that point along the edge of the circle. Again, once we choose a
direction of travel around the circle, the addition is positive
if the flow component agrees, negative if it opposes. In a
purely sideways flow, the arrows on one side point along the
direction of the circle, in the other they point against it, and
the two contributions to the sum cancel. There is no
circulation, and the paddle-wheel does not spin. This
construction is shown in fig. 7.6.
If we wish to represent the circulation in some small region,
about a particular axis, we can do it by drawing an arrow that
points along the axis, whose length is whatever value the sum
takes around the circle perpendicular to it. The only
difficulty lies in assigning a direction when we say we want to
add the components of the flow arrows ``along'' the edge of the
circle, and which direction the corresponding arrow should
point. This can be resolved if we agree to use what is called a
right hand rule.
A person's right hand gives a reliable
and unique way to relate the direction around a circle to a
particular direction down its axis. Suppose the fingers of the
hand wrap around the circle with some orientation. Then the thumb
points in one of the two directions. No matter how the circle
and the hand are turned, the orientation of the thumb always has
the same relation to the direction of the fingers around the
circle. This is shown in fig. 7.6.
We agree that if the flow field adds to a positive number around some circle, we represent that by a positive-length arrow in the direction of the thumb. If we consider the travel around the circle to be the opposite way, the flow components add to a negative sum, but the thumb points in the opposite direction, and the resulting arrow is the same as before. It only depends on the characteristics of the flow and which axis we use to measure the direction of the circulation, and not on our conventions in drawing the arrows.