[Martin Taylor 931219 17:30]
I apologize in advance for continuing this irrelevant thread on vortices, but
several posters have asked me to do so. I'd prefer to get on with talking
about real PCT.
We all agree that vortices and other self-organized structures have no
purpose. As such, they are not PERCEPTUAL control systems. But some people
seem not to find it intuitive that they are structures maintained by negative
feedback, and to resist the notion that they might be. To me, it's a self-
evident proposition that could almost be taken as axiomatic, and the counter-
intuition has been a big surprise to me. So now let's look into it a bit
deeper, and hope that the thread can be terminated before Gary complains that
there is too much energy as well as too much information in my postings.
Let's see if we can work out what happens in a vortex. Very simply, without
recourse to complex concepts like entropy (but I don't think we can avoid
energy). I've never tried this before, and it may not work. Beforehand, I've
always thought about these things in terms of the more general concepts rather
than the mechanics of the specific example.
The basic notion is the same for all self-organized structures. There is a
strong energy flow between some source and some sink. In this flow some
symmetry is broken by an infinitesimal irregularity. The effect of this
broken symmetry is to amplify, initially exponentially, the effect of the
original irregularity, creating a rapidly changing structure. At some point
in this evolution, the forces within the structure come to some equilibrium
state, such that disturbances to the structure are resisted. A disturbance
may add or subtract energy from the structure, but either effect is resisted.
If the negative feedback gain is greater than unity, the structure will be
stable against such disturbances, whether they add energy or remove it.
Contrast this to the ball-in-a-bowl situation, in which a disturbance that
moves the ball up the wall of the bowl is adding energy, and the added energy
is lost to the environmental sink when the ball returns to the bottom of the
bowl.
Apply this thinking to the vortex of water going down a drain, with a
continuing supply of just enough water to maintain a constant head. And we
will suppose that the water comes into the basin in some way with negligible
vorticity and from all directions equally (for example, by spilling in on all
sides over the lip of the basin). If the water comes in slowly, the head will
be small, and the water will flow over the lip of the drain orthogonally, with
no tendency to form a vortex. Any effect that a small disturbance at one point
on the drain has on neighbouring "packets" of water will be dissipated before
it has gone all around the drain. You can try to form a vortex by sloshing the
water to one side, but even though you might get a momentary vortex, it will
not last if the flow is too small.
Now increase the water flow. On the assumption of perfectly symmetrical
arrival of the water at the drain, all the forces parallel to the drain edge
are balanced, and there is no reason for the water to deviate from going
straight over the edge. But, as Bob Clark pointed out, when the basin is
on the Earth and not at the equator, there are Coriolis forces. There is a
very slight lateral force on every packet of water, and it is in the same
direction relative to the drain edge, no matter whether you are looking at
water that has arrived from the North, South, East, or West. This force is
tiny, and in itself would create no measureable affect on the angle at which a
packet of water flows over the drain edge. But it is enough to break the
symmetry. (Almost anything one does to the water or the shape of the drain
will be a stronger symmetry breaker than the Coriolis force, as anyone can
test by actually trying to make bathtub drain vortices go one way or the
other).
To simplify the issue, ignore the Coriolis force, and imagine a drain at the
equator. Water is going straight over the edge in all the radial directions.
Now push the water momentarily aside to the left at one point, say the North,
as it is going over the edge. The push will also push the water leftward at a
point just to the East of North, and the reduction in pressure on the other
side will "pull" the water from the West of North. (It isn't really a pull, of
course, but a differential pressure that results in a net force in that
direction). These effects propagate around the drain.
If that were all there was to it, all you would have would be a simple ball-
in-a-bowl phenomenon. The disturbance would have provided a lateral force
vector that added some energy, the amount depending on how much water was
pushed how far. That energy would dissipate, and the flow would return to its
strictly radial configuration. But that is not all there is to it, because
there is the energy of the flow itself, and when flows curve, things happen.
Some part of the fluid is now moving faster than other parts, and the pressure
is lower in the fast moving part (Bernoulli--that's the heart of how an
airfoil keeps an aircraft in the air). That tends to enhance the bend, and if
the basic flow is fast enough, it _will_ enhance the bend. It's the same kind
of phenomenon that leads to meanders in rivers or ocean currents.
The initial disturbance to the radial flow extracts energy from the main flow
into the lateral flow, so that as the disturbance propagates around the drain
rim, it does not decrease, but increases if the main flow is strong enough.
When the wave arives back at the North point, it is larger than the original
disturbance, and propagates around and around and around... increasing all the
time, until it is limited by the available energy that can be extracted from
the main flow.
At some point, the vortex will have a stable amplitude and shape. The
question now is whether that structure, fed by the main energy flow, and
dissipating through the normal viscous processes, is sustained by negative
feedback of gain greater than unity. Apply the Test. The Test is available to
an outside observer. In its concept, the Test implies that there exists some
perceptual function that defines a CEV (Complex Environmental Variable), but
the only CEVs available to the outside observer are in the observer. The
observer does something that "should" affect the CEV, if it isn't being
stabilized by something (or someone) else, and compares the prediction with
the actual effect on the CEV. There is no way for the observer to tell
whether the stabilization is caused by a simple negative feedback system or by
a negative feedback system that involves a perceptual signal that is compared
with a reference level.
Consider first a disturbance caused by a small sideways jet of water applied
near the drain rim at the North point, that reduces the energy in the vortex
by momentarily reducing the lateral deviation of the incoming water. What
happens? In the previously stable vortex, a wave around the drain rim is
induced, propagating the pressure changes and the reduction in the lateral
deviation of the flow. As it would in any other stream, the main flow at some
other point, say East, augments the bend to equalize the pressures, and the
propagating changes in momentum can even generate oscillations. By the time
the wave has propagated around the drain rim back to the North point, it might
augment or oppose the effect of the small lateral jet, depending on the
various factors of flow rate, viscosity, surface tension, and the like. But
with short propagation times, it will oppose the jet. In this, the effect is
just the same as happens in a conventional control system when acts have
delayed effects of perceptions.
A diagram, for Rick.
-------->- | Main flow (power supply)
> > V
> Gain
> > >
--<-------| -----> Flow down drain
deviation from V
stable vortex shape | vortex
One might at first think that such a system would be liable to go into lateral
oscillations that rhythmically increase and decrease the vortex, but it is
unlikely to do so, because the vortex itself is based on the specific
propagation delay around the rim that occurs with the existing main flow. The
feedback is constrained to be negative by the very fact that the vortex is a
self-organized structure that maintains its own existence against minor
disturbances. If it were not, the vortex would change shape and size in a
random-walk fashion when not overtly disturbed, and would shift permanently
after a sensible disturbance.
The same argument holds, of course, when the disturbance augments the energy
of the main flow. The situation is symmetric. The vortex structure
represents a very small number of degrees of freedom out of those available to
the main flow, and the partitioning of energy into those few degrees of
freedom is what allows us to see it as a "structure." Disturbances may
augment or reduce the energy in those structural degrees of freedom, but so
long as the main flow is unaltered, the energy in the structure will return to
its original value. This is fundamentally different from the ball-in-a-bowl
or even a weight on a spring, for both of which a disturbance must add energy
to the system, energy which is dissipated into heat (or otherwise lost) as the
original state of the system is restored.
I hope this clarifies the issue somewhat.
Martin