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The computer simulations in that SciAm article depict the motion as a big
nasty mess of 3D vortices. Imagine a hot-tub with several turbulent water
jets. Imagine a bunch of smoke-rings which are colliding and
interweaving.
A Wimshurst electrostatic generator gives a good analogy: if there are no
ambient e-fields present, then the Wimshurst generator will never develop
a high voltage no matter how long we turn the crank. But if a small
e-field hits the device, it causes a breaking of symmetry and the machine
can then ramp up its voltage exponentially until it is limited by corona
leakage. A shorted-out self-exciting dynamo is similar: we can crank it
forever, but if there is zero ambient b-field, then the generator won't
run. If a small ambient b-field is present, then the generator can create
a small current, which makes a bigger b-field, which generates a larger
current, and the current grows exponentially until it is limited by
resistive losses.