Suppose you find yourself face-to-face with Young-Mills or Navier-Stokes or a nonlinear PDE or a funky metamaterial or a cloudy day. And you ask yourself, is this thing “turbulent” What does that even mean?
If you were had a serious course on chaos, as Professor Ribeiro had, you must have learned about the coin toss (Bernoulli map). I’ll walk you through this basic example of deterministic chaos, than through the kicked rotor, a neat physical system that is chaotic, and then put infinity of these together to explain what chaos or turbulence looks like in the spacetime.
What emerges is a spacetime which is very much like a big spring mattress that obeys the familiar continuum versions of a harmonic oscillator, the Helmholtz and Poisson equations, but instead of being “springy”, this metamaterial has an unstable rotor at every lattice site, that gives, rather than pushes back. We call this simplest of all chaotic field theories the spatiotemporal cat. There is a QM3 version, ask Boris Gutkin or Tomaž Prosen to tell you about it.
In the spatiotemporal formulation of turbulence there is no evolution in time, there are only a repertoires of admissible spatiotemporal patterns. In other words: throw away your integrators, and look for guidance in clouds' repeating patterns.
That's turbulence. And if you don't know, now you know.
No actual cats, graduate or undergraduate, have shown interest in, or were harmed during this research.