Zeitlin's model for ideal hydrodynamics on the sphere

(Part 1: numerical experiments)

Klas Modin

Collaborator: Milo Viviani

SNS Pisa

\dot\omega = \{\psi,\omega \}, \quad \Delta \psi = \omega

Euler's equations

\dot W = [P,W], \quad \Delta_N P = W

Zeitlin's model

W,P \in \mathfrak{su}(N)

azimuth

elevation

Smooth, randomly generated initial data

azimuth

elevation

Smooth, vanishing angular momentum initial data

azimuth

elevation

...run it faster

Questions:

  • How exactly is Zeitlin's model related to Euler?
     
  • How accurate is it?
     
  • Does it capture the right dynamics?
     
  • Can it give new insights?

at fine scales: not at all!

...but no method is!

Example: canonical splitting

W = W_s + W_r \qquad (\omega = \omega_s + \omega_r)
W_s
W_r

projection onto stabilizer of \(P\)

Entire presentation at CUNY

Zeitlin's model for ideal hydrodynamics on the sphere

By Klas Modin

Zeitlin's model for ideal hydrodynamics on the sphere

CUNY Einstein seminar given 2021-12.

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