Leonhard Euler
Make sense on any Riemannian manifold
Nothing known, not even existence (related to Millenium problem)
Some things known:
inverse energy cascade
[Kraichnan 1967]
Various hypotheses based on statistical mechanics
Holy grail of 2D incompressible hydrodynamics:
Zonal jet and vortex structures on Jupiter
Copyright: NASA, Cassini Imaging Team
Apply \(\operatorname{curl}\) to
Vorticity \(\omega\) transported along \(v\)
Point-vortex dynamics (PVD):
invariant set of weak solutions
Conservation of Casimirs
Idea by Onsager (1949):
Miller (1990) and Robert & Sommeria (1991): (MRS)
Predicts equilibrium of large-scale vortex structures
2D Euler equations are not ergodic
...but perhaps MRS is "generically" correct
Flow ergodic except at "KAM islands"
Poincaré section of finite dimensional Hamiltonian system
Need discretization that:
\(\Rightarrow\) classical methods (FD, FEM, FV) are untrustworthy
(criterion in MRS)
On \(\mathbb{T}^2\) such discretization exists (sine-bracket)
[Zeitlin 1991, McLachlan 1993]
based on quantization theory by Hoppe (1989)
Sine-bracket simulations support MRS on \(\mathbb{T}^2\) [Abramov & Majda 2003, and others]
MRS generally assumed valid also on \(S^2\)
However, "untrustworthy" simulations by Dritschel, Qi, & Marston (2015) contradict MRS
DQM simulation yield persistent unsteadiness
Our mission: construct trustworthy discretization on \(S^2\)
Let \(B\colon\mathfrak{g}\to\mathfrak{g}\)
isospectral flow
Analytic function \(f\) yields first integral
Casimir function
Hamiltonian case
Hamiltonian function
Note: Non-canonical Poisson structure (Lie-Poisson)
\((C_0^\infty(\mathbb S^2),\{\cdot,\cdot\})\) a Poisson algebra
Quantization: projections \(P_N:C^\infty_0(\mathbb S^2) \to \mathfrak g_N\) such that
Lie algebras
expressed through spherical harmonics
[Bordemann, Hoppe, Schaller, Schlichenmaier, 1991]
[Hoppe & Yau, 1998]
banded matrices
Note: corresponds to
\(N^2\) spherical harmonics
\(O(N^2)\) operations
\(O(N^3)\) operations
Isospectral flow \(\Rightarrow\) discrete Casimirs
Aims: numerical integrator that is
What about symplectic Runge-Kutta methods (SRK)?
[M. & Viviani 2019]
Given \(s\)-stage Butcher tableau \((a_{ij},b_i)\) for SRK
Theorem: method is isospectral and Lie-Poisson preserving on any reductive Lie algebra
Same initial conditions as Dritschel, Qi, & Marston (2015)
\(N=501\)
Let's run it fast...
Strong numerical evidence against MRS!
What are "generic" initial conditions?
Our interpretation: sample from Gaussian random fields on \(H^1(S^2)\)
Non-zero angular momentum
\(N=501\)
Observation: large scale motion quasiperiodic
Assumptions for new mechanism:
Known since long: \(k\)-PVD integrable for \(k\leq 3\)
What about the 4-blob formations?
4-PVD on \(S^2\) non-integrable in general, but integrable for zero-momentum [Sakajo 2007]
Aref (2007) on PVD:
"a classical mathematics playground"
"many strands of classical mathematical physics come together"
For generic initial conditions:
Prove things!
M. & Viviani
A Casimir preserving scheme for long-time simulation of spherical ideal hydrodynamics
J. Fluid Mech., vol. 884, 2020