Collaborator: Milo Viviani

Euler equations of ideal hydrodynamics

Leonhard Euler

Motivation for 2D Euler: geophysical hydrodynamics

Vorticity formulation

Apply curl to $v$

Geometry of 2D Euler

Lie-Poisson system on $\mathfrak{X}_\mu(S^2)^* \simeq C^\infty_0(S^2)$

$G=\mathrm{Diff}_\mu(S^2)$

$T_e^*G\simeq\mathfrak g^*$

Casimir functions:

Finite-dim (weak) co-adjoint orbits:

Statistical mechanics theories for 2D Euler

Idea by Onsager (1949):

• Approximate $\omega$ by PV for large $N$
• Find invariant measure and presume ergodicity

Hamiltonian function:

Statistical mechanics theories for 2D Euler

Idea by Onsager (1949):

• Approximate $\omega$ by PV for large $N$
• Find invariant measure and presume ergodicity

Hamiltonian function:

Onsager's observation:

Pos. and neg. strengths $\Rightarrow$ energy takes values $-\infty$ to $\infty$

Statistical mechanics theories for 2D Euler

Idea by Onsager (1949):

• Approximate $\omega$ by PV for large $N$
• Find invariant measure and presume ergodicity

Hamiltonian function:

Onsager's observation:

Pos. and neg. strengths $\Rightarrow$ energy takes values $-\infty$ to $\infty$

$\Rightarrow$ phase volume function $v(E)$ has inflection point

Statistical mechanics theories for 2D Euler

Idea by Onsager (1949):

• Approximate $\omega$ by PV for large $N$
• Find invariant measure and presume ergodicity

Hamiltonian function:

Problems with Onsager's theory

• PV solutions far from smooth ($H^{-1}$ but never $L^p$)
• $\Rightarrow$ No Casimir functions
• But experiments and numerical simulations strongly indicate that Casimirs affect long-time behavior

Statistical mechanics theories for smooth solutions

Miller (1990) and Robert & Sommeria (1991): (MRS)

• Maximize entropy of probability distribution of macroscopic states under energy and Casimir constraints

Is the MRS prediction correct?

A problem for geometric

numerical integration

To test MRS we need to:

• Run long simulations
• Preserve the Casimirs
  (energy + enstrophy alone not enough)
• Preserve the Lie-Poisson structure

(criterion in MRS)

[Abramov & Majda 2003]

A torus is not a sphere

MRS generally assumed valid also on $S^2$

However, non-structure preserving simulations by Dritschel, Qi, & Marston (2015) contradict MRS on $S^2$

DQM simulation yield persistent unsteadiness

2D Euler to isospectral flow via Berezin-Toeplitz quantization

Exists if $M$ compact quantizable Kähler manifold

Idea: approximate Poisson algebra with matrix algebras

Lie-Poisson isospectral flows

Let $B\colon\mathfrak{g}\to\mathfrak{g}$

Analytic function $f$ yields first integral

Hamiltonian case

Explicit B-T quantization on $S^2$

[Hoppe, 1989]

• Express $\omega$ in spherical harmonics expansion $$\omega = \sum_{l=1}^\infty \sum_{m=-l}^l \omega^{lm}Y_{lm}$$
• Truncate at $l_{\it max}=N-1$
• For fixed $m$, linear map between $(\omega^{lm})_{l=1}^{N-m})$ and $m$:th diagonal of $W$
• Gives $N$ linear maps

Discrete $S^2$ Laplacian on $\mathfrak{su}(N)$

• "Magic" formula [Hoppe & Yau, 1998]
  $$\Delta_N W =\frac{N^2-1}{2}\left([X^N,[X^N,W]]- \frac{1}{2}[X_+^N,[X_-^N,W]]- \frac{1}{2}[X_-^N,[X_+^N,W]] \right)$$

Recall

What is $\Delta_N$ and how compute $\Delta_N^{-1}W$ ?

(Naive approach requires $O(N^3)$ operations with large constant)

• $\Delta_N$ admits sparse $LU$-factorization with $O(N^2)$ non-zeros

Spatial discretization obtained!

Note: corresponds to

$N^2$ spherical harmonics

Isospectral flow $\Rightarrow$ discrete Casimirs

Time discretization

Aim: numerical integrator that is

• isospectral, $W_{k}\to W_{k+1}$ an isospectral map
  necessary to preserve Casimirs
   
• symplectic, $W_{k}\to W_{k+1}$ a Lie-Poisson map $\mathfrak{su}(N)^*\to\mathfrak{su}(N)^*$
  necessary to (nearly) preserve energy and phase space structure

What about symplectic Runge-Kutta methods (SRK)?

• Not Lie-Poisson preserving!
• Not isospectral!

Isospectral Symplectic

Runge-Kutta methods

[M. & Viviani 2019]

Given $s$-stage Butcher tableau $(a_{ij},b_i)$ for SRK

Isospectral midpoint method

What now?

Numerical results

Evolution of quantized vorticity with $N=501$

Alignment with

point-vortex dynamics

Other initial conditions

What are "generic" initial conditions?

Our interpretation: sample from Gaussian random fields on $H^{1+\epsilon}(S^2)$

Non-zero angular momentum

$N=501$

Mechanism for long-time behaviour

Observation: large scale motion quasiperiodic

Assumptions for new mechanism:

1. Small formations merge to larger (inverse energy cascade)
2. Well-separated blobs interact approximately by PVD
3. Dynamics is not integrable $\Rightarrow$ blobs continue to merge
4. $k$-PVD integrable $\Rightarrow$ quasi-periodicity prevents further mixing

Integrability of PVD on $S^2$

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"

Predictions for Euler on $S^2$

For generic initial conditions:

1. Momentum small $\Rightarrow$ 4-PVD is KAM-integrable $\Rightarrow$ expect 4 non-steady vortex blobs
    
2. Momentum intermediate $\Rightarrow$ 3-PVD is integrable $\Rightarrow$ expect 3 non-steady vortex blobs
    
3. Momentum large $\Rightarrow$ expect 2 large and 1 small vortex blobs

Thank you!

• M. & Viviani, A Casimir preserving scheme for long-time simulation of spherical ideal hydrodynamics, J. Fluid Mech., 2020
• M. & Viviani, Lie-Poisson methods for isospectral flows, Found. Comp. Math., 2020
• M. & Viviani, Integrability of point-vortex dynamics via symplectic reduction, Arnold Math. J., 2020.
• M. & Viviani. Canonical scale separation in two-dimensional incompressible hydrodynamics, preprint, 2021

• Slides at slides.com/kmodin
• Videos at bitbucket.org/kmodin/euler-sphere-quantization