## Euler equations of ideal hydrodynamics

\dot v + \nabla_v v = -\nabla p
\operatorname{div} v = 0

Leonhard Euler

Make sense on any Riemannian manifold

x
v(x)

## Vorticity formulation

\dot v + \nabla_v v = -\nabla p
\operatorname{div} v = 0

Apply curl to $$v$$

\dot\omega - \{\psi,\omega \} = 0
\Delta\psi = \omega
\dot\omega - L_v\omega = 0
\omega: S^2\to \mathbb{R} \qquad \psi: S^2\to \mathbb{R}

level-sets of $$\omega$$

## 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:

\displaystyle\mathcal C_f(\omega) = \int_{S^2}f(\omega)\mu

\displaystyle\omega = \sum_{k=1}^N \Gamma_k \delta_{q^k}

## 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:

\displaystyle H(x_1,\ldots,x_N) = \sum_{kl} \Gamma_k\Gamma_l G(x_1, x_N)

## 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:

\displaystyle H(x_1,\ldots,x_N) = -\sum_{kl} \Gamma_k\Gamma_l \log |x_1-x_N|

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:

\displaystyle H(x_1,\ldots,x_N) = -\sum_{kl} \Gamma_k\Gamma_l \log |x_1-x_N|

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:

\displaystyle H(x_1,\ldots,x_N) = -\sum \Gamma_k\Gamma_l \log |x_1-x_N|

## 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

## No!

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

## 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)

On $$\mathbb{T}^2$$ such discretization exists (sine-bracket)

[Zeitlin 1991, McLachlan 1993]

based on quantization theory by Hoppe (1989)

[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 $$S^2$$

Our mission:  trustworthy discretization on $$S^2$$

## 2D Euler to isospectral flow via Berezin-Toeplitz quantization

C^\infty(M)\ni f \mapsto T^N_f \in \mathfrak{g}_N

Exists if $$M$$ compact quantizable Kähler manifold

Idea: approximate Poisson algebra with matrix algebras

\displaystyle\dot \omega = \left\{\Delta^{-1}\omega,\omega \right\}
\displaystyle\dot W = [\Delta_N^{-1}W,W]

From 2D Euler

To isospectral

\omega \mapsto W

## Lie-Poisson isospectral flows

\dot W = [B(W),W]

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

isospectral flow

Analytic function $$f$$ yields first integral

C_f(W) := \mathrm{tr}(f(W))

Casimir function

Hamiltonian case

B(W) = \nabla H(W)^\dagger

Hamiltonian function

Note: Non-canonical Poisson structure (Lie-Poisson)

## Explicit B-T quantization on $$S^2$$

C^\infty(\mathbb S^2)\ni \omega\mapsto W \in \mathfrak{su}(N)

[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

Complicated coefficients, expressed by Wigner 3-j symbols of very high order

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

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

banded matrices

\displaystyle\dot W = [\Delta_N^{-1}W,W]

Recall

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

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

$$O(N^2)$$ operations

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

## Spatial discretization obtained!

\displaystyle \dot W = [\Delta_N^{-1}W,W]

Note: corresponds to

$$N^2$$ spherical harmonics

$$O(N^2)$$ operations

$$O(N^3)$$ operations

Isospectral flow $$\Rightarrow$$ discrete Casimirs

\displaystyle C^N_f(W) = \operatorname{tr}(f(W))

## 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!

## Runge-Kutta methods

[M. & Viviani 2019]

\dot W = [B(W),W]

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

\begin{aligned} X_i &= - (W_n + \sum_{j=1}^s a_{ij}X_j)h B(\tilde W_i) \\ Y_i &= hB(\tilde W_i)(W_k + \sum_{j=1}^sa_{ij}Y_j \\ K_{ij} &= h B(\tilde W_i)(\sum_{j'=1}^s(a_{ij'}X_{j'}+a_{jj'}K_{ij'})) \\ \tilde W_i &= W_k + \sum_{j=1}^s a_{ij}(X_j + Y_j + K_{ij} \\ W_{k+1} &= W_k + \sum_{i=1}^s [hB(\tilde W_i),\tilde W_i] \end{aligned}

Theorem: method is isospectral and Lie-Poisson preserving on any reductive Lie algebra

## Isospectral midpoint method

W_k = \big(I-\frac{h}{2}B(\tilde W)\big)\tilde W \big(I + \frac{h}{2}B(\tilde W)\big)
W_{k+1} = \big(I+\frac{h}{2}B(\tilde W)\big)\tilde W \big(I - \frac{h}{2}B(\tilde W)\big)
\dot W = [B(W),W]

# What now?

## Numerical results

Evolution of quantized vorticity with $$N=501$$

Let's run it fast...

Strong numerical evidence against MRS!

## 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 blob vortex dynamics (BVD)
3. Dynamics is not integrable $$\Rightarrow$$ blobs continue to merge
4. $$k$$-BVD integrable $$\Rightarrow$$ quasi-periodicity prevents further mixing

## Integrability of PVD and BVD on $$S^2$$

Known since long: $$k$$-PVD integrable for $$k\leq 3$$

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

## Canonical scale separation

Canonical splitting by stabilizer projection:

W = W_s + W_r\qquad W_s \in \mathrm{stab}(P)

initial time

intermediate time

long time

## Canonical scale separation

Canonical splitting by stabilizer projection:

W = W_s + W_r\qquad W_s \in \mathrm{stab}(P)

wave number

energy

By Klas Modin

# Long-time simulation of spherical hydrodynamics via quantization

Presentation given 2021-09 at the NUMDIFF-16 conference.

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