Long-time simulation of spherical hydrodynamics via quantization

Klas Modin

Collaborator: Milo Viviani

Euler equations of ideal hydrodynamics

\dot v + \nabla_v v = -\nabla p

\operatorname{div} v = 0

Leonhard Euler

Make sense on any Riemannian manifold

v(x)

Motivation for 2D Euler: geophysical fluid dynamics

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

Finite-dim (weak) co-adjoint orbits:

\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

Is the MRS prediction correct?

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

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)

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 \(S^2\)

DQM simulation yield persistent unsteadiness

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

\{f,g\} \to [T^N_f,T^N_g] \quad N\to\infty

\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!

Isospectral Symplectic

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!

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:

Small formations merge to larger (inverse energy cascade)
Well-separated blobs interact approximately by blob vortex dynamics (BVD)
Dynamics is not integrable \(\Rightarrow\) blobs continue to merge
\(k\)-BVD integrable \(\Rightarrow\) quasi-periodicity prevents further mixing

Integrability of PVD and BVD on \(S^2\)

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

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:

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