## Euler equations of ideal hydrodynamics

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

Leonhard Euler

Make sense on any Riemannian manifold

## 3D

Nothing known, not even existence (related to Millenium problem)

Some things known:

[Kraichnan 1967]

Various hypotheses based on statistical mechanics

## Motivation

Holy grail of 2D incompressible hydrodynamics:

• Inner workings of the inverse energy cascade
• Long-time behavior of mean-flow condensates

Zonal jet and vortex structures on Jupiter

## Vorticity formulation

\dot v + \nabla_v v = -\nabla p
\operatorname{div} v = 0
\dot\omega + \{\omega,\psi \} = 0
\Delta\psi = \omega
v = \nabla^\bot\psi

Apply $$\operatorname{curl}$$ to

\dot\omega + L_v\omega = 0

Vorticity $$\omega$$ transported along $$v$$

Point-vortex dynamics (PVD):

invariant set of weak solutions

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

Conservation of Casimirs

\displaystyle \mathcal C_f(\omega) = \int f(\omega)

## Statistical mechanics theories for 2D Euler

Idea by Onsager (1949):

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

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

• Minimize microcanonical entropy under energy and Casimir constraints

Predicts equilibrium of large-scale vortex structures

## 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 simulations to test MRS theory

Need discretization that:

• Preserves the Casimirs
• Preserves the Lie-Poisson structure

$$\Rightarrow$$ 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]

## The torus is not a sphere

MRS generally assumed valid also on $$S^2$$

However, "untrustworthy" simulations by Dritschel, Qi, & Marston (2015) contradict MRS

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

## 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)

## Examples

• Toda lattice (periodic and non-periodic)
Particles interacting pairwise with exponential forces
Connection to numerical linear algebra: flow that diagonalizes matrices, continuous analog of $$QR$$-algorithm

• $$n$$-dimensional free rigid body

• Heisenberg spin chain
Discretization of Landau-Lifschitz equation $\dot{\mathbf s} = \mathbf s\times \Delta \mathbf s,\quad s:S^1\to \mathbf R^3\qquad\phantom{hej}$
• etc.

## via quantization

\displaystyle\dot \omega = \left\{\frac{\delta H}{\delta\omega},\omega \right\}, \quad H(\omega) = \frac{1}{2}\int\omega \psi

$$(C_0^\infty(\mathbb S^2),\{\cdot,\cdot\})$$ a Poisson algebra

Quantization: projections $$P_N:C^\infty_0(\mathbb S^2) \to \mathfrak g_N$$ such that

Lie algebras

\displaystyle\dot \omega = \left\{\frac{\delta H}{\delta\omega},\omega \right\} \quad\Rightarrow\quad \dot W = [\nabla H_N(W)^\dagger,W]

## Explicit construction by Hoppe

P_N:C^\infty_0(\mathbb S^2) \to \mathfrak{su}(N)

expressed through spherical harmonics

• Convergence $$\{\cdot,\cdot\} \to [\cdot,\cdot]_N$$ established ($$L^\alpha$$-convergence)

• "Magic" formula for discrete Laplacian
$\Delta_N W =\frac{N^2-1}{2}\left([X^N,[X^N,W]]- \frac{1}{2}[X_+^N,[X_-^N,W]]- \frac{1}{2}[X_-^N,[X_+^N,W]] \right)$

[Bordemann, Hoppe, Schaller, Schlichenmaier, 1991]

[Hoppe & Yau, 1998]

banded matrices

## 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

Aims: 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

## Numerical results

Same initial conditions as Dritschel, Qi, & Marston (2015)

$$N=501$$

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 $$H^1(S^2)$$

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 $$k\leq 3$$

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

## What's next?

Prove things!

1. Convergence of quantized Euler on $$S^2$$ as $$N\to\infty$$
$$\mathbb T^2$$ by Gallagher (2002) and Abramov & Majda (2003)
2. Estimates on vortex blob dynamics vs. PVD
[Caglioti, Lions, Marchioro, Pulvirenti, ...]
3. Quasi-periodic Euler solutions as perturbations of integrable PDE, infinite-dim KAM theory
[Kuksin, ...]
4. Quantization as approach to "weak diffeomorphisms"
5. Rotating case (Coriolis parameters)
6. ...

By Klas Modin

# Long-time behaviour of 2D spherical ideal hydrodynamics

Presentation given 2019-11 at the Hausdorff Institute in Bonn.

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