Long-time behaviour of 2D spherical ideal hydrodynamics

Klas Modin

PhD project of 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

What is the generic

long-time behaviour?

3D

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

Some things known:

inverse energy cascade

[Kraichnan 1967]

Various hypotheses based on statistical mechanics

2D

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

Copyright: NASA, Cassini Imaging Team

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

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

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

DQM simulation yield persistent unsteadiness

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.

Euler \(\iff\) isospectral flow

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

\{f,g\} \approx [P_N(f),P_N(g)]\quad\text{as}\quad N\to\infty

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!

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

Numerical results

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

\(N=501\)

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 \(H^1(S^2)\)

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

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. ...

Thank you!

M. & Viviani

A Casimir preserving scheme for long-time simulation of spherical ideal hydrodynamics
J. Fluid Mech., vol. 884, 2020