Klas Modin PRO
Mathematician at Chalmers University of Technology and the University of Gothenburg
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:
- Small formations merge to larger (inverse energy cascade)
- Well-separated blobs interact approximately by PVD
- Dynamics is not integrable \(\Rightarrow\) blobs continue to merge
- \(k\)-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:
- 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
What's next?
Prove things!
- Convergence of quantized Euler on \(S^2\) as \(N\to\infty\)
\(\mathbb T^2\) by Gallagher (2002) and Abramov & Majda (2003) - Estimates on vortex blob dynamics vs. PVD
[Caglioti, Lions, Marchioro, Pulvirenti, ...] - Quasi-periodic Euler solutions as perturbations of integrable PDE, infinite-dim KAM theory
[Kuksin, ...] - Quantization as approach to "weak diffeomorphisms"
- Rotating case (Coriolis parameters)
- ...
Thank you!
M. & Viviani
A Casimir preserving scheme for long-time simulation of spherical ideal hydrodynamics
J. Fluid Mech., vol. 884, 2020
- Slides at slides.com/kmodin
- Videos at bitbucket.org/kmodin/euler-sphere-quantization
Long-time behaviour of 2D spherical ideal hydrodynamics
By Klas Modin
Long-time behaviour of 2D spherical ideal hydrodynamics
Presentation given 2019-11 at the Hausdorff Institute in Bonn.
