Klas Modin PRO
Mathematician at Chalmers University of Technology and the University of Gothenburg
Hydrodynamics on the Sphere
and Quantization
Klas Modin
Collaborator: Milo Viviani
SNS Pisa
azimuth
2D Euler on the Sphere
elevation
\dot\omega - \{\psi,\omega \} = 0
azimuth
elevation
2D Euler on the Sphere
\dot\omega - \{\psi,\omega \} = 0
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
Vorticity formulation
\dot v + \nabla_v v = -\nabla p - 2\Omega\times v
\operatorname{div} v = 0
Apply curl to \(v\)
\dot\omega - \{\psi,\omega \} = 0
\Delta\psi = \omega - f
\dot\omega - L_v\omega = 0
\omega: S^2\to \mathbb{R} \qquad \psi: S^2\to \mathbb{R}
level-sets of \(\omega\)
How to discretize Lie-Poisson structure?
Structure Preserving Discretization via Quantization
Vladimir Zeitlin
Classical
Quantized
\omega \in C^\infty
W \in \mathfrak{su}(N)
\{\cdot,\cdot \}
[\cdot,\cdot ]
\dot\omega = \{\psi,\omega \}
\dot W = [P,W]
Discrete Laplace-Beltrami 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]
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))
Spatial convergence
(work in progress)
-
Insight: strong convergence not expected
(proof: spectral theorem)
- \(L^\infty\) weak* convergence established
("right" setting for 2D turbulence [Shnirelman, 1993])
Clues on long-time behavior
Non-zero angular momentum
\(N=501\)
Possible mechanism
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
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
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
Conclusions
- Quantization + Lie-Poisson time-discretization provide superior numerical methods for 2D hydrodynamics
- The quantized Euler equations themselves yield insights up to \(L^\infty\) weak* convergence
THANKS!
References:
- Hoppe, Int. J. Mod. Phys., 1989
- Zeitlin, Phys. D, 1991
- Hoppe and Yau, Comm. Math. Phys., 1998
- M. and Viviani, J. Fluid Mech., 2020
- M. and Viviani, arXiv:2102.01451
Hydrodynamics on the Sphere: Numerical Methods based on Quantization
By Klas Modin
Private
Hydrodynamics on the Sphere: Numerical Methods based on Quantization
Presentation given 2021-05 at the SIAM Conference on Dynamical Systems 2021.
