Klas Modin PRO
Mathematician at Chalmers University of Technology and the University of Gothenburg
Hamiltonian Isospectral Flows and Geophysical Hydrodynamics
Klas Modin
Part of PhD work
Milo Viviani
Chalmers and University of Gothenburg
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
Outline
- Connection between isospectral flows and geophysical hydrodynamics
- New isospectral symplectic methods: Isospectral Symplectic Runge-Kutta (IsoSRK)
- Predictions by statistical mechanics
- Predictions by numerical simulations
Hamiltonian isospectral flows
W˙=[B(W),W]
\dot W = [B(W),W]
Let B:Cn×n→Cn×n
=B(W)W−WB(W)
= B(W)W - WB(W)
isospectral flow
Analytic function f yields first integral
Cf(W):=tr(f(W))
C_f(W) := \mathrm{tr}(f(W))
Casimir function
Hamiltonian case
B(W)=∇H(W)†
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 s˙=s×Δs,s:S1→R3hej - etc.
Euler equations on S2
v˙+∇vv=−∇p−2Ω~×v,divv=0
\dot v + \nabla_v v = -\nabla p - 2\tilde\Omega\times v, \qquad \operatorname{div} v = 0
Apply curl
ω˙+v⋅∇ω=0,ω=curlv
\dot \omega + v\cdot\nabla\omega = 0, \qquad \omega = \operatorname{curl} v
geodesic equation on SDiff(S2)
vorticity formulation
Note: ω transported by v
Helmholtz decomposition ⇒ v=∇⊥ψ
ω˙={ψ,ω},Δψ=ω−2Ω⋅n
\dot \omega = \{\psi,\omega \}, \qquad \Delta\psi = \omega - 2\Omega\cdot\mathbf n
Coriolis force
Stream function
Casimirs: for any f:R→R
Cf(ω)=∫f(ω)
\displaystyle C_f(\omega) = \int f(\omega)
Note: Casimirs strongly affect long-time behavior
Euler ⟺ isospectral
via geometric quantization
ω˙={δωδH,ω},H(ω)=21∫(ω−f)ψ
\displaystyle\dot \omega = \left\{\frac{\delta H}{\delta\omega},\omega \right\}, \quad H(\omega) = \frac{1}{2}\int(\omega-f) \psi
(C0∞(S2),{⋅,⋅}) a Poisson algebra
Quantization: projections PN:C0∞(S2)→gN such that
{f,g}≈[PN(f),PN(g)]asN→∞
\{f,g\} \approx [P_N(f),P_N(g)]\quad\text{as}\quad N\to\infty
Lie algebras
ω˙={δωδH,ω}⇒W˙=[∇HN(W)†,W]
\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*
* [J. Hoppe, PhD thesis, MIT Cambridge 1982]
PN:C0∞(S2)→su(N)
P_N:C^\infty_0(\mathbb S^2) \to \mathfrak{su}(N)
expressed through spherical harmonics
- Convergence {⋅,⋅}→[⋅,⋅]N established (Lα-approximations)
- "Magic" formula for discrete Laplacian
ΔNW=2N2−1([XN,[XN,W]]−21[X+N,[X−N,W]]−21[X−N,[X+N,W]])
[Bordemann, Hoppe, Schaller, Schlichenmaier, 1991]
[Hoppe & Yau, 1998]
banded matrices
Spatial discretization obtained!
W˙=[ΔN−1(W−F),W]N
\displaystyle \dot W = [\Delta_N^{-1}(W-F),W]_N
Note: corresponds to
N2 spherical harmonics
O(N2) operations
O(N3) operations
Isospectral flow ⇒ discrete Casimirs
CfN(W)=tr(f(W))
\displaystyle C^N_f(W) = \operatorname{tr}(f(W))
Time discretization
Aims: numerical integrator that is
-
isospectral, Wk→Wk+1 an isospectral map
necessary to preserve Casimirs
-
symplectic, Wk→Wk+1 a Lie-Poisson map su(N)∗→su(N)∗
necessary to (nearly) preserve energy and phase space structure
What about symplectic Runge-Kutta methods?
- Not Lie-Poisson preserving!
- Not isospectral!
...nevertheless, symplectic Runge-Kutta saves the day...
Isospectral Symplectic
Runge-Kutta methods (IsoSRK)*
*[M. & Viviani, FoCM, 2019]
W˙=[B(W),W]
\dot W = [B(W),W]
Given s-stage Butcher tableau (aij,bi) for SRK
XiYiKijW~iWk+1=−(Wn+j=1∑saijXj)hB(W~i)=hB(W~i)(Wk+j=1∑saijYj=hB(W~i)(j′=1∑s(aij′Xj′+ajj′Kij′))=Wk+j=1∑saij(Xj+Yj+Kij=Wk+i=1∑s[hB(W~i),W~i]
\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
Example: Isospectral midpoint method (IsoMP)
W˙=[B(W),W]
\dot W = [B(W),W]
XKW~Wk+1=−(Wn+21X)hB(W~)=2hB(W~)(X+K)=Wk+21(X−X†+K)=Wk+X−X†+K−K†
\begin{aligned}
X &= - (W_n + \frac{1}{2}X)h B(\tilde W) \\
K &= \frac{h}{2} B(\tilde W)(X+K) \\
\tilde W &= W_k + \frac{1}{2}(X-X^\dagger+K) \\
W_{k+1} &= W_k + X-X^\dagger + K-K^\dagger
\end{aligned}
Application to geophysical hydrodynamics
Controversy in 2D turbulence:
Statistical mechanics suggests steady asymptotic minimizing entropy while preserving the Casimirs
High resolution numerical simulations suggest otherwise
[Robert & Sommeria, 1991]
[Dritschel, Qi, Marston, 2015]
Their numerical method use dissipation and does not preserve all Casimirs
⇒ likely affect asymptotic behavior
Our results (non-rotating)
[M. & Viviani, 2019 (under review)]
Fast-forward
Evolution of vorticity ω
Alignment with
point-vortex dynamics
Results (Rossby-Haurwitz wave)
...compare with Jupiter
THANKS!
- Slides at slides.com/kmodin
- Videos at bitbucket.org/kmodin/euler-sphere-quantization
Hamiltonian Isospectral Flows and Geophysical Hydrodynamics Klas Modin
Hamiltonian Isospectral Flows and Geophysical Hydrodynamics
By Klas Modin
Hamiltonian Isospectral Flows and Geophysical Hydrodynamics
Presentation given 2019-04 in Lund.
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