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2D Euler on the Sphere

Can statistical mechanics be used to explain clustering?

Idea by Onsager (1949):

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

Hamiltonian function:

Onsager's observation:

Pos. and neg. strengths $\Rightarrow$ energy takes values $-\infty$ to $\infty$

$\Rightarrow$ phase volume function $V(E)$ has inflection point

Idea by Onsager (1949):

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

Hamiltonian function:

Can statistical mechanics be used to explain clustering?

Onsager's theory has problems

Quantization yields Lie-Poisson preserving discretizations

Zeitlin (1991)

Classical

Quantized

Zeitlin's equations

vorticity matrix

stream matrix

Hoppe-Yau Laplacian

Lost in translation: dictionary hydrodynamics $\leftrightarrow$ quantum physics

 $\operatorname{SDiff}(\mathbb{S}^2)$

 $\operatorname{SU}(N)$

 $\mathfrak{X}_\mu(\mathbb{S}^2)$

 $C^\infty_0(\mathbb{S}^2)$

 $\mathfrak{su}(N)$

 $\mathfrak{su}(N)^*\simeq \mathfrak{su}(N)$

 $\lVert \cdot\rVert_{L^\infty}$

spectral norm

 $\lVert \cdot\rVert_{L^2}$

Frobenius norm

 $\lVert \cdot\rVert_{H^{-1}}$

 $\operatorname{tr}(PW)^{1/2}$

values of $\omega$

eigenvalues of $W$

$\omega$ zonal

$W$ diagonal

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Long-time dynamics look like

"blob"-vortex dynamics

Are there low dim, near integrable stable invariant manifolds?

1. Small vorticity formations merge to larger
2. Well-separated blobs interact by blob-vortex dynamics (BVD)
3. Dynamics is not integrable $\Rightarrow$ blobs continue to merge
4. $k$-BVD integrable $\Rightarrow$ quasi-periodicity prevents further mixing

Shnirelman (2005)

Question 1: How can merging occur when vorticity is transported?

• Arnold (1966): $\omega(t) = \varphi^t_*\omega(0)$ for $\varphi^t\in\operatorname{SDiff}(\mathbb{S}^2)$
• Izosimov, Khesin, Mousavi  (2016): Orbits of simple Morse functions = Reeb graphs

Euler's equations (2D):

Zeitlin's equations:

• $W(t) = E(t)W(0) E(t)^\dagger$ for $E(t) \in SU(N)$
• Spectral theorem: orbits = diagonal matrices with increasing eigenvalues
• Diagonal = zonal $\rightarrow$ possible to destroy critical points $\rightarrow$ merging possible (but "hard")

Zeitlin's equations describe generalized hydrodynamics

Shnirelman (1993)

Bordemann, Meinrenken, Schlichenmaier (1994)

$W \to \omega$ in weak* sense as $N\to \infty$

Question 2: Is blob vortex

dynamics integrable?

Dirac $\delta_x$

axisymmetric blob $b_x$ centered at $x$

Same $SO(3)$ symmetry!

Known about integrability

Point vortex dynamics on $\mathbb{S}^2$

• 3-PVD is integrable (Sakajo, 1999)
• 4-PVD is integrable for vanishing momentum (Sakajo, 2007)
• 4-PVD non-integrable in general (Bagrets & Bagrets, 1997)

Symplectic reduction theory:

only $SO(3)$ symmetry needed in proof

Blob vortex dynamics on $\mathbb{S}^2$

• 3-BVD is integrable
• 4-BVD is integrable for vanishing momentum

Proposed mechanism under scrutiny

Non-zero angular momentum

Question 3: Are BVD solutions stable within Euler flow?

• Clearly BV profiles change dynamically
• Initially small perturbation
• Remains small?

• Strategy: study dynamics on $(\mathfrak{su}(2)\times \mathfrak{su}(N))^k$

Question 4: Is there a natural way to separate scales?

Underlying notion: quantization yields canonical vorticity splitting

1. Linear subspace $\operatorname{stab}_P = \operatorname{ker}[P,\cdot]\subset \mathfrak{su}(N)$
2. Project $\Pi_P:W\mapsto W_s\in\operatorname{stab}_P$
3. Gives canonical splitting $W = W_s + W_r$

Evolution of canonical components

Enstrophy for canonical components

Enstrophies:

$E = \langle W,W\rangle_F$       $E_s = \langle W_s,W_s\rangle_F$        $E_r = \langle W_r,W_r\rangle_F$

Energy for canonical components

Energies:

$H = \langle W,\Delta_N^{-1}W\rangle_F$       $H_s = \langle W_s,\Delta_N^{-1}W_s\rangle_F$        $H_r = \langle W_r,\Delta_N^{-1}W_r\rangle_F$

Geometry of enstrophy and energy

$W_r \bot W_s$ in enstrophy norm

enstrophy levelset of $W$

We expect Zeitlin's

model to be "grainy"

Canonical components capture "broken line" energy spectrum

Summary: Zeitlin's model might yield useful insights on long-time dynamics for 2D Euler

References:

• M. & Viviani (2021)
  Canonical scale separation in two-dimensional incompressible hydrodynamics
• M. & Viviani (2021)
  Integrability of point-vortex dynamics via symplectic reduction: a survey
• M. & Viviani (2020)
  A Casimir preserving scheme for long-time simulation of spherical ideal hydrodynamics
• M. & Viviani (2020)
  Lie-Poisson methods for isospectral flows

Slides available at: slides.com/kmodin