# Quantized hydrodynamics on the sphere

azimuth

## 2D Euler on the Sphere

elevation

\dot\omega - \{\psi,\omega \} = 0
\Delta\psi = \omega

## 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:

\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

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|

## Quantization yields Lie-Poisson preserving discretizations

Zeitlin (1991)

Classical

Quantized

\omega \in C_0^\infty
W \in \mathfrak{su}(N)
\{\cdot,\cdot \}
[\cdot,\cdot ]
\dot\omega = \{\psi,\omega \}
\dot W = [P,W]

# Zeitlin's equations

\dot W = [P,W],
\Delta_N P = W
P,W \in \mathfrak{su}(N)

vorticity matrix

stream matrix

Hoppe-Yau Laplacian

\displaystyle C^\infty_0(\mathbb{S}^2)\ni\omega \longrightarrow \sum_{l=0}^\infty\sum_{m=-l}^l \omega_{lm} Y_{lm} \longrightarrow \sum_{l=0}^N\sum_{m=-l}^l \omega_{lm} T_{lm} \in \mathfrak{su}(N)

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

Classical Quantum
Lie group
Lie algebra
Phase space
"Strong" norm
Enstrophy norm
Energy norm
Measurables
Singular sol. vortex sheets rank-1 matrices
Axi-symmetry

$$\operatorname{SDiff}(\mathbb{S}^2)$$

$$\operatorname{SU}(N)$$

$$\mathfrak{X}_\mu(\mathbb{S}^2)$$

$$C^\infty_0(\mathbb{S}^2)$$

$$\mathfrak{su}(N)$$

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

$$\lVert \cdot\rVert_{L^\infty}$$

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

azimuth

elevation

positive blobs

negative blobs

## 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)

\mathcal{O}_{\omega_0} = L^\infty \text{ weak* closure of } \omega_0\circ\operatorname{SDiff}(\mathbb{S}^2)

Bordemann, Meinrenken, Schlichenmaier (1994)

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

## dynamics integrable?

Dirac $$\delta_x$$

x
x
\delta_x
b_x
\displaystyle H = -\sum \Gamma_k\Gamma_l \log |x_k-x_l|
\displaystyle H = -\frac{1}{2}\sum \langle b^l_{x_l},\Delta^{-1} b^k_{x_k}\rangle

axisymmetric blob $$b_x$$ centered at $$x$$

Same $$SO(3)$$ symmetry!

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

\Rightarrow

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 $$(\mathfrak{su}(2)\times \mathfrak{su}(N))^k$$
t = 0
t = \epsilon

small w.r.t.

$$H^{-1}$$ or $$L^2$$

or what?

## 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 $$W = W_s + W_r$$
W
W_s
\operatorname{stab}_P
W_r

t = 0
t = \text{large}
W
W_s
W_r

## Enstrophy for canonical components

E
0
E_r
t
E_s

Enstrophies:

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

E = E_s + E_r

## Energy for canonical components

H
0
H_r
t
H_s

Energies:

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

H_s = H + H_r

## Geometry of enstrophy and energy

W
W_s
W_r

$$W_r \bot W_s$$ in enstrophy norm

$$W_r \bot W$$ in energy norm

enstrophy levelset of $$W$$

energy levelset of $$W$$

\frac{1}{N}\lVert W_r\rVert_{E} \leq \lVert W_r \rVert_{H}

Equivalence of norms:

## 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

By Klas Modin

# Quantized hydrodynamics on the sphere

Online-presentation given 2021-10 in the Hamiltonian Seminar Series, University of Toronto and University of Arizona.

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