## Quasi-geostrophic equation

\dot v + \nabla_v v = -\nabla p - 2 \tilde\Omega\times v
\operatorname{div} v = 0

3D Navier-Stokes equation

Shallow water equation

Quasi-geostrophic equation

## Quasi-geostrophic equation

\dot v + \nabla_v v = -\nabla p - 2 \tilde\Omega\times v
\operatorname{div} v = 0

Jule Charney

3D Navier-Stokes equation

Shallow water equation

Quasi-geostrophic equation

## Quasi-geostrophic equation

\dot v + \nabla_v v = -\nabla p - 2 \tilde\Omega\times v
\operatorname{div} v = 0

3D Navier-Stokes equation

Shallow water equation

Quasi-geostrophic equation

horizontal scale $$\gg$$ vertical scale

## Quasi-geostrophic equation

\dot v + \nabla_v v = -\nabla p - 2 \tilde\Omega\times v
\operatorname{div} v = 0

3D Navier-Stokes equation

Shallow water equation

Quasi-geostrophic equation

horizontal scale $$\gg$$ vertical scale

## Quasi-geostrophic equation

\dot v + \nabla_v v = -\nabla p - 2 \tilde\Omega\times v
\operatorname{div} v = 0

3D Navier-Stokes equation

Shallow water equation

Quasi-geostrophic equation

Coriolis + pressure $$\gg$$ inertial forces

## Vorticity formulation

\dot v + \nabla_v v = -\nabla p - 2 \tilde\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 $$\omega$$

## Geometry of QGE

Lie-Poisson system on $$\mathfrak{X}_\mu(S^2)^* \simeq C^\infty_0(S^2)$$

$$G=\mathrm{Diff}_\mu(S^2)$$

$$T_e^*G\simeq\mathfrak g^*$$

Casimir functions:

\displaystyle\mathcal C_f(\omega) = \int_{S^2}f(\omega)\mu

\displaystyle\omega = \sum_{k=1}^N \Gamma_k \delta_{q^k}

## 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

## 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

## for geometric integration

To test MRS we need to:

• Run long simulations
• Preserve the Casimirs
(energy + enstrophy alone not enough)
• Preserve the Lie-Poisson structure

(criterion in MRS)

On $$\mathbb{T}^2$$ such discretization exists (sine-bracket)

[Zeitlin 1991, McLachlan 1993]

based on quantization theory by Hoppe (1989)

Numerical simulations support MRS on $$\mathbb{T}^2$$

[Abramov & Majda 2003]

## A torus is not a sphere

MRS generally assumed valid also on $$S^2$$

However, non-structure preserving simulations by Dritschel, Qi, & Marston (2015) contradict MRS on $$S^2$$

Our mission: construct trustworthy discretization on $$S^2$$

## 2D Euler to isospectral flow via Berezin-Toeplitz quantization

C^\infty(M)\ni f \mapsto T^N_f \in \mathfrak{g}_N

Exists if $$M$$ compact quantizable Kähler manifold

Idea: approximate Poisson algebra with matrix algebras

\displaystyle\dot \omega = \left\{\Delta^{-1}\omega,\omega \right\}
\displaystyle\dot W = [\Delta_N^{-1}W,W]

From 2D Euler

To isospectral

\omega \mapsto W

## 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)

## Explicit B-T quantization on $$S^2$$

C^\infty(\mathbb S^2)\ni \omega\mapsto W \in \mathfrak{su}(N)

[Hoppe, 1989]

• Express $$\omega$$ in spherical harmonics expansion $\omega = \sum_{l=1}^\infty \sum_{m=-l}^l \omega^{lm}Y_{lm}$
• Truncate at $$l_{\it max}=N-1$$
• For fixed $$m$$, linear map between $$(\omega^{lm})_{l=1}^{N-m})$$ and $$m$$:th diagonal of $$W$$
• Gives $$N$$ linear maps

Complicated coefficients, expressed by Wigner 3-j symbols of very high order

~2 weeks to compute coefficients for $$N=1025$$

## Discrete $$S^2$$ Laplacian 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]

Recall

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))

## In progress: spatial convergence

Conjecture [M. & Viviani, 2020]
Fixed interval $$[0,T]$$, constant $$C=C(T,\omega_0)$$ s.t. $\lVert\omega(t,\cdot)-T_N^{-1}(W^N(t))\rVert_\infty \leq C/N^2$

Classical global existence and uniqueness in $$L^\infty$$ setting

$$\Longrightarrow$$ $$L^\infty$$-norm conserved (it's a Casimir)

Toeplitz-Berezin quantization theory gives

\lVert T_N^{-1}([T_N(\omega_1),T_N(\omega_2)]) - \{\omega_1,\omega_2 \} \rVert_{\infty} = O(1/N^2)

## Time discretization

Aim: 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!

## 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 for QGE

Zonal jet and vortex structures on Jupiter

Simulation of unstable Rossby-Haurwitz wave

# Thank you!

M. & Viviani

M. & Viviani
Lie-Poisson methods for isospectral flows

Found. Comp. Math., 2020

By Klas Modin

# Berezin-Toeplitz quantization and Lie-Poisson integrators for quasi-geostrophic equations

Presentation given 2020-06 at the online-only FoCM Workshop "Geometric Integration and Computational Mechanics" June, 15-18, 2020.

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