## 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

## 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

\dot W = [B(W),W]

Let $$B\colon\mathbb{C}^{n\times n}\to\mathbb{C}^{n\times n}$$

= B(W)W - WB(W)

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)

## 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 $\dot{\mathbf s} = \mathbf s\times \Delta \mathbf s,\quad s:S^1\to \mathbf R^3\qquad\phantom{hej}$
• etc.

## Euler equations on $$\mathbb S^2$$

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

Apply $$\operatorname{curl}$$

\dot \omega + v\cdot\nabla\omega = 0, \qquad \omega = \operatorname{curl} v

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

vorticity formulation

Note: $$\omega$$ transported by $$v$$

Helmholtz decomposition $$\Rightarrow$$ $$v = \nabla^\bot \psi$$

\dot \omega = \{\psi,\omega \}, \qquad \Delta\psi = \omega - 2\Omega\cdot\mathbf n

Coriolis force

Stream function

Casimirs: for any $$f:\mathbb R\to\mathbb R$$

\displaystyle C_f(\omega) = \int f(\omega)

Note: Casimirs strongly affect long-time behavior

## via geometric quantization

\displaystyle\dot \omega = \left\{\frac{\delta H}{\delta\omega},\omega \right\}, \quad H(\omega) = \frac{1}{2}\int(\omega-f) \psi

$$(C_0^\infty(\mathbb S^2),\{\cdot,\cdot\})$$ a Poisson algebra

Quantization: projections $$P_N:C^\infty_0(\mathbb S^2) \to \mathfrak g_N$$ such that

Lie algebras

\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]

P_N:C^\infty_0(\mathbb S^2) \to \mathfrak{su}(N)

expressed through spherical harmonics

• Convergence $$\{\cdot,\cdot\} \to [\cdot,\cdot]_N$$ established ($$L^\alpha$$-approximations)

• "Magic" formula for discrete Laplacian
$\Delta_N W =\frac{N^2-1}{2}\left([X^N,[X^N,W]]- \frac{1}{2}[X_+^N,[X_-^N,W]]- \frac{1}{2}[X_-^N,[X_+^N,W]] \right)$

[Bordemann, Hoppe, Schaller, Schlichenmaier, 1991]

[Hoppe & Yau, 1998]

banded matrices

## Spatial discretization obtained!

\displaystyle \dot W = [\Delta_N^{-1}(W-F),W]_N

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))

## Time discretization

Aims: 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

• Not Lie-Poisson preserving!
• Not isospectral!

...nevertheless, symplectic Runge-Kutta saves the day...

## Runge-Kutta methods (IsoSRK)*

*[M. & Viviani, FoCM, 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

## Example: Isospectral midpoint method (IsoMP)

\dot W = [B(W),W]
\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

$$\Rightarrow$$ likely affect asymptotic behavior

## Our results (non-rotating)

[M. & Viviani, 2019 (under review)]

Fast-forward

Evolution of vorticity $$\omega$$

## Results (Rossby-Haurwitz wave)

...compare with Jupiter

By Klas Modin

# Hamiltonian Isospectral Flows and Geophysical Hydrodynamics

Presentation given 2019-04 in Lund.

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