ANU Institutional Update

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�Monday 18 March, 2024

Acknowledgement: Australian Research Council, Simons Foundation

R. L. Dewar ^[1], A. Tassavoli ^[1], D. Muir ^[1], S. Jeyakumar ^[1,3], D. Pfefferlé ^[1,3], N. Bohlsen ^[1],

R. Tang ^[1], K. Duru ^[1], V. Robins ^[1], N. Cross ^[1], S. R. Hudson ^[4], Z. Qu ^[5] , M,. Kraus ^[6], N. Duignan^[7]

Prof. Matthew Hole^[1,2] (matthew.hole@anu.edu.au)

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Mathematical Sciences Institute & School of Computing

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University of Sydney

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Simons Retreat at the ANU: 4-15 Dec. 2023

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Fusion Plasma Theory and Modelling

energetic ion physics

M. J. Hole, R. L. Dewar, A. Tassavoli,

�Students:

S. Jeyakumar, J. Doak, M. Thompson, D. Muir, R. Tang, N. Bohlsen, �T. Malcolm, L. Huang �

MSI Collaborators:

L. Ferrario, K. Duru

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Local: Z. Qu (NTU), �D. Pfefferlé (UWA), �N. Duignan (Sydney)

DG methods and Non-Linear Optimisation for MRxMHD

• Attempt to develop a more capable and robust numerical scheme for the MRxMHD model with Tokamak and stellarator geometry and improve the ability of the current SPEC code to handle complex geometry.

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[R. Tang, K. Duru, D. Muir, M. J.Hole, R. L. Dewar ]

DG methods and Non-Linear Optimisation for MRxMHD�1D illustration

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• A new MHD model is proposed by Dewar, R.L. and Qu, Z.S., 2022. Journal of Plasma Physics, 88(1).
• The existence and robustness of nested flux surfaces is not guaranteed in general, especially in 3D MHD.
• During the optimization, the condition of frozen-in flux surfaces of ideal MHD is relaxed and magnetic reconnection is allowed.
• To attain a “well-posed” optimization problem, a weak version of ideal Ohm’s law is imposed through the augmented Lagrangian method.

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Relaxed magnetohydrodynamics with weak ideal-Ohm’s-law constraint

[A. Tassavoli, S. R. Hudson, Z. Qu, R. L. Dewar, M. J.Hole]

The Hamiltonian minimization

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The global constraints are:

• Magnetic helicity conservation
• Cross helicity conservation
• Entropy conservation

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The weak local constraint is:

• Ideal Ohm’s law

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The hard microscopic constraint is:

• Conservation of mass

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Other microscopic constraints are relaxed

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• The global constraint are imposed by the Lagrange multipliers.
• The IOL constraints is imposed by the augmented Lagrangian method.

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IOL constraint

Euler-Lagrange Equations

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Isothermal ideal gas equation

Hamiltonian minimization in slab geometry

• A simple model in a 1D slab is proposed to apply to this new MHD.
• In this model the variable fields are expanded by Chebyshev polynomials and the system is optimized for the coefficients of expansions.

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Stable and accurate method for simulating anisotropic diffusion in toroidally confined magnetic fields

[D. Muir, K. Duru, M. J. Hole, S. R. Hudson]

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Developing structure-preserving particle methods for different collision operators

• New publication (accepted by Journal of Plasma Physics):

A structure-preserving particle discretisation for the Lenard-Bernstein collision operator (https://arxiv.org/abs/2309.16894)

• Current work: Developing similar methods for the Landau collision operator using the metriplectic approach of Morrison (1986) [1].
• Both methods are being implemented in the Julia package VlasovMethods.jl, along with ideal solvers.

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[1] Morrison, P. J. (1986). A paradigm for joined Hamiltonian and dissipative systems. Physica D: Nonlinear Phenomena, 18(1–3), 410–419. https://doi.org/10.1016/0167-2789(86)90209-5

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Illustration of a Poisson manifold. Surfaces are of constant entropy, where ideal (Hamiltonian) dynamics take place, while collisions carry you across different surfaces through the changing entropy. Image courtesy Michael Kraus, IPP Garching.

[S. Jeyakumar, M. Kraus, D. Pfefferlé, M. J. Hole].

Automated field line classification with persistent homology

Question: Can we determine the class of a field line (stochastic, KAM torus, island chain, etc.) from only the Poincare section automatically?

Answer: Yes, with persistent Homology

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Form sequence of simplicial complexes by progressively connecting points which are further and further apart.

Add triangles as soon as possible (Vietoris-Rips filtration). Homology classes are born and die (display on Persistence Diagram).

[N. Bohlsen, V. Robins ]

Recent Developments

Problem: Previous method for separating islands from topological noise was unrigorous (just looked at lifetime of the class). Can we do better?

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Note: In 2023 Bobrowski and Skraba (A universal null-distribution for topological data analysis) showed empirically that the distribution of multiplicative persistence for any random cloud of points can be transformed to a universal distribution.

This gives us a Null-hypothesis to we can compare the multiplicative persistence of our field line orbits against.

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Idea: Use hypothesis testing to only count statistically significant topological features as islands.

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Different CDFs for death/birth for different random processes

Same CDF after transformation on the data (left-skewed gumbel)

Hypothesis testing with a Bonferroni correction does not count the number of islands in a chain correctly but counts the large islands in a chaotic layer well.

FLR Effects on Ballooning Stability Near Field null

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[N.Cross, M. J. Hole]

The Effect of Pressure Anisotropy on ELMs

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[J. Doak, M. J. Hole]

Anisotropic Pressure Models in the Pedestal Region

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MHD Equilibrium with Neural Models

Grad-Shafranov Equation (with a certain assumptions from the Soloviev profiles):

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Enforcing the equation into a loss term

Neural Models�(e.g., PINN, Neural Operator)

Equilibrium state

Training with respect to the loss terms

Output

Input (R,Z)

[F. Rizqan, C. Gretton, M. J. Hole]

B. Jang et al “Grad-Shafranov equilibria via data-free physics informed neural networks”, https://arxiv.org/abs/2311.13491

[D. Pfefferlé, D. Perella, N. Duignan]