研究成果報告会

2024/5/9

Meta-Hierarchy Dynamics Unit
Research Report [2023/4~2024/3]
Naoki Sato1
1National Institute for Fusion Science
Acknowledgment: M. Yamada2, P. J. Morrison3, D. Pfefferle4, D. Perrella4, N. Duignan52Kyoto University, 3UT Austin, 4Univ. Of Western Australia, 5Univ. of Sydney

Magnetohydrostatics (MHS)MHD Equilibria are described by the MHS equations∇ × 𝑩 × 𝑩 = 𝜇0 ∇𝑃,∇⋅𝑩= 0in Ω,∇𝑃 × 𝒏 = 𝟎on 𝜕Ω.

1Main Results
1. Given a set of nested flux surfaces Ψ 𝒙 , there exists an integration factor 𝜆 𝒙 and asolenoidal vector field 𝑩 such that ∇ × 𝑩 × 𝑩 = 𝜆∇Ψ. Furthermore, thisconfigurations can be interpreted as solutions of anisotropic MHS [1].
2. Dynamical system describing magnetic field turbulence around an MHS solution. Thissystem can be used to rapidly construct MHD equilibria with nested flux surfaces [2].
Papers[1] N. Sato and M. Yamada, J. Math. Phys. 64, 081505 (2023).
[2] N. Sato and M. Yamada, arxiv.org/abs/2311.03095.
Talks[T1] N. Sato and M. Yamada, "A Reduced Ideal MHD System for Nonlinear Magnetic Field
Turbulence in Plasmas with Approximate Flux Surfaces", The Grad conjecture in Fluid
Mechanics and MHD: Theory and Applications, I-ste Namba, Osaka, 28 March 2024 (INVITED).
[T2] N. Sato, "MHS (Magnetohydrostatics)", Simons Hidden Symmetries and Fusion Energy
Collaboration Australian Retreat, 12 December 2023.
[T3] N. Sato and M. Yamada, "On the Grad conjecture in anisotropic MHD", AAPPS-DPP 2023,
F-5-I3, 15/11/2023 (INVITED).
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[1]

Grad Conjecture Workshop (23-26 March 2024, I-Site Namba, Osaka)
Aim & ScopeMagnetohydrostatics (MHS) and the Euler equations are gathering
momentum due to their relevance for practical applications, including thedesign of novel fusion reactors. The workshop mainly consisted of talks
pertaining to MHS and the Euler equations, with both mathematicians
and plasma physicists joining the event. One of the purposes of thisgathering was to discuss the Grad conjecture from a broad perspective,
why it is difficult to solve it, what it means to solve it, and envisage
applications and significance for math and physics.
Grant
流体力学及び電磁流体力学における Grad 予想の数理と応用, 大阪公立大学数学研究所 文部科学省 共同利用・共同研究拠点 「数学・理論物理の協働・共創による 新たな国際的研究・教育拠点」 2023年 - 2024
年, 佐藤直木（研究代表者）, 阿部健, 横山知郎, 吉田善章, 山田道夫
External links• Workshop webpage (slides, video recordings, OCAMI report):
https://sites.google.com/view/grad-conjecture/home?authuser=0• OCAMI webpage:https://www.omu.ac.jp/orp/ocami-en/joint/
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International Speakers
Federico Pasqualotto (UC Berkeley)
Javier Peñafiel-Tomás (ICMAT)
Daniel Peralta-Salas (ICMAT)David Perrella (U. Western Australia)
David Pfefferlé (U. Western Australia)
National Speakers
Ken Abe (Osaka Metropolitan U.)
Yasuhide Fukumoto (Kyushu U.)
Yukimi Goto (Kyushu U.)
Kiori Obuse (Okayama U.)
Koji Ohkitani (Kyoto U.)
Naoki Sato (NIFS)
Ikkei Shimizu (Osaka U.)
Keiichirou Takeda (U. Tokyo)
Tomoo Yokoyama (Saitama U.)

Solution of theG-S eq. in a heart
shaped domain
by D. PfefferléAn Extended Hasegawa-Mima Equation For Nonlinear Drift Wave Turbulence in General Magnetic Configurations

We proposed [3-4] a generalization of the HM equation which accounts for drift wave turbulence in systems
with strong magnetic field and electron density gradients 𝐿∇ log 𝐵 ∼ 𝐿∇ log 𝐴𝑒 ∼ 1:
𝜕𝐴𝑒 ∇⊥ 𝜒𝜆𝐴𝑒 𝜒 − 𝜎∇ ⋅𝜕𝑡𝐵2•••••••𝜒= ∇ ⋅ 𝐴𝑒𝑩 ⋅ ∇ × 𝒗𝑬
𝜒𝜎−1𝒗.
𝑬𝐵2𝒙 ∈ Ω ⊆ ℝ3 , 𝑡 ∈ [0, ∞)𝜎𝜒 𝒙, 𝑡 = 𝜑 𝒙, 𝑡 + 𝒗2𝑬 𝒙, 𝑡 (charged particle energy)2
𝑩 = 𝑩 𝒙 ≠ 𝟎 (static magnetic field of arbitrary geometry)
𝐴𝑒 = 𝐴𝑒 𝒙 (leading order electron spatial density, 𝑛𝑒 = 𝐴𝑒 exp 𝑒𝜑/𝑘𝐵 𝑇𝑒 )∇⊥ = −𝐵 −2 𝑩 × 𝑩 × ∇𝜒𝒗𝑬 = 𝑩 × ∇𝜒/𝐵2𝒗𝑬 = 𝑩 × ∇𝜑/𝐵22HM eq. when
𝑩 = 𝐵0 ∇𝑧,log 𝐴𝑒 = log 𝐴𝑒0 + 𝛽𝑥,𝛽𝐿 ∼ 𝜖Papers[3] N. Sato and M. Yamada, Physica D: Nonlinear Phenomena 459, 134031 (2024).
[4] 佐藤直木, 山田道夫, プラズマ核融合学会誌 99 (5) 183-186 2023年5月.
Talks[T4] N. Sato and M. Yamada, "Guiding Center Derivation, Hamiltonian Structure, and Nonlinear Stability of Steady Solutions of the Generalized
Hasegawa-Mima Equation for Drift Wave Turbulence in Curved Magnetic Fields", 16aA307-9, JPS 2023 fall meeting, Tohoku University, Sendai, 16
September 2023.
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A Collision Operator for Describing Dissipation in Noncanonical Phase Space
Suppose that the distribution function 𝑓 𝒛, 𝑡 with phase space coordinate 𝒛 evolves in time 𝑡 according to𝜕𝑓𝜕𝑓= − 𝑓, ℌ ∗ +,𝜕𝑡
𝜕𝑡 coll3

where ℌ is the Hamiltonian (total energy), ⋅,⋅ ∗ a noncanonical Poisson bracket, and 𝜕𝑓/𝜕𝑡 coll a collision term.
Problem. What is the expression of 𝜕𝑓/𝜕𝑡 coll for binary collisions that occur over short times scales and small
spatial scales compared to the ideal dynamics described by the noncanonical Poisson bracket?
Theorem. The collision term 𝜕𝑓/𝜕𝑡 coll is given by [5]𝜕𝑓𝜕𝜕 log 𝑓 ′
𝜕 log 𝑓′′′ .
= 𝒞 𝑓, 𝑓 =⋅ 𝑓ℐ න 𝑓 Π ℐ−ℐ𝑑𝒛
𝜕𝑡 coll𝜕𝒛𝜕𝒛′𝜕𝒛

4Here, Π is a symmetric covariant 2-tensor (interaction tensor) depending on the type of binary interactions, ℐ thePoisson 2-tensor associated with the Poisson bracket, and the notation 𝑓 ′ = 𝑓 𝒛′ , 𝑡 and ℐ ′ = ℐ 𝒛′ was used.
Papers[5] N. Sato and P. J. Morrison, arXiv:2401.15086 (2024).
Talks[T5] N. Sato and P. J. Morrison, "On the Landau Collision Operator in Noncanonical Phase Space and the resulting modification of the moment
equations governing fluid dynamics", Fluids in Seoul 2024, June E Huh Center for Mathematical Challenges (HCMC), 18 January 2024 (INVITED).
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Current & Future Projects• Enhance International Collaborations• Interact with the math, fluid mechanics, & theoretical physics communities
Current projects
MHS without incompressibility > asymmetric solutions probably exist
Enstrophy conservation in relativistic flows (with K. Takeda)
Numerical simulation of Navier-Stokes equations with Clebsch potentials [6] (with S. Murai and Z. Yoshida)
Statistical mechanics of pair plasmas [7]International collaborations under development
N. Duignan (U. Sydney) > Differential geometric formulation of Nambu dynamics > some new results [8]D. Pfefferlé (UWA), D. Perrella (UWA), and F. Pasqualotto (UC Berkeley) > MHS existence & regularity
H. Abdelhamid (Mansoura U.) > Dynamo theory in XMHD
M. Materassi (CNR) > Metriplectic formalism with stochastic PDEs and fractional diffusion

Papers[6] S. Murai, N. Sato, and Z. Yoshida, doi.org/10.48550/arXiv.2305.16673.
[7] N. Sato, Physics of Plasmas 30, 042503 (2023).
[8] N. Sato, Progress of Theoretical and Experimental Physics 2024, Issue 4, 049201 (2024).

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