用于引导中心动力学、各向异性平衡和准对称的变分框架在仿星器中的应用

中文摘要 AI 辅助

我们提出了一个变分框架，其中引导中心动力学理论、具有旋转变换/各向异性压力的宏观力平衡以及准对称（QS）约束表现为单一结构的不同方面。从引导中心的弗拉索夫 - 麦克斯韦作用出发，通过约束变分得到引导中心动力学方程和麦克斯韦方程。在没有唯象闭合的情况下，动量守恒产生宏观力平衡\(\mathbf{J} \times \mathbf{B}/c = \nabla\cdot\boldsymbol{\Pi}\)，其中\(\mathbf{J}\)是电流密度，\(\mathbf{B}\)是磁场，\(\boldsymbol{\Pi}\)是旋转变换应力张量。我们通过\(f_T \equiv \nabla\psi \cdot \bigl(\nabla B \times \nabla(\mathbf{B} \cdot \nabla B)\bigr) = 0\)将QS与以无坐标形式表示的可积性条件联系起来，其中\(\psi\)是通量表面标签，并展示了该条件如何导致与最近在约束的克鲁斯卡尔 - 库尔斯鲁德变分公式中发现的那些密切相关的各向异性可解性约束。

英文摘要

We present a variational framework in which (i) guiding-center kinetics, (ii) macroscopic force balance with gyrotropic pressure, and (iii) quasisymmetry (QS) appear as facets of a single structure. Starting from a guiding-center Vlasov--Maxwell action, constrained variations yields the guiding-center kinetic equation and Maxwell equations. Without phenomenological closure, momentum conservation yields $\mathbf J \times \mathbf{B}/c = \nabla \cdot Π$. This framework recovers the classical result that, for any equilibrium distribution built on the invariants of the reduced motion, the parallel projection of this balance holds identically, so that pressure anisotropy is a kinematic consequence of magnetic-moment conservation together with the variation of $B$ along field lines, rather than a modeling choice. We connect QS to an integrability condition expressed in coordinate-free form. We then derive an explicit expression for the current crossing the flux surfaces. Its flux-surface average vanishes identically in a QS field, but it vanishes pointwise only if $p_\parallel+p_\perp$ is a flux function. Combined with the parallel identity this forces $p_\parallel = p(ψ) + K(ψ)B^2$ and $p_\perp = p(ψ) - K(ψ)B^2$, consistent with the anisotropic equilibria obtained from a constrained Kruskal--Kulsrud functional, in which $(p_\parallel+p_\perp)/2$ is a flux function [Phys. Rev. E 104, 015213 (2021)]. The structure arises from the kinetic moments without a Lagrange multiplier. For such equilibria the parallel current in a QS field follows in closed form, with a single resonance at $ι= N/M$ in place of the dense set of resonant obstructions of a generic three-dimensional field. Finally, QS is a Noether symmetry, so its breaking is measured by the non-conservation of the associated momentum. We construct a distribution-weighted residual and relate it to existing QS metrics.

发表机构

• Courant Institute School of Mathematics, Computing, and Data Science, New York University（纽约大学库朗数学科学研究所）
• Department of Astrophysical Sciences, Princeton University（普林斯顿大学天体物理科学系）

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