标题： MAST-U上静态前向自由边界Grad-Shafranov问题的多个解 显示英文标题

标题： Multiple solutions to the static forward free-boundary Grad-Shafranov problem on MAST-U

摘要： 广义-沙法诺夫（GS）方程是一个非线性椭圆偏微分方程，它描述了托卡马克等离子体的理想磁流体动力学平衡。 以往的研究已经证明，在理想几何结构中使用简化的等离子体电流密度剖面和边界条件求解GS方程时，存在多个解。 到目前为止，关于在具有更复杂电流密度剖面和积分自由边界条件（通常用于生产级平衡代码）的实际托卡马克几何中是否存在多个平衡状态的问题仍然没有答案。 在本工作中，我们使用经过验证的演化平衡求解器FreeGSNKE和阻尼延续算法，在MAST-U托卡马克几何中发现了静态正向自由边界GS问题的多个解。 通过改变GS方程中的等离子体电流、电流密度剖面系数或线圈电流，我们识别并表征了不同的平衡解，包括深度约束和较浅约束的等离子体状态。 我们认为，积分自由边界条件的限制性质可能阻止了更多平衡状态的存在，该条件在全球范围内将计算边界上的环向通量与内部的环向通量耦合在一起。 最后，我们讨论了这些发现对更广泛的平衡建模的影响，并强调需要探索其他平衡代码和托卡马克中是否存在多个解，以及它们对依赖GS平衡的下游模拟的潜在影响。 显示英文摘要

摘要： The Grad-Shafranov (GS) equation is a nonlinear elliptic partial differential equation that governs the ideal magnetohydrodynamic equilibrium of a tokamak plasma. Previous studies have demonstrated the existence of multiple solutions to the GS equation when solved in idealistic geometries with simplified plasma current density profiles and boundary conditions. Until now, the question of whether multiple equilibria might exist in real-world tokamak geometries with more complex current density profiles and integral free-boundary conditions (commonly used in production-level equilibrium codes) has remained unanswered. In this work, we discover multiple solutions to the static forward free-boundary GS problem in the MAST-U tokamak geometry using the validated evolutive equilibrium solver FreeGSNKE and the deflated continuation algorithm. By varying the plasma current, current density profile coefficients, or coil currents in the GS equation, we identify and characterise distinct equilibrium solutions, including both deeply and more shallowly confined plasma states. We suggest that the existence of even more equilibria is likely prohibited by the restrictive nature of the integral free-boundary condition, which globally couples poloidal fluxes on the computational boundary with those on the interior. We conclude by discussing the implications of these findings for wider equilibrium modelling and emphasise the need to explore whether multiple solutions are present in other equilibrium codes and tokamaks, as well as their potential impact on downstream simulations that rely on GS equilibria.