In this article, we study Beltrami equilibria for plasmas in near the horizon of a spinning black hole, and develop a framework for constructing the magnetic field profile in the near horizon limit for Clebsch flows in the single-fluid approximation. We find that the horizon profile for the magnetic field is shown to satisfy a system of first-order coupled ODEs dependent on a boundary condition for the magnetic field. For states in which the generalized vorticity vanishes (the generalized `superconducting' plasma state), the horizon profile becomes independent of the boundary condition, and depend only on the thermal properties of the plasma. Our analysis makes use of the full form for the time-independent Ampere's law in the 3+1 formalism, generalizing earlier conclusions for the case of vanishing vorticity, namely the complete magnetic field expulsion near the equator of an axisymmetric black horizon assuming that the thermal properties of the plasma are symmetric about the equatorial plane. For the general case, we find and discuss additional conditions required for the expulsion of magnetic fields at given points on the black hole horizon. We perform a length scale analysis which indicates the emergence of two distinct length scales characterizing the magnetic field variation and strength of the Beltrami term, respectively.

中文翻译：

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关于黑洞事件视界附近的贝尔特拉米态

在本文中，我们研究了旋转黑洞近视界等离子体的贝尔特拉米平衡，并开发了一个框架，用于构建单流体近似中 Clebsch 流近视界极限的磁场分布。我们发现磁场的水平剖面显示满足依赖于磁场边界条件的一阶耦合 ODE 系统。对于广义涡度消失的状态（广义的“超导”等离子体状态），地平线轮廓变得与边界条件无关，并且仅取决于等离子体的热特性。我们的分析利用了 3+1 形式主义中与时间无关的安培定律的完整形式，概括了早先的涡度消失情况的结论，即假设等离子体的热特性关于赤道面对称，则在轴对称黑色地平线的赤道附近完全排出磁场。对于一般情况，我们找到并讨论了在黑洞视界上的给定点上驱逐磁场所需的附加条件。我们进行了长度尺度分析，这表明出现了两个不同的长度尺度，分别表征了贝尔特拉米项的磁场变化和强度。