Analytical solutions of the Grad–Shafranov equilibrium equation in simply connected plasma configurations, comprised of toroidal magnetic surfaces and open surfaces connected to electrodes, are reviewed and generalised. The Grad–Shafranov equation is linearised introducing assumptions on plasma current and pressure, which preserve regularity of solutions on the symmetry axis, as required for a simply connected geometry. Particular solutions are found by separation of variables both in cylindrical coordinates and in spherical ones. Equilibria that model local or global features of PROTO-SPHERA plasmas are constructed by combining a few particular solutions.

中文翻译：

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PROTO-SPHERA 平衡的分析研究

回顾和概括了简单连接等离子体配置中的 Grad-Shafranov 平衡方程的解析解，该配置由环形磁表面和连接到电极的开放表面组成。Grad-Shafranov 方程是线性化的，引入了对等离子体电流和压力的假设，这些假设保持了对称轴上解的规律性，这是简单连接几何的要求。通过分离柱坐标和球坐标中的变量来找到特定的解决方案。模拟 PROTO-SPHERA 等离子体的局部或全局特征的平衡是通过结合一些特定的解决方案来构建的。