Rudimentary elements of the Pencil Code were originally developed at the Helmholtz Institute for Supercomputational Physics in Golm, Germany. It began during the Summer School “Tools to Simulate Turbulence on Supercomputers” held 27 August – 21 September 2001 within the premises of the Albert Einstein Institute in Golm; see its annual report (Nicolai 2001). The spatial and temporal discretisation schemes used in the code are described by Brandenburg and Dobler (2002) and Brandenburg (2003), which have become the most commonly quoted sources with reference to the Pencil Code. In the meantime, however, more than seventy people have contributed to the further development of the code,1 such that some revised and more comprehensive references are badly needed. It is towards this end that we present this special issue in Geophysical and Astrophysical Fluid Dynamics (GAFD) to discuss specific applications and their numerical aspects, especially relating to newly emerging research topics. Early applications of the Pencil Code were in dynamo theory of forced turbulence in Cartesian domains (Haugen et al. 2003, 2004a, 2004b). Subsequently, continued code developments have extended its scope inmany different directions. On one hand, cylindrical and spherical geometries have been employed in applications ranging frommean-field and forced turbulence simulations (Mitra et al. 2009, 2010, Kemel et al. 2011) to convectively driven dynamos (Käpylä et al. 2010, 2012, 2013) and circumstellar disks (Lyra and Mac Low 2012, Lyra et al. 2015, 2016). On the other hand, Cartesian geometries have been used inmore complex physical settings ranging from technical applications to a broad spectrum of astrophysical applications. This special issue of GAFD gives insight into some of the problemswith whichmembers of the Pencil Code community are currently concerned. Some of the topics relate to the necessary physical setup, in order to address specific questions in the most appropriate manner, while others concern the solutions to numerical challenges that have been encountered in the process of solving various problems. Specifically, we begin the special issue by addressing the global convection-driven dynamo problem in spherical geometry. The actual implementation of spherical geometry has been described in an earlier paper by Mitra et al. (2009), which involves replacing partial derivatives by covariant derivatives. As an example, the traceless rate-of-strain tensor is generalised by substituting

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介绍

Pencil Code 的基本元素最初是在德国戈尔姆的亥姆霍兹超级计算物理研究所开发的。它始于 2001 年 8 月 27 日至 9 月 21 日在戈尔姆阿尔伯特爱因斯坦研究所举办的“超级计算机湍流模拟工具”暑期学校；参见其年度报告（Nicolai 2001）。Brandenburg and Dobler (2002) 和 Brandenburg (2003) 描述了代码中使用的空间和时间离散化方案，它们已成为参考铅笔代码最常引用的来源。然而，与此同时，有 70 多人为代码的进一步开发做出了贡献，1 因此急需一些经过修订和更全面的参考资料。为此，我们在地球物理和天体物理流体动力学 (GAFD) 中提出了这一特刊，以讨论特定应用及其数值方面，尤其是与新兴研究主题相关的问题。Pencil Code 的早期应用是在笛卡尔域中强制湍流的发电机理论中（Haugen 等人，2003、2004a、2004b）。随后，持续的代码开发将其范围扩展到许多不同的方向。一方面，圆柱和球形几何形状已被用于从平均场和强制湍流模拟（Mitra et al. 2009, 2010, Kemel et al. 2011）到对流驱动的发电机（Käpylä et al. 2010, 2012, 2013） ) 和星周盘 (Lyra and Mac Low 2012, Lyra et al. 2015, 2016)。另一方面，笛卡尔几何已被用于更复杂的物理环境，从技术应用到广泛的天体物理应用。本期 GAFD 特刊深入介绍了 Pencil Code 社区成员目前关注的一些问题。一些主题涉及必要的物理设置，以便以最合适的方式解决特定问题，而其他主题涉及解决各种问题过程中遇到的数值挑战的解决方案。具体来说，我们通过解决球面几何中的全局对流驱动的发电机问题开始本期特刊。Mitra 等人在较早的论文中描述了球面几何的实际实现。(2009)，其中涉及用协变导数替换偏导数。