Plasmas are highly nonlinear and multiscale, motivating a hierarchy of models to understand and describe their behavior. However, there is a scarcity of plasma models of lower fidelity than magnetohydrodynamics (MHD), although these reduced models hold promise for understanding key physical mechanisms, efficient computation, and real-time optimization and control. Galerkin models, obtained by projection of the MHD equations onto a truncated modal basis, and data-driven models, obtained by modern machine learning and system identification, can furnish this gap in the lower levels of the model hierarchy. This work develops a reduced-order modeling framework for compressible plasmas, leveraging decades of progress in projection-based and data-driven modeling of fluids. We begin by formalizing projection-based model reduction for nonlinear MHD systems. To avoid separate modal decompositions for the magnetic, velocity, and pressure fields, we introduce an energy inner product to synthesize all of the fields into a dimensionally consistent, reduced-order basis. Next, we obtain an analytic model by Galerkin projection of the Hall-MHD equations onto these modes. We illustrate how global conservation laws constrain the model parameters, revealing symmetries that can be enforced in data-driven models, directly connecting these models to the underlying physics. We demonstrate the effectiveness of this approach on data from high-fidelity numerical simulations of a three-dimensional spheromak experiment. This manuscript builds a bridge to the extensive Galerkin literature in fluid mechanics and facilitates future principled development of projection-based and data-driven models for plasmas.

中文翻译：

---

用于磁流体动力学的受物理约束的低维模型：第一性原理和数据驱动方法

等离子体是高度非线性和多尺度的，激发了模型的层次结构来理解和描述它们的行为。然而，与磁流体动力学 (MHD) 相比，缺乏保真度较低的等离子体模型，尽管这些简化的模型有望理解关键物理机制、有效计算以及实时优化和控制。通过将 MHD 方程投影到截断模态基础上获得的 Galerkin 模型，以及通过现代机器学习和系统识别获得的数据驱动模型，可以在模型层次结构的较低级别提供这种差距。这项工作利用基于投影和数据驱动的流体建模数十年的进展，为可压缩等离子体开发了降阶建模框架。我们首先将非线性 MHD 系统的基于投影的模型简化形式化。为了避免对磁、速度和压力场进行单独的模态分解，我们引入了能量内积以将所有场合成为尺寸一致的降阶基础。接下来，我们通过霍尔-MHD 方程在这些模式上的伽辽金投影获得解析模型。我们说明了全局守恒定律如何约束模型参数，揭示可以在数据驱动模型中强制执行的对称性，将这些模型直接连接到基础物理。我们证明了这种方法对 3D spheromak 实验的高保真数值模拟数据的有效性。这份手稿为流体力学中广泛的伽辽金文献架起了一座桥梁，并促进了基于投影和数据驱动的等离子体模型的未来原则性发展。