Gyroscopic alignment of a fluid occurs when flow structures align with the rotation axis. This often gives rise to highly spatially anisotropic columnar structures that in combination with complex domain boundaries pose challenges for efficient numerical discretizations and computations. We define gyroscopic polynomials to be three-dimensional polynomials expressed in a coordinate system that conforms to rotational alignment. We remap the original domain with radius-dependent boundaries onto a right cylindrical or annular domain to create the computational domain in this coordinate system. We find the volume element expressed in gyroscopic coordinates leads naturally to a hierarchy of orthonormal bases. We build the bases out of Jacobi polynomials in the vertical and generalized Jacobi polynomials in the radial. Because these coordinates explicitly conform to flow structures found in rapidly rotating systems the bases represent fields with a relatively small number of modes. We develop the operator structure for one-dimensional semi-classical orthogonal polynomials as a building block for differential operators in the full three-dimensional cylindrical and annular domains. The differential operators of generalized Jacobi polynomials generate a sparse linear system for discretization of differential operators acting on the gyroscopic bases. This enables efficient simulation of systems with strong gyroscopic alignment.

翻译：当流体中的流动结构与旋转轴对齐时，会产生陀螺对准现象。这通常导致高度空间各向异性的柱状结构，结合复杂的域边界，对高效数值离散化和计算构成挑战。我们将陀螺多项式定义为在符合旋转对准的坐标系中表达的三维多项式。我们将具有半径依赖边界的原始域重新映射到直圆柱或环形域，从而在此坐标系中构建计算域。我们发现，在陀螺坐标系中表达的体积元素自然导出一系列正交基。我们利用径向的雅可比多项式及广义雅可比多项式构建这些基。由于这些坐标显式符合快速旋转系统中发现的流动结构，因此基函数能以相对较少的模式表示场。我们发展了一维半经典正交多项式的算子结构，作为全三维圆柱和环形域中微分算子的基本构建块。广义雅可比多项式的微分算子生成一个稀疏线性系统，用于对作用于陀螺基的微分算子进行离散化。这实现了对具有强陀螺对准系统的有效模拟。