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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.

Standard spectral codes for full sphere dynamics utilize a combination of spherical harmonics and a suitableradial basis to represent fluid variables. These basis functions have a rotational invariance not present ingeophysical flows. Gyroscopic alignment - alignment of dynamics along the axis of rotation - is ahallmark of geophysical fluids in the rapidly rotating regime. The Taylor-Proudman theorem, resultingfrom a dominant balance of the Coriolis force and the pressure gradient force, yields nearly invariant flows along this axial direction.In this paper we tailor a coordinate system to the cylindrical structures found in rotating spherical flows.This "spherindrical" coordinate system yields a natural hierarchy of basis functions, composed of Jacobi polynomialsin the radial and vertical direction, regular throughout the ball.We expand fluid variables using this basis and utilize sparse Jacobi polynomial algebra to implement all operatorsrelevant for partial differential equations in the spherical setting. We demonstrate the representation power ofthe basis in three eigenvalue problems for rotating fluids.