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A gyroscopic polynomial basis in the sphere

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.