  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.
