Development of efficient algorithms with rigorous analysis for partial differential equations (PDE) on domains and surfaces requires knowledge of regularity of solutions of the PDE. In this talk, for the Navier–Stokes equations rotating spheres we discuss (i) the global regularity for real and complexified time; (ii) a high-order algorithm with stability and convergence analysis; (iii) long time simulation of a benchmark atmospheric flow and justification of a turbulence theory.