  We present a geometric derivation of the quasi-geostrophic equations on the
sphere, starting from the rotating shallow water equations. We utilise
perturbation series methods in vorticity and divergence variables. The
derivation employs asymptotic analysis techniques, leading to a global
quasi-geostrophic potential vorticity model on the sphere without approximation
of the Coriolis parameter. The resulting model forms a closed system for the
evolution of potential vorticity with a rich mathematical structure, including
Lagrangian and Hamiltonian descriptions. Formulated using the Lie-Poisson
bracket reveals the geometric invariants of the quasi-geostrophic model.
Motivated by these geometric results, simulations of quasi-geostrophic flow on
the sphere are presented based on structure-preserving Lie-Poisson
time-integration. We explicitly demonstrate the preservation of Casimir
invariants and show that the hyperbolic quasi-geostrophic equations can be
simulated in a stable manner over long time. We show the emergence of
longitudonal jets, wrapped around the circumference of the sphere in a general
direction that is perpendicular to the axis of rotation.
