  We present a linear analysis of inviscid, incompressible, magnetohydrodynamic
(MHD) shallow water systems. In spherical geometry, a generic property of such
systems is the existence of five wave modes. Three of them (two
magneto-Poincare modes and one magneto-Rossby mode) are previously known. The
other two wave modes are strongly influenced by the magnetic field and
rotation, and have substantially lower angular frequencies; as such, we term
them "magnetostrophic modes". We obtain analytical functions for the velocity,
height and magnetic field perturbations in the limit that the magnitude of the
MHD analogue of Lamb's parameter is large. On a sphere, the magnetostrophic
modes reside near the poles, while the other modes are equatorially confined.
Magnetostrophic modes may be an ingredient in explaining the frequency drifts
observed in Type I X-ray bursts from neutron stars.
