  We derive analytical solutions and dispersion relations of global magnetic
Poincar\'e (magneto-gravity) and magnetic Rossby waves in the approximation of
shallow water magnetohydrodynamics. The solutions are obtained in a rotating
spherical coordinate system for strongly and weakly stable stratification
separately in the presence of toroidal magnetic field. In both cases magnetic
Rossby waves split into fast and slow magnetic Rossby modes. In the case of
strongly stable stratification (valid in the radiative part of the tachocline)
all waves are slightly affected by the layer thickness and the toroidal
magnetic field, while in the case of weakly stable stratification (valid in the
upper overshoot layer of the tachocline) magnetic Poincar\'e and fast magnetic
Rossby waves are found to be concentrated near the solar equator, leading to
equatorially trapped waves. However, slow magnetic Rossby waves tend to
concentrate near the poles, leading to polar trapped waves. The frequencies of
all waves are smaller in the upper weakly stable stratification region than in
the lower strongly stable stratification one.

Astrophysics