  Quasi-biennial oscillations (QBO) are frequently observed in the solar
activity indices. However, no clear physical mechanism for the observed
variations has been suggested so far. Here we study the stability of magnetic
Rossby waves in the solar tachocline using the shallow water
magnetohydrodynamic approximation. Our analysis shows that the combination of
typical differential rotation and a toroidal magnetic field with a strength >
10^5 G triggers the instability of the m=1 magnetic Rossby wave harmonic with a
period of 2 years. This harmonic is antisymmetric with respect to the equator
and its period (and growth rate) depends on the differential rotation
parameters and the magnetic field strength. The oscillations may cause a
periodic magnetic flux emergence at the solar surface and consequently may lead
to the observed QBO in the solar activity features. The period of QBO may
change throughout the cycle, and from cycle to cycle, due to variations of the
mean magnetic field and differential rotation in the tachocline.
