  The influence of a toroidal magnetic field on the dynamics of shallow water
waves in the solar tachocline is studied. A sub-adiabatic temperature gradient
in the upper overshoot layer of the tachocline causes significant reduction of
surface gravity speed, which leads to trapping of the waves near the equator
and to an increase of the Rossby wave period up to the timescale of solar
cycles. Dispersion relations of all equatorial magnetohydrodynamic (MHD)
shallow water waves are obtained in the upper tachocline conditions and solved
analytically and numerically. It is found that the toroidal magnetic field
splits equatorial Rossby and Rossby-gravity waves into fast and slow modes. For
a reasonable value of reduced gravity, global equatorial fast magneto-Rossby
waves (with the spatial scale of equatorial extent) have a periodicity of 11
years, matching the timescale of activity cycles. The solutions are confined
around the equator between latitudes 20-40, coinciding with sunspot activity
belts. Equatorial slow magneto-Rossby waves have a periodicity of 90-100 yr,
resembling the observed long-term modulation of cycle strength, i.e., the
Gleissberg cycle. Equatorial magneto-Kelvin and slow magneto-Rossby-gravity
waves have the periodicity of 1-2 years and may correspond to observed annual
and quasi-biennial oscillations. Equatorial fast magneto-Rossby-gravity and
magneto-inertia-gravity waves have periods of hundreds of days and might be
responsible for observed Rieger-type periodicity. Consequently, the equatorial
MHD shallow water waves in the upper overshoot tachocline may capture all
timescales of observed variations in solar activity, but detailed analytical
and numerical studies are necessary to make a firm conclusion toward the
connection of the waves to the solar dynamo.
